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abstract: 'In this work, we present first report on disturbances over Brazilian atmosphere on 16–17 September, 2015 following the Chile tsunami occurrence. Using all-sky imager and magnetometer located at 2330 km away from the epicenter, the presence of disturbances is noted 1–3 hours after the tsunami beginning time and during time which seismic tremor was also felt in the region. We argue that their presence towards continent at 2000-3000 km away from the epicenter offers another example of similar atmospheric response as those observed during Tohoku-Oki tsunami, 2011. This similarity and their appearance during seismic tremor over the region classify them to be of tsunamigenic and/or seismogenic nature.'
title: 'Airglow and magnetic field disturbances over Brazilian region during Chile tsunami (2015)'
---
Introduction {#Introduction}
============
Geospace monitoring of seismogenic disturbances using non-seismic sensors has gain significant momentum in the last decade. During giant tsunamis that occurred in last ten years, tsunamigenic disturbances in the atmospheric airglow from all-sky imagers, ionospheric electron density from GNSS receivers and geomagnetic disturbances from magnetometers were reported [@Rolland2010; @Rolland2011; @Makela2011; @Galvanetal:2012; @Kheranietall2012; @Astafyeva2013; @Occhipinti2013; @Klausneretal:2014; @Coisson2015].
For the first time, airglow disturbances over Hawaii were observed during Tohoku-Oki tsunami [@Makela2011]. An interesting aspect observed during this tsunami was the presence of tsunamigenic total electron content (TEC) disturbances towards continent opposite to the tsunami propagation direction, covering up to 25$^\circ$ from epicentral distance [@Tsugawa2011; @Galvanetal:2012]. These backward propagating disturbances were simulated and interpreted as owing to the strong atmosphere shaking from the tsunami forcing near the coast [@Kheranietall2012].
Motivated by the presence of backward propagating tsunamigenic disturbances during Tohoku-Oki tsunami, we search for such disturbances in the present study during the Chile tsunami that occurred on 16th of September, 2015 at 22:55 UT, with epicenter located at $31.57^{\circ}$ S and $71.65^{\circ}$ W in $25$ km depth, and accompanied by an earthquake of magnitude $8.3$. We select a Brazilian observatory located at São José dos Campos (SJC - 23.2$^\circ$S, 45.9$^\circ$W) and present the observation from the all-sky imager and magnetometer. The all-sky is a 180$^\circ$ field of view multispectral imaging system that uses four 4-inch diameter interference filter of 2.0 nm bandwidth for OI 557.7 nm emission. The OI 557.7-nm greenline emission has been largely used to investigate wave dynamical processes around the mesopause region. In this kind of images, it is possible to observe gravity waves and tides signatures as well as infer the atomic oxygen profile [@Schubert1999]. The OI 557.7-nm emission volumetric peak rate occurs around 95$\pm$2 km. A fluxgate magnetometer with 0.1 nT resolution belonging to “Universidade do Vale do Paraíba” is used in the present study. This magnetometer also belongs to EMBRACE Magnetometer Network in South America. In Figure \[fig:map\], the epicenter of earthquake and SJC observatory location are shown. Also, it is shown an airglow image over SJC with effective field of view.
![Locations of São José dos Campos observatory (SJC – 23.2$^\circ$S, 45.9$^\circ$W) used to geomagnetic field variations and airglow disturbances. The red square shows the epicenter of the Chilean earthquake of 16th September, 2015. The circle indicates the field of view at a zenith angle of 90$^\circ$ projected at a $100$ km altitude and presents a raw image for OI 557.7-nm emission obtained at 23:05 LT over the field of view. The black dot shows SJC location.[]{data-label="fig:map"}](Map1.jpg "fig:"){width="10cm"}\
\[fig:TTT\]
Results and Discussion {#Results and analyses}
======================
Airglow Disturbances
--------------------
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![Airglow disturbance fronts propagating in Southeast direction. Vertical and horizontal axis are the latitude and longitude in degrees, respectively. The raw image at OI OI 557.7-nm at top and linearized at bottom obtained at 23:05 LT (02:05 UT).[]{data-label="fig:direction"}](O5_SJK_20150916_230517.jpg "fig:"){width="10cm"}\
![Airglow disturbance fronts propagating in Southeast direction. Vertical and horizontal axis are the latitude and longitude in degrees, respectively. The raw image at OI OI 557.7-nm at top and linearized at bottom obtained at 23:05 LT (02:05 UT).[]{data-label="fig:direction"}](direction.png "fig:"){width="10cm"}\
In Figures \[fig:557\]–\[fig:direction\], we present the images on the night of September 16–17, 2015 obtained using all-sky imager. Figure \[fig:557\] shows sequences of images of the OI 557.7-nm emission in which we note the presence of faint band-like airglow disturbances that appear around 02:05 UT (3 hours and 5 minutes after the earthquake event) and last about 30 minutes. These disturbances have wavefront aligned in northwest-southeast and propagated in northeastward direction, *i. e.*, in the opposite direction to the tsunami propagation. From an enlarged image at 02:05 UT in Figure \[fig:direction\], we note that these disturbances have wavelength in between 60 to 100 km. The airglow movie for the night of this reported event and the night before and after are available in the auxiliary material as Movie\_15Sep2015, Movie\_16Sep2015 and Movie\_17Sep2015 for comparison. From these movies, we verify that such disturbances as noted on the night of tsunami are absent on other days.
Band-like airglow disturbances are common feature over Brazilian region, and they are associated to gravity waves/medium-scale traveling ionospheric disturbances (GWs/MSTIDs) [@Medeiros2003; @Wrasse2006; @Candidoetal2008; @Pimenta2008]. Both mesospheric GWs and MSTIDs signatures observed in airglow images by [@Medeiros2003; @Wrasse2006; @Candidoetal2008; @Pimenta2008] are commonly attributed to tropospheric convective processes or to middle latitude instabilities. In addition, they have opposite orientation (northeast to southwest) and propagation direction (northwestward) to those observed here. Therefore, the observed airglow disturbances in Figures \[fig:557\]–\[fig:direction\] are not the convectively-driven GWs/MSTIDs commonly observed in the region.
On the other hand, these disturbances are possibly seismogenic or tsunamigenic. For the Tohoku-Oki tsunami, ionospheric disturbances in TEC were reported to propagate towards backward direction in the form of concentric wavefronts, and to arrive up to 23$^\circ$ from epicentral distance [@Tsugawa2011; @Galvanetal:2012]. The airglow disturbances in Figures \[fig:557\]–\[fig:direction\] offer a similar scenario where their presences are noted at 2000–3000 km ($\sim$18$^\circ$ – 27$^\circ$) from epicentral distance in the backward direction.
For the Tohoku-Oki tsunami, the backward propagating TEC disturbances were simulated and interpreted as owing to the strong atmosphere shaking from the tsunami forcing near the coast [@Kheranietall2012]. Therefore, the observed backward propagating airglow disturbances in the present study are possibly arising from the similar atmospheric shaking.
Geomagnetic Field Disturbances
------------------------------
On Figures \[fig:event\]–\[fig:compHeZ\], geomagnetic field measurements are presented. In Figure \[fig:event\], the H- and Z-components during 16-17 September, 2015 are plotted. In Figure \[fig:compHeZ\], the mean value (black color) of the quietest day of September (2015) and the standard deviation (grey color) between each day to the mean value are plotted. The 16th of September, 2015 is highlighted in blue color.
![Minutely magnetogram data from the H-component (top) and Z-component (bottom) variations over São José dos Campos. The dashed line shows the time of the Chilean earthquake (2015) and the arrows point to the increase of the geomagnetic field over São José dos Campos several minutes after of the event occurrence.[]{data-label="fig:event"}](RawMagneticDATA.jpg "fig:"){width="14cm"}\
In Figure \[fig:event\], we note a magnetic pulse around 24:00 UT, about 40 minutes after the earthquake. Such pulse is not present on previous night. Also, we note the presence of such pulse in Figure \[fig:compHeZ\] on night of tsunami while no such pulse is present in any other quiet days of September, 2015. Moreover, this pulse appears during the time when the seismic tremors were felt in the region. Consequently, these aspects provide further evidence that this magnetic pulse is seismogenic or tsunamigenic.
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Conclusions {#Conclusions}
===========
We document first report on the airglow and geomagnetic field disturbances related to the recently occurred Chile tsunami (2015). Also, these disturbances are observed over the Brazilian sector for the first time. The fact of airglow disturbances could be detected in opposite direction of the tsunami propagating is very encouraging, and it shows that the constant monitoring of the ionosphere and geomagnetic field could play an important role to calibrate seismic/tsunami models, and used in understanding of the physical processes involved in the tsunami propagation.
V. Klausner and C. M. N. Candido wish to thanks their Postdoctoral research for the financial support within the Programa Nacional de Pós-Doutorado (PNPD – CAPES) at UNIVAP and INPE, respectively. We wish to express his sincere thanks to the Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP), for providing financial support through the process numbers 2009/15769-2 and 2012/08445-9, CNPq grants number 457129/2012-3, and FINEP number 01.100661-00 for the partial financial support.
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abstract: 'We propose a precise definition of a continuous time dynamical system made up of interacting open subsystems. The interconnections of subsystems are coded by directed graphs. We prove that the appropriate maps of graphs called [graph fibrations]{} give rise to maps of the appropriate dynamical systems. Consequently surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.'
address: 'Department of Mathematics, University of Illinois, Urbana, IL 61801'
author:
- Lee DeVille and Eugene Lerman
title: Dynamics on networks of manifolds
---
Introduction
============
Given a dynamical system, one is initially interested in studying invariant subsystems; this includes equilibria, periodic orbits, etc. Moreover, constructing projections onto smaller systems as well as conjugacies and semi-conjugacies with simpler systems are generally useful for understanding the qualitative properties of dynamical systems. In this paper, we study the combinatorial properties of dynamical systems defined on networks to present a unified theoretical framework for all of the above objects. In this paper we give a precise definition of a continuous time dynamical system made up of interacting open subsystems. We then exploit the combinatorial aspect of networks to produce maps of dynamical systems out of appropriate maps of graphs called [graph fibrations]{} (q.v.Definition \[graph fibration\]). We show that surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.
Our results shed new light on the existence of invariant subsystems in coupled cell networks proved by Golubitsky, Pivato, Stewart and T[ö]{}r[ö]{}k [@Stewart.Golubitsky.Pivato.03; @Golubitsky.Stewart.Torok.05], since we impose no groupoid symmetries on our systems. In this respect our work is close in spirit to the approach to dynamics on networks advocated by Field [@FieldCD]; in contrast to Field we find it convenient to use the language of category theory. We also find it useful to borrow the notions of open systems and their interconnection from engineering (see, for example [@Brockett; @willems; @Pappas]) and the definition of a graph fibration from computer science [@Vigna1] (see [@VignaFibPage] for a history of the notion and alternative terminologies).
In more detail we construct a category of networks of continuous time systems. A network in our sense consists of
- a finite directed graph $G$ with a set of nodes $G_0$,
- a [*phase space function*]{} ${{\mathcal P}}$ that assigns to each node of the graph an appropriate phase space (which we take to be a manifold),
- a family of open systems $\{w_a\}_{a\in G_0}$ (one for each node $a$ of the graph $G$) consistent in an appropriate way with the structure of the graph, and
- an interconnection map ${\mathscr{I}}$ that turns these open systems into a vector field on the product $\bigsqcap _{a\in G_0} {{\mathcal P}}(a)$ of the phase spaces of the nodes.
Our main result, Theorem \[thm:main\], shows that [ graph fibrations]{} compatible with phase space functions give rise to maps of dynamical systems.
Putting aside all of the technicalities, however, the main idea of the construction of this paper is: given a map between any two graphs, we might like to induce a map on the associated dynamical systems via pullbacks. We observe that in order for this pullback to be well-defined, the map of graphs is required to have certain properties, and these are exactly the properties that make the graph map a [*fibration*]{}, as we will show below. The perhaps surprising conclusion made below is that not only are the definition conditions for a fibration necessary, they are also sufficient — any graph fibration gives rise to a map of dynamical systems, in a sense that can be made precise.
The paper is organized as follows. We start by recalling the definition of a directed multigraph, define the notion of a network of manifolds and the total space of the network. We then recall open systems and their interconnections. We show how a network of manifolds naturally leads to spaces of open systems that can be interconnected. One can think of these spaces as forming an infinite dimensional vector bundle over the set of nodes of the appropriate graph. We then prove our main result, Theorem \[thm:main\]: graph fibrations give rise to maps of dynamical systems.
Definitions and constructions
=============================
Graphs and manifolds {#graphs-and-manifolds .unnumbered}
--------------------
Throughout the paper [*graphs*]{} are finite directed multigraphs, possibly with loops. More precisely, we use the following definition:
\[def:directed graph\] A [graph]{} $ G$ consists of two finite sets $ G_1$ (of arrows, or edges), $ G_0$ (of nodes, or vertices) and two maps ${\mathfrak s},{\mathfrak t}\colon
G_1 \to G_0$ (source, target): $$G = \{{\mathfrak s},{\mathfrak t}\colon G_1 \to G_0\}.$$ We write $ G = \{ G_1{\rightrightarrows}G_0\}$.
The set $ G_1$ may be empty, i.e., we may have $G= \{\emptyset {\rightrightarrows}G_0\}$, making $G$ a disjoint collection of vertices.
A [map of graphs]{} $\varphi\colon A\to B$ from a graph $A$ to a graph $B$ is a pair of maps $\varphi_1\colon A_1\to B_1$, $\varphi_0\colon A_0\to B_0$ taking edges of $A$ to edges of $B$, nodes of $A$ to nodes of $B$ so that for any edge $\gamma$ of $A$ we have $$\varphi_0 ({\mathfrak s}(\gamma)) = {\mathfrak s}(\varphi_1 (\gamma))\quad
\textrm{and} \quad \varphi_0 ({\mathfrak t}(\gamma)) = {\mathfrak t}(\varphi_1 (\gamma)).$$ We often omit the indices 0 and 1 and write $\varphi(\gamma)$ for $\varphi_1 (\gamma)$ and $\varphi(a)$ for $\varphi_0 (a)$.
The collection of (finite directed multi-)graphs and maps of graphs form a category ${\mathsf{Graph}}$.
In order to construct networks from graphs we need to have a consistent way of assigning manifolds to nodes of our graphs. We formalize this idea by making the collection of graphs with manifolds assigned to vertices into a category ${\mathsf{Graph}}/{\mathsf{Man}}$.
\[def.network.mfld\] A [network of manifolds]{} is a pair $(G,{{\mathcal P}})$ where $G$ is a (finite directed multi-)graph and ${{\mathcal P}}\colon G_0 \to {\mathsf{Man}}$ is a function that assigns to each node $a$ of $G$ a manifold ${{\mathcal P}}(a)$. We think of ${{\mathcal P}}$ as an assignment of phase spaces to the nodes of the graph $G$, and for this reason we refer to ${{\mathcal P}}$ as a [*phase space function.*]{}
Networks of manifolds form a category ${\mathsf{Graph}}/{\mathsf{Man}}$. Its objects are are pairs $(G,{{\mathcal P}})$ as above. A morphism $\varphi$ from $(G,{{\mathcal P}})$ to$(G',{{\mathcal P}}')$ is a map of graphs $\varphi\colon G\to G'$ with $${{\mathcal P}}'\circ \varphi = {{\mathcal P}}.$$
Given a category ${\mathscr{C}}$ we denote the opposite category by ${{{\mathscr{C}}}^{\mbox{\sf{\tiny{op}}}}}$, i.e. the category with all of the same objects and all of the arrows reversed. We adhere to the convention that a [ *contravariant functor*]{} from a category ${\mathscr{C}}$ to a category ${\mathscr{D}}$ is a covariant functor $$F\colon {{{\mathscr{C}}}^{\mbox{\sf{\tiny{op}}}}}\to {\mathscr{D}}.$$ Then for any morphism $c\xrightarrow{\gamma}c'$ of ${\mathscr{C}}$ we have $F(c)\xleftarrow{F(\gamma)}F(c')$ in ${\mathscr{D}}$.
Next we recall the notion of a [*product*]{} in a category ${\mathscr{C}}$. We do this to draw a distinction between categorical and Cartesian products of finite families of manifolds which we will exploit throughout the paper.
A [product]{} of a family $\{c_s\}_{s\in S}$ of objects in a category ${\mathscr{C}}$ indexed by a set $S$ is an object $\bigsqcap_{s'\in S}c_{s'}$ of ${\mathscr{C}}$ together with a family of morphisms $\{\pi_s\colon \bigsqcap_{s'\in S}c_{s'} \to c_s\}_{s\in S}$ with the following universal property: given an object $c'$ of ${\mathscr{C}}$ and a family of morphisms $\{f_s\colon c'\to c_s\}_{s\in S}$ there is a unique morphism $f\colon c'\to \bigsqcap_{s\in S}c_s$ with $$\pi_s \circ f = f_s \quad \textrm{ for all }s\in S.$$
If a product exists then it is unique up to a unique isomorphism [@Awodey].
\[empt:cat-prod\]\
Observe that in the category ${\mathsf{Man}}$ of (finite dimensional paracompact smooth) manifolds a product $\bigsqcap_{s\in S}M_{s}$ exists for any family $\{M_s\}_{s\in S}$ of manifolds indexed by a finite set. It can be constructed by ordering the elements of $S$ and taking the corresponding Cartesian product. That is, products of manifolds indexed by a set $S$ can be constructed by writing $S=\{s_1, \ldots,
s_n\}$ and setting $\bigsqcap_{s\in S}M_{s} = \prod_{i=1}^n M_{s_i}$, where the right hand side is the Cartesian product. (This is the more common mathematical definition of “product”.)
However, for our purposes it is better to have a construction of the product that does not involve a choice of ordering of the indexing set in question. This may be done as follows. Given a family $\{M_s\}_{s\in S} $ of manifolds, denote by $\bigsqcup _{s\in S}M_s$ their disjoint union.[^1] Now define $$\bigsqcap_{s\in S}M_{s} := \{x\colon S \to \bigsqcup _{s\in
S}M_s \mid x(s)\in M_s \textrm{ for all }s\in S\}.$$ The projection maps $\pi_s\colon \bigsqcap_{s'\in S}M_{s'}\to M_s$ are defined by $$\pi_s (x) = x(s).$$
We denote $x(s)\in M_s$ by $x_s$ and think of it as $s^{th}$ “coordinate” of an element $x\in \bigsqcap_{s\in S}M_{s}$. Equivalently we may think of elements of the categorical product $\bigsqcap_{s\in S}M_{s} $ as [unordered]{} tuples $(x_s)_{s\in S}$ with $x_s\in M_s$.
Our discussion of categorical versus Cartesian products may seem fussy, but it will be useful to maps between total phase of networks of manifolds from maps of networks in Proposition \[prop2.13\] below.
For a pair $(G,{{\mathcal P}})$ consisting of a graph $G$ and an assignment ${{\mathcal P}}\colon G_0\to {\mathsf{Man}}$, that is, for an object $(G,{{\mathcal P}})$ of ${\mathsf{Graph}}/{\mathsf{Man}}$ we set $${\mathbb{P}}G\equiv {\mathbb{P}}(G,{{\mathcal P}}) := \bigsqcap _{a\in G_0} {{\mathcal P}}(a),$$ the categorical product of manifolds attached to the nodes of the graph $G$ by the [phase space function]{} ${{\mathcal P}}$ and call the resulting manifold ${\mathbb{P}}G$ the [total phase space ]{} of the network $(G,{{\mathcal P}})$.
\[example:1\] Consider the graph
at (0,0) [$G = $]{}; at (2,0) (1) [a]{}; at (4,0) (2) [b]{};
\(1) edge \[bend left\] node [$\beta$]{} (2) (1) edge \[bend right\] node \[below\] [$\alpha$]{} (2) ;
Define ${{\mathcal P}}\colon G_0\to {\mathsf{Man}}$ by ${{\mathcal P}}(a) = S^2$ (the two sphere) and ${{\mathcal P}}(b) = S^3$. Then $${\mathbb{P}}(G, {{\mathcal P}}) = S^2 \times S^3.$$
If $G=\{\emptyset {\rightrightarrows}\{a\}\}$ is a graph with one node $a$ and no arrows, we write $G=\{a\}$. Then for any phase space function ${{\mathcal P}}\colon
G_0 =\{a\} \to {\mathsf{Man}}$ we abbreviate ${\mathbb{P}}(\{\emptyset {\rightrightarrows}\{a\} \},
{{\mathcal P}}\colon \{a\} \to {\mathsf{Man}})$ as ${\mathbb{P}}a$.
\[prop2.13\] The assignment $$(G,{{\mathcal P}}) \mapsto {\mathbb{P}}G := \bigsqcap _{a\in G_0} {{\mathcal P}}(a)$$ of phase spaces to networks extends to a contravariant functor $${\mathbb{P}}\colon {{ ({\mathsf{Graph}}/{\mathsf{Man}})}^{\mbox{\sf{\tiny{op}}}}} \to {\mathsf{Man}}.$$
Suppose $\varphi\colon (G,{{\mathcal P}})\to (G',{{\mathcal P}}')$ is a morphism in ${\mathsf{Graph}}/{\mathsf{Man}}$. That is, suppose $\varphi\colon G\to G'$ is a map of graphs with ${{\mathcal P}}'\circ \varphi = {{\mathcal P}}$. We need to define a map of manifolds $${\mathbb{P}}\varphi\colon {\mathbb{P}}G' \to {\mathbb{P}}G.$$ Since by definition ${\mathbb{P}}G$ is the product $\bigsqcap _{a\in G_0}
{{\mathcal P}}(a)$, the universal property of products implies that in order to define ${\mathbb{P}}\varphi$ it is enough to define a family of maps $$\left\{({\mathbb{P}}\varphi)_a\colon {\mathbb{P}}G' \to {{\mathcal P}}(a)\right\}_{a\in G_0}.$$ For any node $a'$ of $G'$ we have the canonical projection $$\pi'_{a'}\colon {\mathbb{P}}G'\to {{\mathcal P}}' (a').$$ We therefore define $$({\mathbb{P}}\varphi)_a := \pi'_{\varphi_a}\colon {\mathbb{P}}G' \to {{\mathcal P}}' (\varphi(a) ) = {{\mathcal P}}(a)$$ for all $a\in G_0$. By the universal property of ${\mathbb{P}}G= \bigsqcap _{a\in G_0}
{{\mathcal P}}(a)$ this defines the desired map ${\mathbb{P}}\varphi\colon {\mathbb{P}}G'\to {\mathbb{P}}G$.
The universal property of products also implies that the map ${\mathbb{P}}$ on morphisms of ${\mathsf{Graph}}/{\mathsf{Man}}$ as defined above is actually a functor. That is, $${\mathbb{P}}(\psi \circ \varphi) = {\mathbb{P}}\varphi\circ {\mathbb{P}}\psi$$ for any pair $(\psi, \varphi)$ of composable morphisms in ${\mathsf{Graph}}/{\mathsf{Man}}$.
\[ex:1.0\] Suppose $G$ is a graph with two nodes $a, b$ and no edges, $G'$ is a graph with one node $\{c\}$ and no edges, ${{\mathcal P}}'(c)$ is a manifold $M$, $\varphi\colon G\to G'$ is the only possible map of graphs (it sends both nodes to $c$). ${{\mathcal P}}\colon G_0\to {\mathsf{Man}}$ is given by ${{\mathcal P}}(a) = M =
{{\mathcal P}}(b)$ (so that ${{\mathcal P}}'\circ \varphi = {{\mathcal P}}$. Then ${\mathbb{P}}G'\simeq M$, $${\mathbb{P}}G = \{ (x_a, x_b) \mid x_a\in {{\mathcal P}}(a), x_b \in {{\mathcal P}}(b)\} \simeq M\times M$$ and ${\mathbb{P}}\varphi\colon M\to M\times M$ is the unique map with $({\mathbb{P}}\varphi
(x))_a = x$ and $({\mathbb{P}}\varphi (x))_b = x$ for all $x\in {\mathbb{P}}G'$. Thus ${\mathbb{P}}\varphi\colon M\to M\times M$ is the diagonal map $x\mapsto (x,x)$.
Let $(G,{{\mathcal P}})$, $(G', {{\mathcal P}}')$ be as in Example \[ex:1.0\] above and $\psi\colon (G',{{\mathcal P}}')\to (G,{{\mathcal P}})$ be the map that sends the node $c$ to $a$. Then ${\mathbb{P}}\psi\colon {\mathbb{P}}G \to {\mathbb{P}}G'$ is the map that sends $(x_a,
x_b)$ to $x_a$.
\[rmrk:2.16\] If $(G,{{\mathcal P}})$ is a graph with a phase function, that is, an object of ${\mathsf{Graph}}/{\mathsf{Man}}$, and $\varphi\colon H\to G$ a map of graphs then ${{\mathcal P}}\circ
\varphi\colon H\to {\mathsf{Man}}$ is a phase function and $\varphi\colon (H,{{\mathcal P}}\circ
\varphi)\to (G,{{\mathcal P}})$ is a morphism in ${\mathsf{Graph}}/{\mathsf{Man}}$. We then have a map of manifolds $${\mathbb{P}}\varphi\colon {\mathbb{P}}(H, {{\mathcal P}}\circ \varphi)\to {\mathbb{P}}(G,{{\mathcal P}}).$$ Similarly, a commutative diagram $$\xy
(-8, 4)*+{ K} ="1";
(8, 4)*+{ H} ="2";
(0, -4)*+{G}="3";
{\ar@{->}^{ j} "1";"2"};
{\ar@{->}_{ \psi} "1";"3"};
{\ar@{->}^{\varphi} "2";"3"};
\endxy$$ of maps of graphs and a phase space function ${{\mathcal P}}\colon G\to {\mathsf{Man}}$ give rise to the commutative diagram of maps of manifolds $$\xy
(-18, 14)*+{{\mathbb{P}}(K, {{\mathcal P}}\circ \psi) } ="1";
(18, 14)*+{ {\mathbb{P}}(H, {{\mathcal P}}\circ \varphi)} ="2";
(0, -4)*+{{\mathbb{P}}(G,{{\mathcal P}})}="3";
{\ar@{->}_{ {\mathbb{P}}j} "2";"1"};
{\ar@{->}^{ {\mathbb{P}}\psi} "3";"1"};
{\ar@{->}_{{\mathbb{P}}\varphi} "3";"2"};
\endxy.$$
Embeddings and submersions from maps of graphs {#embeddings-and-submersions-from-maps-of-graphs .unnumbered}
----------------------------------------------
As we said in the introduction, the main goal of this paper is to construct maps of dynamical systems from graph fibrations. In Proposition \[prop2.13\] we showed that a map of networks $\varphi\colon
(G,{{\mathcal P}})\to (G', {{\mathcal P}}')$ defines a map of manifolds ${\mathbb{P}}\varphi \colon {\mathbb{P}}(
(G', {{\mathcal P}}') \to {\mathbb{P}}(G,{{\mathcal P}})$. In this subsection we prove that:
1. If the map of graphs $\varphi\colon G\to G'$ is injective on nodes, then ${\mathbb{P}}\varphi$ is a surjective submersion.
2. if the map of graphs $\varphi\colon G\to G'$ is surjective on nodes, then ${\mathbb{P}}\varphi$ is an embedding.
\[Recall that a smooth map between two manifolds is a submersion if its differential is onto at every point. A smooth map between two manifolds is an embedding if it is 1-1, its differential is 1-1 everywhere and it is a homeomorphism onto its image.\] Combined with Theorem \[thm:main\] below, this shows that surjective fibrations of networks of manifolds give rise to invariant dynamical subsystems and injective fibrations give rise to projections of dynamical systems.
\[lemma:2.surj\] Suppose $\varphi\colon (G,{{\mathcal P}})\to (G',{{\mathcal P}}')$ is a map of networks of manifolds such that the map on nodes, $\varphi_0\colon G_0 \to G'_0$, is surjective. Then ${\mathbb{P}}\varphi\colon {\mathbb{P}}G'\to {\mathbb{P}}G$ is an embedding whose image is the “polydiagonal” $$\Delta_\varphi = \{x\in {\mathbb{P}}G\mid x_a = x_b\textrm{ whenever
}\varphi(a) = \varphi(b)\}.$$
Assume first for simplicity that $G'$ has only one vertex $*$ and ${{\mathcal P}}'(*) = M$. Then for any vertex $a$ of $G$ we have $${{\mathcal P}}(a) = {{\mathcal P}}' (\varphi(a)) = {{\mathcal P}}'(*) = M,$$ ${\mathbb{P}}G' = M$ and ${\mathbb{P}}G = M\times \cdots \times M$ ($|G_0|$ copies), where as before $G_0$ is the set of vertices of the graph $G$. In this case the proof of Proposition \[prop2.13\] shows that the map ${\mathbb{P}}\varphi\colon M\to
M^{G_0}$ is of the form $${\mathbb{P}}\varphi (x) = (x,\ldots, x)$$ for all $x\in M$.
In general $${\mathbb{P}}\varphi\colon {\mathbb{P}}G' =\bigsqcap_{a'\in G'_0} {{\mathcal P}}' (a') \to \bigsqcap_{a'\in G'_0} \left( \bigsqcap_{a\in \varphi{^{-1}}(a')} {{\mathcal P}}(a)\right) = {\mathbb{P}}G$$ is the product of maps of the form $${{\mathcal P}}' (a') \to \bigsqcap_{a\in \varphi{^{-1}}(a')} {{\mathcal P}}(a), \quad
x\mapsto (x,\ldots, x).$$
\[lemma:2.inj\] Suppose $\varphi\colon (G,{{\mathcal P}})\to (G',{{\mathcal P}}')$ is a map of networks of manifolds such that the map $\varphi_0\colon G_0 \to G'_0$ on nodes is injective. Then ${\mathbb{P}}\varphi\colon {\mathbb{P}}G'\to {\mathbb{P}}G$ is a surjective submersion.
Since $\varphi\colon G\to G'$ is injective, the set of nodes $G_0'$ of $G'$ can be partitioned as the disjoint union of the image $\varphi(G_0)$, which is a copy of $G_0$, and the complement. Hence $${\mathbb{P}}G' \simeq \bigsqcap_{a\in G_0} {{\mathcal P}}(\varphi(a)) \times
\bigsqcap_{a'\not\in \varphi(G_0)} {{\mathcal P}}' (a') \simeq {\mathbb{P}}G \times
\bigsqcap_{a'\not\in \varphi(G_0)} {{\mathcal P}}' (a').$$ With respect to this identification of ${\mathbb{P}}G'$ with ${\mathbb{P}}G \times
\bigsqcap_{a'\not\in \varphi(G_0)} {{\mathcal P}}' (a')$ the map ${\mathbb{P}}\varphi\colon {\mathbb{P}}G'\to {\mathbb{P}}G$ is the projection $${\mathbb{P}}G \times
\bigsqcap_{a'\not\in \varphi(G_0)} {{\mathcal P}}' (a') \to {\mathbb{P}}G.$$ which is a surjective submersion.
Open systems and their interconnections {#open-systems-and-their-interconnections .unnumbered}
---------------------------------------
Having set up a consistent way of assigning phase spaces to graphs, we now take up continuous time dynamical systems. We start by recalling a definition of an open (control) systems, which is essentially due to Brockett [@Brockett].
A [continuous time control system]{} (or an [open system]{}) on a manifold $M$ is a surjective submersion $p\colon Q\to M$ from some manifold $Q$ together with a smooth map $ F\colon Q \to TM$ so that $$F(q) \in T_{p(q)} M$$ for all $q\in Q $. That is, the following diagram $
\xy (-10, 6)*+{Q}="1";
(6, 6)*+{TM} ="2";
(6,-3)*+{M}="3";
{\ar@{->}_{ p} "1";"3"};
{\ar@{->}^{F} "1";"2"};
{\ar@{->}^{\pi} "2";"3"}; \endxy
$ commutes. Here $\pi\colon TM \to M$ is the canonical projection.
Given a manifold $U$ of “control variables” we may consider control systems of the form $$\label{eq:4}
F\colon M\times U\to TM.$$ Here the submersion $p\colon M\times U \to M$ is given by $$p(x,u) = x.$$ The collection of all such control systems forms a vector space that we denote by ${\mathsf{Control }}(M\times U\to M)$: $${\mathsf{Control }}(M\times U\to M):= \{F\colon M\times U\to TM \mid F(x,u)\in T_x M\}.$$
Now suppose we are given a finite family $\{F_i\colon M_i
\times U_i \to TM_i\}_{i=1}^N$ of control systems and we want to somehow interconnect them to obtain a closed system ${\mathscr{I}}(F_1,
\ldots, F_N)$ that is, a vector field, on the product $\bigsqcap_i M_i$. What additional data do we need to define the interconnection map $${\mathscr{I}}\colon \bigsqcap_i {\mathsf{Control }}(M_i\times U_i \to M_i) \to
\Gamma\, (T(\bigsqcap_i M_i))?$$ The answer is given by the following proposition:
\[prop:3.2\] Given a family $\{p_j \colon M_j
\times U_j \to M_j\}_{j=1}^N$ of projections on the first factor and a family of smooth maps $\{s_j\colon \bigsqcap M_i \to M_j\times U_j\}$ so that the diagrams $$\xy (-10, 6)*+{M_j \times U_j}="1";
(-10,-6)*+{\bigsqcap M_i}="3";
(10,-6)*+{ M_j}="4";
{\ar@{->}^{ p_1} "1";"4"};
{\ar@{->}^{s_j} "3";"1"};
{\ar@{->}_{pr_j} "3";"4"};
\endxy$$ commute for each index $j$, there is an interconnection map ${\mathscr{I}}$ making the diagrams $$\xy
(-30, 6)*+{\bigsqcap_i {\mathsf{Control }}(M_i\times U_i \to M_i)}="1";
(30, 6)*+{\Gamma
(T(\bigsqcap_i M_i))} ="2";
(-30,-6)*+{{\mathsf{Control }}(M_j\times U_j \to M_j)}="3";
(30,-6)*+{{\mathsf{Control }}(\bigsqcap_i M_i \xrightarrow {pr_j} M_j)}="4";
{\ar@{->}^{{\mathscr{I}}} "1";"2"};
{\ar@{->}^{\varpi_j =D(pr_j)\circ -} "2";"4"};
{\ar@{->}^{} "1";"3"};
{\ar@{->}_{{\mathscr{I}}_j} "3";"4"};
\endxy$$ commute for each $j$. The components ${\mathscr{I}}_j$ of the interconnection map ${\mathscr{I}}$ are defined by $ {\mathscr{I}}_j (F_j) := F_j \circ
s_j$ for all $j$.
The space of vector fields $\Gamma (T(\bigsqcap_i M_i))$ on the product $\bigsqcap _i M_i$ is the product of vector spaces ${\mathsf{Control }}(\bigsqcap_i M_i \to M_j)$: $$\Gamma (T(\bigsqcap_i M_i)) = \bigsqcap _j {\mathsf{Control }}(\bigsqcap_i M_i
\xrightarrow{pr_j}M_j).$$ In other words a vector field $X $ on the product $\bigsqcap _i M_i$ is a tuple $X = (X_1, \ldots, X_N)$, where $$X_j := D (pr_j)\circ X.$$ ($D (pr_j)\colon T \bigsqcap M_i \to TM_j)$ denotes the differential of the canonical projection $pr_j\colon \bigsqcap M_i \to M_j$.) Each component $X_j \colon \bigsqcap _i M_i \to TM_i$ is a control system.
To define a map from a vector space into a product of vector spaces it is enough to define a map into each of the factors. We have canonical projections $$\pi_j\colon \bigsqcap_i {\mathsf{Control }}(M_i\times U_i \to M_i) \to
{\mathsf{Control }}(M_j\times U_j\to M_j), \quad j=1,\ldots, N.$$ Consequently to define the interconnection map ${\mathscr{I}}$ it is enough to define the maps $${\mathscr{I}}_j\colon {\mathsf{Control }}(M_j\times U_j\to M_j) \to {\mathsf{Control }}(\bigsqcap_i M_i
\xrightarrow {pr_j} M_j).$$ for each index $j$. We therefore define the maps ${\mathscr{I}}_j\colon {\mathsf{Control }}(M_j\times U_j\to M_j)
\to {\mathsf{Control }}(\bigsqcap_i M_i \xrightarrow {pr_j} M_j)$, $1\leq j\leq N$, by $${\mathscr{I}}_j (F_j) := F_j \circ s_j.$$
\[rmrk:2.21\] It will be useful for us to remember that the canonical projections $$\varpi_j\colon \Gamma (T\bigsqcap M_i))\to {\mathsf{Control }}(\bigsqcap M_i \to M_j)$$ are given by $$\varpi_j (X) = D (pr_j)\circ X,$$ where $D (pr_j)\colon T \bigsqcap M_i \to TM_j$ are the differentials of the canonical projections $pr_j\colon \bigsqcap M_i \to M_j$.
Interconnections and graphs {#interconnections-and-graphs .unnumbered}
---------------------------
We next explain how finite directed graphs whose nodes are decorated with phase spaces, that is, networks of manifolds in the sense of Definition \[def.network.mfld\] give rise to interconnection maps. To do this precisely it is useful to have a notion of an [*input trees*]{} of a directed graph. Given a graph, an input tree $I(a)$ of a vertex $a$ is — roughly — the vertex itself and all of the arrows leading into it. We want to think of this as a graph in its own right, as follows.
\[def:input-tree\] Given a vertex $a$ of a graph $ G$ we define the [*input tree*]{} $I(a)$ to be a graph with the set of vertices $I(a)_0$ given by $$I(a)_0 := \{a\}\sqcup {\mathfrak t}{^{-1}}(a);$$ where, as before, the set ${\mathfrak t}{^{-1}}(a)$ is the set of arrows in $
G$ with target $a$. The set of edges $I(a)_1$ of the input tree is the set of pairs $$I(a)_1 := \{(a, \gamma)\mid \gamma \in G_1, \,\,\,{\mathfrak t}(\gamma) =a \},$$ and the source and target maps $I(a)_1{\rightrightarrows}I(a)_0$ are defined by $${\mathfrak s}(a, \gamma) = \gamma \quad\textrm{and}\quad {\mathfrak t}(a, \gamma) = a.$$ In pictures,
at (2,0) (1) [$\gamma$]{}; at (4,0) (2) [$a$]{};
\(1) edge \[bend left\] node [$(a,\gamma)$]{} (2) ;
\[example:2\] Consider the graph
at (0,-1) [$G = $]{}; at (2,-1) (1) [a]{}; at (4,-1) (2) [b]{};
\(1) edge \[bend left\] node [$\beta$]{} (2) (1) edge \[bend right\] node \[below\] [$\alpha$]{} (2) ;
as in Example \[example:1\]. Then the input tree $I(a)$ is the graph with one node $a$ and no edges: $I(a) = \bullet \,a$. The input tree $I(b)$ has three nodes and two edges:
at (0,0) [$I(b) = $]{}; at (2,1) (1) [$\alpha$]{}; at (2,-1) (2) [$\beta$]{}; at (4,0) (3) [$b$]{};
\(1) edge \[bend left = 10\] node [$(b,\alpha)$]{} (3) (2) edge \[bend right = 20\] node \[below\] [$(b,\beta)$]{} (3) ;
Notice that our definition of input tree “pulls apart” multiple edges coming from a common vertex.
\[remark:xi\] For each node $a$ of a graph $G$ we have a natural map of graphs $$\xi= \xi_a \colon I(a)\to G, \quad \xi_a (a, \gamma) = \gamma,$$ which need not be injective.
\[empt:2.24\] Let $G$ be a graph with a phase space function ${{\mathcal P}}\colon G\to {\mathsf{Man}}$ and let $a$ be a node of $G$. We then have a graph $\{a\}$ with one node and no arrows. Denote the inclusion of $\{a\}$ in $G$ by $\iota_a$ and the inclusion into its input tree $I(a)$ by $j_a$. Then the diagram of maps of graphs $$\xy
(-8, 10)*+{ \{a\}} ="1";
(8, 10)*+{ I(a)} ="2";
(0, -2)*+{G}="3";
{\ar@{->}^{ j_a} "1";"2"};
{\ar@{->}_{ \iota_a} "1";"3"};
{\ar@{->}^{\xi} "2";"3"};
\endxy$$ commutes. By Remark \[rmrk:2.16\] we have a commuting diagram of maps of manifolds $$\xy
(-10, 10)*+{ {\mathbb{P}}\{a\}} ="1";
(10, 10)*+{ {\mathbb{P}}I(a)} ="2";
(0, -5)*+{{\mathbb{P}}G}="3";
{\ar@{<-}^{ {\mathbb{P}}j_a} "1";"2"};
{\ar@{<-}_{ {\mathbb{P}}\iota_a} "1";"3"};
{\ar@{<-}^{{\mathbb{P}}\xi} "2";"3"};
\endxy$$
\[empt:2.25\] Let us now examine more closely the map ${\mathbb{P}}j_a\colon {\mathbb{P}}I(a)\to {\mathbb{P}}a$ in \[empt:2.24\] above. Since the set of nodes $I(a)_0$ of the input tree $I(a)$ is the disjoint union $$I(a)_0 = \{a\} \sqcup {\mathfrak t}{^{-1}}(a),$$ and since $\xi_a (\gamma) = {\mathfrak s}(\gamma)$ for any $\gamma \in {\mathfrak t}{^{-1}}(a) \subset I(a)_0$, we have $${\mathbb{P}}I(a) = {{\mathcal P}}(a) \times \bigsqcap _{\gamma \in {\mathfrak t}{^{-1}}(a)} {{\mathcal P}}({\mathfrak s}(\gamma)).$$ Since $j_a\colon \{a\} \to I(a)_0 = \{a\} \sqcup {\mathfrak t}{^{-1}}(a)$ is the inclusion, $${\mathbb{P}}j_a \colon {\mathbb{P}}I(a) \to {\mathbb{P}}a$$ is the projection $${{\mathcal P}}(a) \times \bigsqcap _{\gamma \in {\mathfrak t}{^{-1}}(a)} {{\mathcal P}}({\mathfrak s}(\gamma)) \to {\mathbb{P}}a.$$ Similarly $${\mathbb{P}}\iota_a\colon {\mathbb{P}}G \to {\mathbb{P}}a$$ is the projection $$\bigsqcap _{b\in G_0} {{\mathcal P}}(b) \to {{\mathcal P}}(a).$$
Putting \[empt:2.24\] and \[empt:2.25\] together we get
\[prop:2.26\] Given a graph $G$ with a phase space function ${{\mathcal P}}\colon G_0 \to {\mathsf{Man}}$, that is, a network $(G,{{\mathcal P}})$ of manifolds, we have commutative diagrams of maps of manifolds $$\xy
(-40, 10)*+{ {\mathbb{P}}I(a)} ="0";
(-10, 10)*+{
{{\mathcal P}}(a)\times \bigsqcap _{\gamma \in {\mathfrak t}{^{-1}}(a)}{{\mathcal P}}({\mathfrak s}(\gamma))} ="1";
(30, 10)*+{ {{\mathcal P}}(a)} ="2";
(-10, -10)*+{\bigsqcap_{b\in G_0} {{\mathcal P}}(b)}="3";
(-30, -10)*+{{\mathbb{P}}G=};
{\ar@{=}^{ } "1";"0"};
{\ar@{->}^(.75){ {\mathbb{P}}j_a} "1";"2"};
{\ar@{->}_{ {\mathbb{P}}\iota_a} "3";"2"};
{\ar@{->}^{{\mathbb{P}}\xi} "3";"1"};
\endxy$$ for each node $a$ of the graph $G$.
\[example:3\] Suppose
at (1,-1) [$G = $]{}; at (2,-1) (1) [a]{}; at (4,-1) (2) [b]{};
\(1) edge \[bend left\] node (2) (1) edge \[bend right\] node \[below\] (2) ;
is a graph as in Example \[example:1\] and suppose ${{\mathcal P}}\colon G_0
\to {\mathsf{Man}}$ is a phase space function. Then $${\mathbb{P}}I(b) \simeq {{\mathcal P}}(a) \times {{\mathcal P}}(a) \times {{\mathcal P}}(b),$$ ${\mathbb{P}}j_b$ is the projection ${{\mathcal P}}(a)
\times {{\mathcal P}}(a) \times {{\mathcal P}}(b) \to {{\mathcal P}}(b),$ and $${\mathsf{Control }}({\mathbb{P}}I(b) \to
{\mathbb{P}}b) = {\mathsf{Control }}({{\mathcal P}}(a) \times {{\mathcal P}}(a) \times {{\mathcal P}}(b) \to {{\mathcal P}}(b)).$$ On the other hand ${\mathbb{P}}I(a) = {{\mathcal P}}(a)$, ${\mathbb{P}}j_a\colon {{\mathcal P}}(a)\to
{{\mathcal P}}(a)$ is the identity map and $${\mathsf{Control }}({\mathbb{P}}I(a)\to {\mathbb{P}}a) = \Gamma (T{{\mathcal P}}(a)),$$ the space of vector fields on the manifold ${{\mathcal P}}(a)$.
Given a network $(G,{{\mathcal P}})$ of manifolds we have a product of vector spaces $${\mathpzc{Ctrl}}(G,{{\mathcal P}}):= \bigsqcap _{a\in G_0} {\mathsf{Control }}( {\mathbb{P}}I(a)\to {\mathbb{P}}a).$$ The elements of ${\mathpzc{Ctrl}}(G,{{\mathcal P}})$ are unordered tuples of $(w_a)_{a\in
G_0}$ of control systems (q.v. \[empt:cat-prod\]). We may think of them as sections of the vector bundle $\bigsqcup _{a\in
G_0}{\mathsf{Control }}( {\mathbb{P}}I(a)\to {\mathbb{P}}a) \to G_0$ over the vertices of $G$.
It is easy to see that Propositions \[prop:3.2\] and \[prop:2.26\] give us
\[thm:3.8\] Given a network $(G,{{\mathcal P}})$ of manifolds, there exists a natural interconnection map $${\mathscr{I}}\colon \bigsqcap_{a\in G_0} {\mathsf{Control }}({\mathbb{P}}I(a)\to {\mathbb{P}}a) \to \Gamma (T{\mathbb{P}}G))$$ with $$\varpi_a \circ {\mathscr{I}}( (w_b)_{b\in G_0}) = w_a \circ {\mathbb{P}}j_a$$ for all nodes $a\in G_0$. Here $\varpi_a\colon \Gamma T{\mathbb{P}}G) \to
{\mathsf{Control }}( {\mathbb{P}}G_0 \xrightarrow{{\mathbb{P}}\iota_a} {\mathbb{P}}a)$ are the projection maps; $\varpi_a = D ({\mathbb{P}}\iota_a)$ (q.v. Remark \[rmrk:2.21\]).
\[example:4\] Consider the graph $G$ as in Examples \[example:1\] and \[example:4\] with a phase space function ${{\mathcal P}}\colon G_0\to {\mathsf{Man}}$. Then the vector field $$X = {\mathscr{I}}(w_a, w_b)\colon {{\mathcal P}}(a)\times {{\mathcal P}}(b) \to T{{\mathcal P}}(a)\times T{{\mathcal P}}(b)$$ is of the form $$X(x,y) = (w_a (x), w_b (x,x, y)) \quad \textrm{ for all } (x,y) \in
{{\mathcal P}}(a)\times {{\mathcal P}}(b).$$
\[example:4’\] Consider the graph
at (1,-1) [$G = $]{}; at (2,-1) (1) [a]{}; at (4,-1) (2) [b]{}; at (6,-1) (3) [c]{};
\(1) edge \[bend left\] node (2) (1) edge \[bend right\] node \[below\] (2) (2) edge node (3) ;
and let ${{\mathcal P}}\colon G_0\to {\mathsf{Man}}$ be a phase space function. Then $$\left({\mathscr{I}}(w_a, w_b, w_c)\right) (x,y,z) =
(w_a (x), w_b (x,x, y), w_c (y,z))$$ for all $(w_a,w_b, w_c) \in {\mathpzc{Ctrl}}(G,{{\mathcal P}})$ and all $(x,y,z)\in {{\mathcal P}}(a)\times {{\mathcal P}}(b) \times {{\mathcal P}}(c)$.
Maps of dynamical systems from graph fibrations
===============================================
Following Boldi and Vigna [@Vigna1] (see also [@VignaFibPage]) we single out a class of maps of graphs called graph fibrations.
\[graph fibration\] A map $\varphi\colon G\to G'$ of directed graphs is a [graph fibration]{} if for any vertex $a$ of $ G$ and any edge $e'$ of $ G'$ ending at $\varphi(a)$ there is a unique edge $e$ of $ G$ ending at $a$ with $\varphi (e) = e'$.
\[example:5\] The map of graphs
at (-2,0) (a1) [a$_1$]{}; at (-2,-2) (a2) [a$_2$]{}; at (0,-1) (bb) [b]{}; at (3,-1) (1) [a]{}; at (5,-1) (2) [b]{}; at (7,-1) (3) [c]{}; at (1.5,-1) (arrow) [$\implies$]{};
(a1) edge node [$\gamma$]{} (bb) (a2) edge node \[below\] [$\delta$]{} (bb) (1) edge \[bend left\] node [$\gamma'$]{} (2) (1) edge \[bend right\] node \[below\] [$\delta'$]{} (2) (2) edge node (3)
;
sending the edge $\gamma$ to $\gamma'$ and the edge $\delta$ to $\delta'$ is a graph fibration.
\[empt:4.2\] Given any maps $\varphi\colon G\to G'$ of graphs and a node $a$ of $G$ there is an induced map of input trees $$\varphi_a\colon I(a) \to I(\varphi(a)).$$ On edges of $I(a)$ the map is defined by $$\varphi (a, \gamma):= (\varphi (a), \varphi (\gamma))$$ (cf. Definition \[def:input-tree\]). Moreover the diagram of graphs $$\xy
(-12, 8)*+{I(a)} ="1";
(12, 8)*+{I(\varphi(a))} ="2";
(-12, -10)*+{G}="3";
(12, -10)*+{G'}="4";
{\ar@{->}^{ \varphi_a} "1";"2"};
{\ar@{->}_{ \xi_a} "1";"3"};
{\ar@{->}^{ \xi_{\varphi(a)}} "2";"4"};
{\ar@{->}^{\varphi} "3";"4"};
\endxy$$ commutes (the map $\xi_a\colon I(a)\to G$ from an input tree to the original graph is defined in Remark \[remark:xi\]).
\[lemma:4.3\] If $\varphi\colon G\to G'$ is a graph fibration then the induced maps $$\varphi_a\colon I(a)\to I(\varphi(a))$$ of input trees defined above are isomorphisms for all nodes $a$ of $G$.
Given an edge $(\varphi(a), \gamma')$ of $I(\varphi(a))$ there is a unique edge $\gamma$ of $G$ with $\varphi(\gamma) = \gamma'$ and ${\mathfrak t}(\gamma) = a$ and consequently $\varphi_a (a,\gamma) =
(\varphi(a),\gamma')$. It follows that $\varphi_a$ is bijective on vertices and edges.
Recall that a map from a network $(G,{{\mathcal P}})$ to a network $(G',{{\mathcal P}}')$ is a map of graphs $\varphi\colon G\to G'$ with the property that $${{\mathcal P}}'\circ \varphi = {{\mathcal P}}.$$
A map of networks $\varphi\colon (G,{{\mathcal P}}) \to (G',{{\mathcal P}}')$ of manifolds is a [fibration]{} if $\varphi\colon G\to G'$ is a graph fibration.
The theorem \[thm:4.4\] below is our reason for singling out fibrations of networks.
\[thm:4.4\] A fibration $\varphi\colon (G,{{\mathcal P}})\to (G',{{\mathcal P}}')$ of networks naturally induces a linear map $$\varphi^*\colon {\mathpzc{Ctrl}}(G',{{\mathcal P}}')\to {\mathpzc{Ctrl}}(G,{{\mathcal P}}).$$
Since $${\mathpzc{Ctrl}}(G,{{\mathcal P}}) = \bigsqcap_{a\in G_0} {\mathsf{Control }}({\mathbb{P}}I(a)\to {\mathbb{P}}a)$$ is a product of vector spaces, the map $\varphi^*$ is uniquely determined by maps from $ {\mathpzc{Ctrl}}(G',{{\mathcal P}}')$ to the factors ${\mathsf{Control }}({\mathbb{P}}I(a)\to {\mathbb{P}}a)$, $a\in G_0$. On the other hand we have canonical projections $$\pi_b\colon {\mathpzc{Ctrl}}(G',{{\mathcal P}}') = \bigsqcap_{c\in G'_0} {\mathsf{Control }}({\mathbb{P}}I(c)\to
{\mathbb{P}}c)\to {\mathsf{Control }}({\mathbb{P}}I(b)\to {\mathbb{P}}b)$$ for all $b\in G_0'$. Hence in order to define the map $\varphi^*$ it is enough to define maps of vector spaces $$\varphi_a^*\colon {\mathsf{Control }}({\mathbb{P}}I(\varphi(a))\to {\mathbb{P}}\varphi(a))\to {\mathsf{Control }}({\mathbb{P}}I(a)\to {\mathbb{P}}a)$$ for all nodes $a$ of the graph $G$. By \[empt:4.2\] the diagram $$\xy
(-12, 8)*+{I(a)} ="1";
(12, 8)*+{I(\varphi(a))} ="2";
(-12, -10)*+{G}="3";
(12, -10)*+{G'}="4";
(0, -22)*+{{\mathsf{Man}}}="5";
{\ar@{->}^{ \varphi_a} "1";"2"};
{\ar@{->}_{ \xi_a} "1";"3"};
{\ar@{->}^{ \xi_{\varphi(a)}} "2";"4"};
{\ar@{->}^{\varphi} "3";"4"};
{\ar@{->}_{{{\mathcal P}}} "3";"5"};
{\ar@{->}^{{{\mathcal P}}'} "4";"5"};
\endxy$$ commutes for each $a\in G_0$. Let $$\varphi|_{\{a\}}\colon \{a\} \to \{\varphi (a)\}$$ denote the restriction of $\varphi\colon G\to G'$ to the subgraph $\{a\}\hookrightarrow G$. It is easy to see that the diagrams $$\xy
(-12, 8)*+{I(a)} ="1";
(12, 8)*+{I(\varphi(a))} ="2";
(-12, -10)*+{\{a\}}="3";
(12, -10)*+{\{\varphi(a)\}}="4";
{\ar@{->}^{ \varphi_a} "1";"2"};
{\ar@{->}_{ j_a} "3";"1"};
{\ar@{->}^{ j_{\varphi(a)}} "4";"2"};
{\ar@{->}^{\varphi|_{\{a\}}} "3";"4"};
\endxy$$ commutes as well. By Lemma \[lemma:4.3\] the map $\varphi_a$ is an isomorphism of graphs. Hence $${\mathbb{P}}\varphi_a\colon {\mathbb{P}}I(a) \to {\mathbb{P}}I(\varphi(a))$$ is an isomorphism of manifolds. Define $$\varphi_a^*\colon {\mathsf{Control }}({\mathbb{P}}I(\varphi(a))\to {\mathbb{P}}\varphi(a))\to {\mathsf{Control }}({\mathbb{P}}I(a)\to {\mathbb{P}}a)$$ by $$\varphi_a^*(F) = D{\mathbb{P}}(\varphi|_{\{a\}}) \circ F \circ ({\mathbb{P}}\varphi_a){^{-1}}$$ for all $F\in {\mathsf{Control }}({\mathbb{P}}I(\varphi(a))$. By the universal property of products this gives us the desired map $\varphi^*$. Moreover the diagrams $$\xy
(-30, 8)*+{{\mathpzc{Ctrl}}(G',{{\mathcal P}}')} ="1";
(30, 8)*+{{\mathpzc{Ctrl}}(G,{{\mathcal P}})} ="2";
(-30, -10)*+{{\mathsf{Control }}({\mathbb{P}}I(\varphi(a))\to {\mathbb{P}}\varphi(a))}="3";
(30, -10)*+{{\mathsf{Control }}({\mathbb{P}}I(a) \to {\mathbb{P}}a)\}}="4";
{\ar@{->}^{ \varphi^*} "1";"2"};
{\ar@{->}_{ \pi_{\varphi(a)}} "1";"3"};
{\ar@{->}^{ \pi_a} "2";"4"};
{\ar@{->}^{\varphi_a^*} "3";"4"};
\endxy$$ commute for all $a\in G_0$.
\[example:6\] We write down an example of the map $\varphi^*$ constructed in Theorem \[thm:4.4\]. Consider the graph fibration $\varphi\colon
G\to G'$, or
at (-2,0) (a1) [a$_1$]{}; at (-2,-2) (a2) [a$_2$]{}; at (0,-1) (bb) [b]{}; at (3,-1) (1) [a]{}; at (5,-1) (2) [b]{}; at (7,-1) (3) [c]{}; at (1.5,-1) (arrow) [$\longrightarrow$]{};
(a1) edge node [$\gamma$]{} (bb) (a2) edge node \[below\] [$\delta$]{} (bb) (1) edge \[bend left\] node [$\gamma'$]{} (2) (1) edge \[bend right\] node \[below\] [$\delta'$]{} (2) (2) edge node (3)
;
as in Example \[example:5\]. Let ${{\mathcal P}}'\colon G_0'\to {\mathsf{Man}}$ be a phase space function. Then $${\mathpzc{Ctrl}}(G',{{\mathcal P}}') = \{(w_a\colon {{\mathcal P}}(a)\to T{{\mathcal P}}(a) ,w_b\colon {{\mathcal P}}'(a)\times {{\mathcal P}}'(a)
\times {{\mathcal P}}'(b)\to T{{\mathcal P}}(b), w_c\colon {{\mathcal P}}(b)\times {{\mathcal P}}(c) \to T{{\mathcal P}}(c))\},
$$ $${\mathpzc{Ctrl}}(G,{{\mathcal P}}'\circ \varphi) = \{(w'_{a_1}\colon {{\mathcal P}}' (a)\to {{\mathcal P}}' (a),
w_{a_2}\colon {{\mathcal P}}' (a)\to {{\mathcal P}}' (a) ,w_b\colon {{\mathcal P}}'(a)\times {{\mathcal P}}'(a) \times
{{\mathcal P}}'(b)\to T{{\mathcal P}}' (b))\}$$ and $$\varphi^* (w_a',w_b',w_c') = (w_a', w_a', w_b').$$
We are now in the position to state and prove the main result of the paper.
\[thm:main\] Let $\varphi\colon (G,{{\mathcal P}})\to (G',{{\mathcal P}}')$ be a fibration of networks of manifolds. Then the pullback map $$\varphi^*\colon {\mathpzc{Ctrl}}(G',{{\mathcal P}}')\to {\mathpzc{Ctrl}}(G,{{\mathcal P}})$$ constructed in Theorem \[thm:4.4\] is compatible with the interconnection maps $${\mathscr{I}}'\colon {\mathpzc{Ctrl}}(G',{{\mathcal P}}')) \to \Gamma (T{\mathbb{P}}G')
\quad \textrm{and} \quad
{\mathscr{I}}\colon ({\mathpzc{Ctrl}}(G,{{\mathcal P}})) \to \Gamma (T{\mathbb{P}}G).$$ Namely for any collection $w'\in {\mathpzc{Ctrl}}(G',{{\mathcal P}}')$ of open systems on the network $(G', {{\mathcal P}}')$ the diagram $$\label{eq:7}
\xy
(-15, 10)*+{T{\mathbb{P}}G'}="1";
(15, 10)*+{T{\mathbb{P}}G}="2";
(-15, -5)*+{{\mathbb{P}}G'}="3";
(15, -5)*+{{\mathbb{P}}G}="4";
{\ar@{->}^{D{\mathbb{P}}\varphi } "1";"2"};
{\ar@{->}^{{\mathscr{I}}'(w') } "3";"1"};
{\ar@{->}_{{\mathscr{I}}(\varphi^*w') } "4";"2"};
{\ar@{->}_{{\mathbb{P}}\varphi} "3";"4"};
\endxy$$ commutes. Consequently $${\mathbb{P}}\varphi\colon ({\mathbb{P}}(G',{{\mathcal P}}'), {\mathscr{I}}' (w'))\to ({\mathbb{P}}(G, {{\mathcal P}}),
{\mathscr{I}}(\varphi^* w'))$$ is a map of dynamical systems.
Recall that the manifold ${\mathbb{P}}G$ is the product $\bigsqcap_{a\in G_0} {\mathbb{P}}a$. Hence the tangent bundle bundle $T{\mathbb{P}}G$ is the product $\bigsqcap_{a\in G_0} T{\mathbb{P}}a$. The canonical projections $$T{\mathbb{P}}G\to T{\mathbb{P}}a$$ are differential of the maps ${\mathbb{P}}\iota_a\colon {\mathbb{P}}G\to {\mathbb{P}}a$ where, as before, $\iota_a\colon \{a\} \hookrightarrow G$ is the canonical inclusion of graphs. Hence by the universal property of products, two maps into $T{\mathbb{P}}G$ are equal if and only if all their components are equal. Therefore, in order to prove that commutes it is enough to show that $$D {\mathbb{P}}\iota_a \circ {\mathscr{I}}(\varphi^*w') \circ {\mathbb{P}}\varphi =
D {\mathbb{P}}\iota_a \circ D {\mathbb{P}}\varphi\circ {\mathscr{I}}'(w')$$ for all nodes $a\in G_0$. By definition of the restriction $\varphi|_{\{a\}}$ of $\varphi\colon G\to G'$ to $\{a\}\hookrightarrow G$, the diagram $$\label{eq:**}
\xy
(-10, 15)*+{\{a\}}="1";
(10, 15)*+{\{\varphi(a)\}}="2";
(-10, 0)*+{ G}="3";
(10, 0)*+{ G'}="4";
{\ar@{->}^{\varphi|_{\{a\}} } "1";"2"};
{\ar@{->}_{\iota_a } "1";"3"};
{\ar@{->}_{\iota_{\varphi(a)} } "2";"4"};
{\ar@{->}_{\varphi} "3";"4"};
\endxy$$ commutes. By the definition of the pullback map $\varphi^*$ and the interconnection maps ${\mathscr{I}}$, ${\mathscr{I}}'$ the diagram $$\label{eq:8}
\xy
(-15, 15)*+{T{\mathbb{P}}a}="1";
(15, 15)*+{T{\mathbb{P}}\varphi(a)}="2";
(-15, -0)*+{{\mathbb{P}}I(a)}="3";
(15, 0)*+{{\mathbb{P}}I(\varphi(a)) }="4";
(-15, -15)*+{{\mathbb{P}}G}="5";
(15, -15)*+{{\mathbb{P}}G'}="6";
{\ar@{->}^{D{\mathbb{P}}\varphi|_{\{a\}} } "2";"1"};
{\ar@{->}_{(\varphi^*w')_a } "3";"1"};
{\ar@{->}^{w'_{\varphi(a)} } "4";"2"};
{\ar@{->}_{{\mathbb{P}}\xi_a } "5";"3"};
{\ar@{->}^{{\mathbb{P}}\xi_{\varphi(a)} } "6";"4"};
{\ar@{->}_{{\mathbb{P}}\varphi_a} "4";"3"};
{\ar@{->}^{{\mathbb{P}}\varphi} "6";"5"};
{\ar@/^{3pc}/^{{\mathscr{I}}(\varphi^*w')_a} "5";"1"};
{\ar@/_{3pc}/_{{\mathscr{I}}' (w')_{\varphi(a)}} "6";"2"};
\endxy$$ commutes as well. We now compute: $$\begin{aligned}
D{\mathbb{P}}\iota_a \circ {\mathscr{I}}(\varphi^*w')\circ {\mathbb{P}}\varphi
&= & ({\mathscr{I}}(\varphi^*w'))_a \circ {\mathbb{P}}\varphi\\
&= & D {\mathbb{P}}(\varphi|_{\{a\}})\circ {\mathscr{I}}' (w')_{\varphi(a)}
\quad \quad \quad \textrm{\quad by (\ref{eq:8})} \\
& = & D {\mathbb{P}}(\varphi|_{\{a\}})\circ D{\mathbb{P}}\iota_{\varphi(a)} \circ {\mathscr{I}}' (w')
\quad \textrm{ by definition of } {\mathscr{I}}' (w')_{\varphi(a)}\\
&=& D {\mathbb{P}}\left(\iota_{\varphi(a)} \circ \varphi|_{\{a\}}
\right) \circ {\mathscr{I}}' (w')
\quad \quad \textrm{ since ${\mathbb{P}}$ is a contravariant functor }\\
& = & D {\mathbb{P}}\left(\varphi \circ \iota_a
\right) \circ {\mathscr{I}}' (w')
\quad \quad \quad \quad \quad\textrm{ by \eqref{eq:**} }\\
&=& D {\mathbb{P}}(\iota_a) \circ D{\mathbb{P}}\varphi
\circ {\mathscr{I}}' (w').\end{aligned}$$ And we are done.
\[example:14\] Consider the graph fibration
at (-2,0) (a1) [a$_1$]{}; at (-2,-2) (a2) [a$_2$]{}; at (0,-1) (bb) [b]{}; at (3,-1) (1) [a]{}; at (5,-1) (2) [b]{}; at (7,-1) (3) [c]{}; at (1.5,-1) (arrow) [$\longrightarrow$]{};
(a1) edge node [$\gamma$]{} (bb) (a2) edge node \[below\] [$\delta$]{} (bb) (1) edge \[bend left\] node [$\gamma'$]{} (2) (1) edge \[bend right\] node \[below\] [$\delta'$]{} (2) (2) edge node (3)
;
as in Examples \[example:5\] and \[example:6\]. Let ${{\mathcal P}}'\colon G_0'\to
{\mathsf{Man}}$ be a phase space function and let ${{\mathcal P}}= {{\mathcal P}}'\circ \varphi$. Then $${\mathbb{P}}G' = {{\mathcal P}}'(a)\times {{\mathcal P}}' (b)\times {{\mathcal P}}' (c),$$ $${\mathbb{P}}G = {{\mathcal P}}'(a)\times {{\mathcal P}}' (a)\times {{\mathcal P}}' (b),$$ $${\mathbb{P}}\varphi (x,y,z) = (x,x,y),$$ and $$D {\mathbb{P}}\varphi (p,q, r) = (p,p,q).$$ For any $w' = (w_a', w_b', w_c') \in {\mathpzc{Ctrl}}(G', {{\mathcal P}}')$, $$\left({\mathscr{I}}' (w')\right) (x,y,z) = (w'_a (x), w_b' (x,x, y), w_c' (y,z)),$$ $$\varphi^* w' = (w_a', w_a', w_b'),$$ $$\left({\mathscr{I}}(\varphi^* w')\right) (x_1,x_2,y) =
(w'_a (x_1), w'_a (x_2), w_b' (x_1,x_2, y))$$ and $$\left({\mathscr{I}}(\varphi^* w')\circ {\mathbb{P}}\varphi\right) (x,y,z) =
(w'_a (x), w'_a (x), w_b' (x,x, y))$$ while $$\left(D{\mathbb{P}}\varphi\circ {\mathscr{I}}' ( w')\right) (x,,y,z) =
D{\mathbb{P}}\varphi (w'_a (x), w_b' (x,x, y, w'_c (y,z)) = (w'_a (x), w'_a (x), w_b' (x,x, y)).$$ Hence $$\left({\mathscr{I}}(\varphi^* w')\circ {\mathbb{P}}\varphi\right) = \left(D{\mathbb{P}}\varphi\circ {\mathscr{I}}' ( w')\right)$$ as expected.
In the Example \[example:14\] above the map $\varphi\colon G\to G'$ is neither injective nor surjective. It can, of course, be factored as a surjection $\psi\colon G\to G''$:
at (-2,0) (a1) [a$_1$]{}; at (-2,-2) (a2) [a$_2$]{}; at (0,-1) (bb) [b]{}; at (3,-1) (1) [a]{}; at (5,-1) (2) [b]{}; at (1.5,-1) (arrow) [$\longrightarrow$]{};
(a1) edge node [$\gamma$]{} (bb) (a2) edge node \[below\] [$\delta$]{} (bb) (1) edge \[bend left\] node [$\gamma'$]{} (2) (1) edge \[bend right\] node \[below\] [$\delta'$]{} (2) ;
followed by an injection $\iota\colon G''\to G$:
at (-2,-1) (a) [a]{}; at (0,-1) (bb) [b]{}; at (3,-1) (1) [a]{}; at (5,-1) (2) [b]{}; at (7,-1) (3) [c]{}; at (1.5,-1) (arrow) [$\longrightarrow$]{};
\(a) edge \[bend left\] node [$\gamma'$]{} (bb) (a) edge \[bend right\] node \[below\] [$\delta'$]{} (bb) (1) edge \[bend left\] node [$\gamma'$]{} (2) (1) edge \[bend right\] node \[below\] [$\delta'$]{} (2) (2) edge node (3)
;
The map ${\mathbb{P}}\psi\colon {\mathbb{P}}G'' \to {\mathbb{P}}G$ is easily seen to be given by $${\mathbb{P}}\psi (x,y) = (x,x,y).$$ It is an embedding, as it should (q.v. Lemma \[lemma:2.surj\]). The map ${\mathbb{P}}\imath\colon {\mathbb{P}}G'\to {\mathbb{P}}G''$ is given by $${\mathbb{P}}\imath (x,y,z) = (x,y).$$ It is a submersion (q.v. Lemma \[lemma:2.inj\]). Since ${\mathbb{P}}$ is a contravariant functor, $${\mathbb{P}}\varphi = {\mathbb{P}}(\imath \circ \psi) = {\mathbb{P}}\psi \circ {\mathbb{P}}\imath.$$ We leave it to the reader to check that for any $w' = (w_a', w_b',
w_c') \in {\mathpzc{Ctrl}}(G', {{\mathcal P}}')$, the map ${\mathbb{P}}\imath$ does project the integral curves of the vector field ${\mathscr{I}}(w')$ to the integral curves of the vector field ${\mathscr{I}}(\imath^* w')$ on ${\mathbb{P}}G''$. Furthermore, ${\mathbb{P}}\psi$ embeds the dynamical system $({\mathbb{P}}G'', {\mathscr{I}}(\imath^* w'))$ into the dynamical system $({\mathbb{P}}G, {\mathscr{I}}(\varphi^* w'))$.
Consider the injective graph fibration $\iota\colon G\to G'$: $$\label{eq:ten}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm,
thick,main node/.style={circle,fill=blue!1,draw,font=\sffamily\bfseries}]
\node[main node] at (2,-1) (1) {1};
\node[main node] at (4,-1) (2) {2};
\node[main node] at (6,-1) (3) {3};
\node at (7.5,-1) (i) {$\hookrightarrow$};
\node[main node] at (9,-1) (10') {10};
\node[main node] at (11,-1) (1') {1};
\node[main node] at (13,-1) (2') {2};
\node[main node] at (15,-1) (3') {3};
\node[main node] at (11.2,0.5) (4') {4};
\node[main node] at (13.4,0.5) (5') {5};
\node[main node] at (15.6,0.5) (6') {6};
\node[main node] at (11.2,-2.5) (7') {7};
\node[main node] at (13.4,-2.5) (8') {8};
\node[main node] at (15.6,-2.5) (9') {9};
\path[every node/.style={font=\sffamily\small}]
(2) edge [bend right] node {} (1)
(1) edge [bend right] node [below] {} (2)
(2) edge node {}(3)
(2') edge [bend right] node {} (1')
(1') edge [bend right] node [below] {} (2')
(2') edge node {}(3')
(1') edge node {}(10')
(1') edge node {}(4')
(1') edge node {}(7')
(2') edge node {}(5')
(2') edge node {}(8')
(3') edge node {}(6')
(3') edge node {}(9')
;
\end{tikzpicture}$$ Choose phase space functions ${{\mathcal P}}, {{\mathcal P}}'$ so that $i\colon (G,{{\mathcal P}})\to (G',{{\mathcal P}}')$ is a map of networks. By Theorem \[thm:main\], for any collection $w'\in {\mathpzc{Ctrl}}(G',{{\mathcal P}}')$ of open systems on the network $(G', {{\mathcal P}}')$ the dynamics in the subsystem $({\mathbb{P}}G, {\mathscr{I}}(i^*w'))$ drives the entire system $({\mathbb{P}}G', {\mathscr{I}}(w'))$. This is intuitively clear from the graph since there are no “feedbacks” from vertices $4, \ldots,10$ back into $1,2,3$.
Concluding remarks {#concluding-remarks .unnumbered}
------------------
Theorem \[thm:main\] allows us to define a category $\mathsf{DSN}$ of continuous time dynamical systems on networks. The objects of this category are triples $$(G, {{\mathcal P}}\colon G_0 \to {\mathsf{Man}}, w\in {\mathpzc{Ctrl}}(G,{{\mathcal P}})),$$ where as before $G$ is a finite directed graph, ${{\mathcal P}}$ is a phase space function and $w=(w_a)_{a\in G_0}$ is a tuple of control systems associated with the input trees of the graph $G$ and the function ${{\mathcal P}}$.
A morphism from $(G',{{\mathcal P}}', w')$ to $(G, {{\mathcal P}},w)$ is a graph fibration $\varphi\colon G\to G'$ (yes, the direction of the map is correct) with ${{\mathcal P}}' \circ \varphi = {{\mathcal P}}$ and $\varphi^*w' = w$. Alternatively we may think of a map from $(G',{{\mathcal P}}', w')$ to $(G, {{\mathcal P}},w)$ as a fibration of networks of manifolds $\varphi\colon (G,{{\mathcal P}}) \to (G',{{\mathcal P}}')$ with $\varphi^*w' = w$.
Theorem \[thm:main\] allows us to define a functor from $\mathsf{DSN}$ to the category $\mathsf{DS}$ of continuous time dynamical systems on manifolds. The objects $\mathsf{DS}$ are pairs $(M,X)$, where $M$ is a manifold and $X$ is a vector field on $M$. A morphism from $(M,X)$ to $(N,Y)$ in this category is a smooth map $f\colon M\to N$ between two manifolds with $Df\circ X = Y\circ f$. The functor $$\mathsf{DSN}\to \mathsf{DS}$$ is defined by $$\left( (G',{{\mathcal P}}', w') \xrightarrow{\varphi} (G,{{\mathcal P}},w)\right)\quad \mapsto
\quad \left(({\mathbb{P}}G', {\mathscr{I}}(w'))\xrightarrow{{\mathbb{P}}\varphi} ({\mathbb{P}}G, {\mathscr{I}}(w)\right).$$
Given a dynamical system on a network $(G,{{\mathcal P}},w)$ we can forget the dynamics. This defines a functor $$\mathsf{DSN} \to ({\mathsf{Man}}/{\mathsf{Graph}})_{\mbox{\sf{\tiny{fib}}}}$$ from the category of dynamical systems on networks to a subcategory of the category of networks of manifolds whose maps are fibrations of networks (hence the subscript $_{\mbox{\sf{\tiny{fib}}}}$). Composing the functor above with the functor ${\mathsf{Man}}/{\mathsf{Graph}}\to {{{\mathsf{Graph}}}^{\mbox{\sf{\tiny{op}}}}}$ forgets all the information except for the graph. These two functors from $\mathsf{DSN}$ to $\mathsf{DN}$ and to $
{{{\mathsf{Graph}}}^{\mbox{\sf{\tiny{op}}}}}$, respectively, allow us to interpret continuous time dynamical systems on networks both as dynamical systems and as graphs.
Acknowledgments {#acknowledgments .unnumbered}
===============
L.D. was supported by the National Science Foundation under grants CMG-0934491 and UBM-1129198 and by the National Aeronautics and Space Administration under grant NASA-NNA13AA91A.
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[^1]: It may be defined by $\bigsqcup _{s\in
S}M_s = \bigcup _{s\in S} M_s\times \{s\}$.
|
---
abstract: 'A common problem in science networks and private wide area networks (WANs) is that of achieving predictable data transfers of multiple concurrent flows by maintaining specific pacing rates for each. We address this problem by developing a control algorithm based on concepts from model predictive control (MPC) to produce flows with smooth pacing rates and round trip times (RTTs). In the proposed approach, we model the bottleneck link as a queue and derive a model relating the pacing rate and the RTT. A MPC based control algorithm based on this model is shown to avoid the extreme window (which translates to rate) reduction that exists in current control algorithms when facing network congestion. We have implemented our algorithm as a Linux kernel module. Through simulation and experimental analysis, we show that our algorithm achieves the goals of a low standard deviation of RTT and pacing rate, even when the bottleneck link is fully utilized. In the case of multiple flows, we can assign different rates to each flow and as long as the sum of rates is less than bottleneck rate, they can maintain their assigned pacing rate with low standard deviation. This is achieved even when the flows have different RTTs. Index Terms — TCP, Congestion Control Algorithm, Model Predictive Control, Low latency, Linux implementation, Experimental evaluation, Simulation analysis'
author:
- 'Taran Lynn$^{1}$ and Dipak Ghosal$^{2}$ and Nathan Hanford$^{3}$[^1][^2][^3][^4]'
bibliography:
- 'reference.bib'
title: ' **Model Predictive Congestion Control for TCP Endpoints** '
---
[^1]: \*This research was supported by NSF grant CNS-1528087
[^2]: $^{1}$Taran Lynn is an alumnus and assistant researcher at the University of California, Davis [[email protected]]{}
[^3]: $^{2}$Dipak Ghosal is with Department of Computer Science at the University of California, Davis [[email protected]]{}
[^4]: $^{3}$Nathan Hanford is a postdoctoral fellow at Lawrence Livermore National Laboratory, Livermore, CA [[email protected]]{}
|
---
abstract: 'Running agent-based models (ABMs) is a burdensome computational task, specially so when considering the flexibility ABMs intrinsically provide. This paper uses a bundle of model configuration parameters along with obtained results from a validated ABM to train some Machine Learning methods for socioeconomic optimal cases. A larger space of possible parameters and combinations of parameters are then used as input to predict optimal cases and confirm parameters calibration. Analysis of the parameters of the optimal cases are then compared to the baseline model. This exploratory initial exercise confirms the adequacy of most of the parameters and rules and suggests changing of directions to two parameters. Additionally, it helps highlight metropolitan regions of higher quality of life. Better understanding of ABM mechanisms and parameters’ influence may nudge policy-making slightly closer to optimal level.'
author:
- Bernardo Alves Furtado
bibliography:
- 'ml.bib'
title: 'Machine Learning simulates Agent-based Model'
---
Problem context
===============
Agent-based models have gained recognition since the original ideas of Schelling [@schelling_models_1969] and the seminal book of Epstein and Axtell [@epstein_growing_1996]. In fact, a number of books have extensively opened up the debate of applications on geography [@heppenstall_agent-based_2012], economics [@hamill_agent-based_2016], social sciences [@helbing_social_2012; @johnson_non-equilibrium_2017] and policy [@geyer_handbook_2015].
Much of the appreciation of ABM comes from its flexibility and high-level case abstraction. However, such adaptability comes with its great weaknesses, that is lack of benchmarking and reproducibility. The alternatives in the space of possibilities is so large that most authors keep expanding their own specific hands-on problem with little attention to comparability. Some efforts have been made towards resolving it [@grimm_standard_2006; @grimm_odd_2010].
Nonetheless, a single ABM can easily run into hundreds of thousands possible combinations of parameters, rules and behaviors. Sensitivity analysis driven by the modelers knowledge of the model and input data is usually applied to each parameter or rule, one at time, *ceteris paribus*. Clearly, this is not an optimal solution. Consider further that, even a model that runs relatively fast, will demand some effort to run a high number of times. Calibration is thus much dependent on the modeler knowledge and control of the object and the ABM.
Given such context, the goal of this paper is to use input from an ABM in the form of configuration parameters and results (sample of 232) to train a set of Machine Learning algorithms having an optimal socioeconomic output as a target. The trained ML algorithms are then used to predict optimal outputs from a random set of parameters (100,000) within a broader number of combinations. The analysis of actual and predict parameters may hint on optimal parameters and enable further understanding of the model. In practice, the ML algorithm simulates the running of the ABM for 100,000 times within seconds.
Besides this introduction, next section makes a brief presentation of the literature. Section 3 explains the methods and step-by-step procedures used and describes the data. Results and considerations conclude the paper.
Machine Learning
================
Machine Learning has been used together with ABM to automatic calibrate time discrete behavior of agents [@torrens_building_2011], as suggested on an early framework by Rand [@rand_machine_2007]. Although there has not been extensive work on associating ML to ABM, the development of ML models and methods *per se* are well established [@hastie_elements_2009; @james_introduction_2015; @tan_introduction_2006]
We exploratorily use four models of classification:
- Random Forest Classification is a “perturb-and-combine technique\[s\][^1] specifically designed for trees. This means a diverse set of classifiers is created by introducing randomness in the classifier construction. The prediction of the ensemble is given as the averaged prediction of the individual classifiers". [(from sklearn documentation)](http://scikit-learn.org/stable/modules/ensemble.html#forest) In the application, we use 10,000 forests using bootstrap, with a maximum depth of 15.
- Support Vector Classification is a “support vector machine constructs a hyper-plane or set of hyper-planes in a high or infinite dimensional space”. The goal of the algorithm is to get the “hyper-plane that has the largest distance to the nearest training data points of any class" [(from sklearn documentation)](http://scikit-learn.org/stable/modules/svm.html#svm-mathematical-formulation). We use the kernel ’poly’ with degree equals to 3.
- Neural Network. Within neural networks we use the Multi-layer Perceptron classifier (MPL) which is a model that “optimizes the log-loss function using LBFGS”, i.e. an “optimizer in the family of quasi-Newton methods". [from sklearn documentation](http://scikit-learn.org/stable/modules/generated/sklearn.neural_network.MLPClassifier.html) Activation method is ‘than‘: the hyperbolic tan function and maximum iteration is restricted to 2,000.
- and a ’Voting System’ which averages the probabilities of the other classifiers in order to make a prediction. We used the ‘soft‘ voting option.
Model and data
==============
We use data from a Spatially-Economic Agent-based Lab (SEAL) [@furtado_policyspace:_2018], that focus on modeling fiscal analysis on municipalities based on three markets: goods, labor and real estate. SEAL uses official input data to enable an empirical model of 46 Brazilian metropolitan regions represented by their ACPs, i.e., their Areas of Concentrated Population. SEAL is an ABM that focus on relatively understanding the mechanisms and influences of parameters and rules on a number of global and local results that include production, unemployment, Quality of Life or commuting demand. Much of the appeal of SEAL is exactly the ability to change parameters and rules and observe the results.
SEAL usually runs with a 2% sample of the population for 20 years, saving and plotting monthly results. A typical run of that size takes between 5 and 60 minutes. Validation has been achieved [@furtado_policyspace:_2018] by comparing fiscal data and its distributions. Robustness was evaluated analyzing parameters, rules, and ACPs’ sensitivity.
All the tests, however, amount for a total of 232 runs of the model [^2]. That includes an average of 7 intervals per parameter, two for boolean parameters (rules) and sometimes just a single run for a specific ACP. In total, we have 67 parameters, being 46 of those ACPs, three boolean, 5 that are freely in the interval $[0, 1]$. Finally, the other parameters have smaller reasonable interval of variance.
It is easy to see that no amount of feasible runs of SEAL would account for a systematic review of the space of possibilities.
Operational details and proposal
--------------------------------
This subsection details the step-by-step used in the analysis. Full code is available on <https://github.com/BAFurtado/MLsimulatesABM>.[^3]
The first step is to organize the data. Every simulation of SEAL generates a configuration file in JSON format in a base directory. Inside such directory, each individual run is stored in folders that are numbered. Those folders keep the results of individual simulations. Additionally, every base directory saves an average results file, containing macro results of all runs for that given configuration of parameters. Thus, the reading of the data implies walking over all the folders containing parameters and results and organizing them into $X$ and $Y$ matrices.
Such summary also applies specific rules to construct the target. We have developed the model so that the modeler may choose the parameters that configure the optimal social case. That includes choosing parameters and percentiles among the available options. We tested the combinations of higher Quality of Life Index (QLI) and GDP index and lower unemployment and GINI index.
Our baseline case is composed of those results in which the QLI are within the **top** 35% of all results and simultaneously on the **bottom** 35% of unemployment. [^4] Thus, our target is given by the cases in which high quality of live with low unemployment is achieved in the last month of the simulation.
The next step is to train the three selected classification methods (Random Forest, SVC and a neural network/MPL) along with a voting system.
Having the machines trained on the simulation results, a larger, random set of parameters is generated so that it mimics a typical JSON configuration file. We used 100,000 parameters configuration. This is a quick process that results in 100,000 samples that contains some boolean parameters, some values drawn from a normal distribution in which the $mu$ is the same of the observed sample and the $sigma$ is the double of the observed one. Finally, we truncate the distributions to have only positive values in order to guarantee valid parameters. Although we could have a larger variation in the generated parameters, the values need to be both comparable to the observed sample and reasonable within the meaning and mechanisms of the ABM model.
Then, the trained models are applied to the simulated configuration parameters and predicted results are produced. Such a procedure, allows for the investigation of the difference in parameters of the configuration files for those cases that reach the target and those that do not.
#### Proposal.
We use optimal targets built upon simulation results along with the originating configuration parameters to train models of classification. The trained models are then applied to a large number of random configuration files to predict results. We analyze the configuration parameters relatively to the results they produced.
Results and considerations
==========================
We ran tests with a training data of 65% of the sample and the rest we used as tests \[table:1\].[^5] The accuracy of the models was relatively good. However, analysis of the confusion matrix indicated that the goodness of fit came mostly from the non-optimal cases that were correctly predicted. In many cases the number of cases classified as optimal that were optimal were the same as the cases classified as optimal that were not so. [^6] We believe that what influenced this difficulty of classification is mostly the fact that the sample has - by construction - a very small number of optimal cases. Ideally, we would need to compensate that with a larger number of samples. But that was exactly the motivation of having the ML in the first place.
Having that in mind, we decided to analyze the one case in which the accuracy was very high and the confusion matrix does not have many optimal cases that were incorrectly classified, i.e., we analyzed the Random Forest Classification results.
[Tree]{} [SVC]{} [MPL]{} [Voting]{}
--------------- ---- ---------- ---- --------- ---- --------- ---- ------------
[Score]{} 0.9878 0.9634 0.9390 0.9756
[Confusion]{} 75 0 75 0 73 2 75 0
[Matrix]{} 1 6 3 4 3 4 2 5
: Results for accuracy and confusion matrix of classification methods tested
\[table:1\]
Most of the parameters have the same behavior from the sample and the random generation parameters. That means, most of them have the higher or lower value for the optimal case, as the sample does.
We consider the two ones that behaved differently of interest, plus the behavior of the boolean rules.
1. *Labor\_market* has a 13% value above the mean for the random generated parameters whereas the value for the sample is 19.8% below the mean for the optimal case. Labor market is a parameter that determines the frequency that the firm enters the market hiring and firing employees. The ML results suggest that the firm should enter the labor market rather more often than not.
2. *Tax on consumption* for the real case on the optimal case is smaller than the average by a factor of 50% whereas the random parameter suggests the optimal case would have a value that is 15% above the average value. This suggests that considering QLI and unemployment for the configuration and mechanisms of the ABM, a higher tax on consumption may benefit the economy.
Those are differences in terms of direction and magnitude. This may suggest either that the model could be better calibrated or that actual cases would need behavior change in order to move the economies in the direction of the parameters to reach better social results.
The boolean parameters for the optimal case confirm and helps reinforce the conclusions drawn in [@furtado_policyspace:_2018]. That is, that the (i) FPM rule, which is a tax transfer scheme for poorer and smaller municipalities, (ii) that the alternative in which the municipalities are together for fiscal purposes and (iii) that the use by the firm of market information to set wages are all socially beneficial.
Finally, we can use the score of the cities’ parameters in the optimal case as a ranking of the cities. The ranking resulting from the sample cases, has Porto Alegre, Rio de Janeiro, Belo Horizonte, Santos, e Jundiaí as top five ACPs. Interestingly enough (for those who knows the cities), the random optimal case ranking brings other relevant ACPs to the top. It reorders the five first ones and it brings Goiânia, Maringá, São Paulo e Ribeirão Preto to the top 9.
All in all, we used a trained ML to simulate parameters for an optimal case of an ABM model. Results suggest that most of the parameters and the boolean rules are well adjusted with two parameters that should be further investigated. Specifically, the samples of different ACPs was very restricted suggesting that further samples should be provided so that the set of ACPs are better represented in the model.
#### Notes and Comments.
This is an initial exploratory exercise delivered as and end-of-course product for a Machine Learning course at Ipea given by Thiago Marzagão. We appreciate the help and the classes.
[^1]: Breiman, ’Arcing Classifiers’, Annals of Statistics 1998.
[^2]: Some runs included more than a single run so that stochasticity can be evaluated. Results, however, are recorded as averages. Even when there is more than a single run, the configuration parameters for a given result does not change. Thus, to be precise, we have results for 232 exclusive parameters configurations
[^3]: Find also additional results and statistics at the repository.
[^4]: Our sample is extremely small - due to the ABM generating process - which led us to choose a rather generous top and bottom percentiles.
[^5]: We also ran tests with normalized data, but the accuracy was lower.
[^6]: This was specially the case for other target optimal cases, such as higher GDP and lower GINI.
|
---
abstract: |
The full nonlinear dissipative quasigeostrophic model is shown to have a unique temporally almost periodic solution when the wind forcing is temporally almost periodic under suitable constraints on the spatial square–integral of the wind forcing and the $\beta$ parameter, Ekman number, viscosity and the domain size. The proof involves the pullback attractor for the associated nonautonomous dynamical system.
[**Key words:**]{} Quasigeostrophic fluid model, dissipative nonautonomous dynamics, almost periodic motion, pullback attractor.
author:
- |
Jinqiao Duan\
*Department of Mathematical Sciences,\
*Clemson University, Clemson, South Carolina 29634, USA.\
*E-mail: [email protected]\
Peter E. Kloeden\
*FB Mathematik, Johann Wolfgang Goethe Universität,\
*D-60054 Frankfurt am Main, Germany\
*E-mail: [email protected]******
title: 'Dissipative Quasigeostrophic Motion under Temporally Almost Periodic Forcing [^1] '
---
[**Short running title:**]{} Almost Periodic Quasigeostrophic Motion
Introduction
============
The barotropic quasigeostrophic (QG) flow model is derived as an approximation of the rotating shallow water equations by an asymptotic expansion in a small Rossby number. The lowest order approximation, which is also the conservation law for the $0$th order potential vorticity, gives the barotropic QG equation ([@Pedlosky], page 234) $$\begin{aligned}
\label{qg}
\Delta \psi_t + J(\psi, \Delta \psi) + \beta \psi_x = \nu \Delta^2
\psi - r
\Delta \psi + f(x,y,t),\end{aligned}$$ where $\psi(x,y,t)$ is the stream function, $\beta$ $>$ $0$ the meridional gradient of the Coriolis parameter, $\nu$ $>$ $0$ the viscous dissipation constant, $r$ $>$ $0$ the Ekman dissipation constant and $f(x,y,t)$ the wind forcing. In addition, $\Delta$ $=$ ${\partial}_{xx}+{\partial}_{yy} $ is the Laplacian operator in the plane and $J(f,g)$ $=$ $f_xg_y -f_yg_x$ is the Jacobian operator.
Equation (\[qg\]) can be rewritten in terms of the relative vorticity ${\omega}(x,y,t)$ $=$ $\Delta \psi(x, y,t)$ as $$\label{eqn}
{\omega}_t + J(\psi, {\omega}) + \beta \psi_x =\nu \Delta {\omega}- r {\omega}+ f(x,y,t) \; ,$$ For an arbitrary bounded planar domain $D$ with area $|D|$ and piecewise smooth boundary this equation can be supplemented by homogeneous Dirichlet boundary conditions for both $\psi$ and ${\omega}= \Delta \psi$, namely, the no-normal flow and slip boundary conditions ([@Pedlosky2], page 34) $$\label{BC}
\psi(x,y,t) = 0, \qquad {\omega}(x,y,t) = 0 \quad \mbox{on} \; \partial
D \; ,$$ together with an appropriate initial condition, $$\label{IC}
{\omega}(x,y,0) = {\omega}_0(x,y) \quad \mbox{on} \; D \; .$$
The global well–posedness (i.e. existence and uniqueness of smooth solutions) of the dissipative model (\[eqn\])–(\[IC\]) can be obtained similarly as in, for example, [@Barcilon; @Dymnikov; @Wu]; see also [@Bennett]. Steady wind forcing has been used in numerical simulations [@Cessi] and Duan [@Duan] has shown the existence of time periodic quasigeostrophic response of time periodic wind forcing by means of a Leray–Schauder topological degree argument and Browder’s principle. In this paper it is assumed that the wind forcing $f(x,y,t)$ is temporally almost periodic and a concept of pullback attraction [@CKS; @KS2] will be used to establish the existence of a unique temporally almost periodic solution of (\[eqn\])–(\[BC\]) under appropriate constraints on the model parameters. The main result is
\[mth\] Assume that $$\frac{r}2 + \frac{\pi \nu}{|D|} > \frac12 \beta \left(\frac{|D|}{\pi}
+1\right)$$ and that the wind forcing $f(x,y,t)$ is temporally almost periodic with its $L^2(D)$–norm bounded uniformly in time $t$ $\in$ $\rit$ by $$||f(\cdot,\cdot,t)|| \leq \sqrt{\frac{\pi r}{|D|}} \;\; \left[\frac{r}2 +
\frac{\pi
\nu}{|D|}
-\frac12 \beta \left(\frac{|D|}{\pi} +1 \right) \right]^{\frac32}.$$ Then the dissipative quasigeostrophic model (\[eqn\])–(\[BC\]) has a unique temporally almost periodic solution that exists for all time $t$ $\in$ $\rit$.
The necessary terminology will be presented as required in the text and proof that follow. Some mathematical preliminaries are stated below, while dissipativity and strong contraction properties of QG flow are established in Section 2. Background material on pullback attraction for nonautonomous systems is presented in Section 3 and that for almost periodicity in Section 4, where it is applied to the QG model under consideration to complete the proof of Theorem \[mth\].
Standard abbreviations $L^2$ $=$ $L^2(D)$, $H^k_0$ $=$ $H^k_0(D)$, $k = 1, 2,\ldots$, are used for the common Sobolev spaces in fluid mechanics [@Temam], with $<\cdot, \cdot>$ and $\| \cdot \|$ denoting the usual scalar product and norm, respectively, in $L^2$. We need the following properties and estimates (see also [@Dymnikov]) of the Jacobian operator $J: H_0^1 \times H_0^1 \rightarrow L^1$. $$\begin{aligned}
\int_D J(f,g) h \,dxdy & = & - \int_D J(f,h) g \,dxdy,
\label{estimate0}
\\[2 ex]
\int_D J(f,g) g \,dxdy & = & 0,
\label{estimate1}
\\[2 ex]
\left|\int_D J(f,g) \,dxdy \right| & \leq & \|\nabla f\| \|\nabla g\|
\label{estimate2}\end{aligned}$$ for all $f$, $g$, $h$ $\in$ $H^1_0$, and $$\begin{aligned}
\left|\int_D J(\Delta f, g) \Delta h \,dxdy \right|
& \leq & \sqrt{\frac{2|D|}{\pi}} \|\Delta f\| \; \|\Delta g\| \;\|\Delta h\|
\label{estimate2.5}\end{aligned}$$ for all $f$, $g$, $h$ $\in$ $H^2_0$, as well as the [*Poincaré inequality*]{} [@Gilbarg-Trudinger] $$\begin{aligned}
\|g\|^2 = \int_D g^2(x,y) \,dxdy \leq \frac{|D|}{\pi} \int_D |\nabla g|^2
\,dxdy
= \frac{|D|}{\pi} \|\nabla g\|^2\end{aligned}$$ for $g$ $\in$ $H^1_0$, and [*Young’s inequality* ]{} [@Gilbarg-Trudinger] $$AB \leq \frac{{\epsilon}}2 A^2 + \frac{1}{2 {\epsilon}}B^2,$$ where $A,B$ are non-negative real numbers and ${\epsilon}>0$.\
Dynamics of dissipative QG flow
===============================
We first show that the equation (\[eqn\]) with boundary conditions (\[BC\]) is a dissipative system in the sense ([@Hale; @Temam], see also [@Duan]) that all solutions ${\omega}(x,y,t)$ approach a bounded set in the phase space $L^2$ as time goes to infinity provided that the $L^2$ norm of the forcing term is uniformly bounded in time and that the system parameters satisfy the inequality of Theorem \[mth\]. Then we show that the system is strongly contracting under the restriction on the magnitude of the $L^2$ norm of the forcing term assumed in Theorem \[mth\].
Dissipativity property
----------------------
Define the solution operator $S_{t,t_0}$ $:$ $L^2$ $\to$ $L^2$ by $S_{t,t_0} {\omega}_0$ $:=$ ${\omega}(t)$ for $t$ $\geq$ $t_0$, where ${\omega}(t)$ is the solution of the QG equations in $L^2$ starting at ${\omega}_0 $ $\in$ $L^2$ at time $t_0$. Since the the dissipative QG equations (\[eqn\])–(\[BC\]) are strictly parabolic, the solution operators $S_{t,t_0}$ exist and are compact for all $t$ $>$ $t_0$; see, for example, [@Temam]. In fact, the $S_{t,t_0}$ are compact in $H^k_0$ for all $k \geq 0$ and so, in particular, $S_{t,t_0}B$ is a compact subset of $L^2$ for each $t$ $>$ $t_0$ and every closed and bounded subset $B$ of $L^2$.\
Multiplying (\[eqn\]) by ${\omega}$ and integrating over $D$, we obtain $$\begin{aligned}
\frac12 \frac{d}{dt}\|{\omega}\|^2 & = & -\nu \|\nabla {\omega}\|^2 -
r \|{\omega}\|^2 + \int_D f(x,y,t){\omega}\,dxdy
\label{estimate3} \\ [2ex]
& & \quad - \int_D J(\psi, {\omega}){\omega}\,dxdy - \beta \int_D \psi_x {\omega}\,dxdy.
\nonumber\end{aligned}$$ Now $\int_D J(\psi, {\omega}){\omega}\,dxdy$ $=$ $0$ by (\[estimate0\]) and from the Young and Poincaré inequalities we have $$\begin{aligned}
\left|\beta \int_D \psi_x {\omega}\,dxdy \right|
& \leq & \frac12 \beta \left(\int_D \psi_x^2 \,dxdy + \int_D {\omega}^2
\,dxdy\right)
\\
& \leq & \frac12 \beta \left(\frac{|D|}{\pi}\int_D {\omega}^2 \,dxdy + \int_D
{\omega}^2 \,dxdy \right),\end{aligned}$$ that is $$\label{estimate4}
\left|\beta \int_D \psi_x {\omega}\,dxdy \right| \leq
\frac12 \beta \left(\frac{|D|}{\pi} +1 \right) \|{\omega}\|^2,$$ and by the Poincaré inequality again we also have $$\label{estimate5}
-\nu \|\nabla {\omega}\|^2 \leq -\frac{\pi \nu}{|D|} \|{\omega}\|^2.$$ Now assume that the square-integral of the wind forcing $f(x,y,t)$ with respect to $(x,y)$ $\in$ $D$ is uniformly bounded in time, i.e. $$\|f(\cdot,\cdot,t)\| \leq M$$ for some positive constant $M$. (This is mild assumption because a temporally almost periodic function is bounded in time, see [@Besicovitch] and later). Then $$\begin{aligned}
\left|\int_D f(x,y,t){\omega}\,dxdy \right|
& \leq & \frac1{2r} \int_D f^2(x,y,t) \,dxdy + \frac{r}2 \int_D {\omega}^2 \,dxdy
\\
& \leq & \frac{M^2}{2r} + \frac{r}2 \|{\omega}\|^2.
\label{estimate6}\end{aligned}$$
Putting (\[estimate4\])–(\[estimate6\]) into (\[estimate3\]) we obtain $$\begin{aligned}
\frac12 \frac{d}{dt}\|{\omega}\|^2 + \alpha \|{\omega}\|^2 \leq \frac{M^2}{2r},\end{aligned}$$ where $$\begin{aligned}
\alpha = \frac{r}2 + \frac{\pi \nu}{|D|}-\frac12 \beta \left(\frac{|D|}{\pi}
+1\right).
\label{alpha}\end{aligned}$$ Then $\alpha$ $>$ $0$ if we assume that $$\begin{aligned}
\frac{r}2 +\frac{\pi \nu}{|D|} > \frac12 \beta \left(\frac{|D|}{\pi}
+1\right),\end{aligned}$$ which is in fact the first constraint of Theorem \[mth\]. Thus, by the Gronwall inequality, we have $$\begin{aligned}
\|{\omega}\|^2 \leq \|{\omega}_0\|^2 e^{-2\alpha t} +
\frac{M^2}{2r\alpha} \left(1 - e^{-2\alpha t}\right) .\end{aligned}$$ Hence all solutions ${\omega}$ enter the closed and bounded set $${\cal B} = \left\{{\omega}: \; \|{\omega}\| \leq \frac{M}{\sqrt{2r\alpha}} \right\}$$ in finite time and stay there. The set ${\cal B}$ is thus an absorbing set of the system and is positively invariant in the sense that $S_{t,t_0} {\cal B}$ $\subset$ ${\cal B}$ for all $t$ $\geq$ $t_0$ and $t_0$ $\in$ $\rit$.\
For later purposes note that the solution operator $S_{t,t_0}$ satisfies $S_{t_0,t_0}$ $=$ $id.$ and $S_{t_2,t_1}\circ S_{t_1,t_0}$ $=$ $S_{t_2,t_0}$ for any $t_0$ $\leq$ $t_1$ $\leq$ $t_2$, that is $\{S_{t,t_0}\ : \: t \geq t_0, t_0
\in \rit\}$ is a nonautonomous process or cocycle mapping. In addition, it follows from existence and uniqueness theory that $(t,t_0,{\omega}_0)$ $\to$ $S_{t,t_0} {\omega}_0$ is continuous. Hence, in particular, when the forcing $f$ is independent of time there exists a global autonomous attractor defined by $${\cal A}_0 = \bigcap_{t\geq 0} S_{t,0}{\cal B},$$ which is a nonempty compact subset of $L^2$, and is invariant under the autonomous semigroup $\{S_{t,0} \ : \: t \geq 0\}$ in the sense that $S_{t,0} {\cal
A}_0 $ $=$ ${\cal A}_0$ for all $t$ $\geq$ $0$.
Strong contraction property
---------------------------
Now consider two trajectories $\omega^{(i)}$ corresponding to initial values $\omega_0^{(i)}$ $\in$ ${\cal B}$, $i$ $=$ $1$ and $2$. Note that these trajectories remain inside ${\cal B}$. Their difference $\delta\omega$ $=$ $\omega^{(1)} - \omega^{(2)}$ satisfies the equation $$\delta\omega_t + J\left(\psi^{(1)}, {\omega}^{(1)}\right) - J\left(\psi^{(2)},
{\omega}^{(2)}
\right) + \beta \delta\psi_x = \nu \Delta \delta{\omega}- r\delta{\omega}.$$ Similarly to the proof above it can be shown from this equation that $$\label{eqes}
\frac12 \frac{d}{dt} \|\delta{\omega}\|^2 + \int_D \delta J \delta \ {\omega}\,dxdy
+\beta \int_D \delta \psi_x \, \delta {\omega}\,dxdy
= -\nu \|\nabla \delta {\omega}\|^2 - r \|\delta {\omega}\|^2$$ where $$\delta J(\psi, {\omega}) := J(\psi^{(1)}, {\omega}^{(1)}) - J(\psi^{(2)}, {\omega}^{(2)}).$$ Now from the properties (\[estimate0\])–(\[estimate2.5\]) of the Jacobian $J$ we have $$\begin{aligned}
\left| \int_D \delta J \delta {\omega}\,dxdy \right|
& = &
\left|\int_D \left(J\left(\psi^{(1)},{\omega}^{(1)} \right) - J\left(\psi^{(2)},
{\omega}^{(2)}\right) \right) \left({\omega}^{(1)}-{\omega}^{(2)}\right) \,dxdy\right|
\\ [2ex]
& = & \left|\int_D J\left(\psi^{(1)},{\omega}^{(1)}\right) {\omega}^{(2)} \,dxdy +
\int_D J\left(\psi^{(2)},{\omega}^{(2)}\right) {\omega}^{(1)} \,dxdy \right|
\\[2ex]
& = & \left| \int_D J\left(\psi^{(1)},{\omega}^{(1)}\right) {\omega}^{(2)}\,dxdy -
\int_D J\left(\psi^{(2)}, {\omega}^{(1)}\right) {\omega}^{(2)} \,dxdy \right|
\\[2ex]
& = & \left|\int_D J\left(\psi^{(1)} - \psi^{(2)}, {\omega}^{(1)}\right)
\left({\omega}^{(1)}
- {\omega}^{(2)}\right)\,dxdy \right|
\\[2ex]
& = & \left|\int_D J\left(\delta \psi, {\omega}^{(1)}\right)\delta {\omega}\,dxdy\right|
\\[2ex]
& = & \left|\int_D J\left( {\omega}^{(1)}, \delta \psi \right)\delta {\omega}\,dxdy\right|
\\[2ex]
& = & \left|\int_D J\left(\Delta \psi^{(1)}, \delta \psi \right)\Delta \delta \psi \,dxdy\right|
\\[2ex]
&\leq &\sqrt{\frac{2|D|}{\pi}} \|\Delta \psi^{(1)}\| \; \|\Delta \delta \psi\| \;\|\Delta \delta \psi\|
\\[2ex]
& = & \sqrt{\frac{2|D|}{\pi}} \|{\omega}^{(1)} \| \; \|\delta {\omega}\|^2 ,\end{aligned}$$ where in the last two steps,we have used (\[estimate2.5\]) with $f= \psi^{(1)}, g= h=\delta \psi$, and the fact $\Delta \delta \psi$ $=$ $\delta \Delta \psi$ $=$ $\delta {\omega}$. Using this and noting that ${\omega}^{(1)}$ is in the positively invariant absorbing set ${\cal B}$ so $\|{\omega}^{(1)}\|$ $\leq$ $M/\sqrt{2r\alpha}$, we have $$\begin{aligned}
\left| \int_D \delta J \delta {\omega}\,dxdy \right|
& \leq &
\sqrt{\frac{2|D|}{\pi}} \|{\omega}^{(1)} \| \; \|\delta {\omega}\|^2 \nonumber
\\[2ex]
& \le &
\sqrt{\frac{2|D|}{\pi}} \frac{M}{\sqrt{2r\alpha}} \|\delta {\omega}\|^2 \nonumber
\\[2ex]
& = &\sqrt{\frac{|D|}{\pi r \alpha}} \; \; M \|\delta {\omega}\|^2.
\label{bigestimate}\end{aligned}$$ Then from equation (\[eqes\]), using (\[bigestimate\]) and (\[estimate4\]), we obtain $$\begin{aligned}
\frac12 \frac{d}{dt} \|\delta{\omega}\|^2
& = & -\nu \|\nabla \delta {\omega}\|^2 - r \|\delta {\omega}\|^2
- \int_D \delta J \delta {\omega}\,dxdy
\\
& & \quad - \beta \int_D \delta \psi_x \delta {\omega}\,dxdy
\\[2 ex]
& \leq & -\nu \|\nabla \delta {\omega}\|^2 - r \|\delta {\omega}\|^2
+ \left|\int_D \delta J \delta {\omega}\,dxdy\right|
\\
& & \quad + \left|\beta \int_D \delta \psi_x \delta {\omega}\,dxdy \right|
\\[2 ex]
& \leq & -\frac{\pi \nu}{|D|} \|\delta {\omega}\|^2 - r \|\delta {\omega}\|^2
+ \sqrt{\frac{|D|}{\pi r \alpha}} \; \; M \|\delta {\omega}\|^2
\\
& & \quad + \frac12 \beta \left(\frac{|D|}{\pi} +1 \right) \|\delta {\omega}\|^2
\\[2 ex]
& \leq & -\gamma \|\delta {\omega}\|^2\end{aligned}$$ where $$\gamma := r + \frac{\pi \nu}{ |D|} -
\frac12 \beta \left(\frac{|D|}{\pi} +1 \right)
- \sqrt{\frac{|D|}{\pi r \alpha}} \; \; M.$$ Note that $\gamma > \alpha - \sqrt{\frac{|D|}{\pi r \alpha}} \; \; M$. Thus, $\gamma > 0$ if we assume that $$\begin{aligned}
\label{parb}
\| f(\cdot,\cdot,t)\| \leq M < \sqrt{\frac{\pi r}{|D|}} \alpha^{\frac32},\end{aligned}$$ for all $t$ $\in$ $\rit$. Here $\alpha$ is defined in (\[alpha\]), so (\[parb\]) holds because of the assumption on the $L^2$ norm of $f$ in Theorem \[mth\]. This gives $$\|\delta {\omega}(t)\|^2 \leq \|{\omega}_0\| e^{-2\gamma t} \to 0 \quad \mbox{as} \ \ t
\to \infty,$$ for solutions starting within the positively invariant absorbing set ${\cal
B}$. This is the desired strong contractive condition. This means there is a unique solution ${\omega}^{*}(t)$ in ${\cal B}$ to which all other solutions converge. This solution ${\omega}^{*}(t)$ can be determined by the pullback convergence to be discussed in the following two Sections.
Nonautonomous dynamical systems
===============================
In order to show existence of temporally almost periodic solutions, we need some results from the theory of nonautonomous dynamical systems. Consider first an autonomous dynamical system on a metric space $P$ described by a group $\theta$ $=$ $\{\theta_t\}_{t \in \rit}$ of mappings of $P$ into itself.
Let $X$ be a complete metric space and consider a continuous mapping $$\Phi : \rit^{+} \times P \times X \to X$$ satisfying the properties $$\Phi(0,p,\cdot) = {\rm id}_X, \qquad
\Phi(\tau +t,p,x) = \Phi(\tau,\theta_t p, \Phi(t,p,x))$$ for all $t$, $\tau$ $\in$ $\rit^{+}$, $p$ $\in$ $P$ and $x$ $\in$ $X$. The mapping $\Phi$ is called a cocycle on $X$ with respect to $\theta$ on $P$.
The appropriate concept of an attractor for a nonautonomous cocyle systems is the [*pullback attractor*]{}. In contrast to autonomous attractors it consists of a family subsets of the original state space $X$ that are indexed by the cocycle parameter set.
\[pba\] A family $\widehat{A}$ $=$ $\{A_p\}_{p \in P}$ of nonempty compact sets of $X$ is called a [pullback attractor]{} of the cocycle $\Phi$ on $X$ with respect to $\theta_t$ on $P$ if it is ${\Phi}$–invariant, i.e. $$\Phi(t,p,A_p) = A_{\theta_t} p \qquad \mbox{for all} \quad t \in \rit^{+},
p \in P,$$ and [pullback attracting]{}, i.e. $$\lim_{t \to \infty} H^{*}_X\left(\Phi(t,\theta_{-t}p,D), A_p\right) = 0
\qquad \mbox{for all} \quad D \in K(X), \ p \in P,$$ where $K(X)$ is the space of all nonempty compact subsets of the metric space $(X,d_X)$.
Here $H^{*}_X$ is the Hausdorff semi–metric between nonempty compact subsets of $X$, i.e. $H^{*}_X(A,B)$ $:=$ $\max_{a \in A} {\rm dist}(a,B)$ $=$ $\max_{a\in A} \min_{b\in B} d_X(A,b)$ for $A$, $B$ $\in$ $K(X)$.
The following theorem combines several known results. See Crauel and Flandoli [@CF], Flandoli and Schmalfu[ß]{} [@FS1], and Cheban [@C1] as well as [@CKS; @KS2] for this and various related proofs.
\[th1\] Let $\Phi$ be a continuous cocycle on a metric space $X$ with respect to a group $\theta$ of continuous mappings on a metric space $P$. In addition, suppose that there is a nonempty compact subset $B$ of $X$ and that for every $D$ $\in$ $K(X)$ there exists a $T(D)$ $\in$ $\rit^{+}$, which is independent of $p$ $\in$ $P$, such that $$\label{fa}
\Phi(t,p,D) \subset B \quad \mbox{for all} \quad t > T(D).$$ Then there exists a unique pullback attractor $\widehat{A}$ $=$ $\{A_p\}_{p \in P}$ of the cocycle $\Phi$ on $X$, where $$\label{pbat}
A_p = \bigcap_{\tau \in \rit^{+}} \overline{\bigcup_{t > \tau \atop t \in
\rit^{+}}
\Phi\left(t,\theta_{-t}p,B\right)}.$$ Moreover, the mapping $p$ $\mapsto$ $A_p$ is upper semicontinuous.
Moreover, in [@CKS] it is shown that the pullback attractor consists of a single trajectory when the cocycle dynamics are in fact strongly contracting.
\[th2\] Suppose that the cocycle $\Phi$ in Theorem \[th1\] is strongly contracting inside the absorbing set $B$. Then the pullback attractor consists of singleton valued sets, i.e. $A_p$ $=$ $\{a^*(p)\}$, and the mapping $p$ $\mapsto$ $a^*(p)$ is continuous.
Almost periodicity
==================
A function $\varphi$ $:$ $\rit$ $\to$ $X$, where $(X,d_X)$ is a metric space, is called [*almost periodic*]{} [@Besicovitch] if for every $\varepsilon$ $>$ $0$ there exists a relatively dense subset $M_{\varepsilon}$ of $\rit$ such that $$d_X \left(\varphi (t+ \tau ), \varphi (t) \right) < \varepsilon$$ for all $t$ $\in$ $\rit$ and $\tau$ $\in M_{\varepsilon }$. A subset $M$ $\subseteq$ $\rit$ is called [*relatively dense*]{} in $\rit$ if there exists a positive number $l$ $\in$ $\rit$ such that for every $a$ $\in$ $\rit$ the interval $[a,a+l]\bigcap \rit$ of length $l$ contains an element of $M$, i.e. $M\bigcap [a,a+l]$ $\ne$ $\emptyset$ for every $a$ $\in$ $\rit$.
The QG solution operators $S_{t,t_0}$ form a cocycle mapping on $X$ $=$ $L^2$ with parameter set $P$ $=$ $\rit$, where $p$ $=$ $t_0$, the initial time, and $\theta_t t_0$ $=$ $t_0+t$, the left shift by time $t$. However, the space $P$ $=$ $\rit$ is not compact here. Though more complicated, it is more useful to consider $P$ to be the closure of the subset $\{\theta_t f, t \in \rit\}$, i.e. the hull of $f$, in the metric space $L^2_{loc}\left(\rit,L^2(D)\right)$ of locally $L^2(\rit)$–functions $f$ $:$ $\rit$ $\to$ $L^2(D)$ with the metric $$d_P(f,g) := \sum_{N=1}^{\infty} 2^{-N} \min\left\{1, \sqrt{\int_{-N}^N
\|f(t)-g(t)\|^2 \, dt} \right\}$$ with $\theta_t$ defined to be the left shift operator, i.e. $\theta_t
f(\cdot)$ $:=$ $f(\cdot+t)$. By a classical result [@Besicovitch; @Sell], a function $f$ in the above metric space is almost periodic if and only if the the hull of $f$ is compact and minimal. Here minimal means nonempty, closed and invariant with respect to the autonomous dynamical system generated by the shift operators $\theta_t$ such that with no proper subset has these properties. The cocycle mapping is defined to be the QG solution ${\omega}(t)$ starting at ${\omega}_0$ at time $t_0$ $=$ $0$ for a given forcing mapping $f$ $\in$ $P$, i.e. $$\Phi(t,f,{\omega}_0) := S_{t,0}^f \ {\omega}_0,$$ where we have included a superscript $f$ on $S$ to denote the dependence on the forcing term $f$. (This dependence is in fact continuous). The cocycle property here follows from the fact that $S_{t,t_0}^f {\omega}_0$ $=$ $S_{t-t_0,0}^{\theta_{t_0}f}\
{\omega}_0$ for all $t$ $\geq$ $t_0$, $t_0$ $\in$ $\rit$, ${\omega}_0$ $\in$ $L^2$ and $f$ $\in$ $P$.
\[th3\] Let the assumptions of Theorem \[mth\] hold. Then the dissipative QG model (\[eqn\])–(\[BC\]) has a unique almost periodic solution ${\omega}^{*}$ defined by $${\omega}^{*}(t) := a^*\left(\theta_t f\right), \qquad t \in \rit,$$ where $\{a^{*}(p)\}$ is the singleton valued pullback attractor–trajectory of the cocycle $\Phi(t,f,{\omega}_0)$ on $L_2(D)$, $P$ is the hull in the metric space $L^2_{loc}\left(\rit,L^2(D)\right)$ of the almost periodic forcing term $f$ and the $\theta_t$ are the left shift operators on $P$.
This is proved as follows. By Theorems \[th1\] and \[th2\] the pullback attractor exists, consists of singleton valued components $\{a^*(p)\}$ and the mapping $p$ $\mapsto$ $a^{*}(p)$ is continuous on $P$. In fact, the mapping $p$ $\mapsto$ $a^{*}(p)$ is uniformly continuous on $P$ because $P$ is compact subset of $L^2_{loc}\left(\rit,L^2(D)\right)$ due to the assumed almost periodicity. That is, for every $\varepsilon$ $>$ $0$ there exists a $\delta(\varepsilon)$ $>$ $0$ such that $\|a^*(p)-a^*(q) \|$ $<$ $\varepsilon$ whenever $d_P(p,q)$ $<$ $\delta$. Now let the point $\bar{p}$ ($=$ $f$, the given temporal forcing function) be almost periodic and for $\delta$ $=$ $\delta(\varepsilon)$ $>$ $0$ denote by $M_{\delta}$ the relatively dense subset of $\rit$ such that $d_P(\theta_{t+\tau}\bar{p},\theta_t \bar{p})$ $<$ $\delta$ for all $\tau$ $\in$ $M_{\delta}$ and $t$ $\in$ $\rit$. From this and the uniform continuity we have $$\|a^*(\theta_{t+\tau} \bar{p}) - a^*(\theta_t \bar{p})\| < \varepsilon$$ for all $t$ $\in$ $\rit$ and $\tau$ $\in$ $M_{\delta(\varepsilon)}$. Hence $t$ $\mapsto$ ${\omega}^*(t)$ $:=$ $a^*(\theta_{t}\bar{p})$ is almost periodic. It is unique as the single-trajectory pullback attractor is the only trajectory that exists and is bounded for the entire time line.
This completes the proof of the main result, Theorem \[mth\].
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[^1]: This work was partly supported by the National Science Foundation Grant DMS-9704345 and the DFG Forschungschwerpunkt “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme". This work was begun during a visit to the Oberwolfach Mathematical Research Institute, Germany.
|
---
abstract: 'This paper summarizes some recent progress and future perspectives in the experimental investigation of the Standard Model (SM) (and physics beyond it) using charged kaon decays, except for the important mode $K^\pm \rightarrow \pi^\pm \nu \overline{\nu}$ which is discussed in detail elsewhere [@pinunu].'
author:
- 'M. S. Sozzi, CERN, Geneva, Switzerland[^1]'
title: |
PHYSICS WITH CHARGED KAONS:\
RECENT AND FUTURE EXPERIMENTS
---
INTRODUCTION
============
Kaons, being the minimal “flavour laboratory”, always had a leading rôle in the discovery and the study of fundamental physics issues which are related to flavour changing transitions induced by weak interactions. Their success as a physics probe is partly due to the mixed blessing of their relatively small mass (on the hadronic scale), and its experimental consequences.
Along the history from the discovery of CP violation [@Cronin] to the evidence of direct CP violation [@DirectCP] [@Cimento] the peculiar system of neutral kaons was central, while charged kaons were deeply exploited for the search of very rare processes probing high energy scales [@rare].
The level of sensitivity at which rare kaon decay searches have been pushed is rather remarkable: several branching ratios at the $10^{-8}$ level have been measured (not to mention the smallest branching ratio ever measured, $BR(K_L \rightarrow e^+ e^-) \simeq 9 \cdot 10^{-12}$), and limits down to $10^{-11}$ are available [@PDG].
Today, great efforts are dedicated to testing the Standard Model by over-constraining the CKM mixing matrix, and it should be noted that also in this respect kaon decays might offer a very powerful probe, complementary and in some cases even superior to the one represented by $B$ mesons [@CKM_KB], for the decays which can be reliably described by the theory. We do not know where new physics will show up first, and even relatively well measured parameters such as the Cabibbo angle might reserve some surprise [@Vus], so that more refined measurements of several kaon decays are now of great interest.
For what concerns CP violation searches, it should be noted that so far the only evidence of this phenomenon in Nature arises from somewhat subtle effects involving coupled neutral states. While after the discovery of direct CP violation we have no reason to doubt that CP asymmetries should be present also in charged particle decays, as predicted by our current paradigm for describing CP violation, such asymmetries - which would be indeed the simplest manifestations of CP violation from a conceptual point of view and entirely due to direct CP violation - have yet to be detected (although admittedly we might be close to this). T-odd correlations, being actively probed in charged K decays through polarization measurements, are also a complementary window on new physics through CP violation (also T violation has only been detected in the state mixing of neutral kaons so far).
One should also mention that precise kaon decay data is needed to further constrain the parameters of the effective theory we use to describe low-energy physics, namely chiral perturbation theory, and that measurements of branching ratios and decay form factors can offer stringent tests of anomalous couplings and universality.
LEPTON FLAVOUR NUMBER VIOLATION
===============================
Several experiments were devoted to searches for lepton flavour number violation (LFV) by searching for rare decays of charged kaons. These measurements allowed to dismiss several models of new physics thanks to their sensitivity to very high mass scales, and represent nowadays an important constraint to models of physics beyond the SM. The results achieved by several dedicated experiments, lately mainly by BNL experiment E865, are rather impressive (see table \[tab:lfv\]). The low limits reached make further progress difficult: kaon flux and physical backgrounds are the limiting factors, and no new dedicated experiments on LFV with kaons are in preparation at this time.
**Mode** **BR limit (90% CL)**
------------------------------------- -----------------------
$K^+ \rightarrow \pi^+ \mu^+ e^-$ $2.8 \cdot 10^{-11}$
$K^+ \rightarrow \pi^+ \mu^- e^+$ $5.2 \cdot 10^{-10}$
$K^+ \rightarrow \pi^- \mu^+ e^+$ $5.0 \cdot 10^{-10}$
$K^+ \rightarrow \pi^- e^+ e^+$ $6.4 \cdot 10^{-10}$
$K^+ \rightarrow \pi^- \mu^+ \mu^+$ $3.0 \cdot 10^{-9}$
: Present limits on LFV charged kaon decays [@PDG].
\[tab:lfv\]
HADRONIC DECAYS AND\
CP VIOLATION
====================
The decay amplitudes for $K^\pm \rightarrow 3\pi$ have been recomputed recently at next-to-leading order in the chiral perturbation expansion [@Bijnens_3pi] [@Prades_3pi], partially accounting for isospin-breaking effects. The corrections with respect to the leading order results are found to be of the order 30%, and the fitted values of the Dalitz plot slopes (see [@PDG] for the definitions) agree rather well with the experimental data, as shown in table \[tab:K3pi\].
**Parameter** **Experiment** $\mathbf{\chi^2/}$**dof** **Theory fit**
-------------------------- ---------------------- --------------------------- ---------------------
$g(\pi^\pm \pi^+ \pi^-)$ $-0.2160 \pm 0.0029$ 2.6 $-0.22 \pm 0.02$
$h(\pi^\pm \pi^+ \pi^-)$ $0.011 \pm 0.004$ 1.3 $0.012 \pm 0.005$
$k(\pi^\pm \pi^+ \pi^-)$ $-0.0093 \pm 0.0022$ 3.2 $0.0054 \pm 0.0015$
$g(\pi^\pm \pi^0 \pi^0)$ $0.628 \pm 0.019$ 6.8 $0.61 \pm 0.05$
$h(\pi^\pm \pi^0 \pi^0)$ $0.045 \pm 0.011$ 1.8 $0.069 \pm 0.018$
$k(\pi^\pm \pi^0 \pi^0)$ $0.003 \pm 0.006$ 5.6 $0.004 \pm 0.002$
\[tab:K3pi\]
New results were obtained recently for the $\pi^\pm \pi^0 \pi^0$ decay mode: a preliminary branching ratio measurement at 1% with small background was obtained by KLOE at DA$\Phi$NE [@KLOE_3pin]: $$BR(K^\pm \rightarrow \pi^\pm \pi^0 \pi^0) = (1.781 \pm 0.013 \pm 0.016)\%$$ from 440 pb$^{-1}$ of integrated luminosity. The constrained kinematics with high-purity pion and muon tagging (obtained from $\pi^\pm \pi^0$ and $\mu^\pm \nu$ decays respectively) are strong points for this experiment, which has good prospects for improving our knowledge of the charged kaon decay parameters. The $8 \cdot 10^{31}$ cm$^{-2}$ s$^{-1}$ peak luminosity obtained in 2002 (leading to a total of 500 pb$^{-1}$ integrated luminosity) is less than an order of magnitude below the design one, which is expected to be reached with steady improvements on the machine.
New results on the Dalitz plot slopes [@KLOE_3pin] [@ISTRA_3pin] helped improving the experimental determination of some parameters, although the consistency of the data remains rather poor, as seen from table \[tab:K3pi\], particularly for what concerns the linear slope of $\pi^\pm \pi^0 \pi^0$ (see also [@KEK_3pin]).
The 30-year long quest for direct CP violation [@Cimento] recently resulted in the definitive experimental evidence of this phenomenon [@DirectCP], ruling out the super-weak *ansatz*. The $\epsilon'/\epsilon$ parameter describing direct CP violation in neutral kaon decays, however, is still under poor theoretical control (to the extent that the experimental value could still be saturated by new physics), so that more measurement of CP violation in different systems are required to really test the present paradigm. Long ago it was suggested that charge asymmetries in $K^\pm$ hadronic decays, such as that for the linear Dalitz plot slope parameter $g$ in the decay to 3 charged pions, related to the kinetic energy distribution of the odd pion $$A_g(\pi^\pm \pi^+ \pi^-) =
\frac{g(\pi^+ \pi^+ \pi^-) - g(\pi^- \pi^- \pi^+)}
{g(\pi^+ \pi^+ \pi^-) + g(\pi^- \pi^- \pi^+)}$$ being in principle free from the amplitude suppression due to the $\Delta I = 1/2$ rule, could be a valid alternative to $\epsilon'/\epsilon$ as a measure of direct CP violation.
From a theoretical point of view, several predictions for the asymmetries in charged kaon decay parameters are available in the literature (see [@Prades_3pi] [@Isidori_review] and references therein), spanning a rather wide range of values. At this time, however, the common understanding is that in the SM such asymmetries, although not fully under theoretical control, are rather tiny for accidental reasons generally below $10^{-4}$ (*i.e.* $A_g(\pi^\pm \pi^+ \pi^-) \sim 0.5 \cdot 10^{-4}$ at most [@Prades_3pi]); searches focus on common decay modes. In some extensions of the Standard Model somewhat larger values, above $10^{-4}$ could be reached [@Shabalin] [@Martinelli].
For $3\pi$ decays the partial decay width asymmetries are suppressed with respect to the ones for the Dalitz plot slopes. Asymmetries are not expected to be significantly larger for other decay modes, such as $K^\pm \rightarrow
\pi^\pm \ell^+ \ell^-$ ($\ell=e,\mu$), the (relatively) larger one ($O(10^{-4}$) being probably expected to be the photon spectrum asymmetry in $K^\pm \rightarrow \pi^\pm \pi^0 \gamma$ decays, which were recently measured with high statistics [@KEK_pi+pi0gamma].
The experimental information on differences in decay parameters for $K^+$ and $K^-$, which would be a sign of direct CP violation, is rather poor at present, as shown in table \[tab:asym\].
**Mode** **Asymmetry** **Notes**
---------------------------------- ---------------------- -----------
$A_\Gamma(\pi^\pm \pi^+ \pi^-)$ $(0.07 \pm 0.12)\%$ meas
$A_g(\pi^\pm \pi^+ \pi^-)$ $(-0.11 \pm 0.34)\%$ meas
$A_h(\pi^\pm \pi^+ \pi^-)$ $(9 \pm 44)\%$ PDG
$A_k(\pi^\pm \pi^+ \pi^-)$ $(9 \pm 20)\%$ PDG
$A_\Gamma(\pi^\pm \pi^0 \pi^0)$ $(0.0 \pm 0.6)\%$ meas
$A_g(\pi^\pm \pi^0 \pi^0)$ $(5.2 \pm 2.8)\%$ PDG
$A_h(\pi^\pm \pi^0 \pi^0)$ $(20 \pm 21\%)$ PDG
$A_k(\pi^\pm \pi^0 \pi^0)$ $(90 \pm 16\%)$ PDG
$A_\Gamma(\mu^\pm \nu)$ $(-0.54 \pm 0.41)\%$ meas
$A_\Gamma(\pi^\pm \pi^0)$ $(0.8 \pm 1.2)\%$ meas
$A_\Gamma(\pi^\pm \pi^0 \gamma)$ $(0.9 \pm 3.3)\%$ meas
$A_\Gamma(\pi^\pm \mu^+ \mu^-)$ $(-2 \pm 12)\%$ meas
: Experimental data on CP-violating asymmetries for $K^\pm$ decay rates ($A_\Gamma$) and Dalitz plot slope parameters ($A_g, A_h, A_k$), from [@PDG], [@HyperCP_3pi], [@HyperCP_pimumu]. In the third column “meas” indicates measurements, and “PDG” naive asymmetries evaluated from PDG values (using inflated errors) when no measurements are available.
\[tab:asym\]
Dalitz plot slope asymmetries have been directly measured only for the $\pi^\pm \pi^+ \pi^-$ mode. A preliminary result was recently obtained, as a byproduct, by the HyperCP experiment at FNAL, devoted to the search for CP violation in hyperon decays. With the 1997 data sample (about $4.2 \cdot 10^7$ $K^+$ and $1.2 \cdot 10^7$ $K^-$ decays, corresponding to 20% of the total collected statistics) the preliminary result [@HyperCP_3pi] $$A_g(\pi^\pm \pi^+ \pi^-) = (-0.22 \pm 0.15 \pm 0.37)\%$$ was obtained (the first error being statistical and the second systematic). The largest systematics (expected to be improved in a full analysis) are given by the effect of residual uncontrolled magnetic fields and Monte Carlo corrections.
Other large differences between the measured parameters for $K^+$ and $K^-$ in different experiments hint at problems in the data; removing a single out-lier measurement all the slope asymmetries become consistent with zero. Clearly, new experiments measuring both kaon charge partners, to better keep systematics under control, are required to improve the situation.
The small SM predictions for direct CP violating asymmetries in charged kaon decays allow for a large window of opportunity in searching for new physics. Two major experimental programs are being carried on to search for direct CP violation by measuring a charge asymmetry in the linear Dalitz plot slopes $g$ of $K^\pm \rightarrow 3\pi$ decays.
The NA48/2 experiment at the CERN SPS [@NA48/2] uses a new beam line delivering simultaneous positive and negative unseparated hadron beams of 60 GeV/$c$ average momentum (with a $\pm$ 5% spread) to the improved NA48 detector. The beams (see fig. \[fig:NA48/2\]), produced at zero degrees by $7 \cdot 10^{11}$ protons per 4.8 s pulse from the SPS every 16.8 s, contain about 5% $K^\pm$, resulting in $2.2(1.3) \cdot 10^6$ $K^+(K^-)$ entering the decay volume per pulse.
{width="145mm"}
A high-rate beam spectrometer based on MICROMEGA chambers (“KABES”) allows the measurement of the incoming particle charge, momentum and direction, complementing the magnetic spectrometer downstream; most of the flux from beam pion decays remains in the beam pipe crossing all detectors. By frequently switching the polarity of the analyzing and beam-line magnets, the experiment can cancel most of the systematics linked to asymmetries of the apparatus. NA48/2 took data for the first time in 2003, and a second run in 2004 will largely increase the statistics, to allow reaching the design error of $1.7 \cdot 10^{-4}$ on $A_g(\pi^\pm \pi^+ \pi^-)$. The experiment also collects a large amount of $K^\pm \rightarrow \pi^\pm
\pi^0 \pi^0$ decays, to reach a similar level of accuracy on the corresponding slope asymmetry.
The OKA experiment [@OKA] is in preparation at Protvino and will be installed on a new RF separated kaon beam line of 15 GeV/$c$ ($\pi$ contamination $<$50%) at the U-70 PS. Only one charge at the time will be available, but the high statistics ($3 \cdot 10^{13}$ protons per pulse giving a flux of $4(1.3) \cdot 10^6$ $K^+(K^-)$ entering the decay volume) is expected to allow reaching an error of $1 \cdot 10^{-4}$ on $A_g(\pi^\pm \pi^+ \pi^-)$.
The magnetic detector which will be used is an evolution of the ISTRA+ and GAMS ones. Several rare decays will also be studied with this setup. Half of the beam line is now ready and the first run is expected in November 2004.
The KLOE experiment, with its source of correlated $K^+ K^-$ pairs, is also expected to contribute to the measurement of charge asymmetries when enough statistics will be available, and its results will be affected by rather different systematics.
SEMI-LEPTONIC DECAYS
====================
The interest in the measurement of semi-leptonic decays of kaons was revived recently due to the present unsatisfactory agreement of experimental data with one of the unitarity constraints on the CKM mixing matrix. The constraint relation $|V_{ud}|^2 + |V_{us}|^2 + |V_{td}|^2 = 1$ is violated at the 2.2 standard deviation level using the 2002 PDG data [@PDG]. Half of the error is due to the knowledge of the $|V_{us}|$ matrix element, which is extracted from hyperon and kaon semi-leptonic decays. A new measurement of the $K^+ \rightarrow \pi^0 e^+ \nu$ ($K^+_{e3}$) branching ratio, obtained in a 1 week special run of experiment E865 at Brookhaven [@E865_ke3], gives a value for $|V_{us}|$ which disagrees with the previous measurements, but is in agreement with the unitarity constraint. Preliminary results from KLOE (with 78 pb$^{-1}$) [@KLOE_kl3] instead confirm previous results and the unitarity problem (see fig. \[fig:Vus\]).
![Recent determinations of the $|V_{us}|$ element of the CKM mixing matrix from semi-leptonic kaon decays.[]{data-label="fig:Vus"}](vus.eps){width="65mm"}
New results are expected by the KLOE, NA48/2 and KTeV experiments, which should clarify the situation.
Another issue in semi-leptonic kaon decays concerns the experimental hints of the presence of scalar and/or tensor anomalous couplings in the form factors of $K^+_{e3}$ decays, which arose several years ago. A recent high statistics ($5.5 \cdot 10^5$ $K^-_{e3}$ decays) measurement by the ISTRA+ experiment at Protvino [@ISTRA_ke3] gave: $$\begin{aligned}
&& f_S/f_+(0) = 0.002^{+0.020}_{-0.022} \pm 0.003 \\
&& f_T/f_+(0) = 0.021^{+0.064}_{-0.075} \pm 0.026 \end{aligned}$$ further supporting the conclusion of the E246 experiment at KEK [@KEK_ke3] which denied the existence of such anomalous couplings.
The E246 experiment also performed an interesting $\mu/e$ universality test by comparing the $\lambda_0$ form factor slope measurement with the value obtained by the $K^+_{\mu3}/K^+_{e3}$ branching fraction ratio.
T-ODD CORRELATIONS
==================
A long standing field of investigation is the search for T-odd correlations, such as the transverse muon polarization $P_T(\mu)$ in $K_{\mu3}$ decays. While for the $K_L$ decay the limit due to SM final state interaction effects was reached in the 70’s, the investigation with $K^+$ decays is still away from such a limit, and the tiny ($< 10^{-5}$) SM effects allow these searches to be a good probe for new physics. The E246 experiment at KEK exploits the semi-muonic decays of stopped $K^+$ and forms a double ratio of events with opposite decay plane orientation to reduce the systematic sensitivity to spurious asymmetries. They recently published a new (null) result [@E246_new] based on the full collected statistics ($8.3 \cdot 10^6$ $K^+_{\mu3}$ stopped decays): $$P_T(\mu) = (-1.12 \pm 2.17 \pm 0.90) \cdot 10^{-3}$$ and for the first time also a result for the $\mu^+ \nu \gamma$ decay [@E246_munug], which are complementary to the former mode for discriminating physics beyond the SM: $$P_T(\mu) = (-0.64 \pm 1.85 \pm 0.10) \cdot 10^{-2}$$ The experiment is now completed: its final sensitivity (for the $K^+_{\mu3}$ mode) is expected to be around $1.5 \cdot 10^{-3}$ in the transverse polarization; the main systematics arise from residual detector misalignments, asymmetric spurious magnetic fields and the large in-plane muon polarization.
Improved experiments on transverse muon polarization have been proposed, which could reach the $\sim 1 \cdot 10^{-4}$ sensitivity.
Another class of T-odd correlations is that involving only three-momenta in four-body final states, such as radiative semi-leptonic decays. As in the previous case, the small SM contributions from final state interactions leaves a large window of opportunity for searches of new physics. In particular, new sources of CP violation with vector or axial couplings are not constrained by the transverse muon polarization results [@Chalov], and will be searched for in the NA48/2 and OKA experiments.
RARE DECAYS
===========
Apart from the very important $K \rightarrow \pi \nu \overline{\nu}$ decays which are not discussed here, the class of FCNC loop-induced decays comprises also $K^\pm \rightarrow \pi^\pm \ell^+ \ell^-$ (where $\ell$ is a charged lepton). These are less interesting than the previous ones (or even their $K_L$ counterparts) since long-distance physics dominates. Still, the value of the ratio of $K^+ \rightarrow \pi^+ \mu^+ \mu^-$ to $K^+ \rightarrow
\pi^+ e^+ e^-$ decays is constrained in a model-independent way in the chiral perturbation expansion, and the experimental situation was unclear recently, with two incompatible experimental results (at 3.4 standard deviations) for the $K^+ \rightarrow \pi^+ \mu^+ \mu^-$ branching ratio, one of which would violate the constraint mentioned above. A new preliminary measurement by the HyperCP experiment [@HyperCP_pimumu] $$BR(K^\pm \rightarrow \pi^\pm \mu^+ \mu^-) =
(9.8 \pm 1.0 \pm 0.5) \cdot 10^{-8}$$ seems to clarify the situation, bringing the experimental value of the above ratio within the limits allowed by the SM.
CP-violating asymmetries are expected to be tiny also in these modes, which makes them unattractive for such searches.
Among other interesting rare decay modes of charged kaons, $\pi^\pm \gamma
\gamma$ should be mentioned, with $\sim 10^{-6}$ branching ratio. This channel is very interesting as a constraint on chiral perturbation theory, since in involves a free $O(p^4)$ parameter and important $O(p^6)$ contributions, complementing the information which can be extracted from the $K_{L,S} \rightarrow \pi^0 \gamma \gamma$ decays. Due to the large ($\times
2 \cdot 10^5$) irreducible background from $\pi^\pm \pi^0$, this mode is rather penalized from the experimental point of view, but high flux experiments are expected to improve its knowledge.
Similarly, the 4-lepton modes $\ell^\pm \nu \ell^+ \ell^-$, two of which were recently observed with much improved statistics and 10-20% background at BNL [@E865_4leptons]: $$\begin{aligned}
&& \hspace{-0.5cm}
BR(K^+ \rightarrow e^+ \nu e^+ e^-) = (2.48 \pm 0.14 \pm 0.14)
\cdot 10^{-8} \\
&& \hspace{-0.5cm}
BR(K^+ \rightarrow \mu^+ \nu e^+ e^-) = (7.06 \pm 0.16 \pm 0.26)
\cdot 10^{-8}\end{aligned}$$ (with $m_{ee}>$ 150 and 145 MeV/$c^2$ respectively), are also good probes of chiral perturbation predictions [@Bijnens_4leptons]. The modes with muon pairs have not been detected yet, but their branching ratios are predicted to be not far below the ones for the electron modes, so that they should be measurable in high flux experiments.
Some more exotic searches have been performed or are in the programs of future charged kaon experiments, such as those for supersymmetric particles or $K^+ \rightarrow \pi^+ \gamma$ decays (!).
$\pi \pi$ INTERACTIONS
======================
It is well known that $K_{e4}$ decays are one of the best places to study low-energy $\pi\pi$ interactions, thanks to the Fermi-Watson theorem; indeed, the asymmetry in the distribution of the angle between the di-pion and the di-lepton planes is sensitive to the $\pi\pi$ scattering phase shifts. The value of the S-wave isoscalar scattering length parameter is a very precise prediction of chiral perturbation theory [@Colangelo]: $a_0^0 = 0.220 \pm 0.005$. Low-energy $\pi\pi$ scattering in the S-wave is particularly sensitive to the size of the $\langle q \overline{q} \rangle$ condensate breaking the chiral simmetry of QCD. The size of this condensate is a free parameter in chiral perturbation theory, and one of the assumptions (giving predictive power to the theory) is that it is large enough to make the lowest order term in the quark masses dominate the mass of light mesons.
Recently, the BNL E865 experiment improved the experimental knowledge of the scattering length parameter with a statistics of $4 \cdot 10^5$ $K^+_{e4}$ decays [@E865_ke4]: $$a_0^0 = 0.216 \pm 0.013 \pm 0.003$$ While this value is in good agreement with the theoretical one, its statistically-dominated precision is still quite larger than the theoretical error, and such an unusual situation in hadronic physics calls for improved experimental investigations.
The NA48/2 experiment at CERN plans to collect more than $1 \cdot 10^6$ $K^+_{e4}$ decays in order to be able to reach an error of $\sim 0.01$ on the $a_0^0$ parameter.
The DIRAC experiment at CERN [@DIRAC] investigates the formation of the electro-magnetically bound state of $\pi^+ \pi^-$ (“pionium”), and from the study of its lifetime expects to be able to measure $|a_0^0 - a_0^2|$ to the 5% level.
Figure \[fig:pipi\] summarizes the current knowledge of the scattering length parameters.
![Recent experimental determinations of the $\pi\pi$ S-wave scattering lengths, compared with theoretical predictions under different assumptions (see [@E865_ke4] for details); the shaded bands are estimates of the expected errors from DIRAC and NA48/2, for illustrative purposes.[]{data-label="fig:pipi"}](pipi.eps){width="65mm"}
THE FUTURE
==========
The present scenario indicates that the future of kaon physics will most likely be dominated by experimental efforts devoted to overconstrain the CKM unitarity triangle, by measuring the very rare decay modes which can be predicted in the cleanest way by the theory, *i.e.* $K \rightarrow
\pi \nu \overline{\nu}$ modes [@pinunu].
**Site** **Energy and statistics** **Beam** **Date**
---------------- -------------------------------------------- -------------------------------------------- ----------
BNL-AGS 6 GeV/$c$ or at rest, $> 10^{12}$ $K^+$ Unseparated ($\pi/K \approx 20$) or 1995+
$E \times B$ separated ($\pi/K < 0.25$)
KEK-PS At rest, $> 10^8$ $K^+$ $E \times B$ separated ($\pi/K \approx 6$) 1996+
LNF-DA$\Phi$NE 100 MeV/$c$, $6 \cdot 10^8$ $K^\pm$ so far $\phi$ factory, pure $K$ 2000+
Protvino-U70 25 GeV/$c$, $> 10^9$ $K^-$ Unseparated ($\pi/K \approx 30$) 2001+
CERN-SPS 60 GeV/$c$, $< 10^{11}$ $K^\pm$ so far Unseparated, ($\pi/K \approx 10$) 2003+
Protvino-U70 12-18 GeV/$c$ Separated ($\pi/K <2$) 2004+
FNAL-MI 22 GeV/$c$ Separated ($\pi/K<0.5$) 2007+
J-PARC-PS 600-700 MeV/$c$ Separated ($\pi/K <0.3$) 2008+
\[tab:world\]
Among new facilities, one where a strong kaon physics program is foreseen is the J-PARC 50 GeV proton synchrotron in construction at Tokai in Japan [@JPARC]. This high-intensity machine ($2 \cdot 10^{14}$ p/3.4 s) is expected to deliver beam to experiments in 2008. Two beam lines are foreseen in the experimental hall (only one at startup), one of which with a low energy ($\sim$ 600 MeV/$c$) separated $K^+$ beam ($\sim 10^7$ $K^+$/s). Several letters of intent for charged kaon experiments have been submitted, including: an improved transverse muon polarization experiment to reach $\sim 1 \cdot 10^{-4}$ sensitivity, an extensive program for the complete measurement of $K^+$ decay modes, a dedicated $K^+_{e3}$ branching ratio measurement, studies of pionium and $\pi K$ atoms, and a $K^+ \rightarrow
\pi^+ \nu \overline{\nu}$ measurement. One can question which of these measurements will remain central in kaon physics by 2008, and the answer clearly depends also on the success of the ongoing experimental efforts.
Considering the availability of charged kaons in the world, one is led to the picture depicted in table \[tab:world\]; the sheer amount and quality of the ongoing and future activities, including the ones which were discussed at this workshop involving upgrades of existing machines and/or experiments, clearly indicate how the field is a very active one.
Playing the seer, one could guess the following possible scenario concerning physics with charged kaons at a time where the LHC is starting: all large branching ratios of the charged kaon will be known with high accuracy, with the universality tests which they imply, and including a consistent value of $|V_{us}|$ from the kaon sector; some CP-violating charge asymmetries will be measured at the $\sim 10^{-4}$ level (and several others at $10^{-3}$), chiral perturbation theory predictions for several rare decays will be put under stringent test, and the experimental results will be closer to the theoretical precision in $\pi\pi$ scattering lengths.
CONCLUSIONS
===========
After the impressive success of the first CP violation studies with $B$ mesons in recent years, also this heavier system is now reaching the stage in which challenging precision measurements are needed, requiring high statistics and a detailed understanding of several hard theoretical issues. It seems therefore worthy to look again with renewed interest at the relative advantages of experiments with kaons, which can offer an alternative and complementary view on some deep unanswered questions of physics.
It is a pleasure to thank the organizers of the workshop (and the conveners too) for the very pleasant and interesting time in the wonderful environment of Alghero.
[99]{}
D. Jaffe, these proceedings.\
See also: L. Passalacqua, these proceedings.
J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay, Phys. Rev. Lett. 13 (1964) 138. J.R. Batley *et al.* (NA48 collaboration), Phys. Lett. B544 (2002), 97.\
A. Alavi-Harati *et al.* (KTeV collaboration), Phys. Rev. D67 (2003) 012005. M.S. Sozzi and I. Mannelli, Riv. Nuovo Cimento 26(3) (2003), 1. For a recent review see *e.g.* L. Littenberg, preprint [hep-ex/0212005]{}, December 2002. K. Hagiwara *et al.* (Particle Data Group), Phys. Rev. D66 (2002), 010001. G. Buchalla, A. Buras, Phys. Rev. D54 (1996) 6782.\
For a recent review see *e.g.*: G.Isidori, preprint `hep-ph/0307014`, July 2003. See *e.g.* G. Isidori, Summary from Working Group VI of the 2nd Workshop on the CKM Unitarity Triangle, IPPP Durham, April 2003 (eConf C0304052), preprint `hep-ph/0311044`, November 2003. J. Bijnens *et al.*, Nucl.Phys. B648 (2003) 317. E. Gámiz *et al.*, preprint `hep-ph/0309172`, October 2003. A. Aloisio *et al.* (KLOE collaboration), preprint `hep-ex/0307054`, July 2003. I.V. Ajinenko *et al.* (ISTRA+ collaboration), Phys. Lett. B567 (2003) 159. Y.-H. Shin *et al.* (KEK E246 collaboration), Eur. Phys. J. C12 (2000) 627. G. D’Ambrosio, G. Isidori, Int. Jou. Mod. Phys. A13 (1998) 1. E.P. Shabalin, preprint `ITEP 8-98` (1998).
G. D’Ambrosio *et al.*, Phys.Lett. B480 (2000) 164. W.-S. Choong, Ph.D. thesis, E.O. Lawrence Berkeley Laboratory, LBNL-47014, October 2000.
R. Batley *et al.* (NA48/2 collaboration), addendum to proposal P253, `CERN/SPSC/P253 add.3` (Geneve) 2000.
V.F. Obraztsov, L.G. Landsberg, Nucl. Phys. (Proc. Suppl.) B99 (2001) 257. S. Shimizu *et al.* (KEK E470 collaboration), Phys.Lett. B554 (2003) 7. A. Sher *et al.* (BNL E865 collaboration), preprint `hep-ex/0305042`, June 2003. See *e.g.*: S. Di Falco (KLOE collaboration), preprint `hep-ex/0311006`, November 2003. I.V. Ajinenko *et al.* (ISTRA+ collaboration), Phys. Lett. B574 (2003) 14. S. Shimizu *et al.* (KEK E246 collaboration), Phys. Lett. B495 (2000) 33. M. Abe *et al.* (KEK E246 collaboration), Nucl. Phys. A721 (2003) 445. V.V. Anisimovsky *et al.* (KEK E246 collaboration), Phys. Lett, B562 (2003) 166. V.V. Braguta *et al.*, Phys.Rev. D68 (2003) 094008. H.K. Park *et al.* (HyperCP collaboration), Phys. Rev. Lett. 88 (2002) 111801. A.A. Poblaguev *et al.* (BNL E865 collaboration), Phys. Rev. Lett. 89 (2002) 061803. J. Bijnens *et al.*, Nucl. Phys. B396 (1993) 81. G. Colangelo *et al.*, Nucl. Phys. B603 (2001) 125. S. Pislak *et al.* (BNL E865 collaboration), Phys.Rev. D67 (2003) 072004. B. Adeva *et al.* (DIRAC collaboration), CERN SPS proposal P 284, `CERN/SPSLC 95-1` (Geneva), December 1994.
M. Furusaka *et al.* (Joint project team of JAERI and KEK), KEK report 99-4, Sep. 1999.
[^1]: On leave from Scuola Normale Superiore, Pisa. E-mail: `[email protected]`
|
---
bibliography:
- 'NeutrinoCR.bib'
---
Astro2020 Science White Paper
Neutrinos, Cosmic Rays and the MeV Band
**Thematic Areas:** $\square$ Planetary Systems $\square$ Star and Planet Formation $\square$ Formation and Evolution of Compact Objects $\square$ Cosmology and Fundamental Physics $\square$ Stars and Stellar Evolution $\square$ Resolved Stellar Populations and their Environments $\square$ Galaxy Evolution Multi-Messenger Astronomy and Astrophysics
**Principal Authors:** Roopesh Ojha (UMBC/NASA GSFC, USA; [email protected]; Phone: +13012862972), Haocheng Zhang (Purdue University, USA; [email protected]), Matthias Kadler (University of Würzburg, Germany; [email protected]), Naoko K. Neilson (Drexel University, USA; [email protected]), Michael Kreter (North-West University, South Africa; [email protected])
**Co-authors:** J. McEnery (NASA GSFC, USA), S. Buson (University of Würzburg, Germany and UMBC, USA), R. Caputo (NASA GSFC, USA), P. Coppi (Yale University, USA), F. D’Ammando (INAF-IRA Bologna, Italy), A. De Angelis (INFN/INAF Padua, Italy), K. Fang (Stanford University), D. Giannios (Purdue University, USA), S. Guiriec (George Washington University, USA), F. Guo (Los Alamos National Lab, USA), J. Kopp (CERN and Johannes Gutenberg University Mainz), F. Krauss (University of Amsterdam, The Netherlands), H. Li (Los Alamos National Lab, USA), M. Meyer (KIPAC/Stanford University, USA), A. Moiseev (NASA GSFC, USA), M. Petropoulou (Princeton University, USA), C. Prescod-Weinstein (University of New Hampshire, USA), B. Rani (NASA GSFC, USA), C. Shrader (NASA GSFC/Catholic University of America, USA), T. Venters (NASA GSFC, USA), Z. Wadiasingh (NASA GSFC, USA).
**Abstract:** The possible association of the blazar TXS 0506+056 with a high-energy neutrino detected by IceCube holds the tantalizing potential to answer three astrophysical questions:\
1. Where do high-energy neutrinos originate?\
2. Where are cosmic rays produced and accelerated?\
3. What radiation mechanisms produce the high-energy $\gamma$-rays in blazars?\
The MeV $\gamma$-ray band holds the key to these questions, because it is an excellent proxy for photo-hadronic processes in blazar jets, which also produce neutrino counterparts. Variability in MeV $\gamma$-rays sheds light on the physical conditions and mechanisms that take place in the particle acceleration sites in blazar jets. In addition, hadronic blazar models also predict a high level of polarization fraction in the MeV band, which can unambiguously distinguish the radiation mechanism. Future MeV missions with a large field of view, high sensitivity, and polarization capabilities will play a central role in multi-messenger astronomy, since pointed, high-resolution telescopes will follow neutrino alerts only when triggered by an all-sky instrument.
Introduction
============
Neutrinos are uniquely efficacious probes because their low cross section and neutrality allow them to travel virtually unhindered through the Cosmos. They are unlikely to be attenuated in contrast to gamma-ray photons, which are subject to $\gamma\gamma$ interactions. Thus they are able to carry information about regions from which photons cannot escape, including some of the most compact as well as the most distant objects and environments in the Universe. In contrast to cosmic rays - which are nuclei mostly of hydrogen - neutrino propagation is not altered by magnetic fields and thus their origin can potentially be determined.
Cosmic rays can reach energies up to $10^8$ TeV ($\gtrsim 1~\rm{EeV}$ is often called ultra-high-energy cosmic rays, UHECRs), orders of magnitude higher than what can be achieved with any particle accelerator on Earth. For over a century, the sources and underlying processes that accelerate cosmic rays have remained a major mystery. High-energy neutrinos are expected if cosmic rays entrained in (for example) a blazar jet interact with particles and radiation. In this sense, neutrinos are not just a messenger but perhaps the message itself.
The detection of astrophysical high-energy neutrinos by the IceCube detector located at the South Pole has opened up an era of multi-messenger astrophysics and a number of candidate counterparts have been proposed [@Ahlers2015]. Blazars, a subclass of AGN whose jets are directed very close to our line of sight with violent $\gamma$-ray variability, have long been of interest as possible sources of UHECRs and high-energy neutrinos. The recent possible ($\sim3\sigma$) association of the blazar TXS0506$+$056 [@ice18a] with a high-energy neutrino lends support to this possibility that relativistic blazar jets may be the source of gamma rays, neutrinos, and cosmic rays.
Observations across the electromagnetic spectrum are crucial to identify and characterize the sources of neutrinos and cosmic rays. The MeV band is particularly salient because the MeV flux is the best proxy for the neutrino flux, as GeV-TeV $\gamma$-rays may be opaque within the source and at large distances due to attenuation. The brightest and most powerful blazars (such as flat-spectrum radio quasars, FSRQs) tend to have their peak emission in the MeV band, making it the most efficient band to probe radiation and particle acceleration in blazars. Polarization signals in the MeV band can distinguish between blazar emission models. Furthermore, current theories suggest that the MeV band is the most important band to constrain the neutrino production through the accompanying cascading pair synchrotron counterpart. Nonetheless, observationally MeV $\gamma$-rays are so far largely unexplored. To understand both electromagnetic and neutrino signatures from AGN jets, often referred to as the multi-messenger approach, *support of observational and theoretical studies of $\gamma$-ray emission will be essential.*
Centrality of the MeV band for neutrino–blazar studies
======================================================
Several attributes of the MeV band make it perfect to look for and characterize blazar counterparts to high-energy neutrinos. Triggered by the detection of individual high-energy IceCube neutrinos in coincidence with gamma-ray blazars in outburst [@Kadler2016; @ice18a], hadronic emission models have experienced a renaissance. These models involve interactions of high-energy protons in the jet with source-internal or external photons [@mannheim93; @Reimer18; @mastetal13; @aharonian00], and can explain the high-energy gamma-ray emission of blazars. There is general agreement that the broadband SED of the candidate neutrino blazars TXS0506+056 and PKS1424$-$418 can be explained with a leptohadronic model in which the leptonic component dominates the GeV band and the hadronic component leads to cascading emission that gets eventually released in the MeV and hard X-ray regime [@Gao17; @Keivani18; @Murase18; @Gao19; @Reimer18]. The down-cascading of gamma-ray photons below the GeV range may explain the mysterious neutrino excess (designated the ‘neutrino flare’) of $\sim 13$ neutrino events detected by IceCube between September 2014 and March 2015, which has been independently
[r]{}[0.65]{} {width="65.00000%"} {width="65.00000%"}
associated with $3 \sigma$ significance with TXS0506+056 during an extended GeV-faint state [@Icecube18b]. In these models, the MeV band is the key to finding the electromagnetic counterparts of the neutrino-brightest blazars in the sky, as it marks the transition from pair synchrotron contribution to inverse Compton (Fig. \[fig:SED\]). It is even possible that a population of relatively faint GeV objects could contribute significantly to the diffuse extragalactic neutrino flux or could even dominate it without being recognized in present-day blazar-neutrino correlation searches, which are largely focusing on *Fermi*LAT-detected sources. In fact, some of the radio-brightest blazars are indeed GeV-faint and cannot be associated with $\gamma$-ray counterparts, even after 9 years of *Fermi*LAT integration [@Lister15]. In addition to closing the most critical gap in the current blazar SED coverage, the large field of view of planned instruments, such as AMEGO, will allow localization of neutrino candidate-associations for a large fraction of all high-energy neutrino events of interest. This capability is essential to direct the observations of telescopes in other bands which typically have a very small field of view.
The maximum possible neutrino production rate of an individual blazar can be calculated based on calorimetric arguments from its spectral energy distribution and the long-term fluence in the keV to GeV range [@Krauss2014], [@Kadler2016]. The GeV gamma-ray flux as measured by LAT is only of limited predictive value for the expected neutrino rate, because of the different possible shapes of blazar SEDs ranging from rather broad SED shapes and flat GeV spectra in the case of high-peaked BLLac objects to MeV-peaked SEDs and steep GeV spectra for low-peaked FSRQs.
[r]{}[0.65]{} {width="65.00000%"}
Neutrino estimates using only the GeV fluxes can differ up to an order of magnitude from the real SED-based estimate [@Krauss2018]. The full high-energy calorimetric output can be determined much more accurately with measurements in the MeV band. This is particularly true for the most luminous flat-spectrum radio quasars, which tend to peak at MeV energies. Studies so far have found that the MeV part of the spectrum is the most important to characterize the most promising neutrino candidate blazars and to constrain the maximum possible neutrino production rate of the overall population of such sources [@Krauss2014; @Krauss2018]. Poisson probabilities for the detection of single or multiple high-energy neutrinos from individual high-fluence blazars are low, of the order of a few percent in the best cases [@Kadler2016]. Consistent with this, associations of neutrinos to individual blazars and results from stacked blazar samples of various characteristics are still very limited even after 10 years of IceCube observations [@Aartsen2017151], [@Aartsen201745]. It is therefore mandatory to observe a large portion of the sky in the most crucial MeV energy band, in order to compile and characterize a statistical sample of reliable blazar-neutrino associations in the next decade.
Centrality of MeV band in the search for sources of UHECRs
==========================================================
The observed high polarization fraction (PF) in radio and optical bands has demonstrated that the blazar low-energy spectral hump is dominated by non-thermal electron synchrotron emission [@scarpa97]. The origin of the high-energy spectral hump from X-ray to $\gamma$-rays, however, has two competing scenarios. One scenario suggests that the same electrons that make the low-energy hump can inverse Compton (IC) scatter low-energy photons to X-rays and $\gamma$-rays. The low-energy photons may come from the low-energy synchrotron itself (synchrotron-self Compton, SSC), or external thermal photons (external Compton, EC) from accretion disk, broad line region, and dusty torus [e.g. @maraschietal92; @dermeretal92; @sikoraetal94; @bloommarscher96; @ghisellinimadau96; @boettcherdermer98]. As the two spectral humps are generally comparable, it requires that the seed photon energy density for the IC should also be similar to the magnetic field energy density that makes the low-energy synchrotron emission. Thus the IC scenario often implies a magnetic field strength of $\lesssim 1~\rm{G}$ [@boettcher13; @cerruti15]. The alternative is the proton synchrotron (PS) scenario. In a high magnetic field ($\gtrsim 10~\rm{G}$), non-thermal protons can efficiently radiate through PS to make X-rays and $\gamma$-rays [e.g., @mannheim93; @muecke03; @mastetal13; @aharonian00; @mueckeprotheroe01]. To match the observed SED, the PS scenario generally requires the acceleration of UHECRs [@boettcher13; @petrodimi15]. In addition to the two scenarios, there may exist a significant secondary synchrotron contribution in the X-ray to MeV $\gamma$-ray bands, originating from the hadronic interactions between non-thermal protons and the low-energy photon field (both low-energy synchrotron and external thermal emission). Hadronic interactions also produce high-energy neutrinos [@petromast15; @Keivani18; @Reimer18]. So far the two scenarios make similar SED fittings [@boettcher13], thus *we need additional constraints to pinpoint the radiation mechanisms in the high-energy spectral hump, and diagnose the radiating particles and magnetic field strength.* Furthermore, theoretical and numerical simulations have shown that both shock and magnetic reconnection can give rise to the non-thermal electrons and protons that make the radiation [@marscher85; @spada01; @larionov13; @zhang16a; @boettcher19; @romanova92; @giannios09; @sironietal15; @petroetal16; @zhang18]. Which mechanism dominates the particle acceleration in AGN jets largely relies on the physical conditions of the acceleration sites, in particular the magnetic field strength and morphology. As the $\gamma$-rays often show fast variability that implies the most fierce particle acceleration, *temporal $\gamma$-ray signatures are vital to probe the extreme particle acceleration in AGN jets.*
[r]{}[0.6]{} {width="60.00000%"}
The largely unexplored MeV $\gamma$-ray band is crucial to advance our knowledge in AGN jet physics. In an IC scenario, MeV bands often mark the spectral transition from SSC to EC, and if there is a secondary synchrotron component from hadronic interactions, it should be significant from X-ray to MeV bands as well [@boettcher13; @cerruti15]. In a PS model, MeV bands can probe the contribution by the secondary synchrotron component, shedding light on the neutrino production. Therefore, both spectral shape and variability in MeV bands can put unprecedented constraints on the AGN jet radiation processes. A very interesting and novel opportunity is MeV polarimetry. Theoretical studies suggest that the IC and PS mechanisms result in very different PF [@zhang13; @zhang16; @paliya18]. Based on detailed spectral fitting of the recent TXS 0506+056 event [@Zhang19], Fig. \[fig:specpol\] demonstrates that *MeV polarimetry can unambiguously distinguish the IC and PS scenarios by the PF*. With the aid of X-ray polarimetry and neutrino detection, we can diagnose the contributions of various radiation mechanisms in the high-energy spectral hump. In particular, if the PS scenario makes the high-energy hump, temporal MeV polarization signatures can probe the magnetic field strength and morphology evolution in the particle acceleration sites, providing novel constraints on the generation of UHECRs. *Next-generation MeV (Compton and pair) instruments with great spectral and temporal sensitivity as well as polarimetry capability will be the best to advance our knowledge of AGN jet physics.*
The fast evolving physical conditions in the blazar emission region and the complicated hadronic processes therein, including the radiative transport, feedback on the non-thermal particles, and neutrino production, prevent the use of simple steady-state analytical models. In the recent decade, first-principle numerical simulations, including magnetohydrodynamic (MHD) and particle-in-cell (PIC) methods, have successfully revealed the time-dependent evolution of fluids and non-thermal particles in the blazar emission region [@Mizuno09; @Spitkovsky08; @Sironi09; @Sironi14; @Guo14]. Advanced Monte-Carlo and ray-tracing methods have been fruitful in reproducing multi-wavelength SEDs, light curves, and optical polarization signatures [@Chen14; @Zhang15; @marscher14]. These new developments have sparked the concept of multi-physics simulations, which aim to self-consistently connect fluid dynamics, particle acceleration, and radiative transfer (preliminary works include, [@tavecchio18; @zhang18; @christie19]). *Multi-physics numerical simulations will be a top priority in the next decade to leverage the multi-messenger studies of AGN jets.*
[r]{}[0.5]{} {width="8"}
MeV and Neutrino Capabilities Required in the Next Decade
=========================================================
MeV Detection
-------------
To achieve the goals discussed above, we need an MeV instrument with high sensitivity, good energy resolution, polarization capability, and a large field of view. For example, AMEGO will have a FOV $>$2.5 sr, covering the entire sky every 3 hours providing an all-sky instrument to match IceCube and other neutrino experiments. A joint detection gives much greater scientific returns with good localization and quick repointing making better background suppression and thus increasing sensitivity. Further, ‘pointed’ telescopes at other bands are likely to only follow neutrino alerts if triggered by large FOV instruments in the MeV band.
IceCube and Neutrino detection capabilities
-------------------------------------------
The IceCube Collaboration has been approved to deploy seven additional strings of photon sensors in the deep, clear Antarctic ice at the bottom center of the existing detector by 2023, forming the IceCube Upgrade. Newly developed photon sensors and new calibration devices will allow IceCube to better model the optical properties of the ice, reducing systematic uncertainties and enhancing IceCube’s already strong contribution to multi-messenger astrophysics via improved reconstruction of the direction of high-energy cascade events for point source searches and enhanced identification of PeV-scale tau neutrinos. This project also provides transitions for construction and detector developments to a proposed IceCube-Gen2 Facility, aimed to be deployed around 2030. Gen2 is a unique, multi-component cosmic neutrino observatory that expands IceCube’s existing energy reach by several orders of magnitude in both directions. An order of magnitude more astrophysical neutrinos are expected to be detected by the expanded in-ice portion of the detector with surface detector components that veto atmospheric background particles efficiently.
In addition to IceCube, several other neutrino detectors, the ARCA instrument on KM3Net, Baikal-GVD, ARA, and GRAND, are expected to be operational in the next decade. Located on different hemispheres and with complementary capabilities, much superior to the already highly successful past decade, the next decade will see a true all-sky coverage of neutrino telescopes. For cosmic rays, AugerPrime and TAx4 plan to operate in the next decade. These will enhance the number of astrophysical sources of interest to be followed up at $\gamma$-rays, and create a better picture of neutrinos, cosmic rays, and the MeV band.
|
---
abstract: 'Fundamental topological phenomena in condensed matter physics are associated with a quantized electromagnetic response in units of fundamental constants. Recently, it has been predicted theoretically that the time-reversal invariant topological insulator in three dimensions exhibits a topological magnetoelectric effect quantized in units of the fine structure constant $\alpha=e^2/\hbar c$. In this Letter, we propose an optical experiment to directly measure this topological quantization phenomenon, independent of material details. Our proposal also provides a way to measure the half-quantized Hall conductances on the two surfaces of the topological insulator independently of each other.'
author:
- 'Joseph Maciejko$^{1,2}$, Xiao-Liang Qi$^{3,1,2}$, H. Dennis Drew$^4$, and Shou-Cheng Zhang$^{1,2}$'
bibliography:
- 'alpha.bib'
title: Topological quantization in units of the fine structure constant
---
Topological phenomena in condensed matter physics are typically characterized by the exact quantization of the electromagnetic response in units of fundamental constants. In a superconductor (SC), the magnetic flux is quantized in units of the flux quantum $\phi_0\equiv\frac{h}{2e}$; in the quantum Hall effect (QHE), the Hall conductance is quantized in units of the conductance quantum $G_0\equiv\frac{e^2}{h}$. Not only are these fundamental physical phenomena, they also provide the most precise metrological definition of basic physical constants. For instance, the Josephson effect in SC allows the most precise measurement of the flux quantum which, combined with the measurement of the quantized Hall conductance, provides the most accurate determination of Planck’s constant $h$ to date [@Mohr2008]. The remarkable observation of such precise quantization phenomena in these imprecise, macroscopic condensed matter systems can be understood from the fact that they are described in the low-energy limit by topological field theories (TFT) with quantized coefficients. For instance, the QHE is described by the topological Chern-Simons theory [@Zhang1992] in $2+1$ dimensions, with coefficient given by the quantized Hall conductance. SC can be described by the topological $BF$ theory [@Hansson2004] with coefficient corresponding to the flux quantum.
More recently, a new topological state in condensed matter physics, the time-reversal (${\cal T}$) invariant topological insulator (TI), has been investigated extensively [@Qi2010; @Moore2009; @Hasan2010]. The concept of TI can be defined most generally in terms of the TFT [@Qi2008] with effective Lagrangian $$\begin{aligned}
\label{Laxion}
\mathcal{L}=\frac{1}{8\pi}\left(\varepsilon{\mathbf{E}}^2-\frac{1}{\mu}{\mathbf{B}}^2\right)
+\frac{\theta}{2\pi}\frac{\alpha}{2\pi}{\mathbf{E}}\cdot{\mathbf{B}},\end{aligned}$$ where ${\mathbf{E}}$ and ${\mathbf{B}}$ are the electromagnetic fields, $\varepsilon$ and $\mu$ are the dielectric constant and magnetic permeability, respectively, and $\theta$ is an angular variable known in particle physics as the axion angle [@Wilczek1987]. Under periodic boundary conditions, the partition function and all physical quantities are invariant under shifts of $\theta$ by any multiple of $2\pi$. Since ${\mathbf{E}}\cdot{\mathbf{B}}$ is odd under ${\cal
T}$, the only values of $\theta$ allowed by ${\cal T}$ are $0$ or $\pi$ (modulo $2\pi$). The second term of Eq. (\[Laxion\]) thus defines a TFT with coefficient quantized in units of the fine structure constant $\alpha\equiv\frac{e^2}{\hbar c}$. The TFT is generally valid for interacting systems, and describes a quantized magnetoelectric response denoted topological magnetoelectric effect (TME) [@Qi2008]. The quantization of the axion angle $\theta$ depends only on the ${\cal T}$ symmetry and the bulk topology; it is therefore universal and independent of any material details. More recently, it has been shown [@Wang2009] that the TFT [@Qi2008] reduces to the topological band theory (TBT) [@Kane2005; @Fu2007; @Moore2007] in the noninteracting limit. Interestingly, the TME is the first topological quantization phenomenon in units of $\alpha$. It can therefore be combined with the two other known topological phenomena in condensed matter, the QHE and SC, to provide a metrological definition of the three basic physical constants, $e$, $h$, and $c$.
The TME has not yet been observed experimentally. An insight into why this is so can be gained by comparing the $3+1$ dimensional TFT (\[Laxion\]) of TI to the $2+1$ dimensional Chern-Simons TFT of the QHE [@Zhang1992]. In $2+1$ dimensions, the topological Chern-Simons term is the only term which dominates the long-wavelength behavior of the system, which leads to the universal quantization of the Hall conductance. On the other hand, in $3+1$ dimensions the topological $\theta$-term in Eq. (\[Laxion\]) and the Maxwell term are equally important in the long wavelength limit. Therefore, one has to be careful when designing an experiment to observe the topological quantization of the TME, in which the dependence on the non-topological materials constants $\varepsilon$ and $\mu$ are removed.
In this Letter, we propose an optical experiment to observe the topological quantization of the TME in units of $\alpha$, *independent of material properties of the TI* such as $\varepsilon$ and $\mu$. This experiment could be performed on any of the available TI materials, such as the Bi$_2$Se$_3$, Bi$_2$Te$_3$, Sb$_2$Te$_3$ family or the recently discovered thallium-based compounds [@chalcogenides]. Consider a TI thick film of thickness $\ell$ with optical constants $\varepsilon_2,\mu_2$ and axion angle $\theta$ deposited on a topologically trivial insulating substrate with optical constants $\varepsilon_3,\mu_3$ (Fig. \[fig:fig1\]).
![(color online). Measurement of Kerr and Faraday angles for a TI thick film of thickness $\ell$ and optical constants $\varepsilon_2,\mu_2$ on a topologically trivial insulating substrate with optical constants $\varepsilon_3,\mu_3$, in a perpendicular magnetic field ${\mathbf{B}}$. (We consider normal incidence in the actual proposal but draw light rays with a finite incidence angle in the figure for clarity.) The external magnetic field can be replaced by a thin magnetic coating on both TI surfaces, as suggested in Ref. [@Qi2008].[]{data-label="fig:fig1"}](fig1.eps){width="0.9\columnwidth"}
The vacuum outside the TI has $\varepsilon=\mu=1$ and trivial axion angle $\theta_\mathrm{vac}=0$. The substrate being also topologically trivial, both interfaces at $z=0$ and $z=\ell$ support a domain wall of $\theta$ giving rise to a surface QHE with half-quantized surface Hall conductance $\sigma_H^s=(n+\frac{1}{2})\frac{e^2}{h}$ with $n\in\mathbb{Z}$ [@Qi2008]. The factor of $\frac{1}{2}$ is a topological property of the bulk and is protected by the ${\cal
T}$ symmetry. On the other hand, the value of $n$ depends on the details of the interface and may thus be different for the two interfaces. To account for this general case we assign $\theta_\mathrm{subs}=2p\pi$ with $p\in\mathbb{Z}$ to the topologically trivial substrate, corresponding to $\sigma_H^{s,0}=\frac{\theta}{2\pi}\frac{e^2}{h}$ on the $z=0$ interface and $\sigma_H^{s,\ell}=(p-\frac{\theta}{2\pi})\frac{e^2}{h}$ on the $z=\ell$ interface. The experiment consists in shining normally incident monochromatic light with frequency $\omega$ on the TI film, and measuring the Kerr angle $\theta_K$ of the reflected light and Faraday angle $\theta_F$ of the transmitted light. However, the effective theory (\[Laxion\]) applies only in the regime $\omega\ll E_g/\hbar$ where $E_g$ is the surface gap [@Qi2008]. Such a surface gap can be opened by a thin magnetic coating on both surfaces of the TI, as first suggested in Ref. [@Qi2008], or by an applied perpendicular magnetic field ${\mathbf{B}}=B\hat{{\mathbf{z}}}$ (Fig. \[fig:fig1\]) through the surface Zeeman effect as well as the exchange coupling between surface electrons and the paramagnetic bulk. We discuss the experimentally simpler case of the external magnetic field. For incident light linearly polarized in the $x$ direction ${\mathbf{E}}_\mathrm{in}=E_\mathrm{in}\hat{{\mathbf{x}}}$, the Kerr and Faraday angles are defined by $\tan\theta_K=E_\mathrm{r}^y/E_\mathrm{r}^x$ and $\tan\theta_F=E_\mathrm{t}^y/E_\mathrm{t}^x$, respectively, with ${\mathbf{E}}_\mathrm{r}=E_\mathrm{r}^x(-\hat{{\mathbf{x}}})+E_\mathrm{r}^y\hat{{\mathbf{y}}}$ and ${\mathbf{E}}_\mathrm{t}=E_\mathrm{t}^x\hat{{\mathbf{x}}}+E_\mathrm{t}^y\hat{{\mathbf{y}}}$ the reflected and transmitted electric fields, respectively (Fig. \[fig:fig1\]). Furthermore, $\theta_K$ and $\theta_F$ are to be measured as a function of $B$. The angles that we discuss in the following are defined as the linear extrapolation of $\theta_K(B)$ and $\theta_F(B)$ as $B\rightarrow 0^+$, in which limit the non-topological bulk contribution to optical rotation is removed [@Qi2008].
The problem of optical rotation at a TI/trivial insulator interface has been studied before [@Qi2008; @Karch2009; @Chang2009]. In general, $\theta_K$ and $\theta_F$ depend on the optical constants $\varepsilon_2,\mu_2$ of the TI. In the thick film geometry considered here, they will also depend in a complicated manner on the optical constants $\varepsilon_3,\mu_3$ of the substrate, the film thickness $\ell$, and the photon frequency $\omega$, due to multiple reflection effects at the two interfaces. It seems therefore dubious that one could extract the exact quantization of the TME from such a measurement. However, we find that these multiple reflection effects can be used for a universal measurement of the TME, with no explicit dependence on $\varepsilon_2,\mu_2,\varepsilon_3,\mu_3,\ell$, and $\omega$.
![(color online). (a) Reflectivity $R$ as a function of photon frequency $\omega$ in units of the characteristic frequency $\omega_\ell$ for a topological insulator Bi$_2$Se$_3$ thick film on a Si substrate; universal function $f(\theta)$ for different values of (b) the substrate dielectric constant $\varepsilon_3$, (c) $p$, the total surface Hall conductance in units of $\frac{e^2}{h}$, and (d) the TI dielectric constant $\varepsilon_2$. The position of the zero crossing is universal and provides an experimental demonstration of the quantized TME.[]{data-label="fig:fig2"}](fig2.eps){width="0.9\columnwidth"}
In Fig. \[fig:fig2\](a) we plot the reflectivity $R\equiv|{\mathbf{E}}_\mathrm{r}|^2/|{\mathbf{E}}_\mathrm{in}|^2$ as a function of photon frequency $\omega$ in units of a characteristic frequency $\omega_\ell\equiv\frac{c}{\sqrt{\varepsilon_2\mu_2}}\frac{\pi}{\ell}$, for $\varepsilon_2=100$, $\varepsilon_3=13$, and $\mu_2=\mu_3=1$, appropriate for a topological Bi$_2$Se$_3$ [@Zhang2009] thin film on a Si substrate [@laforge2009; @butch2010; @JasonHancock]. We observe that minima in $R$ occur when $\omega/\omega_\ell$ is an integer, corresponding to $\ell$ being an integer multiple of $\frac{\lambda_2}{2}$ with $\lambda_2=\frac{2\pi c}{\omega\sqrt{\varepsilon_2\mu_2}}$ the photon wavelength inside the TI. For radiation in the terahertz range this corresponds to $\ell\sim 100$ $\mu$m. When $\omega$ is tuned to any of these minima, we find $$\begin{aligned}
\label{minima1}
\tan\theta_K'=\frac{4\alpha p}{Y_3^2-1+4\alpha^2p^2},
\hspace{5mm}
\tan\theta_F'=\frac{2\alpha p}{Y_3+1},\end{aligned}$$ where $Y_i\equiv\sqrt{\varepsilon_i/\mu_i}$ is the admittance of region $i$, and the prime indicates rotation angles measured at a reflectivity minimum, i.e. for $\omega/\omega_\ell\in\mathbb{Z}$. We see that $\theta_K'$ and $\theta_F'$ are independent of the TI optical constants $\varepsilon_2,\mu_2$. Equation (\[minima1\]) corresponds simply to the results of Ref. \[\] for a *unique* interface with axion domain wall $\Delta\theta=2p\pi$. Moreover, the two angles can be combined [@JiangZhe] to obtain a universal result independent of both TI $\varepsilon_2,\mu_2$ and substrate $\varepsilon_3,\mu_3$ properties, $$\begin{aligned}
\label{pa}
\frac{\cot\theta_F'+\cot\theta_K'}{1+\cot^2\theta_F'}=\alpha p,
\hspace{5mm}p\in\mathbb{Z}.\end{aligned}$$ Since the rotation angles are measured at a reflectivity minimum, Eq. (\[pa\]) has no explicit dependence on $\ell$ or $\omega$ either. Equation (\[pa\]) clearly expresses the topological quantization in units of $\alpha$ solely in terms of experimentally measurable quantities, and is the first important result of this work.
However, neither Eq. (\[minima1\]) nor Eq. (\[pa\]) depend explicitly on the TI axion angle $\theta$, and one may ask whether Eq. (\[pa\]) is at all an indication of nontrivial bulk topology. In fact, Eq. (\[pa\]) describes the topological quantization of the *total* Hall conductance of both surfaces $\sigma_H^{s,\mathrm{tot}}=\sigma_H^{s,0}+\sigma_H^{s,\ell}=p\frac{e^2}{h}$, which holds independently of possible $\mathcal{T}$ breaking in the bulk. In the special case that the two surfaces have the same surface Hall conductance, we have $p=2\sigma_H^{s,0}=\frac{\theta}{\pi}$ and Eq. (\[pa\]) is sufficient to determine the bulk axion angle $\theta$. However, for a TI film on a substrate the two surfaces are generically different and can have different Hall conductance. To obtain the axion angle $\theta$ in the more general case of different surfaces, we propose another optical measurement performed at reflectivity *maxima* $\omega=(n+\frac{1}{2})\omega_l$, $n\in\mathbb{Z}$ \[Fig. \[fig:fig2\](a)\]. We denote by $\theta_K''$ and $\theta_F''$ the Kerr and Faraday angles measured at an arbitrary reflectivity maximum. In contrast to $\theta_K'$ and $\theta_F'$ \[Eq. (\[minima1\])\], these depend on $\varepsilon_2,\mu_2$ as well as on $\varepsilon_3,\mu_3$, $$\begin{aligned}
\label{maxima}
\tan\theta_K''&=\frac{4\alpha
\left[Y_2^2\left(p-\frac{\theta}{2\pi}\right)
-\tilde{Y}_3^2\frac{\theta}{2\pi}\right]}
{\tilde{Y}_3^2-Y_2^4+4\alpha^2
\left[2Y_2^2\frac{\theta}{2\pi}\left(p-\frac{\theta}{2\pi}\right)
-\tilde{Y}_3^2\left(\frac{\theta}{2\pi}\right)^2\right]},
\nonumber\\
\tan\theta_F''&=\frac{2\alpha
\left(p-\frac{\theta}{2\pi}+Y_3\frac{\theta}{2\pi}\right)}
{Y_3+Y_2^2-4\alpha^2\frac{\theta}{2\pi}
\left(p-\frac{\theta}{2\pi}\right)},\end{aligned}$$ where we define $\tilde{Y}_3^2=Y_3^2+4\alpha^2\left(p-\frac{\theta}{2\pi}\right)^2$. More importantly, $\theta_K''$ and $\theta_F''$ depend explicitly on the TI axion angle $\theta$. It is readily checked that Eq. (\[maxima\]) reduces to Eq. (\[minima1\]) in the single-interface limit $\theta=2p\pi$, $Y_2=Y_3$ or $\theta=0$, $Y_2=1$. In general however, from the knowledge of $p$ \[Eq. (\[pa\])\] and either $\theta_K'$ or $\theta_F'$ we can extract $Y_3$ by using Eq. (\[minima1\]) without performing any separate measurement. Moreover, $\theta_K''$ and $\theta_F''$ can be combined to cancel the explicit dependence on the TI properties $\varepsilon_2,\mu_2$. We solve for $Y_2^2$ in Eq. (\[maxima\]) in terms of $\theta_F''$, say, and substitute the resulting expression $Y_2^2=Y_2^2(\theta)$ into the equation for $\theta_K''$ in Eq. (\[maxima\]). The result can be expressed in the form $f(\theta_K',\theta_F',\theta_K'',\theta_F'';p,\theta)=0$ where $f$ is ‘universal’ in the sense that it does not depend explicitly on any material parameter $\varepsilon_i,\mu_i$. Substituting the experimental values of $\theta_K',\theta_F',\theta_K'',\theta_F''$ and $p$ into this expression, we obtain a function of a single variable $f(\theta)$. If we plot $f$ as a function of $\theta$, the zero crossing $f(\theta)=0$ gives the value of the bulk axion angle $\theta$ with no $2\pi$ ambiguity. Plots of the universal function $f$ are given in Fig. \[fig:fig2\](b), (c), and (d) for different values of the material parameters $\varepsilon_2,\varepsilon_3,p$ (setting $\mu_2=\mu_3=1$ without loss of generality) and for a bulk axion angle $\theta=\pi$. The zero crossing point is independent of material parameters and, together with Eq. (\[pa\]), can provide a universal experimental demonstration of the quantization of the TME in the TI bulk. In a *thin* film geometry $\ell\ll\frac{\lambda_2}{2}$ corresponding to $\omega\ll\omega_\ell$, the optical response is always given by the sum of the Hall conductivities of the two surfaces. Therefore, thick films $\ell\geq\frac{\lambda_2}{4}$ to allow destructive interference and reflectivity maxima are essential to the measurement of the bulk TME.
![(color online). (a) Kerr-only measurement setup, with material parameters the same as indicated in Fig. \[fig:fig1\]; (b), (c) and (d): universal function $f_K(\theta)$ for different material parameters \[same as in Fig. \[fig:fig2\](b), (c), (d)\]. As in Fig. \[fig:fig2\], the position of the zero crossing is universal and provides an experimental demonstration of the quantized TME.[]{data-label="fig:fig3"}](fig3.eps){width="0.9\columnwidth"}
Our proposal so far necessitates the measurement of both Kerr and Faraday angles. We now show that it is possible to extract $p$ and $\theta$ from Kerr measurements alone, if the Kerr angle is measured in both directions \[Fig. \[fig:fig3\](a)\]. Indeed, while the Faraday angle is generally independent of the direction of propagation [@LL_EM], the Kerr angle depends on it. Here we exploit this asymmetry of the Kerr angle to extract $p$ and $\theta$. We denote by $\theta_K^{\prime 13}$ and $\theta_K^{\prime\prime 13}$ the Kerr angles defined previously in Eq. (\[minima1\]) and (\[maxima\]), respectively. Conversely, we denote by $\theta_K^{\prime 31}$ and $\theta_K^{\prime\prime
31}$ the Kerr angles for light traveling in the opposite direction, i.e. incident from the substrate \[Fig. \[fig:fig3\](a)\]. As before, the prime and double prime correspond to angles measured at reflectivity minima and maxima, respectively. We find $$\begin{aligned}
\label{thetaK31}
\tan\theta_K^{\prime 31}&=-\frac{4\alpha pY_3}{Y_3^2-1+4\alpha^2p^2},
\\
\tan\theta_K^{\prime\prime 31}&=
\frac{4\alpha Y_3\left[Y_2^2\frac{\theta}{2\pi}-
\gamma\left(p-\frac{\theta}{2\pi}\right)\right]}
{\gamma Y_3^2+4\gamma\alpha^2\left[p^2-\left(\frac{\theta}{2\pi}\right)^2
\right]-Y_2^4-8\alpha^2Y_2^2\left(\frac{\theta}{2\pi}\right)^2},
\nonumber\end{aligned}$$ where we define $\gamma\equiv1+4\alpha^2\left(\frac{\theta}{2\pi}\right)^2$. As previously, $\theta_K^{\prime 13}$ and $\theta_K^{\prime 31}$ can be combined to eliminate $Y_3$ and provide a universal measure of $p\in\mathbb{Z}$, $$\begin{aligned}
\label{pKerr}
\cot\theta_K^{\prime 13}-{\mathop{\mathrm{sgn}}}p
\sqrt{1+\cot^2\theta_K^{\prime 13}(1-\tan^2\theta_K^{\prime 31})}
=2\alpha p,\end{aligned}$$ provided $Y_3^2\equiv\varepsilon_3/\mu_3>1+4\alpha^2 p^2$, which is satisfied in practice for low $p$ since $\alpha^2\sim
10^{-4}$. Furthermore, comparing Eq. (\[thetaK31\]) for $\theta_K^{\prime 31}$ to Eq. (\[minima1\]) for $\theta_K^{\prime 13}$ we see that $Y_3$ is easily obtained as $Y_3=-\cot\theta_K^{\prime 13}\tan\theta_K^{\prime 31}$. Finally, to extract the bulk axion angle $\theta$, we need to solve for $Y_2^2$ in Eq. (\[maxima\]) in terms of $\theta_K^{\prime\prime 13}$, and substitute the resulting expression $Y_2^2=Y_2^2(\theta)$ into the equation for $\theta_K^{\prime\prime 31}$ in Eq. (\[thetaK31\]). The result of this analysis can once again be expressed in the form $f_K(\theta_K^{\prime 13},\theta_K^{\prime
31},\theta_K^{\prime\prime 13},\theta_K^{\prime\prime
31};p,\theta)=0$, where $f_K$ is a ‘universal’ function which only depends on the measured Kerr angles. As before, we substitute into $f_K$ the experimental values of $\theta_K^{\prime
13},\theta_K^{\prime 31},\theta_K^{\prime\prime
13},\theta_K^{\prime\prime 31}$ and $p$ \[obtained from Eq. (\[pKerr\])\] and obtain a function of a single variable $f_K(\theta)$ which crosses zero at the value of the bulk axion angle with no $2\pi$ ambiguity. In Fig. \[fig:fig3\](b), (c) and (d) we plot the universal function $f_K$ for different values of the material parameters $\varepsilon_2,\varepsilon_3,p$ and for a bulk axion angle $\theta=\pi$. The zero crossing point is independent of material parameters and, together with Eq. (\[pKerr\]), provides another means to demonstrate experimentally the universal quantization of the TME in the bulk of a TI.
Recent work [@tse2010] has addressed the similar problem of optical rotation on a TI film, and found interesting and novel results for the rotation angles. However, these results hold only in certain limits which are less general than the ones discussed in this work. First, Ref. [@tse2010] considers a free-standing TI film in vacuum. Most films are grown on a substrate which can affect the physics qualitatively. For instance, the giant Kerr rotation $\theta_K=\tan^{-1}(1/\alpha)\simeq\pi/2$ found in Ref. [@tse2010] is a special case of our Eq. (\[minima1\]) with $p=1$ and $\varepsilon_3/\mu_3=1$. It is dramatically suppressed when $\varepsilon_3/\mu_3-1$ is greater than $\alpha^2\sim 10^{-4}$, which is typically the case in practice. Second, in Ref. [@tse2010] a correction proportional to $\Delta/\epsilon_c$ was introduced to the surface Hall conductance, where $\Delta$ is the ${\cal T}$-breaking Dirac mass and $\epsilon_c$ is a non-universal high-energy cutoff. According to the general bulk TFT of the TI [@Qi2008], the surface Hall conductance is always quantized as long as the surface is gapped and the bulk is ${\cal T}$-invariant (in the $B\rightarrow 0$ limit). Thus we conclude that such a non-universal correction is absent and the requirement $\Delta\ll\epsilon_c$ is not necessary within the TFT approach [@Qi2008]. This difference clearly demonstrates the power of the TFT approach [@Qi2008] in predicting universally quantized topological effects in condensed matter physics.
We acknowledge helpful discussions with A. Fried, T. L. Hughes, A. Kapitulnik, R. Li, R. Maciejko, and especially with J. N. Hancock. J.M. is supported by the Stanford Graduate Foundation. This work is supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515, and the NSF MRSEC at University of Maryland under Grant No. DMR-0520471.
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abstract: 'Westochastically quantize the Born-Infeld field which can hardly be dealtwith by means of the standard canonical and/or path-integral quantization methods. We set a hypothetical Langevin equation in order to quantize the Born-Infeld field, following the basic idea of stochastic quantization method. Numerically solving this nonlinear Langevin equation, we obtain a sort of \`\`particle mass” associated with the gauge-invariant Born-Infeld field as a function of the so-called universal length.'
address: 'Department of Physics, Waseda University, Tokyo 169, Japan'
author:
- 'Hiroshi Hotta, Mikio Namiki and Masahiko Kanenaga'
title: 'Stochastic quantization of Born-Infeld field'
---
Introduction
============
Many years ago Born and Infeld [@bi] presented a nonlinear electromagnetic field with a non-polynomial action including the so-called [*universal length*]{}. One of the most important characteristics of the Born-Infeld field is found in its static solution which has no divergence of static self-energy. Many physicists expected that this might be an example of divergence-free field theory. However, no one could succeed to quantize the field, by means of the standard canonical quantization method, because of the complicated nonlinearity. Even the path-integral quantization can hardly be applied to this field, because we cannot easily manipulate such a non-polynomial action. We have to invent a new quantization method if we want to quantize the Born-Infeld field.
About ten years ago, Parisi and Wu [@pw] proposed a new quantization method, called [*stochastic quantization*]{}, by introducing a hypothetical stochastic process with respect to a new (fictitious) time, say $t$, other than ordinary time, say $x_{0}$. The stochastic process is so designed as to yield quantum mechanics in the infinite $t$-limit (thermal equilibrium limit). This theory starts from a hypothetical Langevin equation for the stochastic process, given by adding the fictitious-time derivative and the random source to the classical equation of motion. Remember that the stochastic quantization can be formulated only on the basis of classical field equation, without resorting to canonical formalism.
Brief review of Born-Infeld field
=================================
The ordinary electromagnetic field is described by the following Lagrangian density $${\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}({\bf E}^{2}-{\bf H}^{2})
\ , \label{eq:eml}$$ where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The corresponding action is given by $${\cal S}=\int d^{4}x {\cal L}\ . \label{eq:ems}$$ Here we have followed the usual notation. Note that we keep the Minkowski metric.
In the case of spherically symmetric static electric field, we can put $${\bf E}=-\nabla A_{0}(r), \qquad {\bf H}=0\ , \label{eq:spsym}$$ and obtain $$A_{0}=\frac{e}{r} \label{eq:se}$$ for a point charge $e$. That this field becomes $\infty$ for $r\rightarrow 0$ is a well-known fact.
Let us introduce the action functional of the Born-Infeld field [@bi] : $${\cal S}_{B}=\int d^{4}x \left[ -b^{2}\sqrt{1-\frac{2}{b^{2}}{\cal L}}\
+ b^{2} \right]\ , \label{eq:bis}$$ where $1/\sqrt{b}$ is a sort of universal constant called [*universal length*]{}, and has the dimension of length in natural unit $\hbar=c=1$. We can easily see that ${\cal S}_{B} \rightarrow {\cal S}$, that is, the Born-Infeld field will become the ordinary electromagnetic field as $b$ tends to $\infty$.
As is well known, all physical quantities can be written only in terms of the dimension of length in natural unit $\hbar=c=1$. Many years ago Heisenberg anticipated that we could formulate a finite field theory, free from field-theoretical divergences, if we could bring a sort of [*universal length*]{} into physics in an appropriate way. Following his idea, Born and Infeld [@bi] proposed to use the above field given by action (\[eq:bis\]). Unfortunately, however, the Heisenberg’s anticipation was not accomplished yet even now.
For the spherically symmetric static field $A_{0}(r)$ (\[eq:spsym\]), the Born-Infeld action (\[eq:bis\]) yields the following equation $$\frac{\partial}{\partial r} \left[ r^{2} \frac{\frac{\partial}{\partial r}
A_{0}(r)}{\sqrt{1-\frac{1}{b^{2}}(\frac{\partial}{\partial r}A_{0}(r))^{2}}}
\right] =0\ , \label{eq:biseq}$$ whose solution is given by $$A_{0}=\frac{e}{r_0}\int_{r/r_{0}}^{\infty} d\xi \frac{1}{\sqrt{1+\xi^{4}}}
\ \rightarrow \ \left\{
\begin{array}{rl}
(e/r) & \quad \mbox{for $r \gg r_{0}$}\ , \\
1.8541 \cdot (e/r_{0}) & \quad \mbox{for $r\rightarrow 0$}\ ,
\end{array}\right. \label{eq:bise}$$ where $r_{0}=\sqrt{|e|/b}$. We surely realize that the Born-Infeld field has finite static self-energy. Of course, the self-energy goes back to the original infinity as $b \rightarrow \infty$.
We are not interested in static field but in wave field propagating to remote places. The Euler-Lagrange equation of ${\cal S}_{B}$ is generally written as $$\frac{\delta {\cal S_{B}}}{\delta A_{\nu}(x)}=\partial_{\mu}
\left[ \frac{F^{\mu \nu}}{\sqrt{1+\frac{1}{2b^{2}}F^{2}}} \right]
=\partial_{\mu} \left[ \frac{F^{\mu \nu}}{\chi} \right]=0\ , \label{eq:bieq1}$$ or $$\partial_{\mu}F^{\mu \nu}=(\partial_{\mu} \ln \chi)F^{\mu \nu}\ ,
\label{eq:bieq2}$$ where $F^{2}=F_{\mu \nu}F^{\mu \nu}$ and $\chi=\sqrt{1+\frac{1}{2b^{2}}F^{2}}$.
The right-hand side of this equation is proportional to $1/b^{2}$, and can be expanded in a power series of $1/b^{2}$. Needless to say, its unperturbed one (for $1/b=0$) is nothing other than the free Maxwell equation $$\partial_{\mu}F^{\mu \nu}=0\ , \label{eq:mxeq}$$ which allows us to use the wave gauge given by $$A_{0}=0\ , \ \ \ \nabla \cdot {\bf A}=0\ .
\label{eq:wg}$$ Consequently, one may naively expect to have the perturbative theory based on (\[eq:bieq2\]) and (\[eq:wg\]). In this case, however, we can hardly develop this kind of the perturbative approach to the quantized Born-Infeld field, because the interaction part includes higher powers of derivative terms. This is the reason why we attempt to develop an unperturbative approach to the quantized Born-Infeld field by means of [*stochastic quantization*]{} [@pw] in the present paper.
Stochastic quantization
=======================
As is well-known, it is convenient to use the Euclidean action ${\cal S}_{E}$ derived by the Wick rotation ($x_{0} \rightarrow -ix_{0}$) from the original Minkowski action, for the purpose of carrying out the Parisi-Wu stochastic quantization. In order to perform [*stochastic quantization*]{} of the Born-Infeld field $A_{\mu}$, we have to introduce the additional dependence on fictitious-time $t$ (other than ordinary-time $x_{0}$) into the field quantities and then to set the basic Langevin equation $$\frac{\partial}{\partial t}A_{\mu}(x,t)=-\frac{\delta {\cal S}_{E}}
{\delta A_{\mu}}|_{A=A(x,t)}+\eta_{\mu}(x,t)
\label{eq:le}$$ for a hypothetical stochastic process of $A_{\mu}(x,t)$ with respect to $t$ [@pw]. Here $\eta_{\mu}(x,t)$ is the Gaussian white-noise field subject to $$\begin{aligned}
<\eta_{\mu}(x,t)>&=&0\ , \label{eq:st1}\\
<\eta_{\mu}(x,t)\eta_{\nu}(x',t')>&=&2\delta_{\mu \nu}
\delta(x-x')\delta(t-t') \label{eq:st2}\end{aligned}$$ for statistical ensemble averages, where we have put $\hbar=1$ for simplicity.
According to the prescription of stochastic quantization [@pw], we can derive the field-theoretical propagaters through the well-known formula $$\begin{aligned}
\lefteqn{D^{A}_{\mu \nu}(x,x')} \nonumber\\
&& =\lim_{t\rightarrow \infty}
\left[<A_{\mu}(x,t)A_{\nu}(x',t)>
- <A_{\mu}(x,t)><A_{\nu}(x',t)> \right]\ , \label{eq:pro}\end{aligned}$$ where $A_{\mu}(x,t)$ as a function of $\eta_{\mu}$ is to be obtained by solving (\[eq:le\]).
Let us decompose $A_{\mu}$ and $\eta_{\mu}$ into their longitudinal and transverse components, $A^{L}_{\mu}$ and $A^{T}_{\mu}$, and $\eta^{L}_{\mu}$ and $\eta^{T}_{\mu}$, given by $$\begin{aligned}
A^{L}_{\mu}&=&\frac{1}{\Box} \partial_{\mu}
\partial_{\nu}A_{\nu}\ \label{eq:al} \\
A^{T}_{\mu}&=&(\delta_{\mu \nu}-\frac{1}{\Box} \partial_{\mu}
\partial_{\nu})A_{\nu}\ ; \ \ \partial_{\mu}A^{T}_{\mu}=0\ , \label{eq:at}\end{aligned}$$ and similar ones for $\eta$’s. Therefore, we can decompose the basic Langevin equation (\[eq:le\]) as follows; $$\begin{aligned}
\frac{\partial}{\partial t}A^{L}_{\mu}(x,t)&=&0+\eta^{L}_{\mu}(x,t)\ , \label{eq:leal} \\
\frac{\partial}{\partial t}A^{T}_{\mu}(x,t)&=&-\frac{\delta {\cal S}_{E}}
{\delta A^{T}_{\mu}}|_{A^{T}=A^{T}(x,t)}+\eta^{T}_{\mu}(x,t)\ . \label{eq:leat}\end{aligned}$$
The absence of drift force in (\[eq:leal\]) is an important reflection of the gauge invariance that ${\cal S}_{E}$ does not depend on $A^{L}_{\mu}$. Consequently, the longitudinal component $A^{L}_{\mu}(x,t)$ makes a random walk around its initial value $A^{L}_{\mu}(x,0)=\frac{1}
{\Box} \partial_{\mu}\phi(x)$, $\phi(x)$ being a scalar field. As was discussed in detail in the case of non-Abelian gauge field [@nooy], we must introduce [*gauge parameter*]{} $\alpha$ by taking average of $\phi$ over random fluctuations around zero ($\overline{\phi(x)\phi(x')}
=\alpha \delta(x-x'))$. Thus we obtain $$D^{A^L}_{\mu \nu}(x,x')=\frac{1}{\Box} \partial_{\mu}
\partial_{\nu}(\alpha \frac{1}{\Box}+2t)\delta(x-x')\ . \label{eq:alpr}$$ Of course, we know that the longitudinal components never appear in gauge-invariant (physical) quantities.
We should also notice that the transverse components and their propagaters are completely decoupled with the longitudinal ones in the present case. This is an important point quite different from the non-Abelian gauge field case. Thus we can safely discard $A^{L}_{\mu}$, and use the wave gauge (\[eq:wg\]), even in the present case, for the purpose of deriving propagaters of the transverse components.
Considering wave propagation along the $z=x_{3}$-axis, we put $$\begin{aligned}
A_{0}&=&0\ , \qquad A_{3}=0\ , \label{eq:a03} \\
A_{1}&=&A_{1}(x_{0},x_{3})\ , \qquad
A_{2}=A_{2}(x_{0},x_{3})\ , \label{eq:a12} \end{aligned}$$ and $$\begin{aligned}
\eta_{0}&=&0\ , \qquad \eta_{3}=0\ , \label{eq:eta03} \\
\eta_{1}&=&\eta_{1}(x_{0},x_{3})\ , \qquad
\eta_{2}=\eta_{2}(x_{0},x_{3})\ . \label{eq:eta12} \end{aligned}$$ Note that $A_{\mu}$ and $\eta_{\mu}$ have no longitudinal components.
In this case, the Euclid action becomes $${\cal S}_{E}=\int dx_{0}dx_{3} \left[ b^{2}\sqrt{1+\frac{1}{b^{2}}{F_{E}^{2}}}
-b^{2} \right]\ ,
\label{eq:es2}$$ where $F_{E}^{2}=(\partial_{0} A_{1})^{2}+(\partial_{3} A_{2})^{2}
+(\partial_{0} A_{2})^{2} +(\partial_{3} A_{1})^{2}$. The corresponding classical field equation is given by $$\frac{\delta S_{E}}{\delta A_{i}(x)}=-\partial_{0} \left[
\frac{\partial_{0} A_{i}}{\sqrt{1+\frac{1}{b^{2}}F_{E}^{2}}} \right]
-\partial_{3} \left[
\frac{\partial_{3} A_{i}}{\sqrt{1+\frac{1}{b^{2}}F_{E}^{2}}} \right]
\ =0, \ \ (i=1,2)\ .
\label{eq:cleqwg}$$ Note that the dimensions of the field and $b$ are different from the original ones: $[A_{i}]=[L^{0}]$ and $[b^{-1}]=[L]$, $[L]$ standing for the dimension of length.
Based on the classical equation, we can set up the basic Langevin equation for stochastic quantization of the Born-Infeld field as follows; $$\begin{aligned}
\frac{\partial}{\partial t}A_{i}(x_{0},x_{3},t)&=&
\partial_{0} \left[
\frac{\partial_{0} A_{i}}{\sqrt{1+\frac{1}{b^{2}}F_{E}^{2}}} \right]
+\partial_{3} \left[
\frac{\partial_{3} A_{i}}{\sqrt{1+\frac{1}{b^{2}}F_{E}^{2}}} \right]
+\eta_{i}(x_{0},x_{3},t)\ , \nonumber \\
(i=1,2)\ , \label{eq:lgeqwg}\end{aligned}$$ where the fluctuating source-field $\eta_{i}$ should have the statistical properties $$\begin{aligned}
&& <\eta_{i}(x_{0},x_{3},t)>_{\eta} = 0\ , \label{eq:tks1} \\
&& <\eta_{i}(x_{0},x_{3},t)\eta_{j}(x_{0}',x_{3}',t')>_{\eta} = 2\delta_{ij}
\delta(x_{0}-x_{0}')\delta(x_{3}-x_{3}')\delta(t-t')\ . \label{eq:tks2}\end{aligned}$$ Consequently, we have to obtain $A_{i}$ as a function (or functional) of $\eta_{i}$, by solving (\[eq:lgeqwg\]), and then to calculate expectation values of physical quantities, by making use of (\[eq:tks1\]) and (\[eq:tks2\]). For example, the field-theoretical propagater of $A_{i}$ is given by the formula $$\begin{aligned}
\Delta_{ij}^{A}(x_{0}-x_{0}',x_{3}-x_{3}') && \equiv
\lim_{t \rightarrow \infty}
\left[ <A_{i}(x_{0},x_{3},t)A_{j}(x_{0}',x_{3}',t)> \right. \nonumber\\
&& \mbox{ } \left. -<A_{i}(x_{0},x_{3},t)><A_{j}(x_{0}',x_{3}',t)> \right].
\label{eq:prgt} \end{aligned}$$
For conventional fields, we can extract real information about the particle mass, ${\cal M}$ and ${\cal M}'$, associated with the field (or the first energy gap) from the asymptotic formulas $$\begin{aligned}
\Delta_{ii}^{A}(0,x_{3})& \stackrel{|x_{3}| \rightarrow \infty}{\longrightarrow} &\mbox{const.} \exp[-{\cal M}|x_{3}|] \ , \label{eq:m1} \\
\Delta_{ii}^{A}(x_{0},0)& \stackrel{|x_{0}| \rightarrow \infty}{\longrightarrow} &\mbox{const.} \exp[-{\cal M}'|x_{0}|] \label{eq:m2}\end{aligned}$$ ($i$: no summation), in which we can put ${\cal M}={\cal M}'$ for the Euclidean symmetry in space-time. Unfortunately in the Born-Infeld case, however, we have no reliable theory to justify the procedure (\[eq:m1\]) and/or (\[eq:m2\]) to give mass. Despite of this situation, we intend to follow the conventional approach to the \`\`particle mass” associated with the (transverse) Born-Infeld field, based on (\[eq:m1\]) and/or (\[eq:m2\]).
Numerical simulation and particle mass
======================================
Needless to say, we know that it is very difficult to solve (\[eq:lgeqwg\]) analytically, so that we are inevitably enforced to deal with it by means of numerical simulation. For this purpose, we first discretize the Langevin equation (\[eq:lgeqwg\]) on an $N\times N$ lattice with spacings $\Delta x_{0}$ and $\Delta x_{3}$ (along time and space directions, respectively). Denoting the Born-Infeld field on the $(k,l\/)$-th lattice point by $A_{i;k,l}(t)$, where $i$ stands for the $i$-th component of the field and $k,l$ for the ordinary time and spatial position, then we write down the discretized Langevin equation as $$\begin{aligned}
\lefteqn{\frac{A_{i;k,l}(t + \Delta t) - A_{i;k,l}(t)}{\Delta t}}
\nonumber\\
&& = \frac{G_{i;k+1,l}(t) - G_{i;k,l}(t) }{\Delta x_0}
+\frac{H_{i;k,l+1}(t) - H_{i;k,l}(t)}{\Delta x_3}
+\sqrt{\frac{2}{\Delta x_{0} {\Delta x_{3}} {\Delta t}}}
N_{i;k,l}(t) \ , \nonumber\\
&& \label{eq:disl1}\end{aligned}$$ where $$\begin{aligned}
G_{i;k,l}(t)
&=& \frac{ \frac{1}{\Delta x_{0}} ( A_{i;k,l}(t) - A_{i;k-1,l}(t) )}
{\sqrt{ 1 + F_{E}^{2} }} \ ,\label{eq:g} \\
H_{i;k,l}(t)
&=& \frac{ \frac{1}{\Delta x_{3}} ( A_{i;k,l}(t) - A_{i;k,l-1}(t) )}
{\sqrt{ 1 + F_{E}^{2} }} \label{eq:h}\end{aligned}$$ with $$\begin{aligned}
F_{E}^{2}
&=&\frac{1}{b^{2}} \left\{ \left[ \frac{A_{1;k,l}(t) - A_{1;k-1,l}(t)}{\Delta x_0} \right]^{2}
+\left[ \frac{A_{2;k,l}(t) - A_{2;k-1,l}(t)}{\Delta x_0} \right]^{2} \right.
\nonumber \\
& & \left. +\left[ \frac{A_{1;k,l}(t) - A_{1;k,l-1}(t)}{\Delta x_3} \right]^{2}
+\left[ \frac{A_{2;k,l}(t) - A_{2;k,l-1}(t)}{\Delta x_3} \right]^{2} \right\} \label{eq:disl3}\end{aligned}$$ for drift terms, and $$\begin{aligned}
& &
< N_{i;k,l}(t) >_{N} = 0,\qquad
< N_{i;k,l}(t) N_{i;k',l'}(t') >_{N}
= \delta_{ij} \delta_{kk'} \delta_{ll'} \delta_{tt'}\ ,\label{eq:disn1}\\
& &(i, j = 1, 2), \qquad (l, k = 1, \ldots, N) \nonumber
%\label{eq:}\end{aligned}$$ for noise terms.
Here let us introduce a scale unit $a$, which has dimension of length, and put relevant quantities in the following way: $$\begin{aligned}
& &{\Delta x_{0}}={\Delta {\tilde x}_{0}}a,\
{\Delta x_{3}}={\Delta {\tilde x}_{3}}a,\
{\Delta t}={\Delta {\tilde t}}{a^{2}},\
b={\tilde b} a^{-1}, \ \nonumber \\
& &A_{i;k,l}(t)={\tilde A}_{i;k,l}(\tilde t),\
G_{i;k,l}(t)={\tilde G}_{i;k,l}(\tilde t){a^{-1}},\ \nonumber \\
& &H_{i;k,l}(t)={\tilde H}_{i;k,l}(\tilde t){a^{-1}},\
N_{i;k,l}(t)={\tilde N}_{i;k,l}(\tilde t),\ \nonumber \\
& &{\cal M}=\tilde{{\cal M}} a^{-1}, \label{eq:mass}\end{aligned}$$ Note that all quantities with tilde are dimensionless. Thus, the equations (\[eq:disl1\]) and (\[eq:disn1\]) are rewritten as $$\begin{aligned}
& &
\frac{{\tilde A}_{i;k,l}({\tilde t} + \Delta {\tilde t})
- {\tilde A}_{i;k,l}(\tilde t)}{\Delta {\tilde t}} \nonumber\\
& &=\frac{{\tilde G}_{i;k+1,l}(\tilde t) - {\tilde G}_{i;k,l}(\tilde t)}
{ \Delta {\tilde x}_0 }
+\frac{{\tilde H}_{i;k,l+1}(\tilde t) - {\tilde H}_{i;k,l}(\tilde t) }
{ \Delta {\tilde x}_3 }
+\sqrt{
\frac{2}{ {\Delta {\tilde x}_{0}}{\Delta {\tilde x}_{3}}{\Delta {\tilde t}} }
} {\tilde N}_{i;k,l}(\tilde t)\ ,\nonumber \\
&& \\
& &< {\tilde N}_{i;k,l}(\tilde t) >_{\tilde N} = 0, \qquad
< {\tilde N}_{i;k,l}(\tilde t)
{\tilde N}_{i;k',l'}({\tilde t}') >_{\tilde N}
= \delta_{ij} \delta_{kk'} \delta_{ll'} \delta_{{\tilde t}{\tilde t}'}\ .
%\label{eq:}\end{aligned}$$ $\tilde G_{i;k,l}$ and $\tilde H_{i;k,l}$ include the dimensionless $\tilde F_{E}^{2}$ in the same way as $G_{i;k,l}$ and $H_{i;k,l}$ depend on $F_{E}^{2}$. Note that only $\tilde F_{E}^{2}$ contains the Born-Infeld parameter $\tilde b$.
In the conventional field theory, this scale unit $a$ can be determined by making use of the renormalization group theory. In the present case, however, we have no reliable theory to determine the scale unit, and then we shall inevitably calculate all quantities (in particular, the \`\`particle mass”) on an arbitrary scale. Only for the sake of simplicity, let us put $a=1$, in order to go on our procedure. For a while from now on, we suppress those tilders which are put on the quantities. Therefore, the \`\`particle mass”, ${\cal M}$, will be given as a dimensionless quantity in this scheme.
In order to solve numerically the above equation and obtain the field-theoretical propagater $\Delta_{ij}(x_{0},x_{3})$ on the above lattice, we should introduce the periodic boundary condition given by $$A_{i;k,l+2l_{c}} = A_{i;k,l}\ ,\ A_{i;k+2k_{c},l}=A_{i;k,l} \ ,
\label{eq:bdc}$$ where $l_{c}, k_{c}$ and $2l_{c}, 2k_{c}$ stand for the lattice center and the period, respectively. Practically, we have used the Langevin-source method (for example, see [@pw]), in which we have performed $5.4 \times 10^6$ iterations for a lattice of $20 \times 20$ sites with $\Delta t= 0.01$ and $l_{c}=k_{c}=10$ (to realize thermal equilibrium), and then use the subsequent $2.0 \times 10^5$ iterations to calculate the field-theoretical propagater.
Figure 1 shows our numerical results of the field-theoretical propagaters $\Delta_{11}(0,x_{3})$ for $b^{-1}=10,20,30,40$ and $50$, together with curves given by the asymptotic formula $${\Delta}_{ii}(X,0)={\Delta}_{ii}(0,X)=C\frac{\cosh {\cal M}|X-x_{c}|}
{\cosh {\cal M}|x_{c}|}\ \ (i: \mbox{no summation}),\ \label{eq:asm}$$ where $x_{c}$ stands for the center of lattice. Also we numerically have got similar field-theoretical propagaters for other directions and/or components. Equation (\[eq:asm\]) is the substitute of (\[eq:m1\]) and/or (\[eq:m2\]) under the boundary condition (\[eq:bdc\]). We have estimated the \`\`particle mass”, ${\cal M}$, associated with the Born-Infeld field, by making use of $\chi^{2}$-fitting based on (\[eq:asm\]). As shown in Table 1, our results are the following: ${\cal M} = 0.0415, 0.0835, 0.1260, 0.1693, 0.2162$, correspondingly to $b^{-1} = 10, 20, 30, 40, 50$, where $\chi^2$ is $4.69 \times 10^{-4}$. We have estimated the statistical fluctuations for the fictitious time as accuracies in Table 1. Here we have put $C={\Delta}_{ii}(0,0)$ $=<A_{i}^{2}>$ ($i$: no summation). Note that $C$ is independent of $i$ due to the space-time uniformity, and that $C$ is gauge-invariant.
Rigorously speaking from the point of view of (\[eq:mass\]), we can only assert that the above ${\cal M}$ is proportional to the \`\`particle mass” associated with the (transverse) Born-Infeld field. We should repeat that we have no renormalization group theory to give the scaling formula in the case of Born-Infeld field. Remember that the problem is still open to questions. In this paper, however, we are talking about the \`\`particle mass” by ${\cal M}$ which is given by (\[eq:asm\]).
$b^{-1}$ 10 20 30 40 50
------------ -------------- -------------- -------------- -------------- --------------
${\cal M}$ 0.0415 0.0835 0.1260 0.1693 0.2162
Accuracy $\pm 0.0062$ $\pm 0.0021$ $\pm 0.0017$ $\pm 0.0017$ $\pm 0.0015$
: The \`\`particle mass”, ${\cal M}$, associated with the Born-Infeld field
Figures 2 and 3 plot the \`\`particle mass”, given by ${\cal M}$ and ${\cal M'}$, as a function of $b^{-1}$. Observe that ${\cal M}={\cal M'}$. It seems that the \`\`particle mass” is proportional to $b^{-1}$, but unfortunately, we do not know what kind of physical implications this fact suggests. Another important point should be that the \`\`particle mass” seems vanishing, in the case of $b^{-1}=0$, as expected from the fact that the Born-Infeld field must go back to the free Maxwell field in this limit. All results are presented in Table 1.
Table 1 also tells us that the (dimensionless) \`\`particle mass” on this scale (with $a=1$) distributes over a very small region.
Table 1 or Figures 2 and 3 can be fitted well by a single formula given by $${\cal M} = \gamma \frac{1}{b}\ , \qquad \gamma = 0.00426 \ .$$ This equation is rewritten in terms of ${\cal M}$ and $b^{-1}$ having dimension as $${\cal M}
= (\frac{\gamma}{a^2}) \frac{1}{b}\ , \qquad \gamma = 0.00426 \ .$$
Here let us try to choose $a=b^{-1}$ as the scale unit, then we obtain $${\cal M} = \gamma b\ , \qquad \gamma = 0.00426 \ ,$$ or $$\tilde{{\cal M}} = \gamma , \qquad \gamma = 0.00426 \ ,$$ in the dimensionless expression. because $\tilde{b} = 1$ in this case. This implies that the constant $\gamma$ is nothing other than the \`\`particle mass”, being independent of the [*universal length*]{}, on the scale adjusted by $a=b^{-1}$.
Finally, let us examine whether our \`\`particle mass” ${\cal M}$ can be regarded as a sort of particle mass in the sense of conventional field theory. For this purpose, we should numerically compute Fourier transform ${\tilde \Delta}_{ii} (k^{2})$ given by $${\tilde \Delta}_{ii}(k^{2})
=\int_{0}^{N} \frac{dx_{0}}{\sqrt{N}}
\int_{0}^{N} \frac{dx_{3}}{\sqrt{N}}
e^{-ik_{0}x_{0}-ik_{3}x_{3}}
\Delta_{ii}(x_{3},x_{0})\ , \label{eq:ft}$$ $N$ standing for lattice size. If ${\cal M}$ meant a sort of particle mass in this sense, we could hardly observe so sharp ${\cal M}$-dependence of ${\tilde \Delta}_{ii} (k^{2})$ as a function of $k^{2}$, for $\sqrt{k^2} \gg {\cal M}=0.0415, 0.0835, 0.1260, 0.1693, 0.2162$.
Figure 4 shows our numerical results (see [@ni] for technical details), in which all curves for various $b$’s are normalized to ${\tilde \Delta}_{ii}(0)=1$. We can clearly observe in Fig. 4 that ${\tilde \Delta}_{ii}(k^{2})$ is almost independent of $k^{2}$, except in its height. That is to say, our anticipation seems justified. For comparison, we put the dashed curve representing a free-propagater (being a substitute of massless free Maxwell field), $\epsilon^{2}/
(k^{2} + \epsilon^{2})$, with a very small mass $\epsilon=0.001 \ll {\cal M}$.
In order to reconfirm this fact, we present Fig. 5, (for $\frac{1}{{\cal M}^2}\Delta_{ii}(k^2)$ versus $k^2$) stressing that all curves overlap each others for larger $k^{2}$. Note that long distance behavior of the propagater in configuration space gives ${\cal M}$, while short distance behavior determines Fourier transforms for larger $k^2$.
Moreover, we compare a special ${\tilde \Delta}_{ii}(k^{2})$ with the corresponding Feynman propagater ${\cal M}^{2}/(k^{2}+{\cal M}^{2})$ for ${\cal M}=0.126$ in Fig. 6, as an example. In this figure one could observe that the difference between them would represent a possible (damping) effect due to the non-linearity of the Born-Infeld field.
Conclusion
==========
Summarizing, we have stochastically quantized the Born-Infeld field, characterized by the so-called [*universal length*]{}, which cannot be dealt with by means of the conventional quantization methods. Even though we can hardly justify the whole procedure theoretically, we have derived the \`\`particle mass” associated with the (transverse) Born-Infeld field, as a function of the [*universal length*]{}, through the conventional formulas to give them.
It would be interesting to observe that we have derived the \`\`particle mass” from a perfectly gauge-invariant field theory. Of course, we can guess that the \`\`particle mass” is produced by introducing the [*universal length*]{} $b^{-1}$ having the dimension of length.
The authors are indebted to Drs. I. Ohba, S. Tanaka, Y. Yamanaka, K. Okano and B. Zheng for many discussions and suggestions.
[99]{} M. Born and L. Infeld, [*Proc. Roy. Soc.*]{} [**150**]{} (1934) 141;\
M. Born and L. Infeld, [*Proc. Roy. Soc.*]{} [**147**]{} (1934) 522;\
M. Born, [*Proc. Roy. Soc.*]{} [**143**]{} (1934) 410.
G. Parisi and S.Y. Wu, [*Sci. Sin.*]{} [**24**]{} (1981) 483; For review articles, see, M. Namiki, [*Stochastic Quantization*]{} (Springer, Heidelberg, 1992); M. Namiki and K. Okano eds., [*Stochastic Quantization*]{} (Prog. Theor. Phys. Supplement No.111, Kyoto, 1993).
M. Namiki, I. Ohba, K. Okano and Y. Yamanaka, [*Prog. Theor. Phys.*]{} [**69**]{} (1983) 1580; and see the above review articles.
P.J. Davis and P. Rabinowitz, [*Methods of Numerical Integration*]{} (Academic Press, London and New York, 1975).
|
---
abstract: 'We explore the effect of pulsars, in particular those born with millisecond periods, on their surrounding supernova ejectas. While they spin down, fast-spinning pulsars release their tremendous rotational energy in the form of a relativistic magnetized wind that can affect the dynamics and luminosity of the supernova. We estimate the thermal and non thermal radiations expected from these specific objects, concentrating at times a few years after the onset of the explosion. We find that the bolometric light curves present a high luminosity plateau (that can reach $10^{43-44}\,$erg/s) over a few years. An equally bright TeV gamma-ray emission, and a milder X-ray peak (of order $10^{40-42}\,$erg/s) could also appear a few months to a few years after the explosion, as the pulsar wind nebula emerges, depending on the injection parameters. The observations of these signatures by following the emission of a large number of supernovae could have important implications for the understanding of core-collapse supernovae and reveal the nature of the remnant compact object.'
author:
- |
K. Kotera$^{1,2}$[^1], E. S. Phinney$^{2}$, and A. V. Olinto$^{3}$\
$^{1}$Institut d’Astrophysique de Paris, UMR7095 - CNRS, Université Pierre & Marie Curie,\
98 bis boulevard Arago F-75014 Paris, France\
$^{2}$California Institute of Technology, Mailcode 350-17, 1200 E California Blvd, Pasadena CA, 91125\
$^{3}$Department of Astronomy & Astrophysics, Enrico Fermi Institute,\
and Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, Illinois 60637, USA.
bibliography:
- 'KPO.bib'
title: Signatures of pulsars in the light curves of newly formed supernova remnants
---
\[firstpage\]
supernovae, superluminous supernovae, pulsars, pulsar winds, ultrahigh energy cosmic rays
Introduction
============
Core-collapse supernovae are triggered by the collapse and explosion of massive stars, and lead to the formation of black holes or neutron stars (see, e.g., [@Woosley02]). In particular, pulsars (highly magnetized, fast rotating neutron stars) are believed to be commonly produced in such events. The observed light curves of core-collapse supernovae present a wide variety of shapes, durations, and luminosities, that many studies have endeavored to model, considering the progenitor mass, explosion energy, radioactive nucleosyntheis, and radiation transfer mechanisms in the ejecta (e.g., [@Hamuy03; @Utrobin08; @Baklanov05; @Kasen09]).
While they spin down, pulsars release their rotational energy in the form of a relativistic magnetized wind. The effects of a central pulsar on the early supernova dynamics and luminosity is usually neglected, as the energy supplied by the star is negligible compared to the explosion energy, for the bulk of their population. Some pioneering works have however sketched these effects [@Gaffet77b; @Gaffet77a; @Pacini73; @Bandiera84; @Reynolds84], notably in the case of SN 1987A [@McCray87; @Xu88]. More recently, [@Kasen10; @Dessart12] discussed that magnetars, a sub-class of pulsars born with extremely high dipole magnetic fields of order $B\sim 10^{14-15}\,$G and millisecond spin periods, could deposit their rotational energy into the surrounding supernova ejecta in a few days. This mechanism would brighten considerably the supernova, and could provide and explanation to the observed superluminous supernovae [@Quimby12].
In this paper, we explore the effects of mildly magnetized pulsars born with millisecond periods (such as the Crab pulsar at birth) on the light curves of the early supernovae ejecta. Such objects are expected to inject their tremendous rotational energy in the supernovae ejecta, but over longer times compared to magnetars (of order of a few years). Indeed, the spin-down and thus the timescale for rotational energy deposition is governed by the magnetization of the star.
We estimate the thermal and non thermal radiations expected from these specific objects, concentrating at times of a few years after the onset of the explosion. We find that the bolometric light curves present a high luminosity plateau (that can reach $10^{43-44}\,$erg/s) over a few years, and that an equally bright TeV gamma-ray emission could also appear after a few months to a few years, from the acceleration of particles in the pulsar wind, depending on the injection parameters. A milder associated X-ray peak (of luminosity $10^{40-42}\,$erg/s) could also be produced around the same time. The observations of these signatures by the following up of a large number of supernovae could have important implications for the understanding of core-collapse supernovae and reveal the nature of the remnant compact object.
These objects also present the ideal combination of parameters for successful production of ultrahigh energy cosmic rays (UHECRs, see [@Blasi00; @Fang12]). The observation of such supernovae could thus be a further argument in favor of millisecond pulsars as sources of UHECRs, and a potential signature of an ongoing UHECR production.
We first give, in Section \[section:properties\], the list of quantities necessary for this analysis in the regimes of interest for the ejecta: optically thin or thick, and present a scheme of the early interaction between the pulsar wind and the supernova ejecta. In Section \[section:lightcurves\], we calculate the bolometric, thermal, and non thermal light curves of our peculiar supernovae. In Section \[section:discussion\], we briefly discuss available observations, and the implications for UHECR production.
Supernova ejecta hosting a millisecond pulsar: properties {#section:properties}
=========================================================
We note $M_{\rm ej}$ and $E_{\rm ej}$ as the mass and initial energy of the supernova ejecta. The pulsar has an inertial momentum $I$, radius $R_*$, initial rotation velocity $\Omega_{\rm i}$ (corresponding initial period $P_{\rm i}=2\pi/\Omega_{\rm i}$), and dipole magnetic field $B$. Numerical quantities are noted $Q_x\equiv Q/10^x$ in cgs units, unless specified otherwise.
Timescales {#subsection:timescales}
----------
![[Contour plot of the bolometric luminosity of supernova+pulsar wind nebula systems at 1yr after explosion (the fraction of wind energy converted into radiation, as defined in Section \[section:lightcurves\] is set to $\eta_\gamma=1$), as a function of initial period $P$ and magnetic field $B$. The various regimes for radiative emissions described in Section \[subsection:timescales\] are represented. The solid lines indicate pulsar spin-down timescale in seconds (Eq. \[eq:tp\]). The red dashed lines represent the pulsar population for which $t_{\rm p}=t_{\rm thin}$, and separate naked and clothed pulsars (see text). The dotted lines represent $t_{\rm p}=t_{\rm d}$.]{}[]{data-label="fig:SNLum"}](fig_SNLum_plateau.pdf){width="\columnwidth"}
For an ordinary core-collapse supernova, the ejecta expands into the circumstellar medium at a characteristic final velocity $$v_{\rm ej}=v_{\rm SN}=\left(2\frac{E_{\rm ej}}{M_{\rm ej}}\right)^{1/2}\sim 4.5\times 10^8\,\mbox{cm\,s}^{-1}\,E_{\rm ej,51}^{1/2}M_{\rm ej,5}^{-1/2} \ ,$$ where $M_{\rm ej,5}\equiv M_{\rm ej}/5\,M_\odot$. After a few expansion timescales $t_{\rm ex} = R_{\rm ej,i}/v_{\rm SN}$, where $R_{\rm ej,i}$ is the radius of the star that led to the explosion, the ejecta enters into a stage of homologous expansion where its size scales as $R=v_{\rm ej}t$ and its internal energy as $E_{\rm int}(t) \sim (E_{\rm ej}/2)(t_{\rm ex}/t)$.\
The ejecta is first optically thick to electron scattering. Noting $\kappa$ and $\rho$ the opacity and density of the supernova envelope, one can estimate the optical depth of the ejecta: $\tau = R\kappa\rho $. Assuming a constant central supernova density profile (see [@Matzner99] and [@Chevalier05] for more detailed modeling of the interior structure of supernovae) $\rho = 3M_{\rm ej}/(4\pi R^3)$, one can define the effective diffusion time (for thermal photons to cross the ejecta): $$\begin{aligned}
t_{\rm d} &\equiv& \left(\frac{M_{\rm ej}\kappa}{4\pi v_{\rm ej}c}\right)^{1/2} \\
&\sim& 1.6\times 10^{6}\,{\rm s}\,M_{\rm ej,5}^{1/2}\kappa_{0.2}^{1/2}\left(\frac{v_{\rm ej}}{2\times 10^{9}\,{\rm cm\,s}^{-1}}\right)^{-1/2}\ ,\end{aligned}$$ with the opacity to electron scattering defined as $\kappa_{0.2}\equiv \kappa/(0.2\, {\rm g^{-1}\,cm}^{2}$) for thermal photons. This sets the timescale of the supernova light curve, under the assumption that the opacity remains constant throughout the ejecta (no ionization effect), and in the absence of pulsar or $^{56}$Ni heating. For more detailed computation of the these timescales, see, e.g., [@Kasen09].\
As the ejecta expands, it reaches a time $t_{\rm thin}$ when it becomes optically thin to electron scattering, for thermal photons ($\tau=1$): $$t_{\rm thin} = \left(\frac{3 M_{\rm ej}\kappa}{4\pi v_{\rm ej}^2}\right)^{1/2} \sim 1.9\times 10^7 \,{\rm s}\left(\frac{v_{\rm ej}}{2\times 10^{9}\,{\rm cm\,s}^{-1}}\right)\ .$$ For the numerical estimates of $v_{\rm ej}$, we are using the final velocity of the ejecta after its modification by the shock at the interface between the pulsar wind and the initial ejecta, for $E_{\rm ej,51}$, $M_{\rm ej,5}$, and $P_{\rm i}=10^{-3}\,$s (see Eq. \[eq:vej\] in Section \[subsection:velocities\]).
The pulsar spins down by electromagnetic energy losses that is transferred to the surrounding environment. The deposition of this energy happens over the spin-down timescale of the pulsar [@Shapiro83]: $$\label{eq:tp}
t_{\rm p} = \frac{9Ic^3}{2B^2R_*^6\Omega_{\rm i}^2} \sim 3.1\times 10^{7}\,{\rm s}\,I_{45}B_{13}^{-2}R_{*,6}^{-6}P_{\rm i,-3}^2\ .$$
We will consider two regimes for the calculation of radiative emissions from the ejecta: optically thin ($t>t_{\rm thin}$), and optically thick ($t<t_{\rm thin}$) for thermal photons. The deposition of pulsar rotational energy will have different effects on the supernova radiative emissions according to the optical depth of the ejecta at time $t_{\rm p}$. Figure \[fig:SNLum\] pictures these various regimes. The red dashed lines represent the pulsar population for which $t_{\rm p}=t_{\rm thin}$: on its left-hand side, most of the rotational energy of the pulsar is injected when the supernova ejecta is optically thin to electron scattering (the pulsar is [*naked*]{}). On the right-hand side of the red dashed line, the pulsar energy can enhance the luminosity of the supernova, as it is injected while the ejecta is still optically thick (the pulsar is [*clothed*]{}).
![Scheme of the structures created by the interaction between the pulsar wind and the SN ejecta in the blast rest mass frame.[]{data-label="fig:shocks"}](fig_shocks.pdf){width="\columnwidth"}
Characteristics of the supernova ejecta and of the embedded pulsar wind nebula {#subsection:velocities}
------------------------------------------------------------------------------
The interaction between the pulsar wind and the supernova ejecta leads to the formation of the following structures, illustrated in Fig. \[fig:shocks\]: a forward shock at the interface of the shocked and unshocked ejecta, and a reverse shock at the interface between the shocked and unshocked wind (commonly called “termination shock”). The shocked material between the forward and the reverse shock constitutes the pulsar wind nebula (PWN, e.g., [@Chevalier77; @Chevalier92; @Gaensler05]).
The pulsar wind carries a total energy: $$E_{\rm p} = \frac{I\Omega_{\rm i}^2}{2} \sim 1.9\times 10^{52}\,{\rm erg}\, I_{45}P_{\rm i,-3}^2\ ,$$ and injects a luminosity (Shapiro & Teukolsky 1983) $$L_{\rm p}(t) = \frac{E_{\rm p}}{t_{\rm p}}\frac{1}{(1+t/t_{\rm p})^2}\ .$$ into the cold supernova ejecta. The evolution of the pulsar luminosity over time, for magnetic dipole spin-down, is represented in Fig. \[fig:Lp\_t\].
![Evolution of the pulsar luminosity $L_{\rm p}$ as a function of time, for magnetic dipole spin-down. The pulsar has a dipole magnetic field of $B=10^{13}\,$G, and period $P_{\rm i}=1, 10, 100\,$ms (increasing thickness). The vertical dashed lines correspond to the spin-down timescale $t_{\rm p}$ for each intial spin period.[]{data-label="fig:Lp_t"}](fig_Lp_B13.pdf){width="49.00000%"}
The characteristic velocity of the ejecta is not affected by the pulsar wind nebula expansion if $E_{\rm p}\ll E_{\rm ej}$. However, if the pulsar input energy overwhelms the initial ejecta energy $E_{\rm p}\gg E_{\rm ej}$, the ejecta is swept up into the shell at a final shell velocity $v_{\rm f}=(2E_{\rm p}/M_{\rm ej})^{1/2}$ [@Chevalier05]. Taking into account these two extreme cases, one can estimate the characteristic ejecta velocity as $$\label{eq:vej}
v_{\rm ej} = v_{\rm SN}(1+E_{\rm p}/E_{\rm SN})^{1/2}\ .$$
For $E_{\rm p}\ll E_{\rm ej}$, the evolution of the pulsar wind nebula takes place in the central part of the SN ejecta, where the density profile is nearly flat, with $\rho\propto t^{-3}(r/t)^{-m}$. We will assume here that $m=0$. For times $t\le t_{\rm p}$ where $L_{\rm p}\sim E_{\rm p}/t_{\rm p}$, the radius of the pulsar wind nebula can then be expressed [@Chevalier77] $$\begin{aligned}
\label{eq:Rpwn}
R_{\rm PWN} &\sim& \left( \frac{125}{99}\frac{v_{\rm ej}^3E_{\rm p}}{M_{\rm ej}t_{\rm p}}\right)^{1/5}\,t^{6/5}, \mbox{ for } t\le t_{\rm p}, \, E_{\rm p}\ll E_{\rm ej}\end{aligned}$$
Beyond the characteristic velocity $v_{\rm SN}$, the density profile of the ejecta steepens considerably, reaching spectral indices $b\gtrsim 5$ (e.g., [@Matzner99]). For $E_{\rm p}\gg E_{\rm ej}$, the pulsar wind nebula expands past this inflection point and its size depends on whether the swept-up shell breaks up by Rayleigh-Taylor instabilities. [@Chevalier05] discusses that if the shell does not break up, the expansion is determined by the acceleration of a shell of fixed mass, thus, $\mbox{for } t\le t_{\rm p},\, E_{\rm p}\gg E_{\rm ej}, \,\mbox{and no shell disruption} $ $$\begin{aligned}
R_{\rm PWN} &=& \left( \frac{8}{15}\frac{E_{\rm p}}{M_{\rm ej}t_{\rm p}}\right)^{1/2}\,t^{3/2} \label{eq:Rpwn_noshelldisr}\\
&\sim &2.2\times 10^{16}\,{\rm cm}\,E_{\rm p,52}^{1/2}M_{\rm ej,5}^{-1/2}t_{\rm p,yr} \quad \mbox{for } t=t_{\rm p}.\end{aligned}$$ Otherwise, the evolution of the nebula is set by pressure equilibrium, and $R_{\rm PWN}\propto t^{(6-b)/(5-b)}$ (for $t<t_{\rm p}, \, E_{\rm p}\gg E_{\rm ej}$). In the following, because the fate of the shell is unclear at this stage, we will use Eq. (\[eq:Rpwn\_noshelldisr\]) as an illustration.
For $t>t_{\rm p}$, $L_{\rm p}$ drops, and the swept-up material tends towards free expansion. One can roughly assume the relation $$\label{eq:Rpwn_tp}
R_{\rm PWN}(t>t_{\rm p}) = R_{\rm PWN}(t_{\rm p})\frac{t}{t_{\rm p}}\ ,$$ where $R_{\rm PWN}(t_{\rm p})$ is the size of the pulsar wind nebula in Eqs. (\[eq:Rpwn\],\[eq:Rpwn\_noshelldisr\]). More detailed modelings of the dynamical evolution of pulsar-driven supernova remnants can be found in [@Reynolds84].
The magnetic field strength in the pulsar wind nebula can then be estimated assuming a fraction of magnetization $\eta_{\rm B}$ of the luminosity injected by the wind (see Fig. \[fig:PWN\]) $$B_{\rm PWN} = \left(8\pi \eta_{\rm B} \int_0^t L_{\rm p}(t'){\rm d}t' \right)^{1/2}R_{\rm {PWN}}(t)^{-3/2}\ .$$ The value of $\eta_B$ could vary between $0.01-1$, according to pulsar wind nebulae.
![Evolution in time of the radius, $R_{\rm PWN}$, and the magnetic field strength, $B_{\rm PWN}$, of a pulsar wind nebula, assuming no shell disruption (Eqs. \[eq:Rpwn\_noshelldisr\] and \[eq:Rpwn\_tp\]) and $\eta_{\rm B}=0.01$, calculated for a SN ejecta with $M_{\rm ej} = 5\,M_\odot$ and $E_{\rm ej}=10^{51}\,$ergs$^{-1}$, embedding a pulsar with dipole magnetic field of $B=10^{13}\,$G and period $P_{\rm i}=1\,$ms. []{data-label="fig:PWN"}](fig_PWN.pdf){width="\columnwidth"}
Bolometric radiation {#section:lightcurves}
====================
In what follows, we calculate the total radiation expected from the supernova ejecta+pulsar wind nebula. The evolution of the ejecta is computed assuming a one zone core-collapse model. This approximation is debatable for times $t\lesssim t_{\rm d}$, as the radiation should be mainly emitted in the central regions, close to the pulsar wind nebula, and not uniformly distributed as the matter over a single shell. This is not expected to be limiting for our study however, as we are most interested in the late-time light curves (a few years after the explosion), when the ejecta starts to become optically thin.
How much energy of the pulsar wind will be transformed into radiation depends on many factors such as the nature of the wind (leptonic, hadronic or Poynting flux dominated), the efficiency of particle acceleration and of radiative processes. In a first step, these conditions can be parametrized by setting a fraction $\eta_\gamma$ of the wind energy $E_{\rm p}$ that is converted to radiative energy (thermal or non thermal) in the pulsar wind nebula.\
Under the one zone model approximation, the radiation pressure dominates throughout the remnant, $P=E_{\rm int}/3V$, with $V$ the volume of the ejecta. The internal energy then follows the law: $$\label{eq:Eintlaw}
\frac{1}{t}\frac{\partial}{\partial t}[E_{\rm int}t] = \eta_\gamma L_{\rm p}(t) -L_{\rm rad}(t)\ .$$ The radiated luminosity $L_{\rm rad}$ depends on the ejecta optical depth: $$\begin{aligned}
\frac{L_{\rm rad}(t)}{4\pi R^2} &=&\frac{E_{\rm int}c}{(4\pi/3)R^3}\quad t>t_{\rm thin}\\
&=&\frac{E_{\rm int}c}{(4\pi/3)\tau R^3} \quad t\le t_{\rm thin}\end{aligned}$$ which yields $$\begin{aligned}
\label{eq:Lrad}
L_{\rm rad}(t) &=&\frac{3}{\beta_{\rm ej}}\frac{E_{\rm int}}t{}\quad t>t_{\rm thin}\\
&=&\frac{E_{\rm int}t}{t_{\rm d}^2} \quad t\le t_{\rm thin}\end{aligned}$$ where we note $\beta_{\rm ej}\equiv v_{\rm ej}/c$. For $t<t_{\rm thin}$, we assumed that the totality of the luminosity $\eta_\gamma L_{\rm p}$ deposited in the ejecta as photons is thermalized, and used the diffusion transport approximation [@Arnett80]. In the optically thin regime, photons do not diffuse and propagate straight out of the ejecta.
{width="49.00000%"} {width="49.00000%"}
Equation (\[eq:Eintlaw\]) yields $$\begin{aligned}
E_{\rm int}(t) &=& \frac{\eta_\gamma E_{\rm p}}{1+3/\beta_{\rm ej}}\left[h_1\left(\frac{t}{t_{\rm p}}\right)-h_2\left(\frac{t}{t_{\rm p}}\right)\right] \quad t>t_{\rm thin}\\
&=& \frac{1}{t} \,e^{-\frac{t^2}{2t_{\rm d}^2}}\, \left[ \int_{t_{\rm ex}}^t \,e^{\frac{x^2}{2t_{\rm d}^2}}\, \frac{\eta_\gamma E_{\rm p}t_{\rm p}x}{(t_{\rm p}+x)^2}{\rm d}x + E_{\rm ej}\,t_{\rm ex}\,e^{\frac{t_{\rm ex}^2}{2t_{\rm d}^2}} \right] \nonumber\\
&&\qquad t\le t_{\rm thin}\ .\label{eq:Eint}\end{aligned}$$ The hypergeometric functions are noted: $$\begin{aligned}
h_1(x) &\equiv& \,_2F_1(1,1+3/\beta_{\rm ej},2+3/\beta_{\rm ej},-x)\\
h_2(x) &\equiv& \,_2F_1(2,1+3\beta_{\rm ej},2+3/\beta_{\rm ej},-x)\, .\end{aligned}$$ Note that $L_{\rm rad}(t)\sim \eta_\gamma L_{\rm p}(t)$ for $t>t_{\rm thin}$.
To calculate the total bolometric radiated luminosity, we add to $L_{\rm rad}(t)$ the contribution of the ordinary core-collapse supernova radiation $L_{\rm SN}(t)$. $L_{\rm SN}(t)$ is calculated following Eq. (5) of Chatzopoulos et al. (2012), assuming an initial luminosity output of $10^{42}\,$erg/s, as is estimated by Woosley & Heger (2002) in their Eq. (41), for $M_{\rm ej}=5\,M_\odot$ and $E_{\rm ej}=10^{51}\,$ergs$^{-1}$. $L_{\rm SN}$ only contributes when $E_{\rm p}<E_{\rm SN}$.\
Fig. \[fig:Lrad\] presents the bolometric luminosity radiated from the ejecta+PWN system for various sets of pulsar parameters. Again, the [@Arnett80] approximation is not necessarily valid for $t<t_{\rm d}$, where the radiation should not be distributed over the whole ejecta. Even with a $\eta_{\gamma}<10\%$, the plateau in the light curve a few years after the explosion is highly luminous, especially for $P=1\,$ms. This high luminosity plateau stems from the injection of the bulk of the pulsar rotational energy a few years after the supernova explosion. The luminosity is quickly suppressed for high $B$ (for magnetar-type objects), due to the fast spin-down. Supernovae embedding isolated millisecond pulsars with standard magnetic field strengths would thus present unique radiative features observable a few years after their birth.
Note however that the luminosity represented here is the bolometric one. The emission should shift from quasi-thermal to high energy after a few years, depending on the evolution of the opacity of the ejecta. The emission at different energies is discussed in the next section.
Thermal/non-thermal emissions {#section:thermal}
=============================
The bolometric radiation calculated in the previous section stems from the re-processing of high energy radiation created at the base of the SN ejecta, in the PWN region. In the standard picture of PWN, high energy particles (leptons and hadrons) are injected at the interface between the pulsar wind and the ejecta, and radiate high energy photons (X rays and gamma rays). These high energy photons can be either thermalized if the medium (the PWN and/or the SN ejecta) is optically thick to these wavelengths, or can escape from the ejecta and be observed as a high energy emission, if the medium they have to propagate through is optically thin. In this section, we calculate in more detail the emission a few years after the explosion, concentrating mainly on the case of a leptonic wind.\
Upstream of the termination shock, the energy of the pulsar wind is distributed between electrons and positrons, ions and magnetic fields. The fraction of energy imparted to particles is not certain, especially at these early times. Near the neutron star, the Poynting flux is likely to be the dominant component of the outflow energy. After many hundreds of years, observational evidence show that the energy repartition at the termination shock of pulsar wind nebulae is dominated by particles (e.g., Arons 2008). The conventional picture is thus that all but $\sim 0.3-1\%$ of the Poynting flux has already been converted into the plasma kinetic energy by the time the flow arrives the termination shock [@Kennel84a; @Kennel84b; @Emmering87; @Begelman92], $\sim 1\%$ appearing to be a level required to reproduce the observed shape of the Crab Nebula [@Komissarov04; @DelZanna04]. How this transfer happens is subject to debate (see, e.g., [@Kirk09]).
Particles and the Poynting flux are injected in the pulsar wind nebula at the termination shock. We will note the energy repartition between electrons and positrons, ions and the magnetic field in the pulsar wind nebula: $L_{\rm p}=(\eta_{\rm e}+\eta_{\rm i}+\eta_{\rm B})L_{\rm p}$. The ratio between $\eta_{\rm i}$ and $\eta_{\rm e}$ is the subject of another debate (see e.g., [@Kirk09]). However, various authors (e.g., [@Gelfand09; @Fang10; @Bucciantini11; @Tanaka11]) seem to fit satisfactorily the observed emissions for various late time pulsar wind nebulae without adding any hadronic injection. We will thus focus on the emission produced for winds dominated by a leptonic component at the termination shock.
Note that if protons are energetically dominant in the wind, [@Amato03] calculated that a large flux of neutrinos, gamma-rays and secondary pairs from p-p pion production should be expected from Crab-like pulsar wind nebulae around a few years after the supernova explosion. They estimate that the synchrotron emission from secondaries will be negligible, while TeV photon and neutrino emission could be detectable by current instruments if such young objects were present in our Galaxy.
Only 1% of the relativistic ions and magnetic fields components of the wind can be converted into thermal energy in the ejecta ([@Chevalier77]). This fraction can be amplified in presence of, e.g., Rayleigh-Taylor mixing, or high energy cosmic ray diffusion into the ejecta.
Pair injection in the PWN
-------------------------
According to the original idea by [@Kennel84a], the pair injection spectrum into the pulsar wind nebula should present a Maxwellian distribution due to the transformation of the bulk kinetic energy of the wind into thermal energy, and a non-thermal power-law tail formed by pairs accelerated at the shock. Hybrid and PIC simulations have shown indeed such a behavior (e.g., [@Bennett95; @Dieckmann09; @Spitkovsky08]). [@Spitkovsky08] finds that 1% of the particles are present in this tail, with 10% of the total injected energy. The bulk of the particle energy would then be concentrated around the kinetic energy upstream of the termination shock $$\begin{aligned}
\label{eq:epsc}
\epsilon_{\rm b} &=& kT_{\rm e} = \gamma_{\rm w} m_{\rm e} c^2\ \\
&\sim& 5\times 10^{11}\,{\rm eV}\, \frac{\gamma_{\rm w}}{10^6}\ ,\end{aligned}$$ with $\gamma_{\rm w}$ the Lorentz factor of the wind. The non-thermal tail would start around $\epsilon_{\rm b}$ and continue up to higher energies with a spectral index $\gtrsim 2$. In practice, from a theoretical point of view, Lorentz factors as high as $\gamma_{\rm w}\sim 10^6$ are difficult to reach, and current simulations are only capable of producing $\gamma_{\rm w}$ of order a few hundreds [@Spitkovsky08; @Sironi09].
However, observationally, various authors ([@Kennel84a], but also more recently, e.g., [@Gelfand09; @Fang10; @Bucciantini11; @Tanaka11]) demonstrated that the non-thermal radiation produced by the injection of either one single power-law or a broken power-law peaking around $\epsilon_{\rm b}\sim 1\,$TeV, and extending up to PeV energies, could fit successfully the observed emission of various young pulsar wind nebulae. Such a high break energy implies either a high Lorentz factor for the wind $\gamma_{\rm w}\sim 10^{5-6}$, or an efficient acceleration mechanism enabling particles to reach $0.1-1$TeV energies. At higher energies, another acceleration mechanism has to be invoked to produce particles up to PeV energies. [@Bucciantini11] discuss that $\epsilon_{\rm b}$ could possibly be viewed as a transition energy between Type II and Type I Fermi acceleration from low at high energies. This would provide a physical explanation to the broken power-law shape, and alleviate the issue of the high wind Lorentz factor. At high energies, acceleration could also happen in the course of reconnection of the striped magnetic field in the wind, at the termination shock [@Lyubarsky03; @Petri07]. However, it is not clear yet whether this process can lead to a non-thermal particle distribution. One can expect additional particle acceleration in the wind itself, via surf-riding acceleration [@Chen02; @Contopoulos02; @Arons02; @Arons03]. This non-thermal component would not necessarily be processed when injected at the shocks if the particle Larmor radii are large compared to the size of the shock.
In the following, we will assume that pairs are injected in the pulsar wind nebula following a broken power-law of the form $$\label{eq:Neinj}
\frac{{\rm d}\dot{N}}{{\rm d}\epsilon}(\epsilon,t) = \frac{\eta_{\rm e}L_{\rm p}(t)}{\epsilon_{\rm b}^2} \,\left\{
\begin{array}{ll}
(\epsilon/\epsilon_{\rm b})^{-\alpha}&\quad \mbox{if} \quad \epsilon_{\rm min}\le \epsilon < \epsilon_{\rm b} \\
(\epsilon/\epsilon_{\rm b})^{-\beta}&\quad \mbox{if} \quad \epsilon_{\rm b}\le \epsilon \le \epsilon_{\rm max}
\end{array}\right.$$ where $\alpha<2<\beta$, $\epsilon_{\rm min}$ and $\epsilon_{\rm max}$ are the minimum and maximum cut-off energies respectively, and $\epsilon_{\rm b}$ the peak of the injection distribution $\epsilon({\rm d}N/{\rm d}\epsilon)\sim 0.1-1$TeV. It is commonly assumed that $\epsilon_{\rm b} \propto\gamma_{\rm w}\propto \sqrt{L_{\rm p}(t)}$, but such an assumption would imply very high wind Lorentz factors ($>10^9$) at early times, that seem incompatible with the simulations and theoretical models discussed above. It is likely that the Lorentz factor experiences a saturation above a certain value, and for simplicity, we will assume that $\epsilon_{\rm b}$ is constant over time. For our purpose of deriving a rough estimate of the fraction of high energy emission that can escape the ejecta at early times, such an approximation will suffice. A thorough calculation of the emission spectrum would require time-dependent energy loss calculations for particles beyond the one zone approximation that we use here. [@DelZanna04; @DelZanna06] have shown indeed that the high energy emission is strongly affected by the details of the flow dynamics just downstream of the termination shock.
![Timescales at play in the radiation emission of a PWN, for the same system as in Fig. \[fig:PWN\], assuming an electron injection break energy $\epsilon_{\rm b}=0.1\,$TeV, and $\eta_B=0.01$. The dynamical timescale, $t_{\rm dyn}$ (black solid), and cooling timescales via synchrotron, $t_{\rm syn}$ (red solid), via self-compton, $t_{\rm IC, syn}$ (blue solid), and via IC off thermal photons (blue long dashed), are compared to thermalization timescales. The thermalization timescales via photoelectric absorption, Compton scattering, and pair production are in dashed lines (in red for the synchrotron photons and blue for the IC photons). Absorption by $\gamma-\gamma$ interaction at high energies are in green lines (dot-dashed for the interaction between IC photons on synchrotron photons, and long dashed for the interaction of IC photons on the thermal photons of the SN ejecta). The gray shaded region corresponds to $t<t_{\rm d}$, where the @Arnett80 approximation is not valid. And the dotted line indicates $t=t_{\rm thin}$. []{data-label="fig:times"}](fig_times.pdf){width="\columnwidth"}
{width="49.00000%"} {width="49.00000%"}
Radiation by accelerated pairs
------------------------------
The bulk of the electron distribution will predominantly radiate in synchrotron and experience inverse Compton (IC) scattering off the produced synchrotron photons. The cooling timescales of these processes, as well as the dynamical timescale $t_{\rm dyn}=R_{\rm PWN}/c$ of the PWN are indicated in Fig. \[fig:times\]. Inverse Compton scattering off the thermal photons of the ejecta and off the Cosmic Microwave Background (CMB) are negligible compared to the former two processes. Figure \[fig:times\] also demonstrates that the cooling timescale of IC scattering off the thermal photons of the SN ejecta is much longer than the timescale for self-comptonization of the synchrotron emission. This estimate includes only the contribution of the thermal photons of the standard supernova ejecta, as the thermalization of the non-thermal components described here happen on larger timescales, in the optically thin regime which is of interest to us.
The synchrotron cooling timescale of accelerated electron reads $$t_{\rm syn} = \frac{3 m_e^2c^3}{4\sigma_{\rm T}\epsilon_{\rm c}U_B}\ ,$$ with $U_B=B_{\rm PWN}^2/8\pi$. At the early stages that we consider here, the characteristic energy of radiating particles is $\epsilon_{\rm c}(t)=\epsilon_{\rm b}$. The characteristic synchrotron radiation frequency can be expressed $$\nu_{\rm c}(t) = 0.29\frac{3e B_{\rm PWN}(t)}{4\pi m_e^3 c^5}\epsilon_{\rm c}^2\ .\label{eq:nu_c}$$ Accelerated electrons also scatter off these synchrotron photons by IC, producing photons at energy $$\begin{aligned}
\nu_{\rm IC} &=& \nu_{\rm KN}\equiv\frac{m_e c^2}{\gamma_e h} \quad \mbox{if } \nu_{\rm c}>\nu_{\rm KN}\ ,\\
&=& \nu_{\rm c}\quad \mbox{if } \nu_{\rm c}\le \nu_{\rm KN}\ ,\end{aligned}$$ with a cooling timescale $$t_{\rm IC, syn} = \frac{3m_e^2c^4}{4\sigma_{\rm T}c\epsilon_{\rm c}U_{\rm syn}}\ ,$$ where $U_{\rm syn}$ is the synchrotron photon energy density. Electrons radiate in synchrotron and self-compton processes with the following power ratio: $P_{\rm IC}/P_{\rm syn}=U_{\rm syn}/U_B$. Assuming that the energy of the accelerated electron population is concentrated in its peak energy $\epsilon_{\rm b}$, this implies synchrotron and IC luminosities of $L_{\rm syn}=\eta_B\eta_{e}/(\eta_B+\eta_{e})L_{\rm p}$ and $L_{\rm IC}=\eta_{e}^2/(\eta_B+\eta_{e})L_{\rm p}$ respectively. Obviously, the value of $\eta_B$ has an impact on the synchrotron emission, but not on the IC emission.
Figure \[fig:rad\] presents the evolution in time of the luminosities $L_{\rm p}$, $L_{\rm syn}$, and $L_{\rm IC}$, as well as the emission frequencies $\nu_{\rm c}$ and $\nu_{\rm IC}$. The IC radiation is mostly emitted at the break energy of the injection of electrons. The synchrotron emission spans from gamma/X-ray (until a few years) to optical wavelengths (after thousands of years). At the time of interest in this study, X-rays are thus mainly emitted between $0.1-100$keV for $\epsilon_{\rm b}=0.1\,$TeV, and around $100\,{\rm keV}-1\,{\rm GeV}$ for $\epsilon_{\rm b}=1\,$TeV.
Thermalization in the ejecta
----------------------------
The X-ray opacity ($\sim 0.1-100\,$keV) is dominated by photoelectric absorption in metals. Above $\sim 100\,$keV, very hard X-rays and gamma-rays experience predominantly Compton scattering, and pair production above $\sim 10\,$MeV. The opacities of these processes for various atomic media are given in Fig. \[fig:kappa\]. At a given time $t$, the optical depth of the ejecta to the characteristic synchrotron photon emission $\nu_{\rm c}$ reads $$\tau_{\rm syn}(t) = v_{\rm ej} t\kappa_{\nu_{\rm c}}(t)\rho(t)\ .$$ The dominant thermalization process for the TeV IC radiation is pair production by $\gamma-\gamma$ interactions (see Fig. \[fig:times\]). The timescale for thermalization via this process is however slightly longer than the dynamical time; hence the ejecta appears mostly optically thin to this radiation (see Fig. \[fig:rad\]). We note $\tau_{\rm IC}$ for the optical depth of the ejecta to the IC emission.
The luminosity in the characteristic energies $h\nu_{\rm c}$ and $h\nu_{\rm IC}$ after thermalization, noted respectively $L_{\rm X}(t)$ and $L_{\gamma}(t)$, and the luminosity in thermal photons, $L_{\rm th}(t)$, are calculated as follows $$\begin{aligned}
L_{\rm X}(t) &=& L_{\rm syn}(\nu_{\rm c},t)\,e^{-\tau_{\rm syn}(t)}\label{eq:Lxgamma}\\
L_{\gamma}(t) &=& L_{\rm IC}(\nu_{\rm IC},t)\,e^{-\tau_{\rm IC}(t)}\label{eq:Lxgamma}\\
L_{\rm th}(t) &=& L_{\rm rad}(t)-L_{\rm X}(t)-L_{\gamma}(t)\ .\end{aligned}$$
Figure \[fig:lumX\] presents the thermal emission (black), X-ray emission (blue dotted) at $h\nu_{\rm c}\sim 0.1-100\,$keV for $\epsilon_{\rm b}=0.1\,$TeV (left panel) and $\sim 100\,{\rm keV}-1\,{\rm GeV}$ for $\epsilon_{\rm b}=1\,$TeV (right panel), and $0.1-1\,$TeV gamma ray emission (red dashed) expected from a SN ejecta with $M_{\rm ej} = 5\,M_\odot$ and $E_{\rm ej}=10^{51}\,$ergs$^{-1}$, embedding a pulsar with dipole magnetic field of $B=10^{13}\,$G and period $P_{\rm i}=1, 3, 10\,$ms (increasing thickness), assuming $\eta_B=0.01$, $\eta_e=1-\eta_B$, and a break energy $\epsilon_{\rm b}=0.1\,$TeV (left) and $\epsilon_{\rm b}=1\,$TeV (right).
A decrease in flux is expected in the thermal component after a few months to years, when the ejecta becomes optically thin to gamma-rays. For a low break energy ($\epsilon_{\rm b}=0.1\,$TeV) the thermal component can then recover, as the X-ray emission vanishes, because of the increase of the ejecta optical depth for lower energy photons. One robust result is that, in both break energy cases, for fast pulsar rotation periods $P_{\rm i}\le 3\,$ms, the associated gamma-ray flux around $0.1-1\,$TeV emerges at a level that should be detectable at a few tens of Mpc, and remains strong over many years.
![Atomic scattering opacities of high energy photons on H, He, C, and a mixture of composition mimicking that of a type II core-collapse supernova ejecta (60% H, 30% He and 10% C). The black dashed line indicates the contribution of Compton scattering in the latter composition case. From http://henke.lbl.gov/optical\_constants/ for $30\,{\rm eV}\le h\nu <1\,{\rm keV}$ and http://physics.nist.gov/cgi-bin/Xcom/ for $1\,{\rm keV}\le h\nu <100\,{\rm GeV}$. []{data-label="fig:kappa"}](fig_kappa.pdf){width="\columnwidth"}
{width="49.00000%"} {width="49.00000%"}
Discussion, conclusion {#section:discussion}
======================
We have estimated the thermal and non thermal radiations expected from supernova ejectas embedding pulsars born with millisecond periods, concentrating at times a few years after the onset of the explosion. The bolometric light curves should present a high luminosity plateau (that can reach $>10^{43}\,$erg/s) over a few years. A more detailed emission calculation considering the acceleration of leptons in the pulsar wind nebula region shows that an X-ray and a particularly bright TeV gamma-ray emission (of magnitude comparable to the thermal peak) should appear around one year after the explosion. This non thermal emission would indicate the emergence of the pulsar wind nebula from the supernova ejecta.
The light curves calculated in this paper are simple estimates, that do not take into account second order effects of radioactive decay of $^{56}$Ni, recombination, etc. (see, e.g., [@Kasen09]). The non-thermal components are also evaluated assuming that all the leptonic energy is concentrated in one energy bin. A more detailed analysis should be conducted, taking into account the shape of the spectra and its evolution in time, in order to get a more accurate representation of the emission, and for a thorough comparison with observational data. Depending on the spectral indices, a non monoenergetic electron injection spectrum could lead to a decrease of the peak luminosity of one order of magnitude.
Our computation of the evolution of the PWN (radius, magnetic field) is also basic, and could benefit from more thorough estimations. Our toy model suffices however in the scope of this study, where the aim is to demonstrate the importance of multi-wavelength follow ups of SN lightcurves. We also assumed a relatively high magnetization $\eta_B$ of the wind at the termination shock, following estimates that reproduce the features of the Crab nebula [@Komissarov04; @DelZanna04].
Several earlier works treat some of the aspects invoked above in more detail. For example, in the context of evolution of pulsar wind nebulae, early works by [@Pacini73; @Rees74; @Bandiera84; @Weiler80; @Reynolds84] take into account the detailed evolution of particle energy distribution and radiation spectrum. Most of these works aim at calculating radiation features of observed plerions, a few hundred of years after the explosion. However, their modeling at earlier times, especially in the work of [@Reynolds84], lays the ground for the more detailed calculations that should be performed in our framework.
The level of synchrotron emission predicted here can thus be viewed as optimistic values. However, the gamma-ray flux that is predicted does not depend on the magnetization, and remains fairly robust to most parameter changes.
Currently, only a handful of supernovae have been followed over a period longer than a year, and no object, except for SN 1987A, has been examined in X-rays or TeV gamma-rays a year after the explosion. Among the objects that have been followed in optical bands, SN 2003ma [@Rest11] has an abnormally bright luminosity at the peak, and a long bright tail over many years. The six type II supernovae followed by [@Otsuka12] present various shapes of light curves, and a cut-off in the thermal emission after a few years. Our study demonstrates that the features in these light curves could also be due to the energy injection from the pulsar, that could compete with the other processes that are more commonly considered, such as the light echo of the peak luminosity, or the radioactive decay of $^{56}$Ni. An associated X-ray and TeV gamma-ray emission emerging around a few months to a year after the explosion would constitute a clear signature of pulsar rotational energy injection. It is also interesting to note that the emergence of a pulsar wind nebula has been recently reported from radio observations for SN 1986J [@Bietenholz10], though over longer timescales than predicted for the objects studied in this paper.
Some authors [@Katz11; @Svirski12] have discussed that shock breakouts from stars surrounded by a thick wind could lead to bright X-ray peak after a few months, similar to the signal discussed in this paper. This degeneracy can be overcome by the observation of the gamma-ray signal, which should be absent in the shock breakout scenario. Detailed analysis of the respective X-ray spectra should also help distinguish the two scenarios.
The follow up of bright type II supernovae over a few years after the explosion in different wave bands would thus reveal crucial information on the nature of the compact remnant. These suveys should be made possible with the advent of optical instruments such as LSST, and the use of powerful instruments for transient event detection, such as the Palomar Transient Factory or Pan-STARR. The bright X-ray signal should be detected by NuSTAR for supternovae out to redshifts $z\sim 0.5$, and the even brighter gamma-ray signal could be observed by HESS2, by the future Cerenkov Telescope Array (CTA), and by HAWK which will be the choice instrument to explore the transient sky at these energies. For CTA, an adequate survey of the sky outside the Galactic plane could spot gamma-ray sources of luminosity $10^{43}\,$erg/s as predicted by this work within a radius of $\sim 150\,$Mpc (G. Dubus, private comm.). Assuming a gaussian pulsar period distribution centered around $300\,$ms as in [@Faucher06], implies that 0.3% of the total population has spin periods $< 6\,$ms. With this estimate, one could find 4 bright sources within 150Mpc. This is consistent with the numbers quoted in early works by [@Srinivasan84; @Bhattacharya90]. These authors estimated the birthrate of Crab-like pulsar-driven supernova remnants to be of order 1 per 240 years in our Galaxy.
Pulsars born with millisecond periods embedded in standard core-collapse supernova ejectas, as described in this paper, are promising candidate sources for ultrahigh energy cosmic rays [@Fang12; @Fang13]. In the framework of UHECRs, an injection of order 1% of the Goldreich-Julian density into ions would suffice to account for the observed flux, assuming that 1% of Type II supernovae give birth to pulsars with the right characteristics to produce UHECRs (i.e., pulsars born with millisecond periods and magnetic fields $B\sim 10^{12-13}\,$G, [@Fang12]). The observation of the peculiar light curves predicted here could thus provide a signature for the production of UHECRs in these objects. Though no spatial correlation between arrival directions of UHECRs and these supernovae is expected (because of time delays induced by deflections in magnetic fields), an indication of the birth rate of these supernovae could already give direct constraints on this source model. Photo-disintegration and spallation of accelerated nuclei in the supernova ejecta could also lead to an abundant high energy neutrino production ([@Murase09] consider such a neutrino production in the case of magnetars instead of fast-rotating pulsars), that could be detected with IceCube, and correlated with the position of identified peculiar supernovae.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank L. Dessart, G. Dubus, M. Lemoine, K. Murase, E. Nakar, and J. Vink for very fruitful discussions. KK was supported at Caltech by a Sherman Fairchild Fellowship, and acknowledges financial support from PNHE. AO was supported by the NSF grant PHY-1068696 at the University of Chicago, and the Kavli Institute for Cosmological Physics through grant NSF PHY-1125897 and an endowment from the Kavli Foundation.
\[lastpage\]
[^1]: E-mail: [email protected]
|
---
abstract: 'The NEMO (NEutrino Mediterranean Observatory) Collaboration has been carrying out since 1998 an evaluation programme of deep sea sites suitable for the construction of the future Mediterranean km$^3$ Čerenkov neutrino telescope. We investigated the seawater optical and oceanographic properties of several deep sea marine areas close to the Italian Coast. Inherent optical properties (light absorption and attenuation coefficients) have been measured as a function of depth using an experimental apparatus equipped with standard oceanographic probes and the commercial transmissometer AC9 manufactured by WETLabs. This paper reports on the visible light absorption and attenuation coefficients measured in deep seawater of a marine region located in the Southern Ionian Sea, 60$\div$100 km SE of Capo Passero (Sicily). Data show that blue light absorption coefficient is about 0.015 m$^{-1}$ (corresponding to an absorption length of 67 m) close to the one of optically pure water and it doe not show seasonal variation.'
address:
- 'Laboratori Nazionali del Sud INFN, Via S.Sofia 62, 95123, Catania, Italy'
- 'Laboratori Nazionali di Frascati INFN, Via Enrico Fermi 40, 00044, Frascati (RM), Italy'
- 'INFN Sezione Catania, Via S.Sofia 64, 95123, Catania, Italy'
- 'INFN Sezione Bari and Dipartimento Interateneo di Fisica Università di Bari, Via E. Orabona 4, 70126, Bari, Italy'
- 'INFN Sezione Bologna and Dipartimento di Fisica Università di Bologna, V.le Berti Pichat 6-2, 40127, Bologna, Italy'
- 'INFN Sezione Genova and Dipartimento di Fisica Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy'
- 'INFN Sezione Napoli and Dipartimento di Scienze Fisiche Università di Napoli, Via Cintia, 80126, Napoli, Italy'
- 'INFN Sezione Pisa and Dipartimento di Fisica Università di Pisa, Polo Fibonacci, Largo Bruno Pontecorvo 3, 56127, Pisa, Italy'
- 'INFN Sezione Roma 1 and Dipartimento di Fisica Università di Roma “La Sapienza”, P.le A. Moro 2, 00185, Roma, Italy'
- 'Dipartimento di Fisica and Astronomia Università di Catania, Via S.Sofia 64, 95123, Catania, Italy'
- 'INAF Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131, Napoli, Italy'
- 'University of Wisconsin, Department of Physics, 53711, Madison, WI, USA'
author:
- 'G. Riccobene'
- 'A. Capone'
- 'S. Aiello'
- 'M. Ambriola'
- 'F. Ameli'
- 'I. Amore'
- 'M. Anghinolfi'
- 'A. Anzalone'
- 'C. Avanzini'
- 'G. Barbarino'
- 'E. Barbarito'
- 'M. Battaglieri'
- 'R. Bellotti'
- 'N. Beverini'
- 'M. Bonori'
- 'B. Bouhadef'
- 'M. Brescia'
- 'G. Cacopardo'
- 'F. Cafagna'
- 'L. Caponetto'
- 'E. Castorina'
- 'A. Ceres'
- 'T. Chiarusi'
- 'M. Circella'
- 'R. Cocimano'
- 'R. Coniglione'
- 'M. Cordelli'
- 'M. Costa'
- 'S. Cuneo'
- 'A. D’Amico'
- 'G. De Bonis'
- 'C. De Marzo'
- 'G. De Rosa'
- 'R. De Vita'
- 'C. Distefano'
- 'E. Falchini'
- 'C. Fiorello'
- 'V. Flaminio'
- 'K. Fratini'
- 'A. Gabrielli'
- 'S. Galeotti'
- 'E. Gandolfi'
- 'A. Grimaldi'
- 'R. Habel'
- 'E. Leonora'
- 'A. Lonardo'
- 'G. Longo'
- 'D. Lo Presti'
- 'F. Lucarelli'
- 'E. Maccioni'
- 'A. Margiotta'
- 'A. Martini'
- 'R. Masullo'
- 'R. Megna'
- 'E. Migneco'
- 'M. Mongelli'
- 'M. Morganti'
- 'T. Montaruli'
- 'M. Musumeci'
- 'C.A. Nicolau'
- 'A. Orlando'
- 'M. Osipenko'
- 'G. Osteria'
- 'R. Papaleo'
- 'V. Pappalardo'
- 'C. Petta'
- 'P. Piattelli'
- 'F. Raffaelli'
- 'G. Raia'
- 'N. Randazzo'
- 'S. Reito'
- 'G. Ricco'
- 'M. Ripani'
- 'A. Rovelli'
- 'M. Ruppi'
- 'G.V. Russo'
- 'S. Russo'
- 'S. Russo'
- 'P. Sapienza'
- 'M. Sedita'
- 'J-P. Schuller'
- 'E. Shirokov'
- 'F. Simeone'
- 'V. Sipala'
- 'M. Spurio'
- 'M. Taiuti'
- 'G. Terreni'
- 'L. Trasatti'
- 'S. Urso'
- 'V. Valente'
- 'P. Vicini'
title: Deep seawater inherent optical properties in the Southern Ionian Sea
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underwater Čerenkov neutrino telescope ,deep seawater optical properties ,light attenuation ,light absorption
95.55.Vj ,29.40.Ka ,92.10.Pt ,07.88.+y
Introduction
============
The construction of km$^3$-scale high energy neutrino telescopes will complement and extend the field of high energy astrophysics allowing the identification of the highest energy cosmic ray sources. The search for astronomical sources of high energy cosmic rays is possible with particles that reach un-deflected the detectors. The observational horizon of high energy cosmic gamma rays and nuclei from ground based detectors is limited to few tens of Mpc by the interaction with cosmic matter and radiation: this should imply the well known GZK cutoff [@GaisserHalzenStanev1995; @LearnedMannheim2000] in the energy distribution of ultra high energy extragalactic cosmic rays. On the contrary, the low cross section of weak interaction allows neutrinos to reach the Earth un-deflected from the farthermost regions of the Universe. Active Galactic Nuclei [@Mannheim1995], Galactic Supernova Remnants [@Protheroe1997], Microquasars [@Distefano2002] and Gamma Ray Bursters [@WaxmanBahcall1997] are some of the most promising candidate of high energy muon-neutrino sources. On the basis of high energy neutrino fluxes, calculated using astrophysical models, neutrino detectors with an effective area of $\simeq$ 1 km$^2$ will be able to collect, in one year, a statistically significant number of events from point-like astrophysical neutrino sources.
Underwater Čerenkov telescopes detect high energy neutrinos indirectly, tracking the Čerenkov light wavefront radiated, in seawater or in ice, by charged leptons originated in Charged Current neutrino interactions. Seawater, therefore, acts as a neutrino target and as a Čerenkov radiator. An undersea location at a depth of more than 3000 m provides an effective shielding for atmospheric muons background and allows the construction of such detectors, usually referred as [*Neutrino Telescopes*]{} [@markov1961]. Two smaller scale neutrino detectors, AMANDA and BAIKAL, have already collected and reported candidate neutrino events [@AMANDAPRL2003; @BAIKALICRC2003]. AMANDA is located in the South Pole icecap [@AMWWW] at a depth between 1400 and 2400 m. The present size is relatively small, about 25000 m$^2$ for TeV muons, compared to IceCube [@IQWWW], the future $km^3$ detector now under construction. BAIKAL NT-200, the pioneer underwater detector, is deployed in the Siberian Lake Baikal at about 1000 m depth and has a detection area close to 10$^4$ m$^2$ for TeV muons [@BAWWW].
In the Northern Hemisphere, the Mediterranean Sea offers several areas with depths greater than 3000 m; few are close to scientific and logistic infrastructures and offer optimal conditions to install an underwater km$^3$ neutrino telescope. The future IceCube and the Mediterranean km$^3$ will complement each other providing a global 4$\pi$ observation of the sky. The long light absorption length of the Antarctic ice is expected to allow good energy resolution, the long light effective scattering length of the Mediterranean seawater should also allow excellent angular resolution. Three collaborations, NESTOR [@NEWWW], ANTARES [@ANWWW] and NEMO [@NMWWW], are presently active in the Mediterranean Sea. NESTOR proposes the installation of a Čerenkov detector, with a tower-shaped geometry, moored a few nautical miles off the south-west tip of the Peloponnese (Greece), at about 4000 m depth. ANTARES is building a detector in the vicinity of Toulon (France) at $\simeq$2450 m depth to possibly detect astrophysical neutrinos and to demonstrate the feasibility of a km$^3$-scale underwater neutrino telescope.
The NEMO Collaboration is active in the design and tests for the Mediterranean km$^3$ neutrino telescope. After a long period of $R\&D$ activity, at present the collaboration is ready to install a prototype station (NEMO [*phase 1*]{}) at 2000 m depth, 25 km offshore the town of Catania, in Sicily ([*Test Site*]{} in Figure \[fig:CapoPasseroMap\]). Since 1998 we have performed more than 25 oceanographic campaigns in the Central Mediterranean Sea in order to characterize and eventually seek an optimal submarine site for the installation of the Mediterranean km$^3$ [@NEMOHamburg2001]. Three areas close to the Italian Coast have been compared, on the basis of two requirements: depth $>$ 3000 m and distance from shore $<$ 100 km. Two of these sites are trenches located in the Southern Tyrrhenian Sea close to the Alicudi and Ustica Islands (at depth $\simeq$ 3500 m). Measurements of deep seawater optical properties were performed by the NEMO Collaboration in these sites and results were published [@Capone2001]. The third site is a submarine plateau, whose average depth is $\simeq$3500 m, located at a distance of 40$\div$100 km South East of Capo Passero, Sicily (see Figure \[fig:CapoPasseroMap\]). In this paper we report on deep seawater optical properties (absorption and attenuation coefficients) measured in the [[*Capo Passero*]{}]{} marine region during a period extending from December 1999 to July 2003. The results refer to two sites located $\simeq$60 km ($36^{\circ}30$’[**N**]{},$15^{\circ}50$’[**E**]{}) and $\simeq$80 km ($36^{\circ}25$’[**N**]{},$16^{\circ}00$’[**E**]{}) offshore [[*Capo Passero*]{}]{}, hereafter indicated respectively as $KM3$ and $KM4$. The programme of characterisation of deep seawater in [[*Capo Passero*]{}]{} site, carried out by the NEMO Collaboration, includes also long term measurements of optical background (due to bioluminescence and $^{40}K$ radioactive decays), water temperature and salinity, deep sea currents, sedimentation rate and bio-fouling. The results of this work are presented elsewhere [@NEMOAppec2003] and will be published soon.
![ Bathymetric chart of the [[*Capo Passero*]{}]{} region. The location of the $KM3$ (square) and $KM4$ (circle) sites and of the NEMO Phase 1 $Test~Site$ (triangle) is shown. The seabed depth is about 3400 m for the [[*Capo Passero*]{}]{} sites and 2000 m for the $Test~Site$.[]{data-label="fig:CapoPasseroMap"}](KM4_col.eps){width="10cm"}
Measurements of water optical properties with the AC9
=====================================================
The propagation of light in water is quantified, for a given wavelength $\lambda$, by the water Inherent Optical Properties (IOP): the absorption $a(\lambda)$, scattering $b(\lambda)$ and attenuation $c(\lambda)= a(\lambda) + b(\lambda) $ coefficients. The light propagation in water can be described by the laws:
$$\begin{aligned}
\label{eq:lambertlaw}
\nonumber I_{a}(x,\lambda) = I_0 (\lambda) e^ {-x \cdot a(\lambda)} \\
\nonumber I_{b}(x,\lambda) = I_0 (\lambda) e^ {-x \cdot b(\lambda)}\\
\nonumber I_{c}(x,\lambda) = I_0 (\lambda) e^ {-x \cdot c(\lambda)}\end{aligned}$$
where $x$ is the optical path traversed by the light and $I_0$($\lambda$) is the source intensity. A complete description of light scattering in water would require the knowledge of another IOP, i.e. the scattering angular distribution, or volume scattering function, $\tilde{\beta}$($\vartheta,\lambda$). Integrating this function over the diffusion angle $\vartheta$ one gets $b(\lambda)$. In this paper we shall report on measurements of $c(\lambda)$ and $a(\lambda)$ for visible light wavelengths performed with a commercial transmissometer, the AC9 manufactured by WETLabs [@WETLabs]. It is worth to mention that the AC9 performs measurements of the attenuation coefficient in a collimated geometry: the angular acceptance of the $c(\lambda)$ channel is $\simeq 0.7^{\circ}$. The reported values of $c(\lambda)$ are not directly comparable with the results often reported by other authors that concern the effective light attenuation length (or light transmission length). This quantity is defined as $c_{eff}(\lambda)= a(\lambda) + ( 1-\langle \cos
(\vartheta)\rangle) \cdot b(\lambda)$, where $\langle \cos
(\vartheta) \rangle$ is the average cosine of the volume scattering function [@Mobley1994].
Water IOPs are wavelength dependent: the light transmission is extremely favoured in the range 350$\div$550 nm [@Mobley1994] where the photomultipliers used in neutrino telescopes to detect Čerenkov radiation reach the highest quantum efficiency. In natural seawater, IOPs are also function of water temperature, salinity and dissolved particulate [@Pope1997; @Kou1993]. The nature of particulate, either organic or inorganic, its dimension and concentration affect light propagation. All these environmental parameters may vary significantly, for each marine site, as a function of depth and time. It is important, therefore, to perform a long term programme of [*in situ*]{} measurements spanning over a long time interval [@Duntley1963]. It is known, indeed, that seasonal effects like the increase of surface biological activity (typically during spring) or the precipitation of sediments transported by flooding rivers, enlarge the amount of dissolved and suspended particulate, worsening the water transparency.
We carried out light attenuation and absorption measurements in deep seawater using an experimental set-up based on the AC9. This device performs attenuation and absorption measurements, independently, using two different light paths and spanning the light spectrum over nine different wavelengths (412, 440, 488, 510, 532, 555, 650, 676, 715 nm) [@Twardowski1999; @Zaneveld1994; @Pegau1997]. The setup designed for deep seawater measurements consists of an AC9, powered by a submersible battery pack, connected to an Idronaut Ocean MK317 CTD (Conductivity, Temperature, Depth) probe. The whole apparatus is mounted inside an AISI-316 stainless-steel cage and it is operated from sea surface down to deep sea, using an electro-mechanical cable mounted on a winch onboard oceanographic research vessels. The same cable is used to transmit the data stream to the ship deck. The DAQ is designed to acquire, about six times per second, water temperature, salinity, $a$($\lambda$) and $c$($\lambda$) ($412<\lambda< 715$ nm). The apparatus is typically deployed at $\sim 0.7$ m/s vertical speed, allowing the acquisition of roughly 10 data samples per metre of depth [@Capone2001; @Balkanov2002]. As an example we show in Figure \[fig:profileKM4\] the profiles, as a function of depth, of salinity (in practical salinity units \[psu\]), temperature (\[$^{\circ}$C\]), $a$($\lambda$=440 nm) and $c$($\lambda$=440 nm) (\[m$^{-1}$\]) measured in two deployments at the $KM4$ site during December 1999. Each plotted point represents the average value over 10 m depth. The two measurements (red dots and black dots), carried out in two consecutive days, are nearly superimposed. The Figure indicates that deep waters in $KM4$ do not show relevant variations of oceanographic and optical properties in the depth interval 2000$\div$3250 m.
![Temperature \[$^{\circ}$C\], salinity \[psu\], attenuation and absorption coefficients \[m$^{-1}$\] for $\lambda=440$ nm as a function of depth, measured during two deployments (red and black dots) of the AC9 in $KM4$ site in December 1999. Results of the measurements are nearly superimposed.[]{data-label="fig:profileKM4"}](SalTempacvsdepth_KM4.eps){width="9cm"}
AC9 Calibration and systematic errors
-------------------------------------
As described in previous papers [@Capone2001; @Balkanov2002], the AC9 measures the difference between the absorption and attenuation coefficients of seawater with respect to the coefficients for pure water. The AC9 manufacturer provides a set of instrument calibration coefficients, that refer to the instrument response to pure water and dry air, used to obtain the absolute values of $a$($\lambda$) and $c$($\lambda$). In order to reduce systematic uncertainties associated to the measurements, during each naval campaign, the AC9 calibration coefficients have been verified several times (before and after each deployment), recording the instrument readings for light transmission in high purity grade nitrogen atmosphere. With this calibration procedure we estimated that systematic errors amount to $\simeq 1.5 \times
10^{-3}$ \[m$^{-1}$\] for the $a(\lambda)$ and $c(\lambda)$ measurements. We performed in each site at least two deployments of the AC9 setup at short time interval (typically less than 1 day).
Comparison of deep sea sites in the Central Mediterranean Sea
=============================================================
The first measurements of IOP in [[*Capo Passero*]{}]{} were carried out in December 1999, in the $KM3$ and $KM4$ sites. A comparison among the vertical profiles of salinity, temperature, $a$(440 nm) and $c$(440 nm) as a function of depth, recorded in the two sites is shown in Figure \[fig:ProfileKM3\_KM4\]. Between 1250 m and 3250 m depth, the water column in the site $KM3$ shows variations of the attenuation coefficients as a function of depth. We attribute this variation of $c$($\lambda$) to extra sources of light scattering, due to particulate present in this site, which is close to the Maltese shelf break. We never observed this effect in $KM4$, a site farther from the Maltese Escarpment. Figure \[fig:ProfileKM3\_KM4\], indeed, shows that optical properties measured in $KM4$ are almost constant as a function of depth (for depth $>1500$ m). Table \[tab:Dec99\_ac\] summarises the values of $a(\lambda)$ and $c(\lambda)$, measured at the $KM3$ and $KM4$ sites, averaged over an interval of about 400 m depth, 150 m above the seabed ($\simeq$3400 m in $KM4$), which is a suitable range for the installation of neutrino telescopes. As explained above two deployments were carried out in each site. Results are reported in the table. During deployments about 10 data acquisitions per metre of depth are recorded, this implies that large statistics is collected with the instrument in a 400 m depth interval allowing to achieve small statistical errors for each absorption or attenuation coefficient. Table \[tab:Dec99\_ac\] does not report the measured values of the $a$(676 nm) coefficient since its value is used in the off-line analysis as a normalization parameter to estimate corrections due to the not perfect reflectivity mirror in the AC9 absorption channel (see reference [@Capone2001]). During the December 1999 sea campaign, the attenuation channel at $\lambda$=555 nm was not properly working therefore the $c$(555 nm) value is not given in Table \[tab:Dec99\_ac\].
![Comparison between temperature, salinity, attenuation and absorption coefficients (at $\lambda=440$ nm) as a function of depth, measured in $KM4$ (red dots) and $KM3$ (blue dots) during December 1999. Two deployments were carried out in each site, typically in a time window of 24 hours.[]{data-label="fig:ProfileKM3_KM4"}](profiles_km3_km4.eps){width="9"}
------------- ---------------------- ---------------------- ---------------------- ---------------------- --
coefficient KM3 KM3 KM4 KM4
$1^{st}$ measurement $2^{nd}$ measurement $1^{st}$ measurement $2^{nd}$ measurement
a412 $0.0168\pm0.0006$ $0.0137\pm0.0004$ $0.0143\pm0.0006$ $0.0149\pm0.0008$
a440 $0.0177\pm0.0005$ $0.0156\pm0.0005$ $0.0159\pm0.0005$ $0.0172\pm0.0007$
a488 $0.0217\pm0.0004$ $0.0209\pm0.0004$ $0.0208\pm0.0004$ $0.0213\pm0.0005$
a510 $0.0370\pm0.0004$ $0.0365\pm0.0004$ $0.0363\pm0.0003$ $0.0374\pm0.0005$
a532 $0.0532\pm0.0004$ $0.0527\pm0.0004$ $0.0528\pm0.0003$ $0.0529\pm0.0005$
a555 $0.0682\pm0.0005$ $0.0683\pm0.0005$ $0.0683\pm0.0004$ $0.0689\pm0.0006$
a650 $0.3557\pm0.0003$ $0.3560\pm0.0003$ $0.3564\pm0.0003$ $0.3581\pm0.0003$
a715 $1.0161\pm0.0003$ $1.0165\pm0.0003$ $1.0167\pm0.0003$ $1.0169\pm0.0003$
c412 $0.0359\pm0.0025$ $0.0336\pm0.0022$ $0.0309\pm0.0017$ $0.0343\pm0.0026$
c440 $0.0335\pm0.0024$ $0.0312\pm0.0022$ $0.0284\pm0.0016$ $0.0292\pm0.0025$
c488 $0.0368\pm0.0024$ $0.0341\pm0.0021$ $0.0309\pm0.0015$ $0.0329\pm0.0023$
c510 $0.0442\pm0.0024$ $0.0417\pm0.0020$ $0.0397\pm0.0014$ $0.0427\pm0.0021$
c532 $0.0546\pm0.0024$ $0.0520\pm0.0020$ $0.0489\pm0.0014$ $0.0514\pm0.0020$
c650 $0.3780\pm0.0024$ $0.3740\pm0.0020$ $0.3719\pm0.0016$ $0.3747\pm0.0022$
c676 $0.4494\pm0.0021$ $0.4508\pm0.0018$ $0.4489\pm0.0011$ $0.4503\pm0.0018$
c715 $1.0209\pm0.0020$ $1.0193\pm0.0018$ $1.0169\pm0.0012$ $1.0190\pm0.0018$
------------- ---------------------- ---------------------- ---------------------- ---------------------- --
: December 1999 data. Average values of $a(\lambda)$ and $c(\lambda)$ (in units of \[m$^{-1}$\]) measured in the $KM3$ and $KM4$ sites, in the depths interval 2850$\div$3250 m. The statistical errors are the RMS of the measured distributions. Two deployments were carried out in each site. The systematic errors associated with the absorption coefficient data, in all the following tables, are of the order of $1.5 \times
10^{-3}[m^{-1}]$.[]{data-label="tab:Dec99_ac"}
Figure \[fig:Dec99\_Lac\] shows the absorption and attenuation lengths ($L_a(\lambda)=1/a(\lambda)$, $L_c(\lambda)=1/c(\lambda)$), as a function of the wavelengths (measured in the depth range 2850$\div$3250 m) in $Ustica$ and $Alicudi$ (see reference [@Capone2001]) and at the $KM3$ and $KM4$ sites. Data presented for each site are the averages over two deployments; the errors are the RMS of the observed distributions. The same Figure also shows that the values of $L_a(\lambda)$ and $L_c(\lambda)$ measured in the region of [[*Capo Passero*]{}]{} are larger than the ones measured in the other sites. In particular the values of $L_a(\lambda)$ are comparable to the ones of optically pure seawater quoted by Smith and Baker [@SmithBaker1981]. These results lead us to the conclusion that in [[*Capo Passero*]{}]{} $KM4$ site the deep seawater optical properties are close to optically pure water ones. Absorption and attenuation coefficients are almost constant for a large interval of depths making this site optimal for the installation of an underwater neutrino telescope. $KM3$ site was not considered a valid choice, in spite of the advantage to be closer to the coast, since the measured water optical properties are not constant along the vertical water column: this effect is supposed to be due to the proximity to the shelf break.
![Average absorption and attenuation lengths measured with the [*AC9*]{} in $Ustica$, $Alicudi$ ([@Capone2001]), [[*Capo Passero*]{}]{} $KM3$ and $KM4$ sites, in the 2850$\div$3250 m depth interval. Statistical errors are plotted. $L_a(\lambda)$ and $L_c(\lambda)$ of optically pure seawater, reported by Smith and Baker [@SmithBaker1981], are indicated by a solid black line.[]{data-label="fig:Dec99_Lac"}](appcapopasseroseasonlalc_alicudi_La_err_stat.eps "fig:"){width="7.5cm"}![Average absorption and attenuation lengths measured with the [*AC9*]{} in $Ustica$, $Alicudi$ ([@Capone2001]), [[*Capo Passero*]{}]{} $KM3$ and $KM4$ sites, in the 2850$\div$3250 m depth interval. Statistical errors are plotted. $L_a(\lambda)$ and $L_c(\lambda)$ of optically pure seawater, reported by Smith and Baker [@SmithBaker1981], are indicated by a solid black line.[]{data-label="fig:Dec99_Lac"}](appcapopasseroseasonlalc_alicudi_Lc_err_stat.eps "fig:"){width="7.5cm"}
Long term study of optical properties at the Capo Passero site
==============================================================
In order to verify the occurrence of seasonal variations of deep seawater IOPs in $KM4$, we are continuously monitoring this site using the experimental setup described above. The data collected during oceanographic campaigns of December 1999, March 2002, May 2002, August 2002 and July 2003 are reported here. In Figure \[fig:seasonKM4profiles\] the profiles of water temperature, salinity, $a$(440 nm) and $c$(440 nm), as a function of depth, are shown. The whole collected data sample consists of: 2 deployments in December 1999 (red dots), 4 deployments in March 2002 (yellow dots), 2 deployments in May 2002 (blue dots), 3 deployments in August 2002 (orange dots), 2 deployments in July 2003 (light blue dots). Seasonal variations are observed only in shallow waters, down to the thermocline depth of about 500 m. At depths greater than 2000 m the $a$(440) and $c$(440) coefficients measured in different seasons are compatible within the instrument experimental error ($\Delta T \simeq 10^{-2}~^{\circ}$C, $\Delta S
\simeq 10^{-2}$ psu, $\Delta a,\Delta c \simeq 2.2\cdot 10^{-3}$ m$^{-1}$).
![Profiles of temperature ($T$), salinity ($S$), attenuation coefficient $c$(440 nm) and absorption coefficient $a$(440 nm) measured in the [[*Capo Passero*]{}]{} $KM4$ site. The profiles refer to the campaigns performed during December 1999 (2 deployments, red dots), March 2002 (4 deployments, yellow dots), May 2002 (2 deployments, blue dots), August 2002 (3 deployments, orange dots) and July 2003 (2 deployments, light blue dots).[]{data-label="fig:seasonKM4profiles"}](seasonprofilesKM4_s_t_440.eps){width="9cm"}
Table \[tab:KM4\_average\_ac\] gives, for each campaign, the weighted average values of the absorption and attenuation coefficients, as a function of wavelength. Weighted average is calculated from the values of $a$($\lambda$) and $c$($\lambda$), measured in each deployment at depths between 2850 and 3250 m. Statistical errors are calculated from the RMS of the observed distributions. In Figure \[fig:Lac\_KM4\_season\] the absorption and attenuation lengths are shown. During December 1999 and March 2002 campaigns the channel $c$(555) was not properly working; the same happened to channels $c$(488) during all the campaigns after May 2002 and to $a$(488) in July 2003. The corresponding data are not reported here.
coefficient December 1999 March 2002 May 2002 August 2002 July 2003
------------- ------------------- ------------------- ------------------- ------------------- -------------------
a412 $0.0145\pm0.0008$ $0.0151\pm0.0014$ $0.0187\pm0.0014$ $0.0205\pm0.0008$ $0.0127\pm0.0017$
a440 $0.0164\pm0.0009$ $0.0166\pm0.0011$ $0.0160\pm0.0016$ $0.0148\pm0.0005$ $0.0126\pm0.0010$
a488 $0.0210\pm0.0005$ $0.0212\pm0.0007$ $0.0189\pm0.0013$ $0.0181\pm0.0003$ $ $
a510 $0.0366\pm0.0007$ $0.0366\pm0.0007$ $0.0377\pm0.0013$ $0.0383\pm0.0005$ $0.0367\pm0.0008$
a532 $0.0528\pm0.0004$ $0.0529\pm0.0006$ $0.0517\pm0.0010$ $0.0502\pm0.0005$ $0.0507\pm0.0006$
a555 $0.0685\pm0.0006$ $0.0683\pm0.0007$ $0.0675\pm0.0008$ $0.0677\pm0.0005$ $0.0673\pm0.0005$
a650 $0.3572\pm0.0009$ $0.3565\pm0.0010$ $0.3610\pm0.0004$ $0.3619\pm0.0004$ $0.3619\pm0.0003$
a715 $1.0168\pm0.0003$ $1.0117\pm0.0014$ $1.0458\pm0.0003$ $1.0457\pm0.0002$ $1.0451\pm0.0003$
c412 $0.0319\pm0.0028$ $0.0331\pm0.0025$ $0.0351\pm0.0033$ $0.0327\pm0.0024$ $0.0334\pm0.0039$
c440 $0.0287\pm0.0021$ $0.0302\pm0.0024$ $0.0281\pm0.0029$ $0.0283\pm0.0023$ $0.0288\pm0.0034$
c488 $0.0315\pm0.0022$ $0.0329\pm0.0027$ $ $ $ $ $ $
c510 $0.0406\pm0.0024$ $0.0414\pm0.0022$ $0.0436\pm0.0027$ $0.0450\pm0.0027$ $0.0459\pm0.0027$
c532 $0.0497\pm0.0022$ $0.0510\pm0.0025$ $0.0577\pm0.0016$ $0.0584\pm0.0024$ $0.0574\pm0.0021$
c555 $ $ $ $ $0.0808\pm0.0029$ $0.0791\pm0.0023$ $0.0761\pm0.0020$
c650 $0.3729\pm0.0024$ $0.3744\pm0.0025$ $0.3851\pm0.0032$ $0.3849\pm0.0034$ $0.3797\pm0.0015$
c676 $0.4493\pm0.0017$ $0.4502\pm0.0015$ $0.4761\pm0.0041$ $0.4740\pm0.0037$ $0.4684\pm0.0022$
c715 $1.0175\pm0.0019$ $1.0469\pm0.0010$ $1.0645\pm0.0032$ $1.0626\pm0.0030$ $1.0652\pm0.0023$
: Weighted average values of $a$(${\lambda}$) and $c$(${\lambda}$) measured in [[*Capo Passero*]{}]{} $KM4$ during different seasons, in the interval of depth 2850$\div$3250 m.[]{data-label="tab:KM4_average_ac"}
![Average absorption and attenuation lengths measured with the [*AC9*]{} in $KM4$, at depth 2850$\div$3250 m in December 1999 (blue circle), March 2002 (light blue square), May 2002 (purple triangle), August 2002 (red upsidedown triangle) and July 2003 (dark yellow star). Statistical errors are plotted. A solid black line indicates the values of $L_a(\lambda)$ and $L_c(\lambda)$ for optically pure seawater reported by Smith and Baker [@SmithBaker1981].[]{data-label="fig:Lac_KM4_season"}](appcapopasseroseasonlalc_La_err_stat.eps "fig:"){width="7.5cm"}![Average absorption and attenuation lengths measured with the [*AC9*]{} in $KM4$, at depth 2850$\div$3250 m in December 1999 (blue circle), March 2002 (light blue square), May 2002 (purple triangle), August 2002 (red upsidedown triangle) and July 2003 (dark yellow star). Statistical errors are plotted. A solid black line indicates the values of $L_a(\lambda)$ and $L_c(\lambda)$ for optically pure seawater reported by Smith and Baker [@SmithBaker1981].[]{data-label="fig:Lac_KM4_season"}](appcapopasseroseasonlalc_Lc_err_stat.eps "fig:"){width="7.5cm"}
Figure \[fig:KM4\_season\_stability\] shows the time dependence of the average values of $L_a$(440 nm) and $L_c$(440 nm) as a function of time. The plotted error bars are statistical errors. The average absorption length, calculated using the values of Table \[tab:KM4\_average\_ac\] weighted with their statistical errors, is $L_a$ ($\lambda$ = 440 nm) = 66.5 $\pm 8.2_{stat} \pm
6.6_{syst}$ m close to the value of optically pure water. The weighted average attenuation length is $L_c$($\lambda$ = 440 nm)= 34.7 $\pm 3.3_{stat} \pm 1.8_{syst}$ m close to published values of ocean waters measured in conditions of collimated beam and detector geometry [@Duntley1963]. The value of $L_c$ measured in [[*Capo Passero*]{}]{} is larger than the one reported by Khanaev and Kuleshov [@Khanaev1993] for the NESTOR site. We remind that other results (by DUMAND [@Bradner1981], NESTOR [@Anassontzis1994] and ANTARES [@Aguilar2004]) have been obtained measuring the *[effective]{} light attenuation in conditions of not collimated geometry, i.e. using a diffused light source and a large area detector; these results therefore deal with the *[effective]{} attenuation coefficients and cannot be directly compared with our results.**
![Average attenuation and absorption lengths at $\lambda$=440 nm measured with the [*AC9*]{} in $KM4$, at depth 2850$\div$3250 m in December 1999 (blue circle), March 2002 (light blue square), May 2002 (purple triangle), August 2002 (red upsidedown triangle) and July 2003 (dark yellow star). The weighted average values of $L_a$(440) and $L_c$(440) are indicated by dashed black lines (see text). Statistical errors are shown.[]{data-label="fig:KM4_season_stability"}](appcapopasserostability_440_stat.eps){width="9cm"}
Conclusions
===========
The NEMO Collaboration measured, as a function of depth, the salinity, temperature and inherent optical properties in several abyssal sites of the central Mediterranean Sea using an experimental apparatus consisting of an AC9 transmissometer and a standard CTD probe. In order to compare the water transparency to Čerenkov light of different sites we have averaged the measured values of $c(\lambda)$ and $a(\lambda)$ in a range of about 400 m, at the depths which are suitable for the deployment of a km$^3$ neutrino telescope. The data of $L_a(\lambda)$ presented for [[*Capo Passero*]{}]{} $KM4$ site are close to the ones reported by Smith and Baker for optically pure seawater [@SmithBaker1981]. For blue light, the average absorption length is $\simeq$ 67 m, the average attenuation length is $\simeq$ 35 m. It is worth to mention that all the measurements reported in this paper have been carried out over an area of about 10 km$^2$ around the reference point of $KM4$. We conclude that optical and oceanographic properties in [[*Capo Passero*]{}]{} $KM4$ site are homogeneous in a large region and constant over the investigated timescale. The measured absolute values of IOP and the homogeneity of the water column, for more than one thousand metres above the seabed, make [[*Capo Passero*]{}]{} $KM4$ an optimal site for the installation of the future Mediterranean km$^3$ underwater neutrino telescope.
Acknowledgements
================
This work has been been conducted in collaboration with: Department of Physical Oceanography INOGS (Trieste), Istituto Sperimentale Talassografico CNR (Messina) and Istituto di Oceanografia Fisica CNR (La Spezia). We thank Captains E. Gentile, V. Lubrano, A. Patané, the officers and the crew of the R/V [*Alliance*]{}, [*Thetis*]{} and [*Urania*]{} for their outstanding experience shown during the sea campaigns.
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|
---
abstract: 'A method of construction of solution for acoustic-gravity waves (AGW) above a wave source, taking dissipation throughout the atmosphere into account (Dissipative Solution above Source, DSAS), is proposed. The method is to combine three solutions for three parts of the atmosphere: an analytical solution for the upper isothermal part and numerical solutions for the real non-isothermal dissipative atmosphere in the middle part and for the real non-isothermal small dissipation atmosphere in the lower one. In this paper the method has been carried out for the atmosphere with thermal conductivity but without viscosity. The heights of strong dissipation and the total absorption index in the regions of weak and average dissipation are found. For internal gravity waves the results of test calculations for an isothermal atmosphere and calculations for a real non-isothermal atmosphere are shown in graphical form. An algorithm and appropriate code to calculate DSAS, taking dissipation due to finite thermal conductivity into account throughout the atmosphere, are developed. The results of test DSAS calculations for an everywhere isothermal atmosphere are given. The calculation results for DSAS for the real non-isothermal atmosphere are also presented. A method for constructing of the 2x2 Green’s matrix fully taking dissipation into account and allowing to find disturbance from some source of AGW in the atmosphere is proposed.'
address:
- 'Institute of Solar-Terrestrial Physics SB RAS, Irkutsk, Russia'
- 'Institute of Solar-Terrestrial Physics SB RAS, Irkutsk, Russia'
author:
- 'I.S. Dmitrienko'
- 'G.V. Rudenko'
title: Waves in vertically inhomogeneous dissipative atmosphere
---
AGW,upper atmosphere,dissipation,TIDs
Introduction {#section1}
============
The study of acoustic-gravity waves (AGW) in the atmosphere has a long history. The basic properties of AGW propagation were summarized even in [@Hook]. However, dissipationless approach limits the applicability of the results. They apply only to the waves in the lower and middle atmosphere, where weakness of dissipation allows to neglect it. High enough in the upper rarefied atmosphere dissipation becomes the determining factor in the behavior AGW. Therefore, to describe the penetration of AGW in the upper atmosphere it is necessary to involve more complex formalism of the dissipative hydrodynamics. In the dissipative hydrodynamics an analytical description of AGW is obtained in the frames of the isothermal approximation [@Lyons; @Yanowitch_a; @Yanowitch_b; @Rudenko_a; @Rudenko_b]. However, isothermal approximation models the real atmosphere properties adequately in its upper layers only. Without the isothermal approximation the wave dissipation was taken into account in [@Vadas]. These researches were performed with use of the WKB approximation, which has its limitations. It enables to take into account the weak dissipation in the lower and middle layers of the atmosphere and does not allow to adequately describe the strong effect of dissipation on the wave in the upper atmospheric layers.
This paper is devoted to an AGW solution fully taking dissipation into account. The solution is constructed as an integrated whole for two fundamentally different physical conditions of the atmosphere, with virtually no dissipation in the lower and middle atmosphere and with substantial dissipation, increasing with the height, in the upper atmosphere. The solution has two parameters: real wave frequency and horizontal wave number.
The absence of the flow of energy towards the Earth in the upper layers of the atmosphere is the only boundary condition for the solution; because of dissipativity of the atmosphere, it coincides with the condition of upward attenuation of the solution (from the Earth). The solution exists at all heights. Formally, it can be taken as a solution of the wave problem with some source on the Earth surface. However, wider physical meaning of this solution is that the part of this solution, with real wave frequency and horizontal wave number, in the region above any source located in the atmosphere (or on the Earth’s surface), describes the vertical structure of the disturbance produced by this source there in this region . Region above a source is a part of the atmosphere, the heights of which exceed the upper limit of the source localization; there are no limits for horizontal coordinates of this region. Formally, the solution extends indefinitely in the upper atmosphere. Starting from some height it decreases against the rarefying atmosphere background due to increase of the effect of dissipation with altitude. Given dissipative character of the solution and its aforementioned physical meaning we call it dissipative solution above source (DSAS).
DSAS is necessary final element for the solution of the problem of the wave propagation from a source located in the atmosphere.
Now there are well-developed methods for description of wave phenomena in the real atmosphere, based on the direct numerical solution of hydrodynamic equations, which include a variety of model sources of climate and anthropogenic character: [@Hickey1997; @Hickey1998; @Walterscheid1990; @Walterscheid2001; @Snively2003; @Snively2005; @Snively2007; @Yu2007_a; @Yu2007_b; @Yu2007_c; @Yu2009; @Kshevetskii2005; @Gavrilov2014]. These methods are based on direct numerical integration of the nonlinear system of partial differential equations in two- or three-dimensional approximations for stratified atmosphere with viscosity and thermal conductivity and wind stratification. However, these methods are inefficient when describing the perturbation at large distance from the localization of the source, because of their resource consumption and rapid accumulation of numerical errors in the need to increase the grid to ensure sufficient spatial resolution. It is well known that the wave disturbances, propagating far from their sources, are important elements of the dynamics of the atmosphere. Due to increase of their relative amplitudes in height, they manifest themselves in the thermosphere as the main cause of the diverse ionospheric disturbances. The transfer of wave energy and momentum over large distances is provided due to presence of the conditions of waveguide propagation in the temperature inhomogeneities in the lower and middle atmosphere [@Pierce1970]. The most effective method to describe the remote signals is to present them as superpositions of the waveguide modes generated by the source (the method of the normal modes). An algorithm, for solving the disturbance propagation problem, based on the method of normal modes, described in detail in [@Pierce1971], where a result of comparison of the calculated and the observed pressure disturbances on the Earth’s surface in the far zone of a nuclear explosion was first presented. In principle, based on the algorithm of [@Pierce1971], synthesizing propagating normal modes, one can calculate the disturbance parameters at any height where the dissipation-free approach, in which [@Pierce1971], built its own algorithm, still holds.
Our main goal is to study the possibility of similar description of the long-range propagation of disturbances in the ionospheric heights. In the ionosphere, according to the basic concept [@Hines1960], the so-called travelling ionospheric disturbances (TIDs) are the result of infiltration of the neutral atmosphere disturbances from the lower IGW (internal gravity wave) waveguide channels. In the upper atmosphere in the tropospheric heights the dissipative effect on the wave processes becomes dominant that makes it impossible to ignore and even to take into account approximately the dissipative terms of the wave equations system. As a result, we have to deal with a significantly more complicate problem compared to that of [@Pierce1971].
As in the dissipation-free case, [@Pierce1971], the key element of the disturbance propagation problem is to find the Green’s function (or matrix), for the construction of which we need two solutions: the lower one (below the source) and the upper one (above the source) satisfying the lower and upper boundary conditions, respectively. In the case, when a source is in the range of altitudes where the dissipation-free approximation is applicable, then finding a solution under source can now be considered as a routine problem. Thus, in this case, the key problem is to find a solution above the source for a realistic model of the atmosphere with a full account of dissipation. It is the problem the present paper focuses on. Our approach to this problem bases on matching a solution of the weakly-dissipative wave problem of second order for the lower part of the atmosphere with a solution of the dissipative wave problem of order, higher than second, for the upper part of the atmosphere. It is different essentially, on the level of severity, from the method proposed in its time in [@Francis1973_a]. In the paper [@Francis1973_a], a solution over the source was obtained with the so-called multi-layer technique, in which the medium in each horizontal layer is considered isothermal with constant coefficients of viscosity and thermal conductivity; so representation of the solution in each such layer is possible in analytical form. When taking dissipation into account, there are, generally speaking, three wave types with three complex values of the square of the vertical wave number, corresponding to (in the weak dissipation case) conventional acoustic-gravity wave (AGW) and two types of purely dissipative oscillations caused by viscosity and thermal conductivity. [@Francis1973_a] represented the solution in each layer only with AGV solutions, and only two conditions of six ones used for matching solutions of adjacent layers. Such an approach is quite justified in the temperate heights, where dissipation is weak, but obviously incorrect in the strong dissipation regions. Thus, in order to solve the problem, Francis artificially lowered the order of the system of equations from the 6th order up to the second order everywhere. Thus, the structure of wave disturbances in the lower part of the atmosphere and the dispersion characteristics of the modes, captured by the inhomogeneities of the lower atmosphere, the method of Francis allows to calculate sufficiently well, but it obviously is not able to give a correct description of disturbances in the upper atmosphere. The results of Francis are widely used in the theoretical researches and in the interpretation of observations of various disturbances, including those in the upper atmosphere [@Shibata1983; @Afraimovich2001; @Vadas2009; @Vadas2012; @Idrus2013; @Heale2014; @Hedlin2014]. Unlike Francis, we use a method of reducing the order of the system of the wave equations up to the second one (in our own version) only for the small-dissipation altitudes where it is justified. Therefore, our method, unlike the method of Francis, gives the possibility to describe disturbances of the upper atmosphere adequately. Another approach to obtain the solution above source is developed in the papers [@Lindzen1970; @Lindzen1971_a; @Lindzen1971_b]. This approach, based on the numerical method of the tridiagonal matrix algorithm for the full system of linear dissipative hydrodynamic equations, can be considered quite strict. The only restriction is the circumstance that the hydrostatic approximation is used, which is valid only for sufficiently long-period oscillations (with frequencies much lower than the Brunt-Väisälä frequency). The results obtained on the basis of this approach played an important role in the description of tidal modes and continue to be relevant until now [@Forbes1979; @Fesen1995; @Gavrilov1995; @Grigorev1999; @Akmaev2001; @Angelatsi2002; @Yu2009].
Thus, our approach to obtaining the solution above source differs from [@Francis1973_a] by correct consideration of dissipation; as for the papers on the base of [@Lindzen1970], it is of more universal character since does not have restrictions caused by usage of the hydrostatic approximation. It should be noted that the dissipation is taken into account in our present paper less complete than in the aforementioned papers. This is due to the fact that we believe it possible, as a first step, consider the viscous dissipation as negligible, taking only the dissipation due to the heat conductivity into account. We use such an approach in connection with the possibility of a simple analytic representation of solutions in heat-conducting thermosphere, which allows us to put the upper boundary conditions more strict at the infinity. Such conditions are more accurate compared with the boundary conditions at some height used in [@Lindzen1970]. It is also a useful approach as giving a clear understanding of dissipation influence on the wave motion in the upper atmosphere.
Because in reality the dissipation, associated with the thermal conductivity, begins its influence at lower altitude then the viscous dissipation does, our model of the atmosphere can give the real picture of the disturbances in the upper atmosphere sufficiently well, at least, to a certain height. As will be discussed below, taking into account the atmosphere viscosity is, in principle, possible in our approach to finding DSAS. It is also feasibly to include, in the model, the atmosphere wind, which is not considered in this paper.
DSAS formally exists for the entire atmosphere up to the Earth and corresponds in this form (for all heights) to the particular case of a source on the Earth’s surface.If such a solution for certain values of the frequency and horizontal wave number $k_x$ satisfies the lower boundary condition, it can be regarded as a solution for a trapped mode.
In general case with the wind, the continuation of DSAS up to the Earth is possible in such cases when the horizontal phase velocity of the wave is large enough so it is not equal to the wind speed at any height. Otherwise, because of the so-called wave destruction (the phenomenon, responsible for the turbulent properties of the real atmosphere), [@Lindzen1981], DSAS will be applied only to a certain height and can be used for cases of sources at sufficiently high altitudes. However, the horizontal phase velocities of the waveguide modes, which are responsible for the distant propagation [@Francis1973_b], are of the order of the speed of sound, so the condition of equality of the phase velocity to the wind speed never is satisfied.
It should also be noted that the feature of our method is absence of the need to include in the equations the additional dissipative terms taking into consideration the impact of turbulence in a model manner, without which none of the previously developed methods can work, because of the inherent numerical instability.
DSAS with a full view of the dissipation is most important for the synthesis of the waveforms at fairly high altitudes, where it is impossible to consider dissipation with any approximate method. Such altitudes are typical for TIDs. To synthesize waveforms at lower altitudes the algorithms for non-dissipative atmosphere are sufficient, for example, the above algorithm of [@Pierce1971]. Note that the wave forms of the disturbances in the upper atmosphere should be less complicated than those in the middle and lower altitudes, due to the dissipative suppression of multiple waveguide modes of high-frequency sound range. As shown by [@Francis1973_b], only one or two major IGW modes can take place at ionospheric heights, due to the waveguide leakage and the weak horizontal attenuation.
Our consideration, as well as the general problem of the synthesis of waveforms by normal modes, requires linearity of disturbance at all altitudes. Taking dissipation into account provides drop with height of the disturbance amplitude, relative to the background parameters, in the upper atmosphere from a certain level. This circumstance enables the selection of source amplitude small enough for the disturbance to be linear throughout. At first glance, it may seem that limitations of the source amplitude, to ensure linearity of the disturbance, have to be very strong and so the real sources unlikely satisfy them - since the dissipative suppression of growth of the relative amplitude of the disturbance with height, as a rule, begins at the altitudes, at which the disturbance amplitude can grow by orders of magnitude compared to the amplitude near the Earth. Indeed, the disturbances have likely to be non-linear just above the typical sources localized in space and time. However, we must take into account the fact that only a small part of the disturbance energy is captured by the low-atmosphere waveguides, capable of transferring the wave energy over long distances. In addition, if the main source spectral power is distributed in the high-frequency range (this is typical, for example, for explosions), then the amplitudes of the low-frequency spectrum range, generating waveguide modes, will be significantly less than the main amplitudes of the source. All these factors quite can lead to linearity of the waveguide modes.
Note, in particular, the fact that TIDs are typically very close to harmonical waves [@Hook] says also in favor of the assumption of linearity of the actual responses in the upper atmosphere. Thus, we can assume that the linear solution above the source can be the basis for solving the propagation problem for disturbances in a stratified atmosphere, if the amplitude of the source is small enough, or the ducting propagation only is of interest.
The solution to this problem for a stratified atmosphere within the dissipationless approximation is well known [@Hook]. In this case, we have the second-order problem, i.e. it is formulated as a second-order differential equation or a set of two first-order differential equations equivalent to it. To solve the dissipationless problem with a source the Green’s matrix composed of two second-order problem solutions is used. One of these solutions must satisfy the upper boundary condition and the other must satisfy the condition on the Earth. It is shown in this paper that one can also limit to a second-order problem in the small dissipation approximation for disturbance of not too small vertical scale. Therefore, the solution of the problem with a source is also obtained in this case by means of the second-order Green’s matrix. But the small dissipation approximation is applicable only in the lower and middle parts of the atmosphere. It is not sufficient in the upper part because of increase with altitude of the dissipation effect. So we have to solve a problem of order higher, then second, there.
But even fully taking dissipation into account, one can retain a formalism of a second-order problem to a significant extent if the source is located in the altitude range where dissipation is small. For this it is enough to replace the solution of the second-order problem in the $2\times 2$ Green’s matrix that satisfies the upper boundary condition with DSAS. The Green’s matrix obtained in such a way we will call the extended Green’s matrix. Note that in the atmosphere mainly occur sources for which the method of the extended Green matrix is applicable. These are sources located in the lower and middle atmosphere: EXPLOSIONS tsunami meteorites anthropogenic sources, meteorological phenomena.
Description of an extended tail formally indefinitely outgoing to the upper atmosphere is the main feature of DSAS. However, if a wave is attenuated too much for its extension to the upper atmosphere, the amplitude of the upper atmospheric tail can be so small that the disturbance with such an amplitude is of no interest. On the other hand, if a wave weakly damps when propagating to the upper atmosphere, the relative amplitude of the tail (the ratio of the amplitude to the background) can be too large in the upper atmosphere for the disturbance to be under the linear approximation. In this connection, we investigated effect of dissipation on the AGV propagation. Some characteristics of the wave dissipative loss are introduced and, then, their values are calculated.
We organize our paper as follows. Equations used further to construct a DSAS are derived in Section 2. We have demonstrated the possibility of obtaining from the hydrodynamic equations both a system of four first-order ODEs taking into account thermal conductivity only and, in more general case, a system of six first-order ODEs assuming, besides consideration of thermal conductivity, atmospheric viscosity. Numerical solutions of such systems can be obtained by the standard Runge-Kutta method, if the initial values of the unknowns are given at the specified height. The possibility of including wind stratification in the model of the atmosphere is indicated. To describe waves in the weakly dissipative (lower) part of the atmosphere we obtained a system of two first-order ODE.
Section 3 gives details on the method for an analytical solution for the upper atmosphere with an isothermal temperature distribution. The theory is based on the possibility of describing wave disturbances in an isothermal atmosphere by one sixth-order differential equation in the general case of simultaneous consideration of viscosity and thermal conductivity or fourth-order equation if we consider the thermal-conductivity model of the atmosphere without viscosity. In this paper, we have constructed DSAS for the case of the thermal-conductivity model of the atmosphere without viscosity. In this case, a fourth-order equation reduces to the generalized hypergeometric differential equation [@Lyons; @Yanowitch_a; @Yanowitch_b; @Rudenko_a; @Rudenko_b] analytical solutions of which are subsequently used for the isothermal part of the atmosphere. In section 3, we also study the effect of dissipation on the AGV propagation. First, the parameter $z_c$, characterizing the heights of the transition to the regime of strong dissipation, is calculated. This parameter is singled out when deriving the hypergeometric differential equation [@Lyons; @Yanowitch_a; @Yanowitch_b; @Rudenko_a; @Rudenko_b]. In the heights below $z_c$, we have approximately classical oscillations corresponding to usual wave types of oscillations without dissipation. In the range of heights higher $z_c$, dissipation completely changes nature of oscillations transforming them into purely dissipative waves. It follows from the equations that $z_c$ depends only on the wave period for any vertical profile of the atmospheric density, so we calculated it as a function of period. The parameter $z_c$, calculated for the isothermal atmosphere, can be used for estimating of the height of the strong dissipation region, depending on wave period, in the real atmosphere. Second, as a dissipation characteristic at the heights below $z_c$, we introduced an index of total vertical dissipation . We calculated it both on the basis of the WKB-approximation solution for the non-isothermal thermal-conductive atmosphere and and the analytical solution for isothermal thermal-conductive atmosphere.
Section 4 gives our method for constructing of DSAS. We have carried out this method in the frames of the thermal-conductivity model of the atmosphere without viscosity. In this case, the complete set of wave equations is presented by four first-order ordinary differential equations which, as we show, can be numerically solved in the upper part of the atmosphere, where dissipation is strong, to some height in the middle atmosphere, where dissipation effect is sufficiently small. There are calculation difficulties not allowing us to use a complete set of wave equations in the lower part of the atmosphere. On the other hand, dissipation is small there. This allowed us to reduce the fourth-order problem in the lower part of the atmosphere to a second-order problem. Thus, DSAS is constructed of the solutions for three height ranges in the direction from the upper part of the atmosphere to the Earth: an analytical solution for the isothermal upper atmosphere (I); a numerical solution of the system of four ODEs in a non-isothermal upper atmosphere until the height of $z_t$ on which the dimensionless kinematic thermal conductivity reaches a sufficient smallness (II); a solution of the system of two ODEs for the small dissipation approximation in the region below $z_t$ up to the Earth’s surface (III). We will further also call the height ranges I, II, III lower middle and upper parts of the atmosphere, respectively.
An analytical solution (I), satisfying the condition of upward attenuation, is a superposition of a superposition of some two solutions, having asymptotics each of which attenuates upward; the coefficients of these solutions are arbitrary. DSAS is continuous at the boundary of the regions (I) and (II) because an analytical solution (I) at this boundary gives four initial values for solving the system of four ODEs numerically in the region (II).
As for the boundary of the regions (II) and (III) then we applied our method for matching the solutions (II) and (III) there. Matching solutions (II) and (III) is important element in constructing DSAS. It is clear that the solutions corresponding to different approximations can not be match arbitrarily closely at the boundary. However, it is also clear that this is not required, the smallness of jumps is enough. Therefore matching the solutions (II) and (III) carries out as follows. We take values of any two components from four components of the solution (II) at the boundary of the regions (II) and (III) (the height $z_t$) as initial values for solving the system of two ODEs numerically in the region (II). For two other components, we provide sufficient smallness of the jump at the height $z_t$,using the condition of minimization of the jumps to find the ratio between the unknown coefficients in the solution (I) . For the minimal jumps would be sufficiently small, a sufficiently small height $z_t$ must be chosen. Influence of $z_t$ values on the values of the jumps is analyzed using special calculations.
We test our method for constructing of DSAS, using an everywhere isothermal model of the atmosphere, in section 5, and we give the result of calculation of DSAS for a real non-isothermal model of the atmosphere in section 6. Finally, in section 7, we give the extended Green’s matrix, allowing to describe the propagation of disturbances in the atmosphere from the various sources.
Basic wave equations in a vertically stratified horizontally homogeneous non-isothermal atmosphere {#section2}
==================================================================================================
In the approximation of the stationary horizontally homogeneous atmosphere, linear disturbances can be represented by a superposition of waves of the form $f\left(z,\omega,\bf
{k}_{\perp} \right)e^{i\omega t+ i\left(\bf {k}_{\perp} \cdot \bf
{r} \right)}$, where: $z$ is the vertical coordinate, $\left(\omega,\bf {k}_{\perp} \right)$ are the frequency and horizontal wave number. As we will show, vertical structure of such waves is described by either a system of $n$ first-order ordinary differential equations with $n$ independent values or one $n$-order ordinary differential equation for one value; $n$ is two, four or six, depending on choice of approximation. Without dissipation, we have the wave problem of the second order $(n=2)$ describing classical wave oscillations, for example, acoustic or IGWs. With taking into consideration dissipation due to thermal conductivity, the order of the wave problem becomes higher $(n=4)$; two more wave solutions arise as a consequence. Adding viscosity leads to $n=6$ and two more “dissipation” solutions. At the problem order $n=2$, it is possible to reduce it to one second-order differential equation [@Ost; @Pon]. At all the orders of the problem, it shall be shown that it reduces to an system of $n$ explicit first-order ordinary differential equations (normal system).
We will give derivations of necessary equations for the problems with different $n$. We will introduce all physical parameters in the form of sums of their undisturbed and disturbed values: $f \to
f_0+f$. We will regard the system of hydrodynamic equations as input: $$\label{eq1}
\begin{array}{l}
\partial_t\rho+\left({\bf v}_{0\perp}\cdot\nabla \right)\rho+\rho_0\nabla\cdot{\bf v}+{\rho^\prime}_0v_z=0, \\
\partial_t T+\left({\bf v}_{0\perp}\cdot\nabla \right)T+(\gamma-1)T_0\nabla\cdot{\bf v}+{T^\prime}_0v_z=\frac{\kappa}{c_v\rho_0}\Delta T, \\
\rho_0\partial_t{\bf v}+\rho_0\left({\bf v}_{0\perp}\cdot\nabla \right){\bf v}+\nabla p-\rho{\bf g}=\nu_1\Delta{\bf v}
+\left(\frac{1}{3}\nu_3+\nu_2 \right)\nabla(\nabla\cdot{\bf v}),\\
p=RT_0\rho+R\rho_0T.
\end{array}$$ System of equations (\[eq1\]) is sequentially mass-continuity equations, heat-balance equation, momentum equations, and equations of state; $ \rho, p, T,{\bf v}$ is the disturbed values of density, pressure, temperature, and velocity, accordingly; disturbed values corresponding to them with the index $_0$ are the functions of only the vertical coordinate $z$, the prime symbol denotes differentiation by $z$; ${\bf g}$ is the free fall acceleration vector $(0,0,-g)$; $R$ is the universal gas constant; $c_v$ is the specific heat at constant volume; $\kappa$ is the dynamic coefficient of thermal conductivity, $\nu_1$ and $\nu_2$ is the dynamic coefficients of the first and second viscosity; the subscript $_\perp$ denotes a horizontal part of the vector. Without loss of generality, we will put the coordinate axis $x$ directed along a horizontal wave vector ${\bf
k}_\perp$: $(k_x,0,0)$. In this case, all the disturbed values do not depend on $y$. In addition, it is easy to show that $y$ disturbed velocity component is determined by its equation independent of other disturbed parameters of the environment. Therefore, we can ignore it, defining the disturbed velocity vector by only two ${\bf v}=(v_x,0,v_z)$. Replacing partial derivatives with respect to values: $\partial_t, \partial_x$ with the factors $-i\omega, ik_x$, respectively, and introducing a special notation for a frequency function in the moving reference system $\Omega(z)=\omega-k_xv_0x(z)$, we modify the input system of equations to: $$\label{eq2}
\begin{array}{l}
-i\Omega\rho+\rho_0\left(ik_xv_x+{v_z}^\prime\right)+{\rho^\prime}_0v_z=0, \\
-i\Omega T+(\gamma-1)T_0\left(ik_xv_x+{v_z}^\prime\right)+{T^\prime}_0v_z=\frac{\kappa}{c_v\rho_0}\left(-k^2_xT+T^{\prime\prime}\right), \\
-i\Omega v_x+ik_x\rho^{-1}_0 p=\nu_1\left(-k^2_xv_x+v_x^{\prime\prime}\right)
+\left(\frac{1}{3}\nu_3+\nu_2 \right)\left(-k^2_xv_x+ik_xv_z^\prime\right),\\
-i\Omega v_z+\rho^{-1}_0 p^\prime+g\rho_0^{-1}\rho=\nu_1\left(-k^2_xv_z^\prime+v_z^{\prime\prime}\right)
+\left(\frac{1}{3}\nu_3+\nu_2 \right)\left(ik_xv_x+v_z^{\prime\prime}\right),\\
p=RT_0\rho+R\rho_0T.
\end{array}$$ We will get the normal systems from (\[eq2\]) in the form: $$\label{eq3}
F'=\hat{A}F,$$ where: $F$ is a vector of $n$ disturbance parameters sufficient to formulate the wave problem in accordance with a chosen physical approximation; $\hat{A}$ is the $n\times n$ matrix whose elements depend on undisturbed environment parameters and the wave parameters $(\omega,k_x)$
Viscous heat-conducting atmosphere with wind $(n=6)$ (Case I) {#section2.1}
-------------------------------------------------------------
For this case, we will choose the following vector $F$ for the system of equations (\[eq3\]): $$\label{eq4}
F=(T',T,v'_x,v_x,v'_z,v_z).$$ To obtain a system of equations of the form (\[eq3\]), we need to have expressions for the second derivatives with respect to z of the values $(T,v_x,v_z)$ through the components of vector $F$. We get the equation for $T''$ directly from the second equation of system (\[eq2\]). The equation for $v'_x$ is obtained from the third equation of system (\[eq2\]), using the fifth and first equations for expression of $p$. Furthermore, differentiating the fifth and first equations of system (\[eq2\]), we find an expression for the combination $p'+\rho g$ containing only $v''_x$ and the vector $F$ components. Substituting this combination in the fourth equation, we get an expression for $v''_z$. As a result, we get for $\hat A$: $$\label{eq5}
\begin{array}{l}
a_{11}=0; a_{12}=-i\Omega\frac{c_v \rho_0}{\kappa}+k^2_x; a_{13}=0; a_{14}=\frac{c_v \rho_0}{\kappa}(\gamma-1)T_0ik_x; \\
a_{15}=\frac{c_v \rho_0}{\kappa}(\gamma-1)T_0; a_{16}=\frac{c_v \rho_0}{\kappa}T'_0;\\
a_{21}=1; a_{22}=0; a_{23}=0; a_{24}=0; a_{25}=0; a_{26}=0;\\
a_{31}=0; a_{32}=\frac{ik_xp_0}{\nu_1T_0}; a_{33}=0; a_{34}=\frac{ik_x^2p_0}{\Omega\nu_1}
-\frac{i\Omega\rho_0}{\nu_1}+(\frac{4}{3}+\frac{\nu_2}{\nu_1})k_x^2;\\
a_{35}=\frac{k_xp_0}{\Omega\nu_1}-(\frac{1}{3}+\frac{\nu_2}{\nu_1})ik_x;a_{36}=\frac{k_xp_0\rho'_0}{\Omega\nu_1\rho_0};\\
a_{41}=0; a_{42}=0; a_{43}=1; a_{44}=0; a_{45}=0; a_{46}=0;\\
a_{51}=\frac{p_0}{\chi T_0};a_{52}=\frac{R\rho'_0}{\chi};
a_{53}=-\frac{ik_x}{\chi}(\frac{ip_0}{\Omega}+\frac{1}{3}\nu_1+\nu_2);
\\
a_{54}=\frac{k_x}{\chi}\left(\frac{\rho_0}{\Omega}(g+RT'_0) +RT_0\left(\frac{\rho_0}{\Omega}\right)^\prime \right)
;\\
a_{55}=\frac{1}{i\chi}\left(\frac{\rho_0}{\Omega}(g+RT'_0)+RT_0\left(\left(\frac{\rho_0}{\Omega}\right)^\prime +\frac{\rho'_0}{\Omega} \right)\right);\\
a_{56}=\frac{1}{i\chi}\left(\frac{\rho'_0}{\Omega}(g+RT'_0)+\Omega\rho_0+i\nu_1k_x^2+RT_0\left(\frac{\rho'_0}{\Omega}\right)^\prime\right);\\
a_{61}=0; a_{62}=0; a_{63}=0; a_{64}=0; a_{65}=1; a_{66}=0.
\end{array}$$ Here $\chi=\frac{4}{3}(nu_1+nu_2)+\frac{ip_0}{\Omega}$, $p_0=R\rho_0T_0$. The variables $p$ and $\rho$ not belonging to the vector $F$ are expressed through its components of the first and fifth equations of system (\[eq2\]): $$\label{eq6}
\begin{array}{l}
\rho=\frac{\rho_0}{i\Omega}(ik_xf_4+f_5)+\frac{\rho'_0}{i\Omega}f_6,\\
p=RT_0\frac{\rho_0}{i\Omega}(ik_xf_4+f_5)+RT_0\frac{\rho'_0}{i\Omega}f_6+R\rho_0f_2.
\end{array}$$ Eqs. (\[eq3\])-(\[eq6\]) describe wave disturbances for the most general case of the model of the atmosphere. With given initial $F$ values, these equations can be used for numerical solving to find corresponding wave disturbances. As we shall see further, numerical obtaining of the solution to (\[eq3\])-(\[eq6\]) is possible only when heights are sufficiently great and correspond to not too high values of undisturbed density. Underneath, the problem is ill-conditioned. We will show later that one can avoid this difficulty, using smallness of dissipation at lower heights to reduce the problem to the second one.
Non-viscous heat-conducting atmosphere approximation $(n=4)$ (Case II) {#section2.2}
----------------------------------------------------------------------
This approximation is main in this paper. As we will not take the wind into account in this paper later, we will draw all the calculations here without it. The formulas taking wind into account, in this and following sections, are obtained by the substitution $\omega\to \Omega$. In the case of non-viscous heat-conducting atmosphere, system of equations (\[eq2\]) becomes: $$\label{eq7}
\begin{array}{l}
-i\Omega\rho+\rho_0\left(ik_xv_x+{v_z}^\prime\right)+{\rho^\prime}_0v_z=0, \\
-i\Omega T+(\gamma-1)T_0\left(ik_xv_x+{v_z}^\prime\right)+{T^\prime}_0v_z=\frac{\kappa}{c_v\rho_0}\left(-k^2_xT+T^{\prime\prime}\right), \\
-i\Omega v_x+ik_x\rho^{-1}_0 p=0,\\
-i\Omega v_z+\rho^{-1}_0 p^\prime+g\rho_0^{-1}\rho=0,\\
p=RT_0\rho+R\rho_0T.
\end{array}$$ To derive (\[eq7\]) in the form of (\[eq3\]), we choose the following vector $F$ : $$\label{eq8}
F=(T',T,p,v_z).$$ We need expressions of the values $T'',p',v'_z$ through the components of $F$. It is easy to get them excluding extra variables $v_x$ and $\rho$ with use of the third and fifth equation, accordingly. The expression for $T''$ is derived from the second equation, using the expression $ik_xv_x+v'_z$ from the first equation; the expression for $p'$, from the fourth equation; the expression for $v'_z$, from the first equation. As a result, we obtain the elements of the matrix $\hat A$ in the following form: $$\label{eq9}
\begin{array}{l}
a_{11}=0;a_{12}=k_x^2-\frac{i\gamma\omega c_v\rho_0}{\kappa};a_{13}=\frac{i(\gamma-1)\omega c_v}{\kappa
R};a_{14}=\frac{c_v\rho_0
T_0}{\kappa}\left(\frac{T'_0}{T_0}-(\gamma-1)\frac{\rho'_0}{\rho_0}\right);\\
a_{21}=1;a_{22}=0;a_{23}=0;a_{24}=0;\\
a_{31}=0;a_{32}=\frac{g\rho_0}{T_0};a_{33}=-\frac{g}{RT_0};a_{34}=i\omega\rho_0;\\
a_{41}=0;a_{42}=-\frac{i\omega}{T_0};a_{43}=-\frac{i\omega}{\rho_0}\left(\frac{p'_0}{p_0g}+\frac{k_x^2}{\omega^2}\right);a_{44}=-\frac{\rho'_0}{\rho_0}.
\end{array}$$ The variables $v_x$ and $\rho$ not belonging to the vector $F$ are expressed through its components from the third and fifth equations of (\[eq7\]): $$\label{eq10}
\begin{array}{l}
v_x=\frac{k_x}{\Omega}f_3,\\
p=\frac{1}{RT_0}f_3-\frac{\rho_0}{T_0}f_2.
\end{array}$$ Eqs. (\[eq3\]), (\[eq8\])-(\[eq10\]) describe wave disturbances for the considered approximation. Also, as in the previous case, these equations are applicable for numerical calculations only for sufficiently large heights
Dissipationless atmosphere approximation, $n=2$ (Case III) {#section2.3}
----------------------------------------------------------
In this case, system of equations (\[eq2\]) takes the following form: $$\label{eq11}
\begin{array}{l}
-i\Omega\rho+\rho_0\left(ik_xv_x+{v_z}^\prime\right)+{\rho^\prime}_0v_z=0, \\
-i\Omega T+(\gamma-1)T_0\left(ik_xv_x+{v_z}^\prime\right)+{T^\prime}_0v_z=0, \\
-i\Omega v_x+ik_x\rho^{-1}_0 p=0,\\
-i\Omega v_z+\rho^{-1}_0 p^\prime+g\rho_0^{-1}\rho=0,\\
p=RT_0\rho+R\rho_0T.
\end{array}$$ We choose the following vector to obtain an equation in the form of (\[eq3\]): $$\label{eq12}
F=(p,v_z).$$ $$\label{eq13}
\begin{array}{l}
a_{11}=\frac{p'_0}{\gamma p_0};
a_{12}=i\omega\rho_0\left(1-\frac{\omega_N^2}{\omega^2}\right);\\
a_{21}=-\frac{i\omega}{\rho_0}\left(\frac{p'_0}{\gamma
p_0g}+\frac{k_x^2}{\omega^2}\right);a_{22}=-\frac{p'_0}{\gamma
p_0}.
\end{array}$$ Here $\omega_N^2=-\frac{g^2}{c_s^2}-\frac{g\rho'_0}{\rho_0}$, $c_s=\sqrt{\gamma RT_0}$ is sound velocity. $T,v_x,\rho$ and $T'$ are expressed through the vector components $F$ (\[eq12\]) as follows: $$\label{eq14}
\begin{array}{l}
T=\frac{\gamma-1}{\gamma R\rho_0}f_1+\frac{T_0\omega_N^2}{i\omega
g}f_2,\\
v_x=\frac{k_x}{\omega}f_1,\\
\rho=\frac{1}{\gamma RT_0}f_1-\frac{\rho_0\omega_N^2}{i\omega
g}f_2,\\
T'=\frac{1}{R\rho_0}f'_1-\frac{T'_0}{\rho_0}\rho-\frac{\rho'_0}{\rho_0}\rho-\frac{T_0}{\rho_0}\rho'.
\end{array}$$ In the last formula, $f'_1$ and $\rho'$ are obtained from Eq. (\[eq3\]), (\[eq12\]), (\[eq13\]). Thus, Eqs. (\[eq3\]), (\[eq12\])-(\[eq14\]) describe wave disturbances in the case of dissipationless atmosphere $(n=2)$.
Weakly dissipative atmosphere approximation, $n=2$ (Case III-a) {#section2.4}
---------------------------------------------------------------
In the region of the atmosphere where dissipation is small, it is possible to use the dissipationless approximation. But it is clear, if we can take dissipation into account, that accuracy of a solution will be higher. It is easy to show that in the case of small dissipation the equations with dissipation can be reduced to a second-order problem, if we exclude from consideration too small-scale waves. For these waves, equations can be obtained from (\[eq7\]) with use of dissipationless relations when deriving of the dissipative terms. In this manner, as in (\[eq3\]), (\[eq13\]) of Case III, we get a set of two first-order differential equations, but it consider small dissipation. In more detail, realization of this method is as follows. From (\[eq7\]), we get the relations: $$\label{eq56}
\begin{array}{l}
p'=a_{11}p+a_{12}v_z+iHR\rho_0s(T''-k_x^2T),\\
v_z'=a_{21}p+a_{22}v_z+\frac{\omega H^2}{T_0}s(T''-k_x^2T)
\end{array}$$ Further for $T$ and $T''$ , we obtain the expressions in the form of linear combinations $p,v_z$ from dissipative equations (\[eq11\]) We have the following from the equation of state: $$\label{eq57}
\begin{array}{l}
T=\frac{1}{R\rho_0}p-\frac{T_0}{\rho_0}\rho,\\
T''=\frac{1}{R\rho_0}p''-\frac{T_0}{\rho_0}\rho''+2\left(\frac{1}{R\rho_0}\right)^\prime
p'-2\left(\frac{T_0}{\rho_0}\right)^\prime
\rho'+\left(\frac{1}{R\rho_0}\right)^{\prime\prime}p-\left(\frac{T_0}{\rho_0}\right)^{\prime\prime}
\rho.
\end{array}$$ We express $\rho,\rho'$ and $\rho''$ from the fourth equation of (\[eq11\]): $$\label{eq58}
\begin{array}{l}
\rho=\frac{i\omega}{g}\rho_0v_z-\frac{1}{g}p'\\
\rho'=\frac{i\omega}{g}\rho_0v'_z+\frac{i\omega}{g}\rho'_0v_z-\frac{1}{g}p'',\\
\rho''=\frac{i\omega}{g}\rho_0v''_z+\frac{2i\omega}{g}\rho'_0v'_z+\frac{i\omega}{g}\rho''_0v_z-\frac{1}{g}p'''.
\end{array}$$ Then $p',p'',p''',v'_z,v''_z$ from (\[eq58\]) are expressed from (\[eq3\]), (\[eq11\]), (\[eq12\]): $$\label{eq59}
\begin{array}{l}
p'=a_{11}p+a_{12}v_z,\\
v_z'=a_{21}p+a_{22}v_z,\\
p''=a_{11}p'+a_{12}v'_z+a'_{11}p+a'_{12}v_z,\\
v_z''=a_{21}p'+a_{22}v'_z+a'_{21}p+a'_{22}v_z,\\
p'''=a_{11}p''+a_{12}v''_z+2a'_{11}p'+2a'_{12}v'_z+a''_{11}p+a''_{12}v_z.
\end{array}$$ Eqs. (\[eq57\])-(\[eq59\]) allow us to reduce system (\[eq56\]) to the normal form (\[eq3\]): $$\label{eq60}
\left(^p_{v_z}\right)^\prime=\tilde{A}\left(^p_{v_z}\right)=\left(^{\tilde{a}_{11}
\tilde{a}_{12}}_{\tilde{a}_{21}
\tilde{a}_{22}}\right)\left(^p_{v_z}\right).$$ The solutions of this system of equations give only ordinary atmospheric waves under small dissipation effect. They do not contain purely “dissipation” (small-scale) solutions intrinsic to the input system of dissipative wave equations with higher order of the differential equations.
Analysis of dissipation influence on the wave propagation in isothermal atmosphere {#section3}
==================================================================================
The model of an isothermal dissipative atmosphere is required for construction DSAS to have the part of DSAS in the isothermal upper atmosphere, besides it allows to see specific properties of wave propagation in the upper atmosphere, with dissipation effect exponentially growing with height. These properties are intrinsic not only to an isothermal model atmosphere, but also to the real upper atmosphere; their analysis is required for correct matching the parts of DSAS in the upper atmosphere and lower atmosphere. The circumstance that it is possible to reduce the wave problem to one differential equation, which is sixth-order in the most general case , is significant for successful analysis of properties of the wave propagation in the case of isothermal dissipative atmosphere.
Consider an isothermal atmosphere with constant coefficients of thermal conductivity and viscosity, which is formally determined in the whole space $z\in[-\infty ,\infty]$: $$\label{eq15}
\begin{array}{l}
p_0(z)=p_0(z_r)e^{-\frac{z-z_{r}}{H}}, \\
\rho_0(z)=\rho_0(z_r)e^{-\frac{z-z_{r}}{H}}, \\
H=\frac{RT_0}{g}, \kappa=const, \nu_1=const, \nu_2=const.
\end{array}$$ Here $H$ is the scale height of the atmosphere; $z_r$ is the reference height with specified undisturbed pressure and density. For the convenience of the wave description, we will exploit a completely dimensionless form of its representation, i.e. coordinates, time, wave parameters, and disturbance function will be represented by corresponding dimensionless values: $$\label{eq16}
\begin{array}{l}
{\bf r}^*\equiv(x^*,y^*,z^*)={\bf r}/H\equiv(x,y,z)/H, t^*=t\sqrt{g/H}, \\
k=k_xH, \sigma=\omega\sqrt{H/g}, \\
n=\rho'/\rho_0, f=p'/p_0, \Theta=T'/T_0, \\
u=v_x/c_s, w=v_y/c_s, v=v_z/c_s.
\end{array}$$ To bring the disturbance equations to dimensionless form, we will use dimensionless expressions of kinematic dissipative values: $$\label{eq17}
\begin{array}{l}
s(z)=\frac{\kappa}{\sigma\gamma c_vH\sqrt{gh}}\rho_0^{-1}, \\
\mu(z)=\frac{\nu_1}{\sigma H\sqrt{gh}}\rho_0^{-1}, \\
q(z)=\frac{\nu_1/3+\nu_2}{\sigma H\sqrt{gh}}\rho_0^{-1}.
\end{array}$$ By applying introduced determinations in Eqs. (\[eq15\])-(\[eq17\]) to system of equations (\[eq1\]), we obtain complete set of equations for disturbances of density, pressure, velocity, and temperature in the form: $$\label{eq18}
\begin{array}{l}
a) \ \ \ \ \Theta=f-n, \\
b) \ \ \ \ -i\sigma n+\Psi-\sqrt{\gamma}v=0, \\
c) \ \ \ \ -i\sigma f-\sqrt{\gamma}v+\gamma\Psi=\sigma\gamma s\Delta^*\Theta, \\
d) \ \ \ \ -i\sigma \sqrt{\gamma}u+ikf=\sqrt{\gamma}\sigma\mu\Delta^* u+ik\sigma q\Psi, \\
e) \ \ \ \ -i\sigma \sqrt{\gamma}v+\dot{f}-f+n=\sqrt{\gamma}\sigma\mu\Delta^* v+\sigma q\dot{\Psi}, \\
f) \ \ \ \ -i\sigma w=\sigma\mu\Delta^* w.
\end{array}$$ Here the dot is the derivative of a function with respect to $z^*$ argument;$\Psi=\sqrt{\gamma}(\dot{v}+iku)$ is the dimensionless divergence of velocity disturbance; $\Delta^*=\frac{d^2}{dz^{*2}}-k^2$is the dimensionless Laplacian. Eq. (\[eq18\]f) describes an independent viscous solution unrelated to the disturbances we are interested in. Thus, from now on we set $w=0$ and will consider the system of five Eqs. (\[eq18\]a)-(\[eq18\]e) with unknowns $(\Theta,n,f,u,v)$. The kinematic dissipative coefficients $s,\mu$ and $q$ are functions which grow exponentially with height. At low heights, where these coefficients may be ignored, set of equations (\[eq18\]) describes ordinary classical acoustic and gravitational oscillations. In the real atmosphere, thermal conductivity exceeds viscosity. Hence with an increase in height thermal-conductivity dissipation does occur first, where the dimensionless function $s$ becomes of order of unity. A corresponding height in the real atmosphere may be determined, using first formula of Eqs. (\[eq17\]), from the condition $$\label{eq19}
\frac{\kappa}{\omega\gamma c_v\left(H(z_c)\right)^2\rho_0(z_c)}=1.$$ Here the height dependence of the scale height of the atmosphere is taken into account. The value of $z_c$ may be found by solving implicit Eq. (\[eq19\]). Reference to Eq. (\[eq19\]) shows that with an increase in frequency of oscillations the critical height should also increase. Next, we will assume for convenience that the point of reference of the dimensionless coordinate $z^*$ corresponds to $z_c$: $$\label{eq20}
z^*=(z-z_c)/H.$$ Accordingly, $p_0(z_r)$ and $\rho_0(z_r)$ in Eq. (\[eq15\]) are determined by values at $z_r=z_c$ and, therefore, are implicit functions of frequency of oscillations too. From Eqs. (\[eq19\]) and (\[eq20\]), expressions for $s,\mu$, and $q$ take the following form: $$\label{eq21}
\begin{array}{l}
s=e^{z^*}, \\
\mu=\frac{\nu_1}{\kappa}c_ve^{z^*}, \\
q=\frac{\nu_1/3+\nu_2}{\kappa}c_ve^{z^*}.
\end{array}$$ System (\[eq18\]a)-(\[eq18\]e) allows reducing to one sixth-order ordinary differential equation with one variable $\Theta$. To derive this equation, we exclude variables in the following sequence. First, we exclude variables $n$ and $f$ from Eqs. (\[eq18\]a), (\[eq18\]b): $$\label{eq22}
n=\frac{1}{i\sigma}(\Psi-\sqrt{\gamma}v), \ \ \
f=\Theta+n=\Theta+\frac{1}{i\sigma}(\Psi-\sqrt{\gamma}v)$$ Using (\[eq22\]), we obtain the following expressions: $$\label{eq23}
\begin{array}{l}
a) \ \ \ \ \Psi=\frac{\sigma}{\gamma-1}(\gamma
s\Delta^*\Theta+i\Theta),\\
b) \ \ \ \ -i\sigma \sqrt{\gamma}u+\frac{k}{\sigma}\sqrt{\gamma}v+\sqrt{\gamma}\sigma\mu\Delta^* u=ik\Theta-ik\left(\sigma q-\frac{1}{i\sigma}\right)\Psi, \\
c) \ \ \ \ -i\sigma \sqrt{\gamma}v+\frac{1}{i\sigma}\sqrt{\gamma}\dot v+\sqrt{\gamma}\sigma\mu\Delta^* v=\dot\Theta-\Theta-\left(\sigma q-\frac{1}{i\sigma}\right)\dot\Psi, \\
d) \ \ \ \ \Psi=\sqrt{\gamma}(\dot{v}+iku), \\
e) \ \ \ \ \theta=\sqrt{\gamma}(\dot u-ikv), \\
f) \ \ \ \ \Delta^* u=(\dot\theta+ik\Psi)/\sqrt{\gamma}, \\
g) \ \ \ \ \Delta^* v=(\dot\Psi+ik\theta)/\sqrt{\gamma}.
\end{array}$$ Here Eq. (\[eq23\]a) results from the subtraction of Eq. (\[eq18\]b) from Eq. (\[eq18\]c) with the use of Eq. (\[eq18\]a); Eqs. (\[eq23\]b) and (\[eq23\]c) result from the substitution of Eq. (\[eq22\]) into Eq. (\[eq18\]d) and Eq. (\[eq18\]e) respectively; Eq. (\[eq23\]d) gives the divergence; Eq. (\[eq23\]e) is a new auxiliary function of current; Eq. (\[eq23\]f) and Eq. (\[eq23\]g) are auxiliary identities evident from the determinations of divergence and current function. Eq. (\[eq23\]a) gives us the divergence through temperature $\Theta$. The current function may also be expressed through $\Theta$ by summing up differentiated Eq. (\[eq23\]c) and Eq. (\[eq23\]b) multiplied by $ik$: $$\label{eq24}
\theta=\frac{\sigma}{k}\frac{1}{1+i\sigma^2\mu}\left\{
\hat{L}\left[\sigma(\mu+q)\Psi-\Theta-\frac{1}{i\sigma}\Psi\right]+i\sigma\Psi\right\},$$ where: $\hat{L}=\Delta^*-\frac{d}{dz^*}$. By differentiating Eq. \[eq23\]b) and subtracting Eq. \[eq23\]c) multiplied by $ik$, we obtain the differential equation expressed through one unknown function $\Theta$: $$\label{eq25}
\left(1-i\hat{L}\mu\right)\theta+k(\mu+q)\Psi-\frac{k}{\sigma}\Theta=0,$$ Or in more detail $$\label{eq26}
\left(1-i\hat{L}\mu\right)\frac{\hat{L}\left[\sigma(\mu+q)\Psi-\Theta-\frac{1}{i\sigma}\Psi\right]+i\sigma\Psi}{1+i\sigma^2\mu}+\frac{k^2}{\sigma}(\mu+q)\Psi-\frac{k^2}{\sigma^2}\Theta=0.$$ Other values are expressed via $\Theta$ and $\Psi$: $$\label{eq27}
\begin{array}{l}
v=\frac{1}{\sqrt\gamma}\frac{\sigma^2}{\sigma^4-k^2}\times \\
\left[\left(1-\frac{d}{dz^*}-\frac{k^2}{\sigma^2}\right)(\Psi+i\sigma\Theta)+i\sigma^2(\mu+q)\left(\frac{d}{dz^*}+\frac{k^2}{\sigma^2}\right)\Psi-k\mu\left(\frac{d}{dz^*}+\sigma^2\right)\theta \right], \\
f=\Theta+\frac{i}{\sigma}(\sqrt{\gamma}v-\Psi), \\
u=\left(\frac{k}{\sigma}f+i\mu\dot{\theta}-k(\mu+q)\Psi\right)\frac{1}{\sqrt\gamma},\\
n=f-\Theta
\end{array}$$ Eq. (\[eq26\]) and Eqs. (\[eq27\]) completely describe the wave disturbance in the chosen approximation. Eq. (\[eq26\]) does not have analytical solutions, but it may be used for analyzing the asymptotic behavior of solutions at large and small $z^*$, or for numerical solution (see [@Rudenko_a; @Rudenko_b]) Unlike the classical dissipationless solution, Eq. (\[eq26\]) allows the solution without an infinite increase in amplitude of relative values of disturbances; i.e. in the whole space, the solution may satisfy the linear approximation [@Rudenko_a].
AGW solution for the heat-conducting isothermal atmosphere {#section3.1}
----------------------------------------------------------
The most interesting possibility of deriving an analytical form of dissipative solutions which describes disturbances of acoustic and gravitational ranges is provided by the approximation of non-viscous heat-conducting atmosphere $(\mu = q = 0 )$. In this case, Eqs. (\[eq26\]) and (\[eq27\]) becomes: $$\label{eq28}
\left[-\hat{L}\left(\Theta+\frac{1}{i\sigma}\Psi\right)+i\sigma\Psi\right]-\frac{k^2}{\sigma^2}\Theta=0;$$ $$\label{eq29}
\begin{array}{l}
\Psi=\frac{\sigma}{\gamma-1}\left(\gamma e^{z^*}\Delta^*\Theta+i\Theta\right), \\
v=\frac{1}{\sqrt\gamma}\frac{\sigma^2}{\sigma^4-k^2}\left(1-\frac{d}{dz^*}-\frac{k^2}{\sigma^2}\right)(\Psi+i\sigma\Theta), \\
f=\Theta+\frac{i}{\sigma}(\sqrt\gamma v-\Psi),\\
u=\frac{k}{\sqrt\gamma\sigma}f, \\
n=f-\Theta.
\end{array}$$
Introducing a new variable $$\label{eq30}
\xi=\exp\left(-z^*+i\pi\frac{3}{2}\right)$$ allows us to represent Eq. (\[eq28\]) in the canonical form of the generalized hypergeometric equation $$\label{eq31}
\left[\xi\prod^2_{j=1}(\delta-a_j+1)-\prod^4_{i=1}(\delta-b_i)\right]\Theta=0,$$ where $\delta=\xi d/d\xi$, $a_{1,2}=\frac{1}{2}\pm iq$, $b_{1,2}=\pm k$, $b_{3,4}=1/2\pm\alpha$ $$\label{eq32}
\begin{array}{l}
q=\sqrt{-\frac{1}{4}+\frac{\gamma-1}{\gamma}\frac{k^2}{\sigma^2}+\frac{\sigma^2}{\gamma}-k^2}, \\
\alpha=\sqrt{\frac{1}{4}+k^2-\sigma^2}.
\end{array}$$ Eq. (\[eq31\]) has two singular points: $\xi=0$ (the regular singular point, $z^*=+\infty$) and $\xi=\infty$ (the irregular singular point, $z^*=-\infty$)). The fundamental system of solutions of Eq. (\[eq31\]) may be expressed by four linearly independent generalized Meijer functions ([@Luke]): $$\label{eq33}
\begin{array}{l}
a) \ \ \ \ \ \Theta_1=G^{4,1}_{2,4}\left(\xi e^{-i\pi}\left|^{a_1,a_2}_{b_1,b_2,b_3,b_4}\right.\right),\\
b) \ \ \ \ \ \Theta_2=G^{4,1}_{2,4}\left(\xi e^{-i\pi}\left|^{a_2,a_1}_{b_1,b_2,b_3,b_4}\right.\right),\\
c) \ \ \ \ \ \Theta_3=G^{4,0}_{2,4}\left(\xi e^{-2i\pi}\left|^{a_1,a_2}_{b_1,b_2,b_3,b_4}\right.\right), \\
d) \ \ \ \ \ \Theta_4=G^{4,0}_{2,4}\left(\xi\left|^{a_1,a_2}_{b_1,b_2,b_3,b_4}\right.\right).
\end{array}$$ Desired meaningful solution of Eq. (\[eq28\]) may be found with use of the known properties of asymptotic behaviors of $\Theta_i$-functions near two singular points in Eq. (\[eq31\]). Near the irregular point $\xi=\infty$ ($z^*=-\infty$), we have: $$\label{eq34}
\begin{array}{l}
a,b) \ \ \ \Theta_{1,2}(\xi)\sim\Theta_{1,2}^{\infty}(\xi)=p_{1,2}\cdot e^{(1/2\mp iq)z^*}=\\
\ \ \ \frac{\Gamma\left(\frac{1}{2}\mp iq+k\right)\Gamma\left(\frac{1}{2}\mp iq-k\right)\Gamma(1\mp iq+\alpha)\Gamma(1\mp iq-\alpha)}
{\Gamma(1\mp 2iq)}e^{\mp\frac{\pi q}{2}-i\frac{\pi}{4}}\cdot e^{(1/2\mp iq)z^*},\\
\\
c,d) \ \ \ \Theta_{3,4}(\xi)\sim\Theta_{3,4}^{\infty}(\xi)=\pi^{1/2}e^{-i\frac{\pi}{8}(1\mp 2)}\cdot e^{z^*/4}e^{\mp\sqrt{2}(1-i)e^{-z^*/2}}.
\end{array}$$ From Eqs. (\[eq34\]a), (\[eq34\]b) follows that, at real $q$, asymptotics of $\Theta_1$ and $\Theta_2$ correspond to two different classical dissipationless waves. Here we will assume that $\Theta_1$ corresponds to an upward propagating wave (from the Earth); $\Theta_2$, to a downward propagating wave (towards the Earth.) Then, for $\sigma$ and $k$ corresponding to acoustic waves, we set $q < 0$; for those corresponding to internal gravity waves, $q> 0$. Asymptotics of $\Theta_3$ and $\Theta_4$ specified by Eqs. (\[eq34\]c), (\[eq34\]d) correspond to dissipative oscillations. One of these asymptotics is function of extremely rapid decrease and the other is function of extremely rapid increase. Decreasing $\Theta_3$ leaves the physical scene very quickly when $z^*$ decreases; and a coefficient of the solution with $\Theta_4$ asymptotic behavior must be chosen equal to $0$.
The behavior of the solutions $\Theta_1$, $\Theta_2$, and $\Theta_3$ nearby the regular point $\xi =0(z^*=+\infty )$ is represented by the following expressions: $$\label{eq35}
\Theta_i\sim\Theta_i^0=\sum^4_{j=1}t_{ij}e^{-b_jz^*} \ \ \
(i=1,2,3).$$ Expressions for $t_{ij}$ are derived from known asymptotics of the Meijer G-function at the regular singular point ([@Luke]). Their explicit expressions are listed in \[A\].
When constructing the meaningful solution, we will assume that there should not be upward growing asymptotic terms $\sim
e^{-b_2z^*}$ and $\sim e^{-b_4z^*}$ (from the Earth). Besides, we set the incident wave amplitude $= 1$. Accordingly, the desired solution is found in the following form $$\label{eq36}
\Theta(\xi)=p_1^{-1}\Theta_1(\xi)+\alpha_2\Theta_2(\xi)+\alpha_3\Theta_3(\xi),$$ where coefficients $\alpha_2$ and $\alpha_3$ are chosen to satisfy the condition of elimination of growing asymptotics near the regular point of Eq. (\[eq31\]): $$\label{eq37}
\begin{array}{l}
p_1^{-1}t_{12}+\alpha_2t_{22}+\alpha_3t_{32}=0 ,\\
p_1^{-1}t_{13}+\alpha_2t_{23}+\alpha_3t_{33}=0 .
\end{array}$$ Solving (\[eq37\]) yields: $$\label{eq38}
\begin{array}{l}
a) \ \ \ \ \ \alpha_2=-p_1^{-1}e^{-2\pi q}\frac{\sin[\pi(\alpha-iq)]\cos[\pi(k-iq)]}{\sin[\pi(\alpha+iq)]\cos[\pi(k+iq)]},\\
b) \ \ \ \ \ \alpha_3=2\pi p_1^{-1}\frac{e^{-\pi q}}{e^{i2\pi\alpha}+e^{i2\pi k}}\left\{\frac{\sin[\pi(\alpha-iq)]}{\sin[\pi(\alpha+iq)]}- \frac{\cos[\pi(k-iq)]}{\cos[\pi(k+iq)]}\right\}.
\end{array}$$ Eqs. (\[eq36\]) and (\[eq38\]a), ((\[eq38\]b) give the analytical expression of the desired meaningful solution. For $|\xi|<1$ ($z^*>0$), solution of Eq. (\[eq36\]) can be expressed through generalized hypergeometric functions $_mF_n$ which, in the region of such values of arguments, are represented by simple convergent power series, suitable for the numerical calculation. Such a representation of solution is obtained from the standard representation of the Meijer $G$-function. $$\label{eq39}
\begin{array}{l}
G^{m,n}_{p,q}\left(y\left|^{a_p}_{b_q}\right.\right)=\sum^m_{h=1}\frac{\prod^m_{j=1}\Gamma(b_j-b_h)^*\prod^n_{j=1}\Gamma(1-a_j-b_h)}{\prod^q_{j=m+1}\Gamma(1-b_h-b_j)\prod^n_{j=n+1}\Gamma(a_j-b_h)}y^{b_h}\times\\
\ \ \ _pF_{q-1}\left(\left.^{1+b_h-a_p}_{1+b_h-b_q^*}\right|(-1)^{p-m-n}y\right).
\end{array}$$ Here (\*) implies that a term with an index equal to $h$ is omitted. With Eq. (\[eq39\]), after the rather lengthy calculations, we can reduce solution of Eq. (\[eq36\]) to the following form: $$\label{eq40}
\begin{array}{l}
\Theta(z^*>0)=\beta_0\beta_1 e^{-kz^*}\times _2F_3\left(\left.^{\frac{1}{2}+k-iq,\frac{1}{2}+k+iq}_{1+2k,\frac{1}{2}+k+\alpha,\frac{1}{2}+k-\alpha}\right|-ie^{-z^*}\right)+ \\
\ \ \ \beta_0\beta_2 e^{-(1/2+\alpha)z^*}\times _2F_3\left(\left.^{1+\alpha-iq,1+\alpha+iq}_{\frac{3}{2}+\alpha-k,\frac{3}{2}+\alpha+k,1+2\alpha}\right|-ie^{-z^*}\right),
\end{array}$$ where\
$\beta_0=\frac{\Gamma\left(\frac{1}{2}+iq+k\right)\Gamma(iq+\alpha)}{\Gamma(2iq)\Gamma\left(\frac{1}{2}+\alpha+k\right)}e^{-i\frac{\pi}{4}-\pi\frac{q}{2}}$, $\beta_1=\frac{\Gamma\left(\frac{1}{2}+\alpha-k\right)\Gamma\left(\frac{1}{2}+iq+k\right)}{\Gamma(1+2k)\Gamma(1-iq+\alpha)}e^{-i\frac{\pi}{2}k}$,\
$\beta_2=\frac{\Gamma\left(-\frac{1}{2}-\alpha+k\right)\Gamma(1+iq+\alpha)}{\left(\frac{1}{2}+\alpha+k\right)\Gamma(1+2\alpha)\Gamma\left(\frac{1}{2}-iq+k\right)}e^{-i\frac{\pi}{2}\left(\frac{1}{2}+\alpha\right)}$.\
Given $z^*\to+\infty$, the generalized hypergeometric functions $F$ in Eq. (\[eq40\]) tend to unity, and the solution $\Theta$ takes a simple asymptotic form with two exponentially decreasing terms.
The found solution describes the incidence of internal gravity or acoustic wave with an arbitrary inclination to the dissipative region $z^*>0$, its reflection from this region, and its penetration to this region with the transformation of it into a dissipative form. The complex coefficient of reflection can be expressed by a simple analytical formula: $$\label{eq41}
K=\alpha_2p_2.$$ In [@Rudenko_a], the reflection coefficient module is shown to take a value of order of unity, if the typical vertical scale of incident wave $q^{-1}\gtrsim 1$. Otherwise, the contribution of the reflected wave exponentially decreases with decreasing vertical scale. This behavior of reflection is similar to the ordinary wave reflection from an irregularity of a medium. In our case, the scale of the irregularity is the scale height of the atmosphere $H$.
It is worth introducing an index characterizing the value of total vertical dissipation index in the region ($z^*<0$): $$\label{eq42}
\eta=|\Theta(0)|.$$ This value can be calculated, using the solution of Eq. (\[eq40\]). In what follows, we will show that the behavior of $\eta$, depending on a vertical wave scale, is similar to the behavior of reflection coefficient. The value $\eta$ also characterizes the capability of a significant part of a wave disturbance to penetrate to the upper region $z^*>0$. It is reasonable that, if $\eta$ is negligible, further wave disturbance propagation may be neglected too.
Classification of AGW by their dissipative properties {#section3.2}
-----------------------------------------------------
The understanding of certain wave effects at heights of the upper atmosphere requires knowing possible regimes of wave propagation which are associated with dissipation effect. Of particular interest first is to determine the typical height from which dissipation assumes a dominant control over the wave process, and the second is the type of wave propagation, depending on the period and spatial scale of disturbance (whether it is an approximately dissipationless propagation or propagation with dominant dissipation.)
### Classification of AGW by the dissipation critical height {#section3.2.1}
The dependence of $z_c$ on oscillation frequency enables a convenient classification of waves only by their oscillation periods. To make such a classification, we will exploit the following model of medium:\
– The vertical distribution of temperature $T_0(z)$ according to the NRLMSISE-2000 distribution with geographic coordinates of Irkutsk for the local noon of winter opposition;\
– $p_0(z)=p_0(0)\exp\left[-\frac{g}{R}\int_0^z\frac{1}{T_0(z')}dz'\right], \ \ \ p_0(0)=1.01 \rm{Pa}$;\
– $\rho_0(z)=\rho_0(0)\exp\left[-\frac{g}{R}\int_0^z\frac{1}{T_0(z')}dz'\right], \ \ \ \rho_0(0)=287.0 \rm{g/m^3}$;\
– $g=9.807\rm{m/s^2}$, $R=287\rm{J/(kg\cdot K)}$, $\kappa=0.026\rm{J/(K\cdot m\cdot s)}$,\
$c_v=716.72\rm{J/(kg\cdot K)}$.\
Solving Eq. (\[eq19\]) for $z_c$ (the plot in Figure \[fig1\]) provides the following useful information: a) the height of transformation of waves of the selected period into dissipative oscillations; b) the height above which waves of the selected period with vertical scales much less than the height of the atmosphere should be heavily suppressed by dissipation (in fact, such waves should not appear at these heights); c) the height limiting the applicability of the WKB approximation for the wave of the selected period.
![The dashed curve shows the height dependence of temperature in the selected model; the thick solid curve indicates the dependence of $z_c$ on the period of oscillations corresponding to IGWs incident from the lower atmosphere; the solid curve indicates the dependence of $z_c$ on the period of oscillations corresponding to acoustic waves incident from the lower atmosphere.[]{data-label="fig1"}](fig1.eps "fig:"){width="1\linewidth"}\
### Classification of AGW by their total vertical dissipation {#section3.2.2}
An important characteristic of a wave with certain wave parameters $\sigma$ and $k$ is the ratio of the amplitude of the wave solution at $z_c$ to the amplitude of the wave incident from minus infinity $\eta$, Eq. (\[eq42\]). This ratio gives useful information about the ability of this wave to have physically meaningful values at heights of order of and higher than $z_c$. A direct calculation of the wave solution for the height region $z<z_c$ from general expression of Eq. (\[eq36\]) is unlikely technically feasible. The first term of asymptotic expansion Eq. (\[eq34\]a) describes the solution only at a sufficiently large distance from $z_c$ and does not reflect the real behavior of the amplitude under the influence of dissipation because this term has the form of ordinary propagating dissipationless wave. Only beginning from $z_c$, we can exactly calculate the solution by power expansion Eq. (\[eq40\]). On the other hand, the characteristic we are interested in may be compared with its values obtained in the WKB approximation for Eq. (\[eq31\]). The use of the WKB approximation also enables us to obtain the qualitative behavior of the solution in the region $z\lesssim
z_c$. For simplicity, consider the case when a WKB estimated value of $\eta$ may be expressed in an analytical form. For this purpose, rewrite Eq. (\[eq31\]) for a new variable m. For this purpose, rewrite Eq. (\[eq14\]) for a new variable $\Theta_{ref}=\Theta e^{-\frac{1}{2}z^*}$ : $$\label{eq43}
\left\{\frac{d^2}{{dz^*}^2}+q^2+\frac{e^{z^*}}{i}\left(\frac{d^2}{{dz^*}^2}-\frac{1}{4}-k^2+\sigma^2
\right)\left[\left(\frac{d}{dz^*}+\frac{1}{2}\right)^2-k^2\right]\right\}\Theta_{ref}=0.$$ In the WKB approximation, Eq. (\[eq43\]) yields a quartic algebraic equation for a complex value of a dimensionless vertical wave number $k_z^*$: $$\label{eq44}
-{k_z^*}^2+q^2+\frac{e^{z^*}}{i}\left(-{k_z^*}^2-\frac{1}{4}-k^2+\sigma^2
\right)\left[\left(ik_z^*+\frac{1}{2}\right)^2-k^2\right]=0.$$ Note that Eq. (\[eq44\]) can be precisely obtained from similar dispersion Eq. (19) from [@Vadas] by excluding terms comprising first and second viscosities. For simplicity, consider IGW continuum waves with low frequencies $\sigma$ and $k_z^*\gg
1$. In this case, Equation $$\label{eq45}
-{k_z^*}^2+q^2+\frac{e^{z^*}}{i}\left({k_z^*}^2+k^2\right)^2=0$$ gives four simple roots, one of which corresponding to upward propagating IGW wave (from the Earth) at $z^*\rightarrow -\infty$ will be interesting for us: $$\label{eq46}
k_z^*=-\sqrt{\frac{2K^{2}}{\left( \sqrt{4ie^{z^*}K^{2}+1}+1\right)
}-k^{2}},$$ Where $K=\sqrt{q^2+k^2}$ is the full vector of a dissipationless wave. The total vertical dissipation index at $z^*$ may be presented in the form: $$\label{eq47}
\eta_{WKB}=\sqrt{\frac{q}{\left|k_z^*(z^*)\right|}}e^{-{\rm
Im}\left[\int^{z^*}_{\infty}k_z^*(z^{*'})dz^{*'}\right]}=\sqrt{\frac{q}{\left|k_z^*(z^*)\right|}}e^{-\Gamma}.$$ Expression of Eq. (\[eq47\]) may be used for estimating the total vertical dissipation index in the real atmosphere. In the isothermal atmosphere approximation, the integral in the exponent may be presented in the analytical form: $$\label{eq48}
\Gamma={\rm Re}\left[i\sqrt{2}K\left(2in{\rm
Ln}\frac{a+in}{a-in}+b{\rm
Ln}\frac{b-a}{b+a}-2a-i\frac{\pi}{2\sqrt{2}}\right)\right],$$ where $n=\frac{k}{\sqrt{2}K}$; $a=\sqrt{\frac{1}{1+\sqrt{1+4iKe^{z^*}}}-n^2}$; $b=\sqrt{\frac{1}{2}-n^2}$. In addition to the value determined by Formula of Eq. (\[eq47\]), the value of the total vertical dissipation index versus the disturbance amplitude at the given lower height $z_n^*$ may also be useful: $$\label{eq49}
\eta_{WKB}(z^*,z_n^*)=\eta_{WKB}(z^*)/\eta_{WKB}(z_n^*).$$ The total vertical dissipation index $\eta_{WKB}(0)$ expressed via Eq. (\[eq48\]) is comparable with similar value in Eq.(\[eq42\]) obtained from the analytical wave solution (Figure \[fig2\]); as function of $z^*$, this value qualitatively describes the amplitude of the wave solution in the isothermal atmosphere below $z_c$ ($z^*<0$) (Figure \[fig3\]).
![The solid line is the dependence of $\eta$ on vertical length of the incident wave; the dashed line is the same for $\eta_{WKB}(0)$; the dash-dotted line is the dependence of horizontal wavelength on vertical length of the incident wave for the selected wave solutions (the left axis). The frequency $\sigma$ is constant (0.14 of the Vaisala-Brent frequency $\sigma_{VBF}$).[]{data-label="fig2"}](fig2.eps "fig:"){width="1\linewidth"}\
![The height dependence of $\eta_{WKB}$ and $\eta_{WKB,rel}$ at given wave parameters.[]{data-label="fig3"}](fig3.eps "fig:"){width="1\linewidth"}\
Figure \[fig2\] demonstrates convergence of the total vertical dissipation index dependences at decreasing vertical wave scales. This comparison shows propriety of the analytical characteristic of dissipation, $\eta$. The fact that the introduced characteristic of dissipation is determined without limitations for any wave parameters warrants its use as a universal wave characteristic. On the other hand, the obtained convergence of $\eta$ and $\eta_{WKB} $ in the large-scales region may justify using the WKB approximation for real vertically long-wave disturbances to obtain amplitude estimates, though formally this approximation is not applicable.
Figure \[fig3\] illustrates the quite predictable typical behavior of the vertical distribution of wave amplitudes below $z_c$. It is obvious that the dissipation effect begins several scales of the height of the atmosphere to the critical height. The additional characteristic $\eta_{WKB,rel}=e^{\frac{1}{2}z^*}\eta_{WKB}$ demonstrates the required dissipation-provoked suppression of the exponential growth of the amplitude of relative oscillations. The suppression of the exponential growth provides, in turn, a possibility of satisfying the linearity of disturbance at all heights.
![The levels of $\eta$ in the plane of $k$ and $\sigma$.[]{data-label="fig4"}](DSAS_Ist_4.eps "fig:"){width="1\linewidth"}\
![The levels of $q$ in the plane of $k$ and $\sigma$.[]{data-label="fig5"}](DSAS_Ist_5.eps "fig:"){width="1\linewidth"}\
A general picture of distribution of the total vertical dissipation index for internal gravity and acoustic waves is given in Figure \[fig4\], which shows levels of constant values $\eta$ in the plane of wave parameters $(k,\sigma )$. For convenience of comparison, Figure \[fig5\] presents levels of constant values of the vertical wavenumber $q$ in the same plane. Figure \[fig4\] and Figure \[fig5\] classify waves of weak, moderate, and strong total vertical dissipation according to $k$, $\sigma$ and allow us to estimate the possibility of their penetration to heights above $z_c$.
Construction of DSAS for a heat-conducting non-isothermal atmosphere {#section4}
====================================================================
The analysis in the preceding section allows us to start constructing DSAS. It is clear that our DSAS is bound to be a certain combination of solutions having non-growing asymptotics $\sim e^{-kz^*}$ and $\sim e^{-(\frac{1}{2}+\alpha)z^*}$ in the upper atmosphere, satisfying an isothermal condition. The wave solution found by us (Eq. (\[eq40\]) for an everywhere isothermal atmosphere is a sum of two terms being a subset of fundamental solutions of Eq. (\[eq28\]) in their turn. The sum of these solutions does not contain the asymptotics (\[eq34\]d) growing extremely rapidly downward (towards the Earth) which should not be present in a physical solution. Further we will use the notations $..^k$ and $..^\alpha$ for any component of disturbance with a corresponding asymptotics. Then we can write (\[eq40\]) in the following form: $$\label{eq50}
\Theta=\Theta^{(k)}+\Theta^{(\alpha)},$$ $$\label{eq51}
\begin{array}{l}
\Theta^{(k)}=\beta_0\beta_1 e^{-kz^*}\times _2F_3\left(\left.^{\frac{1}{2}+k-iq,\frac{1}{2}+k+iq}_{1+2k,\frac{1}{2}+k+\alpha,\frac{1}{2}+k-\alpha}\right|-ie^{-z^*}\right), \\
\Theta^{(\alpha)}=\beta_0\beta_2 e^{-(1/2+\alpha)z^*}\times _2F_3\left(\left.^{1+\alpha-iq,1+\alpha+iq}_{\frac{3}{2}+\alpha-k,\frac{3}{2}+\alpha+k,1+2\alpha}\right|-ie^{-z^*}\right),
\end{array}$$ Our DSAS should be also presented by a sum of solutions coinciding in the upper isothermal part $(I)$ of the atmosphere with analytical solutions $..^k$ and $..^\alpha$ (Eqs. (\[eq51\])) We will also use $..^k$ and $..^\alpha$ to denote such solutions, consisting of isothermal parts and their continuations into the middle non-isothermal part $(II)$. However, it is clear that since the downward continuations of the asymptotic isothermal solutions are different in the everywhere isothermal atmosphere and in the real atmosphere, the functions $..^k$ and $..^\alpha$ should be included in DSAS for the real atmosphere in a combination other than (\[eq50\]). In the case of the real atmosphere, when a necessary combination of solutions $..^k$ and $..^\alpha$ is unknown, we have to proceed from the consideration that the right combination of solutions $..^k$ and $..^\alpha$ at heights corresponding to small values of the parameter $s<<1$, should yield, according to Formulas (\[eq3\]), (\[eq8\])-(\[eq10\]) of Case II, a solution close to a solution of “dissipationless” problem ((\[eq3\]), (\[eq12\])-(\[eq14\]) of Case III . This condition will be the condition of matching solutions in the upper dissipative atmosphere and lower small-dissipation atmosphere. In its physical sense, this condition is equivalent to $\Theta_4$ elimination in the case of the everywhere isothermal atmosphere. Really, to satisfy the matching condition, the right combination of solutions $..^k$ and $..^\alpha$ for the real atmosphere should not contain the rapidly growing downward solutions, like ((\[eq34\]d), as is the case of the solution for an everywhere isothermal atmosphere ((\[eq40\]). As for the part of the solution for the real atmosphere that is fast decreasing downward, like ((\[eq34\]s), it is negligibly small when $s<<1$, with the natural assumption that the contribution of this part of the solution is equal to the rest of the solution at the heights of $s\sim 1$. Having the right combination of solutions $..^k$ and $..^\alpha$, in the upper height range limited by a selected height with the small parameter $s$, we get a solution satisfying both the upper boundary condition, since it is a combination of $..^k$ and $..^\alpha$, and the condition of matching solutions in the upper dissipative atmosphere and lower small-dissipation atmosphere, due to the selection the coefficients of the combination of $..^k$ and $..^\alpha$. Then, using the $p$ and $v_z$ values of the solution for the region $s>s_t$ at a selected height $s_t$, corresponding to a small value of the parameter $s$,as initial conditions for continuation the solution downward, we can find a DSAS in the lower atmosphere up to the Earth’s surface.
Thus, we single out three height ranges: $$\label{eq52}
\begin{array}{l}
R^I:z\in [z_I,\infty] \\
R^{II}:z\in [z_t,z_I] \\
R^{III}:z\in [0,z_t]
\end{array}$$ There $z_I$ is the minimum height of the isothermal range in a considered model of the atmosphere; $z_t$, the height corresponding to the chosen small parameter $s$ $(s=s_t<<1)$. In the region $II$, we solve the Cauchy problem for the system of equations of Case II. In this connection, we are guided by the reversibility of the numerical solution of the Cauchy problem of Case II in the height range $II$, when choosing $s_t$. It is evident that numerical instability associated with presence of rapid asymptotic increase of the solution (similar to that in the isothermal case (\[eq34\]d) under too small values of the parameter $s$ inevitably compromises the reversibility of the Cauchy problem. The numerical research conducted by us has shown that $0.05$ was the optimal value for $s_t$ satisfying the condition of reversibility. Lower this threshold, the numerical errors manifest, rapidly growing to infiniteness with the further reduction of $s_t$.
In the region $I$, we calculate two solutions $..^k$ and $..^\alpha$ by analytical Formulas (\[eq51\]), (\[eq29\]). Using analytical expressions of the vectors $F$ for the solutions $..^k$ and $..^\alpha$ at the height of $z_I$ as boundary conditions of the wave problem of Case II, accordingly, we solve two Cauchy problems numerically in the region $II$. After the solutions $..^k$ and $..^\alpha$ are obtained in the region $I\cup II$, we start searching their right combinations. Without loss of generality, we present the desired combination in the form of $..^{(k)}+W..^{(\alpha)}$, where $W$ is the desired value. In accordance with our assumption, the values $T,T^\prime$ should satisfy the “small-dissipation” or even “dissipationless” equations of Case III at the height $z_t$ with good accuracy. Designating these values as $T_{III}=T_{III}(p,v_z)$, $T^\prime_{III}=T^\prime_{III}(p,v_z)$ we can introduce a vector $$\label{eq53}
\left(\begin{array}{c}
H(T^{(k)^\prime}-T_{III}^{(k)^\prime} )+WH(T^{(\alpha)^\prime}-T_{III}^{(\alpha)^\prime} )\\
(T^{(k)}-T_{III}^{(k)} )+WH(T^{(\alpha)}-T_{III}^{(\alpha)} )
\end{array}\right)=
\left(\begin{array}{c}
a_1+Wa_2 \\
b_1+Wb_2
\end{array}\right).$$ Minimization of the vector norm ((\[eq53\]) as a positive definite complex-valued bilinear form $$\label{eq54}
(a_1+Wa_2)\overline{(a_1+Wa_2)}+(b_1+Wb_2)\overline{(b_1+Wb_2)}$$ gives the following formula for the complex coefficient $W$: $$\label{eq55}
W=\frac{a_1 \overline{a_2}+b_1 \overline{b_2}}{a_2
\overline{a_2}+b_2 \overline{b_2}}.$$ The line denotes a complex conjugate value. Formulas (\[eq3\]), (\[eq14\]), (\[eq53\]), and (\[eq55\]) give the desired coefficient $W$ through the values of $(p,v_z)^{(k),(\alpha)}$ at the height $z_t$. If, obviously, the minimum value of the norm (\[eq53\]) is close to zero, we have smallness of the jumps of all the components of the solution defined by Eqs. (\[eq14\]) The calculations have showed, predictably, that matching in accordance with our procedure is provided with precision of the order $s_t$. Thus, having defined $W$, we found the desire solution in the region $I\cup II$.
We finally complete the DSAS solving the Cauchy problem for the system of the form (\[eq60\]) in the lower region $III$ ( with use of the dissipationless or small-dissipation approximations). The role of initial values for the solution in the region $III$ is played by values of the solution for the region $I\cup II$ taking at its lower boundary.
Testing of DSAS code on everywhere isothermal atmosphere model {#section5}
==============================================================
We carried out the test calculations of DSAS for the model of an everywhere isothermal atmosphere, using a code developed exactly in accordance with the above described DSAS method for a non-isothermal atmosphere. The dimensionless wave parameters $\sigma=0.0616$, $k=0.1289$ are chosen. If a typical atmospheric height scale is taken $H=28 \ km$, these parameters will correspond to an oscillation with the period $T=90.84 \ min$ and horizontal wavelength $\lambda_{gor}=2\pi/k_x=1365 \ km$.
Test A: {#test-a .unnumbered}
-------
Figure 6 shows all wave component obtain numerically for the model of an everywhere isothermal atmosphere. We use wave components notations in accordance with the notations in Section \[section3\].
![The example of a numerical DSAS for an isothermal atmosphere.[]{data-label="fig6"}](DSAS_Ist_6.eps "fig:"){width="1\linewidth"}\
The frames $(a)$ and $(b)$ of Figure \[fig6\] show height dependences the modules of the wave component in the normal and logarithmic scales, accordingly. The frame $(c)$ of Figure \[fig6\] represents phases of the same components. In this case, we chose the height of transfer from an analytical solution to a numerical one equal to the critical height $z_I^*=z_c^*$. Any choice of $z_I^*>0$ leads to one and the same result. The parameters $s_t$ and $z_t^*$ height corresponding to it are given in the right lower quadrant in Figure \[fig6\]. Solving a Cauchy problem in the reverse order from the height $z_t^*$ of Case II, all the height dependences are reproduced with very good acuracy in the region $I\cup II$. We do not give results of these tests because they coincide with the results of Figure \[fig6\]. Figure \[fig6\] clearly convinces of sufficient smoothness of the wave solution in all its components. The measures of the discontinuity of the wave components $\Theta$ and $n$ at the height $z_t^*$ are displayed in the right lower quadrant in Figure \[fig6\]. The notation $\delta_{[x]}$ denotes a value $$\label{eq61}
\delta_{[x]}=2\frac{|X(+z_t^*)-X(-z_t^*)|}{|X(+z_t^*)+X(-z_t^*)|}.$$ As we expected, the values of these quantities are small and have the order of $s_t$.
Test B: {#test-b .unnumbered}
-------
We carried out a test based on the fact that the analytical solution of Eq. (\[eq40\]) gives us a ready solution in the region $I$. We used this solution for setting the Cauchy problem in region $II$. We do not give a corresponding DSAS because it is visually identical with what was given in Figure \[fig6\]. But in this case, we got some better measures of the discontinuity of a solution: $\delta_{[\Theta]}=0.0272$; $\delta_{[n]}=0.0264$.
Test C: {#test-c .unnumbered}
-------
In the region $II\cup III$ we do not have the possibility of calculating a wave solution in an analytical form because of the difficulties caused by formal divergence of the series of the hypergeometric functions in Eq. (\[eq40\]). We can use only the asymptotics of the analytical solution at $z^*\to-\infty$ for tests. Figure \[fig7\] presents the comparison results of our DSAS and the asymptotics of the analytical solution. We chose $\texttt{Re}\Theta_{ref}=\texttt{Re}\left(\Theta
e^{-\frac{1}{2}z^*}\right)$ as a value to be tested (solid line).
![The analysis of a numerical DSAS in the lower asymptotic region.[]{data-label="fig7"}](DSAS_Ist_7.eps "fig:"){width="1\linewidth"}\
For the asymptotics of the analytical solution, $\texttt{Re}\Theta_{ref}$ is derived from Formulas (\[eq34\]a,b), (\[eq36\]) and (\[eq41\]): $\texttt{Re}\Theta_{ref}=\texttt{Re}(e^{-iqz^*}+Ke^{iqz^*})$ (a dashed line.) In our case, the complex coefficient $K$ is sufficiently small, $K=-0.0055-i0.0439$.
Especially for comparison, we made calculations both taking into account small dissipation in $III$ and without taking it into consideration in this region. Furthermore, we, also for comparison, used two values of $s_t$. The results are shown in different frames of Figure \[fig7\]. The calculations for frames $(a)$, $(c)$, and $(b)$, $(d)$ are distinguished by the fact that $(a)$, $(c)$ in the region $III$ take small dissipation into account in accordance with (\[eq56\])-(\[eq60\]); $(b)$, $(d)$, the dissipationless approximation (\[eq3\]), (\[eq12\])-(\[eq14\]). Frames $(a)$ and $(b)$ of Figure \[fig7\] give results for the cases with the least value $s_t$, $s_t=0.05$; frames $(c)$ and $(d)$, with a large value of the parameter $s_t$, $s_t=0.2$. In all cases, we clearly see that the numerical DSAS coincides with the analytical one up phase. Comparing $(a)$ and $(b)$, one can find only hardly noticeable advantage of result $(a)$. We can see more clearly a positive effect of taking small dissipation into account in region $III$ in Frame $(c)$ in comparison to $(d)$. The solution discontinuity indices most clearly show advantage of taking small dissipation into account:\
the frame $(a)$ of Figure \[fig7\] – $\delta_{[\Theta]}=0.0272,
\ \delta_{[n]}=0.0264$;\
the frame $(b)$ of Figure \[fig7\] – $\delta_{[\Theta]}=0.1245,
\ \delta_{[n]}=0.1200$;\
the frame $(c)$ of Figure \[fig7\] – $\delta_{[\Theta]}=0.3760,
\ \delta_{[n]}=0.3400$;\
the frame $(d)$ of Figure \[fig7\] – $\delta_{[\Theta]}=0.4792,
\ \delta_{[n]}=0.4408$.\
The last equalities convince us of appropriateness of our small dissipation correction even at the least allowed values of the parameter $s_t$ and sufficiency of $0.05$ value for $s_t$ as well.
DSAS for a non-isothermal atmosphere {#section6}
====================================
We used the model of the atmosphere introduced in Section \[section3.2.1\] with the height distribution of undisturbed temperature shown in Figure \[fig1\]. The calculation is carried out for the period $T=90.84 \ min$ and horizontal wavelength $\lambda_{hor}=1365 \ km$. The selected wave parameters $T$ and $\lambda_{hor}$ correspond to the dimensionless $\sigma$ and $k$ used in tests in the isothermal model. The following condition is used to norm the solution: $\texttt{Max}(\sqrt{v_x^2+v_z^2})=50 \
m/sec$. The calculation results are shown in Figure \[fig8\]. They clearly convince of sufficient smoothness of DSAS in all its components at the height of $z_t$. The discontinuity indices $\delta_{[\Theta]}=0.129, \ \delta_{[n]}=0.091$, approximately $5$ times higher than in an isothermal case, but remain sufficiently small. In the upper atmosphere above $z_c$, height dependences of the solution components are close in their nature to dependences of the isothermal model. As in the case of the isothermal model, we compare the received solution to the solution of the dissipationless problem. To do so, we solve the Cauchy problem for Eqs. (\[eq3\]), (\[eq12\]), (\[eq13\]) in the reverse direction from the Earth to the height of $z_c$. We do not show the received solution due to its too small graphical difference from the solution in Figure \[fig8\].
![The example of a DSAS for a non-isothermal atmosphere.[]{data-label="fig8"}](DSAS_P_I_8.eps "fig:"){width="1\linewidth"}\
Green’s matrix with DSAS (extended) {#section7}
===================================
As it is well known, the problem on evolution of disturbance from some source can be solved without consideration of dissipation, using the $2\times 2$ Green’s matrix. However, the dissipationless solution has the disadvantage that it cannot adequately reflect the height structure in the upper atmospheric layers; therefore, one should use a higher order problem than the second one there. But one can achieve the goal of a height-structure description in the upper atmosphere and retain the formalism of the second-order problem to a significant extent at the same time. It is possible to do this in that case when, as it mainly occurs in the atmosphere, a source of disturbance is not too high, so that it does not enter the region of strong dissipation. For the problem on evolution of disturbance from some source with consideration of dissipation we will introduce the $2\times 2$ Green’s modified matrix, which we will call extended Green’s matrix. Let us start with the Green’s matrix for the weakly dissipative problem. The inhomogeneous weakly dissipative problem can be written in the form $$\label{eq60n}
\left(\begin{array}{c} p \\
v_z
\end{array}\right)^\prime=
\left(\begin{array}{cc}
\tilde{a}_{11} & \tilde{a}_{12} \\
\tilde{a}_{21} & \tilde{a}_{22}
\end{array}\right)
\left(\begin{array}{c} p \\
v_z
\end{array}\right)
+
\left(\begin{array}{c} f_p \\
f_v
\end{array}\right).$$ The homogeneous part of (\[eq60n\]) is got in Section \[section2\] (\[eq60\]), the functions $ \left(\begin{array}{c} f_p \\
f_v
\end{array}\right)$ describe a source. They have zero values above $z_{max}$. Eqs. (\[eq60n\]) is a second-order problem, and the corresponding Green’s matrix has the order $2\times 2$. For it construction $F^u=(p^u,v_z^u)$, we will use notations : is the weakly dissipative solution satisfying the upper boundary condition ( the upper solution); $F^l=(p^l,v_z^l)$, the weakly dissipative solution satisfying the lower boundary condition ( the lower solution). For the region $z<a$, where $a>z_t$, the Green’s function can be written in the form: $$\label{eq62n}
\begin{array}{l}
G_{11}=\frac{1}{W(z')}\times\left\{ \begin{array}{c}
p^u(z)v_z^l(z'), \ 0<z'<z \\
p^l(z)v_z^u(z'), \ z<z'<a
\end{array}
\right.; \\ G_{12}=\frac{1}{W(z')}\times\left\{ \begin{array}{c}
v_z^u(z)v_z^l(z'), \ 0<z'<z \\
v_z^l(z)v_z^u(z'), \ z<z'<a
\end{array}
\right.;
\\ G_{21}=\frac{-1}{W(z')}\times\left\{ \begin{array}{c}
p^u(z)p^l(z'), \ 0<z'<z \\
p^l(z)p^u(z'), \ z<z'<a
\end{array}
\right.; \\ G_{22}=\frac{-1}{W(z')}\times\left\{ \begin{array}{c}
v_z^u(z)p^l(z'), \ 0<z'<z \\
v_z^l(z)p^u(z'), \ z<z'<a
\end{array}
\right. .
\end{array}$$ Here $W(z')=v_z^l()z')p^u(z')-v_z^u()z')p^l(z')$ The solution of (\[eq60n\]) is $$\label{eq61n}
\begin{array}{l}
p(z)=\int\limits_{0}^{a}G_{11}(z,z')f_p(z')dz'+\int\limits_{0}^{a}G_{21}(z,z')f_v(z')dz',
\\
v_z(z)=\int\limits_{0}^{a}G_{12}(z,z')f_p(z')dz'+\int\limits_{0}^{a}G_{22}(z,z')f_v(z')dz'
\end{array}$$ It is clear that, if we use DSAS as the upper solution then we will get a solution of the dissipation problem, the homogeneous part of which is given by formulae (\[eq3\]), (\[eq8\]), (\[eq9\]) and the source is from (\[eq60n\]). The disturbance in the upper atmospheric layers is:\
$F(z)=F^u(z)\int\limits_{0}^{z_{max}}\frac{1}{W(z')}[v_z^l(z')f_p(z')-p^l(z')f_v(z')]dz'$
Conclusion {#section7 .unnumbered}
==========
In this paper, we proposed a method for obtaining a wave solution above a source for the real atmosphere (DSAS). The method of construction of DSAS was successfully tested with the everywhere isothermal atmosphere model.
The most essential elements of the test are: calculation of jumps at the matching point to have demonstrated their sufficient smallness and comparison of the numerical solution with an asymptotics of an analytical solution in the region of small dissipation to have demonstrated their coincidence. We have also given the results of DSAS calculations for a real non-isothermal atmosphere.
DSAS itself describes height structure of disturbance above any source provided that it is in the lower part of the atmosphere. In this paper, it is shown that one can construct an expanded Green’s matrix $2\times 2$ with use of DSAS and a solution for the lower weekly dissipative part of the atmosphere, which, as we have shown, can be found analogously to the dissipationless one. The expanded Green’s matrix let us to find the disturbance produced by some source in the all height ranges. Íerewith, the DSAS amplitude is completely determined.
The specific feature of DSAS is describing of dissipative tail, formally indefinitely extended to the isothermal part of the atmosphere. The essential circumstance is that the exponential increase of wave values, relative to background, is changed by their slow decrease, due to dissipation effect. This allows the wave to remain in the frames of the linear description, in whole range of its existence, on condition that its amplitude in the lower part of the atmosphere is sufficiently small. It is important that the method proposed has prospects of the further development in the framework of a more complex model of the atmosphere. In particularly, including vertically stratified wind is possible. The necessary equations for such models are written in this paper. It is also possible to take more complete wave dissipation into account, including not only heat conductivity but viscosity as well.
In this case, the part of DSAS in the upper height range, where dissipation is not small, is described, as we shown in this paper, by the system of six ODEs. Respectively, this system has six independent solutions. For the asymptotic isothermal region, the solutions taking into account both heat conductivity and viscosity were obtain in [@Rudenko_b]. A combination of three such isothermal solutions will satisfy the upper boundary condition. The way to ensure sufficient smoothness of DSAS is the same as in this paper.
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{#A}
$t_{11}=\frac{\Gamma \left( -2k\right) \Gamma \left(
\frac{1}{2}+\alpha -k\right) \Gamma \left( \frac{1}{2}-\alpha
-k\right) \Gamma \left( \frac{1}{2}+k-iq\right) }{\Gamma \left(
\frac{1}{2}-iq-k\right) }e^{i\frac{\pi}{2} k}$,
$t_{12}=\frac{\Gamma \left( 2k\right) \Gamma \left(
\frac{1}{2}+\alpha +k\right) \Gamma \left( \frac{1}{2}-\alpha
+k\right) \Gamma \left( \frac{1}{2}-k-iq\right) }{\Gamma \left(
\frac{1}{2}-iq+k\right) }e^{-i\frac{\pi}{2} k}$,
$t_{13}=\frac{\Gamma \left( k-\frac{1}{2}-\alpha \right) \Gamma
\left( -k-\frac{1}{2}-\alpha \right) \Gamma \left( -2\alpha
\right) \Gamma \left( 1+\alpha -iq\right) }{\Gamma \left(
-iq-\alpha \right) }e^{i\frac{\pi}{2} \left( \frac{1}{2}+\alpha
\right) }$,
$t_{14}=\frac{\Gamma \left( k-\frac{1}{2}+\alpha \right) \Gamma
\left( -k-\frac{1}{2}+\alpha \right) \Gamma \left( 2\alpha \right)
\Gamma \left( 1-\alpha -iq\right) }{\Gamma \left( -iq+\alpha
\right) }e^{i\frac{\pi}{2} \left( \frac{1}{2}-\alpha \right) }$,
$t_{21}=\frac{\Gamma \left( -2k\right) \Gamma \left(
\frac{1}{2}+\alpha -k\right) \Gamma \left( \frac{1}{2}-\alpha
-k\right) \Gamma \left( \frac{1}{2}+k+iq\right) }{\Gamma \left(
\frac{1}{2}+iq-k\right) }e^{i\frac{\pi}{2} k}$,
$t_{22}=\frac{\Gamma \left( 2k\right) \Gamma \left(
\frac{1}{2}+\alpha +k\right) \Gamma \left( \frac{1}{2}-\alpha
+k\right) \Gamma \left( \frac{1}{2}-k+iq\right) }{\Gamma \left(
\frac{1}{2}+iq+k\right) }e^{-i\frac{\pi}{2} k}$,
$t_{23}=\frac{\Gamma \left( k-\frac{1}{2}-\alpha \right) \Gamma
\left( -k-\frac{1}{2}-\alpha \right) \Gamma \left( -2\alpha
\right) \Gamma \left( 1+\alpha +iq\right) }{\Gamma \left(
iq-\alpha \right) }e^{i\frac{\pi}{2} \left( \frac{1}{2}+\alpha
\right) }$,
$t_{24}=\frac{\Gamma \left( k-\frac{1}{2}+\alpha \right) \Gamma
\left( -k-\frac{1}{2}+\alpha \right) \Gamma \left( 2\alpha \right)
\Gamma \left( 1-\alpha +iq\right) }{\Gamma \left( iq+\alpha
\right) }e^{i\frac{\pi}{2} \left( \frac{1}{2}-\alpha \right) }$,
$t_{31}=\frac{\Gamma \left( -2k\right) \Gamma \left(
\frac{1}{2}+\alpha -k\right) \Gamma \left( \frac{1}{2}-\alpha
-k\right) }{\Gamma \left( \frac{1}{2}+iq-k\right) \Gamma \left(
\frac{1}{2}-iq-k\right) }e^{-i\frac{\pi}{2} k}$,
$t_{32}=\frac{\Gamma \left( 2k\right) \Gamma \left(
\frac{1}{2}+\alpha +k\right) \Gamma \left( \frac{1}{2}-\alpha
+k\right) }{\Gamma \left( \frac{1}{2}+iq+k\right) \Gamma \left(
\frac{1}{2}-iq+k\right) }e^{i\frac{\pi}{2} k}$,
$t_{33}=\frac{\Gamma \left( k-\frac{1}{2}-\alpha \right) \Gamma
\left( -k-\frac{1}{2}-\alpha \right) \Gamma \left( -2\alpha
\right) }{\Gamma \left( iq-\alpha \right) \Gamma \left( -iq-\alpha
\right) }e^{-i\frac{\pi}{2} \left( \frac{1}{2}+\alpha \right) }$,
$t_{34}=\frac{\Gamma \left( k-\frac{1}{2}+\alpha \right) \Gamma
\left( -k-\frac{1}{2}+\alpha \right) \Gamma \left( 2\alpha \right)
}{\Gamma \left( iq+\alpha \right) \Gamma \left( -iq+\alpha \right)
}e^{-i\frac{\pi}{2} \left( \frac{1}{2}-\alpha \right) }$.
|
---
abstract: 'Using the combined CLEO II and CLEO II.V data sets of 9.1 fb$^{-1}$ at the $\Upsilon(4S)$, we measure properties of $\psi$ mesons produced directly from decays of the $B$ meson, where “$B$” denotes an admixture of $B^+$, $B^-$, $B^0$, and $\bar{B^0}$, and “$\psi$” denotes either $J/{\psi(1S)}$ or ${\psi(2S)}$. We report first measurements of $\psi$ polarization in $B \to \psi \mbox{(direct)} X$: $\alpha_{{\psi(1S)}} = -0.30^{+0.07}_{-0.06}\pm 0.04$ and $\alpha_{{\psi(2S)}} = -0.45^{+0.22}_{-0.19} \pm 0.04$. We also report improved measurements of the momentum distributions of $\psi$ produced directly from $B$ decays, correcting for measurement smearing. Finally, we report measurements of the inclusive branching fraction for $B \to \psi X$ and $B \to {\chi_{c1}}X$.'
author:
- 'S. Anderson'
- 'V. V. Frolov'
- 'Y. Kubota'
- 'S. J. Lee'
- 'S. Z. Li'
- 'R. Poling'
- 'A. Smith'
- 'C. J. Stepaniak'
- 'J. Urheim'
- 'Z. Metreveli'
- 'K.K. Seth'
- 'A. Tomaradze'
- 'P. Zweber'
- 'S. Ahmed'
- 'M. S. Alam'
- 'L. Jian'
- 'M. Saleem'
- 'F. Wappler'
- 'E. Eckhart'
- 'K. K. Gan'
- 'C. Gwon'
- 'T. Hart'
- 'K. Honscheid'
- 'D. Hufnagel'
- 'H. Kagan'
- 'R. Kass'
- 'T. K. Pedlar'
- 'J. B. Thayer'
- 'E. von Toerne'
- 'T. Wilksen'
- 'M. M. Zoeller'
- 'H. Muramatsu'
- 'S. J. Richichi'
- 'H. Severini'
- 'P. Skubic'
- 'S.A. Dytman'
- 'J.A. Mueller'
- 'S. Nam'
- 'V. Savinov'
- 'S. Chen'
- 'J. W. Hinson'
- 'J. Lee'
- 'D. H. Miller'
- 'V. Pavlunin'
- 'E. I. Shibata'
- 'I. P. J. Shipsey'
- 'D. Cronin-Hennessy'
- 'A.L. Lyon'
- 'C. S. Park'
- 'W. Park'
- 'E. H. Thorndike'
- 'T. E. Coan'
- 'Y. S. Gao'
- 'F. Liu'
- 'Y. Maravin'
- 'I. Narsky'
- 'R. Stroynowski'
- 'M. Artuso'
- 'C. Boulahouache'
- 'K. Bukin'
- 'E. Dambasuren'
- 'K. Khroustalev'
- 'G. C. Moneti'
- 'R. Mountain'
- 'R. Nandakumar'
- 'T. Skwarnicki'
- 'S. Stone'
- 'J.C. Wang'
- 'A. H. Mahmood'
- 'S. E. Csorna'
- 'I. Danko'
- 'Z. Xu'
- 'G. Bonvicini'
- 'D. Cinabro'
- 'M. Dubrovin'
- 'S. McGee'
- 'A. Bornheim'
- 'E. Lipeles'
- 'S. P. Pappas'
- 'A. Shapiro'
- 'W. M. Sun'
- 'A. J. Weinstein'
- 'G. Masek'
- 'H. P. Paar'
- 'R. Mahapatra'
- 'H. N. Nelson'
- 'R. A. Briere'
- 'G. P. Chen'
- 'T. Ferguson'
- 'G. Tatishvili'
- 'H. Vogel'
- 'N. E. Adam'
- 'J. P. Alexander'
- 'K. Berkelman'
- 'F. Blanc'
- 'V. Boisvert'
- 'D. G. Cassel'
- 'P. S. Drell'
- 'J. E. Duboscq'
- 'K. M. Ecklund'
- 'R. Ehrlich'
- 'L. Gibbons'
- 'B. Gittelman'
- 'S. W. Gray'
- 'D. L. Hartill'
- 'B. K. Heltsley'
- 'L. Hsu'
- 'C. D. Jones'
- 'J. Kandaswamy'
- 'D. L. Kreinick'
- 'A. Magerkurth'
- 'H. Mahlke-Krüger'
- 'T. O. Meyer'
- 'N. B. Mistry'
- 'E. Nordberg'
- 'J. R. Patterson'
- 'D. Peterson'
- 'J. Pivarski'
- 'D. Riley'
- 'A. J. Sadoff'
- 'H. Schwarthoff'
- 'M. R. Shepherd'
- 'J. G. Thayer'
- 'D. Urner'
- 'B. Valant-Spaight'
- 'G. Viehhauser'
- 'A. Warburton'
- 'M. Weinberger'
- 'S. B. Athar'
- 'P. Avery'
- 'L. Breva-Newell'
- 'V. Potlia'
- 'H. Stoeck'
- 'J. Yelton'
- 'G. Brandenburg'
- 'A. Ershov'
- 'D. Y.-J. Kim'
- 'R. Wilson'
- 'K. Benslama'
- 'B. I. Eisenstein'
- 'J. Ernst'
- 'G. D. Gollin'
- 'R. M. Hans'
- 'I. Karliner'
- 'N. Lowrey'
- 'M. A. Marsh'
- 'C. Plager'
- 'C. Sedlack'
- 'M. Selen'
- 'J. J. Thaler'
- 'J. Williams'
- 'K. W. Edwards'
- 'R. Ammar'
- 'D. Besson'
- 'X. Zhao'
bibliography:
- 'text.bib'
date: 'May 5, 2002'
title: 'Measurements of Inclusive $B \to \psi$ Production'
---
Inclusive production of $\psi$ is currently understood in the framework of Non-Relativistic QCD (NRQCD) effective field theory [@NRQCD]. In 1995, measurements of prompt $\psi$ production at the Tevatron [@CDFproduction; @CDFproduction2] ruled out the then-dominant Color Singlet Model (CSM); in contrast, NRQCD calculations [@NRQCD-CDFproduction] could accommodate the relatively large production rate. However, measurements of the polarization of these prompt $\psi$ [@CDFpolarization] deviated from NRQCD calculations at high $p_T$. The precision of these calculations is limited by the knowledge of the process-independent, long-distance matrix elements (LDME’s), which also appear in NRQCD calculations of $\psi$ production in $B$ decays. The polarization of $\psi$ produced from $B$ decays [@Ma00alpha; @Fleming97] is sensitive to the color-octet LDME’s; however, these calculations have been done only to leading order (LO). The momentum distribution of $\psi$ produced in $B$ decays [@Beneke00; @Beneke97; @Palmer97] is also sensitive to the dominant color octet terms, particularly at low $p_\psi$. Additionally, the low-momentum region would also be affected by the existence of an intrinsic charm component in $B$ mesons [@Chang01]. Finally, the inclusive branching fraction ${\cal B}(B \to \psi X)$ [@Beneke97; @Ma00; @Ko96] constrains a sum of LDME’s. This Letter reports measurements of these three properties of $\psi$ production in $B$ decays, which could significantly improve the knowledge of the non-perturbative parameters of NRQCD.
Our analysis [@MyThesis] is based on 9.7 million $B\overline{B}$ events ($9.1 \, \hbox{fb}^{-1}$) produced on the $\Upsilon(4S)$ resonance. Additionally, $4.4 \, \hbox{fb}^{-1}$ of data collected slightly below the $\Upsilon(4S)$ resonance were used to subtract the small $(\approx 2\%)$ contribution of continuum ($e^+ e^- \to q\overline{q}, q \in \{ u,d,c,s\} $) ${\psi(1S)}$ production. The $e^+ e^-$ collisions were delivered by the Cornell Electron Stoarge Ring (CESR) and detected with two configurations of the CLEO detector, CLEO II [@CLEOII] and CLEO II.V [@CLEOIIV].
We select events that have spherical energy distributions and are likely to be hadronic. We reconstruct $\psi$ candidates in the dilepton modes $\psi \to \mu^+ \mu^-$ and $\psi \to e^+(\gamma) e^-(\gamma)$. The selection criteria were chosen with a goal of high detection efficiency. In the di-muon channel, at least one of the muon candidates must penetrate at least 3 interaction lengths into the iron of the solenoid return yoke; if only one candidate satisfies this, then the other candidate must leave a shower in the crystal calorimeter which is consistent with that of a minimum ionizing particle. In the di-electron channel, we use shower information from the crystal calorimeter and measurements of specific ionization from the drift chamber to identify electron candidates. We also attempt to recover up to one Bremsstrahlung photon for each electron candidate. To do this, we select the most collinear shower within a five-degree cone around the initial electron direction; furthermore, the shower must not be associated with any track, and, when combined with any other shower in the event, must not result in an invariant mass consistent with a $\pi^0$. The PDG 2001 [@PDG2001] branching fractions are used to combine results from the electron and muon channels, except for ${\cal B}({\psi(2S)}\to \mu^+ \mu^-)$, which we assume by lepton universality to be equal to ${\cal B}({\psi(2S)}\to e^+ e^-)$, with an uncertainty of 20% of itself; this is consistent with recent measurements [@BaBarPT].
About 30% of ${\psi(1S)}$ from $B$ decays have intermediate parents of ${\psi(2S)}$ or ${\chi_{c1}}$ [@PDG2001]. Our measurements of directly produced ${\psi(1S)}$ are obtained by subtracting the contributions of these “feed-down” ${\psi(1S)}$ from the inclusive ${\psi(1S)}$ sample. For every event with a ${\psi(1S)}$ candidate within $^{+25}_{-50}$ MeV of the nominal mass, we search for these intermediate parents through the decay chains ${\chi_{c1}}\to {\psi(1S)}\gamma$ and ${\psi(2S)}\to {\psi(1S)}\pi^+ \pi^-$. We reconstruct $(M_{\ell^+ \ell^- \pi^+ \pi^-}-M_{\ell^+ \ell^-})$ and $(M_{\ell^+ \ell^- \gamma}-M_{\ell^+ \ell^-})$, which have better resolution than the reconstructed ${\psi(2S)}$ and ${\chi_{c1}}$ invariant masses themselves. In the ${\psi(2S)}\to {\psi(1S)}\pi^+ \pi^-$ decay chain, we reduce low-momentum pion background by requiring $M_{\pi^+\pi^-} > 0.45$ GeV, which has a efficiency of about 85%, from Monte Carlo simulation. This decay mode is also used to improve the statistics in our measurements of the inclusive branching fraction and the ${\psi(2S)}$ momentum distribution in $B \to {\psi(2S)}X$. We do not reconstruct the related decay ${\psi(2S)}\to {\psi(1S)}\pi^0 \pi^0$, and argue that the properties of ${\psi(1S)}$ from this decay are identical to those of ${\psi(1S)}$ from ${\psi(2S)}\to {\psi(1S)}\pi^+ \pi^-$; the kinematic difference in the momentum distribution is small compared to the experimental resolution, and the isospin state of the $\pi\pi$ state has no bearing on the polarization of the ${\psi(1S)}$.
The CLEO Monte Carlo simulation, based on GEANT [@GEANT], is used to obtain the invariant mass lineshape for signal events and to estimate the detection efficiency. In these simulated signal events, one of the $B$ mesons decays via one of the decay chains listed above. For each decay chain, we generate two samples of events; one with all $\psi$ longitudinally polarized, the other with all $\psi$ transversely polarized. We find that the detection efficiency varies slightly as a function of $\psi$ momentum and polarization.
The procedure and results for the inclusive branching fraction and momentum distribution measurements are as follows. We divide the data into partitions in $p_\psi$, the momentum of the $\psi$ candidate, using a binsize of 0.1 GeV/$c$. For each partition, the invariant mass distribution of $\psi$ candidates is fit to a sum of a signal lineshape, obtained from the Monte Carlo simulation, and a cubic polynomial background. The average $\chi^2$ of the fits is consistent with the number of degrees of freedom, thus justifying our choice of the above parametrization. We repeat this procedure using signal Monte Carlo events, binning in generated $\psi$ momentum, to obtain detection efficiencies as a function of $p_\psi$. The data is then corrected for detection efficiency bin by bin; this minimizes the effect of any discrepancy between the true $p_\psi$ distribution and that generated by the Monte Carlo simulation. Similarly, we fit the invariant mass distributions of $(M_{\ell^+ \ell^- \pi^+ \pi^-}-M_{\ell^+ \ell^-})$ and $(M_{\ell^+ \ell^- \gamma}-M_{\ell^+ \ell^-})$, to extract efficiency-corrected yields of feed-down ${\psi(1S)}$. We thus obtain momentum distributions of ${\psi(1S)}$ and ${\psi(2S)}$ which have been corrected for detection efficiency, ${\psi(1S)}$ feed-down, and continuum background. The yields are then normalized by $n_B \times {\cal B}(\psi \to \ell^+ \ell^- (\pi^+ \pi^-))$, where $n_B$ is the number of $B$ and $\overline{B}$ mesons in the data; the uncertainties in these quantities are reflected in our results as an overall scale factor error. Inclusive branching fractions are obtained by summing the normalized momentum distributions over all bins. Finally, the Monte Carlo simulation is used to obtain a matrix which correlates the momentum of the $\psi$ as generated to the momentum as measured; by inverting this matrix and applying it to the observed momentum distribution, we are able to deconvolve the effects of detector measurement smearing from the distribution.
We investigate the possible sources of systematic error; for each source, we make an appropriate modification to the measurement procedure and observe the deviation of the resulting yield relative to the nominal procedure. The deviations are then combined to obtain final systematic errors. The sources of error are grouped as follows: (1) Monte Carlo simulation of track and shower finding, electron and muon identification, $\psi$ polarization, global event and kinematic cuts, (2) invariant mass fit procedure; (3) branching fractions of unmeasured modes; and (4) overall scale factor.
The results for the inclusive branching fractions are given in Table \[tab:BF\] and the momentum distributions are shown in Figure \[fig:p\]. Our branching fraction results are consistent with previous published measurements [@CLEOBigB] as well as preliminary measurements [@BaBarBF], and are limited by systematic errors. The combined error for ${\cal B}(B \to {\psi(1S)}(\mbox{direct}) X)$ is smaller than the error of the PDG 2001 average [@PDG2001] by a factor of two. However, the theoretical uncertainties in the NRQCD calculations of the branching fractions are such that the improved accuracy of this measurement is unlikely to further constrain the NRQCD LDME’s. The momentum distributions reported here are the first to subtract the [*measured*]{} distributions of feed-down and continuum $\psi$, correct for detector measurement smearing, and analyze systematic errors for each bin individually. Figure 1(b) is also the first to show the momentum distribution of multibody ($\ge 3$-body) decays in $B \to {\psi(2S)}X$ production; these decays account for much of the total $B \to {\psi(2S)}X$ production, as is also the case with $B \to {\psi(1S)}X$ [@Beneke97]. It should be possible to update previous phenomenological studies of the $\psi$ momentum distribution with these improved measurements.
Decay Branching Fraction (%)
-------------------------------------- --------------------------------------- --
$B \to {\psi(1S)}X$ $1.121 \pm 0.013 \pm 0.040 \pm 0.013$
$B \to {\psi(1S)}(\mbox{direct}) X$ $0.813 \pm 0.017 \pm 0.036 \pm 0.010$
$B \to {\chi_{c1}}X \to {\psi(1S)}X$ $0.119 \pm 0.008 \pm 0.009 \pm 0.001$
$B \to {\chi_{c1}}X$ $0.435 \pm 0.029 \pm 0.031 \pm 0.026$
$B \to {\psi(2S)}X \to {\psi(1S)}X$ $0.189 \pm 0.010 \pm 0.018 \pm 0.002$
$B \to {\psi(2S)}X$ $0.316 \pm 0.014 \pm 0.023 \pm 0.016$
: Inclusive branching fraction results. The errors shown are (in order) statistical, systematic, and due to an overall scale factor uncertainty. \[tab:BF\]
![\[fig:p\] Momentum distributions of (a) ${\psi(1S)}$ and (b) ${\psi(2S)}$ produced directly from $B$ decays. There is an additional overall scale uncertainty of $1.2\%$ for ${\psi(1S)}$ and $5.1\%$ for ${\psi(2S)}$ which is not depicted in the plots. The histograms show the contributions of two-body $B \to \psi X$ decays, where the lineshapes are obtained from Monte Carlo simulation and the normalizations are from previous determinations of exclusive branching fractions [@PDG2001; @Bellepsik1; @CLEOpsi2k]. ](1){height="5.208in" width="5.0625in"}
The polarization parameter $\alpha$ is equal to ($+1$, 0, $-1$) for a population of (transversely, randomly, longitudinally) polarized $\psi$. For $\psi \to \ell^+ \ell^-$ decays, it is determined experimentally by measuring the decay angle $\theta$, which is defined as the angle between the $\ell^+$ direction in the $\psi$ rest frame and the $\psi$ direction in the $B$ rest frame. The ${\cos\theta}$ distribution for a population of $\psi$ is proportional to $(1 + \alpha \cos^2\theta)$. The angular distribution is obtained in a similar manner as the momentum distributions: the dataset is partitioned into 5 equal bins in ${\cos\theta}$ between $-1$ and $1$; for each partition, we fit the invariant mass distribution to find the signal yield. In addition to measuring the polarization of direct ${\psi(1S)}$ and ${\psi(2S)}$ for all momenta, we also extract $\alpha_{{\psi(1S)}}$ in 3 coarse momentum bins.
At CLEO, $B$ mesons are produced with a small boost in the $\Upsilon(4S)$ (lab) frame, the direction of which is unknown. The boost of the $B$ results in a smeared measurement of ${\cos\theta}$; directly fitting for $\alpha$ using the measured ${\cos\theta}$ distribution would yield a biased result. However, this kinematic smearing is accurately modeled by the Monte Carlo simulation. Our procedure is to generate Monte Carlo events in two sets; one with all $\psi$ generated longitudinally, the other with all transverse. The measured ${\cos\theta}$ distribution from the data is then fit to a sum of the reconstructed ${\cos\theta}$ distributions from the polarized Monte Carlo sets. This procedure correctly accounts for both the boost smearing and detection efficiency. Since the efficiency also depends on $p_\psi$, we must ensure that the Monte Carlo distributions of generated $p_\psi$ match those of Figure \[fig:p\]; this is accomplished through a rejection technique. Because the observed ${\cos\theta}$ distributions are not directly corrected for detection efficiency, the observed feed-down distributions are corrected only for the efficiency of detecting the additional particles needed to reconstruct the ${\psi(2S)}$ or ${\chi_{c1}}$. The final feed-down and continuum-subtracted angular distributions are shown in Figure \[fig:angular\]. The systematic error study included the previously mentioned sources of bias; additionally, we have investigated the possible systematic error arising for the methods for feed-down subtraction and fitting for $\alpha$.
The final polarization results are listed in Table \[tab:alpha\]; these are the first results for the polarization of ${\psi(1S)}$ and ${\psi(2S)}$ from $B \to \psi\mbox{(direct)} X$. For comparison, we measure $\alpha = -0.35 \pm 0.03$ (statistical error only) for ${\psi(1S)}$ from $B \to {\psi(1S)}\mbox{(all)} X$. Our result for $\alpha_{{\psi(1S)}}$ is about $4$ standard deviations from zero; this measurement therefore strongly disfavors the color evaporation model of charmonium production [@Fritzsch77], which predicts zero net polarization, independent of the production mechanism. When next-to-leading-order calculations become available, these results also have the potential to significantly constrain the long-distance matrix elements of NRQCD.
We gratefully acknowledge the effort of the CESR staff in providing us with excellent luminosity and running conditions. This work was supported by the National Science Foundation, the U.S. Department of Energy, the Research Corporation, and the Texas Advanced Research Program.
![\[fig:angular\] Decay angle distributions of (a) ${\psi(1S)}$ and (b) ${\psi(2S)}$ from $B \to \psi\mbox{(direct)}X$, summed over all $p_\psi$. The points represent the data, showing $1\sigma$ statistical errors. In both figures, the fit result (solid histogram) is the sum of a longitudinal component (dashed histogram) and a transverse component. ](2){height="2.7855in" width="5.0625in"}
$\psi$ Meson $p_{\psi}$ (GeV/$c$)
-------------- ---------------------- --------- -------------------- ------------
${\psi(1S)}$ $0.0 - 2.0$ $-0.30$ $^{+0.07}_{-0.06}$ $\pm 0.04$
${\psi(2S)}$ $0.0 - 1.6$ $-0.45$ $^{+0.22}_{-0.19}$ $\pm 0.04$
${\psi(1S)}$ $0.0 - 0.8$ $+0.32$ $^{+0.33}_{-0.27}$ $\pm 0.15$
${\psi(1S)}$ $0.8 - 1.4$ $-0.37$ $^{+0.09}_{-0.09}$ $\pm 0.04$
${\psi(1S)}$ $1.4 - 2.0$ $-0.52$ $^{+0.08}_{-0.07}$ $\pm 0.03$
: Polarization of ${\psi(1S)}$ and ${\psi(2S)}$ from $B \to \psi\mbox{(direct)}X$ over the full momentum range (top two values) and for ${\psi(1S)}$ in three momentum ranges. The errors are statistical and systematic. \[tab:alpha\]
|
---
abstract: 'In a non-ideal classical Coulomb one-component plasma (OCP) all thermodynamic properties are known to depend only on a single parameter – the coupling parameter $\Gamma$. In contrast, if the pair interaction is screened by background charges (Yukawa OCP) the thermodynamic state depends, in addition, on the range of the interaction via the screening parameter $\kappa$. How to determine in this case an effective coupling parameter has been a matter of intensive debate. Here we propose a consistent approach for defining and measuring the coupling strength in Coulomb and Yukawa OCPs based on a fundamental structural quantity, the radial pair distribution function (RPDF). The RPDF is often accessible in experiments by direct observation or indirectly through the static structure factor. Alternatively, it is directly computed in theoretical models or simulations. Our approach is based on the observation that the build-up of correlation from a weakly coupled system proceeds in two steps: First, a monotonically increasing volume around each particle becomes devoid of other particles (correlation hole), and second (upon further increase of the coupling), a shell structure emerges around each particle giving rise to growing peaks of the RPDF. Using molecular dynamics simulation, we present a systematic study for the dependence of these features of the RPDF on $\Gamma$ and $\kappa$ and derive a simple expression for the effective coupling parameter.'
author:
- 'T. Ott'
- 'M. Bonitz'
- 'L. Stanton'
- 'M. S. Murillo'
title: '[Coupling Strength in Coulomb and Yukawa One-Component Plasmas]{}'
---
Introduction
============
Strongly coupled or strongly correlated systems are abundant in many fields of physics. In plasma physics, they have come into the focus in recent times due to the increasing availability of experimental realizations. In these experiments, the mutual repulsion of the like-charged particles is comparable to or even greater than their thermal kinetic agitation. This leads to the emergence of complex phenomena such as shear waves [@Liu2010a; @Ott2013], solidification [@Hartmann2010], cooperative behavior [@Coueedel2014] and anomalous transport [@Ott2009b; @Feng2010]. Because of these rich physics, there is strong aspiration to reach ever stronger degrees of correlation in the experiments.
However, despite the central role of the coupling strength, it is often difficult to assess from experimental data as it requires detailed knowledge about the system state. In addition, the role of Debye-screened interaction is often neglected when statements about the coupling strength are made. This leads to difficulties in comparing the degree of correlation across experiments which include dusty plasmas [@Bonitz2010a], ultracold neutral plasmas [@Killian2007], ions in traps [@Dantan2010], and warm dense matter setups.
The situation is even more complex in a system containing multiple species such as two-component plasmas. Here, in principle, one has to distinguish the coupling strength of the two components as well as the inter-species coupling. In high density plasmas, such as warm dense matter, where the light component (i.e., the electrons) may be quantum degenerate and weakly coupled, the heavy component may be classical and strongly coupled. In addition, partial ionization maybe relevant making the analysis of structural and thermodynamic properties challenging.
In this work, we give a structrual definition of the coupling strengths in nonideal plasmas, focusing exclusively on one-component plasmas. There, the coupling strength is typically given in terms of a coupling parameter $g$ of the form $$\begin{aligned}
g &= \frac{\langle E_\textrm{NN}\rangle}{k_BT}, \label{eq:coupling_g}\end{aligned}$$ where $E_\textrm{NN}$ is a measure of the typical nearest-neighbor interaction. Strong coupling is associated with $g>1$. For Coulomb systems with the interaction potential $V(r)=Q/r$, the coupling takes the form $$\begin{aligned}
\Gamma=\frac{Q^2/a}{k_BT}, \label{eq:coulomb_g}\end{aligned}$$ where $Q$ is the particle charge and $a=\left[3/\left (4 n \pi \right) \right]^{1/3}$ is the Wigner-Seitz radius, a measure of the nearest-neighbor distance. Evaluation of $\Gamma$ for a given experimental situation thus requires the measuring of $T$ and $Q$ separately (alongside the density $n$). The importance of the coupling parameter for a model one-component Coulomb plasma (OCP) arise from the observation that its mean energy and all thermodynamic quantities do not depend on density and temperature separately but only via $\Gamma$.
In Yukawa systems, the interaction potential takes the form $V(r)=Q/r \times \exp(-\kappa r/a)$, where $\kappa$ is related to the inverse of the screening length $\lambda$ by $\kappa=a/\lambda$. In a classical plasma $\lambda$ is given by the Debye length whereas, in a strongly degenerate quantum plasma, it is given by the Thomas-Fermi screening length. In general, $\lambda$ is defined by the static long-wavelength limit of the longitudinal polarization function, see, e.g., Ref. [@Bonitz1998]. It is customary to give the coupling strength of Yukawa systems in terms of $\Gamma$ as per Eq. together with the inverse screening length $\kappa$. However, since $Q^2/a$ is not the true measure of the potential energy of a Yukawa system (due to the missing screening [@Lyon2013]), $\Gamma$ carries no immediate physical significance for these systems. This raises the question of how to compare Yukawa and Coulomb systems on the one hand and Yukawa systems of different screening on the other.
We thus face two intricacies when using $\Gamma$ as a measure of the coupling strength: 1) It requires the measurement of $Q$ and $T$ separately, and 2) it is only physically meaningful for Coulomb systems. The goal of this work is to alleviate these problems by finding a one-to-one mapping between the structure of Coulomb and Yukawa systems to $\Gamma$ \[Eq. \]. This allows one to infer $\Gamma$ from the structure alone, i.e., without knowledge of $Q$ or $T$. Furthermore, by using structural features to define the coupling strength, it is possible to give an effective coupling parameter [$\Gamma^\textrm{eff}$]{}for Yukawa systems [@Murillo2008; @Ott2011b; @Hartmann2005]. For a Yukawa system with a given screening length, this parameters equals the value of $\Gamma$ of the corresponding Coulomb system with the most similar structure, i.e., the most meaningful comparison system. A similar approach as been used in Ref. [@Clerouin2013].
The path taken here towards this goal is the following: Using Langevin Dynamics simulation, we obtain reference data for the structure of Coulomb and Yukawa systems in the form of the radial pair distribution function (RPDF) $g(r)$. To uniquely relate the shape of this function to the physical degree of non-ideality we use two of its properties: the width of the correlation hole and the height of the first peak of $g(r)$.
Comparing these experimentally accessible quantities to the reference data allows one to infer $\Gamma$. Carefully optimized fit formulas are derived which connect the structure to the coupling strength and allow for an interpolation to values not covered in the reference data.
![Radial pair distribution function of a Coulomb system at $\Gamma=1;4;40;100$. \[fig:gofr\_visual\]](gofr_visual.eps)
Methods and Simulation
======================
We use Langevin dynamics simulation for $N=8192$ particles to obtain the equilibrium properties of the Coulomb and Yukawa One-Component plasma. The Langevin equation reads $$\begin{aligned}
m \ddot{{{\boldsymbol{r}}}}_i ={\boldsymbol{F}}_i - m \bar \nu {\boldsymbol{v}}_i + {\boldsymbol{y}}_i \, \qquad i=1\dots N\, \label{eq:langevin} \end{aligned}$$ where $\bar \nu$ is the friction coefficient, ${\boldsymbol{F}}_i$ is the repulsive interaction force between the particles and ${\boldsymbol{y}}_i(t)$ is a Gaussian white noise with zero mean and the standard deviation $
\langle y_{\alpha,i}(t_0)y_{\beta,j}(t_0+t)\rangle=2k_BTm\bar \nu\delta_{ij}\delta_{\alpha\beta}\,\delta(t)
$ \[$\alpha,\beta\in \{x,y,z\}$\]. We use a constant friction coefficient of $\bar \nu = 0.1\omega_p$, where $\omega_p= [ 4\pi Q^2 n/m]^{1/2}$ is the nominal (Coulomb) plasma frequency. We vary $\kappa$ between $0$ and $2$ in steps of $0.2$ and vary $\Gamma$ to cover the entire liquid phase (i.e., from $\Gamma=1$ up to close to the phase transition temperature).
To assess the structural state of the system, we use the radial pair distribution defined as $$\begin{aligned}
g({\boldsymbol{r}}) = \frac{1}{Nn} \left\langle \sum_{i,j=1 \atop i \neq j}^N \delta({\boldsymbol{r}}- {\boldsymbol{r}}_{ij}) \right\rangle, \label{eq:pdf}\end{aligned}$$ where $r_{ij}=\vert {\boldsymbol{r}}_i - {\boldsymbol{r}}_j\vert$ and the averaging is over time. The RPDF is the most simple structural quantity of a many-particle system and describes the relative occurrence frequency of a particular pair distance $r$ in the system. In many setups, it is experimentally accessible through direct optical monitoring (e.g., in dusty plasmas) or indirectly through the measurement of the static structure factor (e.g., through scattering measurements). There are also various theoretical approaches to calculate the RPDF including the Hypernetted Chain Approach [@Ng1974; @Bruhn2011] and simulations [@Brush1966].
For a system of non-interacting particles, $g(r)\equiv 1$ and any correlation effects will manifest themselves in deviation of $g(r)$ from unity. One of the two main RPDF characteristics of strongly coupled systems is the correlation void at small values of $r$ which reflects the mutual repulsion of particles at small distances. The second feature is the emergence of a series of peaks in $g(r)$ related to the formation of shell-like structures of first, second, etc. neighbors around any particular particle.
![[$g_\textrm{max}$]{}and [$r_{1/2}$]{}as a function of $\Gamma$ for a Coulomb system. Note that in the left part of the figure, the $\Gamma$-axis is scaled logarithmically. \[fig:gmaxr05\]](gmaxr05.eps)
The build-up of correlation from an uncorrelated system manifests itself in two subsequent steps (Fig. \[fig:gofr\_visual\]): First, the correlation void grows rapidly as the correlation increases. After this process, upon further increase of the coupling strength, the shell structure emerges and becomes gradually more pronounced. Notably, the size increase of the correlation void with the coupling strength is rapid only at small values of the coupling, while the growth of the peak structure is most prominent at larger coupling values. This complementary development leads us to use both features of the RPDF in our subsequent analysis (see Fig. \[fig:gofr\_visual\]). Specifically, we assess the size of the correlation void as the value of ${\ensuremath{r_{1/2}}\xspace}$ defined by $$\begin{aligned}
g(a\cdot {\ensuremath{r_{1/2}}\xspace}) = 0.5, \end{aligned}$$ i.e., the dimensionless distance at which $g(r)$ has risen to half its asymptotic limit. For the peak structure, we use the height of the first peak in the RPDF, i.e., its global maximum [$g_\textrm{max}$]{} [^1].
As an example, we show in Fig. \[fig:gmaxr05\] both [$r_{1/2}$]{}and [$g_\textrm{max}$]{}as a function of $\Gamma$ for a Coulomb system (note that in the left part of Fig. \[fig:gmaxr05\], the $\Gamma$-axis is scaled logarithmically). At small $\Gamma$, [$r_{1/2}$]{}varies rapidly and [$g_\textrm{max}$]{}varies only slowly while for larger $\Gamma$, the inverse is true. However, since both [$r_{1/2}$]{}and [$g_\textrm{max}$]{}are strictly monotonic functions of $\Gamma$ at given $\kappa$, they both uniquely define the whole structural composition of the system and thus the complete thermodynamic equilibrium properties of the plasma. The switching between [$r_{1/2}$]{}and [$g_\textrm{max}$]{}to define the coupling strength is thus only a matter of practicality.
Structure and Coupling Strength
===============================
We now establish a one-to-one mapping between the structure of a system and its physical coupling strength. In doing so, we assume that the screening length $\kappa$ is known in a given experimental setup. This is crucial, because in this first step, we express the coupling of a Yukawa system in the customary way through the nominal Coulomb coupling parameter $\Gamma$. Since this is not the actual physical coupling strength of a Yukawa system, systems with the same value of $\Gamma$ but different $\kappa$ exhibit different degrees of structural correlation. Conversely, a given structure can correspond to a multitude of $\{\Gamma, \kappa\}$ pairs, so that without knowledge of $\kappa$, these systems cannot be distinguished. If, however, $\kappa$ is known, then the structure \[i.e. the shape of $g(r)$\] uniquely defines the coupling value $\Gamma$.
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First, we consider systems whose correlation is large enough that a peak structure has formed and the first peak in the RPDF is clearly visible (trivially, [$g_\textrm{max}$]{}tends to unity when the coupling strength is lowered). The relation between [$g_\textrm{max}$]{}and $\Gamma$ at a given $\kappa$ is shown in the top graph of Fig. \[fig:graphrel\]. The unique correspondence between $\Gamma$ and [$g_\textrm{max}$]{}(at given $\kappa$) is clear from these data. One also sees that a given peak height can correspond to several values of $\Gamma$, depending on $\kappa$, or, conversely, that for a given $\Gamma$, systems with higher $\kappa$ have lower peak heights and are thus less strongly coupled. This is a reflection of the fact that in systems with large values of $\kappa$, the increased screening reduces the interaction between the particles. To use [$g_\textrm{max}$]{}as a reliable indicator of $\Gamma$, the dependence $\Gamma({\ensuremath{g_\textrm{max}}\xspace})$ must be sensitive to a variation of [$g_\textrm{max}$]{}. This corresponds to those parts of the curves in Fig. \[fig:graphrel\] which have a sufficiently high slope, i.e., ${\ensuremath{g_\textrm{max}}\xspace}\gtrsim 1.4$.
For less strongly coupled systems, $\Gamma$ is only a weak function of [$g_\textrm{max}$]{}, since the peak develops slowly as the coupling is increased. Instead, the build-up of correlation at these small coupling values is indicated by the growth of the correlation void [$r_{1/2}$]{}. The dependence of $\Gamma$ on [$r_{1/2}$]{}is shown in the bottom graph of Fig. \[fig:graphrel\]. It is clear that [$r_{1/2}$]{}is a sensitive indicator of $\Gamma$ in the range ${\ensuremath{r_{1/2}}\xspace}\lesssim 1.3$ and can be used to deduce $\Gamma$ at a given $\kappa$. The complementary nature of [$g_\textrm{max}$]{}and [$r_{1/2}$]{}is readily observable from the two graphs of Fig. \[fig:graphrel\]: While [$g_\textrm{max}$]{}is sensitive at higher values of $\Gamma$, [$r_{1/2}$]{}is sensitive at small values of $\Gamma$. There is an overlap around $\Gamma\approx 30\ldots 50$ in which both methods are usable and give identical results. We stress that the use of both [$g_\textrm{max}$]{}and [$r_{1/2}$]{}is made to maximize the sensitivity of the measurement during both stages of the correlation build-up.
We now develop approximate fit formulas to the data of Fig. \[fig:graphrel\] in which the *relative* mean-squared deviation is minimized [@Schmidt2009]. For $\Gamma({\ensuremath{g_\textrm{max}}\xspace})$, the data is well described by the polynomial $$\begin{aligned}
\Gamma(g_\textrm{max},\kappa) &= a_1(\kappa) + a_2(\kappa) g_\textrm{max} + a_3(\kappa) g_\textrm{max}^2, \nonumber\\
& \hspace{3.5cm}1.4<{\ensuremath{g_\textrm{max}}\xspace}<2.4\label{eq:gamma_gmax}, \end{aligned}$$ where $a_i(\kappa)$ is given by the values in Table \[tab:gmax\_fit\]. From the errors given in Table \[tab:gmax\_fit\], it is clear that Eq. is an excellent fit to the data. To simplify the usage of Eq. and interpolate to intermediate values of $\kappa$, the functional form of $a_i(\kappa)$ can be further approximated by $$\begin{aligned}
a_1(\kappa) &= 22.40 - 7.88 \kappa + 9.68 \kappa^2 \nonumber\\
a_2(\kappa) &= -70.09 + 20.28 \kappa - 32.48 \kappa^2 \label{eq:gamma_gmax_fit}\\
a_3(\kappa) &= 52.60 - 12.71 \kappa + 23.73 \kappa^2 \nonumber.
\end{aligned}$$
Use of Eqs. in yields an maximum error of $3.3\%$ and an average error of $1.45\%$ over all numerical data.
[m[1cm]{}m[1cm]{}m[1.5cm]{}m[1cm]{}cc]{} $\kappa$ & $a_1$ & $a_2$ & $a_3$ & $\Delta_\textrm{max}$ (%) & $\Delta_\textrm{avg}$ (%)\
0.0 & 22.864 & -68.942 & 51.209 & 0.22 & 0.09\
0.2 & 18.640 & -64.525 & 50.289 & 0.20 & 0.06\
0.4 & 21.858 & -69.515 & 52.682 & 0.16 & 0.06\
0.6 & 22.704 & -72.805 & 55.285 & 0.22 & 0.07\
0.8 & 23.608 & -77.265 & 59.012 & 0.22 & 0.08\
1.0 & 23.765 & -82.112 & 63.732 & 0.24 & 0.07\
1.2 & 26.843 & -91.755 & 70.828 & 0.18 & 0.07\
1.4 & 28.297 & -101.537 & 79.289 & 0.29 & 0.08\
1.6 & 35.852 & -120.384 & 91.765 & 0.19 & 0.09\
1.8 & 37.093 & -135.352 & 105.181 & 0.27 & 0.07\
2.0 & 47.348 & -163.976 & 124.738 & 0.30 & 0.08\
The dependence $\Gamma({\ensuremath{r_{1/2}}\xspace})$ is approximated by the following relation, $$\begin{aligned}
\Gamma({\ensuremath{r_{1/2}}\xspace}, \kappa) &= b_1(\kappa) \exp\big({b_2 {\ensuremath{r_{1/2}}\xspace}^3}\big) + b_3(\kappa),\nonumber\\
& \hspace{2.5cm}\Gamma\geq 1 \textrm{ ~and~ } {\ensuremath{r_{1/2}}\xspace}<1.3\label{eq:gamma_r05},\end{aligned}$$ where $b_2=1.575$ is a constant and the other fit parameters are given in Table \[tab:r05\_fit\] alongside the maximum and average deviation from the numerical data. For $b_i$, the following approximation can be given: $$\begin{aligned}
b_1(\kappa) &= 1.238 - 0.280 \kappa + 0.644 \kappa^2 \nonumber\\
b_2\phantom{(\kappa)} &= 1.575 \label{eq:gamma_r05_fit}\\
b_3(\kappa) &= -0.931 + 0.422 \kappa - 0.696 \kappa^2 \nonumber,\end{aligned}$$ which, in combination with Eq. , give a maximum error of $5.59$% and an average error of $1.68$% over all numerical data.
[@m[1cm]{}m[1cm]{}m[1cm]{}cc@]{} $\kappa$ & $b_1$ & $b_3$ & $\Delta_\textrm{max}$ (%) & $\Delta_\textrm{avg}$ (%)\
0.0 & 1.200 & -0.873 & 2.46 & 1.77\
0.2 & 1.211 & -0.882 & 2.80 & 1.65\
0.4 & 1.254 & -0.909 & 2.38 & 1.34\
0.6 & 1.336 & -0.978 & 2.01 & 1.06\
0.8 & 1.453 & -1.073 & 1.62 & 0.92\
1.0 & 1.608 & -1.210 & 1.15 & 0.51\
1.2 & 1.811 & -1.400 & 1.66 & 0.46\
1.4 & 2.071 & -1.651 & 1.63 & 0.54\
1.6 & 2.400 & -2.985 & 3.93 & 1.22\
1.8 & 2.800 & -2.405 & 3.61 & 1.03\
2.0 & 3.307 & -2.940 & 1.24 & 0.38\
Thus, in conclusion, we have developed a measure of the coupling strength based solely on structural features of the system. From this follow two complementary methods to infer the value of $\Gamma$ from the structure of the system. Figure \[fig:graphrel\] shows the relationship between [$g_\textrm{max}$]{}and $\Gamma$ and between [$r_{1/2}$]{}and $\Gamma$ and can be used directly to obtain $\Gamma$ from the RPDF at a given $\kappa$. Equations and provide a more convenient means and allow the interpolation to intermediate screening lengths while only incurring a small error. Together these methods provide a non-invasive measurement method for $\Gamma$ for both Coulomb and Yukawa systems. The only knowledge required is the density $n$ (to obtain $a$) and either the peak height [$g_\textrm{max}$]{}or the correlation void size [$r_{1/2}$]{}, both of which are generally much easier to measure than the charge state $Q$ and the kinetic temperature $T$.
Effective coupling strength
===========================
After having addressed the problem of the definition and measurement of the coupling parameter $\Gamma$ in the previous section, we now turn to the question of how a unified effective coupling parameter [$\Gamma^\textrm{eff}$]{}can be defined which carries physical significance not only for Coulomb but also for Yukawa systems. This problem has been considered before, especially for two-dimensional Yukawa systems [@Ott2011b; @Ikezi1986; @Totsuji2001; @Vaulina2002a; @Hartmann2005] but also for three-dimensional systems, based, e.g., on the packing fraction [@Murillo2008].
![Sketch of the graphical definition of [$\Gamma^\textrm{eff}$]{}: A Yukawa system with $\Gamma=300$ and $\kappa=2$ has an effective structural coupling of ${\ensuremath{\Gamma^\textrm{eff}}\xspace}=120$. \[fig:gmax\_sketch\]](gmax_sketch.eps)
Here, our goal is to give a definition based on the structural information contained in [$g_\textrm{max}$]{}and [$r_{1/2}$]{}, answering, in essence, the question “Given a Yukawa system with a known nearest-neighbor correlation (i.e., a given [$g_\textrm{max}$]{}or [$r_{1/2}$]{}), what is the structurally most similar Coulomb system?”
This question can be answered by a graphical solution based on Fig. \[fig:graphrel\] whose principle is sketched in Fig. \[fig:gmax\_sketch\]: The value of [$g_\textrm{max}$]{}or [$r_{1/2}$]{}for the known Yukawa system is projected down on the corresponding $\Gamma({\ensuremath{g_\textrm{max}}\xspace})$ curve for $\kappa=0$ and the corresponding $\Gamma$, which is now equivalent to [$\Gamma^\textrm{eff}$]{}, is read off. For situations in which both $\Gamma$ and $\kappa$ of a Yukawa system are known (e.g., in simulations), the corresponding [$g_\textrm{max}$]{}or [$r_{1/2}$]{}can be obtained directly from Fig. \[fig:graphrel\] as well.
For a formulaic solution to the question posed above, one needs to invert the relation $\Gamma({\ensuremath{g_\textrm{max}}\xspace},\kappa)$ \[Eq. \] to yield ${\ensuremath{g_\textrm{max}}\xspace}(\Gamma,\kappa)$ and obtain [$\Gamma^\textrm{eff}$]{}as $$\begin{aligned}
{\ensuremath{\Gamma^\textrm{eff}}\xspace}(\Gamma,\kappa) & \stackrel{.}{=} \Gamma({\ensuremath{g_\textrm{max}}\xspace}(\Gamma,\kappa),\kappa=0). \label{eq:geffgmax}\end{aligned}$$
The same procedure yields, *mutatis mutandis*, the complementary definition $$\begin{aligned}
{\ensuremath{\Gamma^\textrm{eff}}\xspace}(\Gamma,\kappa) & \stackrel{.}{=} \Gamma({\ensuremath{r_{1/2}}\xspace}(\Gamma,\kappa),\kappa=0). \label{eq:geffr05}\end{aligned}$$
![The scaling function $f(\kappa)$ as a function of $\kappa$. The bounds of the gray shaded area show the linearization of Eq. (upper bound) and Eq. (lower bound). Also shown are the intuitive scaling function $\exp(-\kappa)$ and the scaling function for two-dimensional Yukawa systems [@Hartmann2005]. \[fig:geff\_scaling\]](geff_scaling.eps)
Note that both Eq. and carry the same limitations for their application as Eqs. \[$1.4<{\ensuremath{g_\textrm{max}}\xspace}<2.4$\] and \[$\Gamma\geq 1 \textrm{ ~and~ } {\ensuremath{r_{1/2}}\xspace}<1.3$\], respectively. Taken together, they provide a definition of [$\Gamma^\textrm{eff}$]{}over a range of coupling strengths equivalent to a Coulomb system with $\Gamma=1\ldots 150$, i.e., over practically the whole strongly coupled liquid regime (crystallization of a Coulomb system occurs at $\Gamma=172$).
A further simplification can be introduced by noticing that Eqs. and are very well approximated by their respective linearizations. In addition, for a given $\kappa$, the two linearizations of these equations coincide within 3% with their joint average, which leads us to the following simple definition of [$\Gamma^\textrm{eff}$]{}: $$\begin{aligned}
{\ensuremath{\Gamma^\textrm{eff}}\xspace}(\Gamma,\kappa) &= f(\kappa) \cdot \Gamma, \hspace{1cm} 0\leq\kappa\leq 2,\nonumber\\
& \hspace{2.8cm}1\leq{\ensuremath{\Gamma^\textrm{eff}}\xspace}\leq 150 \label{eq:geff}\end{aligned}$$ where the scaling function is given by $$\begin{aligned}
f(\kappa) &= 1 - 0.309 \kappa^2 + 0.0800 \kappa^3.\label{eq:geff_scaling}\end{aligned}$$ The scaling function has been found as a least-square fit to the average of the linearizations. In this way, the definition is the most accurate representation of ${\ensuremath{\Gamma^\textrm{eff}}\xspace}(\Gamma,\kappa)$ valid for the whole liquid range in which $1<{\ensuremath{\Gamma^\textrm{eff}}\xspace}<150$ since it incorporates both linearizations of and [^2].
![$\kappa-\Gamma$ phase diagram for Yukawa systems. The symbols indicate the melting transition [@Hamaguchi1997]. The solid line marks a constant effective coupling parameter ${\ensuremath{\Gamma^\textrm{eff}}\xspace}=172$. Note that, for larger $\kappa$ the phase diagram is more complex due to the existence of an additional fcc-lattice phase (not shown) [@Hamaguchi1997].\[fig:geff\_melting\]](geff_melting.eps)
The dependence of $f(\kappa)$ on $\kappa$ is shown in Fig. \[fig:geff\_scaling\] together with the intuitive scaling function $\exp(-\kappa)$ which follows from straightforward application of Eq. for a Yukawa system. Clearly, such a simple approach fails to capture the true structural coupling described by [$\Gamma^\textrm{eff}$]{}. Figure \[fig:geff\_scaling\] also shows the corresponding scaling function for a two-dimensional Yukawa system as derived by Hartmann *et al.* [@Hartmann2005] which is valid for ${\ensuremath{\Gamma^\textrm{eff}}\xspace}\gtrsim 40$ and where $\kappa=(\lambda\sqrt{\pi n})^{-1}$ is the two-dimensional definition of the screening strength. This comparison shows that the nominal screening of a Yukawa system has a stronger effect on the effective coupling in two dimensions than it does in three dimensions.
Finally, we consider the liquid-solid phase transition of Coulomb and Yukawa fluids. Since this transition occurs when the the ratio of potential and kinetic energy exceeds a threshold value, one expects an effective structural coupling to be an indicator of the phase change. Figure \[fig:geff\_melting\] shows the value $\Gamma_\textrm{m}$ at which the phase transition occurs [@Hamaguchi1997] together with a constant effective coupling parameter of ${\ensuremath{\Gamma^\textrm{eff}}\xspace}=172$ according to our definition . Evidently, even though [$\Gamma^\textrm{eff}$]{}defined in this work has only been validated in the regime $1\leq{\ensuremath{\Gamma^\textrm{eff}}\xspace}\leq 150$, the phase transition is well described by this effective coupling value. This indicates that [$\Gamma^\textrm{eff}$]{}captures the actual physical coupling of Yukawa systems up to the phase transition.
At the transition itself, one expects a sudden change in [$g_\textrm{max}$]{}as was observed in the two-dimensional case [@Ott2011b] which signifies the re-ordering of the system into a body-centered cubic (bcc) crystal. A similar behavior is expected to appear at the transition from the bcc phase to the face-centered cubic which occurs in Yukawa systems at higher screening [@Hamaguchi1997; @Chugunov2003]. These questions are beyond the present analysis and will be studied elsewhere.
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In addition, it is well known that the phase transition is closely connected to the properties of the static structure factor $S(k)$. More precisely, the Hansen-Verlet criterion [@Hansen1969] states that the phase transition occurs when the maximum peak [$S_\textrm{max}$]{}of $S(k)$ exceeds a threshold value of typically $2.85$. In order to connect the Hansen-Verlet criterion with the short-range definition of [$\Gamma^\textrm{eff}$]{}at hand, we have performed additional calculations in the Hypernetted Chain Approximation (HNC) to obtain both [$S_\textrm{max}$]{}and [$g_\textrm{max}$]{}as a function of $\Gamma$ and $\kappa$. Figure \[fig:hnc\] shows the results of these calculations. While the HNC calculations do not extend all the way to the phase transition, one can see that both [$g_\textrm{max}$]{}and [$S_\textrm{max}$]{}depend in a qualitatively identical way on $\Gamma$. This is not a trivial result since $S(k)$ is related to an integral over $g(r)$ and thus depends on the complete $r$-dependence of the pair distribution function. We conclude that the phase transition is indicated by a critical value of ${\ensuremath{g_\textrm{max}}\xspace}$ (and thus of [$\Gamma^\textrm{eff}$]{}) in the same way as it is indicated by a critical value of [$S_\textrm{max}$]{}by the Hansen-Verlet criterion.
Summary
=======
Particle correlations are a central issue in a wide range of plasma conditions and experiments. Despite the field’s growing importance, there is lack of a clear, unified language when talking about the degree of correlation or the strength of coupling. In this work, we have proposed an approach based on the static structure of the system to define the degree of correlation as well as a simple way to measure the system’s correlation from its structural properties. Our methodology is applicable to both unscreened, pure Coulomb systems as well as screened Yukawa systems with a Debye length corresponding to $\kappa\leq 2$, which encompasses almost all situations of interest.
From an experimentalist’s point of view, with the approach presented here, it suffices to have knowledge of the particle density $n$ and the radial pair distribution function (or the static structure function) to infer the coupling parameter $\Gamma$, instead of the measurement of the particle charge $Q$, the kinetic temperature $T$ and the particle density $n$.
An assessment of the coupling strength based on the structure of the system furthermore allows one to make meaningful comparisons between Coulomb and Yukawa systems and between Yukawa systems with different screening. The common denominator is the effective coupling parameter [$\Gamma^\textrm{eff}$]{}, which corresponds to the equivalent value of $\Gamma$ for a Coulomb system with the most similar nearest-neighbor structure. We have derived a definition of this effective coupling parameter [$\Gamma^\textrm{eff}$]{}by considering the structural features of the respective systems during all stages of correlation build-up. Our definition is thus valid for the whole range of the strongly coupled liquid.
We also briefly remark on the need of knowing the screening length in order to apply our method to Yukawa systems. It is, in principle, possible to infer the value of the screening length non-invasively from the dynamics of the system, in much the same way as we have inferred the value of the coupling strength from the statics of the system (see Ref. [@Ott2011b] for two-dimensional systems). Other methods include direct plasma measurements or the observation of self-excited waves [@Nunomura2000; @Nunomura2002]. The assumptions in this work, therefore, pose no principal limitation on the applicability of the methodology presented.
The approach presented here for a one-component plasma is directly extendable to multicomponent plasmas. A detailed analysis of this generalization will be subject of a forthcoming paper.
This work is supported by the Deutsche Forschungsgemeinschaft via SFB-TR 24 (project A7) and the North-German Supercomputing Alliance (HLRN) via grant shp00006.
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[^1]: Additional properties of the RPDF will be the subject of a forthcoming paper.
[^2]: This is in contrast to the definitions of [$\Gamma^\textrm{eff}$]{}for two-dimensional given in Refs. [@Ott2011b; @Hartmann2005] which only consider [$g_\textrm{max}$]{}and are thus only valid for more strongly coupled systems.
|
\[section\] \[lemma\][Theorem]{} \[lemma\][Definition]{} \[lemma\][Corollary]{} \[lemma\][Problem]{} \[lemma\][Proposition]{}
A closed model structure for $n$-categories, internal $\underline{Hom}$, $n$-stacks and generalized Seifert-Van Kampen {#a-closed-model-structure-for-n-categories-internal-underlinehom-n-stacks-and-generalized-seifert-van-kampen .unnumbered}
======================================================================================================================
Carlos SimpsonCNRS, UMR 5580, Université Paul Sabatier, 31062 Toulouse CEDEX, France.
[**.Introduction**]{}
The purpose of this paper is to develop some additional techniques for the weak $n$-categories defined by Tamsamani in [@Tamsamani] (which he calls [*$n$-nerves*]{}). The goal is to be able to define the internal $Hom(A,B)$ for two $n$-nerves $A$ and $B$, which should itself be an $n$-nerve. This in turn is for defining the $n+1$-nerve $nCAT$ of all $n$-nerves conjectured in [@Tamsamani], which we can do quite easily once we have an internal $Hom$. It is essentially clear [*a priori*]{} that we cannot just take an internal $Hom$ on all of the $n$-nerves of Tamsamani, and in fact some simple examples support this: any strict $n$-category may be considered in an obvious way as an $n$-nerve i.e. a presheaf of sets over $\Delta ^n$ satisfying certain properties, but the morphisms of the resulting presheaves are the same as the strict morphisms of the original strict $n$-categories; on the other hand one can see that these strict morphisms are not enough to reflect all of the “right” morphisms. [^1] Our strategy to get around this problem will be based on the idea of [*closed model category*]{} [@Quillen]. We will construct a closed model category containing the $n$-nerves of Tamsamani. Then we can simply take as the “right” $n$-nerve of morphisms, the internal $Hom(A,B)$ whenever $A$ and $B$ are [*fibrant*]{} objects in the closed model category (all objects will be cofibrant in our case). This strategy is standard practice for topologists.
As usual, in order to define a closed model category we first have to enlarge the class of objects under consideration. Instead of $n$-nerves as defined by Tamsamani we look at $n$-pre-nerves (i.e. presheaves of sets over the cartesian product of $n$ copies of the standard simplicial category) which satisfy the constancy condition—C1 in Tamsamani’s definition of $n$-nerve—and call these [*$n$-precats*]{} (this notion being in between the pre-nerves and nerves of [@Tamsamani], we take a different notation). An $n$-precat may be interpreted as a presheaf on a certain quotient $\Theta^n$ of $\Delta ^n$, in particular we obtain a category $PC_n$ of objects closed under all limits, with internal $Hom$ etc. We follow the method of constructing a closed model category developed by Jardine-Joyal [@Jardine] [@Joyal] in the case of simplicial presheaves. The cofibrations are essentially just monomorphisms (however we cannot—and don’t—require injectivity for top-degree morphisms, just as sets or categories with monomorphisms are not closed model categories [@Quillen]). The main problem is to define a notion of weak equivalence. Our key construction is the construction of an $n$-nerve $Cat(A)$ for any $n$-precat $A$, basically by throwing in freely all of the elements which are required by the definition of nerve [@Tamsamani] (although to make things simpler we use here a definition of nerve modified slightly to “easy nerve”). Then we say that a morphism of $n$-precats $A\rightarrow B$ is a [*weak equivalence*]{} if $Cat(A)\rightarrow Cat(B)$ is an exterior equivalence of $n$-nerves in the sense of [@Tamsamani]. The fibrant morphisms are characterized in terms of cofibrations and weak equivalences by a lifting property, in the same way as in [@Jardine].
One new thing that we obtain in the process of doing this is the notion of pushout. The category of $n$-precats is closed under direct limits and in particular under pushouts. Applying the operation $Cat$ then gives an [*$n$-categorical pushout*]{}: if $A\rightarrow B$ and $A\rightarrow C$ are morphisms of $n$-nerves then the categorical pushout is $Cat(B\cup ^AC)$.
The main lemma which we need to prove (Lemma \[pushout\] below) is—again just as in [@Jardine]—the statement that a pushout by a trivial cofibration (i.e. a cofibration which is a weak equivalence) is again a trivial cofibration. After that the rest of the arguments needed to obtain the closed model structure are relatively standard following [@Jardine] when necessary.
Once the closed model structure is established, we can go on to define internal $Hom $ and construct the $n+1$-nerve $nCAT$. Using these we can, in principal, define the notion of $n$-stack. Our discussion of $n$-stacks is still at a somewhat speculative stage in the present version of the paper, because there are several slightly different notions of a family of $n$-categories parametrized by a $1$-category ${{\cal X}}$ and ideally we would like to—but don’t yet—know that they are all the same (as happens for $1$-stacks).
The notion of categorical pushout which we developed as a technical tool actually has a geometric consequence: we obtain a generalized Seifert-Van Kampen theorem (Theorem \[svk\] below) for the Poincaré $n$-groupoids $\Pi _n(X)$ of a space $X$ which were defined by Tamsamani ([@Tamsamani] §2.3 ff). If $X$ is covered by open sets $U$ and $V$ then $\Pi _n(X)$ is equivalent to the category-theoretic pushout of $\Pi _n(U)$ and $\Pi _n(V)$ along $\Pi _n(U\cap V)$. We define the [*nonabelian cohomology*]{} of $X$ with coefficients in a fibrant $n$-precat $A$ as $H(X,A):= Hom (\Pi _n(X), A)$. The generalized Seifert-Van Kampen theorem implies a Mayer-Vietoris statement for this nonabelian cohomology.
There are many possible approaches to the notion of $n$-category and, without pretending to be exhaustive, I would like to point out some of the other possibilities here for comparison. —One of the pioneering works in the search for an algebraic approach to homotopy of spaces is the notion of $Cat ^n$-groups of Brown and Loday. This is what is now known as the “cubical” approach where the set of objects can itself have a structure for example of $n-1$-category, so it isn’t quite the same as the approach we are looking for (commonly called the “globular” case). —Gordon, Powers and Street have intensively investigated the cases $n=3$ and $n=4$ [@Gordon-Power-Street], following the path set out by Benabou for $2$-categories [@Benabou]. —In [@Grothendieck] A. Grothendieck doesn’t seem to have hit upon any actual definition but gives a lot of nice intuition about $n$-categories. —On p. 41 of [@Grothendieck] starts a reproduction of a letter from Grothendieck to Breen dated July 1975, in which Grothendieck acknowledges having recieved a proposed definition of non-strict $n$-category from Breen, a definition which according to [*loc. cit*]{} “...has certainly the merit of existing...”. It is not clear whether this proposed construction was ever worked out. —In [@Street], R. Street proposes a definition of weak $n$-category as a simplicial set satisfying a certain variant of the Kan condition where one takes into account the directions of arrows. —Kapranov and Voevodsky in [@Kapranov-Voevodsky] construct, for a topological space $X$, a “Poincaré $\infty$-groupoid” which is a strictly associative $\infty$-groupoid but where the arrows are invertible only up to equivalence. This of course raises the question to know if strictly associative $n$-categories would be a sufficient class to yield the correct $n+1$-category $nCAT$. As pointed out in the footnote above, one wonders in particular whether there is a closed model structure to go along with these strict $n$-categories. —In his recent preprint [@Batanin] M. Batanin develops some ideas towards a definition of weak $\infty$-category based on operads. In the introduction he mentions a letter from Baez and Dolan to Street dating to November 29, 1995 which contains some ideas for a definition of weak $n$-category; and he states that Makkai, Hermida and Power have worked on the idea contained in this letter. —M. Rosellen told me in September 1996 that he was working on a version using the theory of operads (cf [@Adams] for example). Just as our current effort is based on Segal’s delooping machine, there should probably be an $n$-category machine analogous to any of the other various delooping machines, and in fact the problems are almost identical: the basic problem of doing $n$-categories comes down to doing delooping while keeping track of the non-connected case and not requiring things to be invertible up to homotopy (cf the last section of [@Tamsamani] for some arguments relating $n$-categories and delooping machines). —J. Baez and J. Dolan have developed their theory originating in the letter refered to above, a definition of $n$-categories based on operads, in a preprint [@BaezDolanIII] of February 1997. In this preprint they discuss operads, give their definition of $n$-category and of certain morphisms of $n$-categories, and define the homotopy category of $n$-categories which they conjecture to be equivalent to the homotopy category for other definitions such as the category $Ho-n-Cat$ mentionned in [@Tamsamani].
The main problem which needs to be accomplished in any of these points of view is to obtain an $n+1$-category (hopefully within the same point of view) $nCAT$ parametrizing the $n$-categories of that point of view. This is the main thing we are doing here for Tamsamani’s point of view. As far as I know, the present one is the first precise construction of the $n+1$-category $nCAT$.
Once several such points of view are up and running, the comparison problem will be posed: to find an appropriate way to compare different points of view on $n$-categories and (one hopes) to say that the various points of view are equivalent and in particular that the various $n+1$-categories $nCAT$ are equivalent via these comparisons. It is not actually clear to me what type of general setup one should use for such a comparison theory, although the first thing to try would be to explore a theory of “internal closed model category”, a closed model category with internal $Hom$: any reasonable point of view on $n$-categories should probably yield an internal closed model category $n--C$ (such as the $PC_n$ we obtain below) and furthermore $nCAT$ should be an object in $(n+1)--C$. Comparison between the theories might then be possible using a version of Quillen’s adjoint functor approach [@Quillen]. We give an indication of how to start on comparison in §11 by sketching how to obtain a functor from any internal closed model category containing $Cat$, to the our closed model category of $n$-categories.
Having a good theory of $n$-categories should open up the possibility to pursue any of the several programs such as that outlined by Grothendieck [@Grothendieck], the generalization to $n$-stacks and $n$-gerbs of the work of Breen [@Breen], or the program of Baez and Dolan in topological quantum field theory [@BaezDolan]. Once the theory of $n$-stacks is off the ground this will give an algebraic approach to the “geometric $n$-stacks” considered in [@geometricN].
We clarify the pretentions to rigor of the various sections of this paper. §§2–7 are supposed to be a first version of something precise and correct (although at the time of this first version I haven’t checked all of the details in a very thorough way). The same holds for §9 on Seifert-Van Kampen. On the other hand, the discussion of §8 on $n$-stacks is blatantly speculative; and the discussion of §10 on nonabelian cohomology is very incomplete.
[*Acknowledgements:*]{} I would specially like to thank Z. Tamsamani and A. Hirschowitz. This work follows up the definition and original work on $n$-nerves done by Z. Tamsamani in his thesis [@Tamsamani]. Much of what is done in the present paper was suggested by discussions with Tamsamani. More recently in preparing some joint work with A. Hirschowitz on universal geometric $n$-stacks related to Brill-Noether, Hirschowitz asked repeatedly for an algebraic approach to $n$-stacks which would be more natural than the approach passing through presheaves of topological spaces or simplicial presheaves. The present work owes much to A. Hirschowitz’s questions and suggestions, as well as to his perseverance in asking for an algebraic approach to $n$-stacks.
I would also like to thank R. Brown for pointing out the importance of the notion of push-out and Seifert-Van Kampen, and G. Maltsiniotis and A. Bruguières for helpful discussions.
[*Ce papier est dedié à Nicole, Chloé et Léo.*]{}
[**.Preliminaries**]{}
Let $\Delta$ be the standard category of ordered finite sets. Let $\Theta ^n$ be the quotient of the cartesian product $\Delta ^n$ obtained by identifying all of the objects $(M, 0, M')$ for fixed $M = (m_1,\ldots , m_k)$ and variable $M'= (m'_1, \ldots , m'_{n-k-1})$. The object of $\Theta ^n$ corresponding to the class of $(M,0,M')$ with all $m_i >0$ will be denoted $M$. The object $(1, \ldots , 1)$ ($k$ times) will be denoted $1^k$. We permit concatenation in our notation for objects, thus $M, m$ denotes the object $(m_1, \ldots , m_k , m)$ (when this makes sense, that is when $k<n$). The class of $(0,\ldots , 0)$ will be denoted by $0$.
We give the explicit construction of $\Theta ^n$. If $M=(m_1,\ldots , m_k)$ and $M' = (m'_1,\ldots , m'_l)$ then set $M,0$ equal to the concatenation of $M$ with $(0,\ldots , 0)$ in $\Delta ^n$ and similarly for $M',0$. We define an equivalence relation on morphisms $\varphi = (\varphi _1, \ldots , \varphi _n)$ from $M, 0$ to $M', 0$ by saying $\varphi \sim \varphi '$ whenever there exists $j$ such that $\varphi _i = \varphi '_i$ for $i\leq j$ and $\varphi _j: m_j \rightarrow m'_j$ factors through the object $0\in \Delta$ (which is the one-point set). This equivalence relation is compatible with composition so we obtain a category $\Theta ^n$ by taking as morphisms the quotient of the morphisms in $\Delta ^n$ by this equivalence relation. There is an obvious projection from $\Delta ^n$ to $\Theta ^n$.
We assume familiarity with [@Tamsamani]. An $n$-precat is a presheaf of sets on $\Theta ^n$. This corresponds to an $n$-prenerve in Tamsamani’s notation (i.e. presheaf of sets on $\Delta ^n$) which satisfies his axiom C1 in the definition of $n$-nerve. Let $PC_n$ denote the category of $n$-precats. An $n$-precat is an [*$n$-category*]{} (or [*$n$-nerve*]{} in the notation of Tamsamani [@Tamsamani] which is the sense which we will always assign to the terminology “$n$-category” below) if it satisfies certain additional conditions [@Tamsamani]. We give an easier version of these conditions which we call an [*easy $n$-category*]{}. We start with the notion of easy equivalence between two easy $n$-categories—this is not circular because the notion of easy $n$-category will only use the notion of easy equivalence for morphisms of $n-1$-categories. If $A$ and $B$ are easy $n$-categories then a morphism $f: A\rightarrow B$ (of $n$-precats, i.e. of presheaves on $\Theta ^n$) is an [*easy equivalence*]{} if for all $v\in B_{1^k}$ (called a $k$-morphism of $B$) and all $a,a' \in
A_{1^{k-1}}$ with $s(v)=f(a)$ and $t(v)=f(a')$ and $s(a)=s(a')$ and $t(a)=t(a')$ (here $s$ and $t$ denote the morphisms “source” and “target” from $T_{1^{k}}$ to $T_{1^{k-1}}$ for any $n$-precat $T$), there exists $u\in
A_{1^k}$ with $s(u)= a$ and $t(u)= a'$ and $f(u)=v$. A [*marked easy equivalence*]{} is the data of a morphism $f$ together with choices $u(a,a', v)$ in every situation as above.
The reader is cautioned that we will still need Tamsamani’s notion of equivalence (which he calls “équivalence extérieure” [@Tamsamani] §1.3) for our closed model category structure below. The notion of easy equivalence is mainly just used when it is an ingredient in the notion of $n$-category.
Before giving the definition of easy $n$-category we introduce the following notation. If $T$ is an $n$-precat then for any $M= (m_1, \ldots , m_k)$ we denote by $T_{M/}$ the $n-k$-precat obtained by restricting $T$ to the subcategory of objects of $\Theta^n$ of the form $(M,M')$ for variable $M'$. This differs from the notation of Tamsamani who called this just $T_M$; our notation with a slash is necessitated by the notation $M$ for objects of $\Theta ^n$. (Sorry about these slight notational changes but is is much easier for us to use $\Theta ^n$ for what will be done below).
With these notations, an $n$-precat $A$ is an [*easy $n$-category*]{} if: —for each $m$, $A_{m/}$ is an easy $n-1$-category; and —the morphisms $$A_{m/} \rightarrow A_{1/} \times _{A_0} \ldots \times _{A_0} A_{1/}$$ are easy equivalences. Note here that $A_0$ is set which is the fiber over the object $0\in \Theta ^n$ (which exits slightly from our notational convention; it is the class of objects $0, M'$ of $\Delta ^n$ but here there is no “$M$” to put into the notation so we put “$0$” instead).
A [*marked easy $n$-category*]{} is an easy $n$-category provided with the addional data of markings for the $A_{m/}$ and markings for the easy equivalences going into the definition. These two conditions amount (recursively) to saying that we have markings for all of the morphisms of the form $$A_{M,m/} \rightarrow A_{M,1/} \times _{A_M} \ldots \times _{A_M} A_{M,1/}.$$
The notion of marking as we have defined above actually makes sense for any $n$-precat, and an $n$-precat with a marking is automatically an easy $n$-category. For this reason, arbitrary inverse limits of marked easy $n$-categories (indexed by systems of morphisms which preserve the markings) are again marked easy $n$-categories.
Suppose $A$ is an $n$-precat. We define the [*marked easy $n$-category generated by $A$*]{} denoted $Cat(A)$ by $$Cat (A)= \lim _{\leftarrow , {{\cal C}}} T$$ where the limit is taken over the category ${{\cal C}}$ whose objects are triples $(T,\mu ,f)$ with $(T, \mu )$ a marked easy $n$-category ($\mu$ denotes the marking) and $f: A\rightarrow T$ is a morphism of $n$-precats. The morphisms of ${{\cal C}}$ are morphisms of $n$-precats (i.e. morphisms of presheaves on $\Theta ^n$) required to preserve $f$ and the marking $\mu$. By the principle given in the previous paragraph, this inverse limit is again a marked easy $n$-category.
The construction $Cat(A)$ is the key to the rest of what we are going to say. The description of $Cat(A)$ given above is one of cutting it down to size. There is also a creative description. In order to explain this we first discuss certain push-outs of $n$-precats. An object of $\Theta ^n$ represents a presheaf (i.e. $n$-precat). If $M$ is an object we denote the $n$-precat represented by $M$ as $h(M)$. A morphism of $n$-precats $h(M)\rightarrow A$ is the same thing as an element of $A_M$. Note that direct limits exist in the category of $n$-precats (as in any category of presheaves). In particular push-outs exist.
We construct the following standard $n$-precats. Let $M= (m_1, \ldots , m_l)$ with $l\leq n-1$, and let $m\geq 1$ (although by the remark below we could also restrict to $m\geq
2$). Let $-1 \leq k \leq n-l-1$. We will state the constructions by universal properties (although we give an explicit construction later). Note that these universal properties admit solutions because we work in the category of presheaves over a given category $\Theta ^n$ so the necessary limits exist.
Define $\Sigma = \Sigma (M, [m], \langle k,k+1\rangle )$ to be the universal $n$-precat with $k$-morphisms $a,b$ in $\Sigma _{M,m/}$ (i.e. $a,b\in \Sigma _{M,m, 1^k}$) and a $k+1$-morphism $$v= (v_1, \ldots , v_m) \in (\Sigma _{M,1/} \times _{\Sigma _{M,0}}
\ldots \times _{\Sigma _{M,0}}\Sigma _{M,1/} )_{1^{k+1}}$$ such that $s(a)=s(b)$, $t(a)=t(b)$, and such that the images of $a$ and $b$ by the usual map to the product of $\Sigma _{M,1/}$ are $s(v)$ and $t(v)$ respectively. Note that $h=h(M,m, 1^{k+1})$ is the universal $n$-precat with a $k+1$-morphism $u$ in $\Sigma _{M,m/}$ (i.e. $u\in \Sigma
_{M,m,1^{k+1}}$). Note that setting $a$ to $s(u)$, $b$ to $t(u)$ and $v$ to the image of $u$ by the usual map to the product, we obtain (by the universal property of $\Sigma$) a morphism $$\varphi = \varphi (M, [m], \langle k,k+1\rangle ) :
\Sigma (M, [m], \langle k,k+1\rangle )
\rightarrow h(M,m,1^{k+1})$$ We will show below that $\varphi$ is a cofibration, [^2]at the same time giving an explicit construction of $\Sigma$ as a pushout of representable presheaves. Before doing that, we mention the modifications to the above definition necessary for the boundary cases $k=-1$ and $k=n-l-1$.
For $k=-1$, $\Sigma (M, [m], \langle -1,0\rangle )$ is the universal $n$-precat with an object $$v= (v_1, \ldots , v_m) \in \Sigma _{M,1} \times _{\Sigma _{M,0}}
\ldots \times _{\Sigma _{M,0}}\Sigma _{M,1} .$$ As $h(M,m,0)$ is the universal $n$-precat with an object $u\in h_{M,m}$ we have an object $v$ as above for $h$ (the image of $u$ by the usual map) so we obtain $\Sigma \rightarrow h$.
For $k=n-l-1$, $\Sigma (M, [m], \langle n-l-1,n-l\rangle )$ is the universal $n$-precat with $a,b\in \Sigma _{M,m, 1^{n-l-1}}$ such that $s(a)=s(b)$ and $t(a)=t(b)$ and such that $a$ and $b$ map to the same elements of $$(\Sigma _{M,1/} \times _{\Sigma _{M,0}}
\ldots \times _{\Sigma _{M,0}}\Sigma _{M,1/} )_{1^{n-l-1}} .$$ Note that $h=h(M, m, 1^{n-l})$ is normally speaking not defined because the length of the multiindex $(M,m,1^{n-l})$ is $n+1$. Thus we formally define this $h$ to be equal to $h(M,m, 1^{n-l-1})$ and take the elements $a=b$ equal to the canonical $n-l-1$-morphism in $h_{M,m/}$. This gives a morphism $\Sigma \rightarrow h$.
We will now give an explicit construction of $\Sigma$ and use this to show that $\Sigma \rightarrow h$ is a cofibration. (The boundary cases will be left to the reader). In general the universal $n$-precat with a collection of elements with certain equalities required, is a quotient of the disjoint union of the representable $n$-precats corresponding to the elements we want, by identifying pairs of morphisms from the representable $n$-precats corresponding to the elements which need to be equal. We do this in several steps. First, the universal $n$-precat $\Upsilon = \Upsilon (M, [m], 1^k)$ with element $$v= (v_1, \ldots , v_m) \in (\Upsilon _{M,1/} \times _{\Sigma _{M,0}}
\ldots \times _{\Upsilon _{M,0}}\Upsilon _{M,1/} )_{1^{k}}$$ is constructed as the quotient of the disjoint union of $m$ copies of $h(M,1,1^k)$ making $m-1$ identifications over pairs of maps $$h(M,1,1^k) \leftarrow h(M) \rightarrow h(M,1,1^k).$$ This is the same as taking the pushout of the diagram $$h(M,1,1^k) \leftarrow h(M) \rightarrow h(M,1,1^k) \leftarrow \ldots
\leftarrow h(M) \rightarrow h(M,1, 1^k).$$ Now $\Sigma (M, [m], \langle k,k+1\rangle )$ is the quotient of the disjoint union $$h(M, m, 1^k)^a\sqcup h(M, m, 1^k)^b \sqcup \Upsilon (M, [m], 1^{k+1})$$ by the following identifications (the superscripts $a$ and $b$ in the above notation are there to distinguish between the two components, which correspond respectively to choosing $a$ and $b$). There are two maps (dual to $s$ and $t$) $s^{\ast}, t^{\ast}: h(M,m,1^{k-1})\rightarrow h(M,m, 1^k)$, and we identify over the pairs of morphisms $$h(M, m, 1^k)^a \stackrel{s^{\ast}}{\leftarrow}
h(M,m, 1^{k-1})\stackrel{s^{\ast}}{\rightarrow} h(M, m, 1^k)^b$$ and $$h(M, m, 1^k)^a \stackrel{t^{\ast}}{\leftarrow}
h(M,m, 1^{k-1})\stackrel{t^{\ast}}{\rightarrow} h(M, m, 1^k)^b.$$ Then we also identify over the pairs of maps $$h(M,m,1^k)^a\leftarrow \Upsilon (M, [m], 1^{k})
\stackrel{s^{\ast}}{\rightarrow}\Upsilon (M, [m], 1^{k+1})$$ and $$h(M,m,1^k)^b\leftarrow \Upsilon (M, [m], 1^{k})
\stackrel{t^{\ast}}{\rightarrow}\Upsilon (M, [m], 1^{k+1})$$ where the left maps are induced by the collection of principal morphisms $1\rightarrow m$. The result of all these identifications is $\Sigma (M, [m], \langle k,k+1\rangle )$. This gives an explicit construction for those wary of just defining things by the universal property.
We now show that the morphism $\varphi : \Sigma \rightarrow h$ is a cofibration in the sense of §3 below, i.e. injective on all levels except the top one.
We will say that a diagram of $n$-precats of the form $$A{\stackrel{\displaystyle \rightarrow}{\rightarrow}}B \rightarrow C$$ (where the two compositions are the same) is [*semiexact*]{} if the morphism from the coequalizer of the two arrows to $C$ is a cofibration in the sense of §3 below. Our above construction gives $\Sigma$ as the coequalizer of $$\Upsilon ' \sqcup \Upsilon ' \sqcup h' \sqcup h' {\stackrel{\displaystyle \rightarrow}{\rightarrow}}\Upsilon \sqcup h^a \sqcup h^b$$ where $$\Upsilon = \Upsilon (M, [m], 1^{k+1}), \;\;\;
\Upsilon ' = \Upsilon (M, [m], 1^{k}),$$ $$h^a (\mbox{resp.}\,\, h^b) := h(M,m,1^k),\;\;\; h' := h(M,m, 1^{k-1}).$$ We would like to prove that $$\Upsilon ' \sqcup \Upsilon ' \sqcup h' \sqcup h' {\stackrel{\displaystyle \rightarrow}{\rightarrow}}\Upsilon \sqcup h^a \sqcup h^b \rightarrow h(M, m, 1^{k+1})$$ is semiexact. To prove this it suffices (by a simple set-theoretic consideration) to show that $$h' \sqcup h' {\stackrel{\displaystyle \rightarrow}{\rightarrow}}h^a \sqcup h^b \rightarrow h(M, m, 1^{k+1})$$ $$\Upsilon ' {\stackrel{\displaystyle \rightarrow}{\rightarrow}}\Upsilon \sqcup h^a \rightarrow h(M, m, 1^{k+1})$$ and $$\Upsilon ' {\stackrel{\displaystyle \rightarrow}{\rightarrow}}\Upsilon \sqcup h^b \rightarrow h(M, m, 1^{k+1})$$ are semiexact.
The first statement follows from the claim that for any $M= (m_1, \ldots ,
m_l)$, $$h(M) \sqcup h(M){\stackrel{\displaystyle \rightarrow}{\rightarrow}}h(M, 1) \sqcup h(M, 1) \rightarrow h(M,1,1)$$ is semi-exact. Let $P= (p_1, \ldots , p_n)$ be an element of $\Theta ^n$ (some of the $p_j$ may be zero). A morphism $P\rightarrow (M, 1)$ corresponds to a collection of morphisms $f_i:p_i
\rightarrow m_i$ for $i\leq l$ and $f_{l+1}:p_{l+1} \rightarrow 1$, up to equivalence. The equivalence relation is obtained by saying that if one of the morphisms factors through $0$ then the subsequent ones don’t matter. The first thing to note is that the two morphisms $(I, s), (I,t): h(M,
1)\rightarrow h(M, 1, 1)$ are injective, as follows directly from the above description. Now suppose that $f, g: P\rightarrow (M, 1)$ are two morphisms such that $(I,s)\circ f = (I, t)\circ g$. Since $s: 0\rightarrow 1$ composed with anything is different from $t: 0\rightarrow 1$ composed with the same, this means that one of the $f_i$ must factor through $0$ for $i\leq l+1$, and that $f_j=g_j$ for $j\leq i$. If it is the case for $i\leq l$ then $f$ and $g$ both come from the morphism $(f_1, \ldots , f_l): P \rightarrow M$, which is equivalent to $(g_1, \ldots , g_l)$, via either one of the morphisms $M\rightarrow (M, 1)$. If $i=l+1$ then $f_{l+1}=g_{l+1}$ factors through one of the two morphisms $0\rightarrow 1$, so $f$ and $g$ both come from the morphism $(f_1, \ldots , f_l)=(g_1, \ldots , g_l)$ via the morphism $M\rightarrow (M, 1)$ corresponding to the morphism $0\rightarrow 1$ occuring above. Thus $f$ and $g$ are equivalent in the coequalizer, giving the claim for this paragraph and thus the first of our semiexactness statements.
For the next semiexactness statement, we first note that $\Upsilon \rightarrow
h(M, m, 1^{k+1})$ is cofibrant. In fact we can describe $\Upsilon$ as a subsheaf of $h(M,m, 1^{k+1})$ as follows. For $P= (p_1 , \ldots , p_n)$ the morphisms from $P$ to $(M,m, 1^{k+1})$ are the sequences of morphisms $f= (f_1, \ldots , f_{l+k +2})$ with $f_i : p_i \rightarrow m_i$ (or taking values in $m$ or $1$ as appropriate depending on $i$). Such a morphism is contained in $\Upsilon _P$ if and only if the morphism $f_{l+1}: p_{l+1}\rightarrow m$ factors through one of the principal morphisms $1\rightarrow m$ (we leave to the reader to verify that $\Upsilon$ is equal to this subsheaf). Suppose $f\in \Upsilon _P$ and $g= (g_1, \ldots , g_{l+k+1})\in
h^a_P$, projecting to the same element of $h(M,m,1^{k+1})_P$. Note that $g$ projects to the element $(g_1, \ldots , g_{l+k+1}, s)$ where $s: 0\rightarrow
1$ denotes the source map (or really its dual but for purposes of the present argument we omit the dual notation). In particular $f$ is equivalent to $(g_1, \ldots , g_{l+k+1}, s)$, which implies that (up to changing $f$ and $g$ in their equivalence classes) $g_{l+1}$ factors through one of the principal maps $1\rightarrow m$ and $f_{l+k+2}=s$. This exactly means that $f$ comes from $\Upsilon '_P \rightarrow \Upsilon _P$ and $g$ from $\Upsilon ' _P \rightarrow h^a_P$. Thus $f$ and $g$ are equivalent in the coequalizer, giving the second of our semiexactness statements.
The proof of the third semiexactness statement is the same as that of the second (although $s$ above would be replaced by $t$). This completes the proof that the standard morphisms $\Sigma \rightarrow h$ are cofibrations (modulo the boundary cases which we have left to the reader).
An $n$-precat $A$ is an easy $n$-category if and only if every morphism $\Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A$ extends to a morphism $h(M, m, 1^{k+1})\rightarrow A$. A marked easy $n$-category is an $n$-precat $A$ together with choice of extension for every morphism $\Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A$. Finally, we say that a [*partially marked $n$-precat*]{} is an $n$-precategory provided with a distinguished subset $\mu$ of the set of all morphisms of the form $f:\Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A$, and for each such morphism, a chosen extension $f^{\mu}$ to $h(M, m, 1^{k+1})$.
If $(A, \mu )$ is a partially marked $n$-precat, then we define a new partially marked $n$-precat $Raj(A, \mu )$ by taking the pushout via $\varphi (M, [m], \langle k,k+1\rangle )$ for all morphisms $$\Sigma (M,[m], \langle k,k+1\rangle )\rightarrow A$$ which are not in the subset $\mu$ of marked ones.
[*Remark:*]{} In the above notations, if $m=1$ then $$\Sigma (M, [1], \langle k,k+1\rangle ) =
\Upsilon (M, [1], 1^{k+1}) = h(M, 1 , 1^{k+1})$$ so the pushout by $\varphi (M, [1], \langle k,k+1\rangle )$ is trivial and we can ignore these cases if we like in the previous notation (and also in the notion of marking).
If $A$ is an $n$-precat then the marked easy $n$-category $Cat(A)$ is obtained by iterating infinitely many times (i.e. over the first countable ordinal) the operation $(A', \mu ') \mapsto Raj(A', \mu ')$, starting with $(A,
\emptyset )$.
[*Proof:*]{} If $(B, \nu )$ is a marked easy $n$-category and $(A', \mu ') \rightarrow
(B, \nu
)$ is a morphism compatible with the partial marking of $A'$, then there is a unique extension to a morphism $Raj( A', \mu ' )\rightarrow B$ compatible with the partial marking of $Raj(A', \mu ')$. It follows that if we set $Cat '(A)$ equal to the result of the iteration described in the lemma, then there is a unique morphism $Cat '(A)\rightarrow B$ compatible with the partial marking of $Cat '(A)$ and extending the given morphism $A\rightarrow B$. But $Cat '(A)$ is fully marked. By the universal property of $Cat (A)$ this implies that $Cat (A)=Cat '(A)$. [$/$$/$$/$]{}
We will often have a need for the following construction. If $A$ is an $n$-precat then iterate (over the first countable ordinal) the operation $(A', \mu ') \mapsto Raj (A', \emptyset )$. Call this $BigCat(A)$. Another way to describe this consruction is that we throw in an infinite number of times the pushouts of all of the required diagrams (which is in some sense a more obvious way to obtain an $n$-category). There is an obvious morphism $Cat (A) \rightarrow BigCat (A)$. One of the advantages of the $BigCat$ construction is that $BigCat(A) \cong BigCat (BigCat(A))$ (although the natural maps are not this isomorphism). More generally, we will use the terminology “reordering” below to indicate that a sequence of pushouts can be done in any order (subject to the obvious condition that the things over which the pushouts are being done exist at the time they are done!), which yields isomorphisms such as $BigCat(A) \cong BigCat (BigCat(A))$.
If $B\leftarrow A \rightarrow C$ is a diagram of $n$-categories, then we define the [*category-theoretic pushout*]{} to be $Cat (B\cup ^AC)$. It is again an $n$-category. We will also often use just the pushout of $n$-precats, i.e. the pushout of presheaves over $\Theta ^n$.
[**.The closed model category structure**]{}
We now come to the first main definition. A morphism $A\rightarrow B$ of $n$-precats (that is, a morphism of presheaves on $\Theta ^n$) is a [*weak equivalence*]{} if the induced morphism $Cat(A)\rightarrow Cat(B)$ is an exterior equivalence of $n$-categories in the sense of Tamsamani ([@Tamsamani] §1.3). Note in particular that we don’t require it to be an easy equivalence—which would be too strong a condition.
The second main definition is relatively easy: we would like to say that a morphism $A\rightarrow B$ of $n$-precats is a cofibration if it is a monomorphism of presheaves on $\Theta ^n$. However, this doesn’t work out well at the top degree (for example, the category of sets with isomorphisms as weak equivalences and injections as cofibrations, is not a closed model category [@Quillen]). Thus we leave the top level alone and say that a morphism $A\rightarrow B$ of $n$-precats is a [*cofibration*]{} if for every $M= (m_1,
\ldots , m_k)$ with $k <n$, the morphism $A_M \rightarrow B_M$ is injective. A cofibration which is a weak equivalence is called a [*trivial cofibration*]{}.
The third definition which goes along automatically with these two is that a morphism $A\rightarrow B$ of $n$-precats is a [*fibration*]{} if it satisfies the lifting property for trivial cofibrations, that is if every time $U\hookrightarrow V$ is a trivial cofibration and $U\rightarrow A$ and $V\rightarrow B$ are morphisms inducing the same $U\rightarrow B$ then there exists a lifting to $V\rightarrow A$ compatible with the first two morphisms.
We recall from [@Quillen] the definition of [*closed model category*]{}, as well as from [@QuillenAnnals] an equivalent set of axioms.
\[cmc\] The category of $n$-precats with the weak equivalences, cofibrations and fibrations defined above, is a closed model category.
[*Some lemmas*]{}
The proof of Theorem \[cmc\] is by induction on $n$. Thus we may assume that the theorem and all of the lemmas contained in the present section and §§4-6 are true for $n'$-precats for all $n'<n$. In view of this, we state all (or most) of the lemmas before getting to the proofs.
Our proof will be modelled on the proof of Jardine that simplicial presheaves on a site form a closed model category [@Jardine]. The main lemma that we need (which corresponds to the main lemma in [@Jardine]) is
\[pushout\] Suppose $A\rightarrow B$ is a trivial cofibration and $A\rightarrow C$ is any morphism. Let $D = B\cup ^AC$ be the push-out of these two morphisms (the push-out of $n$-precats). Then the morphism $C\rightarrow D$ is a weak equivalence.
This lemma speaks of push-out of $n$-precats. Applying the construction $Cat$ we obtain a notion of push-out of $n$-categories: if $A\rightarrow B$ and $A\rightarrow C$ are morphisms of $n$-categories (i.e. morphisms of the corresponding $n$-precats) then define the [*push-out $n$-category*]{} to be $$Cat ( B\cup ^AC ).$$
If $A\rightarrow B$ is an equivalence of $n$-categories then $$C\rightarrow Cat ( B\cup ^AC )$$ is an equivalence of $n$-categories.
We will come back to push-out below in the section on Siefert-Van Kampen.
Going along with the previous lemma is something that we would like to know:
\[equiv\] If an $n$-precat $A$ is an $n$-category in the sense of [@Tamsamani] then the morphism $A\rightarrow Cat(A)$ (resp. the morphism $A\rightarrow BigCat(A)$) is an equivalence of $n$-categories in the sense of [@Tamsamani].
Another lemma which is an important technical point in the proof of everything is the following. An $n$-precat $A$ can be considered as a collection $\{
A_{m/}\}$ of $n-1$-precats (functor of $\Delta$ and the first element is a set). We obtain the collection $\{
Cat(A_{m/})\}$ which is a functor from $\Delta$ to the category of $n-1$-precats. Divide by the equivalence relation setting the $0$-th element to a constant $n-1$-precat, in this way we obtain a new $n$-precat denoted $Cat _{\geq 1}
(A)$.
\[partialCat1\] Suppose that $A$ and $B$ are $n$-precats and $f: A\rightarrow B$ is a morphism which induces an equivalence on the $n-1$-categories $Cat (A_{m/})\rightarrow
Cat (B_{m/})$. Then $Cat(A)\rightarrow Cat(B)$ (resp. $BigCat(A)\rightarrow
BigCat(B)$) is an equivalence.
\[partialCat\] The morphism $$Cat (A) \rightarrow Cat (Cat _{\geq 1}(A))$$ is an equivalence of $n$-categories.
[*Proof:*]{} The morphism $A \rightarrow Cat _{\geq 1}(A)$ satisfies the hypotheses of the previous lemma so the corollary follows from the lemma. [$/$$/$$/$]{}
\[coherence\] For any $n$-precat $A$, the morphism $A\rightarrow Cat(A)$ (resp. $A\rightarrow
BigCat(A)$) is a weak equivalence.
The closed model structure that we already have by induction for $n-1$-precats allows us to deduce some things about $n$-categories. Let $HC_{n-1}$ denote the localization of $PC_{n-1}$ by inverting the set weak equivalences, which is also (see \[htytype\] below) the localization of $n-1$-categories by inverting the set of equivalences. We know from the closed model structure [@Quillen] that this is equivalent to the category of fibrant (and automatically cofibrant) objects where we take as morphisms, the homotopy classes of morphisms. We also know that a morphism in $PC_{n-1}$ is a weak equivalence if and only if it projects to an isomorphism in $HC_{n-1}$ ([@Quillen], Proposition 1, p. 5.5). In particular by \[equiv\] in degree $n-1$ we know that a morphism of $n-1$-categories is an equivalence if and only if it projects to an isomorphism in $HC_{n-1}$.
Suppose $A$ is an $n$-category. For $x,y\in A_0$ we have an $n-1$-category $A_1(x,y)$ which we could denote by $Hom_A(x,y)$. Let $LHom_A(x,y)$ denote the image of this object in the localization $HC_{n-1}$. On the other hand, the truncation $T^{n-1}A$ is a $1$-category. We claim that for $x$ fixed, the mapping $y\mapsto LHom_A(x,y)$ is a functor from $T^{n-1}A$ to $HC_{n-1}$. Similarly we claim that for $y$ fixed the mapping $x\mapsto
LHom_A(x,y)$ is a contravariant functor from $T^{n-1}A$ to $HC_{n-1}$. These claims give some meaning at least in a homotopic sense to the notion of “composition with $f: y\rightarrow z$” as a map $LHom _A(x,y)\rightarrow
LHom_A(x,z)$.
We prove the first of the two claims, the proof for the second one being identical. Note that these arguments are generalizations of what is mentionned in [@Tamsamani] Proposition 2.2.8 and the following remark. Suppose $f\in
A_1(y,z)$. Then let $A_2(x,y,f)$ be the homotopy fiber of $A_2(x,y,z)\rightarrow
A_1(y,z)$ over the object $f$ (this is calculated by replacing the above map by a fibrant map and taking the fiber). The condition that $A_2$ be equivalent to $A_1\times _{A_0}A_1$ implies that this homotopy fiber maps by an equivalence to $A_1(x,y)$. On the other hand it maps to $A_1(x,z)$ and this diagram gives a morphism $LHom _A(x,y)\rightarrow LHom _A(x,z)$ in the localized category $H_{n-1}$. We just have to check that this morphism is independent of the choice of $f$ in its equivalence class. For this we use Proposition \[intervalK\] below (there is no circularity because we are discussing $n-1$-categories here). If $f$ is equivalent to $g$ as elements of the $n-1$-category $A_1(y,z)$ then let $K$ denote the $n-1$ category given by \[intervalK\]; there is a morphism $K\rightarrow A_1(y,z)$ sending $0$ to $f$ and $1$ to $g$, and since $K$ is a contractible object (weakly equivalent to $\ast$) this proves that the homotopy fibers over $f$ or $g$ are equivalent to the homotopy fiber product with $K$; we have a single map from here to $A_1(x,z)$ so our two maps induced by $f$ and $g$ are homotopic.
Associativity is given by a similar argument using $A_3$ which we omit.
Once we have our functors $T^{n-1}A\rightarrow HC_{n-1}$ we obtain the following type of statement: suppose $f$ is an equivalence between $u$ and $x$, then composition with $f$ induces an equivalence $LHom_A(x,y)\cong LHom
_A(u,y)$ (and similarly for composition in the second variable).
\[remark\] If $$A\stackrel{f}{\rightarrow }B \stackrel{g}{\rightarrow } C$$ is a pair of morphisms of $n$-categories such that any two of $f$, $g$ or $g\circ f$ are equivalences in the sense of [@Tamsamani] then the third is also an equivalence.
If $f: A\rightarrow B$ and $g:B\rightarrow A$ are two morphisms of $n$-categories such that $fg$ is an equivalence and $gf$ is the identity then $f$ and $g$ are equivalences.
[*Proof:*]{} The fact that composition of equivalences is an equivalence is [@Tamsamani] Lemme 1.3.5. The statement concluding that $f$ is an equivalence if $g$ and $g\circ f$ are, is a direct consequence of Tamsamani’s interpretation of equivalence in terms of truncation operations ([@Tamsamani] Proposition 1.3.1).
For the conclusion for $g$, note first of all that on the level of truncations $T^nA\rightarrow T^nB \rightarrow T^nC$ the fact that $f$ and $gf$ are isomorphisms of sets implies that $g$ is an isomorphism of sets. This gives essential surjectivity. Now suppose $x,y$ are objects of $B$. Choose objects $u,v$ of $A$ and equivalences $f(u)\sim x$ and $f(v)\sim y$. Then composition with these equivalences induces an isomorphism in the localized category $HC_{n-1}$ between $LHom _B(x,y)$ and $LHom_B(f(u),
f(v))$ (see the discussion preceeding this lemma). The image under $g$ of this isomorphism is the same as composition with the images of the equivalences, so we have a diagram $$\begin{array}{ccc}
LHom _B(x,y) & \rightarrow & LHom _C(g(x), g(y)) \\
\downarrow &&\downarrow \\
LHom _B(f(u), f(v)) & \rightarrow & LHom _C(gf(u), gf(v))
\end{array}$$ in the category $HC_{n-1}$. The horizontal arrows are the localizations of the arrows $g:B_1(x,y)\rightarrow C_1(g(x), g(y))$ etc., and the vertical arrows are composition with our chosen equivalences, isomorphisms in $HC_{n-1}$. On the other hand, the bottom arrow fits into a diagram $$LHom _A(u,v)
LHom _B(f(u), f(v)) \rightarrow LHom _C(gf(u), gf(v))$$ where the first arrow induced by $f$ is an isomorphism, and the composed arrow induced by $gf$ is an isomorphism; thus the bottom arrow of the previous diagram is an isomorphism therefore the top arrow is an isomorphism in $H_{n-1}$. This implies that the morphism $B_1(x,y)\rightarrow C_1(g(x), g(y))$ is an equivalence of $n-1$-categories. This is what we needed to prove to complete the proof that $g$ is an equivalence.
We turn now to the second paragraph of the lemma: suppose $f$ and $g$ are morphisms of $n$-categories such that $fg$ is an equivalence and $gf$ is the identity. The corresponding fact for sets shows that $T^nf$ and $T^ng$ are isomorphisms between the sets of equivalence classes of objects $T^nA$ and $T^nB$. Suppose $x,y\in A_0$. Note that $gf(x)=x$ and $gf(y)=y$. We obtain morphisms $$A_1(x,y) \stackrel{f}{\rightarrow} B_1(fx,fy)
\stackrel{g}{\rightarrow} A_1(gfx,gfy)=A_1(x,y)
\stackrel{f}{\rightarrow} B_1(fgfx, fgfy) = B_1(fx,fy).$$ We have again that $gf$ is the identity on $A_1(x,y)$ and $fg$ is an equivalence on $B_1(fx,fy)$. Inductively by our statement for $n-1$-categories, the morphism $f: A_1(x,y)\rightarrow B_1(fx,fy)$ is an equivalence. This implies that $f: A\rightarrow B$ is an equivalence and hence that $g$ is an equivalence (by the first paragraph of the lemma). [$/$$/$$/$]{}
\[htytype\] The localized category of $n$-precats modulo weak equivalence is equivalent to the category $Ho-n-Cat$ of $n$-categories localized by equivalence defined in [@Tamsamani].
[*Proof:*]{} The functor $Cat$ sends weak equivalences to equivalences (by Lemma \[equiv\] together with Lemma \[remark\]). Thus it induces a functor $c$ on localizations. Let $i$ be the functor induced on localizations by the inclusion of $n$-categories in $n$-precats. The natural transformation $A\rightarrow
Cat(A)$ gives a natural isomorphism $1 \cong c\circ i$ of functors on the localization of $n$-categories. On the other hand, Lemma \[coherence\] says that the same natural transformation induces a natural isomorphism $1\cong
i\circ c$ of functors on the localization of $n$-precats. [$/$$/$$/$]{}
We will prove lemmas \[pushout\], \[equiv\] and \[partialCat1\] all at once in one big induction on $n$. Thus we may assume that they hold for $n' <
n$. All lemmas from here until the end of the big induction presuppose that we know the inductive statement for $n'<n$.
[*Remark on the passage between $Cat$ and $BigCat$ in \[equiv\] and \[partialCat1\]:*]{} The statements for $Cat$ and $BigCat$ are equivalent. Take Lemma \[equiv\] for example. If $A\rightarrow Cat(A)$ is an equivalence for any $n$-category $A$ then $BigCat(A)$ can be constructed as the iteration over the first countable ordinal of the operation $A' \mapsto Cat (A')$ (and starting at $A$). The morphisms at each stage in the iteration are equivalences, so it follows that the morphism $A\rightarrow BigCat(A)$ is an equivalence. On the other hand, suppose we know that $A\rightarrow BigCat(A)$ is an equivalence for any $n$-category $A$. Then $Cat(A)\rightarrow BigCat(Cat(A))$ is an equivalence, but by reordering $BigCat(Cat(A))=BigCat(A)$. Thus the hypothesis also gives that $A\rightarrow BigCat(Cat(A))$ is an equivalence. Lemma \[remark\] then implies that $A\rightarrow Cat(A)$ is an equivalence. We obtain the required statement concerning \[partialCat1\] by using the fact that $Cat(A)\rightarrow BigCat(A)$ (resp. $Cat(B)\rightarrow BigCat(B)$) is an equivalence—note that our proof of \[partialCat1\] comes after our proof of \[equiv\] below—and applying \[remark\].
[*A simplified point of view*]{}
We started to see, in the proof of Lemma \[remark\], a simplified or “derived” point of view on $n$-categories. We will expand on that a bit more here. When we use the statements of the above lemmas for $n-1$-categories, they may be considered as proved in view of our global induction. The homotopy or localized category $HC_{n-1}$ of $n-1$-precats modulo weak equivalence, also equal to the localization of Tamsamani’s $(n-1)-Cat$ by equivalences, admits direct products. There is a functor $T^{n-1}: HC_{n-1}
\rightarrow Sets$ related to the inclusion $i: Sets \subset HC_{n-1}$ by morphisms $iT^{n-1}(X)\rightarrow X$ and $T^{n-1}iS \cong S$ (the first is only well defined in the localized category). Thus if $A\times B\rightarrow C$ is a morphism in $HC_{n-1}$ then we obtain the map $A\times iT^nB \rightarrow C$.
Note that $HC_{n-1}$ admits fibered products over objects of the form $i(S)$ for $S$ a set, since these are essentially just direct products. (However the homotopy category does not admit general fibered products nor, dually, does it admit pushouts.)
We can define the notion of $HC_{n-1}$-category, as simply being a category in the category $HC_{n-1}$ such that the object object is a set. Applying the functor $T^{n-1}$ yields a category, and this category acts on the morphism objects of the previous one, using the above remark.
If $A$ is an $n$-category then taking $A_0$ as set of objects and using the object $LHom_A(x,y)$ as morphism object in $HC_{n-1}$ we obtain an $HC_{n-1}$-category which we denote $HC_{n-1}(A)$. We can write $$Hom _{HC_{n-1}(A)}(x,y) := LHom_A(x,y)$$ which in turn is, we recall, the image of $A_1(x,y)$ in the localization of the category of $n-1$-precats. This is what we used in the proof of Lemma \[remark\] above. The truncation operation $T^{n-1}$ applied to $HC_{n-1}(A)$ gives the $1$-category $T^{n-1}A$. We obtain again the action of this category on the morphism objects in $HC_{n-1}(A)$.
In the next section we will be interested in the notion of [*$HC_{n-1}$-precategory*]{}, a functor $F:\Delta \rightarrow HC_{n-1}$ sending $0$ to a set. An $HC_{n-1}$-precategory $F$ is an $HC_{n-1}$-category if and only if the usual morphisms $$F_p\rightarrow F_1\times _{F_0} \ldots \times _{F_0}F_1$$ are isomorphisms. If $A$ is an $n$-precat then let $HC_{n-1}(A)$ denote the $HC_{n-1}$-precategory which to $p\in \Delta$ associates the image of $A_{p/}$ in the localized category $HC_{n-1}$.
Here is a small remark which is sometimes useful.
\[HCequivCat\] Suppose $f:A\rightarrow B$ is a morphism of $n$-categories and suppose that $HC_{n-1}(A)\rightarrow HC_{n-1}(B)$ is an equivalence in $HC_{n-1}Cat$. Then $f$ is an equivalence.
[$/$$/$$/$]{}
We can also make a similar statement for $HC_{n-1}$-precats under the condition of requiring an isomorphism on the set of objects.
\[HCequivPreCat\] Suppose $f:A\rightarrow B$ is a morphism of $n$-precats and suppose $HC_{n-1}(A)\rightarrow HC_{n-1}(B)$ is an isomorphism of functors $\Delta \rightarrow HC_{n-1}Cat$. Then $f$ is a weak equivalence.
[*Proof:*]{} This is just a restatement of Lemma \[partialCat1\] (in particular it is not available for use in degree $n$ until we have proved \[partialCat1\] below). [$/$$/$$/$]{}
It would have been nice to be able to have an operation on $HC_{n-1}$-precategories which, when applied to $HC_{n-1}(A)$ yields $HC_{n-1}(Cat(A))$. This doesn’t seem to be possible (although I don’t have a counterexample) because the construction we discuss in the next section relies heavily on pushouts but these don’t exist in $HC_{n-1}$. If this had been possible we would have been able to formulate a notion of weak equivalence for $HC_{n-1}$-precats and in particular we would have been able to give a stronger formulation in the previous lemma.
We end this discussion by pointing out that information is lost in passing from $A$ to $HC_{n-1}(A)$. (See the next paragraph for some counterexamples but I don’t have counterexamples for all of the nonexistence statements which are made.) Let $HC_{n-1}Cat$ (resp. $HC_{n-1}PreCat$) denote the categories of $HC_{n-1}$-categories (resp. $HC_{n-1}$-precategories). The functors $n-Cat\rightarrow HC_{n-1}Cat$ and $n-Cat \rightarrow HC_n$ do not enter into a commutative triangle with a morphism between $HC_n$ and $HC_{n-1}Cat$ in either direction. The only thing we can say is that there is an obvious notion of equivalence between two $HC_{n-1}$-categories, and if we let $Ho-HC_{n-1}-Cat$ denote the category of $HC_{n-1}$-categories localized by inverting these equivalences, then there is a factorization $$n-CAT \rightarrow HC_n \rightarrow Ho-HC_{n-1}-Cat$$ but the second arrow in the factorization is not an isomorphism. In particular, when we pass from $A$ to $HC_{n-1}(A)$ we lose information. Nonetheless, it may be helpful especially from an intuitive point of view to think of an $n$-category in terms of its associated object $HC_{n-1}(A)$ which is a category in the homotopy category of $n-1$-categories.
The topological analogy of the above situation (which can be made precise using the Poincaré groupoid and realization constructions [@Tamsamani]—thus providing some counterexamples to support some of the the nonexistence statements made in the previous paragraph) is the following: if $X$ is a space then for each $x,y\in X$ we can take as $h(x,y)$ the space of paths from $x$ to $y$ viewed as an object in the homotopy category $Ho(Top)$. We obtain a category in $Ho(Top)$. If $X$ is connected it is a groupoid with one isomorphism class, thus essentially a group in $Ho(Top)$. This group is just the loop space based at any choice of point, viewed as a group in $Ho(Top)$. It is well known ([@Adams] [@Tanre]) that this object does not suffice to reconstitute the homotopy type of $X$, thus our functor from $Top$ to the category of groupoids in $Ho(Top)$ does not yield a factorization of the localization functor $Top\rightarrow Ho(Top)$. On the other hand, since there is no way to canonically choose a collection of basepoints for an object in $Ho(Top)$, there probably is not a factorization in the other direction either.
[*Another simplified point of view*]{}
We now give another set of remarks relating the present approach to $n$-categories with the usual standard ideas. This is based on the following observation. The proof of the lemma is based on some ideas from the next section so the reader should look there before trying to follow the proof. We have put the lemma here for expository reasons.
\[fibrantpieces\] If $A$ is a fibrant $n$-precat then the $A_{p/}$ are fibrant $n-1$-precats.
[*Proof:*]{} Fix objects $x_0, \ldots , x_p\in A_0$. We show that $A_{p/}(x_0, \ldots ,
x_p)$ is fibrant. Suppose $U\rightarrow V$ is a trivial cofibration of $n-1$-precats. Let $B$ (resp. $C$) be the $n$-precat with objects $0, \ldots , p$ and such that $B_{q/}(i_0, \ldots , i_q)$ (resp. $C_{q/}(i_0, \ldots , i_q)$) is the disjoint union of $U$ (resp. $V$) over all morphisms $f:q\rightarrow p$ such that $f(q)=i_q$. Then (as can be seen by the discussion of the next section) $B\rightarrow C$ is a trivial cofibration. A morphism $B\rightarrow A$ (resp. $C\rightarrow A$) is the same thing as a morphism $U\rightarrow A_{p/}(x_0,\ldots , x_p)$ (resp. $V\rightarrow
A_{p/} (x_0,\ldots , x_p)$). It follows immediately that if $A$ is fibrant then $A_{p/}(x_0,\ldots , x_p)$ has the required lifting property to be fibrant. [$/$$/$$/$]{}
Now we can use the closed model category structure on $PC_{n-1}$ to analyze the collection of $A_{p/}$ when $A$ is fibrant. Recall that morphisms in the localized category between fibrant and cofibrant objects are represented by actual morphisms [@Quillen]. Thus the morphism $$A_{2/} \rightarrow A_{1/} \times _{A_0} A_{1/}$$ which is an equivalence, can be inverted and then followed by the projection to the third edge of the triangle to give $$A_{1/} \times _{A_0} A_{1/}\rightarrow A_{2/} \rightarrow A_{1/}.$$ We get a morphism “composition” $$m:A_1(x,y)\times A_1(y,z) \rightarrow A_1(x,z)$$ which represents the composition $$LHom_A(x,y)\times LHom_A(y,z)\rightarrow
LHom_A(x,z)$$ of the previous “simplified point of view”. Of course our composition morphism $m$ is not uniquely determined but depends on the choice of inversion of the original equivalence. In particular $m$ will not in general be associative. However $A_{3/}$ gives a homotopy in the sense of Quillen between $m(m(f,g),h)$ and $m(f, m(g,h))$. This can be turned into a homotopy in the sense of the $n-1$-categories of morphisms (an exercise left to the reader).
[**.Calculus of “generators and relations”**]{}
For the proofs of \[equiv\] and \[partialCat1\] we need a close analysis of an operation which when iterated yields $BigCat$. This analysis will lead us to a point of view which generalizes the idea of generators and relations for an associative monoid. At the end we draw as a consequence one of the main special lemmas needed to treat the special case \[specialcase\] in the proof of \[pushout\].
The overall goal of this section is to investigate the operation $A\mapsto
Cat(A)$ in the spirit of looking at the simplicial collection of $n-1$-precats $A_{p/}$ as a functor from $\Delta$ to our closed model category in degree $n-1$. We would like to understand the transformation which this functor undergoes when we apply the operation $Cat$ to $A$.
We first describe a general type of operation which we often encounter. Suppose $A$ is an $n$-precat and suppose $A_{m/}\rightarrow B$ is a cofibration of $n-1$-precats provided with a morphism $\pi :B\rightarrow A_0 \times \ldots
\times A_0$ making the composition $$A_{m/}\rightarrow
B\rightarrow A_0 \times \ldots \times
A_0$$ equal to the usual morphism (there are $m+1$ factors $A_0$ in the product). We can alternatively think of this as a collection of cofibrations $$A_{m/} (x_0, \ldots , x_m) \rightarrow B(x_0,\ldots , x_m)$$ for all sequences of objects $x_i \in A_0$. Then we define the cofibration of $n$-precats $$A\rightarrow {{\cal I}}(A; A_{m/}\rightarrow B)$$ as follows (the projection $\pi$ is part of the data even though it is not contained in the notation). For any $p$, ${{\cal I}}(A; A_{m/}\rightarrow
B)_{p/}$ is the multiple pushout of $A_{p/}$ and $A_{m/}\rightarrow B$ over all morphisms $A_{m/} \rightarrow A_{p/}$ coming from morphisms $p\rightarrow m$ which do not factor through $0$. Functoriality is defined as follows: if $q\rightarrow p$ is a morphism then for any $f:p\rightarrow m$ such that the composition $q\rightarrow m$ doesn’t factor through $0$, we define the morphism of functoriality on the part of the pushout corresponding to $f$ as the identity in the obvious way; on the other hand, if $f: p\rightarrow m$ is a morphism such that the composition $q\rightarrow m$ factors through $0\rightarrow m$ then we obtain (from the projection $\pi$) a morphism $B\rightarrow A_0$ extending the morphism $A_{m/}\rightarrow A_0$ and so that part of the pushout is sent into the image of $A_0$ in $A_{q/}$.
We call $A\rightarrow {{\cal I}}(A; A_{m/}\rightarrow B)$ the [*pushout of $A$ induced by $A_{m/}\rightarrow B$*]{}. Using Lemma \[pushout\] in degree $n-1$ we find that if $A_{m/}\rightarrow
B$ is a trivial cofibration then the morphisms $$A_{p/} \rightarrow {{\cal I}}(A; A_{m/}\rightarrow B)_{p/}$$ are trivial cofibrations.
This operation occurs notably in the process of doing $Cat$ or $BigCat$ to $A$. Fix $m\geq 1$, $M=(m_1, \ldots , m_l)$, $m'$ and $k$. Let $$\Sigma := \Sigma (m, M, [m'], \langle
k,k+1 \rangle ),$$ and $$\varphi := \varphi (m, M, [m'], \langle k,k+1 \rangle ):\Sigma
\rightarrow h(m,M,m', 1^{k+1}).$$ Suppose again $a: \Sigma\rightarrow A$ is a morphism and let $C$ be the pushout $n$-precat of $A$ and $\varphi$ over $a$. In this case note that $\Sigma$ and $h(m,M,m',
1^{k+1})$ are pushouts of diagrams of objects entirely within the category $(m, \Theta ^{n-1})$ of objects of the form $(m,M')$. The restriction of $A$ to this category is just $A_{m/}$. Let $\psi : A_{m/}\rightarrow F$ be the pushout $n-1$-precat of $\varphi$ over $a$ considered in this way. Then $C = {{\cal I}}(A; A_{m/}\rightarrow F)$ is the pushout of $A$ induced by $\psi$ (note that $\psi$ admits a projection $\pi$ in an obvious way). The proof is that $h(m,M,m', 1^{k+1})$ has exactly the same description as a pushout of $\Sigma$.
Suppose $A$ is an $n$-precat. Define a new $n$-precat $Fix(A)$ by iterating the above operation of pushout by all standard cofibrations $\varphi (m, M, [m'], \langle k,k+1 \rangle )$, over all possible values of $m$, $M$, $m'$ and $k$, and repeating this operation a countable number of times. By reordering, $Fix(A)$ may be seen as obtained from $A$ by a sequence of standard pushouts of the form $$A' \rightarrow {{\cal I}}(A', A'_{m/} \rightarrow BigCat(A'_{m/})).$$ In particular it is clear that each $A_{p/}\rightarrow Fix(A)_{p/}$ is a trivial cofibration. On the other hand it is also clear that the $Fix(A)_{p/}$ are $n-1$-categories (they are obtained by iterating operations of the form, taking $BigCat$ then taking a bunch of pushouts then taking $BigCat$ and so on an infinite number of times—and such an iteration is automatically an easy $n-1$-category).
In order to get to $Cat(A)$ or $BigCat(A)$ we need another type of operation which relates the different $A_{m/}$. Suppose $A$ is an $n$-precat, fix $m\geq 2$ and suppose that we have a diagram $$A_{m/}
\stackrel{f}{\rightarrow} Q \stackrel{g}{\rightarrow} A_{1/} \times _{A_0}
\ldots \times _{A_0}A_{1/}$$ with the first arrow cofibrant. Then we define the pushout $A\rightarrow {{\cal J}}(A; f,g)$ as follows. ${{\cal J}}(A; f,g)_{p/}$ is the multiple pushout of $A_{m/}\rightarrow A_{p/}$ and $A_{m/} \rightarrow Q$ over all maps $p\rightarrow m$ not factoring through any of the principal maps $1\rightarrow m$. The morphisms of functoriality are defined in the same way as for the construction ${{\cal I}}$ using the map $g$ here.
[*Remark:*]{} This pushout changes the object over $1\in \Delta$ because there are morphisms $1\rightarrow m$ (the faces other than the principal ones) which don’t factor through the principal face maps.
The remaining of our standard pushouts which are not covered by the operation ${{\cal I}}$ are covered by this operation ${{\cal J}}$. Fix some $m\geq 2$ and $k$. Write $$\Sigma\;\;\; \mbox{for} \;\;\; \Sigma ([m], \langle
k,k+1 \rangle ),$$ and $$\varphi := \varphi ([m], \langle k,k+1 \rangle ):\Sigma
\rightarrow h(m, 1^{k+1}).$$ We have a diagram of $n-1$-precats $$\Sigma _{m/}
\stackrel{f(m,k)}{\rightarrow} h(1^{k+1}) \stackrel{g(m,k)}{\rightarrow}
\Sigma _{1/} \times _{\Sigma
_0} \ldots \times _{\Sigma _0} \Sigma _{1/},$$ and via this diagram $$h(m, 1^{k+1}) = {{\cal J}}(\Sigma ; f(m,k),g(m,k)).$$ It follows that if $A$ is an $n$-precat and $\Sigma \rightarrow A$ is a morphism then the standard pushout $B$ of $A$ along $\varphi$ is of the form $B = {{\cal J}}(A; f,g)$ for appropriate maps $f$ and $g$ induced by the above ones.
We need to have some information about decomposing and commuting the operations ${{\cal I}}$ and ${{\cal J}}$. Suppose $$A_{m/} \stackrel{\varphi}{\rightarrow}
P \stackrel{\pi}{\rightarrow} A_0 \times \ldots \times
A_0$$ is a morphism. Let $$\eta :A_{1/}\rightarrow B$$ denote the multiple pushout of $A_{1/}$ by $\varphi$ over all of the principal morphisms $1\rightarrow m$ (with projection $\nu : B\rightarrow A_0\times
A_0$). We obtain a factorization $$A_{m/} \stackrel{\varphi}{\rightarrow}
P \stackrel{\psi}{\rightarrow} B\times _{A_0} \ldots \times _{A_0} B$$ and we have $${{\cal I}}(A; \varphi , \pi )= {{\cal J}}({{\cal I}}(A; \eta ,\nu ); \varphi , \psi ).$$ In this way we turn an operation of the form ${{\cal I}}$ for $m$ into an operation of the form ${{\cal I}}$ for $1$ followed by an operation of the form ${{\cal J}}$ for $m$.
We define a type of operation combining operations of the form ${{\cal J}}$ for $m$ with operations of the form ${{\cal I}}$ for $1$. However, we would like to keep track of certain sub-$n-1$-precats of $A_{m/}$ and $A_{1/}$. So we say that an [*$(m,1)$-painted $n$-precat*]{} (or just [*painted $n$-precat*]{} in the current context where $m$ is fixed) is an $n$-precat $A$ together with cofibrations of $n-1$-precats $A^{\ast}_{m/} \rightarrow
A_{m/}$ and $A^{\ast}_{1/} \rightarrow A_{1/}$. We require a lifting of the standard morphism to $$A^{\ast}_{m/} \rightarrow A^{\ast}_{1/} \times _{A_0} \ldots \times _{A_0}
A^{\ast}_{1/}.$$
Suppose $(A,A^{\ast}_{m/},A^{\ast}_{1/})$ is a painted $n$-precat, and suppose that we have morphisms $$A^{\ast}_{1/}\stackrel{\eta}{\rightarrow} B\stackrel{\nu}{\rightarrow}A_0\times
A_0$$ and $$A^{\ast}_{m/} \stackrel{\varphi}{\rightarrow}
P \stackrel{\psi}{\rightarrow} B\times _{A_0} \ldots \times _{A_0} B$$ compatible with the previous lifting of the standard morphism to the painted parts. Let $\eta '$, $\nu '$, $\varphi '$ and $\psi '$ be obtained by taking the pushouts of the above with $A_{1/}$ or $A_{m/}$. Then we define a new painted $n$-precat $${{\cal J}}'(A; \eta , \nu ; \varphi , \psi )
:= {{\cal J}}({{\cal I}}(A; \eta ',\nu '); \varphi ', \psi '),$$ with painted parts $(P, B)$ replacing $(A_{m/}^{\ast}, A_{1/}^{\ast})$. This operation now behaves well under iteration: the composition of two such operations is again an operation of the same form. Furthermore, our operations ${{\cal I}}$ and ${{\cal J}}$ coming from standard trivial cofibrations can be interpreted as operations of the above type if the original $\Sigma \rightarrow A$ sends the arrows $(a,b, v_i)$ into the painted parts $A_{m/}^{\ast}, A_{1/}^{\ast}$. These operations are exactly designed to do two things: replacing the painted parts by their associated $n$-categories; and getting the standard map to being an equivalence. In particular, starting with $A^{\ast}_{1/} = A_{1/} $ and $A^{\ast}_{m/} = A_{m/}$, there is a sequence of operations coming from standard trivial cofibrations (concerning only $m$ and $1$) such that, when interpreted as operations on painted $n$-precats, combine into one big operation of the form ${{\cal J}}'$ where $\eta : A^{\ast}_{1/}
\rightarrow B$ is a trivial cofibration to an $n-1$-category, and where the morphism $$P \stackrel{\psi}{\rightarrow} B\times _{A_0} \ldots \times _{A_0} B$$ is an equivalence of $n$-categories.
Going back to the original definition of the operation ${{\cal J}}'$ in terms of ${{\cal J}}$ and ${{\cal I}}$ we find that an appropriate sequence of trivial cofibrations can be reordered into an operation of the form $A\mapsto A' = {{\cal I}}(A, \eta ,
\nu )$ followed by ${{\cal J}}(A'; f,g)$ for $$A'_{m/}\stackrel{f}{\leftarrow} {{\cal G}}[m](A)
\stackrel{g}{\rightarrow} A'_{1/}\times _{A_0} \times \ldots \times _{A_0}
A'_{1/}$$ where $g$ is a weak equivalence. Recall that the morphism $A\rightarrow A'$ coming before the operation ${{\cal J}}$ has the property that the $A_{p/}\rightarrow A'_{p/}$ are weak equivalences. Note also that we can assume that $A'_{1/}$ is an $n-1$-category, because it is equal to $B$—there are no morphisms $1\rightarrow 1$ other than the identity and those which factor through $0$—and $B$ can be chosen to be an $n-1$-category. (This paragraph is the conclusion we want; the discussion of painted $n$-precats was just a means to arrive here and will not be used any further below.)
Let $Gen [m](A)$ denote the result of the previous operation, which we can thus write as $$Gen [m](A)= {{\cal J}}(A'; A'_{m/}\stackrel{f}{\rightarrow} {{\cal G}}[m](A)
\stackrel{g}{\rightarrow} A'_{1/}\times _{A_0} \times \ldots \times _{A_0}
A'_{1/}).$$ The pushouts chosen as above may be assumed to contain, in particular, all of the standard pushouts of the second type for $m$.
Put $Gen _1(A)=Fix(A)$ and $Gen _i (A):= Fix(Gen[i](Gen _{i-1}(A)))$ for $i\geq 2$. Let $Gen(A)$ be the inductive limit of the $Gen _i(A)$. Finally, iterate the operation $A'\mapsto Gen(A')$ a countable number of times. It is clear that, by reordering, this yields $BigCat(A)$, since on the one hand all of the necessary pushouts occur, whereas on the other hand only the standard pushouts are used.
[*Proofs of \[equiv\] and \[partialCat1\]*]{}
The above description yields immediately the proofs of these two lemmas.
[*Proof of \[equiv\]:*]{}
Suppose $A$ is an $n$-precat such that $A_{1/}$ is an $n-1$-category and such that $(\ast )$ for all $m$ the morphisms $$A_{m/} \rightarrow A_{1/}\times _{A_0} \ldots \times _{A_0} A_{1/}$$ are weak equivalences of $n-1$-precats. Fix $m$ and apply the operation $Gen [m](A)$. Let $A'$ be the intermediate result of doing the preliminary operations ${{\cal I}}$. The morphism $$f:A'_{m/} \rightarrow {{\cal G}}[m](A)$$ is a trivial cofibration of $n$-precats, using: (1) the fact that ${{\cal G}}[m](A)\rightarrow
A'_{1/}\times _{A_0} \ldots \times _{A_0} A'_{1/}$ is a weak equivalence; (2) the fact that $A'_{1/}$ are $n$-categories equivalent to $A_{1/}$, and noting that direct products of $n$-categories (or fibered products over sets) preserve equivalences; and (3) the hypothesis that $A_{m/}$ is weakly equivalent to the product of the $A_{1/}$, again coupled with the fact that $A'_{m/}$ is weakly equivalent to $A_{m/}$ because the operations ${{\cal I}}$ preserve the weak equivalence type of the $A_{p/}$.
It now follows from the definition of the operation ${{\cal J}}(A' ; f,g)$ that the morphisms $$A_{p/} \rightarrow A'_{p/} \rightarrow
{{\cal J}}(A' ; f,g)_{p/}$$ are weak equivalences. Thus (under the hypothesis $(\ast )$ above) the morphism $A\rightarrow Gen [m](A)$ induces weak equivalences $A_{p/}\rightarrow Gen [m](A)_{p/}$. The same holds always for the operation $Fix$, and iterating these we obtain the conclusion (under hypothesis $(\ast
)$) that $$A_{p/}\rightarrow BigCat(A)_{p/}$$ are weak equivalences.
In the hypotheses of \[equiv\], $A$ is an $n$-category, so $A_{1/}$ is an $n-1$-category and hypothesis $(\ast )$ is satisfied. It follows from above that the morphisms $$A_{p/} \rightarrow BigCat(A)_{p/}$$ are weak equivalences of $n-1$-precats, but since both sides are $n-1$-categories this implies that they are equivalences of $n-1$-categories (using Lemma \[equiv\] in degree $n-1$). Therefore $A\rightarrow BigCat(A)$ is an equivalence of $n$-categories, completing the proof of \[equiv\]. [$/$$/$$/$]{}
[*Proof of \[partialCat1\]:*]{}
Suppose $A\rightarrow B$ is a morphism of $n$-precats which induces weak equivalences of $n-1$-precats $A_{m/} \rightarrow B_{m/}$ for all $m$. Replacing $A$ by $Fix(A)$ and $B$ by $Fix(B)$ conserves the hypothesis. We show that replacement of $A$ by $Gen [m](A)$ and $B$ by $Gen[m](B)$ conserves the hypothesis. Let $A'$ (resp. $B'$) denote the intermediate $n$-precats used in the definition of $Gen[m](A)$ (resp. $Gen[m](B)$). These are the results of applying operations ${{\cal I}}$ to $A$ and $B$, and we may assume as in the previous proof that $A'_{1/}$ and $B'_{1/}$ are $n-1$-categories. We do these operations in a canonical way so as to preserve morphisms $A'\rightarrow
B'$ and $Gen[m](A)\rightarrow Gen[m](B)$. Note that $A_{1/}\rightarrow A'_{1/}$ is a weak equivalence and the same for $B$. Our hypothesis now implies that the morphism $A'_{1/}\rightarrow B'_{1/}$ is an equivalence of $n-1$-categories. Therefore $$A'_{1/} \times _{A'_0} \ldots \times
_{A'_0}A'_{1/} \rightarrow
B'_{1/} \times _{B'_0} \ldots \times _{B'_0}B'_{1/}$$ is a weak equivalence.
As before $$Gen [m](A)= {{\cal J}}(A'; A'_{m/}\stackrel{f_A}{\rightarrow} {{\cal G}}[m](A)
\stackrel{g_A}{\rightarrow} A'_{1/}\times _{A_0} \times \ldots \times _{A_0}
A'_{1/}),$$ with $g_A$ being a weak equivalence. Similarly $$Gen [m](B)= {{\cal J}}(B'; B'_{m/}\stackrel{f_B}{\rightarrow} {{\cal G}}[m](B)
\stackrel{g_B}{\rightarrow} B'_{1/}\times _{B_0} \times \ldots \times _{B_0}
B'_{1/}),$$ with $g_B$ a weak equivalence. It follows immediately that $${{\cal G}}[m] (A) \rightarrow {{\cal G}}[m](B)$$ is a weak equivalence and hence (using the descripition of ${{\cal J}}$ as well as the fact that weak equivalences on the components induce weak equivalences of pushouts, which we know by induction for $n-1$-precats) the morphism $Gen [m](A)\rightarrow Gen [m](B)$ induces weak equivalences $$Gen [m](A)_{p/}\rightarrow Gen [m](B)_{p/}.$$ This shows that the operation $Gen[m]$ preserves the hypothesis of \[partialCat1\]. Taking limits we get that $Gen$ and finally that $BigCat$ preserve the hypothesis: we get that for all $p$, $$BigCat (A)_{p/} \rightarrow BigCat (B)_{p/}$$ is a weak equivalence. This implies that $BigCat(A)\rightarrow BigCat(B)$ is an equivalence of $n$-categories, finishing the proof of \[partialCat1\]. [$/$$/$$/$]{}
[*$1$-free ordered precats*]{}
Suppose $A$ is an $n$-precat. We say that $A$ is [*$1$-free ordered*]{} if there is a total order on the set $A_0$ of objects (which we suppose for simplicity to be finite) such that the following properties are satisfied: (FO1)—for any sequence $x_0,\ldots , x_n$ which is out of order (i.e. some $x_i$ is strictly bigger than $x_{i+1}$), $A_{m/}(x_0,\ldots , x_m)=\emptyset$; (FO2)—for any sequence $x_0, \ldots , x_n$ with $x_{i-1}\leq x_i$ the morphism $$A_{m/}(x_0,\ldots , x_m)\rightarrow A_{1/}(x_0,x_m)$$ is a weak equivalence; and (FO3)—for any stationary sequence $A_{m/}(x,\ldots , x)$ is weakly equivalent to $\ast$.
Properties (FO1) and (FO2) properties are preserved under the operation $BigCat$. Indeed, the standard cofibrations $\Sigma \rightarrow h$ go between $n$-precats which satisfy these conditions, and these conditions are preserved by pushouts over diagrams of morphisms which are order-respecting (i.e. morphisms respecting $\leq$) between $1$-free ordered $n$-precats. Note that if $A$ satisfies (FO1) then any morphism $\Sigma \rightarrow A$ must respect the order.
The condition (FO3) is preserved by pushouts of morphisms which are strictly order-preserving, and also by the operation $Fix$ (which is essentially the same as $Cat _{\leq 1}$). If $A$ is $1$-free ordered, using (FO1) and (FO3) we get that in order to obtain $Cat(A)$ it suffices to use the operation $Fix$ and pushouts for trivial cofibrations $\Sigma \rightarrow h(m, 1^{k+1})$ via morphisms $\Sigma \rightarrow A$ which are strictly order preserving. Thus $BigCat(A)$ again satisfies (FO3). Furthermore, note that these trivial cofibrations do not change the homotopy type for adjacent objects, so if $x,y$ are adjacent in the ordering then $$A_{1/}(x,y)\rightarrow BigCat(A)_{1/}(x,y)$$ is a weak equivalence. We obtain the following conclusion:
\[freeness\] Suppose $A$ is a $1$-free ordered $n$-precat with finite object set. For two objects $x,y\in A_0$ let $x=x_0, \ldots , x_m = y$ be the maximal strictly increasing ordered sequence going from $x$ to $y$. Let $A':= BicCat(A)$. Then the morphisms $$A'_{1/}(x,y)\leftarrow A'_{m/}(x_0,\ldots , x_m) \rightarrow
A_{1/}(x_0,x_1)\times _{A_0} \ldots \times _{A_0} A_{1/}(x_{m-1}, x_m)$$ are weak equivalences. In particular if $A\rightarrow B$ is a morphism of $1$-free ordered $n$-precats (with finite object sets) preserving the ordering, inducing an isomorphism on object sets and inducing equivalences $A_{1/}
(x,y)\rightarrow B_{1/}(x,y)$ for all pairs of adjacent objects $(x,y)$ then $A\rightarrow B$ is a weak equivalence.
[*Proof:*]{} In the diagram $$A'_{1/}(x,y)\leftarrow A'_{m/}(x_0,\ldots , x_m) \rightarrow
A'_{1/}(x_0,x_1)\times _{A_0} \ldots \times _{A_0} A'_{1/}(x_{m-1}, x_m)$$ the left arrow is a weak equivalence by property (FO2) for $A'$ (we have shown that that property is preserved under passage from $A$ to $A'$); and the right arrow is a weak equivalence because $A'$ is an $n$-category. The product on the right may be replaced by that appearing in the statement of the lemma since $A_{1/}(x_{i-1},x_{i})\rightarrow A'_{1/}(x_{i-1},x_{i})$ is an equivalence because $x_{i-1}$ and $x_i$ are adjacent. This gives the first statement of the lemma. For the second statement, note that (using the same notation for objects of $A$ and $B$, and using the notations $A'$ and $B'$ for associated $n$-categories) the first statement implies that $A'_{1/} (x,y)\rightarrow
B'_{1/}(x,y)$ is a weak equivalence for any pair of objects $x,y$. This implies that $A'\rightarrow B'$ is an equivalence of $n$-categories. [$/$$/$$/$]{}
[*Characterization of weak equivalence*]{}
We close this section by mentioning a proposition which gives a sort of uniqueness for the notion of weak equivalence.
\[general\] Suppose $F: PC_n \rightarrow PC_n$ is a functor with natural transformation $i_A: A\rightarrow F(A)$ such that: (a)—for all $A$, $F(A)$ is an $n$-category; (b)—if $A$ is an $n$-category then $i_A$ is an iso-equivalence of $n$-categories (recall that this means an equivalence inducing an isomorphism on sets of objects); and (c)—for any $n$-precat $A$ the morphism $F(i_A): F(A) \rightarrow F(F(A))$ is an equivalence of $n$-categories. Then for any $n$-precat $A$ the morphism $A\rightarrow F(A)$ is a weak equivalence.
[*Proof:*]{} Put $F'(A):= Cat (F(A))$. It is a marked easy $n$-category. We have a morphism $k_A := Cat (i_A):Cat(A)\rightarrow F'(A)$. Letting $j_A: A\rightarrow Cat(A)$ denote the inclusion and $i'_A$ the map $A\rightarrow F'(A)$, note that $k_Aj_A = i'_A$.
The functor $F'$ again satisfies the properties (a), (b) and (c) above. For property (c) note that the map $F'(i'_A)$ is obtained by applying $Cat$ to the composed map $$F(A) \stackrel{F(i_A)}{\rightarrow} F(F(A))\stackrel{F(j_{F(A)})}{\rightarrow}
F(Cat(F(A))),$$ but the first map is an equivalence by hypothesis and the second is an equivalence because of the diagram $$\begin{array}{ccc}
F(A) & \rightarrow & F(F(A)) \\
\downarrow && \downarrow \\
Cat (F(A)) & \rightarrow & F(Cat (F(A)))
\end{array}$$ where the top arrow is $i_{F(A)}$ which is an equivalence by (b) (but it is different from $F(i_A)$!), the left vertical arrow is the equivalence $i_{F(A)}$ (by (a)) and the bottom arrow is the equivalence $i_{Cat(F(A))}$ again an equivalence by (b).
We have the following diagram: $$\begin{array}{ccccc}
Cat(A) & \rightarrow & Cat(Cat(A)) &&\\
\downarrow & & \downarrow &&\\
F' (A) & \rightarrow & F'(Cat(A)) & \rightarrow & F'(F'(A))
\end{array} .$$ The morphism on the top is $Cat(j_A)$, and the vertical morphisms are $k_A$ and $k_{Cat(A)}$ respectively. The morphisms on the bottom are $F'(j_A)$ and $F'(k_A)$ respectively. The diagram comes from naturality of $k$. The top arrow $Cat(j_A)$ and the middle vertical arrow $k_{Cat(A)}$ are equivalences, as is the composition along the bottom $F'(i'_A)$. On the other hand, all of the arrows are identities on the sets of objects. Thus, when morphisms are equivalences they are in fact iso-equivalences, and in particular equivalences on the level of the $n-1$-categories $(\cdot )_{p/}$. The closed model structure for $n-1$ implies that a morphism of $n-1$-categories is a weak equivalence (hence an equivalence) if and only if it projects to an isomorphism in the localized category. Look at the images of the above diagram in the localized category of $n-1$-precats after applying the operation $(\cdot )_{p/}$. The equivalences that we know show that the bottom left arrow goes to an arrow which has a left and right inverse. It follows that it goes to an invertible arrow, i.e. an isomorphism, in the localized category. Its left inverse, the left vertical arrow, must also go to an isomorphism. This implies that for each $p$, the map $Cat(A)_{p/} \rightarrow F'(A)_{p/}$ is an equivalence. By definition then $A\rightarrow F(A)$ is a weak equivalence. [$/$$/$$/$]{}
[**.Compatibility with products**]{}
The goal of the present section is to prove the following theorem.
\[ce\] Suppose $A$ and $B$ are $n$-precats. Then the morphism $$A\times B \rightarrow Cat(A)\times Cat(B)$$ is a weak equivalence.
Before getting to the proof, we give some corollaries.
\[ProdInterval\] Suppose $B$ is an $n$-precat. Let $\overline{I}$ be the $1$-category with two isomorphic objects denoted $0$ and $1$, considered as an $n$-precat. Then the morphisms $$Cat(B)\stackrel{i_0, i_1}{\rightarrow } Cat (B\times \overline{I})
\stackrel{p}{\rightarrow} Cat(B)$$ are equivalences of $n$-categories, where $i_0$ and $i_1$ come from the inclusions $0\rightarrow \overline{I}$ and $1\rightarrow \overline{I}$ and $p$ comes from the projection on the first factor.
[*Proof:*]{} Note that the morphism $\overline{I} \rightarrow Cat(\overline{I})$ is a weak equivalence by Lemma \[equiv\]. Thus Theorem \[ce\] says that $B\times \overline{I} \rightarrow Cat (B)\times \overline{I}$ is a weak equivalence. On the other hand, the morphism $Cat(B)\times \overline{I} \rightarrow Cat(B)$ is a weak equivalence, so $B\times \overline{I} \rightarrow Cat(B)$ is a weak equivalence. The morphism $B\rightarrow Cat(B)$ is of course a weak equivalence, so Lemma \[remark\] implies that the two morphisms $$B\stackrel{i_0, i_1}{\rightarrow } B\times \overline{I}
\stackrel{p}{\rightarrow} B$$ are weak equivalences, which is the same statement as the corollary. [$/$$/$$/$]{}
\[forInternalHom\] Suppose $A\rightarrow A'$ is a weak equivalence. Then for any $B$, $A\times B\rightarrow A'\times B$ is a weak equivalence.
[*Proof:*]{} By Theorem \[ce\] we have that $$A\times B \rightarrow Cat(A)\times Cat(B), \;\;\;
A'\times B \rightarrow Cat(A')\times Cat(B)$$ are weak equivalences. By hypothesis, the map $Cat(A)\rightarrow Cat(A')$ is an equivalence of $n$-categories. It follows (from any of several characterizations of equivalences of $n$-categories, see for example [@Tamsamani] Proposition 1.3.1) that $Cat(A)\times Cat(B)\rightarrow
Cat(A')\times Cat(B)$ is an equivalence, which gives the corollary. [$/$$/$$/$]{}
[*Proof of Theorem \[ce\]*]{}
We start by making some preliminary reductions. First we claim that it suffices to prove that if $B$ is any $n$-precat and $\Sigma = \Sigma (M,[m], \langle k , k+1 \rangle )$ and $h= h(M,m, 1^{k+1})$ then the morphism $\Sigma \times B \rightarrow h\times B$ is a weak equivalence. Suppose that we know this statement. Then, noting that the proof of \[pushout\] below doesn’t use Theorem \[ce\] in degree $n$ in the case of a pushout by a trivial cofibration which is an isomorphism on objects (which is the case for $\Sigma \times B \rightarrow h\times B$) we obtain that for any $A$ and any morphism $\Sigma \rightarrow A$ the morphism $$A\times B \rightarrow A\times B \cup ^{\Sigma \times B} h\times B$$ is a weak equivalence. The morphism $A\times B \rightarrow Cat(A)\times Cat(B)$ is obtained by iterating operations of this form (either on the variable $A$ or on the variable $B$ which works the same way). Therefore it would follow from the hypothesis of our claim that the morphism of \[ce\] is a weak equivalence.
We are now reduced to proving that $\Sigma \times B \rightarrow h\times B$ is a weak equivalence. In the previous notations if $M$ has length strictly greater than $0$ then the $\Sigma _{p/} \rightarrow h_{p/}$ are weak equivalences. Thus $$(\Sigma \times B)_{p/} = \Sigma _{p/} \times B_{p/}
\rightarrow h_{p/} \times B_{p/} = (h\times B)_{p/}$$ is a weak equivalence, and by Lemma \[partialCat1\] it follows that $\Sigma
\times B \rightarrow h\times B$ is a weak equivalence. Thus we are reduced to treating the case where $M$ has length zero, that is $$\Sigma = \Sigma ([m], \langle k , k+1 \rangle )\;\;\;
\mbox{and} \;\;\; h= h(m, 1^{k+1}).$$ Let $\Sigma ^{nu}=\Sigma ^{nu}([m], 1^{k+1}) $ denote the pushout of $m$ copies of $h(1, 1^{k+1})$ over the standard $h(0)$. We claim that it suffices to prove that $$\Sigma ^{nu} \times B \rightarrow h\times B$$ is a weak equivalence. This claim is proved by induction on $k$. Note that $\Sigma$ is obtained from $\Sigma ^{nu}$ by a sequence of standard cofibrations over $\Sigma ' \rightarrow h'$ which are for smaller values of $k$. Assuming that we have treated all of the cases $\Sigma ^{nu}\times B\rightarrow h\times
B$ and assuming our present claim for smaller values of $k$, we obtain that $\Sigma ^{nu}\times B \rightarrow \Sigma \times B$ is a trivial cofibration. It follows from Lemma \[remark\] that $\Sigma \times B \rightarrow h\times B$ will be a trivial cofibration.
We are now reduced to proving that the morphism $$\Sigma ^{nu}([m], 1^k) \rightarrow h(m, 1^k)$$ induces a weak equivalence $\Sigma ^{nu}\times B \rightarrow h\times B$ (we have changed the indexing $k$ here from the previous paragraphs). Our next reduction is based on the following observation. Suppose we know this statement for $n$-precats $A$, $B$ and $C$ with cofibrations $A\rightarrow B$ and $A\rightarrow C$. Let $P:= B\cup ^AC$. The morphisms $\Sigma ^{nu}\times A \rightarrow h\times A$ (resp. $\Sigma ^{nu}\times B \rightarrow h\times B$, $\Sigma
^{nu}\times C \rightarrow h\times
C$) are weak equivalences inducing isomorphisms on objects. As remarked above, the proof of \[pushout\] below doesn’t use Theorem \[ce\] in degree $n$ when concerning pushouts by weak equivalences which are isomorphisms on objects. On the other hand, the morphism $\Sigma ^{nu}\times P\rightarrow h\times P$ may be obtained as a successive coproduct by these previous morphisms; thus it is a weak equivalence.
We may apply this observation to infinite iterations of cofibrant pushouts. But note that any $n$-precat $B$ may be expressed as an iterated pushout of representable objects $h(M)$ over the boundaries $\partial h(M) \rightarrow
h(M)$. The boundaries are in turn iterated pushouts over representable objects. >From the remark of the previous paragraph, it follows that we are reduced to proving that $\Sigma ^{nu}\times B\rightarrow h\times B$ is a weak equivalence when $B$ is a representable object. We will write $B=h(u , M)$, distinguishing the first variable from the rest.
We next define the following operation. Suppose $C$ is an $n-1$-precat and suppose $D$ is a $1$-precat; then we define $D\oplus C$ to be the $n$-precat with $(D\oplus C)_0:= D_0$ and for any $p$, $(D\oplus C)_{p/}$ is the union for $f\in D_p$ of $C$ when $f$ is not totally degenerate and $\ast$ when $f$ is totally degenerate (we say that $f\in D_p$ is [*totally degenerate*]{} if it is in the image of the morphism $D_0 \rightarrow D_p$). The morphisms of functoriality for $p\rightarrow q$ are obtained by projecting $C$ to the final $n-1$-precat $\ast$ for elements $f\in D_q$ going to totally degenerate elements in $D_p$.
This notation is useful because $h(u , M)= h(u ) \oplus h(M)$. Thus $h(m, 1^k) = h(m)\oplus h(1^k)$; and finally $$\Sigma ^{nu}([m], 1^k)= \Sigma ^{nu} ([m]) \oplus h(1^k).$$
The operation $\oplus$ is compatible with pushouts: if $B\leftarrow A \rightarrow C$ is a diagram of $n-1$-precats then for any $1$-precat $X$, $$X\oplus (B\cup ^AC) = (X\oplus B) \cup ^{X\oplus A} (X\oplus C).$$ On the other hand, if $C\rightarrow C'$ is a weak equivalence then $X\oplus C \rightarrow X\oplus C'$ is a weak equivalence (by applying \[partialCat1\]). The object $h(M)$ is weak equivalent to a pushout of objects of the form $h(1^k)$, with the pushouts being over boundaries which are themselves weak equivalent to pushouts of objects of the same form. Since, as we have seen above, changing things by pushouts or by weak equivalences (which are isomorphisms on objects) in the second variable, preserves the statement in question. Thus it suffices to treat, in the previous notations, the case $M=1^j$. We have now boiled down to the basic case which needs to be treated: we must show that $$\Sigma ^{nu}([m], 1^k) \times h(u, 1^j) \rightarrow h(m, 1^k)\times h(u,1^j)$$ is a weak equivalence.
To do this, use the standard subdivision of the product of simplices $h(m)\times h(u)$ into a coproduct of simplices identified over their boundaries. [^3] In this last part of the proof there was an error in version 1: on p. 31 the line claiming that “$B^{(a,b)} = B^{<(a,b)} \cup ^{B^{\hat{x}}}
B^x$” was not true. Furthermore the notation of this part of the proof was relatively difficult to follow. Thus we rewrite things in the present version 2.
This error and its correction were found during discussions with R. Pellissier, so I would like to thank him.
The basic idea remains the same as what was said in version 1. The objects of $h(m, 1^k)\times h(u, 1^j)$ may be denoted by $(i,j)$ with $i=0,\ldots , m$ and $j=0,\ldots , u$. These should be arranged into a rectangle. The problem is to understand the composition as we go from $(0,0)$ to $(m,u)$. There are many different paths (i.e. sequences of points which are adjacent on the grid and where $i$ and $j$ are nondecreasing) and the goal is to say that the composition along the various different paths is the same. The reason is that when one changes the path by the smallest amount possible at a single square, that is to say changing “up then over” to “over then up”, the composition doesn’t change. This elementary step was done correctly in the original proof (see in v1 the statements that the morphisms $A^{\hat{x}}\rightarrow B^{\hat{x}}$ and $A^x\rightarrow
B^x$ are weak equivalences). Then one has to put these elementary steps together to conclude the desired statement for the big rectangular grid. This requires an inductive argument along the lines of what was done in v1 but in v1 the induction wasn’t organized correctly and was based on a mistaken claim as pointed out above.
So let’s rewrite things and hope for the best! Put $$A:= \Sigma ^{nu}([m], 1^k) \times h(u, 1^j)$$ and $$B:= h(m, 1^k)\times h(u,1^j).$$ Note that $A$ is a coproduct of things of the form $h(1,1^k) \times h(u, 1^j)$. We want to show that the morphism $A\rightarrow B$ is a weak equivalence. The precats $A$ and $B$ share the same set of objects which we denote by $Ob$, equal to the set of pairs $(i,j)$ with $0\leq i
\leq m$ and $0 \leq j\leq u$. Suppose $S \subset Ob$ is a subset of objects. We denote by $A\{ S\} $ (resp. $B\{ S\}$) the full sub-precat of $A$ (resp. $B$) whose object-set is $S$. By “full sub-precat” we mean that for any sequence $x_0, \ldots , x_k$ in $S$, $$A\{ S\} _{k/}(x_0,\ldots , x_k):= A _{k/}(x_0,\ldots , x_k)$$ and the same for $B\{ S\}$. We will use this for subsets $S$ of the form “notched sub-rectangle plus a tail”. By a “sub-recatangle” we mean a subset of the form $$S' = \{ (i,j) : \;\;\; 0 \leq i \leq m',\;\;\ 0\leq j \leq u' \}$$ and by a “tail” we mean a subset of the form $$S'' = \{ (i_k, j_k) : \;\;\; 0 \leq k \leq r \}$$ where $i_k \leq i_{k+1} \leq i_k + 1$ and $j_k \leq j_{k+1} \leq j_k +
1$. A tail $S''$ that goes with a rectangle $S'$ as above, is assumed to have $i_0= m'$, $j_0 = u'$, $i_r = m$, $j_r=u$. In other words, the tail is a path going from the upper corner of $S'$ to the upper corner of $Ob$, and the path goes by steps of at most one in both the $i$ and the $j$ directions.
Finally, a “notched sub-rectangle” is a subset of the form $$S' = \{ (i,j) : \;\;\; (0 \leq i \leq m' \, \mbox{and} \, 0 \leq j
\leq u'-1 ) \, \mbox{or} \, (v'\leq i \leq m' \, \mbox{and}
j=u')\} .$$ We call $(m',u', v')$ the [*parameters*]{} of $S'$, and if necessary we denote $S'$ by $S'(m',u',v')$. Note that $u'\geq 1$ here, and $0\leq v' \leq m'$. A rectangle with $u'=0$ may be considered as part of a tail; thus, modulo the initial case where all of $S$ is a tail, which will be treated below, it is safe to assume $u' \geq 1$. If $v'=0$ then $S'$ is just the rectangle of size $m'
\times u'$. If $v'=m'$ then $S'$ is a rectangle of size $m'\times
(u'-1)$, plus the first segment of a tail going from $(m', u'-1)$ to $(m',u')$. Thus if $S''$ is a tail from $(m',u')$ to $(m,u)$, we get that $$S'(m',u',0) \cup S''$$ is a rectangle of size $m'\times u'$ plus a tail, whereas $$S'(m',u',m') \cup S''$$ is a rectangle of size $m'\times (u'-1)$ plus a tail.
We prove by induction on $(m',u')$ that if $R$ is of the form $R=S'\cup S''$ for $S'$ a rectangle of size $m'\times u'$ and $S''$ a tail from $(m',u')$ to $(m,u)$, then $A\{ R\} \rightarrow B\{ R\}$ is a weak equivalence. We treat the case of $m',u'$ and suppose that it is known for all strictly smaller rectangles (i.e. for $(m'', u'')\neq (m',u')$ with $m'' \leq m'$ and $u''\leq u'$) and all tails. In the current part of the induction we assume that $u' \geq 1$. The case $u'=0$ (which is really the case of a tail $S''$ going all the way from $(0,0)$ to $(m,u)$) will be treated below.
In particular, we know that $$A \{ S'(m',u',m') \cup S'' \} \rightarrow B \{ S'(m',u',m') \cup S'' \}$$ is a weak equivalence (cf the above description of $S'(m',u',m')$). Now we treat the case where $S$ is a notched rectangle plus tail of the form $$S = S'(m',u',v') \cup S''$$ for $0\leq v' \leq m'$. For an $S$ of this notched form, we claim again that $A\{ S\} \rightarrow B\{ S\}$ is a weak equivalence. We prove this by descending induction on $v'$, the initial case $v'=m'$ being obtained above from the case of rectangles of size $m'\times (u'-1)$. Thus we may fix $v'< m'$ and assume that it is known for $$\overline{S}= S'(m',u',v'+1) \cup S'',$$ in other words we may assume that $A\{ \overline{S}\} \rightarrow B\{
\overline{S}\}$ is a weak equivalence.
We analyze how to go from $\overline{S}$ to $S$. Note that $S$ has exactly one object more than $\overline{S}$, the object $$x := (v',u').$$ Let $S^x$ denote the subset of objects $(i,j)\in S$ such that either $(i,j)\leq x$ or $(i,j) \geq x$. Here we define the order relation by $$(i,j) \leq (k,l) \Leftrightarrow i\leq k \;\; \mbox{and}\;\; j \leq l.$$ With respect to this order relation, note that for a sequence of objects $(x_0, \ldots , x_p)$, we have that $B_{p/} (x_0,\ldots , x_p)$ is nonempty, only if $x_0\leq x_1 \leq \ldots \leq x_p$. (The same remark holds [*a fortiori*]{} for $A$ because of the map $A\rightarrow B$.) In particular, if $(x_0,\ldots , x_p)$ is a sequence of objects of $S$ such that $B_{p/} (x_0,\ldots , x_p)$ is nonempty and such that some $x_a=x$ then all of the $x_b$ are in $S^x$.
Let $\overline{S}^x = \overline{S} \cup S^x$. We claim $(\ast )$ that $$A\{ S\} = A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} } A\{ S^x \} ,$$ and similarly that $$B\{ S\} = B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} } B\{ S^x \} .$$ These are the statements that replace the faulty lines of the proof in version 1. To prove the claim, suppose $(x_0, \ldots , x_p)$ is a sequence of objects of $S$. It suffices to show that $$B\{ S\}_{p/}(x_0,\ldots ,x_p)$$ is the pushout of $$B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p)$$ and $$B\{ S^x\}_{p/}(x_0,\ldots ,x_p)$$ over $$B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p)$$ (the proof is the same for $A$, we give it for $B$ here).
If none of the objects $x_a$ is equal to $x$, then either the sequence stays inside $\overline{S}^x$, in which case: $$B\{ S\}_{p/}(x_0,\ldots ,x_p) =
B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p)=
B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p)=
B\{ S^x\}_{p/}(x_0,\ldots ,x_p);$$ or else the sequence doesn’t stay inside $\overline{S}^x$, in which case $$B\{ S\}_{p/}(x_0,\ldots ,x_p) =
B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p)$$ but $$B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p)=
B\{ S^x\}_{p/}(x_0,\ldots ,x_p)=\emptyset .$$ In both of these cases one obtains the required pushout formula. On the other hand, if some $x_a$ is equal to $x$, then either the sequence doesn’t stay inside $S^x$ in which case all terms are empty (cf the above remark), or else it stays inside $S^x$ in which case $$B\{ S\}_{p/}(x_0,\ldots ,x_p) = B\{ S^x\}_{p/}(x_0,\ldots ,x_p)$$ but $$B\{ \overline{S}\}_{p/}(x_0,\ldots ,x_p)=
B\{ \overline{S}^x\}_{p/}(x_0,\ldots ,x_p)= \emptyset .$$ Again one obtain the required pushout formula. This proves the claim $(\ast )$.
We now note that both $S^x$ and $\overline{S}^x$ are of the form, a rectangle of size $v'\times (u'-1)$, plus a tail. The first step of the tail for $S^x$ goes from $(v',u'-1)$ to $x=(v',u')$. The next step goes to $(v'+1,u')$. The first step of the tail for $\overline{S}^x$ goes from $(v',u'-1)$ directly to $(v'+1, u')$. Both tails continue horizontally from $(v'+1,u')$ to $(m',u')$ and then continue as the tail $S''$ from there to $(m,u)$. By our induction hypothesis, we know that $$A\{ S^x\} \rightarrow B\{ S^x\}$$ and $$A\{ \overline{S}^x\} \rightarrow B\{ \overline{S}^x\}$$ are weak equivalences. Recall from above that we also know that $A\{ \overline{S}\} \rightarrow B\{ \overline{S}\}$ is a weak equivalence. These morphisms from full sub-precats of $A$ to full sub-precats of $B$ are all isomorphisms on sets of objects, so by the remark at the start of the proof of Theorem \[ce\], we can use Lemma \[pushout\] for pushouts along these morphisms. A standard argument shows that the morphism $$A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\}
\rightarrow
B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} }B\{ S^x\}$$ is a weak equivalence. For clarity we now give this standard argument. First, $$A\{ \overline{S}\} \rightarrow
A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\}$$ is a trivial cofibration (which again induces an isomorphism on sets of objects). One can verify that the morphism $$A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\}
\rightarrow B\{ \overline{S}\}$$ is a cofibration. It is a weak equivalence by Lemma 3.8. Thus it is a trivial cofibration inducing an isomorphism on sets of objects. Similarly, the morphism $$B\{ \overline{S}^x\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\}
\rightarrow B\{ S^x\}$$ is a trivial cofibration inducing an isomorphism on sets of objects. Thus the pushout morphism (pushing out by these two morphisms at once) $$(A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\} )
\cup ^{B\{ \overline{S}^x\} }
(B\{ \overline{S}^x\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} )$$ $$\rightarrow
B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} }B\{ S^x\}$$ is a weak equivalence. Finally, note that $$(A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }B\{ \overline{S}^x\} )
\cup ^{B\{ \overline{S}^x\} }
(B\{ \overline{S}^x\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} )$$ $$= (A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} )
\cup ^{A\{ \overline{S}^x\} } B \{ \overline{S}^x\}$$ so the morphism from $A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\} $ to this latter, is also a weak equivalence. Putting these all together we have shown that the morphism $$A\{ S\} = A\{ \overline{S}\} \cup ^{A\{ \overline{S}^x\} }A\{ S^x\}$$ $$\rightarrow
B\{ \overline{S}\} \cup ^{B\{ \overline{S}^x\} }B\{ S^x\}
= B\{ S\}$$ is a weak equivalence. This completes the proof of the inductive step for the descending induction on $v'$, so we obtain the result for $v'=0$, in which case $S$ is a rectangle of size $m'\times u'$ plus a tail; in turn, this gives the inductive step for the induction on $(m',u')$.
After all of this induction we are left having to treat the initial case $u'= 0$, where all of $S$ is a tail going from $(0,0)$ to $(m,u)$. This part of the proof is exactly the same as in version 1: what we call the “tail” here corresponds to the sequence which was denoted $x$ in version 1. If $S$ is a tail, then $A\{ S\} $ and $B\{ S\}$ are $1$-free ordered $n$-precats, so by Lemma \[freeness\] it suffices to check that for two adjacent objects $x,y$ in the sequence corresponding to $S$, the morphisms $$A_{1/}(x,y)\rightarrow B_{1/}(x,y)$$ are weak equivalences. In fact these morphisms are isomorphisms. If $x=(i,j)$ and $y=(i,j+1)$ then $$A_{1/}(x,y) = B_{1/}(x,y) = h(1^j);$$ if $x=(i,j)$ and $y=(i+1,j)$ then $$A_{1/}(x,y) = B_{1/}(x,y) = h(1^k);$$ and if $x=(i,j)$ and $y=(i+1,j+1)$ then $$A_{1/}(x,y) = B_{1/}(x,y) = h(1^k)\times h(1^j).$$ Thus the criterion of \[freeness\] implies that $A\{ S\}\rightarrow
B\{ S\}$ is a weak equivalence. This completes the initial case of the induction.
Combining this initial step with the inductive step that was carried out above, we obtain the result in the case where $S=Ob$ is the whole rectangle of size $m\times u$; in this case $A\{
S\} = A$ and $B\{ S\} = B$, so we have completed the proof that $$A=\Sigma ^{nu}([m], 1^k) \times h(u, 1^j) \rightarrow h(m, 1^k)\times h(u,1^j)=B$$ is a weak equivalence. This completes the proof of Theorem \[ce\]. [$/$$/$$/$]{}
The above proof went basically along the same lines as the proof of version 1, but here we met [*all*]{} possible tails going from $(0,0)$ to $(m,u)$ along the way in our induction, whereas in version 1 only some of the tails were met. This should have been taken as an indication that the proof in version 1 was incorrect. I would like again to thank R. Pellissier for an ongoing careful reading which turned up this problem. His reading has also turned up numerous other problems in the exposition or organisation of the argument (the reader has no doubt noticed!); however, I have chosen in the present version 2 to make only a minimalist correction of the above problem.
[**.Proofs of the remaining lemmas and Theorem \[cmc\]**]{}
We can assume \[equiv\], \[partialCat1\] and the corollary \[partialCat\] for degree $n$ also.
[*Proof of \[coherence\]:*]{} We have to prove that the morphism $f:Cat (A) \rightarrow Cat(Cat(A))$ is an equivalence. Note that this is not the same morphism as the standard inclusion, rather it is the morphism induced by $A\rightarrow Cat (A)$. In particular, \[coherence\] is not just an immediate corollary of \[equiv\]. To obtain the proof, note that $Cat(A)$ is marked, so we have a morphism $$r:Cat (Cat(A))\rightarrow Cat (A)$$ of marked easy $n$-categories, inducing the identity on the standard map $i:Cat (A) \rightarrow Cat (Cat (A))$. The morphism $r$ is an equivalence because the standard map $i$ is an equivalence by \[equiv\]. On the other hand, the morphism of $n$-precats $A\rightarrow Cat (A)$ induces the morphism $f$ of marked easy $n$-categories. We obtain a morphism $r\circ f$ of marked easy $n$-categories $Cat (A)\rightarrow Cat (A)$ extending the standard map $A\rightarrow Cat (A)$. By the universal property of $Cat (A)$, $r\circ f$ is the identity. Thus (applying our usual Lemma \[remark\]) the morphism $f$ is an equivalence.
[*Proof of \[pushout\]:*]{} We first treat the following special case of \[pushout\]. Say that a morphism of $n$-categories $A\rightarrow B$ is an [*iso-equivalence*]{} if it is an equivalence and an isomorphism on objects. This is equivalent to the condition that for all $m$, the morphism $A_{m/}\rightarrow B_{m/}$ is an equivalence of $n-1$-categories.
\[isopushout\] Suppose $A\rightarrow B$ is an iso-equivalence of $n$-categories and $A\rightarrow C$ is a morphism of $n$-categories. Then the morphism $$C \rightarrow Cat (B\cup ^AC)$$ is an equivalence (in fact, even an iso-equivalence).
[*Proof:*]{} By our inductive hypotheses, the morphisms $$C_{m/} \rightarrow Cat (B_{m/} \cup ^{A_{m/}}C_{m/})$$ are equivalences of $n-1$-categories. Setting $D= B\cup ^AC$ we have $$D_{m/}=B_{m/} \cup ^{A_{m/}}C_{m/}$$ and note that $D_0=C_0$. Hence the morphism of $n$-precats $$C\rightarrow Cat _{\geq 1}(D)$$ is an equivalence on the level of each $C_{m/}$. But the condition of being an $n$-category depends only on the equivalence type of the $C_{m/}$, in particular $Cat _{\geq 1}(D)$ is an $n$-category (in this special case only—this is not a general principle!). Note in passing that the morphism $$C \rightarrow Cat _{\geq 1}(D)$$ is an equivalence of $n$-categories. By Lemma \[equiv\] in degree $n$, the morphism $$Cat _{\geq 1}(D)\rightarrow Cat (Cat _{\geq 1}(D))$$ is an equivalence. On the other hand, by Lemma \[partialCat\] in degree $n$ the morphism $$Cat (D) \rightarrow Cat (Cat _{\geq 1}(D))$$ is an equivalence. By Lemma \[remark\] at the start of the proof of all of the lemmas, this shows that $C\rightarrow Cat(D)$ is an equivalence. This proves Lemma \[isopushout\]. [$/$$/$$/$]{}
The next lemma is the main special case which has to be treated by hand. This proof uses Corollary \[ProdInterval\], which in turn is where we use the full simplicial structure of $\Delta$. It seems likely (from some considerations in topological examples) that the following lemma would not be true if we looked only at functors from $\Delta_{\leq k}$ to $n$-precats.
\[specialcase\] Suppose $\overline{I}$ is the category with two objects and exactly one isomorphism between them. Let $0$ denote one of the objects. Then for any $n$-category $A$ and object $c\in A_0$, if $0\rightarrow A$ denotes the corresponding morphism, the push-out morphism $$A \rightarrow Cat (\overline{I} \cup ^0 A)$$ is an equivalence.
[*Proof:*]{} Let $\overline{I}^2$ denote $\overline{I}\times \overline{I}$. There is a morphism $h:\overline{I}^2\rightarrow
\overline{I}\times \overline{I}$ equal to the identity on $\overline{I}\times \{
0\}$ and on $\{ 0\} \times \overline{I}$, and sending $\overline{I}\times \{
1\}$ to $(0, 1)$. Let $B:= \overline{I} \cup ^0A$. Then $$B\times \overline{I} = \overline{I}^2 \cup ^{\{ 0\} \times \overline{I}}C\times
\overline{I}.$$ Using $h$ on the first part of this pushout we obtain a map $$f:B\times \overline{I}\rightarrow B\times \overline{I}$$ such that $f|_{B\times \{ 0\}}$ is the identity and $f|_{B\times \{ 1\}}$ is the projection $B\rightarrow A$ obtained from the projection $\overline{I}\rightarrow \{ 0\}$. By Corollary \[ProdInterval\], the morphisms $$i_0:Cat (B)\times \{ 0 \} \rightarrow Cat (B \times \overline{I})$$ and $$i_1:Cat (B)\times \{ 1 \} \rightarrow Cat (B \times \overline{I})$$ are equivalences of $n$-categories. Next, the morphism $f$ induces a morphism $$g:Cat (B\times \overline{I})\rightarrow Cat (B)$$ such that the composition with $i_0$ is the identity $Cat(B)\rightarrow Cat(B)$ and the composition with $i_1$ is the factorization $Cat(B)\rightarrow
Cat(A)\rightarrow Cat(B)$. Looking at $g\circ i_0$ we conclude that $g$ is an equivalence of $n$-categories. Therefore (since $i_1$ is an equivalence) the composition $g\circ i_1$ is an equivalence of $n$-categories. Now we have morphisms $Cat(A)\rightarrow Cat(B)$ and $Cat(B)\rightarrow Cat(A)$ such that the composition in one direction is the identity, and the composition in the other direction is an equivalence of $n$-categories. This implies that the two morphisms are equivalences of $n$-categories by Lemma \[remark\]. [$/$$/$$/$]{}
In preparation for the next corollary, we discuss a sort of “versal semi-interval” $\overline{J}$. Ideally we would like to have an $n$-precat which is weakly equivalent to $\ast$, containing two objects $0$ and $1$ such that for any $n$-category (easy, perhaps) $A$ with two equivalent objects $a$ and $b$, there exists a morphism from our “interval” to $A$ taking $0$ to $a$ and $1$ to $b$. I didn’t find an easy way to make this construction. The problem is somewhat analogous to the problem of finding a canonical inverse for a homotopy equivalence, solved in a certain topological context in [@flexible] but which seems quite complicated to put into action here in view of the fact that our $n$-category $A$ might not be fibrant (we don’t yet have the closed model structure!). Thus we will be happy with a cruder version. Let $\overline{J}$ be the universal easy $n$-category with two objects $0$ and $1$ and a “marked inner equivalence” $u :0\rightarrow
1$. The quasi-inverse of $u$ will be denoted by $v$. The marking means a structure of choice of morphism whenever necessary for the definition of inner equivalence, as well as a choice of diagram (i.e. a partial marking in the sense defined at the start of the paper) whenever necessary for things to make sense. In practice this means that we start with objects $0$ and $1$, add the morphisms $u$ and $v$, add the diagrams over $2\in \Delta$ mapping to $(u,v)$ and $(v,u)$, and (letting $w$ and $y$ denote the compositions resulting from these diagrams) add (inductively by the same construction for $n-1$-categories) equivalences between $w$ and $e$ (resp. $y$ and $e$) where $e$ denote the identities. Let $\overline{L}\subset \overline{J}$ be the full-sub-$n$-category whose object set is $\{ 0\}$. The morphism $\overline{L}\rightarrow \overline{J}$ is automaticallly an equivalence since it is an isomorphism on morphism $n-1$-categories and is essentially surjective since by construction $1\in \overline{J}$ is equivalent to $0$. By the universal property of $\overline{J}$ we obtain a morphism $\overline{J}\rightarrow \overline{L}$ sending $u$ to $e$ and $v$ to $w$, sending our $2$-diagrams to degenerate diagrams in $\overline{L}$ and sending our homotopies to the corresponding homotopies in $\overline{L}$. The composition $$\overline{L}\rightarrow
\overline{J}\rightarrow \overline{L}$$ is the identity. On the other hand we have an obvious map $\overline{J}\rightarrow \overline{I}$, so we obtain a map $$\overline{J}\rightarrow \overline{L}\times \overline{I}.$$ This map is compatible with the inclusions of $\overline{L}$ and hence is an equivalence of $n$-categories. It is also an isomorphism on objects so it is an iso-equivalence.
\[specialJ\] Let $\overline{L}\subset \overline{J}$ be as above. Then for any $n$-category $A$ and morphism $\overline{L}\rightarrow A$, the push-out morphism $$A \rightarrow Cat (\overline{J} \cup ^{\overline{L}} A)$$ is an equivalence.
[*Proof:*]{} Let $B= (\overline{L}\times \overline{I}) \cup ^{\overline{L}} A$ and $C= \overline{J} \cup ^{\overline{L}}
A$ The morphism $\overline{J}\rightarrow (\overline{L}\times \overline{I})$ is an iso-equivalence so it satisfies the hypothesis of \[partialCat1\]. By \[pushout\] for $n-1$-precats, the morphism $C\rightarrow B$ also satisfies the hypothesis of \[partialCat1\]. Therefore the morphism $Cat(C)\rightarrow Cat(B)$ is an equivalence. It suffices to show that $A\rightarrow Cat(B)$ is an equivalence. For this, note that $$\overline{L} \cup ^0\overline{I} \rightarrow \overline{L}\times \overline{I}$$ is a weak equivalence by Lemma \[specialcase\]. Similarly $$A \rightarrow A \cup ^0\overline{I}$$ is a weak equivalence, and $$B = (A\cup ^0\overline{I}) \cup ^{(\overline{L} \cup ^0\overline{I})}
\overline{L}\times \overline{I}$$ so composing these two gives that $A\rightarrow B$ is a weak equivalence. [$/$$/$$/$]{}
Suppose $A\rightarrow B$ is a cofibrant equivalence of $n$-categories such that the objects of $A$ form a subset of the objects of $B$ whose complement has one object. Then the push-out morphism $$C \rightarrow Cat (B \cup ^A C)$$ is an equivalence.
[*Proof:*]{} We may replace $B$ by $Cat(B)$ since the morphism $B\rightarrow Cat(B)$ is a sequence of standard pushouts, so the corresponding morphism on pushouts of $C$ is also a sequence of standard pushouts so the conclusion for $Cat(B)$ implies the conclusion for $B$ (by Lemma \[remark\]). Thus we may assume that $B$ is an easy $n$-category.
Let $A'$ be the full sub-$n$-category of $B$ consisting of the objects of $A$. The pushout of $C$ from $A$ to $A'$ is a weak equivalence by Lemma \[isopushout\]. Thus we may assume that $A=A'$.
Let $b$ denote the single new object of $B$. It is equivalent to an object $a\in A$. By the universal property of $\overline{J}$ there is a morphism $\overline{J}\rightarrow B$ sending $0$ to $a$ and $1$ to $b$. Since $A$ is now a full sub-$n$-category of $B$, this morphism sends $\overline{L}$ to $A$. Let $E$ denote the push-out $$E := Cat (A \cup ^{\overline{L}} \overline{J}).$$ By the prevoius corollary, $A\rightarrow E$ is an equivalence. Our morphism $\overline{J} \rightarrow B$ gives a morphism $$E \rightarrow B$$ (use the marking of $B$ to go from $Cat(B)$ back to $B$) and this is an equivalence since $$A\rightarrow B \rightarrow Cat(B) \;\;\; \mbox{and}\;\;\; A \rightarrow E$$ are equivalences. But the morphism $E\rightarrow B$ induces an isomorphism on objects. Now we have $$C \cup ^A (A \cup ^{\overline{L}} \overline{J})= C \cup
^{\overline{L}}\overline{J}$$ so $$C \rightarrow Cat (C \cup ^A (A \cup ^{\overline{L}} \overline{J}))$$ is an equivalence by the previous corollary. It is obvious from the construction of $Cat$ (resp. $BigCat$) via pushouts, together with the reordering of these pushouts, that $$BigCat (C \cup ^A (A \cup ^{\overline{L}} \overline{J})) =
BigCat (C \cup ^ACat (A \cup ^{\overline{L}}
\overline{J})) =BigCat(C\cup ^AE).$$ Thus (since taking $BigCat$ is equivalent to taking $Cat$ by Lemma \[equiv\] —which we now know—and the reordering principle) $$C\rightarrow Cat(C \cup ^AE)$$ is an equivalence. Now $$Cat (C\cup ^AE) \rightarrow Cat (C \cup ^A B)$$ is an equivalence because $E_{p/}\rightarrow B_{p/}$ is an equivalence so by \[pushout\] in degree $n-1$, $$(C\cup ^AE )_{p/} \rightarrow (C \cup ^A B)_{p/}$$ is an equivalence and by Lemma \[partialCat1\] we get the desired statement. Combining, we get that $$C\rightarrow Cat (C \cup ^A B)$$ is an equivalence. [$/$$/$$/$]{}
[*Proof of Lemma \[pushout\]:*]{} Suppose $A\rightarrow B$ is a cofibration of $n$-categories which is an equivalence. By applying the previous corollary inductively (adding one object at a time) we conclude that the push-out is an equivalence.
Finally we treat the case where $A$, $B$ and $C$ are only $n$-precats rather than $n$-categories. If $\Sigma \rightarrow h$ is one of our standard pushout diagrams and if $\Sigma \rightarrow A$ is a morphism then $$(B\cup ^{\Sigma} h) \cup ^{A\cup ^{\Sigma} h}(C\cup ^{\Sigma} h) = (B\cup
^AC)\cup ^{\Sigma}h.$$ This implies that $$BigCat(B)\cup ^{BigCat(A)}BigCat(C)$$ is obtained by a collection of standard pushouts from $B\cup ^AC$, so in particular (by reordering) $$BigCat(BigCat(B)\cup ^{BigCat(A)}BigCat(C))=BigCat(B\cup ^AC).$$ Now our hypothesis is that $BigCat(A)\rightarrow BigCat(B)$ is an equivalence (note also that it is a cofibration since $A\rightarrow B$ is a cofibration). By our proof of \[pushout\] for the case of $n$-categories (and the equivalence between $Cat$ and $BigCat$ which we now know by \[equiv\]) we conclude that $$BigCat(C)\rightarrow BigCat(BigCat(B)\cup ^{BigCat(A)}BigCat(C))$$ is an equivalence, which is to say that $$BigCat(C)\rightarrow BigCat(B\cup ^AC)$$ is an equivalence. Thus $C\rightarrow B\cup ^AC$ is a weak equivalence. [$/$$/$$/$]{}
[*Remark:*]{} The semi-interval $\overline{J}$ we have constructed above is not contractible (i.e. equivalent to $\ast$). However for some purposes we would like to have such an object. We have the following fact (which is not used in the proof of Theorem \[cmc\] in degree $n$—but which we put here for expository reasons):
\[intervalK\] There is an $n$-category $K$ such that $K\rightarrow \ast$ is an equivalence, together with objects $0,1\in K$ such that if $A$ is an $n$-category and if $a,b$ are two equivalent objects of $A$ then there is a morphism $K\rightarrow A$ sending $0$ to $a$ and $1$ to $b$.
[*Proof:*]{} Since this proposition is not used in degree $n$ in the proof of Theorem \[cmc\], we can apply Theorem \[cmc\]. We say that $K$ is [*contractible*]{} if the morphism $K\rightarrow \ast$ is an equivalence. In view of the versal property of $\overline{J}$, it suffices to construct a contractible $K$ with objects $0,1$ and a morphism $K\rightarrow \overline{J}$ sending $0$ to $0$ and $1$ to $1$. From the original discussion of $\overline{J}$ we have an equivalence $\overline{J}\rightarrow \overline{L}\times \overline{I}$. Using the closed model structure, factor the constant morphism as $$\overline{I}\rightarrow M \rightarrow \overline{L}\times \overline{I}$$ into a composition of a trivial cofibration followed by a fibration. Note that $M$ is contractible. Set $$K:=\overline{J}\times _{\overline{L}\times \overline{I}}M.$$ The morphism $\overline{J} \rightarrow \overline{L}\times
\overline{I}$ is an isomorphism on objects, so for each $p$, $\overline{J}_{p/} \rightarrow (\overline{L}\times \overline{I})_{p/}$ is an equivalence of $n-1$-categories. Note also that $M _{p/}\rightarrow (\overline{L}\times \overline{I})_{p/}$ are fibrations. By the fact that weak equivalences are stable under fibrant pullbacks for $n-1$-categories (Theorem \[properness\]), we have that $$K_{p/}= \overline{J}_{p/}\times _{(\overline{L}\times \overline{I})_{p/}}M
_{p/} \rightarrow M_{p/}$$ are weak equivalences, which in turn implies that $K\rightarrow M$ is a weak equivalence. In particular, $K$ is contractible. Since the morphism $$M \rightarrow \overline{L}\times \overline{I}$$ is surjective on objects, there are objects $0,1\in K$ mapping to $0,1\in
\overline{J}$. This completes the construction. [$/$$/$$/$]{}
\[FibImpliesCat\] If $A$ is a fibrant $n$-precat then $A$ is automatically an easy $n$-category.
[*Proof:*]{} To show this it suffices to show that the morphisms $$\varphi :\Sigma (M, [m], \langle k,k+1 \rangle ) \rightarrow h(M,[m], 1^{k+1})$$ are trivial cofibrations. But $\varphi$ is a cofibration which is the first step in an addition of arbitrary pushouts of our standard morphisms $\varphi$, so by reordering of these pushouts the above inclusion extends to an isomorphism $$BigCat (\Sigma (M, [m], \langle k,k+1 \rangle ) )\cong
BigCat (h(M,[m], 1^{k+1})).$$ Since $Cat (B) \rightarrow BigCat(B)$ is an equivalence by Lemma \[equiv\] applied to $Cat(B)$ plus reordering, this implies that the morphism $\varphi$ above is a trivial cofibration. By the definition of fibrant, $A$ must then satisfy the extension property to be an easy $n$-category. [$/$$/$$/$]{}
[*The proof of Theorem \[cmc\]*]{}
We follow the proof of Jardine-Joyal that simplicial presheaves form a closed model category, as described in [@Jardine]. The proof is based on the axioms CM1–CM5 of [@QuillenAnnals].
[*Proof of CM1:*]{} The category of $n$-precats is a category of presheaves so it is closed under finite (and even arbitrary) direct and inverse limits.
[*Proof of CM2:*]{} Given composable morphisms $$X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z,$$ if any two of $f$ or $g$ or $g\circ f$ are weak equivalences then the same two of $Cat(f)$, $Cat(g)$ or $Cat(g\circ
f)$ are equivalences of $n$-categories in the sense of [@Tamsamani] and by Lemma \[remark\] the third is also an equivalence; thus the third of our original morphisms is a weak equivalence.
[*Proof of CM3:*]{} This axiom says that “the classes of cofibrations, fibrations and weak equivalences are closed under retracts”. Jardine [@Jardine] doesn’t actually discuss the retract condition other than to say that it is obvious in his case, and a look at Quillen yields only the conclusion that the diagram on p. 5.5 of [@Quillen] for the definition of retract is wrong (that diagram has no content related to the word “retract”, it just says that one arrow is the composition of three others). Thus—since I am not sufficiently well acquainted with other possible references for this—we are reduced to speculation about what Quillen means by “retract”. Luckily enough, this speculation comes out to be non-speculative in the end. We say that $f: A\rightarrow B$ is a [*weak retract*]{} of $g:X\rightarrow Y$ if there is a diagram $$\begin{array}{ccccc}
A&\stackrel{i}{\rightarrow}&X&\stackrel{r}{\rightarrow}&A\\ \downarrow
&&\downarrow && \downarrow\\
B&\stackrel{j}{\rightarrow}&Y&\stackrel{s}{\rightarrow}&B \end{array}$$ such that $r\circ i=1_A$ and $s\circ j=1_B$. There is also another notion which we call [*strong retract*]{} obtained by using the same diagram but with the arrows going in the opposite direction on the bottom. It turns out that if $f$ is a strong retract of $g$ then $f$ is also a weak retract of $g$: the strong retract condition can be stated as the condition $j\circ f \circ r = g$ (along with the retract conditions $ri=1$ and $sj=1$). Applying $s$ on the left we obtain $fr=sg$ and applying $i$ on the right we obtain $jf=gi$, these two conditions giving the weak retract condition. Thus for our purposes, if we can show that the classes of maps in question are closed under weak retract, this implies that they are also closed under strong retract, and we don’t actually care which of the two definitions was intended in [@Quillen]!
We start out, then, with the condition that $f$ is a weak retract of $g$ using the notations of the diagram given above. If $g$ satisfies any lifting property then $f$ satisfies the same lifting property, using the retractions. This shows that if $g$ is a fibration then $f$ is a fibration. Furthermore, if $g$ is a cofibration then it is injective over any object $M=(m_1,\ldots , m_k)$ with $k<n$. It follows from the retractions that $f$ satisfies the same injectivity conditions (one has the same diagram of retractions on the values of all of the presheaves over the object $M$). Thus $f$ is a cofibration.
Suppose $g$ is a weak equivalence, we would like to show that $f$ is a weak equivalence. Replacing the whole diagram by $Cat$ of the diagram, we may assume that $A$, $B$, $X$, and $Y$ are $n$-categories and $g$ is an equivalence. Suppose $x,y$ are objects of $A$. Then denoting by the same letters their images in $B$, $X$ and $Y$ we obtain morphisms of $n-1$-categories $f_1(x,y): A_{1/}(x,y)\rightarrow B_{1/}(x,y)$ and $g_1(x,y): X_{1/}(x,y)\rightarrow Y_{1/}(x,y)$ such that $g$ is a retract of $f$ in the category of arrows of $n-1$-precats. Furthermore $g_1(x,y)$ is an equivalence of $n-1$-categories. It follows by induction (since we may assume CM3 known for $n-1$-categories) that $f_1(x,y)$ is an equivalence of $n-1$-categories. In order to prove that $f$ is an equivalence we have to prove that it is essentially surjective. Suppose $w$ is an object of $B$. Then $i(w)$ is an object of $Y$ so by essential surjectivity of $g$ there is an object $u$ of $X$ with an equivalence $e:g(u)\cong i(w)$ (i.e. a pair of elements $e'\in Y_1(g(u),i(w))$ and $e'' \in Y_1(g(u),i(w))$ such that their compositions, which are well defined in the truncation $T^{n-1}Y_1(
g(u),g(u))$ and $T^{n-1}Y_1(
i(w),i(w))$ are the identities in these truncations). Applying the retractions $r$ and $s$ we obtain an element $r(u)\in A_0$ and an equivalence $s(e)$ between $fr(u)$ and $si(w)=w$. This proves essential surjectivity of $f$, completing the verification of CM3.
[*Proof of CM4:*]{} The first part of CM4 is exactly Lemma \[pushout\]. The second part follows from the first by the same trick as used by Jardine ([@Jardine] pp 64-65) and ascribed by him to Joyal [@Joyal].
[*Proof of CM5(1):*]{} For our situation, the cardinal $\alpha$ refered to in Jardine is the countable infinite one $\omega$. Suppose $A\rightarrow C$ is a trivial cofibration. We claim that if $B$ is an $\omega$-bounded subobject of $C$ (by this we mean a sub-presheaf over $\Theta^n$) then there is an $\omega$-bounded subobject $B_{\omega}\subset C$ as well as an $\omega$-bounded subobject $A_{\omega}\subset A\times _BB_{\omega}$ such that $B\subset
B_{\omega} \subset C$ and such that $A_{\omega} \rightarrow
B_{\omega}$ is a trivial cofibration. (Note that in our situation cofibrations are not necessarily injective morphisms of presheaves, so $A_{\omega}$ is not necessarily equal to $A\times _BB_{\omega}$ the latter of which could be uncountable).
To prove the claim, note that for a given element in $B_M$ for some $M$, the statement that it is contained in an $n$-precat which is weakly equivalent to $A$ can in principal be written out explicitly involving only a countable number of elements of various $A_{M'}$ and $B_{M'}$. Iterate this operation starting with all of the elements of $B$ and repeatedly applying it to all of the new elements that are added. The iteration takes place a countable number of times, and each time we add on a countable union of countable objects. At the end we arrive at $A_{\omega} \subset B_{\omega}$ which is an $\omega$-bounded trivial cofibration.
Using this claim, the rest of Jardine’s arguments of ([@Jardine], Lemmas 2.4 and 2.5) work and we obtain the statement that every morphism $f:X\rightarrow
Y$ of $n$-precats can be factored as $f=p\circ i$ where $i$ is a trivial cofibration and $p$ is fibrant—[@Jardine] Lemma 2.5, which is CM5(1). Note that the only sentence in Jardine’s argument which needs further verification is the fact that filtered colimits of trivial cofibrations are again trivial cofibrations; and this holds in our case too.
[*Proof of CM5(2):*]{} We have to prove that any morphism $f$ may be factored as $f=q\circ j$ where $q$ is a fibrant weak equivalence and $j$ a cofibration.
It suffices to construct a factorization $f=q\circ j$ with $j$ a cofibration and $q$ a weak equivalence, for then we can apply CM5(1) to factor $q$ as a product of a trivial cofibration and a fibration, the latter of which is automatically also a weak equivalence by CM2. Thus we now search for $f=q\circ j$ with $q$ a weak equivalence and $j$ a cofibration.
The reader may wish to think about this in the case of $1$-categories to get an idea of what is happening and to see why this part is actually easy modulo some small details: we multiply the number of objects in each isomorphism class in the target category to have the morphism injective on the sets of objects.
If $f:A\rightarrow B$ is a morphism of $n$-precats then we define a canonical factorization $A\rightarrow N(A,B)\rightarrow B$ in the following way. Let $L(A)$ denote the $1$-category (considered as an $n$-category) whose set of objects is equal to $A_0$ and which has exactly one morphism between any pair of objects. Note that $L(A)\rightarrow \ast$ is a weak equivalence. The tautological map $A_0 \rightarrow L(A)_0$ lifts to a unique map of $n$-precats $t:A\rightarrow L(A)$. Set $N(A,B):= L(A) \times B$ with the diagonal map $(t,f) :A\rightarrow N(A,B)$ and the second projection $p: N(A,B)\rightarrow B$. Note that $p$ is a weak equivalence (by an appropriate generalization of Corollary \[ProdInterval\]) and $(t,f)$ is injective on objects.
Now suppose by induction that we have constructed for every morphism $f':A'\rightarrow B'$ of $n-1$-precats a factorization $A'\rightarrow M(A',B')\rightarrow B'$ as a composition of a weak equivalence and a cofibration, functorial in $f'$.
(To start the induction for $n=0$ we set $M(A',B'):= B'$ recalling that all morphisms are cofibrations in this case.)
Suppose $f: A\rightarrow C$ is a morphism of $n$-precats such that $A_0 \hookrightarrow C_0$ is injective. Define a presheaf on $\Delta
\times
\Theta ^{n-1}$ denoted $P(A,C)$, with factorization $A\rightarrow
P(A,C)\rightarrow C$ as follows. Put $$P(A,C)_{p/} := M(A_{p/}, C_{p/}).$$ By functoriality this is a functor from $\Delta ^o$ to $n-1$-precats, and we have a factorization $$A_{p/} \rightarrow P(A,C)_{p/} \rightarrow C_{p/}.$$ The second morphisms in the factorization are equivalences, and the first morphisms are cofibrations. The only problem is that $P(A,C)_{0/}$ is not a set: it is an $n$-category which is equivalent to the set $C_0$.
For any $p$ there is a morphism $\psi _p:P(A,C)_{0/}\rightarrow P(A,C)_{p/}$ which, because it is a section of any one of the morphisms back to $0$, is a cofibration and in fact even injective in the top degree. If $p\rightarrow
q$ is a morphism in $\Delta$ then $P(A,C)_{q/} \rightarrow P(A,C)_{p/}$ composed with $\psi _q$ is equal to $\psi _p$. Hence if we set $$Q(A,C)_{p/}:= P(A,C)_{p/} \cup ^{P(A,C)_{0/}} C_0$$ then $Q(A,C)_{p/}$ is functorial in $p\in \Delta$. Now $Q(A,C)_{0/}= C_0$ is a set rather than an $n-1$-precat so $Q(A,C)$ descends to a presheaf on $\Theta ^n$. We have a morphism $A\rightarrow Q(A,C)$ projected from the above morphism into $P(A,C)$. We also have a morphism $Q(A,C)\rightarrow C$ because the composed morphism $$P(A,C)_{0/} \rightarrow P(A,C)_{p/} \rightarrow C_{p/}$$ factors through the unique morphism $C_0 \rightarrow C_{p/}$. The composition of these morphisms is the morphism $A\rightarrow C$. We next claim that the second morphism $Q(A,C)\rightarrow C$ is a weak equivalence. It suffices by Lemma \[partialCat1\] to prove that $Q(A,C)_{p/}\rightarrow C_{p/}$ are weak equivalences, but the facts that $P(A,C)_{0/}\rightarrow C_0$ is a weak equivalence and $P(A,C)_{0/}\rightarrow
P(A,C)_{p/}$ a cofibration imply (inductively using the closed model structure for $n-1$-precats) that $P(A,C)_{p/}\rightarrow Q(A,C)_{p/}$ is a weak equivalence. Now $P(A,C)_{p/} \rightarrow C_{p/}$ being a weak equivalence implies that $Q(A,C)_{p/}\rightarrow C_{p/}$ is a weak equivalence. This proves the claim.
Finally we claim that $A\rightarrow Q(A,C)$ is a cofibration. It suffices to prove that the $A_{p/}\rightarrow Q(A,C)_{p/}$ are cofibrations. We know by the inductive hypothesis that $A_{p/} \rightarrow P(A,C)_{p/}$ are cofibrations. By the pushout definition of $Q(A,C)_{p/}$ and using the fact that $P(A,C)_{0/}$ is a sub-presheaf of $P(A,C)_{p/}$, it suffices to prove that the map $$A_{p/} \times _{P(A,C)_{p/}} P(A,C)_{0/} \rightarrow C_0$$ is cofibrant. In fact we show below that for any $M$ of length $<n-1$, $$A_{p,M} \times _{P(A,C)_{p,M}} P(A,C)_{0,M} = A_0\subset A_{p,M}$$ which implies what we want, since we have assumed that $A_0\rightarrow C_0$ is injective. Note that the notation $P(A,C)_{0,M}$ means $(P(A,C)_{0/})_M$, and we don’t have in this case that this is a constant $n-1$-category so the usual rule saying that $P(A,C)_{0,M}$ should be equal to $P(A,C)_{0,0}$ doesn’t apply.
Fix any one of the maps $e:p\rightarrow 0 \rightarrow p$. This gives a map $A_{p/} \rightarrow A_{p/}$ whose image is automatically $A_0$. This implies that the fixed subsheaf of the endomorphism $e$ is equal to $A_0$. The endomorphism acts compatibly on $P(A,C)_{p/}$ and the fixed point subsheaf there is $P(A,C)_{0/}$. For any $M$ of length $<n-1$ we have an inclusion $A_{p,M}
\hookrightarrow P(A,C)_{p,M}$. This is compatible with the endomorphisms $e$ on both sides, so the intersection of $A_{p,M}$ with the fixed point set $P(A,C)_{0,M}\subset P(A,C)_{p,M}$ is the fixed point set $A_{0}\subset
A_{p,M}$. This shows the statement of the previous paragraph.
This completes the proof that $A\rightarrow Q(A,C)\rightarrow C$ is a factorization of the desired type, when $A_0 \rightarrow C_0$ is injective on objects. Note also that $Q(A,C)$ is functorial in the morphism $A\rightarrow C$. Suppose now that $A\rightarrow B$ is any morphism. We put $$M(A, B):= Q(A, N(A,B)).$$ We have the factorization $A\rightarrow N(A,B) \rightarrow B$ with the first arrow injective on objects and the second arrow a weak equivalence, discussed at the start of the proof. The first arrow is then factored as $A\rightarrow
M(A,B) \rightarrow N(A,B)$ with the first arrow a cofibration and the second arrow a weak equivalence. The factorization $A\rightarrow M(A,B)\rightarrow B$ therefore has the desired properties, and furthermore it is functorial in $A\rightarrow B$ (this is needed in order to continue with the induction on $n$). This completes the proof of CM5(2).
[$/$$/$$/$]{}
We refer to [@Quillen] for all of the consequences of Theorem \[cmc\].
Recall also that a closed model category is said to be [*proper*]{} if it satisfies the following two axioms: (1) If $A\rightarrow B$ is a weak equivalence and $A\rightarrow
C$ a cofibration then $C\rightarrow B\cup ^AC$ is a weak equivalence; (2) If $B\rightarrow A$ is a weak equivalence and $C\rightarrow
A$ a fibration then $B\times _AC\rightarrow C$ is a weak equivalence.
\[properness\] The closed model category $PC_n$ satisfies axiom [**Pr**]{}(1); and it satisfies axiom [**Pr**]{}(2) for equivalences $B\rightarrow A$ between $n$-categories; however it doesn’t satisfy axiom [**Pr**]{}(2) in general.
[*Proof:*]{} We will prove stability of weak equivalences under coproducts. Suppose $A\rightarrow B$ is a cofibration, and suppose $A\rightarrow C$ is a weak equivalence. We would like to show that $B\rightarrow B\cup ^AC$ is a weak equivalence. For this we use a version of the “mapping cone”. Recall that $\overline{I}$ is the category with two isomorphic objects $0,1$ and no other morphisms. The morphism $B\times \{ 1\} \rightarrow B \times \overline{I}$ is a trivial cofibration, so $$B\cup ^AC\rightarrow D:= (B\cup ^AC)\cup ^{B\times \{ 1\} } B\times \overline{I}$$ is a trivial cofibration. It follows that the projection $D\rightarrow
B\cup ^AC$ deduced from $B\times \overline{I}\rightarrow B$ is a weak equivalence. Let $$E:= (B\times \{ 0\} )\cup ^{A\times \{ 0\} }A \times \overline{I}$$ and note that $B\times \{ 0\} \rightarrow E$ is a weak equivalence (since it is pushout by the trivial cofibration $A\times \{ 0\} \rightarrow
A\times \overline{I}$) hence $E\rightarrow B\times \overline{I}$ is a trivial cofibration. Thus the morphism $$E \cup ^{A\times \{ 1\} } C \rightarrow B\times \overline{I} \cup
^{A\times \{ 1\} } C$$ is a trivial cofibration. But note that $B\times \overline{I} \cup
^{A\times \{ 1\} } C =D$ so $$E \cup ^{A\times \{ 1\} } C \rightarrow D$$ is a weak equivalence. Finally, $$E \cup ^{A\times \{ 1\} } C = B\times \{ 0\} \cup ^{A\times \{ 0\}}
(A\times \overline{I} \cup ^{A\times \{ 1\} } C)$$ and the morphism $$A\times \{ 0\}\rightarrow
A\times \overline{I} \cup ^{A\times \{ 1\} } C$$ is a weak equivalence because it projects to $A\rightarrow C$ which is by hypothesis a weak equivalence. Therefore the map $$B\times \{ 0\} \rightarrow E \cup ^{A\times \{ 1\} } C$$ is a weak equivalence, and from above $B\times \{ 0\} \rightarrow D$ is a weak equivalence. Following by the projection $D\rightarrow B\cup ^AC$ which we have seen to be a weak equivalence, gives the standard map $B\rightarrow B\cup ^AC$ which is therefore a weak equivalence. This proves the first half of properness.
We now prove the second statement, proceeding as usual by induction on $n$. Factoring $B\rightarrow A$ into a cofibration followed by a fibration and treating the fibration, we can assume that the morphism is a cofibration (note that a fibration which is a weak equivalence, over an $n$-category, is again an $n$-category so the hypotheses are preserved). Let $A'\subset A$ be the full sub-$n$-category consisting of the objects which are in the image of $B_0$. Let $C':= A'\times _AC$. The morphism $B\rightarrow A'$ is an iso-equivalence so the $B_{p/}\rightarrow A'_{p/}$ are equivalences of $n-1$-categories. The morphisms $C'_{p/} \rightarrow
A'_{p/}$ are fibrant, so by the inductive hypothesis $$(C'\times _{A'} B)_{p/} = C'_{p/} \times _{A'_{p/}} B_{p/} \rightarrow C'_{p/}$$ are equivalences. This implies that $$C\times _{A} B = C'\times _{A'} B \rightarrow C'$$ is an equivalence. Now $C'$ is a full sub-$n$-category of $C$ (meaning that for any objects $x_0,\ldots , x_m$ of $C'$ the morphism $C'_{m/}(x_0,\ldots , x_m)\rightarrow C_{m/}(x_0,\ldots , x_m)$ is an isomorphism), so to prove that $C'\rightarrow C$ is an equivalence it suffices to prove essential surjectivity. Suppose $x$ is an object of $C$. It projects to an object $y$ in $A$ which is equivalent to an object $y'$ coming from $A'$. By \[intervalK\] there is a morphism $K\rightarrow A$ sending $0$ to $y$ and $1$ to $y'$. The object $x$ provides a lifting to $C$ over $\{ 0\}$, so by the condition that $C\rightarrow A$ is fibrant there is a lifting to $K\rightarrow C$ sending $0$ to $x$ and $1$ to an object lying over $y'$. In particular $1$ goes to an object of $C'$. This shows that $x$ is equivalent to an object of $C'$, the essential surjectivity we needed.
We now sketch an example showing why axiom [**Pr**]{}(2) can’t be true in general. Let $A$ be the category $I^{(2)}$ with objects $0,1,2$ and one morphism $i\rightarrow j$ for $i\leq j$ ($i,j=0,1,2$). Let $B$ be the sub-$1$-precat obtained by removing the morphism $0\rightarrow 2$ (it is the pushout of two copies of $I$ over the object $1$). The morphism $B\rightarrow
A$ is a weak equivalence. Let $C$ be a $1$-category with three objects $x_0,
x_1, x_2$ and morphisms from $x_i$ to $x_j$ only when $i\leq j$. There is automatically a unique morphism $C\rightarrow A$ sending $x_i$ to $i$. One can see that this morphism is fibrant. We can choose $C$ so that the composition morphism $$C_1(x_0,x_1)\times C_1(x_1,x_2)\cong C_2(x_0, x_1, x_2)\rightarrow C_1(x_0,x_2)$$ is not an isomorphism. Let $D:= Cat(B\times _AC)$. There is a unique morphism $D\rightarrow C$ extending the second projection morphism, and this morphism takes $D_1(x_0,x_1)$ (resp. $D_1(x_1,x_2)$) isomorphically to $C_1(x_0,x_1)$ (resp. $C_1(x_1,x_2)$). However, the composition morphism for $D$ is an isomorphism $$D_1(x_0,x_1)\times D_1(x_1,x_2)\stackrel{cong}{\rightarrow} D_1(x_0,x_2).$$ Thus the morphism $D_1(x_0,x_2)\rightarrow C_1(x_0,x_2)$ is not an isomorphism; thus $D\rightarrow C$ is not an equivalence and $B\times _AC\rightarrow C$ is not a weak equivalence. [$/$$/$$/$]{}
As one last comment in this section we note the following potentially useful fact.
\[automatic\] If $f:A\rightarrow A'$ is a fibrant morphism of $m$-precats then it is again fibrant when considered as a morphism of $n$-precats for $n\geq
m$.
[*Proof:*]{} Suppose $m< n$. Define the [*brutal truncation*]{} denoted $\beta \tau _{\leq m}$ from $n$-precats to $m$-precats as follows. If $B$ is an $n$-precat then put $$\beta\tau _{\leq n-1}(B)_M := B_M$$ for $M= (m_1, \ldots , m_k)$ with $k <m$ whereas for $M$ of length $m$ put $$\beta\tau _{\leq n-1}(B)_M:=B_M /\langle B_{M, 1} \rangle$$ where $\langle B_{M, 1} \rangle$ denotes the equivalence relation on $B_M$ generated by the image of $B_{M,1}\rightarrow B_M\times B_M$. This should not be confused with the “good” truncation operation $T^{n-m}$ of [@Tamsamani], as in general they will not be the same (however they are equal in the case of $n$-groupoids). We claim that brutal truncation is compatible with the operation $BigCat$, that is $$BigCat (\beta \tau _{\leq m} B) = \beta \tau _{\leq m} (BigCat(B)).$$ To prove this claim, we note two things: (1) that if $\Sigma \rightarrow h$ is one of our standard cofibrations for $n$-precats then $\beta \tau _{\leq m}\Sigma \rightarrow \beta \tau _{\leq m}h$ is a standard cofibration for $m$-precats; and (2) that any map $\beta \tau _{\leq m}\Sigma \rightarrow
\beta \tau _{\leq m}B$ comes from a map $\Sigma \rightarrow B$ or—in the top degree case—at least from a map $\Sigma \rightarrow BigCat(B)$.
By reordering we find that the two sides of the above equation are the same, which gives the claim. Next we claim that brutal truncation preserves weak equivalences. From the previous claim it suffices to note that it preserves equivalences of $n$-categories, and this follows from the fact that brutal truncation of $1$-categories takes equivalences to isomorphisms of sets.
Finally, it is immediate from the definitions that brutal truncation takes cofibrations of $n$-precats to cofibrations of $m$-precats (using of course the fact that there is no injectivity on the top degree morphisms for cofibrations of $m$-precats).
Suppose $A$ is an $m$-precat, and let $Ind^n_m(A)$ denote $A$ considered as an $n$-precat (for this we simply set $Ind^n_m(A)_{M,M'}:= A_M$ for $M$ of degree $m$ and any $M'$ or for $M$ of degree $<m$ and empty $M'$). Then (speaking of absolute $Hom$ here rather than internal $Hom$ as in the next section) $$Hom (\beta \tau _{\leq m} B, A) = Hom (B, Ind^n_m(A)),$$ in other words $\beta \tau _{\leq m}$ and $Ind^n_m$ are adjoint functors.
We can now prove the lemma. If $f$ is a fibrant morphism of $m$-precats and $B\rightarrow C$ is a trivial cofibration of $n$-precats then $\beta\tau _{\leq
m}B\rightarrow \beta\tau _{\leq
m}C$ is a trivial cofibration of $m$-precats, so $f$ has the lifting property for this latter. By adjointness $Ind^n_m(f)$ has the lifting property for $B\rightarrow C$. Therefore $Ind ^n_m(f)$ is fibrant. [$/$$/$$/$]{}
[**.Internal $\underline{Hom}$ and $nCAT$**]{}
Recall the result of Corollary \[forInternalHom\]: that direct product with any $n$-precat preserves weak equivalences. Direct product also preserves cofibrations, so it preserves trivial cofibrations. This property is not a standard property of any closed model category, it is one of the nice things about our present situation which allows us to obtain the right thing by looking at internal $Hom$ of $n$-precats.
\[hom\] Suppose $A$ is an $n$-precat and $B$ is a fibrant $n$-precat. Then the internal $
\underline{Hom}(A,B)$ of presheaves over $\Theta ^n$ is a fibrant easy $n$-category. Furthermore if $B'\rightarrow B$ is a fibrant morphism then $\underline{Hom} (A, B')\rightarrow \underline{Hom} (A, B)$ is fibrant. Similarly if $A\rightarrow A'$ is a cofibration and if $B$ is fibrant then $\underline{Hom}(A', B)\rightarrow \underline{Hom}(A,B)$ is fibrant.
[*Proof:*]{} Note that it suffices to prove that $\underline{Hom} (A,B)$ is fibrant, for \[FibImpliesCat\] then shows that it is an easy $n$-category. A morphism $S\rightarrow \underline{Hom} (A,B)$ is the same thing as a morphism $S\times
A\rightarrow B$. Suppose $S\rightarrow T$ is a trivial cofibration. Then $S\times A \rightarrow T \times A$ is a trivial cofibration. It follows immediately from the definition of $B$ being fibrant that any map $S\times A \rightarrow B$ extends to a map $T \times A \rightarrow B$. Thus $\underline{Hom} (A, B)$ is fibrant. Similarly if $B'\rightarrow B$ is fibrant then any map $T\times A\rightarrow B$ with lifting $S\times A\rightarrow B'$ admits a compatible lifting $T\times A\rightarrow B$. Thus $\underline{Hom} (A,B')\rightarrow \underline{Hom}(A, B)$ is fibrant.
Suppose $A\rightarrow A'$ is cofibrant, and $B$ fibrant. We show that $\underline{Hom} (A',B)\rightarrow \underline{Hom}(A, B)$ satisfies the lifting property to be fibrant. Say $S\rightarrow S'$ is a trivial cofibration, and suppose we have maps $S'\rightarrow
\underline{Hom}(A, B)$ and lifting $S\rightarrow \underline{Hom}(A', B)$. These are by definition maps $S'\times A \rightarrow B$ and $S\times
A'\rightarrow B$ which agree over $S\times A$. These give a morphism $$f:S\times A' \cup ^{S\times A} S ' \times A \rightarrow B.$$ The morphism $$g:S\times A' \cup ^{S\times A} S ' \times A \rightarrow S' \times A'$$ is a cofibration. Lemma \[pushout\] applied to the trivial cofibration $S\times A\rightarrow S'\times A$ implies that the morphism $$S\times A'\rightarrow S\times A' \cup ^{S\times A} S '\times A$$ is a weak equivalence. On the other hand the morphism $S\times A' \rightarrow S'\times A'$ is a weak equivalence by \[forInternalHom\], so by Lemma \[remark\] the morphism $g$ is a weak equivalence. Thus the fact that $B$ is fibrant means that our morphism $f$ extends to a morphism $S'\times A' \rightarrow B$, and this gives exactly the desired lifting property for the last statement of the theorem. [$/$$/$$/$]{}
\[stability\] Suppose $A\rightarrow A'$ is a weak equivalence, and $B$ fibrant. Then $\underline{Hom} (A', B)\rightarrow \underline{Hom} (A, B)$ is an equivalence of $n$-categories.
If $B\rightarrow B'$ is an equivalence of fibrant $n$-precats then $\underline{Hom}(A,B)\rightarrow
\underline{Hom}(A,B')$ is an equivalence.
Suppose $A\rightarrow B$ and $A\rightarrow C$ are cofibrations. Then $$\underline{Hom} (B\cup ^AC, D) = \underline{Hom} (B, D)
\times _{\underline{Hom}(A,D)}\underline{Hom}(C,D).$$
[*Proof:*]{} The last statement is of course immediate, because for any $S$ we have $(B\cup ^AC)\times S = (B\times S ) \cup ^{(A\times S)} (C\times S)$. We treat the other statements.
Suppose first that $A\rightarrow A'$ is a trivial cofibration. Suppose that $S\rightarrow T$ is any cofibration. Suppose we have maps $T\rightarrow \underline{Hom}(A, B)$ lifting over $S$ to $S\rightarrow
\underline{Hom}(A', B)$. We claim that the lifting extends to $T$; then the characterization of weak equivalences in ([@Quillen] §5, Definition 1, Property M6, part (c)) will imply that $\underline{Hom}(A', B)\rightarrow
\underline{Hom}(A,B)$ is a weak equivalence. To prove the claim, note that our data correspond to a morphism $$T\times A \cup ^{S\times A} S\times A' \rightarrow B.$$ The morphism $S\times A \rightarrow S\times A'$ is a trivial cofibration, so the morphism $$T\times A \rightarrow T\times A \cup ^{S\times A} S\times A'$$ is a trivial cofibration, and since $T\times A \rightarrow T\times A'$ is a weak equivalence we get that the morphism $$T\times A \cup ^{S\times A} S\times A'
\rightarrow T\times A'$$ is a trivial cofibration. The fibrant property of $B$ implies that our map extends to a map $T\times A' \rightarrow B$, so we get the required lifting to $T\rightarrow \underline{Hom}(A', B)$. This implies that $\underline{Hom}(A',B)\rightarrow \underline{Hom}(A,B)$ is a fibrant weak equivalence.
Next we treat the case of any weak equivalence $A\rightarrow A'$. Let $C$ be the $n$-precat pushout of $A\times 0\rightarrow A\times \overline{I}$ and $A\times 0 \rightarrow A' \times 0$. Since $0\rightarrow \overline{I}$ is a trivial cofibration, the various morphisms $$A\hookrightarrow C, \;\; A' \hookrightarrow C, \;\; C \rightarrow A'$$ (the first sending $A$ to $A\times 1$, the second sending $A'$ to $A' \times 0$ and the third coming from the projection $A\times \overline{I} \rightarrow A'$) are all weak equivalences. We have a composable pair of morphisms $$\underline{Hom} (A', B) \rightarrow \underline{Hom} (C, B) \rightarrow
\underline{Hom} (A', B)$$ composing to the identity, and where the second arrow is an equivalence by the previous paragraph since $A'\rightarrow C$ is a trivial cofibration. Therefore the first arrow is an equivalence. Next, the morphism $\underline{Hom} (C, B)\rightarrow
\underline{Hom} (A,B)$ obtained from the trivial cofibration $A\rightarrow C$ (going to $A\times 1$) is an equivalence, so the composed map $\underline{Hom}
(A', B)\rightarrow \underline{Hom} (A,B)$ is an equivalence. This is the map induced by our original $A\rightarrow A'$. This completes the proof of the first part of the lemma.
We now turn to the second part and treat first a fibrant weak equivalence $B\rightarrow B'$. Note first that such a morphism satisfies the lifting property for any cofibrations (this is the other half of CM4 which comes from Joyal’s trick). We prove that $\underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B')$ satisfies lifting for any cofibration (which as above implies that it is a fibrant weak equivalence). Suppose $S\rightarrow T$ is a cofibration. and $T\rightarrow \underline{Hom} (A,B')$ is a map with lifting over $S$ to a map $S\rightarrow \underline{Hom}
(A,B)$. These correspond to maps $T\times A \rightarrow B'$ and lifting to $S\times A \rightarrow B$. The morphism $S\times A \rightarrow
T\times A $ is a cofibration so by the lifting property of $B\rightarrow B'$ for any cofibration, there is a lifting to $T\times A\rightarrow B$ compatible with the given map on $S$. This establishes the necessary lifting property to conclude that $\underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B')$ is a fibrant equivalence.
Next suppose that $i:B\rightarrow B'$ is a trivial cofibration of fibrant $n$-precats. The lifting property for $B$ lets us choose a retraction $r:
B'\rightarrow B$ such that $ri= 1_B$. Let $p:B' \cup ^BB' \rightarrow B'\rightarrow
B'$ be the projection which induces the identity on both of the components $B'$. Note that $B'\rightarrow B' \cup _BB'$ is a trivial cofibration by \[pushout\] so the projection $p$ is a weak equivalence (using \[remark\]). Choose a factorization $$B' \cup ^BB' \rightarrow P \rightarrow B'$$ with the first morphism cofibrant and the second morphism fibrant (whence $P$ fibrant itself); and both morphisms weak equivalences. Let $q: B' \cup
^BB'\rightarrow B'$ be the morphism inducing the retraction $r$ on the first copy of $B'$ and the identity on the second copy. Since $B'$ is fibrant this extends to a morphism we again denote $q: P\rightarrow B'$. The result of the previous paragraph implies that the morphism $$p:\underline{Hom}(A, P)\rightarrow \underline{Hom}(A, B')$$ is an equivalence, which implies that either of the two morphisms $$j_0, j_1:\underline{Hom}(A, B')\rightarrow \underline{Hom}(A, P)$$ (refering to the two inclusions $j_0,j_1: B' \rightarrow P$) are equivalences. Now we have that the composition $$\underline{Hom}(A, B')\stackrel{j_1}{\rightarrow} \underline{Hom}(A, P)
\stackrel{q}{\rightarrow} \underline{Hom}(A, B')$$ is the morphism induced by $qj_1= 1_{B'}$ thus it is the identity. The fact that the morphism induced by $j_1$ (the first of the above pair) is an equivalence implies that the morphism induced by $q$ (the second in the above sequence) is an equivalence. But since the morphism induced by $j_0$ is an equivalence, we get that the morphism induced by $qj_0=ir$ is an auto-equivalence of $\underline{Hom}(A,B')$. The morphism induced by $ri=1_B$ is of course the identity. The last part of Lemma \[remark\] now implies that the morphism $$i: \underline{Hom}(A,B)\rightarrow \underline{Hom}(A,B')$$ is an equivalence. This proves the statement in case of a trivial cofibration.
Finally note that any equivalence of $n$-categories $B\rightarrow B'$ decomposes as a composition $B\rightarrow C \rightarrow B'$ where the first arrow is a trivial cofibration and the second a fibration and weak equivalence. Note that $C$ is fibrant since by hypothesis $B'$ is fibrant. Thus our two previous discussions apply to give that the two morphisms $$\underline{Hom}(A,B)\rightarrow \underline{Hom}(A,C)
\rightarrow \underline{Hom}(A,B')$$ are equivalences, their composition is therefore an equivalence. This completes the proof of the first paragraph of Lemma \[stability\]. [$/$$/$$/$]{}
For any fibrant $n$-categories $A$, $B$ and $C$ we have composition morphisms $$\underline{Hom} (A, B) \times \underline{Hom}(B,C) \rightarrow
\underline{Hom}(A,C),$$ which are associative.
Define a simplicial $n$-category $nCAT$ by setting $nCAT_0 $ equal to a set of representatives for [*isomorphism*]{} classes of fibrant $n$-categories, and by setting $$nCAT_m(A_0, \ldots , A_m):= \underline{Hom} (A_0, A_1) \times \ldots \times
\underline{Hom} (A_{m-1},
A_m),$$ with simplicial structure given by the above compositions. Since $nCAT_0$ is a set considered as $n$-precat, this simplicial $n$-precat (presheaf on $\Delta \times \Theta ^{n}$) descends to a presheaf on $\Theta ^{n+1}$, in other words it is an $n+1$-precat. The composition gives the necessary conditions in the first degree, and in higher degrees the fact that $\underline{Hom} (A,B)$ are $n$-categories completes what we need to know to conclude that $nCAT$ is an $n+1$-category.
If $A$ and $B$ are $n$-categories but not necessarily fibrant then let $Fib(A)$ and $Fib(B)$ be their fibrant replacements (given by the above construction for Theorem \[cmc\] for example). We call these the [*fibrant envelopes*]{}. The “right” $n$-category of morphisms from $A$ to $B$ is $\underline{Hom}(Fib(A),
Fib(B))$. We will sometimes use the notation $$HOM (A, B):= \underline{Hom}(Fib(A), Fib(B)).$$ We obtain an $n+1$-category equivalent to $nCAT$ by taking all $n$-categories as objects and taking the $HOM (A,B)$ as morphism $n$-categories.
[*Question:*]{} Describe the fibrant envelope of the $n+1$-category $nCAT$. This would be important if one wants to consider weak morphisms $A\rightarrow
nCAT$ as families of $n$-categories indexed by $A$ in a meaningul way.
We have almost proved Conjecture ([@Tamsamani] between 1.3.6 and 1.3.7) on the existence of $nCAT$. We just have to check that the truncation of $nCAT$ down to a $1$-category is equivalent to the localization of the category of $n$-categories by the subcategory of morphisms which are equivalences.
In Corollary \[htytype\] above, we have seen that the localization in question is equal to the localization of $PC_n$ by the weak equivalences. We now know that $PC_n$ is a closed model category, and Quillen shows in this case that the $Hom$ in the localized category is equal to the set of [*homotopy classes*]{} of morphisms between fibrant and cofibrant objects. In our (second) definition above, we took $nCAT$ to be the category of fibrant (and automatically cofibrant) $n$-precats. The $Hom$ in the localized category is thus the set of homotopy classes of maps. On the other hand, the truncation of $nCAT$ down to a $1$-category is obtained by replacing the $\underline{Hom}(A,B)$ $n$-categories by their sets of equivalence classes of objects. Thus, to prove the conjecture we simply need to show that for $A$ and $B$ fibrant, the set of equivalence classes of objects in the $n-1$-category $\underline{Hom} (A,B)$ is equal to the set of homotopy classes of maps from $A$ to $B$. Note that the objects of $\underline{Hom} (A,B)$ are again just the maps from $A$ to $B$ so we are reduced to showing the following lemma.
If $A$ and $B$ are fibrant $n$-precats then two morphisms $f,g: A\rightarrow B$ are homotopic in the sense of [@Quillen] if and only if the corresponding elements of the $n$-category $\underline{Hom}(A,B)$ are equivalent.
[*Proof:*]{} Suppose $f$ and $g$ are homotopic ([@Quillen] p. 0.2). Then there is an object $A'$ with morphisms $i,j: A\rightarrow A'$ each inducing a weak equivalence, with a projection $p:A'\rightarrow A$ such that the compositions $pi$ and $pj$ are the identity, and a morphism $h: A'\rightarrow B$ such that $hi=f$ and $hj=g$. We may assume that $A'$ is fibrant. Then we obtain pullback morphisms on the $\underline{Hom}$ $n$-categories and in particular, two morphisms $$i^{\ast}, j^{\ast} :\underline{Hom} (A', B) \rightarrow \underline{Hom} (A, B)$$ and a morphism $$p^{\ast}: \underline{Hom} (A, B)\rightarrow \underline{Hom} (A', B)$$ which are weak equivalences by Lemma \[stability\]. These induce isomorphisms on the sets of equivalence classes which we denote $T^n\underline{Hom}
(A,B)$ etc., so we have $$T^ni^{\ast}, T^nj^{\ast} :T^n\underline{Hom} (A', B) \cong T^n
\underline{Hom} (A, B)$$ and $$T^np^{\ast}:T^n \underline{Hom} (A, B)\cong T^n \underline{Hom} (A', B).$$ Here, as before, we have that $T^ni^{\ast} \circ T^np^{\ast}$ and $T^n j^{\ast}\circ T^np^{\ast}$ are equal to the identity. This implies that $T^ni^{\ast} = T^nj^{\ast}$ and hence that their applications to the class of $h$ give the same equivalence class. The results are respectively the classes of $f$ and of $g$, hence $f$ is equivalent to $g$.
Conversely suppose $f$ and $g$ are equivalent as objects in $\underline{Hom}(A,B)$. Then by Proposition \[intervalK\] there is a contractible $K$ with $0,1\in K$ and a morphism $K\rightarrow \underline{Hom}(A,B)$ taking $0$ to $f$ and $1$ to $g$. This yields (by the universal property of the internal $\underline{Hom}$) a morphism $h:A\times
K\rightarrow B$. This morphism together with the various others gives a homotopy from $f$ to $g$. [$/$$/$$/$]{}
[([@Tamsamani] Conjecture 1.3.6-7)]{} The $n+1$-category $nCAT$ yields when truncated down to a $1$-category the localization $Ho-n-Cat$ of [@Tamsamani].
[$/$$/$$/$]{}
[**.$n$-stacks**]{}
We can give a preliminary discussion of the notion of $n$-stack, following the lines that are already well known for simplicial presheaves and even $n$-stacks of $n$-groupoids (approached via topological spaces in [@flexible], discussed for $n=2,3$ in [@Breen]). Our present discussion will be incomplete, basically for the following reason: if ${{\cal X}}$ is a $1$-category, there are several natural types of objects which represent the idea of a family of $n$-categories indexed (contravariantly) by ${{\cal X}}$, and we would like to know that all of these notions are equivalent. The main possible versions are: 1. A functor ${{\cal X}}^o\rightarrow nCAT$, which if we take the second point of view on $nCAT$ presented above, is the same thing as a presheaf of fibrant $n$-categories over ${{\cal X}}$; 2. A weak functor from ${{\cal X}}^o$ to $nCAT$, in other words a functor from ${{\cal X}}^o$ to $Fib(nCAT)$ or (what is basically the same thing) an element of $HOM({{\cal X}}^o, nCAT)$ i.e. a morphism in $(n+1)CAT$; 3. A “fibered $n$-category over ${{\cal X}}$”, which would be a morphism of $n$-categories ${{\cal F}}\rightarrow {{\cal X}}$ (note that a $1$-category considered as an $n$-category is automatically fibrant by \[automatic\]) satisfying some condition analogous to the definition of fibered $1$-category—I haven’t written down this condition (note however that it is distinct from the condition that the morphism be fibrant in the sense we use in this paper).
Here is what I currently know about the relationship between these points of view. From (1) one automatically gets (2) just by composing with the morphism $nCAT \rightarrow Fib(nCAT)$ to the fibrant envelope. From (2) one should be able to get (3) by pulling back a universal fibered $n+1$-category over $Fib(nCAT)$. To construct this universal object, first construct a universal $n+1$-category ${{\cal U}}\rightarrow nCAT$ (with fibers the $n$-categories being parametrized—in particular this morphism is relatively $n$-truncated) then replace the composed morphism ${{\cal U}}\rightarrow Fib(nCAT)$ by a fibrant morphism. Finally, from (3) one should be able to get (1) by applying the “sections functor”: if ${{\cal F}}\rightarrow {{\cal X}}$ is a fibered $n$-category then define $\Gamma ({{\cal X}}, {{\cal F}})$ to be the $n$-category fiber (calculated in the correct homotopic sense) over $1_{{{\cal X}}} \in HOM ({{\cal X}}, {{\cal X}})$ of $$HOM ({{\cal X}}, {{\cal F}})\rightarrow HOM ({{\cal X}}, {{\cal X}}).$$ Require now that ${{\cal F}}\rightarrow {{\cal X}}$ be a fibrant morphism (if this doesn’t come into the condition of being fibered already). Then $$X\in {{\cal X}}\mapsto \Gamma ({{\cal X}}/X, {{\cal F}}\times _{{{\cal X}}} ({{\cal X}}/X ))$$ should be strictly functorial in the variable $X$ yielding a presheaf ${{\cal X}}^o
\rightarrow nCAT$ which is notion (1). The condition of being a fibered category should imply (as it does in the case $n=1$) that the morphism $$\Gamma ({{\cal X}}/X, {{\cal F}}\times _{{{\cal X}}} ({{\cal X}}/X ))\rightarrow {{\cal F}}_X:= {{\cal F}}\times
_{{{\cal X}}} \{ X\}$$ be an equivalence of $n$-categories (one might even try to take this condition as the definition of being fibered but I’m not sure if that would work). Finally we would like to show that doing these three constructions in a circle results in an essentially equivalent object.
The previous paragraph is for the moment speculative, the main questions left open being the definition of “fibered $n$-category” and the construction of the universal family. However, for the rest of this section we will discuss the theory of $n$-stacks supposing that the above equivalences are known. Denote by $\int$ the operation going from (1) to (3).
Suppose ${{\cal X}}$ is a site. There are a couple of different ways of approaching the notion of $n$-stack over ${{\cal X}}$. Our first definition will be modelled on what was done in [@flexible]. A fibered $n$-category ${{\cal F}}\rightarrow {{\cal X}}$ is an [*$n$-stack*]{} if for any $X\in {{\cal X}}/X$ and any sieve ${{\cal B}}\subset {{\cal X}}/X$ the morphism $$\Gamma ({{\cal X}}/X, {{\cal F}}\times _{{{\cal X}}} ({{\cal X}}/X ))\rightarrow
\Gamma ({{\cal B}}, {{\cal F}}\times _{{{\cal X}}} {{\cal B}})$$ is an equivalence of $n$-categories. If ${{\cal F}}\rightarrow {{\cal X}}$ is a fibered $n$-category then we (should be able to) construct the [*associated $n$-stack*]{} by iterating $n+2$ times the operation $$L({{\cal F}}):= \int \left( X\mapsto \lim _{\rightarrow , {{\cal B}}\subset {{\cal X}}/X}
\Gamma ({{\cal B}}, {{\cal F}}\times _{{{\cal X}}} {{\cal B}}) \right) .$$ This conjecture is based on the corresponding result for flexible sheaves in [@flexible].
The second main type of approach is to combine the theory of simplicial presheaves of Jardine-Joyal-Brown-Gersten (cf [@Jardine]) with the discussion in the present paper to obtain a closed model structure for the category of presheaves of $n$-precats over ${{\cal X}}$. In this case the fibrant condition would imply the condition of being an $n$-stack. To give the definitions (without proving that we get a closed model category) it suffices to define weak equivalence—the cofibrations being just the maps which over each object of ${{\cal X}}$ are cofibrations of $n$-precats, and the fibrations then being defined by the lifting property for trivial cofibrations. (As usual the main problem would then be to prove that pushout by a trivial cofibration is again a trivial cofibration—and for this we could probably just combine the proofs of Jardine/Joyal [@Jardine] and the present paper.) If $A\rightarrow B$ is a morphism of presheaves of $n$-precats over ${{\cal X}}$ then we obtain a morphism $Cat(A)\rightarrow Cat(B)$ of presheaves of easy $n$-categories (where $Cat(A)(X):= Cat(A(X))$). We will say that $A\rightarrow
B$ is a weak equivalence if $Cat(A)\rightarrow Cat(B)$ is a weak equivalence of presheaves of $n$-categories, notion which we now define. Let $T$ denote Tamsamani’s truncation operation [@Tamsamani] which is functorial so it extends to presheaves of $n$-categories. A morphism of presheaves of $n$-categories $A\rightarrow B$ is a [*weak equivalence in top degree*]{} if for every $n$-morphism of $B$ and lifting of its source and target to $n-1$-morphisms in $A$, there exists a unique lifting to an $n$-morphism in $A$. Now we say that a morphism $A\rightarrow B$ of presheaves of $n$-categories is a [*weak equivalence*]{} if for every $k$ the morphism $$Sh(T^kA)\rightarrow Sh(T^kB)$$ is a weak equivalence in top degree, where $Sh$ denotes the stupid sheafification operation (i.e. sheafify each of the presheaves $A_M$).
In this point of view, if $A$ is a presheaf of $n$-categories over ${{\cal X}}$ then we define the [*associated $n$-stack*]{} to be the fibrant object equivalent to $A$ in the previous presumed closed model category.
[*$n$-categories as $n-1$-stacks*]{}
Heuristically we can define a structure of [*site*]{} on $\Delta$ where the coverings of an object $m$ are the collections of morphisms $\lambda _i \hookrightarrow m$ where $\lambda _i = \{ a_i , a_i + 1, \ldots , b_i\}$ such that $a_{i+1}=b_i$. If $A$ is an $n$-precat then the collection $\{ A_{p/}\}$ may be thought of as a presheaf of $n-1$-precats over $\Delta $. The condition to be an $n$-category is that this should be a presheaf of $n-1$-categories which should satisfy descent for coverings, i.e. it should be an $n-1$-stack of $n-1$-categories over this site. The construction $Cat$ is essentially just finding the $n-1$-stack associated to an $n-1$-prestack by an operation similar to that described in [@flexible]. The main problems above are caused by the fact that this site doesn’t admit fiber products. It might be a good idea to replace this site by its associated topos, the category of categories, which would lead to the yoga: [*that an $n$-category is an $n-1$-stack over the topos of categories.*]{}
It might be possible, by treating $n$-stacks at the same time as $n$-categories, to simplify the arguments of the present paper by recursively defining $n$-categories as $n-1$-stacks. I haven’t thought about this any further.
[**.The generalized Seifert-Van Kampen theorem**]{}
Our closed model category structure allows us (with a tiny bit of extra work) to obtain the analogue of the Siefert-Van Kampen theorem for the Poincaré $n$-groupoid of a topological space $\Pi _n(X)$ defined by Tamsamani ([@Tamsamani], §2.3).
\[svk\] If $X$ is a space covered by open subsets $X = U\cup V$ then (setting $W:=
U\cap V$) $\Pi _n (X)$ is equivalent to the category-theoretic pushout of the diagram $$\Pi _n (U) \leftarrow \Pi _n (W) \rightarrow \Pi _n (V).$$
In order to prove this theorem we recall Tamsamani’s realization functor from $n$-precats to topological spaces ([@Tamsamani] §2.5). There is a covariant functor $R: \Delta ^n \rightarrow Top$ which associates to $M= (m_1, \ldots , m_n)$ the product $R^{m_1}\times
\ldots \times R^{m_n}$ where $R^m$ denotes the usual topological $m$-simplex. If $A: (\Delta ^n)^o\rightarrow Sets$ is a presheaf of sets then Tamsamani defines the [*realization of $A$*]{} in the standard way combining $R$ and $A$. We denote this $\langle R, A\rangle$ because it is a sort of pairing of functors. If $A$ is an $n$-precat in our notations then pull it back to a presheaf on $\Delta ^n$ and apply the realization (we still denote this as $\langle R, A\rangle$). The functor $\langle R, \cdot
\rangle$ obviously preserves pushouts.
[*Caution:*]{} The realization functor does not preserve cofibrations. It takes injective morphisms of presheaves over $\Theta^n$ to cofibrations of spaces, but the cofibrations which are not injective in the top degree are taken to non-injective morphisms.
Recall Tamsamani’s Proposition 3.4.2(ii):
If $A\rightarrow B$ is an equivalence of $n$-categories then $$\langle R, A \rangle \rightarrow \langle R, B \rangle$$ is an equivalence of spaces.
[*Proof:*]{} The proof of [@Tamsamani] for $n$-groupoids using induction on $n$ also works for $n$-categories. [$/$$/$$/$]{}
Say that a morphism $X\rightarrow Y$ of topological spaces is an [*$n$-weak equivalence*]{} if it is an isomorphism on $\pi _0$ and for any choice of basepoint in $X$ it is an isomorphism on $\pi _i$ for $i\leq n$. This is equivalent to saying that it induces a weak equivalence on the Postnikov tower up to stage $n$.
\[equivtoequiv\] The realization functor $\langle R, \cdot \rangle$ from $n$-precats to $Top$ takes weak equivalences to $n$-weak equivalences of topological spaces.
[*Proof:*]{} Realization takes our standard trivial cofibrations $\Sigma \rightarrow h$ to homotopy equivalences of topological spaces. This is essentially the content of the constructions of retractions in the proof of Theorem 2.3.5 (that $\Pi _n(X)$ is an $n$-nerve) of [@Tamsamani].
For all except the upper boundary cases, the standard trivial cofibrations are taken to cofibrations of topological spaces. Pushout by the injective standard trivial cofibrations becomes pushout by a trivial cofibration of spaces, whence a homotopy equivalence. In order to deal with the upper boundary cases we introduce the following notation: $$\rangle R, A \langle _{q}$$ denotes the $q$-skeleton of the realization of $A$, that is the realization taken over all $M$ with $\sum m_i \leq q$. Then, if $q\leq n-1$ the functor $\rangle R, \cdot \langle _{q}$ takes cofibrations to cofibrations of spaces.
Suppose $\varphi : \Sigma \rightarrow h$ is a standard trivial cofibration in one of the boundary cases. Using the notation of §2 we can write $\Sigma$ as the coequalizer of $$h' \sqcup h' \sqcup \Upsilon ' \rightarrow h^a \sqcup h^a$$ (the component $\Upsilon$ which appears on the right in the general case disappears in the upper boundary case). The map $\Sigma \rightarrow h$ is given by the map $h^a \sqcup h^b\rightarrow h$ which in this case is two times the identity (because $h(M, m, 1^{k+1})$ which doesn’t exist is replaced by $h:= h(M, m, 1^k)= h^a=h^b$). The cells in $\langle R,\Sigma \rangle $ of dimension $n$ are automatically of the form $h(1^l, 1, 1^k)$ for maps $1^l\rightarrow M$ and $1\rightarrow m$. There are two such which are identified whenever $1\rightarrow m$ is not one of the principal morphisms. The cells coming from principal $1\rightarrow m$ occur only once in the realization of $\Sigma$ already. It follows (since any non-principal $1\rightarrow m$ is a path which is homotopic to a concatenation of principal $1\rightarrow m$) that the $n$-cells which are identified are homotopic.
Note that on the level of cells of dimension $<n$ the morphism $\Sigma
\rightarrow h$ is an isomorphism. In particular, pushout via $\varphi$ over any $\Sigma \rightarrow A$ preserves the $n-1$-skeleton of the realization, and the $n$-cells which are identified are homotopic. In this boundary case the pushout by $\varphi$ is surjective, in particular it is surjective on $n+1$-cells. A surjective morphism of cell complexes which is an isomorphism on $n-1$-skeleta and which only identifies $n$-cells which are homotopic (relative the $n-1$-skeleton) is an $n$-weak equivalence. This completes the proof that pushout by any of our standard trivial cofibrations $\varphi$ induces an $n$-weak equivalence.
It follows by construction of the operation $Cat$ that for any $n$-precat $A$ the morphism $$\langle R, A \rangle \rightarrow \langle R, Cat(A) \rangle$$ is an $n$-weak equivalence of spaces. Now we can complete the proof: if $A\rightarrow B$ is a weak equivalence then by definition $Cat(A)\rightarrow Cat(B)$ is an equivalence of $n$-categories so in the diagram $$\begin{array}{ccc}
\langle R, A \rangle & \rightarrow & \langle R, B \rangle \\
\downarrow && \downarrow \\
\langle R, Cat(A) \rangle & \rightarrow & \langle R, Cat(B) \rangle
\end{array}$$ the vertical arrows are $n$-weak equivalences from the previous argument and the bottom arrow is a weak equivalence by the proposition, so the top arrow is an $n$-weak equivalence of spaces. [$/$$/$$/$]{}
\[groupoid\] If $A\rightarrow B$ and $A\rightarrow C$ are morphisms of $n$-groupoids with one being a cofibration, then the category-theoretic pushout $Cat(B\cup ^AC)$ is an $n$-groupoid.
[*Proof:*]{} We say that an $n$-precat is a [*pre-groupoid*]{} if its associated $n$-category is a groupoid. We prove that the pushout of pre-groupoids is again a pre-groupoid, and we proceed by induction on $n$ so we may assume this is known for $n-1$-pre-groupoids.
Suppose now that $A$, $B$ and $C$ are $n$-groupoids with morphisms as in the statement of the lemma. Then the $A_{p/}$, $B_{p/}$ and $C_{p/}$ are $n-1$-groupoids, and $$(B\cup ^AC)_{p/} = B_{p/}\cup ^{A_{p/}}C_{p/}.$$ In particular, $(B\cup ^AC)_{p/}$ are $n-1$-pre-groupoids. The process of going from this collection of $n-1$-precats to the collection corresponding to $Cat
(B\cup ^AC)$ as described in §4, uses only iterated pushouts by the various $(B\cup ^AC)_{p/}$ in various combinations. Since we know by induction that pushouts of $n-1$-pre-groupoids are again $n-1$-pre-groupoids, it follows that $Cat(B\cup ^AC)_{p/}$ are $n-1$-groupoids. It suffices now to show that the truncation of $Cat(B\cup ^AC)$ down to a $1$-category is a groupoid. But this truncation is the same as the brutal truncation since we know that the $Cat(B\cup ^AC)_{p/}$ are $n-1$-groupoids. On the other hand, brutal truncation commutes with the operations $Cat$ and pushout, therefore the truncation of $Cat(B\cup ^AC)$ is the pushout of the truncations of $A$, $B$ and $C$ which are groupoids. Finally, the $1$-category pushout of groupoids is again a groupoid, so $Cat(B\cup ^AC)$ is a groupoid.
To complete the proof it remains to be seen that the pushout of $n$-pregroupoids is a pregroupoid. Suppose $A$, $B$ and $C$ are $n$-pre-groupoids. Then by reordering $$BigCat(B\cup ^AC) = BigCat(Cat(B) \cup ^{Cat(A)}Cat(C)).$$ Thus $BigCat(B\cup ^AC)$ is the category-theoretic pushout of $n$-groupoids so by the previous argument it is an $n$-groupoid. This shows that $B\cup ^AC$ is an $n$-pre-groupoid, completing the proof of the induction step. [$/$$/$$/$]{}
[*Proof of Theorem \[svk\]:*]{} Note first of all that Tamsamani’s proof that $\Pi_n(X)$ is an $n$-category ([@Tamsamani] Theorem 2.3.6) actually shows that it is an easy $n$-category. With the notations of the theorem, we have a diagram of easy $n$-categories $$\begin{array}{ccc}
\Pi _n(W) & \rightarrow & \Pi _n(U) \\
\downarrow && \downarrow \\
\Pi _n(V) & \rightarrow & \Pi _n(X).
\end{array}$$ Let $A$ be the pushout $n$-precat of the upper and left arrows. We have a morphism $A\rightarrow \Pi _n(X)$, and hence (non-uniquely) $Cat(A)\rightarrow
\Pi _n(X)$ since the latter is an easy $n$-category. The realization of $A$ is the pushout of the realizations of $\Pi _n(U)$ and $\Pi _n(V)$ over $\Pi
_n(W)$. These last realizations are $n$-weak equivalent to $U$, $V$ and $W$ respectively ([@Tamsamani] 3.3.4), so the realization of $A$ is $n$-weak equivalent to the pushout of $U$ and $V$ over $W$, in other words to $X$. Thus the morphism $A\rightarrow \Pi _n(X)$ induces an $n$-weak equivalence on realizations. On the other hand we have seen above that $A\rightarrow Cat(A)$ induces an $n$-weak equivalence on realizations. Thus the morphism $Cat(A)\rightarrow \Pi _n(X)$ induces an $n$-weak equivalence on realizations. Lemma \[groupoid\] implies that $Cat(A)$ is an $n$-groupoid. Applying the functor $\Pi _n$ again and using Proposition 3.4.4 of [@Tamsamani] we conclude that $Cat(A)\rightarrow \Pi _n(X)$ is an equivalence of $n$-groupoids. This proves the theorem. [$/$$/$$/$]{}
[**.Nonabelian cohomology**]{}
If $A$ is a fibrant $n$-category and $X$ a topological space then define the [*nonabelian cohomology of $X$ with coefficients in $A$*]{} to be $H(X, A): = \underline{Hom} (\Pi _n(X),
A)$. It is an $n$-category. This satisfies Mayer-Vietoris: if $U,V\subset X$ and $W=U\cap V$ then $$m:H(X, A) \rightarrow H(U,A)\times _{H(W,A)}H(V,A)$$ is an equivalence of $n$-categories (where the fiber product is understood to be the homotopic fiber product obtained by replacing one of the morphisms with a fibrant morphism). To see this, note that if $A$ is fibrant then for any cofibration $B\rightarrow C$ the morphism $\underline{Hom}(C,A)\rightarrow
\underline{Hom}(B,A)$ is fibrant. To prove this claim it suffices to remark that if $S\rightarrow T$ is a trivial cofibration then $$S\times C \cup ^{S\times B} T\times B \rightarrow T
\times C$$ is a trivial cofibration, now apply the universal property of the internal $\underline{Hom}$ to obtain the lifting property in question.
In particular, note that $H(U,A)\rightarrow H(W,A)$ and $H(V,A)\rightarrow
H(W,A)$ are fibrations (since open inclusions induce cofibrations of $\Pi _n$). Thus the above fiber product is the homotopic fiber product.
We now prove that the Mayer-Vietoris map $m$ is an equivalence of $n$-categories. It is the same as the map $$\underline{Hom} (\Pi _n(X), A)\rightarrow \underline{Hom} (\Pi _n(U)\cup
^{\Pi _n(W)}\Pi _n(V), A).$$ But we have seen in \[svk\] that the morphism $$\Pi _n(U)\cup ^{\Pi _n(W)}\Pi _n(V)\rightarrow \Pi _n(X)$$ is a trivial cofibration. Thus, to complete the proof it suffices to note (from \[stability\]) that for any trivial cofibration $B\rightarrow C$ the morphism $\underline{Hom} (C,A)\rightarrow \underline{Hom} (B,A)$ is an equivalence of $n$-categories.
If we take cohomology of a CW complex $X$ with coefficients in a fibrant groupoid $A$ then $H(X,A)$ is equivalent to $\Pi _n(\underline{Hom} _{Top}(X,
\langle R,A\rangle )$. To prove this note that $\Pi _n$ is adjoint to the realization $\langle R, A \rangle$, which implies on the level of internal $\underline{Hom}$ that for any $n$-precat $B$ and space $U$, $$\underline{Hom}(B, \Pi _n(U)) = \Pi _n (\underline{Hom}_{Top}(\langle R,
B\rangle , U)$$ where $\underline{Hom}_{Top}$ denotes the compact-open mapping space. Corolary \[equivtoequiv\] and the adjointness imply that for any space $U$, $\Pi _n(U)$ is fibrant. On the other hand, Tamsamani proves in [@Tamsamani] §3 that for any $n$-groupoid $A$ the morphism $$A\rightarrow \Pi _n (\langle R, A \rangle )$$ is an equivalence of $n$-groupoids. Thus if $A$ is a fibrant $n$-groupoid we have an equivalence $$\underline{Hom}(\Pi _n(X), A) \rightarrow
\underline{Hom}(\Pi _n(X), \Pi _n (\langle R, A \rangle ))
= \Pi _n \underline{Hom}_{Top}(\langle R, \Pi _n(X) \rangle ,
\langle R, A \rangle ).$$ On the other hand, again from [@Tamsamani] §3 we know that $\langle R, A \rangle $ is $n$-truncated, and there is a space $W$ and diagram $$X \leftarrow W \rightarrow \langle R, \Pi _n(X) \rangle$$ where the left morphism is a weak homotopy equivalence and the right morphism induces an isomorphism on homotopy groups in degrees $\leq n$. Thus, under the assumption that $X$ is a CW complex (which allows us to obtain weak equivalences when we apply $\underline{Hom}_{Top}$) we obtain a diagram of weak homotopy equivalences $$\underline{Hom}_{Top}(X ,
\langle R, A \rangle )
\rightarrow
\underline{Hom}_{Top}(W ,
\langle R, A \rangle )
\leftarrow
\underline{Hom}_{Top}(\langle R, \Pi _n(X) \rangle ,
\langle R, A \rangle ).$$ Combining with the above we get a diagram of equivalences $$\underline{Hom}(\Pi _n(X), A)\rightarrow
\Pi _n \underline{Hom}_{Top}(W ,
\langle R, A \rangle )
\leftarrow
\Pi _n \underline{Hom}_{Top}(X ,
\langle R, A \rangle ).$$ Thus the nonabelian cohomology with coefficients in an $n$-groupoid coincides with the approach using topological spaces.
Of course even the nonabelian cohomology with coefficients in an $n$-category $A$ which isn’t a groupoid doesn’t really give a new homotopy invariant since all of the information is contained in the Poincaré $n$-groupoid $\Pi _n(X)$. However, it might give some interesting special cases to study.
Once the theory of $n$-stacks gets off the ground, we should be able to interpret $H(X,A)$ as the $n$-category of global sections of the $n$-stack associated to the constant presheaf $U\mapsto A$ over the site $Site(X)$ of disjoint unions of open subsets of $X$.
More generally if ${{\cal X}}$ is any site and $\underline{A}$ is a presheaf of $n$-categories over ${{\cal X}}$ (or a fibered $n$-category over ${{\cal X}}$) then the $n$-category of global sections of the $n$-stack associated to $\underline{A}$ is the [*nonabelian cohomology*]{} $H({{\cal X}}, \underline{A})$.
We now treat the example mentionned in the footnote of the introduction. Suppose $G$ is a group and $U$ an abelian group, and let $A$ (resp. $B$) be the strict $1$-category with one object and group of automorphisms $G$ (resp. the strict $n$-category with one arrow in each degree $<n$ and group $U$ of automorphisms in degree $n$). Let $A'$ (resp. $B'$) be fibrant replacements for $A$ and $B$. We would like to show that the set $T^n\underline{Hom}(A',B')$ is equal to the group cohomology $H^n(G, U)$. For the moment, the only way I see to do this is to pass by topology using the Seifert-Van Kampen theorem. Let $X=K(G,1)$ and $Y= K(U,n)$. Then $\Pi _n(X)$ is equivalent to $A$ and $\Pi
_n(Y)$ is equivalent to $B$. Similarly in the other direction $\langle R, B'
\rangle $ is equivalent to $\langle R , B \rangle$ which in turn is equivalent to $Y$. By the above discussion $H(X, B')$ is equivalent to $\Pi
_n(\underline{Hom} (X, Y))$. The truncation $T^nH(X,B')$ is thus equal to $\pi
_0(\underline{Hom} (X,Y))$ which (as is well-known) is $H^n(X, U)= H^n(G, U)$. But by definition $H(X,B')=\underline{Hom} (\Pi _n(X), B')$ which is equivalent to $\underline{Hom} (A', B')$. This gives the desired statement.
The above argument is clearly not ideal, since we are looking for a purely algebraic approach to these types of problems. It seems likely that the algebraic techniques of [@Quillen] with appropriate small additional lemmas would permit us to give a purely algebraic proof of the result of the previous paragraph.
[**.Comparison**]{}
As pointed out in the introduction, there are many different theories of weak $n$-categories in the process of becoming reality, and this will pose the problem of comparison. As an initial step we give a construction of functors modeled on Tamsamani’s Poincaré $n$-groupoid construction. We denote our “Poincaré $n$-category” functors by $\Upsilon_n$ to avoid confusion with the $\Pi _n$ (specially on the fact that the $\Upsilon_n$ will not take images in $n$-groupoids).
This section is only a sketch, with many details of proofs missing. In particular the following proposed set of axioms for internal model categories is a preliminary attempt only.
Suppose ${{\cal C}}$ is a closed model category with the following additional properties: (IM1)—${{\cal C}}$ admits an internal $\underline{Hom}$; (IM2)—If $A$ and $B$ are fibrant and cofibrant objects then $\underline{Hom}
(A,B)$ is fibrant; (IM3)—If $A\rightarrow A'$ is a cofibration (resp. trivial cofibration) of fibrant and cofibrant objects, and if $B'\rightarrow B$ is a fibration (resp. trivial fibration) of fibrant and cofibrant objects, then $\underline{Hom}(A', B')\rightarrow \underline{Hom}(A,B)$ is a fibration (resp. a trivial fibration); (IM4)—Internal $\underline{Hom}$ takes cofibrant pushout in the first variable (resp. fibrant fiber product in the second variable) to fiber product.
We call ${{\cal C}}$ an [*internal closed model category*]{}.
Suppose now that ${{\cal C}}$ is an internal closed model category with an inclusion $i:Cat\subset {{\cal C}}$ having the following properties: (a)—$i(\emptyset )$ is the initial object and $i(\ast )$ is the final object of ${{\cal C}}$; (b)—$i$ is compatible with disjoint union; (c)—$i$ takes values in the fibrant and cofibrant objects of ${{\cal C}}$; (d)—$i$ takes the internal $\underline{Hom}$ in $Cat$ to the internal $\underline{Hom}$ of ${{\cal C}}$. (e)—Let $I$ denote the category with two objects $0,1$ and one non-identity morphism from $0$ to $1$. Let $I^{(m)}$ denote the symmetric product of $m$ copies of $I$, which is the category with objects $0,\ldots , m$ and one morphism from $i$ to $j$ when $i\leq j$. Then we require that the morphism from the ${{\cal C}}$-pushout of the diagram $$i(I) \leftarrow \ast \rightarrow i(I) \leftarrow \ast \ldots \ast \rightarrow
i(i)$$ to $i(I^{(m)})$ be a cofibrant weak equivalence in ${{\cal C}}$.
[*Remark:*]{} Our closed model categories $PC_n$ are internal, and (for $n\geq 1$) have functors $i: Cat\hookrightarrow PC_n$ satisfying properties (a)–(e) above.
These seem to be reasonable properties to ask of any closed model category representing a theory of $n$-categories (or $\infty$-categories). However, some of the properties are of a rather technical nature so it is possible that some technically slightly different approach to comparison would be needed—the present section is just a first attempt.
Suppose $({{\cal C}}, i)$ is an internal closed model category with inclusion $i$ having the above properties. Let ${{\cal C}}_f$ denote the subcategory of fibrant objects. Then for any $n$ we define a functor $\Upsilon _n: {{\cal C}}\rightarrow
n-Cat \subset PC_n$, which we call the “Poincaré $n$-category” functor. These functors will have the property that they take weak equivalences in ${{\cal C}}_f$ to equivalences of $n$-categories, and will be compatible with direct products (hence with fiber products over sets).
The definition is by induction. First of all, $\Upsilon _0(X)$ is defined bo be equal to the set of homotopy classes of maps $\ast\rightarrow X$. Then, supposing that we have defined $\Upsilon _{n-1}$ we define for any $X\in {{\cal C}}_f$ the simplicial object $U$ of ${{\cal C}}$ by: $U_0$ is the set (note that sets are categories so $i$ gives an inclusion of sets into ${{\cal C}}$) of morphisms $\ast\rightarrow X$; and for $x_0,\ldots ,
x_m\in U_0$, $U_m(x_0,\ldots , x_m)$ is the fiber of $$\underline{Hom}(I^{(m)}, X)\rightarrow \underline{Hom}(\{ 0, \ldots , m\} , X)$$ over the point $(x_0, \ldots , x_m)$. Then $U_m$ is the disjoint union of the $U_m(x_0,\ldots , x_m)$ over all sequences of $x_i \in U_0$.
Axiom IM4 and condition (e) imply that the usual morphism $$U_m \rightarrow U_1\times _{U_0} \ldots \times _{U_0}U_1$$ is a weak equivalence.
With the above notation set $$\Upsilon _n(X)_{m/}:= \Upsilon _{n-1}(U_m).$$ This simplicial $n-1$-category is an $n$-category since $\Upsilon _{n-1}$ is compatible with direct products and preserves weak equivalences. Note that $\Upsilon _n$ is obviously compatible with direct products.
One has to prove that $\Upsilon _n$ preserves weak equivalences (we leave this out for now).
[*Examples*]{}
Tamsamani’s functor $\Pi _n$ is essentially an example of the functor functor $\Upsilon _n$ for ${{\cal C}}= Top$ and $i: Cat \rightarrow Top$ the realization functor. Our definition of $\Upsilon _n$ is a generalization of the definition of ([@Tamsamani] §3).
For ${{\cal C}}= PC_{n'}$ we obtain the functor $\Upsilon _n$. If $n=n'$ it is essentially the identity; for $n< n'$ is is the truncation $T^{n'-n}$; and for $n>n'$ it is the induction $Ind ^{n'}_n$. Note however that the induction doesn’t preserve pushouts, so $\Upsilon _n$ will not necessarily preserve pushouts in general (where by pushouts here we mean the replacement of pushouts by weak-equivalent objects of ${{\cal C}}_f$).
For any given theory ${{\cal C}}$ of $n$-categories satisfying the above properties, one would like to check that the functor $\Upsilon_n$ is an equivalence of homotopy theories in the sense of [@Quillen] (or at least that it induces an isomorphism of localized categories). If $n$ is correctly chosen to correspond to the level of ${{\cal C}}$ then one would try to show that $\Upsilon _n$ preserves pushouts.
There are examples of ${{\cal C}}$ which are not equivalent to $PC_n$, such as $Top$ or, for example, the category of “Segal categories”, i.e. simplicial spaces whose first object is a set and which satisfy Segal’s condition (cf [@Tamsamani] §3). Even if we look at Segal categories whose elements are $n-1$-truncated, the functor $\Upsilon _n$ will go into the $n$-precats $A$ whose $A_{p/}$ are $n-1$-groupoids, in particular $\Upsilon _n$ will not be essentially surjective.
Similarly, one can imagine looking at a theory ${{\cal C}}$ of $n$-categories with extra structure. For example $Top$ is basically the theory of $n$-categories where the $i$-morphisms have essential inverses. Baez and Dolan propose another type of extra structure of “adjoints” rather than inverses, in relation to topological quantum field theory [@BaezDolan]. It is possible in this case that ${{\cal C}}$ would again be an internal closed model category and that we would have a functor $i$. The resulting functor $\Upsilon_n$ would then be essentially the functor of “forgetting the extra structure” and taking the underlying $n$-category.
A more fundamental example of the above phenomenon will be the closed model category of $n$-stacks. This retracts onto that of $n$-categories: the inclusion being the constant stack functor and the morphism $\Upsilon _n$ being the global section functor. Of course in this situation we don’t expect $\Upsilon _n$ to be an equivalence of theories. This example shows that more is needed than just the above axioms for ${{\cal C}}$ in order to prove that the composition in the other direction is the identity.
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[^1]: The simplest example which shows that the strict morphisms are not enough is where $G$ is a group and $V$ an abelian group and we set $A$ equal to the category with one object and group of automorphisms $G$, and $B$ equal to the strict $n$-category with only one $i$-morphism for $i<n$ and group $V$ of $n$-automorphisms of the unique $n-1$-morphism; then for $n=1$ the equivalence classes of strict morphisms from $A$ to $B$ are the elements of $H^1(G,V)$ so we would expect to get $H^n(G,V)$ in general, but for $n>1$ there are no nontrivial strict morphisms from $A$ to $B$.
[^2]: The definition, from §3 below, is that a morphism $A\rightarrow B$ of $n$-precats is a [*cofibration*]{} if for every $M= (m_1,
\ldots , m_k)$ with $k <n$, the morphism $A_M \rightarrow B_M$ is injective.
[^3]: This is basically the only place in the paper where we really use the fact that we have taken the category $\Delta$ and not some other category such as the semisimplicial category or a truncation of $\Delta$ using only objects $m$ for $m\leq m_0$. One can see for example that the statement \[ce\] for $1$-precats is no longer true if we try to replace $\Delta$ by the semisimplicial category throwing out the degeneracy maps—this example comes down to saying that the product of two free monoids on two sets of generators is not the free monoid on the product of the sets of generators.
|
**[Odderon Effects in pp Collisions:\
Predictions for LHC Energies]{}**
C. Merino$^*$, M. M. Ryzhinskiy$^{**}$, and Yu. M. Shabelski$^{**}$\
$^*$ Departamento de Física de Partículas, Facultade de Física,\
and Instituto Galego de Física de Altas Enerxías (IGFAE),\
Universidade de Santiago de Compostela,\
Santiago de Compostela 15782,\
Galiza, Spain\
E-mail: [email protected]
$^{**}$ Petersburg Nuclear Physics Institute,\
Gatchina, St.Petersburg 188350, Russia\
e-mail: [email protected]\
e-mail: [email protected] 0.9 truecm
*Lecture given by Yu.M. Shabelski at the XLIII PNPI Winter School on Physics*
*St.Petersburg, Russia*
0.3 truecm
*February 2009*
0.9 truecm
A b s t r a c t
We consider the possible contribution of Odderon (Reggeon with $\alpha_{Odd}(0) \sim 1$ and negative signature) exchange to the differences in the total cross sections of particle and antiparticle, to the ratios of real/imaginary parts of the elastic $pp$ amplitude, and to the differences in the inclusive spectra of particle and antiparticle in the central region. The experimental differences in total cross sections of particle and antiparticle are compatible with the existence of the Odderon component but such a large Odderon contribution seems to be inconsistent with the values of Re/Im ratios. In the case of inclusive particle and antiparticle production the current energies and/or accuracy of the experimental data don’t allow a clear conclusion. It is expected that the LHC will finally solve the question of the Odderon existence.
Introduction
============
The Odderon is a singularity in the complex $J$-plane with intercept $\alpha_{Odd} \sim 1$, negative $C$-parity, and negative signature. Thus its zero flavour-number exchange contribution to particle-particle and to antiparticle-particle interactions, e.g., to $pp$ and $\bar{p}p$ total cross sections, has opposite signs. In QCD the Odderon singularity is connected [@BLV] to the colour-singlet exchange of three reggeized gluons in $t$-channel. The theoretical and experimental status of Odderon has been recently discussed in refs. [@Nic; @Ewe]. The possibility to detect Odderon effects has also been investigated in other domains as $\pi p \to \rho N$ reaction [@Cont], charm photoproduction [@BMR], and diffractive production of pion pairs [@Hag; @Ginzb].
The difference in the total cross sections of antiparticles and particles interactions with nucleon targets are numerically small and decrease rather fast with initial energy, so the Odderon coupling should be very small with respect to the Pomeron coupling. However, several experimental facts favouring the presence of the Odderon contribution exist. One of them is the difference in the $d\sigma/dt$ behaviour of elastic $pp$ and $\bar{p}p$ scattering at $\sqrt{s} =$ 52.8 GeV and $\vert t \vert = 1 - 1.5$ GeV$^2$ presented in references [@Nic; @Bre]. The behaviour of $pp$ and $\bar{p}p$ elastic scattering at ISR and SPS energies was analyzed in [@JSS]. Also the differences in the yields of baryons and antibaryons produced in the central (midrapidity) region and in the forward hemisphere in meson-nucleon and in meson-nucleus collisions, and in the midrapidity region of high energy $pp$ interactions [@ACKS; @BS; @AMS; @Olga; @SJ3; @AMS1; @MRS], can also be significant in this respect. The question of whether the Odderon exchange is needed for explaining these experimental facts, or they can be described by the usual exchange of a reggeized quark-antiquark pair with $\alpha_{\omega}(t) = \alpha_{\omega}(0) + \alpha_{\omega}'t$ ($\omega$-Reggeon exchange) is a fundamental one.
The detailed description of all available data on hadron-nucleon elastic scattering with accounting for Regge cuts results in $\alpha_{\omega}(0) = 0.43$, $\alpha_{\omega}' = 1$ GeV$^{-2}$ [@VLLTM], and the simplest power fit $$\Delta \sigma_{hp}\ = \sigma^{tot}_{\bar{h}p} - \sigma^{tot}_{hp} =
\sigma_R\cdot (s/s_0)^{\alpha_R(0) - 1}$$ for experimental points of $\bar{p}p$ and $pp$ scattering starting from $\sqrt{s} = 5$ GeV gives the value $\alpha_R = 0.424 \pm 0.015$ [@ShSh]. The accounting for Regge cut contributions of the type $RP$, $RPP$, $RP...P$, and $Rf$, $RfP$, $RfP...P$ slightly decrease (from $\alpha_R = 0.424 \pm 0.015$ to $\alpha_R = 0.364 \pm 0.015$) [@ShSh] the effective value of $\alpha_R$. Thus any process with exchange of a negative signature object with effective intercept $\alpha_{eff} > 0.7$ could be considered as an Odderon contribution, while if $\alpha_{eff} \leq 0.5$ one could say that there is no room for the Odderon contribution.
In this paper we carry out this anlysis for the case of high energy $hp$ collisions. In Section 2 we study the Regge pole contributions from the data on $\bar{p}p$, $pp$, $\pi^{\pm}p$, and $K^{\pm}p$ total cross sections. In Section 3 we consider the possible Odderon effect on the ratios of real/imaginary parts of the elastic $pp$ amplitude. In Section 4 we compare the experimental ratios of $\bar{h}$ to $h$ inclusive production in the midrapidity (central) region for $pp$ collisions to predictions from double-Reggeon diagrams. In Sections 5, 6, and 7 we compare these experimental data to the theoretical predictions of the Quark–Gluon String Model (QGSM). Our predictions for the Odderon effects at LHC energies are presented in Section 8.
Regge-pole analysis of total $hp$ and $\bar{h}p$ cross sections
===============================================================
Let us start from the analysis of high energy elastic particle and antiparticle scattering on the proton target. Here the simplest contribution is the one Regge-pole $R$ exchange corresponding to the scattering amplitude $$A(s,t) = g_1(t)\cdot g_2(t)\cdot \left(\frac{s}{s_0}\right)^{\alpha_R(t) - 1}\cdot \eta(\Theta) \;,$$ where $g_1(t)$ and $g_2(t)$ are the couplings of a Reggeon to the beam and target hadrons, $\alpha_R(t)$ is the $R$-Reggeon trajectory, and $\eta(\Theta)$ is the signature factor which determines the complex structure of the scattering amplitude ($\Theta$ equal to +1 and to $-1$ for reggeon with positive and negative signature, respectively): $$\eta(\Theta) = \left\{ \begin{array}{ll} i - \tan^{-1}(\frac{\pi \alpha_R}2) & \Theta = +1 \\
i + \tan({\frac{\pi \alpha_R}2}) & \Theta = -1 \;, \end{array} \right.$$ so the amplitude $A(s,t=0)$ becomes purely imaginary for positive signature and purely real for negative signature when $\alpha_R \to 1$.
The contribution of the Reggeon exchange with positive signature is the same for a particles and its antiparticle, but in the case of negative signature the two contributions have opposite signs, as it is shown in Fig. 1.
-.2cm {width=".7\hsize"} -.5cm
The difference in the total cross section of high energy particle and antiparticle scattering on the proton target is $$\Delta \sigma^{tot}_{hp} = \sum_{R(\Theta=-1)} 2\cdot Im\,A(s,t=0) =
\sum_{R(\Theta=-1)} 2\cdot g_1(0)\cdot g_2(0)\cdot \left(\frac{s}{s_0}\right)^{\alpha_R(0) - 1}
\cdot Im\,\eta(\Theta=-1) \;.$$
The experimental data for the differences of $\bar{p}p$ and $pp$ total cross sections are presented in Fig. 2. Here we use the data compiled in ref. [@CERN] by presenting at every energy the experimental points for $pp$ and $\bar{p}p$ by the same experimental group and with the smallest error bars. At ISR energies (last three points in Fig. 2) we present the data in ref. [@Car] as published in their most recent version.
.2cm ![ Experimental differences of $\bar{p}p$ and $pp$ total cross sections at $\sqrt{s} > 8$ GeV (left panel) and at $\sqrt{s} > 13$ GeV (right panel) together with their fit by Eq. (1) (solid curves), fit of [@DL] Eq. (5) (dashed curves) and fit by Eq. (6) (dash-dotted curves).](sigma_ppbar.eps "fig:"){width=".9\hsize"} -.5cm
In the left panel of Fig. 2 our fit to the experimental data with Eq. (1) starting from $\sqrt{s} > 8$ GeV is presented (solid line). For this fit we obtain the value of ${\alpha_R} = 0.43 \pm 0.017$ with $\chi^2 =$ 33.3/15 ndf. This result is in good agreement with [@ShSh], where the experimental points at energies $\sqrt{s} > 5$ GeV were included in the fit, and it only slightly differs from the general fit of all $hp$ total cross sections [@DL] which results in $$\Delta \sigma_{pp} = 42.31 \cdot s^{-0.4525} {\rm (mb)} \;,$$ and it starts from $\sqrt{s} > 10$ GeV. This last fit is also shown in Fig. 2 by a dashed line. The values of the parameters for the two fits, together with the $\chi^2$ values, are presented in Table 1. It is needed to note that the fit in [@DL] was aimed at the total $\bar{p}p$ and $pp$ cross sections and not specifically at their differences, so the not that good values of $\chi^2$ for this fit are not very significant.
As one can see in Table 1, the Eq. (1) fit can only describe the experimental difference in the total $\bar{p}p$ and $pp$ cross sections when starting from high enough energies. When starting at lower energies other Regge poles, as well as other contributions, can contribute, but their contribution becomes negligible at higher energies. Thus the values of the parameters in Eq. (1) can be different in different energy regions. To check the stability of the parameter values, we present in the right panel of Fig. 2 the same experimental data as in the left panel, but at $\sqrt{s} > 13$ GeV. Here we obtain ${\alpha_R} = 0.62 \pm 0.05$ with $\chi^2 =$ 8.3/10 n.d.f., i.e. now the description of the data is better, with the value of ${\alpha_R}$ significantly increasing. This indicates that it is reasonable to account for two contributions to $\Delta \sigma_{pp}$, the first one corresponding to the well-known $\omega$-reggeon and the second one corresponding to a possible Odderon exchange: $$\Delta \sigma_{hp}\ = \sigma_{\omega}\cdot (s/s_0)^{\alpha_{\omega}(0) - 1} +
\sigma_{Odd}\cdot (s/s_0)^{\alpha_{Odd}(0) - 1} \:.$$
Parameterization $\sigma_R (mb)$ $\alpha_R(0)$ 0.5cm $\chi^2/ndf$
------------------------------------------ ----------------- ------------------ --------------------
$p^{\pm}p, \sqrt{s} > 8$ GeV (Eq. (1)) $75.4 \pm 6.1$ $0.43 \pm 0.017$ 35.1/15
$p^{\pm}p, \sqrt{s} > 13$ GeV (Eq. (1)) $25.5 \pm 7.1$ $0.625 \pm 0.05$ 8.8/10
$p^{\pm}p, \sqrt{s} > 8$ GeV (Eq. (5)) 42.31 (fixed) 0.5475 (fixed) 92.3/17
$p^{\pm}p, \sqrt{s} > 13$ GeV (Eq. (5)) 42.31 (fixed) 0.5475 (fixed) 34.5/12
$\pi^{\pm}p, \sqrt{s} > 8$ GeV (Eq. (1)) $9.51 \pm 1.89$ $0.51 \pm 0.04$ 17.2/20
$\pi^{\pm}p, \sqrt{s} > 8$ GeV (Eq. (5)) 8.46 (fixed) 0.5475 (fixed) 26.3/22
$K^{\pm}p, \sqrt{s} > 8$ GeV (Eq. (1)) $28.0 \pm 3.7$ $0.45 \pm 0.03$ 15.4/18
$K^{\pm}p, \sqrt{s} > 8$ GeV (Eq. (5)) 8.46 (fixed) 0.5475 (fixed) 50.1/20
Table 1: [The Regge-pole fits of the differences in $\bar{h}p$ and $hp$ total cross sections by using Eq. (1) and Eq. (5).]{}
The accuracy of the available experimental points is not good enough for the determination of the values of the four parameters in Eq. (6), so by sticking to the idea of existence of the Odderon, we have fixed the value of $\alpha_{Odd}(0)$ close to one (we take $\alpha_{Odd}(0) = 0.9$[^1]), thus obtaining the fit shown by a dash-dotted curve both in the left panel and in the right panel of Fig. 2 with the values of the parameters presented in Table 2.
Energy $\sigma_{\omega} (mb)$ $\alpha_{\omega}(0)$ $\sigma_{Odd} (mb)$ $\alpha_{Odd}(0)$ $\chi^2$/ndf
-------------------------------- ------------------------ ---------------------- --------------------- ------------------- --------------
$p^{\pm}p, \sqrt{s} > 8$ GeV $165 \pm 37$ $0.19 \pm 0.06$ $2.65 \pm 0.45$ 0.9 (fixed) 26.7/14
$p^{\pm}p, \sqrt{s} > 13$ GeV $450 \pm 119$ $-0.09 \pm 0.03$ $3.61 \pm 0.09$ 0.9 (fixed) 5.8/9
$p^{\pm}p, \sqrt{s} > 8$ GeV $172 \pm 7$ $0.15 \pm 0.09$ $5.16 \pm 0.32$ 0.8 (fixed) 27.6/14
$p^{\pm}p, \sqrt{s} > 13$ GeV $450 \pm 164$ $-0.16 \pm 0.05$ $7.38 \pm 0.66$ 0.8 (fixed) 6.1/9
$\pi^{\pm}p, \sqrt{s} > 8$ GeV $20.8 \pm 42.6$ $0.25 \pm 0.60$ $0.52 \pm 0.69$ 0.9 (fixed) 16.7/19
$K^{\pm}p, \sqrt{s} > 8$ GeV $23. \pm 11.2$ $0.52 \pm 0.86$ $-0.55 \pm 1.72$ 0.9 (fixed) 15.3/17
Table 2: [The double Regge-pole fit to the differences in $\bar{h}p$ and $hp$ total cross sections using Eq. (6).]{}
From the results of this fit for $\sqrt{s} > 8$ GeV one can see that an Odderon contribution with $\alpha_{Odd}(0) \sim 0.9$ is in agreement with the experimental data, the values of $\chi^2$/ndf for parametrization by Eq. (6) being smaller that those in the case of Eq. (1). The contributions of Odderon and $\omega$-reggeon to the differences in $\bar{p}p$ and $pp$ total cross sections would be approximately equal at $\sqrt{s} \sim 25$$-$$30$ GeV. The fit with Eq. (6) at $\sqrt{s} > 13$ GeV qualitatively results in the same curve as the fit at $\sqrt{s} > 8$, but now the errors in the values of the parameters are very large.
Such large value of $\alpha_{Odd}(0)$ ($\alpha_{Odd}(0) \sim 0.9$) with a rather large Odderon coupling should necessarily reflect in a large value of the ratio $$\rho = \frac{Re A(s,t=0)}{Im A(s,t=0)}\; ,$$ but this could be in disagreement with the existing experimental data, as it will be discussed in the next section. In any case, this problem fades away when considering smaller values of $\alpha_{Odd}(0)$. For this reason in Table 2 we present our fits for the differences in $\bar{h}p$ and $hp$ total cross sections by using Eq. (6) with a fixed value $\alpha_{Odd}(0) = 0.8$. The new curves are very close to those of the $\alpha_{Odd}(0) = 0.9$ fit, but now the values of $\chi^2$/ndf are slightly increased.
In the left panel of Fig. 3 the experimental data for the differences of $\pi^- p$ and $\pi^+ p$ total cross sections taken from [@CERN] are shown, together with the power fit of Eq. (1) (solid line), the fit in ref. [@DL] (dashed curve), and the double Reggeon fit of Eq. (6) with a value $\alpha_{Odd}(0) = 0.9$ (dash-dotted curve).
![ The experimental differences in $\pi^- p$ and $\pi^+ p$, left panel (and in $K^- p$ and $K^+ p$, right panel) total cross sections, together with their fits by Eq. (1) (solid curves), by ref. [@DL] (dashed curves), and by Eq. (6) (dash-dotted curves).](sigmas_pi.eps "fig:"){width=".9\hsize"} -.3cm
Since the Odderon corresponds to a three-gluon exchange, it can not contribute to the difference in $\pi^- p$ and $\pi^+ p$ total cross sections, what is consistent with our results, the values of $\chi^2$/ndf being practically the same for the solid and dash-dotted curves in Fig. 3, while the value of $\sigma_{Odd}$ is compatible with zero.
Similar results for the differences in $K^- p$ and $K^+ p$ total cross sections are shown in the right panel of Fig. 3. Here, the double Reggeon fit is again compatible with a zero Odderon contribution.
Needless to say, the presented results do not prove the Odderon existence in $pp$ scattering, We can only say that the assumption of the presence of an Odderon contribution is consistent with the experimental data on total $pp$ and $\bar{p}p$ cross sections. In any case, a more detailed analysis is needed, especially concerning the experimental error bars for the differences in $pp$ and $\bar{p}p$ cross sections. Thus, we have considered independent experimental values of the $pp$ and $\bar{p}p$ cross sections, but the experimental error bars in their differences would be decreased if both would be measured with the same experimental equipment.
Odderon contribution to the ratio Re/Im parts of elastic $pp$ amplitude
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As one can see from Eqs. (2) and (3) the Odderon exchange generates a large real part of the elastic $pp$ amplitude which is proportional to $\tan({\frac{\pi \alpha_R}2})$. The singularity at $\alpha_R = 1$ should be compensated by the smallness of the corresponding coupling. In the normalization, where $Im A_{hp} = \sigma^{tot}_{hp}$, one has $$Re\hspace{0.05cm}A_{Odd} = \frac12 (\sigma^{tot}_{\bar{p}p} - \sigma^{tot}_{pp})_{Odd}
\cdot\tan\left({\frac{\pi \alpha_R}2}\right)\; ,$$ and the additional contribution by the Odderon to the total $\rho = Re A_{pp}/Im A_{pp}$ value it would be equal to $$\rho_{Odd} = \frac{Re A_{Odd}}{\sigma^{tot}_{pp}} \;.$$
Results in Table 2 and in Fig. 2 show that the possible Odderon contribution to the difference in the total $pp$ and $\bar{p}p$ cross sections is of the order of the positive signature (mainly Pomeron) contribution at $\sqrt{s} \sim 25$$-$$30$ GeV and of about one half of the positive signature contribution at $\sqrt{s} \sim 10$ GeV. So, in the case of $\alpha_{Odd}(0) = 0.9$ the value of $Re A_{Odd}$ in Eq. (8) can be $Re A_{Odd}\sim 3$-$4$ mb, what would result in an additional $\rho_{Odd} = 0.07$-$0.1$ contribution to the total $Re A_{pp}/Im A_{pp}$ ratio. This additional Odderon contribution would be in disagreement with the experimental data presented in Table 3.
In fact, the experimental data in refs. [@Vor; @Ama] are in good agreement with the theoretical estimations based on the dispersion relations without Odderon contribution [@Grein], so the hypothetical Odderon contribution could be as much of the order of the experimental error bars. The same situation appears at the CERN-SPS energy [@Aug]. On the other hand, the experimental points [@Faj; @Ber] allows some room for the presence of the Odderon contribution. It is necessary to keep in mind that the theoretical predictions also have some “error bars", e.g. the predictions for UA4 energy presented in [@Aug] are between $\rho = 0.12$ [@Col] and $\rho = 0.15$ [@DL1].
Let us note that the level of disagreement of the theoretical estimations on $\rho_{Odd}$ with experimental data decreases when decreasing the value of $\alpha_{Odd}$.
Experiment $\rho(s)$ Theory
------------------------------ -------------------- --------------------------
$\sqrt{s} = 13.7$ GeV [@Vor] $-0.092 \pm 0.014$ $-0.085$ [@Grein]
$\sqrt{s} = 13.7$ GeV [@Faj] $-0.074 \pm 0.018$ $-0.085$ [@Grein]
$\sqrt{s} = 15.3$ GeV [@Faj] $-0.024 \pm 0.014$ $-0.060$ [@Grein]
$\sqrt{s} = 16.8$ GeV [@Vor] $-0.040 \pm 0.014$ $-0.047$ [@Grein]
$\sqrt{s} = 16.8$ GeV [@Faj] $ 0.008 \pm 0.017$ $-0.047$ [@Grein]
$\sqrt{s} = 18.1$ GeV [@Faj] $-0.011 \pm 0.019$ $-0.04$ [@Grein]
$\sqrt{s} = 19.4$ GeV [@Faj] $ 0.019 \pm 0.016$ $-0.033$ [@Grein]
$\sqrt{s} = 21.7$ GeV [@Vor] $-0.041 \pm 0.014$ $-0.02$ [@Grein]
$\sqrt{s} = 23.7$ GeV [@Vor] $-0.028 \pm 0.016$ $-0.007$ [@Grein]
$\sqrt{s} = 30.6$ GeV [@Ama] $ 0.042 \pm 0.011$ 0.03 [@Grein]
$\sqrt{s} = 44.7$ GeV [@Ama] $ 0.062 \pm 0.011$ 0.062 [@Grein]
$\sqrt{s} = 52.9$ GeV [@Ama] $ 0.078 \pm 0.010$ 0.075 [@Grein]
$\sqrt{s} = 62.4$ GeV [@Ama] $ 0.095 \pm 0.011$ 0.084 [@Grein]
$\sqrt{s} = 546$ GeV [@Ber] $ 0.24 \pm 0.04 $ 0.10$-$0.15 [@Ber]
$\sqrt{s} = 541$ GeV [@Aug] $ 0.135 \pm 0.015$ 0.12$-$0.15 [@Col; @DL1]
Table 3: [Experimental data for the ratio Re/Im parts of elastic $pp$ amplitude at high energies together with the corresponding theoretical estimations.]{}
Regge-pole analysis of inclusive particle and antiparticle production in the central region
===========================================================================================
The inclusive cross section of the production of a secondary $h$ in high energy $pp$ collisions in the central region is determined by the Regge-pole diagrams shown in Fig. 4 [@AKM]. The diagram with only Pomeron exchange (Fig. 4a) is the leading one, while the diagrams with one secondary Reggeon $R$ (Figs. 4b and 4c) correspond to corrections which disappear with the increase of the initial energy.
-2.5cm {width=".45\hsize"} -.3cm
The inclusive production cross section of hadron $h$ with transverse momentum $p_T$ corresponding to the diagram shown in Fig. 4b has the following expression: $$F(p_T,s_1,s_2,s) = \frac{1}{\pi^2 s} g^{pp}_{R}\cdot g^{pp}_{P}\cdot g^{hh}_{RP}(p_T)\cdot
\left(\frac{s_1}{s_0}\right)^{\alpha_{R}(0)}\cdot\left(\frac{s_2}{s_0}\right)^{\alpha_{P}(0)} \;,$$ where $$\begin{aligned}
s_1 & = & (p_a + p_h)^2 = m_T\cdot s^{1/2}\cdot e^{-y^*} \\ \nonumber
s_2 & = & (p_b + p_h)^2 = m_T\cdot s^{1/2}\cdot e^{y^*} \;,\end{aligned}$$ with $s_1\cdot s_2 = m^2_T\cdot s$ [@Kar], and the rapidity $y^*$ defined in the center-of-mass frame.
The contribution of diagram in Fig. 4c differs from Eq. (10) in the change of $s_1$ by $s_2$ and viceversa, and in the contribution of the diagram in Fig. 4a is obtained from Eq. (10) by changing the Reggeon $R$ by Pomeron $P$.
Let us consider the $R$-Reggeon in Fig. 4 as the effective sum of all amplitudes with negative signature, so its contribution to the inclusive spectra of secondary protons and antiprotons has the opposite sign. In the midrapidity region, i.e. at $y^*=0$, the ratios ($\langle m_T \rangle \simeq 1$ GeV) of $p$ and $\bar{p}$ yields integrated over $p_T$ can be written as $$\frac{\bar{p}}p = \frac{1 - r_-(s)}{1 + r_-(s)} \;,$$ where $r_-(s)$ is the ratio of the negative signature ($R$) to the positive signature ($P$) contributions: $$r_-(s) = c_1\cdot \left(\frac{s}{s_0}\right)^{(\alpha_{R}(0) - \alpha_{P}(0))/2} \;,$$ and $c_1$ is a normalization constant.
The theoretical fit by Eq. (12) to the experimental data [@Gue; @Agu; @BRA; @PHO; @PHE; @STAR] on the ratios of $\bar{p}$ to $p$ production cross sections at $y^*=0$ is presented in Fig. 5. Here we have used four experimental points from RHIC, obtained by BRAHMS, PHOBOS, PHENIX, and STAR Collaborations, and we present both $\bar{p}/p$ and $1 - \bar{p}/p$ as functions of initial energy. The obtained values of the parameters $c_1$ and $\alpha_{R}(0) - \alpha_{P}(0)$ are presented in Table 4.
{width=".9\hsize"}
Parameterization $ c_1 $ $ \alpha_R(0)-\alpha_P(0)$ $\chi^2$/ndf
---------------------------------- ----------------- ---------------------------- --------------
$\bar{p}/p$ (Eq. (13), Fig. 5) $4.4 \pm 1.1$ $-0.71 \pm 0.07$ 4.3/8
$K^-/K^+$ (Eq. (13), Fig. 6) $2.8 \pm 2.6$ $-0.90 \pm 0.27$ 2.0/8
$\bar{p}/p$ (Eq. (14), Fig. 7) $4.0 \pm 0.7$ $-0.79 \pm 0.04$ 15.0/8
$K^-/K^+$ (Eq. (14), Fig. 7) $2.3 \pm 0.8$ $-0.99 \pm 0.12$ 10.0/7
$\pi^-/\pi^+$ (Eq. (14), Fig. 7) $0.44 \pm 0.12$ $-0.98 \pm 0.11$ 34.5/7
Table 4: [The Regge-pole fit of the experimental ratios of $\bar{h}p$ and $hp$ total cross sections by using Eqs. (12) and (13), and by using Eqs. (12) and (15).]{}
The value of difference of $\alpha_R(0)-\alpha_P(0)$ obtained in the fit seems to be too large for allowing the presence of an Odderon contribution.
The corresponding fit of the experimental data [@Gue; @Agu; @BRA; @PHO; @PHE; @STAR] on the ratios of $K^-$ to $K^+$ production cross sections at $y^*=0$ is presented in Fig. 6, again for $K^-/K^+$ and $1 - K^-/K^+$ as functions of the initial energy. The values of the parameters obtained in the fit are also presented in Table 3. The value of $\alpha_R(0)-\alpha_P(0)$ obtained in the $K^-/K^+$ is compatible with the value obatined in the $\bar{p}/p$ fit.
{width=".9\hsize"} -.3cm
It is needed to note that both fits in Figs. (5) and (6) are in fact normalized to the experimental point in ref. [@Agu], since the error bar of this point is several times smaller than those of the other considered experimental data.
The ratios of $\pi^-$ over $\pi^+$ production cross sections in midrapidity region $y^*=0$ differ from unity only at moderate energies where different processes can contribute. At higher energies, where the applicability of Regge-pole asymptotics seems to be reasonable, these ratios are very close to one, so they can not be used in our analysis.
Though the experimental points for antiparticle/particle yield ratios obtained by different Collaborations at RHIC energy $\sqrt{s} = 200$ GeV are in reasonable agreement with each other (see Figs. 5 and 6), the BRAHMS Collaboration results [@BRA1] are of special interest because they were obtained not only at $y^*=0$, but also at different values of non-zero rapidity $y^*$, and they can then provide some additional information.
Thus we present in the right panels of Figs. 5 and 6 the results of the fit to the same experimental data [@Gue; @Agu] at $\sqrt{s} < 70$ GeV, but only considering BRAHMS Collaboration experimental point at RHIC energy (dashed curves). In the case of the $\bar{p}$ to $p$ ratio the result of this fit is practically the same as with all four RHIC points (solid curve in Fig. 5). However in the case of the $K^-$ to $K^+$ ratio the fit with only the BRAHMS Collaboration experimental point (dashed curve) significantly differs from the solid curve, meaning that the energy dependence of the $K^-$ to $K^+$ experimental ratio is very poorly known.
For the case of inclusive production at some rapidity distance $y^* \neq 0$ from the c.m.s. the quantitiy $r_-(s,y^*)$ in Eq. (12) takes the form: $$r_-(s,y^*) = \frac{c_1}2\cdot \left(\frac{s}{s_0}\right)^{(\alpha_{R}(0) - \alpha_{P}(0))/2}\cdot
\left(e^{y^* (\alpha_{R}(0) - \alpha_{P}(0))} +
e^{-y^* (\alpha_{R}(0) - \alpha_{P}(0))} \right) \;.$$
In Fig. 7 we present the fit to the experimental rapidity distribution ratios $\bar{p}/p$ (left panel), $K^-/K^+$ (right panel), and $\pi^-/\pi^+$ (lower panel) at $\sqrt{s} = 200$ GeV [@BRA] by using Eq. (14). The values of parameters obtained in this fit are in agreement with those in the fits of Figs. 5 and 6 (see Table 3), so we can arguably claim that in the framework of Regge-pole phenomenology one gets a model independent description of the rapidity dependence of the $\bar{p}/p$ and $K^-/K^+$ ratios by using the values of the parameters that were obtained in the description of the energy dependence of these ratios at $y^* = 0$ using Eq. (12).
.4cm ![ Ratios of the inclusive cross sections $\bar{p}$ to $p$ (left panel), $K^-$ to $K^+$ (right panel), and $\pi^-$ to $\pi^+$ (lower panel) in $pp$ collisions at $\sqrt{s} = 200$ GeV [@BRA] as function of the c.m. rapidity, together with their fit by Eq. (14) (solid curves).](ratio_rapidity_all.eps "fig:"){width=".85\hsize"} -.3cm
The values of $\alpha_R(0)-\alpha_P(0)$ for the $K^-/K^+$ and $\pi^-/\pi^+$ ratios are the same and they seem to be larger than the value for the $\bar{p}/p$ ratio. This situation is qualitatively similar to that of the differences in the total cross sections considered in Section 2.
However, the fit $\bar{p}/p$ ratios provide values of $\alpha_R(0)-\alpha_P(0)$ significantly larger than those one could expect if the Odderon contribution was present.
Inclusive spectra of secondary hadrons in the Quark-Gluon String Model
======================================================================
The ratios of inclusive production of different secondaries can also be analyzed in the framework of the Quark-Gluon String Model (QGSM) [@KTM; @KaPi; @Sh], which allows us to make quantitative predictions at different rapidities including the target and beam fragmentation regions. In QGSM high energy hadron-nucleon collisions are considered as taking place via the exchange of one or several Pomerons, all elastic and inelastic processes resulting from cutting through or between Pomerons [@AGK].
Each Pomeron corresponds to a cylindrical diagram (see Fig. 8a), and thus, when cutting one Pomeron, two showers of secondaries are produced as it is shown in Fig. 8b. The inclusive spectrum of a secondary hadron $h$ is then determined by the convolution of the diquark, valence quark, and sea quark distributions $u(x,n)$ in the incident particles with the fragmentation functions $G^h(z)$ of quarks and diquarks into the secondary hadron $h$. These distributions, as well as the fragmentation functions are constructed using the Reggeon counting rules [@Kai]. Both the diquark and the quark distribution functions depend on the number $n$ of cut Pomerons in the considered diagram.
{width=".6\hsize"} -.5cm
For a nucleon target, the inclusive rapidity or Feynman-$x$ ($x_F$) spectrum of a secondary hadron $h$ has the form [@KTM]: $$\frac{dn}{dy}\ = \
\frac{x_E}{\sigma_{inel}}\cdot \frac{d\sigma}{dx_F}\ =\ \sum_{n=1}^\infty
w_n\cdot\phi_n^h (x)\ ,$$ where the functions $\phi_{n}^{h}(x)$ determine the contribution of diagrams with $n$ cut Pomerons and $w_n$ is the relative weight of this diagram. Here we neglect the contribution of diffraction dissociation processes which is very small in the midrapidity region.
For $pp$ collisions $$\begin{aligned}
\phi_{pp}^h(x) &=& f_{qq}^{h}(x_+,n)\cdot f_q^h(x_-,n) +
f_q^h(x_+,n)\cdot f_{qq}^h(x_-,n)
\nonumber\\
&&\hspace*{3.5cm}+\ 2(n-1)f_s^h(x_+,n)\cdot f_s^h(x_-,n)\ ,
\\
x_{\pm} &=& \frac12\left[\sqrt{4m_T^2/s+x^2}\ \pm x\right] ,\end{aligned}$$ where $f_{qq}$, $f_q$, and $f_s$ correspond to the contributions of diquarks, valence quarks, and sea quarks, respectively.
These functions are determined by the convolution of the diquark and quark distributions with the fragmentation functions, e.g. for the quark one can write: $$f_q^h(x_+,n)\ =\ \int\limits_{x_+}^1u_q(x_1,n)\cdot G_q^h(x_+/x_1) dx_1\ .$$ The diquark and quark distributions, which are normalized to unity, as well as the fragmentation functions, are determined by the corresponding Regge intercepts [@Kai].
At very high energies both $x_+$ and $x_-$ are negligibly small in the midrapidity region and all fragmentation functions, which are usually written [@Kai] as $G^h_q(z) = a_h (1-z)^{\beta}$, become constants that are equal for a particle and its antiparticle (this would correspond to the limit $r_-(s) \to 0$ in Eq. (12)): $$G_q^h(x_+/x_1) = a_h \ .$$ This leads, in agreement with [@AKM], to $$\frac{dn}{dy}\ = \ g_h \cdot (s/s_0)^{\alpha_P(0) - 1}
\sim a^2_h \cdot (s/s_0)^{\alpha_P(0) - 1} \,,$$ that corresponds to the only one-Pomeron exchange diagram in Fig. 4a, the only diagram contributing to the inclusive density in the central region (AGK theorem [@AGK]) at asymptotically high energy. The intercept of the supercritical Pomeron $\alpha_P(0) = 1 + \Delta$, $\Delta = 0.139$ [@Sh], is used in the QGSM numerical calculations.
The baryon as a 3q+SJ system
============================
In the string models, baryons are considered as configurations consisting of three connected strings (related to three valence quarks) called string junction (SJ) [@Artru; @IOT; @RV; @Khar]. The colour part of a baryon wave function reads as follows [@Artru; @RV] (see Fig. 9):
-1.5cm {width=".5\hsize"} -.8cm
$$\begin{aligned}
&&B\ =\ \psi_i(x_1)\cdot\psi_j(x_2)\cdot\psi_k(x_3)\cdot J^{ijk}(x_1, x_2, x_3, x)
\,,
\\
&& J^{ijk}(x_1, x_2, x_3, x) =\ \Phi^i_{i'}(x_1,x)\cdot\Phi_{j'}^j(x_2,x)\cdot
\Phi^k_{k'}(x_3,x)\cdot\epsilon^{i'j'k'} \,,
\\
&& \Phi_i^{i'}(x_1,x) = \left[ T\cdot\exp \left(g\cdot\int\limits_{P(x_1,x)}
A_{\mu}(z) dz^{\mu}\right) \right]_i^{i'} \,,\end{aligned}$$
where $x_1, x_2, x_3$, and $x$ are the coordinates of valence quarks and SJ, respectively, and $P(x_1,x)$ represents a path from $x_1$ to $x$ which looks like an open string with ends at $x_1$ and $x$. Such a baryon structure is supported by lattice calculations [@latt].
This picture leads to some general phenomenological predictions. In particular, it opens room for exotic states, such as the multiquark bound states, 4-quark mesons, and pentaquarks [@RV; @DPP1; @RSh]. In the case of inclusive reactions the baryon number transfer to large rapidity distances in hadron-nucleon and hadron-nucleus reactions can be explained [@ACKS; @BS; @AMS; @Olga; @SJ3; @AMS1] by SJ diffusion.
The production of a baryon-antibaryon pair in the central region usually occurs via $SJ$-$\overline{SJ}$ (according to Eqs. (21) and (22) SJ has upper color indices, whereas antiSJ ($\overline{SJ}$) has lower indices) pair production which then combines with sea quarks and sea antiquarks into, respectively, $B\bar{B}$ pair [@RV; @VGW], as it is shown in Fig. (10). In the case of $pp$ collisions the existence of two SJ in the initial state and their diffusion in rapidity space lead to significant differences in the yields of baryons and antibaryons in the midrapidity region even at rather high energies [@ACKS; @AMS].
-3.5cm {width=".6\hsize"} -1.8cm
The quantitative theoretical description of the baryon number transfer via SJ mechanism was suggested in the 90’s and used to predict [@KP1] the $p/\bar{p}$ asymmetry at HERA energies then experimentally observed [@H1]. Also in ref. [@Bopp] was noted that the $p/\bar{p}$ asymmetry measured at HERA can be obtained by simple extrapolation of ISR data. The quantitative description of the baryon number transfer due to SJ diffusion in rapidity space was obtained in [@ACKS] and following papers [@ACKS; @BS; @AMS; @Olga; @SJ3; @AMS1; @MRS].
In the QGSM the differences in the spectra of secondary baryons and antibaryons produced in the central region appear for processes which present SJ diffusion in rapidity space. These differences only vanish rather slowly when the energy increases.
To obtain the net baryon charge, and according to ref. [@ACKS], we consider three different possibilities. The first one is the fragmentation of the diquark giving rise to a leading baryon (Fig. 11a). A second possibility is to produce a leading meson in the first break-up of the string and a baryon in a subsequent break-up [@Kai; @22r] (Fig. 11b). In these two first cases the baryon number transfer is possible only for short distances in rapidity. In the third case, shown in Fig. 11c, both initial valence quarks recombine with sea antiquarks into mesons $M$ while a secondary baryon is formed by the SJ together with three sea quarks.
{width=".55\hsize"} -0.1cm
The fragmentation functions for the secondary baryon $B$ production corresponding to the three processes shown in Fig. 11 can be written as follows (see [@ACKS] for more details): $$\begin{aligned}
G^B_{qq}(z) &=& a_N\cdot v_{qq} \cdot z^{2.5} \;,
\\
G^B_{qs}(z) &=& a_N\cdot v_{qs} \cdot z^2\cdot (1-z) \;,
\\
G^B_{ss}(z) &=& a_N\cdot\varepsilon\cdot v_{ss} \cdot z^{1 - \alpha_{SJ}}\cdot (1-z)^2 \;,\end{aligned}$$ for Figs. 11a, 11b, and 11c, respectively, and where $a_N$ is the normalization parameter, and $v_{qq}$, $v_{qs}$, $v_{ss}$ are the relative probabilities for different baryons production that can be found by simple quark combinatorics [@AnSh; @CS].
The fraction $z$ of the incident baryon energy carried by the secondary baryon decreases from Fig. 11a to Fig. 11c, whereas the mean rapidity gap between the incident and secondary baryon increases. The first two processes can not contribute to the inclusive spectra in the central region, but the third contribution is essential if the value of the intercept of the SJ exchange Regge-trajectory, $\alpha_{SJ}$, is large enough. At this point it is important to stress that since the quantum number content of the $SJ$ exchange matches that of the possible Odderon exchange, if the value of the $SJ$ Regge-trajectory intercept, $\alpha_{SJ}$, would turn out to be large and it would coincide with the value of the Odderon Regge-trajectory, $\alpha_{SJ}\simeq 0.9$, then the $SJ$ could be identified to the Odderon, or to one baryonic Odderon component.
Let’s finally note that the process shown in Fig. 11c can be very naturally realized in the quark combinatorial approach [@AnSh] through the specific probabilities of a valence quark recombination (fusion) with sea quarks and antiquarks, the value of $\alpha_{SJ}$ depending on these specific probabilities.
The contribution of the graph in Fig. 11c has in QGSM a coefficient $\varepsilon$ which determines the small probability for such a baryon number transfer to occur.
At high energies the SJ contribution from Eq. (26) to the inclusive cross section of secondary baryon production at large rapidity distance $\Delta y$ from the incident nucleon can be estimated as $$\frac{1}{\sigma}\cdot\frac{d\sigma^B}{dy} \sim a_B\cdot\varepsilon\cdot
e^{(1 - \alpha_{SJ}) \Delta y} \;,$$ where $a_B = a_N\cdot v^B_{ss}$. The baryon charge transferred to large rapidity distances can be determined by integration of Eq. (27), so being of the order of $$\langle n_B \rangle \sim \frac{a_B\cdot\varepsilon}{(1 - \alpha_{SJ})} \;.$$ It is then clear that the value $\alpha_{SJ} \geq 1$ should be excluded due to the violation of baryon-number conservation at asymptotically high energies.
Comparison of the QGSM predictions to the experimental data
===========================================================
With the value $\alpha_{SJ}=0.5$ used to obtain the first QGSM predictions [@ACKS] different values of $\varepsilon$ in Eq. (26) were needed for the correct description of the experimental data at moderate and high energies. A better solution was found in ref. [@BS], where it was shown that all experimental data can be described with the value $\alpha_{SJ}=0.9$ and only one value of $\varepsilon$. This large value of $\alpha_{SJ}$ allows to describe the preliminary experimental data of H1 Collaboration [@H1] on asymmetry of $p$ and $\bar{p}$ production in $\gamma p$ interactions at HERA with a rather small change in the description of the data at moderate energies. A similar analysis presented in ref. [@Olga] for midrapidity asymmetries of $\bar{\Lambda}/\Lambda$ produced in $pp$, $pA$, $\pi p$, and $ep$ interactions also shows that the value $\alpha_{SJ}=0.9$ is slightly favoured, mainly due to the H1 Collaboration point [@H1a].
Here we compare the results of QGSM predictions with all available experimental data on the $\bar{p}/p$ ratios presented in Fig. 5. To obtain these predictions we use the values of the probabilities $w_n$ in Eq. (15) that are calculated in the frame of Reggeon theory [@KTM], and the values of the normalization constants $a_{\pi}$ (pion production), $a_K$ (kaon production), $a_{\bar{N}}$ ($B\bar{B}$ pair production), and $a_N$ (baryon production due to SJ diffusion) that were determined [@KTM; @KaPi; @Sh] from the experimental data at fixed target energies.
To compare the QGSM results obtained with different values of $\alpha_{SJ}$ in Eq. (26) all curves should be normalized at the same arbitrary point. This can be done in two different ways, either to use the experimental points at the highest energy, or to use the experimental point with the smallest error bar.
In the first case one takes into account that Reggeon theory is an asymptotic theory, and so numerically small (say of the order of 10%) disagreements with the experimental data at fixed target energies are not very important. This first possibility was used in [@BS; @AMS1] to obtain a rather good description of high energy data (HERA, RHIC, and ISR), and a reasonable agreement with the data at fixed target energies. However, the value of $\chi^2$ calculated here from the data at fixed target energies is not that good.
As example we present in Table 5 the calculated yields of different secondaries produced in $pp$ collisions at RHIC energy $\sqrt{s} = 200$ GeV [@AMS1].
----------------------- ------------------- ----------------------- ---------------------------
Particle
$\varepsilon = 0$ $\varepsilon = 0.024$ STAR Collaboration [@abe]
$\pi^+$ 1.27
$\pi^-$ 1.25
$K^+$ 0.13 $0.14 \pm 0.01$
$K^-$ 0.12 $0.14 \pm 0.01$
$p$ 0.0755 0.0861
$\overline{p}$ 0.0707
$\Lambda$ 0.0328 0.0381 $0.0385 \pm 0.0035$
$\overline{\Lambda}$ 0.0304 $0.0351 \pm 0.0032$
$\Xi^-$ 0.00306 0.00359 $0.0026 \pm 0.0009$
$\overline{\Xi}^+$ 0.00298 $0.0029 \pm 0.001$
$\Omega^-$ 0.00020 0.00025 \*
$\overline{\Omega}^+$ 0.00020 \*
----------------------- ------------------- ----------------------- ---------------------------
$^* dn/dy(\Omega^- + \overline{\Omega}^+) = 0.00034 \pm 0.00019$
Table 5: [The QGSM results ($\alpha_{SJ} = 0.9$) for midrapidity yields $dn/dy$ ($\vert y \vert < 0.5$) integrated over $p_T$ for different secondaries at RHIC energies. The results for $\varepsilon = 0.024$ are presented only when different from the case $\varepsilon = 0$.]{}
The second way was used in [@MRS], where it was possible to describe all the data with good $\chi^2$, except for the $\bar{p}p$ and $\bar{\Lambda}\Lambda$ asymmetries at HERA (see discussion below). In this case the experimental value of $\bar{p}/p$ ratio at $\sqrt{s}$ = 27.4 GeV [@Agu] was chosen as normalization. To do so it was necessary to slightly change the fragmentation function of $uu$ and $ud$ diquarks into secondary antiproton, which now has the form: $$G^{\bar{p}}_{uu}(z) = G^{\bar{p}}_{ud}(z) =
a_{\bar{N}}\cdot (1 - z)^{\lambda - \alpha_R + 4(1-\alpha_B)}\cdot (1 + 3z) \;,$$ with a smaller value of $a_{\bar{N}}$, $a_{\bar{N}} = 0.13$, and an additional factor $(1 + 3z)$ with respect to the expression in ref. [@ACKS]. For all other quark distributions and fragmentation functions the same expressions as in ref. [@ACKS] have been taken. The quality of the description of the $\bar{p}$ inclusive spectra with fragmentation function of Eq. (29) is shown to be even better than in previous papers (see Fig. 12). Here, however, the calculated inclusive density of antiprotons at RHIC energy decreases in comparison with the value presented in Table 5.
![ The experimental spectra of $\bar{p}$ produced in $pp$ collisions at 400 GeV/c [@Agu] (left panel) and 175 GeV/c [@Bren] (right panel), together with their QGSM description.](bf2sept.eps "fig:"){width=".48\hsize"} ![ The experimental spectra of $\bar{p}$ produced in $pp$ collisions at 400 GeV/c [@Agu] (left panel) and 175 GeV/c [@Bren] (right panel), together with their QGSM description.](bf3sept.eps "fig:"){width=".48\hsize"} -.3cm
The ratio of $p$ to $\bar{p}$ yields at $y^*=0$ calculated with the QGSM and with the fragmentation function in Eq. (29) is shown in the left panel of Fig. 13. The results with $\alpha_{SJ} = 0.9$ and $\varepsilon = 0.024$, $\alpha_{SJ} = 0.6$ and $\varepsilon = 0.057$, and $\alpha_{SJ} = 0.5$ and $\varepsilon = 0.0757$ are presented by dashed ($\chi^2$/ndf=21.7/10), dotted ($\chi^2$/ndf=12.2/10), and dash-dotted ($\chi^2$/ndf=11.1/10) curves, respectively. Thus the most probable value of $\alpha_{SJ}$ from the point of view of $\chi^2$ analyses is $\alpha_{SJ} = 0.5 \pm 0.1$.
![ The experimental ratios of $\bar{p}$ to $p$ production cross sections in high energies $pp$ collisions at $y^*=0$ [@Gue; @Agu; @BRA; @PHO; @PHE; @STAR] (left panel) and as the functions of rapidity at $\sqrt{s} = 200$ GeV [@BRA] (right panel), together with their fits by Eqs. (12), (13), and (14) (solid curves), and by the QGSM description (dashed, dotted, and dash-dotted curves).](ratio_pp_all.eps){width=".9\hsize"}
The calculated ratios of $\bar{p}$ to $p$ yields as function of rapidity are shown in the right panel of Fig. 12. In accordance with the experimental conditions [@BRA] we use here the value $\langle p_T \rangle$ = 0.9 GeV/c both for secondary $p$ and $\bar{p}$. We also present here the calculations with $\alpha_{SJ} = 0.9$ and $\varepsilon = 0.024$, $\alpha_{SJ} = 0.6$ and $\varepsilon = 0.057$, and $\alpha_{SJ} = 0.5$ and $\varepsilon = 0.0757$ by dashed ($\chi^2$/ndf=72.8/10), dotted ($\chi^2$/ndf=18.6/10), and dash-dotted ($\chi^2$/ndf=17.0/10) curves, the most probable value of $\alpha_{SJ}$ being again $0.5 \pm 0.1$.
Predictions for LHC
===================
We present in Tables 6 and 7 our predictions for antibaryon/baryon ratios in midrapidity region at energies $\sqrt{s} = 900$ GeV and $\sqrt{s} = 14$ TeV by considering three different scenarios for the SJ contribution: not SJ contribution ($\varepsilon = 0$), SJ contribution with $\alpha_{SJ} = 0.5$, and SJ contribution with $\alpha_{SJ} = 0.9$.
The first remarkable feature of our prediction is that practically equal $\bar{B}/B$ ratios are predicted for baryons with different strangeness content, the small differences presented in Tables 6 and 7 being inside the of the accuracy of our calculations.
At $\sqrt{s} = 900$ GeV, probably the lowest energy used for testing at LHC as intermediate energy between RHIC and LHC energies, we expect the values of $\bar{B}/B$ ratios to be about 0.96 in the case of $\alpha_{SJ} = 0.5$ (which corresponds to the well known $\omega$-reggeon exchage at lower energies), and of about 0.90 in the case of $\alpha_{SJ} = 0.9$ (Odderon contribution).
-------------------------------- ------------------- --------------------- ---------------------
Ratio
$\varepsilon = 0$ $\alpha_{SJ} = 0.5$ $\alpha_{SJ} = 0.9$
$\overline{p}/p$ 0.981 0.955 0.892
$\overline{\Lambda}/\Lambda$ 0.976 0.949 0.887
$\overline{\Xi}^+/\Xi^-$ 0.992 0.965 0.909
$\overline{\Omega}^+/\Omega^-$ 1.0 0.965 0.907
-------------------------------- ------------------- --------------------- ---------------------
Table 6: [The QGSM predictions for antibaryon/baryon yields ratios in $pp$ collisions in midrapidity region ($\vert y \vert < 0.5$) at $\sqrt{s} = 900$ GeV. Three different scenarios for the SJ contribution are considered: not SJ contribution ($\varepsilon = 0$), SJ contribution with $\alpha_{SJ} = 0.5$ ($\omega$-reggeon exchange), and SJ contribution with $\alpha_{SJ} = 0.9$ (Odderon exchange).]{}
For $\sqrt{s} = 14$ TeV the $\bar{B}/B$ ratios are predicted to be larger than 0.99 for $\alpha_{SJ} = 0.5$ and of about 0.96 for $\alpha_{SJ} = 0.9$, so an experimental accuracy of the order of $\sim 1$ % in the measurement of these ratios will be needed to discriminate between the two values of $\alpha_{SJ}$.
-------------------------------- ------------------- --------------------- ---------------------
Ratio
$\varepsilon = 0$ $\alpha_{SJ} = 0.5$ $\alpha_{SJ} = 0.9$
$\overline{p}/p$ 0.998 0.994 0.957
$\overline{\Lambda}/\Lambda$ 0.992 0.993 0.957
$\overline{\Xi}^+/\Xi^-$ 0.999 0.995 0.966
$\overline{\Omega}^+/\Omega^-$ 1.0 0.995 0.967
-------------------------------- ------------------- --------------------- ---------------------
Table 7: [The QGSM predictions for antibaryon/baryon yields ratios in $pp$ collisions in midrapidity region ($\vert y \vert < 0.5$) at $\sqrt{s} = 14$ TeV. Three different scenarios for the SJ contribution are considered: not SJ contribution ($\varepsilon = 0$), SJ contribution with $\alpha_{SJ} = 0.5$ ($\omega$-reggeon exchange), and SJ contribution with $\alpha_{SJ} = 0.9$ (Odderon exchange).]{}
Our predictions for the $\bar{B}/B$ ratios at intermediate LHC energies $\sqrt{s} = 5.5$ TeV and $\sqrt{s} = 10$ TeV, calculated with Odderon contribution $\alpha_{SJ} = 0.9$, are presented in Table 8.
Ratio $\sqrt{s} = 5.5$ TeV $\sqrt{s} = 10$ TeV
-------------------------------- ---------------------- ---------------------
$\overline{p}/p$ 0.942 0.953
$\overline{\Lambda}/\Lambda$ 0.941 0.953
$\overline{\Xi}^+/\Xi^-$ 0.954 0.962
$\overline{\Omega}^+/\Omega^-$ 0.954 0.962
Table 8: [The QGSM predictions for antibaryon/baryon yields ratios in $pp$ collisions in midrapidity region ($\vert y \vert < 0.5$) for energies $\sqrt{s} = 5.5$ TeV and $\sqrt{s} = 10$ TeV, obtained by considering one Odderon contribution with $\alpha_{SJ} = 0.9$.]{}
The midrapidity densities of produced secondaries are more model dependent than the antiparticle/particle ratios. We present in Table 9 the QGSM predictions at LHC energies $\sqrt{s} = 5.5$ TeV, $\sqrt{s} = 10$ TeV, and $\sqrt{s} = 14$ TeV, obtained by considering one Odderon contribution with $\alpha_{SJ} = 0.9$ and $\varepsilon = 0.024$. Baryon densities estimated without any Odderon contribution are smaller, and they practically coincide with the antibaryon densities at the same energy.
Particle $\sqrt{s} = 5.5$ TeV $\sqrt{s} = 10$ TeV LHC ($\sqrt{s} = 14$ TeV)
----------------------- ---------------------- --------------------- ---------------------------
$\pi^+$ 2.23 2.43 2.54
$\pi^-$ 2.23 2.43 2.54
$K^+$ 0.22 0.24 0.25
$K^-$ 0.22 0.24 0.25
$p$ 0.161 0.176 0.185
$\overline{p}$ 0.152 0.168 0.177
$\Lambda$ 0.0784 0.0868 0.0914
$\overline{\Lambda}$ 0.0738 0.0827 0.0875
$\Xi^-$ 0.0094 0.0106 0.0113
$\overline{\Xi^+}$ 0.0089 0.0102 0.0109
$\Omega^-$ 0.00076 0.00088 0.00094
$\overline{\Omega^+}$ 0.00073 0.000845 0.000912
Table 9: [The QGSM results for midrapidity yields $dn/dy$ ($\vert y \vert < 0.5$) of different secondaries at LHC energies $\sqrt{s} = 5.5$ TeV, $\sqrt{s} = 10$ TeV, and $\sqrt{s} = 14$ TeV, obtained by considering one Odderon contribution with $\alpha_{SJ} = 0.9$.]{}
Discussion and conclusions
==========================
From the general point of view, our approach, in which the Odderon effects in soft interactions are generated by $t$-channel SJ exchange, coincides with those in Refs. [@BLV; @IOT; @RV; @Khar; @KP1; @KP2]. Really, the two-Pomeron diagram in Fig. 4a leads by definition to equal yields of baryons and antibaryons in the central region, because the Pomeron has vacuum quantum numbers. All the difference in the production of baryons and antibaryons has to come from the diagrams Figs. 4b and/or 4c, where the Reggeon $R$ has negative C-parity and negative signature, so it contributes to the baryon and antibaryon production with opposite signs. In string models, such contribution is identified with SJ diffusion in rapidity space in the inelastic amplitude of particle production, and, due to unitarity, these squared inelastic amplitudes result in the diagrams of Figs. 4b, or 4c. If $\alpha_{SJ} \leq 0.6$ the $R$-reggeon with negative signature should be identified to the $\omega$-Reggeon, while in the case of $\alpha_{SJ} \geq 0.7-0.8$ it should be correspond to the Odderon. From this point of view, the hard Odderon represents the triple-gluon exchange [@BLV; @KP1; @KP2], whereas the soft Odderon consists of a SJ-antiSJ exchange.
Some differences between our the numerical results and those presented in [@BLV; @IOT; @RV; @Khar; @KP1; @KP2] can be connected to the processes (diagrams) which are considered in the calculations by each approach, and to the values of the corresponding model parameters. Also, the values of $\alpha_{SJ}(0)$ extracted from the experimental data depend on the energy region and on the experimental points included into the analysis.
We neglect the possibility of interactions between Pomerons (the so-called enhancement diagrams) in our calculations. Such interactions are already very important in the cases of heavy ion [@CKT] and nucleon-nucleus [@MPS] collisions at RHIC energies, and their contribution should increase with energy. However, we estimate that the inclusive density of secondaries produced in $pp$ collisions at LHC energies is not large enough to affected by these effects.
In the case of heavy ion collisions, the effects of SJ diffusion ($\bar{B}B$ asymmetry and others) could be increased by the contribution to the inclusive spectrum of a secondary baryon of diagrams in which several incident nucleons interact with the target to the inclusive spectrum of a secondary baryon. The detailed study of this contribution will be a requirement of further analysis.
The experimental data on the differences in particle and antiparticle total cross sections with a proton target have been considered in Section 2. As discussed there, a possible Odderon contribution can be present in this case of $\bar{p}p$ and $pp$ total cross sections, while such an Odderon contribution shoud turn out to be zero for meson-proton total cross sections. With the Odderon corresponding to a reggeized three-gluon exchange in $t$-channel this last feature naturally appears.
However one has to note that the possibility of an Odderon exchange contribution in $\bar{p}p$ and $pp$ is supported by the data of $\bar{p}/p$ scattering obtained at ISR energies ($\sqrt{s} \geq 30$ GeV), where experimental data (at so high energies) do not exist for $K^{\pm}p$ and $\pi^{\pm}p$ exist.
On the other hand, the main part of experimental data for the ratios of real/imaginary parts of elastic $pp$ amplitude, including the ISR data, are in agreement, either with the absence of any Odderon contribution, or with the presence of a very small Odderon contribution, if the value of $\alpha_{Odd}$ is close to one. The exceptions to this fact are the FNAL data [@Faj] and the oldest CERN-SPS experimental point [@Ber] that allows some room for the Odderon contribution to be present.
Thus it seems that the ISR data on the differences of particle and antiparticle total interaction cross sections and the data on the ratios of real/imaginary parts of elastic $pp$ amplitude are not completely consistent with each other.
In the case of the inclusive production of particles and antiparticles in central (midrapidity) region in $pp$ collisions we could not see any contribution by the Odderon. All experimental data starting from those at fixed target energies are consistent with a value $\alpha_R(0) \simeq 0.5$, a little larger than the conventional value of $\alpha_{\omega}(0) \simeq 0.4$, but still too small for the Odderon contribution to be there. On top of that, even at RHIC energies the available energy for the possible Odderon exchange is $\sqrt{s}\simeq 15$$-$$20$ GeV, what is perhaps too small, since we did not saw any Odderon contribution at such energies in the case of the differences of particle and antiparticle total interaction cross sections. Actually, the only evidence for the Odderon exchange with $\alpha_{Odd}(0) \simeq 0.9$ in inclusive reactions are two experimental points for $\bar{B}B$ production asymmetry by the H1 Collaboration [@H1; @H1a]. The first point [@H1] (for $\bar{p}/p$ ratio) is until now not published, and the second one [@H1a] (for $\bar{\Lambda}/\Lambda$ ratio) shows a very large error bar, but, on the other hand, only for these two points the kinematics would allow the energy of the Odderon exchange to be large enough, $\sqrt{s} \simeq 100$ GeV.
One has to expect that the LHC data will make the situation more clear. The QGSM predictions for the deviation of $\bar{B}/B$ ratios from unity due to SJ contribution with $\alpha_{SJ}(0)\simeq 0.9$ have been already published [@AMS1] (see also Tables 6, 7, 8, and 9 in section 8), and they allow deviations from unity on the level of $\sim 4$% at $\sqrt{s} = 14$ TeV, while for smaller values of $\alpha_{SJ}(0)$ these ratios should be close to one.
[**Acknowledgements**]{}
We are grateful to A.B. Kaidalov for the idea of providing this analysis and discussions, and to Ya.I. Azimov and M.G. Ryskin for useful discussions and comments. We also thank Y. Foka and P. Christakoglou for sharing with us their insight on the experimental requirements and conditions at LHC.
This paper was supported by Ministerio de Educación y Ciencia of Spain under the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) and project FPA 2005–01963, by Xunta de Galicia and, in part, by grants RFBR-07-02-00023 and RSGSS-1124.2003.2.
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[^1]: The value of $\alpha_{Odd}(0)$ can [*a priori*]{} be larger than one.
|
---
abstract: 'We present the results of a systematic study of the constraints on neutrino neutral-current non-standard interactions (NSI) that can be obtained from the analysis of the COHERENT spectral and temporal data. First, we consider for the first time the general case in which all the ten relevant neutral-current NSI parameters are considered as free. We show that they are very weakly constrained by the COHERENT data because of possible cancellations between the up and down quark contributions. However, the up-down average parameters are relatively well constrained and the strongest constraints are obtained for an optimal up-down linear combination of NSI parameters. We also consider the case in which there are only NSI with either up or down quarks, and we show that the LMA-Dark fit of solar neutrino data is excluded at $5.6\sigma$ and $7.2\sigma$ respectively, for NSI with up and down quark. We finally present the tight constraints that can be obtained on each NSI parameter if it is the dominant one, assuming that the effects of the others are negligible.'
author:
- 'C. Giunti'
date: 1 September 2019
title: 'General COHERENT Constraints on Neutrino Non-Standard Interactions'
---
Introduction {#sec:introduction}
============
Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) has been observed recently for the first time in the COHERENT experiment [@Akimov:2017ade], many years after its prediction [@Freedman:1973yd; @Freedman:1977xn; @Drukier:1983gj]. Several analyses of the COHERENT data provided interesting information on nuclear physics [@Cadeddu:2017etk; @Papoulias:2019lfi; @Huang:2019ene; @Papoulias:2019txv; @Khan:2019mju; @Cadeddu:2019eta], neutrino properties and interactions [@Coloma:2017ncl; @Liao:2017uzy; @Kosmas:2017tsq; @Denton:2018xmq; @AristizabalSierra:2018eqm; @Cadeddu:2018dux; @Dutta:2019eml; @Papoulias:2019txv; @Khan:2019mju; @Cadeddu:2019eta], and weak interactions [@Cadeddu:2018izq; @Huang:2019ene; @Papoulias:2019txv; @Khan:2019mju; @Cadeddu:2019eta]. In particular, several authors constrained the parameters of neutrino non-standard interactions (NSI) [@Coloma:2017ncl; @Liao:2017uzy; @Kosmas:2017tsq; @Denton:2018xmq; @Papoulias:2019txv; @Khan:2019mju]. However, in those studies the constraints have been derived by considering only one or two of the NSI parameters as non-vanishing or by considering only NSI interactions with either up or down quarks [@Coloma:2017ncl]. In this paper we present for the first time the general COHERENT constraints on the relevant NSI parameters obtained with a fit of the COHERENT data in which all the NSI parameters are considered as free.
Our calculations implement the improved quenching factor in Ref. [@Collar:2019ihs] and use both the spectral and temporal information given in the COHERENT data release [@Akimov:2018vzs]. In particular, as already shown in Refs. [@Cadeddu:2018dux; @Cadeddu:2019eta], the combined spectral and temporal information of the COHERENT data allows us a better determination of the different interactions of $\nu_{e}$ and $\nu_{\mu}$ than the spectral data alone, which are used in some analyses, or the total number of event data, which is used in the simplest analyses. This is due to the fact that in the Oak Ridge Spallation Neutron Source muon neutrinos are produced from $\pi^+$ decays at rest ($\pi^+\to \mu^+ + \nu_\mu$) and arrive at the COHERENT detector as a prompt monochromatic signal with energy $ ( m_{\pi}^2 - m_{\mu}^2 ) / 2 m_{\pi} \simeq 29.8 \, \text{MeV} $, within about $1.5 \, \mu\text{s}$ after protons-on-target. On the other hand, muon antineutrinos and electron neutrinos are produced by $\mu^{+}$ decays at rest ($\mu^{+} \to e^{+} + \nu_{e} + \bar\nu_{\mu}$) and arrive at the detector with continuous spectra extending up to $( m_{\mu} - m_{e} ) / 2 \simeq 52.8 \, \text{MeV}$ in a longer time interval of about $10 \, \mu\text{s}$ after protons-on-target.
Since previous studies that obtained the constraints on NSI assuming interactions with either up or down quarks did not consider the complete spectral and temporal information of the COHERENT data and used the old quenching factor in Ref. [@Akimov:2017ade], we present also the updated values of these constraints. This is particularly interesting for testing the LMA-Dark [@Miranda:2004nb] fit of solar neutrino data [@Coloma:2017ncl].
Finally, we also present the constraints on each individual NSI parameter considered as the only non-vanishing one. This is the simplest approach for the analysis of the data and has been adopted by some authors. Although it is a very special case, it is physically possible if for some reason one of the NSI parameters is much larger than the other ones, whose effects are negligible in the analysis of the COHERENT data.
The plan of the paper is as follows. In Section \[sec:nsi\] we review the contribution of NSI to coherent neutrino-nucleus elastic scattering, setting our conventions and notation. In Sections \[sec:general\] we present the general constraints on NSI from the COHERENT data. In Section \[sec:ud\] we present the constraints on NSI from the COHERENT data assuming only interactions with either up or down quarks. In Section \[sec:dominant\] we present the constraints on each individual NSI parameter considered as the only effectively non-vanishing one. At the end, in Section \[sec:conclusions\] we summarize our results.
NSI in CE$\nu$NS {#sec:nsi}
================
We consider neutral-current neutrino non-standard interactions generated by a heavy mediator and described by the effective four-fermion interaction Lagrangian (see the reviews in Refs [@Ohlsson:2012kf; @Miranda:2015dra; @Farzan:2017xzy; @Dev:2019anc]) $$\mathcal{L}_{\text{NSI}}^{\text{NC}}
=
- 2 \sqrt{2} G_{\text{F}}
\sum_{\alpha,\beta=e,\mu,\tau}
\left( \overline{\nu_{\alpha L}} \gamma^{\rho} \nu_{\beta L} \right)
\sum_{f=u,d}
\varepsilon_{\alpha\beta}^{fV}
\left( \overline{f} \gamma_{\rho} f \right)
,
\label{lagrangian}$$ where $G_{\text{F}}$ is the Fermi constant. The parameters $\varepsilon_{\alpha\beta}^{fV}$ describe the size of non-standard interactions relative to standard neutral-current weak interactions. From the hermiticity of the Lagrangian, we have $\varepsilon_{\alpha\beta}^{fV} = \varepsilon_{\beta\alpha}^{fV*}$.
The differential cross section for coherent elastic scattering of a $\nu_{\alpha}$ with energy $E$ and a nucleus $\mathcal{N}$ with $Z$ protons, $N$ neutrons, and mass $M$, is given by (see Ref. [@Barranco:2005yy]) $$\dfrac{d\sigma_{\nu_{\alpha}\text{-}\mathcal{N}}}{d T}
(E,T)
=
\dfrac{G_{\text{F}}^2 M}{\pi}
\left(
1 - \dfrac{M T}{2 E^2}
\right)
Q_{\alpha}^2
,
\label{cs}$$ where $T$ is the nuclear recoil kinetic energy and $$\begin{aligned}
Q_{\alpha}^2
=
\null & \null
\left[
\left( g_{V}^{p} + 2 \varepsilon_{\alpha\alpha}^{uV} + \varepsilon_{\alpha\alpha}^{dV} \right)
Z
F_{Z}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)
+
\left( g_{V}^{n} + \varepsilon_{\alpha\alpha}^{uV} + 2 \varepsilon_{\alpha\alpha}^{dV} \right)
N
F_{N}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)
\right]^2
\nonumber
\\
\null & \null
+
\sum_{\beta\neq\alpha}
\left[
\left( 2 \varepsilon_{\alpha\beta}^{uV} + \varepsilon_{\alpha\beta}^{dV} \right)
Z
F_{Z}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)
+
\left( \varepsilon_{\alpha\beta}^{uV} + 2 \varepsilon_{\alpha\beta}^{dV} \right)
N
F_{N}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)
\right]^2
,
\label{Qalpha2}\end{aligned}$$ with $$g_{V}^{p}
=
\dfrac{1}{2} - 2 \sin^2\!\vartheta_{W}
,
\qquad
g_{V}^{n}
=
- \dfrac{1}{2}
.
\label{gV}$$ Here $\vartheta_{W}$ is the weak mixing angle, given by $\sin^2\!\vartheta_{W} = 0.23857 \pm 0.00005$ at low energies [@Tanabashi:2018oca].
In Eq. (\[Qalpha2\]), $F_{Z}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)$ and $F_{N}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)$ are, respectively, the form factors of the proton and neutron distributions in the nucleus, that depend on the three-momentum transfer $|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}| \simeq \sqrt{2 M T}$. They are given by the Fourier transforms of the nuclear proton and neutron distributions and describe the loss of coherence for $|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}| R_{p} \gtrsim 1$ and $|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}| R_{n} \gtrsim 1$, where $R_{p}$ and $R_{n}$ are the corresponding rms radii. It has been shown in Ref. [@Cadeddu:2017etk] that different parameterizations of the form factors are practically equivalent in the analysis of COHERENT data. Therefore, we consider only the Helm parameterization [@Helm:1956zz] $$F(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)
=
3
\,
\dfrac{j_{1}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}| R_{0})}{|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}| R_{0}}
\,
e^{- |{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2 s^2 / 2}
,
\label{ffHelm}$$ where $
j_{1}(x) = \sin(x) / x^2 - \cos(x) / x
$ is the spherical Bessel function of order one, $s = 0.9 \, \text{fm}$ [@Friedrich:1982esq] is the surface thickness and $R_{0}$ is related to the rms radius $R$ by $ R^2 = 3 R_{0}^2 / 5 + 3 s^2 $. For the rms radii of the proton distributions of $^{133}\text{Cs}$ and $^{127}\text{I}$ we adopt the values determined with high accuracy from muonic atom spectroscopy [@Fricke:1995zz]: $$R_{p}({}^{133}\text{Cs}) = 4.804 \, \text{fm}
,
\qquad
R_{p}({}^{127}\text{I}) = 4.749 \, \text{fm}
.
\label{Rp}$$ On the other hand, there is no separate measurement of the rms radii of the neutron distributions of $^{133}\text{Cs}$ and $^{127}\text{I}$. The average neutron rms radius of CsI has been obtained from the COHERENT data assuming the absence of non-standard effects [@Cadeddu:2017etk; @Papoulias:2019lfi; @Huang:2019ene; @Papoulias:2019txv; @Khan:2019mju; @Cadeddu:2019eta]. Taking into account also atomic parity violation (APV) experimental results [@Cadeddu:2018izq; @Cadeddu:2019eta], the most precise determination of the average neutron rms radius of CsI from experimental data is [@Cadeddu:2019eta] $$R_{n} = 5.04 \pm 0.31 \, \text{fm}
.
\label{Rn}$$ Taking into account the uncertainties, this value is compatible with the predictions of nuclear models (see Table I in Ref. [@Cadeddu:2017etk]). Since there are already ten NSI parameters to be determined by the analysis of the COHERENT data, it is practically advantageous to consider fixed values of the neutron rms radii of $^{133}\text{Cs}$ and $^{127}\text{I}$, instead of considering them free as in Refs. [@Cadeddu:2018dux; @Cadeddu:2019eta], where a smaller number of other parameters have been constrained. Hence, we adopt the values $$R_{n}({}^{133}\text{Cs}) = 5.01 \, \text{fm}
,
\qquad
R_{n}({}^{127}\text{I}) = 4.94 \, \text{fm}
,
\label{RnRMF}$$ obtained with the relativistic mean field (RMF) NL-Z2 [@Bender:1999yt] nuclear model calculation in Ref. [@Cadeddu:2017etk], that are in good agreement with the average value (\[Rn\]). We take into account the form factor uncertainties with a 5% contribution to $\sigma_{\alpha_{\text{c}}}$ in the least-square functions (\[chi-spe\]) and (\[chi-tim\]), following the COHERENT prescription [@Akimov:2017ade].
One can note that the NSI contributions of up and down quarks can cancel in $Q_{\alpha}^2$. A total cancellation happens for $$\left[ 2 + \frac{N}{Z} \, \frac{F_{N}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)}{F_{Z}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)} \right]
\varepsilon_{\alpha\beta}^{uV}
+
\left[ 1 + 2 \, \frac{N}{Z} \, \frac{F_{N}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)}{F_{Z}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)} \right]
\varepsilon_{\alpha\beta}^{dV}
=
0
.
\label{canc1}$$ However, in practice only a partial cancellation is possible, because: 1) the ratio $F_{N}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)/F_{Z}(|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2)$ depends on $|{\ensuremath{\hskip-1pt\vec{\hskip1ptq}}}|^2$; 2) in the scattering on different nuclei, as $^{133}\text{Cs}$ and $^{127}\text{I}$ in the case of the COHERENT experiment, $Z$ and $N$ (and the corresponding form factors) are different. Nevertheless, since the proton and neutron form factors are not very different for a heavy nucleus and in the COHERENT case of scattering on CsI $(N/Z)_{^{133}\text{Cs}} \simeq 1.418$ is not very different of $(N/Z)_{^{127}\text{I}} \simeq 1.396$, the cancellation can be strong. This means that in practice coherent elastic neutrino-nucleus scattering is not sensitive to small values of the NSI couplings to $u$ and $d$ quarks if they are both considered as free parameters. As we will see in Section \[sec:general\], there is only a sensitivity to very large values of the NSI couplings to $u$ and $d$ quarks, for which the residuals of the cancellations are significant.
Considering $(N/Z)_{^{133}\text{Cs}} \simeq (N/Z)_{^{127}\text{I}} \simeq 1.4$ and neglecting the form factors, the cancellation relation (\[canc1\]) becomes $$\varepsilon_{\alpha\beta}^{dV}
\simeq
- \frac{3.4}{3.8} \, \varepsilon_{\alpha\beta}^{uV}
\simeq
- 0.89 \, \varepsilon_{\alpha\beta}^{uV}
.
\label{canc2}$$ We will see in Section \[sec:general\] that the allowed regions of the NSI parameters obtained from COHERENT data have a slope close to that in Eq. (\[canc2\]) and lie close to the corresponding line. This means that there is no indication of a significant NSI signal in the COHERENT data, taking into account the uncertainties.
From Eq. (\[canc2\]) one can see that the cancellation between the $u$ and $d$ couplings occurs when one is almost equal to the opposite of the other. This means that the COHERENT data are practically not sensitive to small values of the difference between $\varepsilon_{\alpha\beta}^{uV}$ and $\varepsilon_{\alpha\beta}^{dV}$, but can probe small values of linear combinations of $\varepsilon_{\alpha\beta}^{uV}$ and $\varepsilon_{\alpha\beta}^{dV}$ that are proportional to a value close to their average. Hence, in Section \[sec:general\] we present also the constraints on the up-down averages $$\overline{\varepsilon}_{\alpha\beta}^{V}
=
\frac{1}{2} \left( \varepsilon_{\alpha\beta}^{uV} + \varepsilon_{\alpha\beta}^{dV} \right)
.
\label{ave}$$
General COHERENT constraints on NSI {#sec:general}
===================================
We performed two analyses of the COHERENT data: one of the spectral data only, and one of the joint spectral and temporal data. In this way we can evidence the improvements obtained by adding the temporal information, that has been previously used only in Refs. [@Cadeddu:2018dux; @Dutta:2019eml; @Cadeddu:2019eta].
For the analysis of the COHERENT spectral data only, we considered the least-squares function $$\chi^2_{\text{S}}
=
\sum_{i=4}^{15}
\left(
\dfrac{
N_{i}^{\text{exp}}
-
\left(1+\alpha_{\text{c}}\right) N_{i}^{\text{th}}
-
\left(1+\beta_{\text{c}}\right) B_{i}
}{ \sigma_{i} }
\right)^2
+
\left( \dfrac{\alpha_{\text{c}}}{\sigma_{\alpha_{\text{c}}}} \right)^2
+
\left( \dfrac{\beta_{\text{c}}}{\sigma_{\beta_{\text{c}}}} \right)^2
+
\left( \dfrac{\eta_{\text{c}}-1}{\sigma_{\eta_{\text{c}}}} \right)^2
.
\label{chi-spe}$$ Here, for each energy bin $i$, $N_{i}^{\text{exp}}$ is the experimental event number, $N_{i}^{\text{th}}$ is the theoretical event number that depends on the NSI parameters through the cross section (\[cs\]), $B_{i}$ is the estimated number of background events, and $\sigma_{i}$ is the statistical uncertainty. We considered only the 12 energy bins from $i=4$ to $i=15$ of the COHERENT spectrum, because they cover the recoil kinetic energy of the new Chicago-3 quenching factor measurement [@Collar:2019ihs], where the value of the quenching factor and its uncertainties are more reliable. In Eq. (\[chi-spe\]), $\alpha_{\text{c}}$, $\beta_{\text{c}}$, and $\eta_{\text{c}}$ are nuisance parameters which quantify, respectively, the systematic uncertainties of the signal rate, of the background rate, and of the quenching factor, with corresponding standard deviations $\sigma_{\alpha_{\text{c}}} = 0.12$, $\sigma_{\beta_{\text{c}}} = 0.25$ [@Akimov:2017ade], and $\sigma_{\eta_{\text{c}}} = 0.05$ [@Collar:2019ihs]. The value of $\sigma_{\alpha_{\text{c}}}$ has been obtained by summing in quadrature a 5% signal acceptance uncertainty, a 5% neutron form factor uncertainty, and a 10% neutron flux uncertainty, estimated by the COHERENT collaboration [@Akimov:2017ade].
For the analysis of the joint COHERENT spectral and temporal data, we considered the least-squares function $$\begin{aligned}
\chi^2_{\text{ST}}
=
\null & \null
2
\sum_{i=4}^{15}
\sum_{j=1}^{12}
\left[
\left( 1 + \alpha_{\text{c}} \right) N_{ij}^{\text{th}}
+
\left( 1 + \beta_{\text{c}} \right) B_{ij}
+
\left( 1 + \gamma_{\text{c}} \right) N_{ij}^{\text{bck}}
-
N_{ij}^{\text{C}}
\vphantom{
\ln\!\left(
\frac{ N_{ij}^{\text{C}} }{
\left( 1 + \alpha_{\text{c}} \right) N_{ij}^{\text{th}}
+
\left( 1 + \beta_{\text{c}} \right) B_{ij}
+
\left( 1 + \gamma_{\text{c}} \right) N_{ij}^{\text{bck}}
}
\right)
}
\right.
\nonumber
\\
\null & \null
\hspace{2cm}
\left.
+
N_{ij}^{\text{C}}
\ln\!\left(
\frac{ N_{ij}^{\text{C}} }{
\left( 1 + \alpha_{\text{c}} \right) N_{ij}^{\text{th}}
+
\left( 1 + \beta_{\text{c}} \right) B_{ij}
+
\left( 1 + \gamma_{\text{c}} \right) N_{ij}^{\text{bck}}
}
\right)
\right]
\nonumber
\\
\null & \null
+
\left( \frac{\alpha_{\text{c}}}{\sigma_{\alpha_{\text{c}}}} \right)^2
+
\left( \frac{\beta_{\text{c}}}{\sigma_{\beta_{\text{c}}}} \right)^2
+
\left( \frac{\gamma_{\text{c}}}{\sigma_{\gamma_{\text{c}}}} \right)^2
+
\left( \dfrac{\eta_{\text{c}}-1}{\sigma_{\eta_{\text{c}}}} \right)^2
,
\label{chi-tim}\end{aligned}$$ that allows us to take into account time-energy bins with few or zero events. In Eq. (\[chi-tim\]), $i$ is the index of the energy bins, $j$ is the index of the time bins, $N_{ij}^{\text{th}}$ are the theoretical predictions that depend on the NSI parameters, $N_{ij}^{\text{C}}$ are the coincidence (C) data, which contain signal and background events, $B_{ij}$ are the estimated neutron-induced backgrounds, and $N_{ij}^{\text{bck}}$ are the estimated backgrounds obtained from the anti-coincidence (AC) data given in the COHERENT data release [@Akimov:2018vzs]. The nuisance parameters $\alpha_{\text{c}}$, $\beta_{\text{c}}$, and $\eta_{\text{c}}$ are the same as in the least-square function in Eq. (\[chi-spe\]), that we used in the analysis of the time-integrated COHERENT data. The additional nuisance parameter $\gamma_{\text{c}}$ and its uncertainty $\sigma_{\gamma_{\text{c}}} = 0.05$ quantify the systematic uncertainty of the background estimated from the AC data.
The general marginalized constraints on the NSI parameters obtained with the analyses of the spectral and the joint spectral and temporal COHERENT data are listed in Table \[tab:general\]. Note that the bounds on the off-diagonal NSI parameters are given only for their absolute values, because a change in sign or phase does not have any effect in the upper bounds, as a consequence of the structure of $Q_{\alpha}^2$ in Eq. (\[Qalpha2\]).
From Table \[tab:general\], one can see that, as explained in Section \[sec:nsi\], the marginalized bounds on the individual NSI parameters $\varepsilon_{\alpha\beta}^{uV}$ and $\varepsilon_{\alpha\beta}^{dV}$ are very weak, because of the possible cancellations of their effects. On the other hand, the up-down averages $\overline{\varepsilon}_{\alpha\beta}^{V}$ in Eq. (\[ave\]) are relatively well constrained, with upper values smaller or close to unity for reasonable values of the confidence level. However, these constraints are larger than may be expected, for example from the analysis in Ref. [@Liao:2017uzy] that found the 90% CL bounds $ -0.16 \leq \overline{\varepsilon}_{ee}^{V} \leq 0.33 $ and $ -0.11 \leq \overline{\varepsilon}_{ee}^{V} \leq 0.26 $. The reason why the constraints on the averages $\overline{\varepsilon}_{\alpha\beta}^{V}$ cannot be so tight is that these averages do not satisfy well the cancellation constraint in Eq. (\[canc2\]), especially when the individual up and down NSI parameters are large and with opposite signs. Therefore, we consider also the optimal up-down linear combinations $$\widetilde{\varepsilon}_{\alpha\beta}^{V}
=
\dfrac{3.4 \, \varepsilon_{\alpha\beta}^{uV} + 3.8 \, \varepsilon_{\alpha\beta}^{dV}}{7.2}
.
\label{tilde}$$ Table \[tab:general\] shows that the constraints for these optimal up-down linear combinations are strong. In the joint COHERENT spectral and temporal data analysis the absolute values of all the optimal up-down linear combinations of NSI parameters are smaller than 0.35 at $3\sigma$. The larger values of the constraints on the averages $\overline{\varepsilon}_{\alpha\beta}^{V}$ are due to the fact that $\varepsilon_{\alpha\beta}^{uV}$ and $\varepsilon_{\alpha\beta}^{dV}$ can be large and opposite yielding a very small value of $\widetilde{\varepsilon}_{\alpha\beta}^{V}$ that corresponds to a much larger value of $\overline{\varepsilon}_{\alpha\beta}^{V}$. For example, if we consider $\varepsilon_{ee}^{uV}=10$, that is allowed within $1\sigma$ by the limits in Table \[tab:general\], we have $\varepsilon_{ee}^{uV}=0.2$, that is allowed within $1\sigma$, for $\varepsilon_{ee}^{dV}=-8.6$, that is also allowed within $1\sigma$. In this case $\overline{\varepsilon}_{ee}^{V}=0.7$ and this value must be allowed within $1\sigma$, in agreement with Table \[tab:general\] and contrary to the limits in Ref. [@Liao:2017uzy].
It is clear from Table \[tab:general\] that the analysis of the joint COHERENT spectral and temporal data is more powerful in constraining the NSI parameters than the analysis of the spectral data only. Therefore, in the rest of this Section we discuss only the results of the joint spectral and temporal data analysis.
Figures \[fig:eeu-eed\]–\[fig:mtu-mtd\] show the allowed regions in the planes $(\varepsilon_{\alpha\beta}^{uV},\varepsilon_{\alpha\beta}^{dV})$ with $\alpha=e,\mu$ and $\beta=e,\mu,\tau$. For the off-diagonal NSI parameters, the allowed regions are marginalized in the real planes. From these figures, one can see that all the allowed regions are approximately parallel to the line in Eq. (\[canc2\]), in agreement with the expectation. The enlargements in the right panels show that the preferred values of the NSI parameters are close to the approximate cancellation line (\[canc2\]), indicating that there is no indication of NSI effects within the uncertainties.
Near the origin the boundaries of the $1\sigma$, $2\sigma$, and $3\sigma$ allowed regions can be parameterized with the lines $$\varepsilon_{\alpha\beta}^{dV}
=
a \, \varepsilon_{\alpha\beta}^{uV} + b
,
\label{boundaries}$$ with the values of the parameters $a$ and $b$ given in Table \[tab:ab\]. One can see that the slopes $a$ are all very close to the one in Eq. (\[canc2\]), as expected.
Figure \[fig:ave\] shows the marginal allowed regions in different planes of the up-down average NSI parameters $\overline{\varepsilon}_{\alpha\beta}^{V}$ in Eq. (\[ave\]), that are relatively well constrained by the COHERENT data. However, as shown in Table \[tab:general\] and Figure \[fig:tilde\], the strongest constraints are obtained for the optimal up-down linear combinations $\widetilde{\varepsilon}_{\alpha\beta}^{V}$ in Eq. (\[tilde\]).
COHERENT constraints on NSI with either up or down quarks {#sec:ud}
=========================================================
In this Section we consider the possibility that neutrino NSI with nuclei are dominated by either up or down quarks, with the subdominant quark NSI having negligible effects. The results of the analysis assuming interactions with up quark only are given in Table \[tab:up\], and those obtained assuming interactions with down quark only are given in Table \[tab:down\]. From these tables one can see that with these assumptions the NSI parameters are well determined to be smaller than one at more than $3\sigma$ in both the spectral and the joint spectral and temporal analyses. Since the joint spectral and temporal analysis is more restrictive, we present only the corresponding correlated allowed regions in different planes of the NSI parameters in Figure \[fig:5u\] for interactions with up quarks only and in Figure \[fig:5d\] for interactions with down quarks only.
Figures \[fig:5u\] and \[fig:5d\] show the comparison of the COHERENT allowed regions in the $(\varepsilon_{ee}^{uV},\varepsilon_{\mu\mu}^{uV})$ and $(\varepsilon_{ee}^{dV},\varepsilon_{\mu\mu}^{dV})$ planes, respectively, with the solar LMA-Dark [@Miranda:2004nb] allowed regions reported in Ref. [@Coloma:2017ncl]. One can see that the allowed regions are incompatible at more than $3\sigma$, disfavoring the LMA-Dark fit of solar neutrino data more than in Ref. [@Coloma:2017ncl], where the analysis of the COHERENT data was performed considering only the total number of events.
Combining the $\chi^2$’s of the analyses of COHERENT spectral data with the marginal $\Delta\chi^2$’s of the LMA-Dark and LMA fits of solar neutrino data in Fig. 1 of Ref. [@Coloma:2017ncl], we found a $\chi^2_{\text{min}}$ difference between LMA-Dark and LMA of $24.5$ and $46.5$, respectively, for NSI with up and down quark. Therefore, LMA-Dark is excluded at $4.9\sigma$ and $6.8\sigma$, respectively, for NSI with up and down quark, for one degree of freedom. These exclusions are already much stronger than the $3.1\sigma$ and $3.6\sigma$ obtained in Ref. [@Coloma:2017ncl]. We further improved the comparison between LMA-Dark and LMA by considering the COHERENT spectral and temporal data, that lead to the exclusion of LMA-Dark at $5.6\sigma$ ($\Delta\chi^2_{\text{min}} = 31.3$) and $7.2\sigma$ ($\Delta\chi^2_{\text{min}} = 52.6$), respectively, for NSI with up and down quark.
One can note that our allowed region from COHERENT data in the $(\varepsilon_{ee}^{uV},\varepsilon_{\mu\mu}^{uV})$ plane has a different shape than that in Figure 2 of Ref. [@Coloma:2017ncl], that has a hole around about $(0.2,0.2)$. The only explanation that we found of this difference is that in Ref. [@Coloma:2017ncl], in spite of the statements in the text, the allowed region was not calculated marginalizing over the remaining off-diagonal NSI parameters of the interaction with up quarks. Indeed, a hole in the allowed region of $(\varepsilon_{ee}^{uV},\varepsilon_{\mu\mu}^{uV})$ appears around about $(0.2,0.2)$ if only the first line in Eq. (\[Qalpha2\]) is considered and corresponds to its suppression. Neglecting $g_{V}^{p} \simeq 0.023$ and the form factors, for $\varepsilon_{\alpha\alpha}^{dV}=0$ the first line in Eq. (\[Qalpha2\]) vanishes for $$\varepsilon_{\alpha\alpha}^{uV}
\simeq
\dfrac{N}{2 \left( 2 Z + N \right)}
\simeq
0.21
\qquad
(\alpha=e,\mu)
,
\label{uhole}$$ considering the average CsI values $Z=54$ and $N=76$. If one does not consider the effects of the off-diagonal NSI parameters of the interaction with up quarks in the second line of Eq. (\[Qalpha2\]) the cross section is suppressed around $(\varepsilon_{ee}^{uV},\varepsilon_{\mu\mu}^{uV}) \simeq (0.2,0.2)$ generating a hole in the allowed region. However, appropriate values of the off-diagonal NSI parameters of the interaction with up quarks can compensate the suppression of the first line in Eq. (\[Qalpha2\]), filling the hole. That is why our allowed region in Figure \[fig:5u\] has no hole.
As a further check, we present in the left panel of Figure \[fig:spe2\] the allowed region in the $(\varepsilon_{ee}^{uV},\varepsilon_{\mu\mu}^{uV})$ plane that we obtained assuming that only these two NSI parameters are non-vanishing and fitting the COHERENT spectral data alone. One can see that there is a hole around $(0.2,0.2)$ and the shape is similar to that in Ref. [@Coloma:2017ncl]. The left panel in Figure \[fig:tim2\] shows that the allowed region reduces significantly in the analysis of the joint COHERENT spectral and temporal data, increasing the tension with the LMA-Dark fit of solar neutrino data.
The allowed region in the $(\varepsilon_{ee}^{uV},\varepsilon_{\mu\mu}^{uV})$ plane assuming only these two non-vanishing NSI parameters was obtained also in Ref. [@Khan:2019mju] by fitting the COHERENT spectral data. Our allowed region in the left panel of Figure \[fig:spe2\] is more stringent than that in the right panel of Fig. 7 of Ref. [@Khan:2019mju], probably because in Ref. [@Khan:2019mju] the quenching factor was approximated with a constant, whereas we implemented the $T$-dependent quenching factor function in Ref. [@Collar:2019ihs].
For completeness, we present in the right panels of Figure \[fig:spe2\] and \[fig:tim2\] also the allowed regions in the $(\varepsilon_{ee}^{dV},\varepsilon_{\mu\mu}^{dV})$ plane that we obtained assuming that only these two NSI parameters are non-vanishing. In this case the hole corresponding to the suppression of the first line in Eq. (\[Qalpha2\]) occurs for $$\varepsilon_{\alpha\alpha}^{dV}
\simeq
\dfrac{N}{2 \left( Z + 2 N \right)}
\simeq
0.18
\qquad
(\alpha=e,\mu)
,
\label{dhole}$$ considering the average CsI values $Z=54$ and $N=76$. The figures show also the strong tension with the LMA-Dark fit of solar neutrino data.
COHERENT constraints on dominant individual NSI parameters {#sec:dominant}
==========================================================
In this Section we present the results of the analyses of COHERENT data assuming that only one of the NSI parameters is dominant and the others have negligible effects. The allowed intervals for each parameter are listed in Table \[tab:1\]. Figure \[fig:1\] shows the $\Delta\chi^2 = \chi^2 - \chi^2_{\text{min}}$ for each parameter. One can see that for the diagonal NSI parameters and some confidence levels there are disconnected allowed intervals, because the $\Delta\chi^2$ is not parabolic, but has a local central maximum. This occurs for the same reason of the hole in the two-dimensional plots in Figures \[fig:spe2\] and \[fig:tim2\], because there is a cancellation in the first line of Eq. (\[Qalpha2\]) that suppresses the cross section and gives a bad fit of the data. Indeed, the local central maximum occurs at a value of about $0.2$, in agreement with Eqs. (\[uhole\]) and (\[dhole\]).
Table \[tab:1\] and Figure \[fig:1\] show that the individual NSI parameters are better determined with the analysis of the joint COHERENT spectral and temporal data than with the analysis of the spectral data alone. The resulting bounds are more stringent than those obtained recently in Refs. [@Papoulias:2019txv; @Khan:2019mju].
In particular, the diagonal $\nu_{\mu}$ NSI parameters $\varepsilon_{\mu\mu}^{uV}$ and $\varepsilon_{\mu\mu}^{dV}$ are well constrained in two disconnected intervals at $3\sigma$ with the joint spectral and temporal analysis. In general, the constraints on the $\nu_{\mu}$ NSI parameters are more stringent than those on the $\nu_{e}$ NSI parameters because there are two $\nu_{\mu}$ fluxes, one from $\pi^{+}$ decay and one, of $\bar\nu_{\mu}$, from $\mu^{+}$ decay, whereas there is only one flux of $\nu_{e}$ from $\mu^{+}$ decay. One can also note that $\varepsilon_{e\mu}^{uV}$ and $\varepsilon_{e\mu}^{dV}$ are more constrained than the other off-diagonal NSI parameters, because they contribute to all the interactions of $\nu_{\mu}$, $\bar\nu_{\mu}$, and $\nu_{e}$. The less constrained off-diagonal NSI parameters are $\varepsilon_{e\tau}^{uV}$ and $\varepsilon_{e\tau}^{dV}$, that contribute only to the interactions of the $\nu_{e}$ flux.
Conclusions {#sec:conclusions}
===========
In this work we performed a systematic study of the constraints on neutrino neutral-current non-standard interactions that can be obtained from the analysis of the COHERENT spectral and temporal data. We have shown that the joint analysis of the COHERENT spectral and temporal data gives more information on the NSI parameters than the analysis of the COHERENT spectral data alone. This is a general feature for quantities that depend on the neutrino flavor, as already emphasized in Refs. [@Cadeddu:2018dux; @Cadeddu:2019eta].
First, we considered for the first time the general case in which all the ten neutral-current NSI parameters are considered as free. We have shown that in this case the analysis of the COHERENT data give very weak constraints the NSI parameters, because the contributions of the NSI parameters with up and down quarks can almost entirely cancel each other. The up-down average parameters in Eq. (\[ave\]) are relatively well constrained, but the strongest constraints are obtained for the optimal up-down linear combination of NSI parameters in Eq. (\[tilde\]).
We also considered the case in which there are only NSI with either up or down quarks, that was considered in Ref. [@Coloma:2017ncl] in order to test the LMA-Dark fit of solar neutrino data. In this case, we obtained very stringent constraints on the NSI parameters, that exclude LMA-Dark at $5.6\sigma$ and $7.2\sigma$ respectively, for NSI with up and down quark. These exclusions are much stronger than the $3.1\sigma$ and $3.6\sigma$ obtained in Ref. [@Coloma:2017ncl].
We finally considered also the case of only one dominant NSI parameter, assuming that the effects of the others is negligible. This is the simplest analysis that has been performed recently also by other authors [@Papoulias:2019txv; @Khan:2019mju]. We obtained more stringent constraints on each individual NSI parameter through the joint analysis of the COHERENT spectral and temporal data.
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10.1103/PhysRevC.60.034304),
------------------------------------------- ---------- --------------------- --------------------- --------------------- ---------- --------------------- --------------------- ---------------------
Best Fit $1\sigma$ $2\sigma$ $3\sigma$ Best Fit $1\sigma$ $2\sigma$ $3\sigma$
$\varepsilon_{ee}^{uV}$ 0 $ -29 \div 29 $ $ -38 \div 39 $ $ -47 \div 47 $ 0 $ -15 \div 15 $ $ -27 \div 26 $ $ -35 \div 35 $
$\varepsilon_{ee}^{dV}$ 0 $ -26 \div 26 $ $ -35 \div 35 $ $ -42 \div 43 $ 0 $ -13 \div 14 $ $ -24 \div 24 $ $ -31 \div 32 $
$\varepsilon_{\mu\mu}^{uV}$ 0 $ -25 \div 25 $ $ -32 \div 32 $ $ -39 \div 40 $ 0 $ -11 \div 11 $ $ -19 \div 19 $ $ -26 \div 25 $
$\varepsilon_{\mu\mu}^{dV}$ 0 $ -23 \div 23 $ $ -29 \div 30 $ $ -36 \div 36 $ 0 $ -10 \div 10 $ $ -17 \div 18 $ $ -22 \div 24 $
$|\varepsilon_{e\mu}^{uV}|$ 0 $<19$ $<25$ $<30$ 0 $<9$ $<16$ $<21$
$|\varepsilon_{e\mu}^{dV}|$ 0 $<17$ $<22$ $<27$ 0 $<8$ $<14$ $<19$
$|\varepsilon_{e\tau}^{uV}|$ 0 $<29$ $<38$ $<47$ 0 $<15$ $<26$ $<35$
$|\varepsilon_{e\tau}^{dV}|$ 0 $<26$ $<35$ $<43$ 0 $<14$ $<24$ $<31$
$|\varepsilon_{\mu\tau}^{uV}|$ 0 $<25$ $<32$ $<39$ 0 $<11$ $<19$ $<25$
$|\varepsilon_{\mu\tau}^{dV}|$ 0 $<23$ $<29$ $<36$ 0 $<10$ $<17$ $<23$
$\overline{\varepsilon}_{ee}^{V}$ $0.1$ $ -1.2 \div 1.4 $ $ -1.7 \div 1.9 $ $ -2.1 \div 2.2 $ $0.1$ $ -0.7 \div 0.8 $ $ -1.1 \div 1.3 $ $ -1.6 \div 1.7 $
$\overline{\varepsilon}_{\mu\mu}^{V}$ $0$ $ -1.0 \div 1.2 $ $ -1.3 \div 1.5 $ $ -1.7 \div 1.9 $ $0.1$ $ -0.4 \div 0.6 $ $ -0.8 \div 1.0 $ $ -1.1 \div 1.3 $
$|\overline{\varepsilon}_{e\mu}^{V}|$ $0.1$ $ < 0.9 $ $ < 1.1 $ $ < 1.4 $ $0.1$ $ < 0.4 $ $ < 0.7 $ $ < 1.0 $
$|\overline{\varepsilon}_{e\tau}^{V}|$ $0.2$ $ < 1.3 $ $ < 1.8 $ $ < 2.2 $ $0.1$ $ < 0.8 $ $ < 1.2 $ $ < 1.6 $
$|\overline{\varepsilon}_{\mu\tau}^{V}|$ $0.5$ $ < 1.1 $ $ < 1.5 $ $ < 1.8 $ $0.1$ $ < 0.6 $ $ < 0.9 $ $ < 1.2 $
$\widetilde{\varepsilon}_{ee}^{V}$ $0.09$ $ -0.17 \div 0.35 $ $ -0.23 \div 0.42 $ $ -0.29 \div 0.48 $ $0.09$ $ -0.01 \div 0.20 $ $ -0.10 \div 0.29 $ $ -0.17 \div 0.35 $
$\widetilde{\varepsilon}_{\mu\mu}^{V}$ $0.09$ $ -0.15 \div 0.34 $ $ -0.20 \div 0.38 $ $ -0.24 \div 0.43 $ $0.10$ $ -0.04 \div 0.23 $ $ -0.08 \div 0.27 $ $ -0.11 \div 0.30 $
$|\widetilde{\varepsilon}_{e\mu}^{V}|$ $0.04$ $ < 0.18 $ $ < 0.22 $ $ < 0.25 $ $0.01$ $ < 0.10 $ $ < 0.14 $ $ < 0.17 $
$|\widetilde{\varepsilon}_{e\tau}^{V}|$ $0.05$ $ < 0.27 $ $ < 0.33 $ $ < 0.38 $ $0.02$ $ < 0.11 $ $ < 0.19 $ $ < 0.26 $
$|\widetilde{\varepsilon}_{\mu\tau}^{V}|$ $0.13$ $ < 0.24 $ $ < 0.30 $ $ < 0.34 $ $0.06$ $ < 0.13 $ $ < 0.17 $ $ < 0.21 $
------------------------------------------- ---------- --------------------- --------------------- --------------------- ---------- --------------------- --------------------- ---------------------
----------------------------------------------------------- ---- --------- --------- --------- --------- --------- ---------
$a$ $b$ $a$ $b$ $a$ $b$
$(\varepsilon_{ee}^{uV},\varepsilon_{ee}^{dV})$ LB $-0.90$ $-0.02$ $-0.90$ $-0.10$ $-0.91$ $-0.15$
UB $-0.90$ $0.36$ $-0.90$ $0.43$ $-0.90$ $0.50$
$(\varepsilon_{\mu\mu}^{uV},\varepsilon_{\mu\mu}^{dV})$ LB $-0.90$ $-0.05$ $-0.91$ $-0.08$ $-0.91$ $-0.11$
UB $-0.92$ $0.41$ $-0.91$ $0.43$ $-0.92$ $0.47$
$(\varepsilon_{e\mu}^{uV},\varepsilon_{e\mu}^{dV})$ LB $-0.92$ $-0.17$ $-0.91$ $-0.20$ $-0.91$ $-0.24$
UB $-0.92$ $0.18$ $-0.91$ $0.21$ $-0.91$ $0.23$
$(\varepsilon_{e\tau}^{uV},\varepsilon_{e\tau}^{dV})$ LB $-0.91$ $-0.19$ $-0.91$ $-0.27$ $-0.91$ $-0.32$
UB $-0.92$ $0.19$ $-0.91$ $0.27$ $-0.90$ $0.32$
$(\varepsilon_{\mu\tau}^{uV},\varepsilon_{\mu\tau}^{dV})$ LB $-0.90$ $-0.22$ $-0.90$ $-0.26$ $-0.91$ $-0.29$
UB $-0.91$ $0.23$ $-0.91$ $0.26$ $-0.92$ $0.30$
----------------------------------------------------------- ---- --------- --------- --------- --------- --------- ---------
-------------------------------- ---------- --------------------- --------------------- --------------------- ---------- --------------------- --------------------- ---------------------
Best Fit $1\sigma$ $2\sigma$ $3\sigma$ Best Fit $1\sigma$ $2\sigma$ $3\sigma$
$\varepsilon_{ee}^{uV}$ $0.02$ $ -0.18 \div 0.56 $ $ -0.23 \div 0.62 $ $ -0.28 \div 0.66 $ $0.20$ $ 0.01 \div 0.38 $ $ -0.08 \div 0.47 $ $ -0.15 \div 0.54 $
$\varepsilon_{\mu\mu}^{uV}$ $0.18$ $ -0.08 \div 0.44 $ $ -0.13 \div 0.51 $ $ -0.17 \div 0.56 $ $0.20$ $ -0.06 \div 0.45 $ $ -0.10 \div 0.48 $ $ -0.13 \div 0.52 $
$|\varepsilon_{e\mu}^{uV}|$ $0.04$ $ < 0.22 $ $ < 0.26 $ $ < 0.29 $ $0.04$ $ < 0.19 $ $ < 0.23 $ $ < 0.26 $
$|\varepsilon_{e\tau}^{uV}|$ $0.16$ $ < 0.37 $ $ < 0.42 $ $ < 0.47 $ $0.04$ $ < 0.19 $ $ < 0.28 $ $ < 0.35 $
$|\varepsilon_{\mu\tau}^{uV}|$ $0.04$ $ < 0.26 $ $ < 0.32 $ $ < 0.37 $ $0.12$ $ < 0.26 $ $ < 0.29 $ $ < 0.32 $
-------------------------------- ---------- --------------------- --------------------- --------------------- ---------- --------------------- --------------------- ---------------------
-------------------------------- ---------- --------------------- --------------------- --------------------- ---------- --------------------- --------------------- ---------------------
Best Fit $1\sigma$ $2\sigma$ $3\sigma$ Best Fit $1\sigma$ $2\sigma$ $3\sigma$
$\varepsilon_{ee}^{dV}$ $0.17$ $ -0.16 \div 0.52 $ $ -0.20 \div 0.56 $ $ -0.25 \div 0.60 $ $0.18$ $ 0.01 \div 0.34 $ $ -0.07 \div 0.42 $ $ -0.14 \div 0.49 $
$\varepsilon_{\mu\mu}^{dV}$ $0.17$ $ -0.07 \div 0.41 $ $ -0.12 \div 0.47 $ $ -0.16 \div 0.51 $ $0.18$ $ -0.06 \div 0.41 $ $ -0.08 \div 0.44 $ $ -0.12 \div 0.47 $
$|\varepsilon_{e\mu}^{dV}|$ $0.04$ $ < 0.20 $ $ < 0.23 $ $ < 0.26 $ $0.04$ $ < 0.17 $ $ < 0.21 $ $ < 0.24 $
$|\varepsilon_{e\tau}^{dV}|$ $0.16$ $ < 0.34 $ $ < 0.38 $ $ < 0.43 $ $0.04$ $ < 0.17 $ $ < 0.25 $ $ < 0.31 $
$|\varepsilon_{\mu\tau}^{dV}|$ $0.04$ $ < 0.24 $ $ < 0.30 $ $ < 0.33 $ $0.11$ $ < 0.23 $ $ < 0.26 $ $ < 0.29 $
-------------------------------- ---------- --------------------- --------------------- --------------------- ---------- --------------------- --------------------- ---------------------
-------------------------------- ------------------ ------------------ ----------------- ------------------ ------------------ ------------------
$1\sigma$ $2\sigma$ $3\sigma$ $1\sigma$ $2\sigma$ $3\sigma$
$\varepsilon_{ee}^{uV}$ $ $ $-0.21\div0.60$ $ $-0.08\div0.47$ $-0.15\div0.53$
\left( \left( \left(
\begin{array}{c} \begin{array}{c} \begin{array}{c}
-0.09\div0.03 -0.15\div0.17 -0.02\div0.18
\\ \\ \\
0.36\div0.48 0.23\div0.54 0.21\div0.41
\end{array} \end{array} \end{array}
\right) \right) \right)
$ $ $
$\varepsilon_{\mu\mu}^{uV}$ $ $ $-0.15\div0.53$ $ $ $
\left( \left( \left( \left( \left(
\begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c}
-0.06\div0.03 -0.10\div0.08 -0.03\div0.03 -0.07\div0.06 -0.11\div0.10
\\ \\ \\ \\ \\
0.37\div0.44 0.31\div0.49 0.37\div0.42 0.33\div0.46 0.29\div0.49
\end{array} \end{array} \end{array} \end{array} \end{array}
\right) \right) \right) \right) \right)
$ $ $ $ $
$|\varepsilon_{e\mu}^{uV}|$ $<0.13$ $<0.17$ $<0.22$ $<0.09$ $<0.14$ $<0.19$
$|\varepsilon_{e\tau}^{uV}|$ $<0.21$ $<0.29$ $<0.36$ $<0.12$ $<0.21$ $<0.28$
$|\varepsilon_{\mu\tau}^{uV}|$ $<0.16$ $<0.22$ $<0.28$ $<0.12$ $<0.18$ $<0.23$
$\varepsilon_{ee}^{dV}$ $ $ $-0.19\div0.55$ $ $-0.07\div0.43$ $-0.13\div0.48$
\left( \left( \left(
\begin{array}{c} \begin{array}{c} \begin{array}{c}
-0.09\div0.03 -0.14\div0.15 -0.02\div0.17
\\ \\ \\
0.33\div0.43 0.21\div0.49 0.18\div0.37
\end{array} \end{array} \end{array}
\right) \right) \right)
$ $ $
$\varepsilon_{\mu\mu}^{dV}$ $ $ $-0.13\div0.48$ $ $ $
\left( \left( \left( \left( \left(
\begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c}
-0.05\div0.02 -0.09\div0.07 -0.03\div0.02 -0.06\div0.05 -0.09\div0.09
\\ \\ \\ \\ \\
0.33\div0.40 0.28\div0.44 0.33\div0.38 0.30\div0.41 0.27\div0.45
\end{array} \end{array} \end{array} \end{array} \end{array}
\right) \right) \right) \right) \right)
$ $ $ $ $
$|\varepsilon_{e\mu}^{dV}|$ $<0.12$ $<0.16$ $<0.20$ $<0.08$ $<0.13$ $<0.17$
$|\varepsilon_{e\tau}^{dV}|$ $<0.19$ $<0.26$ $<0.33$ $<0.11$ $<0.19$ $<0.25$
$|\varepsilon_{\mu\tau}^{dV}|$ $<0.15$ $<0.20$ $<0.25$ $<0.11$ $<0.16$ $<0.21$
-------------------------------- ------------------ ------------------ ----------------- ------------------ ------------------ ------------------
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|
---
author:
- 'J. Iglesias-Páramo'
- 'A. Boselli'
- 'L. Cortese'
- 'J.M. Vílchez'
- 'G. Gavazzi'
date: 'Received ......; accepted .........'
subtitle: 'The H$\alpha$ Luminosity Functions'
title: 'A Deep H$\alpha$ Survey of Galaxies in the Two Nearby Clusters Abell 1367 and Coma'
---
=-.15in
Introduction
============
The strong morphology segregation observed in rich clusters of galaxies (Dressler, 1980) testifies the fundamental role played by the environment on the evolution of galaxies. Which physical mechanisms are responsible for such transformations is however still matter of debate. Several processes might alter the evolution of cluster galaxies. Some of them refer to the interaction of the galaxies with the intracluster medium (Gunn & Gott, 1972) and others account for the effects of gravitational interactions produced by the gravitational potential of the cluster (Merritt, 1983) or by galaxy-galaxy interactions (Moore et al. 1996, 1998, 1999). All these mechanisms can produce strong perturbations in the galaxy morphology with the formation of tidal tails, dynamical disturbances which appear as asymmetries in the rotation curves (Dale et al. 2001) and significant gas removal (Giovanelli & Haynes 1985; Valluri & Jog 1990).
Some of these processes are expected to produce changes in the star formation rates of galaxies in clusters. Several studies have addressed the issue of the influence of the cluster environment on the SFR of disk galaxies, however no agreement has been established so far: whereas some authors proposed similar or even enhanced star formation in cluster spirals than in the field (Donas et al. 1990, 1995; Moss & Whittle 1993, Gavazzi & Contursi 1994; Moss et al. 1998; Gavazzi et al. 1998; Moss & Whittle 2000), some others claim quenched SFRs in cluster spirals (Kennicutt 1983; Balogh et al. 1998; Hashimoto et al. 1998). This discrepancy could arise from non-uniformity of the adopted methods (UV vs. H$\alpha$ vs. \[O[ii]{}\] data) or from real differences in the studied clusters (Virgo, Coma, Abell 1367, clusters from Las Campanas Redshift Survey, clusters at $z > 0.18$).\
In particular, an enhanced fraction of spirals with circumnuclear H$\alpha$ emission was found in the highest density regions of some nearby clusters (Moss et al. 1998; Moss & Whittle 2000), whereas no such difference was found for galaxies with diffuse emission. The compact H$\alpha$ emission seems associated with ongoing interactions of galaxies, but numerical simulations by Bekki (1999) showed that mergers between clusters and subclusters might produce central starbursts in cluster spirals.
Existing studies of the H$\alpha$ properties of galaxies in clusters suffer from various biases: the photoelectric data by Kennicutt et al. (1984) and Gavazzi et al. (1991, 1998) are based on samples of galaxies selected on the basis of their optical properties, independent of their H$\alpha$ properties. On the other hand, the objective-prism surveys by Moss et al. (1988, 1998) and Moss & Whittle (2000) are H$\alpha$ selected but they are too shallow to allow a determination of the H$\alpha$ luminosity function as deep as desired.\
With the aim of obtaining a reliable determination of the current SFR in nearby clusters of galaxies and to study the spatial distribution of the star formation regions, we undertook a deep imaging survey of a one degree $\times$ one degree area of the Coma and Abell 1367 clusters.\
Our work provides the first deep and complete study of galaxies in clusters based on their H$\alpha$ emission properties.\
This paper is arranged as follows: Section 2 contains a description of the observations, of the data reduction and the detection procedures. The H$\alpha$ data are presented in Section 3. The H$\alpha$ luminosity function and a brief discussion on the contribution of both clusters to the local star formation rate density are presented in Section 4. Conclusions are presented in Section 5. Comments on the most interesting objects as well as the H$\alpha$ images of the detected galaxies are given in the Appendix.
Observations and Data Reduction
===============================
$$
[lllll]{} Field & R.A. & Dec. & Exp. & Filter\
& (J2000) & (J2000) & sec &\
\
Coma 1 & 12:59:24.75 & $+$27:58:49.89 & $3\times 1200$ & \[S[ii]{}\]\
& & & $3\times 300$ & $r'$\
Coma 2 & 13:01:24.45 & $+$27:58:52.12 & $1200$ & \[S[ii]{}\]\
& & & $300$ & $r'$\
\
Coma 3 & 13:01:24.26 & $+$28:28:52.12 & $3\times
1200$ & \[S[ii]{}\]\
& & & $3\times 300$ & $r'$\
Coma 4 & 12:59:24.57 & $+$28:58:49.89 & $3\times
1200$ & \[S[ii]{}\]\
& & & $3\times 300$ & $r'$\
\
A1367 1 & 11:41:35.83 & $+$19:58:21.44 & $3\times
1200$ & \[S[ii]{}\]\
& & & $3\times 300$ & $r'$\
A1367 2 & 11:43:35.61 & $+$19:58:20.73 & $3\times
1200$ & \[S[ii]{}\]\
& & & $3\times 300$ & $r'$\
\
A1367 3 & 11:45:35.40 & $+$19:58:20.10 & $3\times
1200$ & \[S[ii]{}\]\
& & & $3\times 300$ & $r'$\
A1367 4 & 11:43:35.56 & $+$19:28:20.73 & $3\times
1200$ & \[S[ii]{}\]\
& & & $3\times 300$ & $r'$\
$$
The observations were carried out with the Wide Field Camera (WFC) at the Prime Focus at the INT 2.5m telescope located at Observatorio de El Roque de los Muchachos (La Palma), on April 26th and 28th 2000, under photometric conditions. The average seeing ranged from 1.5 to 2 arcsecs during both nights. Given the mean velocity of the galaxies in the two clusters under study, $6555~\pm~684$ km sec$^{-1}$ and $6990~\pm~821$ km sec$^{-1}$ for Abell 1367 and Coma respectively[^1] (Fadda et al. 1996), the narrow-band \[S[ii]{}\] filter ($\lambda_{0} = 6725$Å, $\delta \lambda
\approx 80$Å) was used to isolate the H$\alpha$ line and the $r'$ Sloan-Gunn broad-band filter ($\lambda_{0} =
6240$Å, $\delta \lambda \approx 1347$Å) to recover the continuum. Figure \[filters\] shows the transmitance profiles of both filters. Given the width of the \[S[ii]{}\] filter, the \[N[ii]{}\]$\lambda\lambda$6548,6584Å lines are included in the high transmitance pass-band of this filter, so in what follows we will refer to the combined H$\alpha$ $+$ \[N[ii]{}\] flux and equivalent width, as H$\alpha$ flux and equivalent width respectively.
The WFC is composed by a science array of four thinned AR coated EEV 4K$\times$2K devices, plus a fifth acting as autoguider. The pixel scale is 0.333 arcsec pixel$^{-1}$, giving a total field of view of about $34\times 34$ arcmin$^{2}$. Given the arrangement of the detectors, a squared area of about $11 \times 11$ arcmin$^{2}$ is lost at the top right corner of the field. The top left corner of detector \#3 is also lost because of filter vignetting.
Four fields near the center of each cluster were observed. Three different exposures, slightly dithered to remove cosmic rays, were obtained for each position in each filter, except for the second exposure of the Coma cluster where only one exposure per filter was obtained. Figure \[map\] shows our surveyed area. Our observations cover mainly the North-East region of the Coma cluster as described by Colless & Dunn (1996), coinciding with the central part of the Godwin catalog of the Coma cluster (Godwin et al. 1983). One of our fields of Abell 1367 (number 1 in Figure \[map\]) is not covered by the Godwin catalog (Godwin & Peach 1982). For comparison the X-ray contour maps of the two clusters (White et al. (1993) for Coma, and Donnelly et al. (1998) for Abell 1367) are plotted in the figure. The galaxies detected in H$\alpha$ are marked with filled dots. The diary of the observations is presented in Table \[log\].
The data reduction was carried out using standard tools in the IRAF[^2] environment. The astrometric solution was found with the USNO[^3] catalog of stars. The accuracy of this solution was found to be better than 3 arcsecs throughout the frames. Several exposures of standard spectrophotometric stars were taken during both nights. The chip-to-chip differential responses were derived by direct comparison of the photometry measured for the objects, non-saturated stars and galaxies, present in the overlapping regions. Zero-points and extinction coefficients were derived from the calibration equations. Overall, our photometric uncertainty is less than 10%.
In order to properly subtract the continuum from the H$\alpha$ frames, we scaled the counts of the continuum frames until (unsaturated) stars and elliptical galaxies reached an average H$\alpha +$ \[N[ii]{}\] equivalent width of 0 Å. The net H$\alpha +$ \[N[ii]{}\] photometry of the selected galaxies was performed using the QPHOT command of the APPHOT package in IRAF. Aperture photometry was carried out, in both the ON-band and continuum frames, for each galaxy and subtracted to get the net H$\alpha +$ \[N[ii]{}\] fluxes.
Object Selection
================
We made extensive use of the NASA Extragalactic Database (NED) to search for known galaxies in the area covered by the observations. We measured the H$\alpha +$ \[N[ii]{}\] fluxes for all galaxies with known radial velocities, thus up to $r'
\approx 15.5$ for Abell 1367 and $r' \approx 16.5$ for Coma.
The narrow band filter used did not cover the whole velocity interval of the clusters. In order to avoid large uncertainties in the determination of fluxes and equivalent widths, we measured only galaxies for which the filter transmitance was larger than 0.5.
Visual inspection of the net H$\alpha +$ \[N[ii]{}\] frames allowed us to identify faint galaxies with non negligible net H$\alpha +$ \[N[ii]{}\] emission. For these galaxies, there is no estimate of their velocities in NED. A population of faint galaxies ($r' \geq 17.2$) showed up, most of them belonging to Abell 1367. Their H$\alpha +$ \[N[ii]{}\] fluxes are low ($-15.53 < \log
F(\mbox{H}\alpha + \mbox{[N{\sc ii}]}) < -13.83$ erg sec$^{-1}$ cm$^{-2}$) but their H$\alpha$ equivalent width is in the range $9 <
EW(\mbox{H}\alpha + \mbox{[N{\sc ii}]}) < 418$Å. The search was performed for all objects visible on the NET-frames, but, in order to avoid spurious detections, we considered only objects with $F(\mbox{H}\alpha + \mbox{[N{\sc ii}]})/\sigma_{flux} > 5$ as reliable detections (see Column (9) of Tables \[ha\_list\_ab\] and \[ha\_list\_coma\] for the definition of $\sigma_{flux}$). Since for some of them, their redshift is unknown, both the H$\alpha$ fluxes and equivalent widths were computed assuming that their velocity coincides with the average velocity of the cluster[^4]$^{,}$[^5].
In total 41 and 22 H$\alpha$ emitting galaxies were detected in Abell 1367 and Coma respectively. These are listed in Tables \[ha\_list\_ab\] and \[ha\_list\_coma\], arranged as follows:
- Col (1): Galaxy designation.
- Col (2): CGCG name (Zwicky et al, 1961-1968).
- Col (3): Other source designation.
- Col (4, 5): Celestial coordinates (J2000).
- Col (6): Radial velocity in km sec$^{-1}$. Galaxies flagged with $\dagger$ are those detected in H$\alpha$ but with unknown redshift. For these objects, the mean velocity of the cluster was adopted. Galaxies flagged with $\dagger\dagger$ are those for which a spectroscopic follow-up was carried out in order to measure their redshifts (Gavazzi et al 2002, in prep).
- Col (7): Magnitude in the $r'$ band.
- Col (8): Log of the H$\alpha +$\[N[ii]{}\] flux, in erg sec$^{-1}$ cm$^{-2}$.
- Col (9): Error ($\sigma_{flux}$) of the H$\alpha +$\[N[ii]{}\] flux. The uncertainty includes three contributions: the Poisson photon counts error, the uncertainty on the background and the photometric uncertainty, which is assumed as 10% of the net flux. These errors were determined separately on the ON and OFF-band frames, and combined using the standard error propagation.
- Col (10): H$\alpha +$\[N[ii]{}\] equivalent width in Å.
- Col (11): Error in the H$\alpha +$\[N[ii]{}\] equivalent width, computed similarly to $\sigma_{flux}$ (Column (9)) except that the error on the absolute flux scale does not affect the equivalent width.
Figure \[ha\_ew\] shows the histograms of H$\alpha$ fluxes and equivalent widths of the emitting galaxies.
In order to check the quality of the photometry, we compared our fluxes and equivalent widths with those taken from the literature (see Table \[ha\_comp\]). Figure \[haflux\_comp\] shows the plots of the H$\alpha$ fluxes and equivalent widths reported in other works $vs$ ours. The linear regressions found for both plots are the following: $$\log F(\mbox{H}\alpha+\mbox{[N{\sc ii}]})_{this~work} =$$ $$0.34(\pm 0.91) + 1.03(\pm 0.07) \times \log
F(\mbox{H}\alpha+\mbox{[N{\sc ii}]})_{literature}$$ $$\log EW(\mbox{H}\alpha+\mbox{[N{\sc ii}]})_{this~work} =$$ $$0.24(\pm 0.11) + 0.85(\pm 0.07) \times \log
EW(\mbox{H}\alpha+\mbox{[N{\sc ii}]})_{literature}$$ Both plots show a discordant point, which corresponds with galaxy CGCG 097-114. This galaxy was measured by Kennicutt et al. (1984), Moss et al. (1988), Moss et al. (1998) and Gavazzi et al. (1998). There is agreement in the flux between our data and that of Kennicutt et al. (1984) but not in the equivalent width. The measurements by Moss et al. (1988, 1998) are consistent with each other, but not with other sources. The equivalent width by Gavazzi et al. (1998) is fairly consistent with ours.
Discussion
==========
The H$\alpha$ Luminosity Function
---------------------------------
The H$\alpha$ luminosity functions were computed separately for the two clusters under study from the measured fluxes. $H_{0} = 50$ km sec$^{-1}$ Mpc$^{-1}$ is assumed to allow a direct comparison with Gallego et al. (1995).
H$\alpha$ fluxes were corrected for \[N[ii]{}\] contamination and dust extinction. The first correction is the one proposed by Gavazzi et al. (2002, in prep.), based on the relationship found between the $H$ band luminosities and the \[N[ii]{}\]/H$\alpha$ ratio. After a empirical relationship between the $H$ and $r'$ magnitudes for the galaxies in common in both samples the correction was finally given by: $$\log [\mbox{N{\sc ii}}]/\mbox{H}\alpha = 1.26 - 0.19 \times r' + 0.70 \times \log D$$ $D$ being the distance of the galaxies in Mpc.
The morphological type dependent dust extinction correction was taken from Boselli et al. (2001). For galaxies with known morphological type (from NED or other sources), the correction was taken to be $$\begin{array}{ll}
A(\mbox{H}\alpha) = & \mbox{1.1 mag, for type Scd or earlier} \\
& \mbox{0.6 mag, for type Sd or later} \\
\end{array}$$ For unclassified galaxies we adopted $M_{B} = -18.25$ as the statistical limiting magnitude for galaxy types intermediate between Scd and Sd, from Sandage et al. (1985).
The contribution of active nuclei to the H$\alpha$ detections is negligible because no relevant point-like nuclear features were detected in the H$\alpha$ frames.
In order to normalize the luminosity function to a proper volume, angular radii of 3 and 4 degrees were assumed for Abell 1367 and Coma respectively (Gavazzi et al. 1995), corresponding to linear sizes of 4.6 and 6.5 Mpc. The clusters were assumed spherically symmetric, thus the surveyed volume corresponds to the intersection between the solid angle covered by our observations and the sphere containing the clusters.
A statistical correction was applied to account for the incomplete velocity coverage of the adopted \[S[ii]{}\] filter. Figure \[velo\_haflux\] shows the flux distribution of galaxies with known redshift $versus$ their radial velocities. The dashed line represents the gaussian distributions of velocities described in Section 2. The shaded regions correspond to the velocity ranges excluded from the filter transmitance window for each cluster. We estimate that about 20% of the velocity distribution for Abell 1367 and 11% for Coma are not within the transmitance window of the narrow band filter. We also corrected in a consistent way the effects of the velocity distribution of the H$\alpha$ emitting galaxies with unknown redshift. The correction was performed as follows: first, we randomly distributed the velocities of these galaxies following the gaussian probability density function with mean velocities and dispersions as described in Section 2. New H$\alpha$ fluxes were derived for these galaxies, according to the values of the transmitance of the \[S[ii]{}\] filter, for the randomly chosen velocities. If the assigned velocity of any of these galaxies gave a transmitance $<$50%, the object was discarded. The final correction was performed by assuming that the relationship, if any, between the radial velocities of the galaxies and the H$\alpha$ fluxes should be symmetric with respect to the mean velocity of the cluster. We repeated this procedure ten times in order to estimate the statistical uncertainties induced by this effect on the luminosity function. Thus, H$\alpha$ luminosity functions were computed with ten different flux distributions for each cluster.
The functional form assumed for the LF is the Schechter (1976) function: $$\phi(L)dL = \phi^{*}(L/L^{*})^{\alpha}\mbox{exp}(-L/L^{*})d(L/L^{*})$$ The size of the bins was taken to be $\delta\log L = 1.0$ in order to minimize the statistical errors. Table \[counts\] shows the number counts per luminosity interval for both clusters, averaged over the different random distributions of velocities for the galaxies with unknown redshift.
Table \[fit\] lists the obtained best fitting Schechter parameters of the upper and lower envelopes for each cluster, as well as the parameters for the average LFs finally adopted.
The upper and lower envelope H$\alpha$ LFs of the two clusters are given in Figure \[lf\_field\]. Shaded regions between the envelopes show the range of uncertainty of the H$\alpha$ LF for each cluster. The points correspond to the mean values listed in Table \[counts\], and the error bars show their typical poissonian uncertainties. As reference, we plot the H$\alpha$ LFs of field galaxies obtained by Gallego et al. (1995), Tresse & Maddox (1998) and Sullivan et al. (2000). The lines are truncated at the completeness limits of each sample.
$$
[lcc]{} $\log L(\mbox{H}\alpha)$ &\
erg sec$^{-1}$ & Abell 1367 & Coma\
38.8 & 8 & 2\
39.8 & 18 & 11\
40.8 & 13 & 8\
41.8 & 1 & 1\
$$\end{table}
\begin{table}[b]
\caption[]{Best fitting parameters for the upper and lower envelopes
corresponding to Abell~1367 and Coma. Also, the average adopted parameters for
the H$\alpha$ LF are listed.
}
\label{fit}$$
[lccc]{} & $\log \phi^{*}$ & $\alpha$ & $\log L^{*}$\
& Mpc$^{-3}$ & & erg sec$^{-1}$\
\
Upper envelope & $-0.06$ & $-0.94$ & 41.37\
Lower envelope & $+0.20$ & $-0.72$ & 41.21\
Average & $+0.06$ & $-0.82$ & 41.30\
\
Upper envelope & $-0.09$ & $-0.70$ & 41.24\
Lower envelope & $-0.04$ & $-0.53$ & 41.21\
Average & $-0.07$ & $-0.60$ & 41.23\
$$
Disregarding non-completeness effects, which should only affect our lowest luminosity bins, the LFs of the two clusters are in fair agreement. The apparent difference with the field LFs is mainly in the normalization since the density of galaxies is several orders of magnitude larger in clusters than in the field. Beside the normalization, the shape of the cluster LFs appears steeper at the bright end and flatter at the faint end. The former derives from undersampling at high luminosity (due to small volume coverage in the two clusters we do not detect any object with $F(H\alpha) \geq
10^{42}$ erg sec$^{-1}$ as opposed to Gallego et al. 1996).
The slope of the fitted LFs appear different among clusters and field at the faint end. However the data points, within the completeness limits of each survey, appear in full agreement among each other, as shown in Fig. \[pru\_lf\].
In this Figure we scaled the cluster LFs in such a way that they match the field LF at $\log L(\mbox{H}\alpha) \approx 41$ erg sec$^{-1}$. Above $\log L(\mbox{H}\alpha)\approx 40$ erg sec$^{-1}$, where all the samples are complete, there is consistency between the field and the cluster datasets. Nothing can be said for fainter luminosities because the field samples are incomplete or present rather poor statistics, opposite to the present cluster survey which is complete to $\log L(\mbox{H}\alpha) \approx 39$ erg sec$^{-1}$. Deeper H$\alpha$ surveys of the field are necessary to assess if the differences at the faint luminosity end are significant.
The Virgo Cluster
-----------------
It is instructive to compare the H$\alpha$ LF of A1367 and Coma with that of the Virgo cluster. Given its large angular size, performing a complete H$\alpha$ survey of this cluster would be prohibitive. However H$\alpha$ observations of most of the brightest galaxies (230 objects brighter than B=16 mag) are available (Boselli & Gavazzi 2002; Gavazzi et al. 2002). Using these data we construct a “pseudo” H$\alpha$ LF by transforming the B band LF into an H$\alpha$ one after having shown that H$\alpha$ luminosity and $M_B$ are found proportional one-another.
Figure \[virgo\_rel\] shows the H$\alpha$ luminosity $vs$ the absolute $M_B$ magnitude relationship. Distances are estimated according to the Virgo cluster group membership, as defined in Gavazzi & Boselli (1999). The best fit to the data gives a slope of 0.37, consistent with 0.40 (i.e. a slope of 1 in a luminosity-luminosity plot). For simplicity we adopted this last value, because it allows to transform the observed B band Schechter function into an H$\alpha$ LF of the same functional form. Therefore we adopt: $$\label{havb}
\log L(\mbox{H}\alpha) = -0.40 \times M_{B} + 33.12$$ Combining this relationship with the $B$ band luminosity function for spirals and irregulars in the Virgo core obtained by Sandage et al. (1985) we obtain an H$\alpha$ LF: $$\phi'(L) = 1.07 \times (L/10^{41.2})^{-0.8}~\mbox{exp}[-(L/10^{41.2})]$$
Figure \[lf\_clusters\] shows the Virgo H$\alpha$ LF together with the ones obtained for Abell 1367 and Coma. The shaded region reflects the scatter in the relationship between H$\alpha$ luminosities and $M_B$ found in Virgo. The shape of the Virgo LF appears consistent with that of Abell 1367 and Coma, despite the different nature of the three clusters, Virgo being unrelaxed and spiral rich, Abell 1367 relaxed and spiral rich, and Coma relaxed and spiral poor.
Star Formation Rates in Clusters
--------------------------------
The total star formation rate per unit volume for clusters is derived by integrating the best fitting Schechter functions over the whole range of luminosities. To be consistent with Gallego et al. (1995), we convert the H$\alpha$ luminosities to star formation per unit time using: $$L(\mbox{H}\alpha) = 9.40 \times 10^{40}
\frac{\mbox{SFR}}{M_{\odot}~\mbox{yr}^{-1}}\mbox{erg~sec}^{-1}$$ Total integrated SFRs of 2.20 and 1.36 $M_{\odot}$ yr$^{-1}$ Mpc$^{-3}$ are obtained for Abell 1367 and Coma respectively, i.e. more than two orders of magnitude higher than the value of 0.013 $M_{\odot}$ yr$^{-1}$ Mpc$^{-3}$ reported for the local Universe by Gallego et al. (1995).
The estimate of the contribution of the clusters to the total SFR per unit volume of the local Universe, is obtained by taking into account the local spatial density of clusters. For Abell type 2 clusters, like Abell 1367 and Coma, this value was reported to be $1.84 \times
10^{-5}$ Mpc$^{-3}$ (Bramel et al. 2000), although this number is affected by large uncertainties. We conclude that the typical contribution of Abell type 2 clusters to the SFR per unit volume is about $3.3 \times
10^{-5}$ $M_{\odot}$ yr$^{-1}$, that is 0.25% of the total SFR in the local Universe.
Similarly, by integrating the Virgo H$\alpha$ luminosity function, we obtain a total H$\alpha$ luminosity density of $1.56 \times
10^{41}$ erg sec$^{-1}$ Mpc$^{-3}$, which gives a SFR of 1.65 $M_{\odot}$ yr$^{-1}$ Mpc$^{-3}$. Taking into account that the Virgo cluster is classified as Abell type 1 (Struble & Rood 1982), and assuming the spatial density for clusters of this type (Bramel et al. 2000) of $8.46 \times 10^{-4}$ Mpc$^{-3}$, we obtain that the contribution of type 1 clusters is $1.40 \times
10^{-3}$ $M_{\odot}$ yr$^{-1}$ Mpc$^{-3}$, corresponding to 10.8% of the total SFR density in the local Universe.
Conclusions
===========
We have carried out an H$\alpha$ imaging survey of the central 1 deg$^{2}$ of the nearby clusters Abell 1367 and Coma. Significant H$\alpha$ emission is found associated with 41 galaxies in Abell 1367 and 22 in Coma. These data are used to estimate, for the first time, the H$\alpha$ luminosity function of 2 nearby clusters of galaxies. These LFs are found consistent with the H$\alpha$ luminosity function derived for the Virgo cluster, despite their different nature. The typical Schechter parameters: $\phi^{*} \approx 10^{0.00\pm0.07}$ Mpc$^{-3}$, $L^{*} \approx
10^{41.25\pm0.05}$ erg sec$^{-1}$ and $\alpha \approx -0.70\pm0.10$ are obtained.
The best fitting parameters of the cluster LFs are significantly different from those found for field galaxies, in particular at the faint end where the cluster slope is shallower than the extrapolated slope of the field LF. However it must be stressed that the steep slope found in the field is based on relatively high luminosity points and no data are available below $\log L(\mbox{H}\alpha) \approx 40$ erg sec$^{-1}$ i.e. where the cluster LFs begin to flatten out. After re-normalizing the cluster data on the field ones, the two sets of data points are found consistent within the completeness limit of the field samples. Until a deeper field LF will be available it is impossible to establish whether the apparent underabundance of low luminosity objects in clusters is a real evolutionary effect or it is an artifact due to incompleteness.
By computing the total SFR per unit volume of the cluster galaxies, and taking into account the cluster density in the local Universe, we estimate that the contribution of type 2 and type 1 clusters is about 0.25% and 10.8% respectively of the SFR per unit volume of the local Universe.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The INT is operated on the island of La Palma by the ING group, in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofísica de Canarias.
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Schweizer, F. 1980, Ap.J., 237, 303
Seguin, P. & Dupraz, C. 1996, A.& A., 310, 757
Struble, M. F. & Rood, H. J. 1982, A.J., 87, 7
Sullivan, M., Treyer, M. A., Ellis, R. S., Bridges, T. J., Milliard, B., & Donas, J. ; 2000, M.N.R.A.S., 312, 442
Tresse, L. & Maddox, S. J. 1998, Ap.J., 495, 691
Valluri, M. & Jog, C. J. 1990, Ap.J., 357, 367
de Vaucouleurs, G., de Vaucouleurs, A., Corwin, J. R., Buta, R. J., Paturel, G., & Fouque, P. 1991, Third reference catalog of Bright galaxies, 1991, New York : Springer-Verlag
White, S. D. M., Briel, U. G., & Henry, J. P. 1993, M.N.R.A.S., 261, L8
Zwicky, F., Herzog, E., Karpowicz M., Kowal, C., Wild P. 1961-1968 “Catalog of Galaxies and Clusters of Galaxies”, Pasadena, California Institute of Technology (CGCG)
$$
[lccccccccrr]{} Name & CGCG & Other & R.A. & Dec. & $v_{r}$ & $r'$ & $F_{\alpha}$ & $\Delta_{f}$ & W$_{\alpha}$ & $\Delta_{\scriptsize{\mbox{W}}}$\
114024+195747 & — & — & 11 40 24.90 & +19 57 47.7 & 6749 & 15.48 & $-$13.71 & 0.04 & 14 & 1\
114038+195437 & — & — & 11 40 38.96 & +19 54 37.4 & 6500$\dagger\dagger$ & 17.35 & $-$14.09 & 0.05 & 36 & 3\
114107+200251 & — & — & 11 41 07.79 & +20 02 51.3 & 6500$\dagger$ & 18.91 & $-$14.60 & 0.05 & 43 & 4\
114110+201117 & — & — & 11 41 10.47 & +20 11 17.7 & 6500$\dagger$ & 17.57 & $-$13.95 & 0.04 & 56 & 2\
114112+200109 & — & — & 11 41 12.81 & +20 01 09.9 & 6500$\dagger$ & 19.44 & $-$14.80 & 0.07 & 38 & 5\
114141+200230 & — & — & 11 41 41.20 & +20 02 30.5 & 6500$\dagger$ & 17.37 & $-$14.26 & 0.06 & 26 & 4\
114142+200054 & — & — & 11 41 42.57 & +20 00 54.9 & 6500$\dagger$ & 17.33 & $-$14.36 & 0.07 & 19 & 3\
114149+194605 & — & — & 11 41 49.79 & +19 46 05.1 & 6500$\dagger$ & 17.52 & $-$14.37 & 0.05 & 23 & 2\
114156+194207 & — & — & 11 41 56.69 & +19 42 07.8 & 6500$\dagger$ & 19.77 & $-$15.10 & 0.07 & 36 & 5\
114157+194329 & — & — & 11 41 57.90 & +19 43 29.4 & 6500$\dagger$ & 20.21 & $-$15.29 & 0.05 & 34 & 3\
114158+194149 & — & — & 11 41 58.05 & +19 41 49.6 & 6500$\dagger$ & 19.46 & $-$15.02 & 0.06 & 38 & 4\
114158+194205 & — & — & 11 41 58.10 & +19 42 05.9 & 6500$\dagger$ & 20.30 & $-$15.27 & 0.04 & 49 & 2\
114158+194900 & — & — & 11 41 58.26 & +19 49 00.9 & 6500$\dagger$ & 20.70 & $-$15.53 & 0.07 & 32 & 4\
114202+194348 & — & — & 11 42 02.30 & +19 43 48.5 & 6500$\dagger$ & 20.83 & $-$15.35 & 0.05 & 70 & 6\
114202+192648 & — & — & 11 42 02.96 & +19 26 48.2 & 6500$\dagger$ & 19.54 & $-$14.66 & 0.06 & 32 & 4\
114214+195833 & 097-062 & PGC036330 & 11 42 14.55 & +19 58 33.6 & 7815 & 14.51 & $-$13.19 & 0.04 & 28 & 1\
114215+200255 & 097-063 & PGC036323 & 11 42 15.70 & +20 02 55.2 & 6102 & 15.36 & $-$13.69 & 0.04 & 13 & 1\
114218+195016 & — & — & 11 42 18.08 & +19 50 16.1 & 6476 & 15.79 & $-$14.24 & 0.04 & 6 & 1\
114239+195808 & — & — & 11 42 39.23 & +19 58 08.0 & 7345 & 16.95 & $-$13.89 & 0.04 & 40 & 1\
114240+195716 & — & — & 11 42 40.36 & +19 57 16.6 & 6500$\dagger$ & 17.68 & $-$14.70 & 0.08 & 13 & 2\
114256+195757 & 097-073 & PGC036382 & 11 42 56.67 & +19 57 57.7 & 7275 & 15.50 & $-$12.81 & 0.04 & 86 & 1\
114313+193645 & — & — & 11 43 13.08 & +19 36 45.8 & 6500$\dagger\dagger$ & 17.27 & $-$14.06 & 0.05 & 30 & 3\
114313+200015 & 097-079 & PGC036406 & 11 43 13.93 & +20 00 15.6 & 7000 & 16.50 & $-$12.69 & 0.04 & 130 & 2\
114341+200135 & — & — & 11 43 41.62 & +20 01 35.3 & 6500$\dagger$ & 17.08 & $-$14.15 & 0.06 & 25 & 3\
114348+195812 & 097-087 & UGC06697 & 11 43 48.59 & +19 58 12.8 & 6725 & 14.22 & $-$12.19 & 0.04 & 81 & 2\
114348+201456 & — & — & 11 43 48.92 & +20 14 56.0 & 6146 & 15.86 & $-$12.95 & 0.04 & 137 & 1\
114349+195833 & — & — & 11 43 49.87 & +19 58 33.2 & 7542 & 16.11 & $-$13.99 & 0.04 & 19 & 2\
114355+192743 & — & — & 11 43 55.71 & +19 27 43.9 & 6500$\dagger$ & 18.72 & $-$14.67 & 0.07 & 27 & 4\
114358+201105 & 097-092 & PGC036478 & 11 43 58.17 & +20 11 05.6 & 6373 & 14.71 & $-$13.10 & 0.04 & 30 & 1\
114358+200433 & 097-091 & NGC3840 & 11 43 58.81 & +20 04 33.0 & 7368 & 13.92 & $-$12.86 & 0.07 & 25 & 4\
114400+200144 & 097-097 & NGC3844 & 11 44 00.86 & +20 01 44.5 & 6834 & 13.62 & $-$13.41 & 0.04 & 5 & 1\
114430+195718 & — & — & 11 44 30.41 & +19 57 18.8 & 6500$\dagger$ & 20.23 & $-$14.38 & 0.04 & 418 & 19\
114447+194624 & 097-114 & NGC3860B & 11 44 47.88 & +19 46 24.6 & 8293 & 15.33 & $-$13.24 & 0.05 & 40 & 4\
114454+194733 & — & — & 11 44 54.22 & +19 47 33.2 & 6500$\dagger\dagger$ & 20.27 & $-$13.99 & 0.06 & 103 & 17\
114454+194635 & 097-125 & PGC036589 & 11 44 54.99 & +19 46 35.8 & 8271 & 14.50 & $-$13.00 & 0.05 & 24 & 2\
114454+200101 & — & — & 11 44 54.71 & +20 01 01.5 & 6500$\dagger\dagger$ & 16.17 & $-$14.41 & 0.04 & 6 & 1\
114503+195002 & — & — & 11 45 03.38 & +19 50 02.7 & 6500$\dagger$ & 17.90 & $-$14.76 & 0.07 & 9 & 1\
114506+195801 & 097-129E & NGC3861B & 11 45 06.91 & +19 58 01.6 & 6009 & 14.64 & $-$13.38 & 0.06 & 19 & 2\
114513+194523 & — & — & 11 45 13.86 & +19 45 23.0 & 6500$\dagger\dagger$ & 15.60 & $-$13.86 & 0.04 & 12 & 1\
114518+200009 & — & — & 11 45 18.00 & +20 00 09.5 & 6500$\dagger$ & 17.54 & $-$14.28 & 0.06 & 22 & 3\
114603+194712 & 097-143B & — & 11 46 03.68 & +19 47 12.9 & 7170 & 15.80 & $-$14.93 & 0.05 & 1 & 1\
$$
Objects with unknown redshift but detected in the net H$\alpha$ frames
Objects for which we have measured the redshift. It will appear in a subsequent paper.
$$
[lccccccccrr]{} Name & CGCG & Other & R.A. & Dec. & $v_{r}$ & $r'$ & $F_{\alpha}$ & $\Delta_{f}$ & W$_{\alpha}$ & $\Delta_{\scriptsize{\mbox{W}}}$\
125757+280343 & — & FOCA610 & 12 57 57.73 & +28 03 43.3 & 8299 & 15.23 & $-$13.37 & 0.05 & 22 & 2\
125805+281433 & 160-055 & NGC4848 & 12 58 05.67 & +28 14 33.2 & 7049 & 14.04 & $-$12.54 & 0.05 & 34 & 2\
125845+284133 & — & FOCA353 & 12 58 45.64 & +28 41 33.1 & 7001$\dagger\dagger$ & 17.21 & $-$14.02 & 0.06 & 35 & 5\
125845+283235 & — & FOCA399 & 12 58 45.80 & +28 32 35.3 & 7001$\dagger$ & 17.76 & $-$13.83 & 0.04 & 101 & 4\
125856+275002 & 160-212 & FOCA600 & 12 58 56.55 & +27 50 2.7 & 7378 & 15.12 & $-$13.84 & 0.05 & 3 & 1\
125902+280656 & 160-213 & FOCA498 & 12 59 02.14 & +28 06 56.4 & 9436 & 15.15 & $-$13.32 & 0.06 & 28 & 3\
125907+275118 & 160-219 & IC3960 & 12 59 07.97 & +27 51 18.0 & 6650 & 14.50 & $-$14.12 & 0.05 & 2 & 1\
125923+282919 & — & FOCA361 & 12 59 23.13 & +28 29 19.0 & 7001$\dagger\dagger$ & 15.75 & $-$13.98 & 0.04 & 10 & 1\
130006+281500 & — & FOCA371 & 13 00 06.42 & +28 15 0.9 & 7259 & 17.04 & $-$14.48 & 0.07 & 6 & 1\
130037+280327 & 160-252 & FOCA388 & 13 00 37.99 & +28 03 27.6 & 7840 & 14.68 & $-$12.93 & 0.08 & 41 & 4\
130037+283951 & — & — & 13 00 37.24 & +28 39 51.6 & 7001$\dagger\dagger$ & 16.86 & $-$14.64 & 0.05 & 6 & 2\
130040+283113 & — & FOCA242 & 13 00 40.75 & +28 31 13.4 & 8901 & 15.80 & $-$13.11 & 0.05 & 68 & 6\
130056+274727 & 160-260 & FOCA445 & 13 00 56.03 & +27 47 27.7 & 7985 & 13.11 & $-$12.76 & 0.07 & 11 & 2\
130114+283118 & — & FOCA195 & 13 01 14.99 & +28 31 18.5 & 8426 & 17.02 & $-$14.04 & 0.05 & 29 & 3\
130125+284036 & 160-098 & FOCA137 & 13 01 25.04 & +28 40 36.9 & 8762 & 14.41 & $-$13.21 & 0.04 & 18 & 1\
130127+275957 & — & GMP2048 & 13 01 27.17 & +27 59 57.0 & 7558 & 15.64 & $-$14.35 & 0.04 & 4 & 1\
130128+281515 & — & — & 13 01 28.63 & +28 15 15.9 & 7001$\dagger$ & 20.41 & $-$14.96 & 0.04 & 107 & 6\
130130+283328 & — & FOCA158 & 13 01 30.85 & +28 33 28.0 & 7001$\dagger\dagger$ & 16.76 & $-$13.95 & 0.06 & 24 & 2\
130140+281456 & — & GMP1925 & 13 01 40.97 & +28 14 56.6 & 7001$\dagger$ & 19.33 & $-$14.43 & 0.07 & 132 & 36\
130158+282114 & — & — & 13 01 58.43 & +28 21 14.8 & 7001$\dagger$ & 19.81 & $-$14.39 & 0.04 & 278 & 8\
130212+281023 & — & FOCA218 & 13 02 12.00 & +28 10 23.0 & 8950 & 16.09 & $-$13.41 & 0.05 & 30 & 2\
130212+281253 & 160-108 & FOCA204 & 13 02 12.55 & +28 12 53.0 & 8177 & 14.93 & $-$13.29 & 0.04 & 25 & 1\
$$\end{table}
\twocolumn
\newpage
\onecolumn
\begin{table}[t]
\caption[]{Comparison of the H$\alpha$ fluxes and equivalent widths with
data from the literature, for the objects in common.}
\label{ha_comp}$$
[lcccccccccccc]{} CGCG & & & & & &\
& $\log F$ & $EW$ & $\log F$ & $EW$ & $\log F$ & $EW$ & $\log F$ & $EW$ & $\log F$ & $EW$ & $\log F$ & $EW$\
097-062 & -13.19 & 28 & – & – & -12.93 & 58 & -13.10 & 45 & – & – & – & 34\
097-073 & -12.81 & 86 & -12.84 & – & — & — & -12.84 & 80 & – & – & -12.76 & 108\
& & & & & & & & & & & -12.75 & 94\
097-079 & -12.69 & 130 & -12.54 & — & — & — & -12.64 & 145 & -12.64 & 131 & -12.66 & 137\
097-087 & -12.19 & 81 & -12.22 & 64 & -12.43 & 84 & -12.19 & 61 & – & – & -12.22 & 74\
097-092 & -13.10 & 30 & -13.06 & – & -12.95 & 30 & – & – & – & – & – & 27\
097-091 & -12.86 & 25 & -12.92 & 17 & -12.86 & 21 & -12.74 & 23 & – & – & – & –\
097-114 & -13.24 & 40 & -12.82 & 79: & -12.82 & 60 & -13.20 & 4 & – & – & – & 48\
097-125 & -13.00 & 24 & -13.13 & 29 & -13.04 & 26 & – & – & – & – & – & 21\
097-129E & -13.38 & 19 & – & – & – & – & – & – & – & – & -13.38 & 18\
160-252 & -12.93 & 41 & – & – & – & – & -12.93 & 35 & – & – & – & –\
160-055 & -12.54 & 34 & – & – & – & – & -12.65 & 23 & – & – & -12.51 & 34\
160-260 & -12.76 & 11 & – & – & – & – & – & – & – & – & -13.03 & 8\
160-098 & -13.21 & 18 & – & – & – & – & – & – & – & – & -13.15 & 20\
$$
Moss et al. 1988
Moss et al. 1998
Kennicutt et al. 1984
Gavazzi et al. 1991
Gavazzi et al. 1998
Comments on Individual Objects
==============================
The field around galaxies CGCG 97-114, CGCG 97-120 and CGCG 97-125 deserves special attention. Each one of them shows H$\alpha$ emission. However CGCG 97-120 is not listed in Table \[ha\_list\_ab\] because its radial velocity, 5595 km sec$^{-1}$, (de Vaucouleurs, 1991) is out of the transmitance window of the on-band filter. Instead, CGCG 97-114 and CGCG 97-125 have very similar velocities lying in our filter range: 8293 km sec$^{-1}$ (de Vaucouleurs, 1991) and 8271 km sec$^{-1}$ (Haynes et al, 1997) respectively. If these galaxies form a sub-group within Abell 1367 ($<V> = 6500$ km sec$^{-1}$), this entity must have a very high velocity dispersion. The group coincides in position with the X-ray cluster center (Donnelly et al., 1998). CGCG 97-120 is a very H[i]{} deficient galaxy (def(H[i]{}) = 0.9, Chincarini et al., 1983), indicating that it has passed through the cluster center, while CGCG 97-125 is not (def(H[i]{}) = $-0.21$, Haynes et al., 1997). No H[i]{} measurement is available for CGCG 97-114,because it lies close (in space and in velocity) to CGCG 97-125.
In between CGCG 97-120 and CGCG 97-125 we found two emission-line dwarf galaxies 114454+194733 and 114451+194713 (labeled as F.O. but not included in our list because its S/N ratio is lower than 5) (see Figure \[fo18\]) which were already revealed by Sakai et al. (2001). For both galaxies a spectroscopic follow-up gave velocities about 8100 km sec$^{-1}$, consistent with those of CGCG 97-114 and CGCG 97-125, indicating that they are probably associated with these galaxies.
In addition, our net frame shows the presence of some H$\alpha$ emission bridging CGCG 97-114 and CGCG 97-125. Two bright H$\alpha$ knots are detected near CGCG 97-125 followed by a filamentary structure near the bright star between the two galaxies. This structure probably indicates a tidal interaction between the two galaxies, consistently with the shells of 1’ diameter seen in the off-band image around CGCG 97-125. In fact N-body simulations (Quinn, 1984; Seguin, 1996) confirm Schweizer (1980) suggestion that shells around early type galaxies result from galaxy collisions. It is not yet possible to decide whether 114454+194733 and F.O. are tidal dwarfs of CGCG 97-125 or patches of star-forming gas stripped from it.
[^1]: for Abell 1367, we use the average of the redshifts reported in this paper.
[^2]: Image Reduction and Analysis Facility, written and supported at the National Optical Astronomy Observatory
[^3]: United States Naval Observatory
[^4]: the transmitance peak of the filter is 0.85, and the transmitance for the H$\alpha$ lines at the averages velocities of both clusters corresponds to 0.82 and 0.76 for Coma and Abell 1367 respectively
[^5]: Contamination due the H$\beta$ or \[O[iii]{}\] emission lines of background objects is not important. For some of these objects the redshift was measured and only less than 10% were found to be non members of the cluster
|
---
abstract: 'The Mössbauer technique is proposed as an alternative experimental procedure to be used in the detection of Coherent Elastic $\nu$-Nucleus Scattering (CENNS). The $Z^{0}$-boson exchange interaction is considered as a perturbation on the nuclear mean-field potential. This causes a change in the valence neutron quantum states in the $^{57}$Fe nucleus of the Mössbauer detector, which is a typical isotope used in Mössbauer spectroscopy. The transferred energy causes a perturbation at the valence neutron level, modifying the location of the isomeric peak of the Mössbauer electromagnetic resonance. We calculate the CENNS isomeric shift correction and show that this quantity is able to be detected with enough precision. Therefore, the difference between the Mössbauer isomeric shift in the presence of a reactor neutrino beam and without the neutrinos flux is pointed out as a figure of merit to manifest CENNS. In this work, we show that the CENNS correction of the isomeric shift is of $\approx 10^{-7}$eV, which is greater than the $10^{-10}$eV resolution of the technique.'
author:
- '$^1$$^,$$^2$C. Marques'
- '$^1$G. S. Dias'
- '$^3$H. S. Chavez'
- '$^2$S. B. Duarte'
title: 'Searching evidences of Neutrino-Nucleus Coherent Scattering with Mössbauer Spectroscopy'
---
\[sec:level1\]Introduction
==========================
The search for Coherent Elastic $\nu$-Nucleus Scattering (CENNS) is considered an experimentally challenging task as an introduction of the weak neutral current in the context of the Standard Model (SM) [@freedman; @giunt; @brice]. An observation has been intensively pursued by many experimental collaborations in the last four decades [@drukier; @formaggio; @giunt; @brice; @collar] developing a great effort to detect this weak process with largest predicted cross section.
However, only recently the COHERENT collaboration detected undoubtedly the process [@akimov] using an efficient scintillator detector. The experiment, located at Oak Ridge National Laboratory, uses a Spallation Neutron Source with an extremely intense neutron beam. In the scattering of these neutrons by a mercury target, a secondary pion beam is produced. The pion decay generates the intense neutrino flux ($\approx 10^{11}/$s) used in the experiment, with energy in the range of $16$ to $53$ MeV [@akimov]. The experiment accumulates fifteen months of photoelectrons, which was produced in accordance to the SM prediction, when the pulsed neutrino flux is scattered by $14.6$ kg crystal made of CsI doped with Sodium atoms. The experimental setup was properly structured to prevent any contamination from external sources of neutrons and neutrinos, like atmospheric or solar and galactic neutrinos.
The present work proposes the study of the CENNS using Mössbauer nuclear spectroscopy(MS) [@greenwood; @gruverman]. Note that the main characteristic of Mössbauer technique is the recoilless interaction of the electromagnetic radiation with the nucleus.
We consider that the nucleus has a fixed position in the local minimum of the crystal potential lattice and that the effect of the transferred energy is to induce a perturbation in the quantum state of the valence neutron in the atomic nuclei. The volume of the nucleus is modified, with the change in electronic density of K-electron inside the nucleus. The magnitude of this change can be measured by the isomeric displacement in a typical Mössbauer experiment. We assume that only the valence neutron in $^{57}$Fe is excited with the $Z^{0}$ nuclear absorption. Note that excitations of nucleons inside the core are Pauli blocked.
This Letter is organized as follows: In Section II we summarize the main characteristics of CENNS; in Section III we present our proposal to use the Mössbauer technique to observe CENNS and, in Section IV, we calculate the isomeric shift correction when a neutrino reactor flux are present. Section V summarizes our main conclusions.
The Main Characteristic of CENNS
================================
The CENNS was proposed theoretically by Freedman [@freedman]in 1974. A Feynman diagram of this weak process is shown in Fig. \[diagram\]. The effective Lagrangian to describe the process is given by $$L = G_{F} L^{\mu}J_{\mu},$$ where $G_{F}$ is the Fermi constant, $L^{\mu}$ the lepton current, and $J_{\mu}$ is the hadron current inside the nucleus.
Experimental efforts have been developed in the detection of CENNS, some of them represented by large scientific collaborations namely, COHERENT [@coherent; @akimov], CONNIE [@connie] and TEXONO [@tex], among others. As mentioned before, after decades of searching, only in the last year the COHERENT Collaboration [@akimov] announced the first irrefutable detection of CENNS.
The CENNS coherence, as it is well known, requires $qR \ll 1$, with $q$ being the transferred momentum and $R$ the nuclear radius. This implies that the wavelength of neutrinos will be comparable to the nuclear radius. Detailed discussions about the phenomena can be found in Refs. [@freedman; @mosel; @krauss; @formaggio; @brice] and references therein. We stress the fact that the cross section of this process has the largest value $(\sigma \approx 10^{-38}$ cm$^{2})$ at least four orders of magnitude larger than other neutrino interactions in the same low-energy regime [@formaggio].
![Feymann diagram of the CENNS process.[]{data-label="diagram"}](diagg.eps)
The Freedman differential cross section for this process is [@freedman; @giunt; @kate] $$\frac{d\sigma_{\rm CENNS}}{dT}=\frac{G_{F}^{2}}{4\pi}Q_{w}^{2}M_{A}F^{2}(q^{2})\bigg(1 - \frac{M_{A}T}{2E_{\nu}^{2}}\bigg) ,
\label{cross}$$ where $T$ is the transferred energy to the nucleus, ${A}$ is the target mass, $E_{\nu}$ is the neutrino energy and $Q_{w}= N -Z(1-4$sin$^{2}\theta_{w})$ is the weak charge, which depends on the number of neutrons ($N$) and protons ($Z$). Here $\theta_{w}$ is the Weinberg angle satisfying sin$^{2}\theta_{w}\approx 1/4$ and the proton contribution is negligible. The last fact made CENNS a very sensitive probe to nuclear neutron density [@balant]. The form factor $F(q^{2})\to 1$ as $q\to 0$ define the coherence condition.
We note that as a weak process, the interaction involved in the CENNS should be many order of magnitude greater than the gravitational phenomena. Even so, the MS was successfully choose as an appropriated technique to measure the gravitational red shift of light by Pound and Rebka [@pound; @snider], at the end of the fifties. Thus, we hopefully expected that MS can detect CENNS when properly employed.
The Mössbauer Technique Applied to Detect CENNS
===============================================
One of the main characteristics of the MS is that the nuclei in the absorber material of the machine are recoilless when interacting with photons that came from the source decay. This condition is fundamental for the resonant radiation absorption in the MS. This feature was preserved for the CENNS because the energy is in the same range of the one of the photon. The transferred momentum by the $Z^{0}$ exchange is assumed to be transmitted to the valence neutron, slightly modifying the neutron distribution in the nuclear surface and promoting a typical isomeric shift correction in the MS experiment. In addition, it is straightforward to show that the recoilless nuclei in the source and absorber is consistent with the resonant radiation absorption and with coherent neutrino scattering by the nuclei.
The fraction $f$ of the recoilless nuclei in $Z^{0}$ exchange between neutrino and nuclei in the CENNS can be analyzed similarly to the case of the gamma radiation interaction. It can be shown that the fraction of recoilless events can be put in the form of Debye-Waller factor [@marf], which, for the CENNS, takes the form $$f=\exp{\left(-\frac{T^{2}}{Mc^{2}\hbar\omega}\right)} ,$$ where $T=E_{\nu}^{2}/2Mc^{2}$ is the energy transferred by the $Z^{0}$ to target nucleus of mass $M$. Here $\hbar\omega \approx 10^{-3} $ eV for Fe, Co etc, is the order of magnitude of energy lattice vibrations. In the range of neutrino energies below $\approx 50$ MeV the recoilless $f$ factor is essentially unity. Therefore, we argue that this small energy fraction is accommodated by the neutron levels.
Isomeric Shift Correction due to the CENNS Interaction
======================================================
We consider that the energy transfer to the valence neutron is done in a perturbative way, taking into account the first-order perturbation series in a shell model picture. We assume that the perturbed neutron wave functions acquire a small projection in the next shell model state of the unperturbed system. This picture allows us to consider the problem as a two-level system, with the neutron state fluctuating between the two unperturbed state levels. In the present case we will focus on $^{57}$Fe because it is the most common in the literature, but many other nuclei can be studied with this technique, e.g., La, Te, Cd and Sm [@greenwood]. The unperturbed $^{57}$Fe valence neutron is at a state of definite angular momentum, given by the common distribution of the neutron and proton content [@weiss] in the conventional nuclear shell model – its wave function is regular at the origin (typically a Bessel function). Thus the perturbed valence neutron states, after the $Z^{0}$ interaction are $$\begin{aligned}
\Phi^{+}= \frac{-\lambda j_{3/2}(kr)}{\sqrt{1+\lambda^{2}}}+\frac{j_{5/2}(kr)}{\sqrt{1+\lambda^{2}}},\\
\Phi^{-}= \frac{j_{3/2}(kr)}{\sqrt{1+\lambda^{2}}}+\frac{\lambda j_{5/2}(kr)}{\sqrt{1+\lambda^{2}}}.
\label{pert}\end{aligned}$$
The $\lambda$ parameter in the above equations appears in the perturbative treatment and is associated to the ratio between the energy transferred in the CENNS interaction [@kuchiev]. Explicitly we have, $$\lambda = \lambda(E_{\nu})=\frac{3E_{\nu}^{2}}{8Mc^{2}(E_{5/2}-E_{3/2})}.
\label{lambda}$$ The term $(E_{5/2}-E_{3/2})$ is the difference between the energy of the non-perturbed states of the valence neutron. In $^{57}$Fe is this is 14.4 KeV.
In the context of the shell model for Woods-Saxon potential with spin orbit term [@samuel; @weiss; @charm; @kuksa], the two states of valence neutron for the $^{57}$Fe are represented by $j_{3/2}(kr)$ and $j_{5/2}(kr)$. We have used for the wave number of the valence neutrons $k \approx 0.5$ fm$^{-1}$.
The isomeric shift can be calculated [@greenwood] as being $$\delta I^{*}_{s}=\frac{4\pi Ze^{2}R^{2}}{5}\Bigl(\frac{R_{exc}-R_{gs}}{R}\Bigr)[\psi_{l=0}^{2}(0)_{a}-\psi_{l=0}^{2}(0)_{s}] ,
\label{deltas}$$ where $Z$ is the number of protons in the nucleus, $R$ is the mean radius of the charge distribution and $R_{exc/gs}$ for the $Z^{0}$ excited and ground-state nuclear radius, respectively, and the $\psi 's$ is the $s$ electrons wave functions, evaluated at the origin [@marf; @greenwood]. In the literature, the difference $(R_{exc}-R_{gs})$ in conventional gamma absorption is reported to be of order $10^{-3}R$ [@marf; @greenwood]. Our estimative for $\frac{R_{exc}-R_{gs}}{R}$, calculated using the perturbed neutron wave functions $\Phi_{+/-}$ above is $\approx 10^{-4}R$. With this result and Eq. \[deltas\], we can obtain the correction at isomeric displacement induced by $Z^{0}$ and we see that it is only one order of magnitude smaller than the typical already measured values for the characteristic $\gamma$ measurements. This value for $\delta_{I^{*}_{s}}$ is perfectly solved with the MS technique accuracy, namely $10^{-10}$ eV [@pound; @snider; @greenwood]. Consequently, we point out that if we take subtraction of a MS measurement without the neutrino flux and other result of identical measure with the reactor neutrino beam, we would reveal the contribution of the CENNS interactions.
Conclusions
===========
In this work we develop some arguments that supports that MS could be a suitable technique to see the effect of the neutral interaction between neutrinos and nucleus.
The correction obtained for this isomeric contribution at the MS experiment, e.g., $\sim 10^{-7}$ eV for $^{57}$Fe, appears to be greater than the energy resolution of this technique, which is $\sim 10^{-10}$ eV , and we argue that in future this technique could be suitable to integrate the neutrino experimental search plants.
S. B. Duarte acknowledge financial support from CNPq. The authors are also grateful to Pedro Cavalcanti Malta, Gustavo Pazzini de Brito, José A. Helayël-Neto, Hélio da Motta and Arthur M. Kós for the encouragement and for clarifying discussions during the development of this work.
[99]{} Daniel Z. Freedman; Coherent effects of a weak neutral current, Physical Review D, V.9 $N^{0}$ 5 Mar. 1974 . Giunt.C and Studenikin A.; Neutrino Eletromagnetic Interactions: A Window to New Physics, Rev. Mod. Phys.87. 2015 531. S. J. Brice et al, A new Method for measuring Coherent Elastic Neutrino Nucleous Scattering at an Off-Axis High-Energy Beam Target; arXiv:1311.5958v1 , Nov.2013. Krauss, Lawrence M., Sheldon L. Glashow, and David N. Schramm. “Antineutrino astronomy and geophysics.” Nature 310.5974 (1984): 191-198. A. Drukier, L. Stodolsky; Principles and applications of a neutral-current detector for neutrino physics and astronomy. PRD, Vol30, Number 11 Dec.1984. J. A Formaggio, G. P. Zeller; From eV to EeV: Neutrino cross sections across energy scales, Reviews of Modern Physics, Sep.2012. J.I. Collar, N. E. Fields, M. Hai, T. W. Hossbach, J. L. Orrel, C. T. Overmann, G. Perumpilly, B. Scholz, arXiv: 1047.7524v2\[physics.ins-det\] Aug. 2014. A. Drukier, L. Stodolsky; Principles and applications of a neutral-current detector for neutrino physics and astronomy. PRD, Vol30, Number 11 Dec.1984. Akimov, D., Albert, J. B., An, P., Awe, C., Barbeau, P. S., Becker, B., ... Cervantes, M. (2017). Observation of coherent elastic neutrino-nucleus scattering. Science, 357(6356), 1123-1126. Gruverman, Irwin J. Mössbauer Effect Methodology: Proceedings of the Fifth Symposium on Mössbauer Effect Methodology New York City, February 2, 1969. Springer Science and Business Media, 2013. Greenwood, Norman Neill. Mössbauer spectroscopy. Springer Science Business Media, 2012. Akimov, D., et al. “The COHERENT Experiment at the Spallation Neutron Source.” arXiv preprint arXiv:1509.08702 (2015). CONNIE: arXiv:1608.01565v3 \[physics.ins-det\] 10 Oct 2016. Singh, V. Wong, H.T. Phys. Atom. Nuclei (2006) 69: 1956. https://doi.org/10.1134/S1063778806110214 U. Mosel, O. Lalakulich; Neutrino-nucleus Interactions, arXiv: 1211.1977 vI\[nucl-th\] Nov. 2012. Scholberg, K. (2012). Prospects for Measuring Neutrino–Nucleus Coherent Scattering with a Stopped-Pion Neutrino Source. Nuclear Physics B-Proceedings Supplements, 229, 505. Balantekin, A. B. ($2017$). Facets of Neutrino-Nucleus Interactions. arXiv preprint arXiv$:1711.03667$. Pound, Robert V., and Glen A. Rebka Jr. “Resonant absorption of the $14.4-$keV $\gamma$ ray from $0.10-\mu$sec Fe$ 57$.” Physical Review Letters 3.12 (1959): 554. Pound, R. V., and J. L. Snider. “Effect of gravity on nuclear resonance.” Physical Review Letters 13.18 (1964): 539. Marfunin, Arnold Sergeevic. Spectroscopy, luminescence and radiation centers in minerals. Springer Science Business Media, $2012$. Blatt, John Markus, and Victor Frederick Weisskopf. Theoretical nuclear physics. Springer Science Business Media, 2012. Kuchiev, M. Yu, and V. V. Flambaum. “Influence of perturbations on the electron wavefunction inside the nucleus.” Journal of Physics B: Atomic, Molecular and Optical Physics 35.19 (2002): 4101. Introduction to Nuclear Physics, Samuel S. M. Wong; Prentice Hall, Englewood Cliffs, New Jersey 07632. Jonker, M., et al. “Experimental study of differential cross sections in charged-current neutrino and antineutrino interactions.” Phys. Lett., B 109.1/2 (1982): 133-140. Hamzavi, M., and A. A. Rajabi. “Generalized nuclear Woods-Saxon potential under relativistic spin symmetry limit.” ISRN High Energy Physics 2013 (2013).
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abstract: 'We observed two-photon phase super-resolution in an unbalanced Michelson interferometer with classical Gaussian laser pulses. Our work is a time-reversed version of a two-photon interference experiment using an unbalanced Michelson interferometer. A measured interferogram exhibits two-photon phase super-resolution with a high visibility of $97.9\%\pm0.4\%$. Its coherence length is about 22 times longer than that of the input laser pulses. It is a classical analogue to the large difference between the one- and two-photon coherence lengths of entangled photon pairs.'
author:
- Kazuhisa Ogawa
- Shuhei Tamate
- Hirokazu Kobayashi
- Toshihiro Nakanishi
- Masao Kitano
title: 'Time-reversed two-photon interferometry for phase super-resolution'
---
Introduction {#sec:5_0}
============
Entangled photon pairs generated by spontaneous parametric down-conversion (SPDC) have peculiar characteristics that have never been seen in classical optics. The time-frequency correlation is one such characteristic. Two time-frequency correlated photons tend to be detected simultaneously, and the sum of their frequencies is constant. The two-photon coherence length of time-frequency entangled photon pairs is much larger than that of individual photons. By utilizing these properties, various two-photon interference phenomena have been observed, such as automatic dispersion cancellation [@PhysRevA.45.6659; @PhysRevLett.68.2421], nonlocal interference [@PhysRevLett.62.2205; @PhysRevLett.65.321], and two-photon phase super-resolution in an unbalanced Michelson interferometer [@PhysRevLett.66.1142; @PhysRevLett.86.5620].
The demonstration of these quantum optical phenomena suffers from the low efficiency of generating entangled photon pairs; because of a low output signal, long-term stability is required for this demonstration. To avoid this difficulty, various studies have been proposed to simulate the quantum optical phenomena by using classical light [@PhysRevLett.98.223601; @kaltenbaek2008quantum; @Lavoie2009; @PhysRevLett.102.243601; @PhysRevA.81.063836; @PhysRevA.84.051803; @mazurek2013dispersion]. These studies employ the *time-reversal method*, in which they use the time-reversed version of the conventional schemes using entangled photon pairs. Because of the time-reversal symmetry of quantum unitary dynamics, a time-reversed experiment reproduces the same result as conventional experiments. The time-reversal method replaces the generation of entangled photon pairs with the detection of correlated photon pairs. Thus we can realize quantum optical phenomena by using only classical light.
The previous investigations employing the time-reversal method realized a time-frequency correlation using chirped-pulse interferometry (CPI) [@kaltenbaek2008quantum; @Lavoie2009; @PhysRevLett.102.243601; @PhysRevA.84.051803; @mazurek2013dispersion], which requires a pair of oppositely chirped laser pulses and sum-frequency generation (SFG) to mimic the time-frequency correlation.
In this paper, we observe two-photon phase super-resolution in an unbalanced Michelson interferometer with classical unchirped laser pulses. We apply the time-reversal method to a two-photon interference experiment using an unbalanced Michelson interferometer [@PhysRevLett.66.1142; @PhysRevLett.86.5620]. In contrast to CPI, our method does not require chirped laser pulses. Our scheme extracts the time-frequency correlation by using SFG followed by a bandpass filter. SFG only converts two photons that arrive simultaneously into a sum-frequency photon, and the bandpass filter extracts sum-frequency photons with a constant frequency. In the measured interferogram, we observe a classical counterpart of the large difference between the one- and two-photon coherence lengths of entangled photon pairs. Owing to the simplicity of our experimental setup, we achieved high-visibility two-photon interference with a high degree of efficiency.
This paper is organized as follows. In Sec. \[sec:theory\], we introduce a theory of conventional and time-reversed methods for observing two-photon phase super-resolution in an unbalanced Michelson interferometer. In Sec. \[sec:experiment\], we demonstrate time-reversed experiments for observing two-photon phase super-resolution. We also experimentally confirm a classical counterpart of the large difference between the one- and two-photon coherence lengths of entangled photon pairs. In Sec. \[sec:conclusion\], we summarize the findings of our study and discuss the advantages of our method.
Theory {#sec:theory}
======
We first represent the theory of the time-reversal method, which is based on the time-reversal symmetry of quantum unitary dynamics. For unitary operator $\hat{U}$ and pair of states $\ket{\mathrm{i}}$ and $\ket{\mathrm{f}}$, relation $$\begin{aligned}
|\bracketii{\mathrm{f}}{\hat{U}}{\mathrm{i}}|^2=|\bracketii{\mathrm{i}}{\hat{U}^{-1}}{\mathrm{f}}|^2\label{eq:5}\end{aligned}$$ holds. The relation is easily derived from conjugate transposition $\bracketii{\mathrm{f}}{\hat{U}}{\mathrm{i}}=\bracketii{\mathrm{i}}{\hat{U}^\dag}{\mathrm{f}}^*$ and unitarity $\hat{U}^\dag=\hat{U}^{-1}$. The left-hand side of Eq. (\[eq:5\]) corresponds to the success probability of the projection to final state $\ket{\mathrm{f}}$ for the system evolved with $\hat{U}$ from initial state $\ket{\mathrm{i}}$. We call this the *time-forward process*. By utilizing Eq. (\[eq:5\]), we can exchange the roles of $\ket{\mathrm{i}}$ and $\ket{\mathrm{f}}$, if we can physically realize time evolution $\hat{U}^{-1}$. We call the evolution from initial state $\ket{\mathrm{f}}$ by $\hat{U}^{-1}$ projected to final state $\ket{\mathrm{i}}$ the *time-reversed process*. It has the same success probability as the corresponding time-forward process.
In the following subsections, we illustrate the time-forward and time-reversed processes of the experiment for observing two-photon phase super-resolution in an unbalanced Michelson interferometer.
Time-forward process {#sec:time-forward-method}
--------------------
The schematic setup of the time-forward experiment is shown in Fig. \[fig:exp\_int\_jun\](a). The input is narrow-band pump photons with center frequency $2\omega_0$. The nonlinear optical crystal for SPDC converts the input photons into time-frequency entangled photon pairs with center frequency $\omega_0$. After passing through the interferometer, the photon pairs are simultaneously detected by the two detectors.
The measured interference pattern varies depending on optical-path difference $z$ of the interferometer. Due to the large difference between one- and two-photon coherence lengths $l_1$ and $l_2$ of the entangled photon pairs, one-photon interference occurs for $|z|< l_1$, but two-photon interference occurs for $l_1<|z|<l_2$. In the latter case, we observed two-photon phase super-resolution.
When $|z|<l_1$, the coincidence counting probability per photon pair is given by $$\begin{aligned}
p(z)=\frac{1}{2}\left[\frac{1+\cos(\omega_0z/c)}{2}\right]^2,\label{eq:3}\end{aligned}$$ where $c$ is the speed of light in a vacuum. This interference pattern is the square of the one-photon interference pattern.
On the other hand, when $l_1<|z|<l_2$, the two photons from the shorter and longer arms (S and L) are substantially separated in time. By setting the time window of the coincidence counting shorter than the arrival-time difference between photons S and L, the photon pairs can be extracted from the same arms (SS and LL) by coincidence counting. Since $l_1\ll l_2$ in time-frequency entangled photon pairs, photon pairs SS and LL interfere with each other and generate two-photon interference fringes. The coincidence counting probability per photon pair is given by $$\begin{aligned}
p(z)= \frac{1}{\,8\,}\frac{1+\cos(2\omega_0z/c)}{2},\label{eq:4}\end{aligned}$$ which indicates two-photon phase super-resolution with perfect visibility. The maximum value of Eq. (\[eq:4\]) is a quarter of the maximum value of Eq. (\[eq:3\]).
![(color online). Schematic setups for observing two-photon phase super-resolution. Michelson interferometer is composed of a 50/50 beam splitter (BS) and mirrors (M1 and M2). M1 is displaced by $z/2$ to provide optical-path difference $z$ between the two arms of the interferometer. (a) Time-forward process. This setup is composed of a nonlinear optical crystal for SPDC, an unbalanced Michelson interferometer, and two detectors for coincidence counting. The dashed-line box describes the possible states of a photon pair. Letters S and L denote the photons from the shorter and longer arms of the interferometer, respectively. When $z$ is large enough, only photon pairs SS and LL contribute to the coincidence counts. (b) Time-reversed process. This setup is composed of an unbalanced Michelson interferometer, a nonlinear optical crystal for SFG, a bandpass filter, and a detector. When $z$ is large enough, two pulses from different arms are individually converted into sum-frequency pulses by SFG. After the bandpass filter stretches the pulse widths, these pulses interfere with each other. This interference corresponds to that of photon pairs SS and LL in the time-forward process. []{data-label="fig:exp_int_jun"}](fig_1.eps){width="8.5cm"}
Time-reversed process {#sec:time-reversed-method}
---------------------
We next present the time-reversal counterpart of the experiment described in the previous subsection. Figure \[fig:exp\_int\_jun\](b) shows the schematic setup. The input light is an unchirped coherent laser pulse with center frequency $\omega_0$ and coherence length $l_1'=c\tau_1$, where $\tau_1$ is the pulse duration. We describe the complex electric field amplitude of the input light as $E_1(t)=f(t)\ee^{-\ii\omega_0 t}$, where $f(t)$ is the pulse envelope function. The field amplitude after passing through the interferometer is given by $$\begin{aligned}
E_2(t)&=\frac{1}{2}[E_1(t)+E_1(t+z/c)]\no\\
&= \frac{1}{2}[f(t)+f(t+z/c)\ee^{-\ii\omega_0z/c}]\ee^{-\ii\omega_0 t},\end{aligned}$$ where $z$ is the optical-path difference of the interferometer. The nonlinear optical crystal for SFG converts the field amplitude into $$\begin{aligned}
E_3(t)&= \alpha E_2(t)^2\no\\
&= \frac{\alpha}{4}[f(t)+f(t+z/c)\ee^{-\ii\omega_0z/c}]^2\ee^{-\ii 2\omega_0 t},\end{aligned}$$ where $\alpha$ is a constant characterizing the SFG efficiency. These pulses pass through the bandpass filter followed by a detector. The bandpass filter narrows the light’s bandwidth and broadens its pulse duration. Assuming $g(t)$ denotes an envelope function of the input light, the effect of a bandpass filter is represented by convolution integral $(g*h)(t)$, where $h(t)$ is the Fourier transform of the transmission spectrum function of the bandpass filter. Especially when the bandwidth of the transmission spectrum is narrow enough, that is, time width $\tau_2$ of $h(t)$ is much larger than that of input envelope function $g(t)$, envelope function $g(t)$ is approximated as unnormalized delta function $a[g]\delta(t)$, where coefficient $a[g]$ is defined as $a[g]:=\int^\infty_{-\infty}\dd tg(t)$. Thus the field amplitude after the bandpass filter is described as $(g*h)(t)\approx a[g]h(t)$. Coherence length $l'_2=c\tau_2$ of the converted pulses is usually much longer than $l'_1$.
Owing to time-reversal symmetry, we expect to observe a similar interferogram as in the time-forward process. As calculated below, interference of the pump light occurs when $|z|<l'_1$, but the interference of the sum-frequency light occurs when $l'_1<|z|<l'_2$.
When $|z|< l_1'$, $f(t)$ and $f(t+z/c)$ almost overlap, $f(t)\approx f(t+z/c)$. Thus we obtain $$\begin{aligned}
E_3(t)\approx \frac{\alpha}{4}f(t)^2(1+\ee^{-\ii\omega_0z/c})^2\ee^{-\ii 2\omega_0 t}.\end{aligned}$$ The bandpass filter converts $f(t)^2$ into widely spread envelope function $a[f^2]h(t)$. The detected signal is described as $$\begin{aligned}
I(z)&\approx \int\dd t\left|\frac{\alpha}{4}a[f^2]h(t)(1+\ee^{-\ii\omega_0z/c})^2\ee^{-\ii 2\omega_0 t}\right|^2\no\\
&=|\alpha|^2\left[\frac{1+\cos(\omega_0z/c)}{2}\right]^2\int\dd t\left|a[f^2]h(t)\right|^2,\label{eq:2}\end{aligned}$$ which reproduces the same interference pattern as the one-photon interference expressed as Eq. (\[eq:3\]) in the time-forward process. This interference is the square of the white-light interference of the input laser pulses.
On the other hand, when $l_1'<|z|<l_2' $, $f(t)$ and $f(t+z/c)$ only slightly overlap, $f(t)f(t+z/c)\approx0$. Thus we obtain $$\begin{aligned}
E_3(t)\approx \frac{\alpha}{4}[f(t)^2+f(t+z/c)^2\ee^{-\ii2\omega_0z/c}]\ee^{-\ii 2\omega_0 t}.\end{aligned}$$ The bandpass filter converts $f(t)^2$ and $f(t+z/c)^2$ into widely spread envelope functions $a[f^2]h(t)$ and $a[f^2]h(t+z/c)$, respectively. If $l'_2\gg|z|$, $h(t)$ and $h(t+z/c)$ greatly overlap and are approximated as $h(t)\approx h(t+z/c)$. The detected signal is described as $$\begin{aligned}
I(z)&\approx \int\dd t\left|\frac{\alpha}{4}a[f^2]h(t)(1+\ee^{-\ii2\omega_0z/c})\ee^{-\ii 2\omega_0 t}\right|^2\no\\
&=\frac{|\alpha|^2}{4}\frac{1+\cos(2\omega_0z/c)}{2}\int\dd t\left|a[f^2]h(t)\right|^2,\end{aligned}$$ which reproduces the same interference pattern as the two-photon interference expressed by Eq. (\[eq:4\]) in the time-forward process. In this case, the white-light interference of the input pulsed light does not occur due to the large optical path difference of the interferometer. After the bandpass filter broadens the pulse widths of the sum-frequency light, these pulses interfere with each other. We call this *sum-frequency light interference*, which exhibits a classical analogue to two-photon phase super-resolution with perfect visibility. We can also see that the maximum intensity of the sum-frequency light interference fringes is a quarter of that of the white-light interference fringes.
As seen from the above discussion, the interferogram’s shape in the time-reversed process resembles that in the time-forward process. The difference between $l'_1$ and $l'_2$ is a classical analogue to the large difference between one- and two-photon coherence lengths $l_1$ and $l_2$.
Experiments and results {#sec:experiment}
=======================
![(color online). Time-reversed experimental setup for observing two-photon phase super-resolution in an unbalanced Michelson interferometer. A coherent laser pulse from a femtosecond fiber laser enters the Michelson interferometer. To observe the white-light interference fringes, optical-path difference $z$ of the interferometer is adjusted to about zero and the output light is detected by photodiode PD1. For observing the sum-frequency light interference, $z$ is adjusted to about 100$\upmu$m and the output light is converted into sum-frequency light by a BBO crystal. A grating and a slit, both of which function as a bandpass filter, pass the sum-frequency light within a narrow band (0.039nm) before the light is detected by a photodiode PD2.[]{data-label="fig:exp_int_detail"}](fig_2.eps){width="7cm"}
We demonstrate a time-reversed experiment for observing two-photon phase super-resolution. The experimental setup is shown in Fig. \[fig:exp\_int\_detail\]. We used a femtosecond fiber laser (Menlo Systems, T-Light 780) with a center wavelength of 782nm, a pulse duration of 74.5fs FWHM, and an average power of 54.1mW. The coherence length of the laser pulse was $l_1'=c\tau_1=$ 22.3$\upmu$m, where $\tau_1$ is the pulse duration of 74.5fs FWHM. The Michelson interferometer is composed of a 50/50 nonpolarizing beam splitter for ultrashort pulses (BS) and silver mirrors (M1 and M2). M1 can be translated by several $\upmu$m by a piezoelectric actuator. The output beam from the interferometer is introduced to the flipper mirror (FM). To observe the white-light interference, optical-path difference $z$ of the interferometer is adjusted to about zero. The output beam of the interferometer is reflected onto FM and detected by a photodiode (PD1; Thorlabs, SM05PD1A). On the other hand, for observing sum-frequency light interference, $z$ is adjusted to about 100$\upmu$m. FM is removed, and the beam is focused by a lens into a 1mm-length $\upbeta$-barium borate (BBO) crystal for a collinear type-I SFG. The sum-frequency beam is then collimated by another lens and filtered to pass a 0.039-nm bandwidth centered around 391nm by a 3,600lines/mm aluminium-coated diffraction grating followed by a slit. The optical power is measured by a GaP photodiode (PD2; Thorlabs, PDA25K).
![(color online). Measured interference fringes: (a) white-light and (b) sum-frequency light. From the fitted curves in solid lines, visibilities are estimated to be $99.1\%\pm0.2\%$ and $97.9\%\pm0.4\%$, respectively. Relative displacement $z'$ of M1 depends nonlinearly on piezo voltage $V$. Assuming that $z'$ is approximated by a quadratic function of $V$, we fitted $z'(V)$ to the measured white-light interference fringes (a). Calibrated function $z'(V)$ is also applied to the sum-frequency light interference fringes (b). Note that $z'$ is the relative displacement of M1 from where optical-path difference $z$ is about zero in (a) and about 100$\upmu$m in (b). []{data-label="fig:graph_int"}](fig_3.eps){width="6.5cm"}
Figure \[fig:graph\_int\] shows the measured interference fringes for (a) white-light and (b) sum-frequency light as functions of relative displacement $z'$ of M1. The period of the sum-frequency interference is half of the period of the white-light interference. This is the classical analogue to the two-photon phase super-resolution. The visibilities of the interference fringes were (a) $99.1\%\pm0.2\%$ and (b) $97.9\%\pm0.4\%$, respectively. The maximum optical power of the sum-frequency light interference signal was 2.8$\upmu$W, which corresponds to about $10^{13}$photons/s. This count rate is about $10^{11}$ times higher than the two-photon interference signal in the previous time-forward experiments for observing two-photon phase super-resolution in an unbalanced Michelson interferometer [@PhysRevLett.66.1142; @PhysRevLett.86.5620].
Next we measured an interferogram detected by PD2 for a wide range of $z$, which is shown in Fig. \[fig:4\]. This experiment confirms the difference between the coherence lengths of the white-light and sum-frequency light interference $l'_1$ and $l'_2$. In this measurement, M1 is moved by a DC servo motor instead of a piezoelectric actuator in the previous experiment. We also changed the bandwidth filtered by the grating and the slit to 0.093 nm. For $z$ near zero, we observed a light signal proportional to the square of the white-light interference signal. The measured coherence length of the white-light interference pattern was $23.3\pm0.4$$\upmu$m FWHM, which is in good agreement with theoretical coherence length $l_1'=$ 22.3$\upmu$m. On the other hand, when $|z|$ is larger than about 100$\upmu$m, the light signal exhibits two-photon phase super-resolution. The measured coherence length of the sum-frequency light interference was $510\pm20$$\upmu$m FWHM, which is about 22 times larger than that of the white-light interference. The theoretical coherence length is calculated to be $l'_2=(4\ln2/\pi)\lambda^2/\Delta\lambda=1.5$mm, where $\lambda=391$nm is the central wavelength of the sum-frequency light and $\Delta\lambda=0.093$nm is the bandwidth filtered by the grating and the slit. The measured maximum value of the white-light interference signal is 3.8 times larger than that of the sum-frequency light interference signal. The slight shortage compared with the theoretical value of 4 is due to a misalignment of the interferometer. This interferogram indicates that our experiment demonstrated a classical analogue to the large difference between the one- and two-photon coherence lengths of entangled photon pairs.
![(color online). Interferogram detected by PD2 for wide range of $z$. When $z$ is about zero, the light signal is proportional to the square of the white-light interference signal. On the other hand, when $|z|$ is larger than about 100$\upmu$m, the light signal exhibits two-photon phase super-resolution. The coherence length of the sum-frequency interference is about 22 times larger than that of the white-light interference. This difference of coherence lengths is a classical analogue to the large difference between one- and two-photon coherence lengths of entangled photon pairs. []{data-label="fig:4"}](fig_4.eps){width="6.5cm"}
Summary and Discussion {#sec:conclusion}
======================
We observed two-photon phase super-resolution in an unbalanced Michelson interferometer with classical unchirped laser pulses. We applied the time-reversal method to a conventional two-photon interference experiment using an unbalanced Michelson interferometer. The measured interferogram of the experiment exhibits sum-frequency light interference with about 22 times longer coherence length than that of the input laser light. It is a classical analogue to the large difference between the one- and two-photon coherence lengths of entangled photon pairs.
Kaltenbaek *et al.* [@PhysRevLett.102.243601] first observed a classical analogue to the large difference between one- and two-photon coherence lengths using chirped-pulse interferometry. They used a pulsed light source with an average power of 2.8W to observe the white-light and sum-frequency light interferences with visibilities of $87.1\%\pm0.2\%$ and $84.5\%\pm0.5\%$, respectively. The maximum optical power of the sum-frequency light interference was about 3.5$\upmu$W. In our experiment the visibilities of the white-light and sum-frequency light interference were $99.1\%\pm0.2\%$ and $97.9\%\pm0.4\%$, respectively. The maximum optical power of the sum-frequency light interference was 2.8$\upmu$W, which is comparable to that of the experiment by Kaltenbaek *et al.*, in spite of our low-power pulsed light source (average power 54.1mW). Such high visibility and high efficiency are due to the simplicity of our experimental setup.
Some previous investigations using coincidence counting for detecting correlated $N$ photons demonstrated $N$-photon phase super-resolution [@PhysRevLett.98.223601; @PhysRevA.81.063836]. Coincidence counting, however, cannot detect frequency-correlated photons. For this reason, these previous investigations using coincidence counting did not demonstrate the quantum optical phenomena induced by the frequency correlation of photons, such as the difference between one- and two-photon coherence lengths.
We also mention the relation between our experiments and optical lithography. Phase super-resolution using entangled photons has been applied to sub-Rayleigh resolution lithography [@PhysRevLett.85.2733; @PhysRevLett.87.013602], which is called quantum lithography. Sub-Rayleigh resolution lithography was also proposed and demonstrated by a classical optical setup with multiphoton absorption [@bentley2004nonlinear; @PhysRevLett.96.163603]. Pe’er *et al.* [@Peer2004] demonstrated sub-Rayleigh resolution lithography using two-photon absorption. Their experiment resembles ours except it used a Young-like interferometer and detected frequency-correlated photons by two-photon absorption, suggesting that some findings about the time-reversal method can be utilized for optical lithography.
Our study revealed that two-photon phase super-resolution can be realized in a simple classical system without chirped laser pulses. This simplification enabled us to achieve high-visibility interference with high efficiency. We expect this technique to open up new practical applications of quantum optical technologies.
This research was supported by the Global COE Program “Photonics and Electronics Science and Engineering” at Kyoto University and Grants-in-Aid for Scientific Research No. 22109004 and No. 25287101. One of the authors (S.T.) was supported by JSPS (Grant No. 224850).
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[(Version: 2016/01/19)]{}
[**Continuous-state branching processes**]{}
[**in Lévy random environments[^1]**]{}
[Hui He, Zenghu Li and Wei Xu[^2]]{}
[School of Mathematical Sciences, Beijing Normal University,]{}
[Beijing 100875, People’s Republic of China]{}
[E-mails: [email protected], [email protected], [email protected]]{}
*Abstract.* A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than $-1$. We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey’s condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcricital CBRE-process with immigration.
*Mathematics Subject Classification (2010)*: Primary 60J80, 60K37; Secondary 60H20, 60G51
*Key words and phrases*: continuous-state branching process, random environment, Lévy process, transition semigroup, backward stochastic equation, survival probability, immigration, ergodicity.
Introduction
============
Galton-Watson processes in random environments (GWRE-processes) were introduced by Smith (1968) and Smith and Wilkinson (1969) as extensions of classical Galton-Watson processes (GW-processes). Those extensions possess many interesting new properties such as phase transitions of survival probabilities. For instance, that different regimes for the survival probability arise in the subcritical regime. For recent results on the speed of decay of the survival probability, the reader may refer to Afanasy’ev et al. (2005, 2012), Guivarc¡¯h and Liu (2001) and Vatutin (2004) and the references therein.
Let $\{W(t): t\ge 0\}$ be a Brownian motion and $\{S(t): t\ge 0\}$ a Brownian motion with drift. We assume the two processes are independent. By the Yamada-Watanabe theorem, for any constants $c\ge 0$, $\sigma\ge 0$ and $b\in \mbb{R}$ there is a unique positive solution to the stochastic differential equation: \[eq1.1\] dX(t) = dW(t) - bX(t)dt + X(t)dS(t). The solution $\{X(t): t\ge 0\}$ is called a continuous-state branching diffusion in random environment (CBRE-diffusion). The environment here is determined by the process $\{S(t): t\ge 0\}$. It was proved in Kurtz (1978) that the CBRE-diffusion arises as the limit of a sequence of suitably rescaled GWRE-processes; see also Helland (1981). A diffusion approximation of the GWRE-process was actually conjectured by Keiding (1975). It turns out that the CBRE-diffusion is technically more tractable than the GWRE-process. In the particular case of $\sigma=0$, the CBRE-diffusion reduces to the well-known Feller branching diffusion, which belongs to an important class of positive Markov processes called continuous-state branching processes (CB-processes); see Feller (1951), Jiřina (1958) and Lamperti (1967a, 1967b).
In the work of Böinghoff and Hutzenthaler (2012), it was shown that the survival probability of the CBRE-diffusion can be represented explicitly in terms of an exponential functional of the environment process $\{S(t): t\ge 0\}$. Based on the representation, Böinghoff and Hutzenthaler (2012) gave an exact characterization for the decay rate of the survival probability of the CBRE-diffusion in the critical and subcritical cases. The results of Böinghoff and Hutzenthaler (2012) are more complete than the corresponding results for the GWRE-processes in the sense that they calculated the accurate limiting constants. In addition, they characterized the CBRE-process conditioned to never go extinct and established a backbone construction for the conditioned process. See also Hutzenthaler (2011) for some related results.
Continuous-state branching processes with immigration (CBI-processes), which generalize the CB-processes, were introduced by Kawazu and Watanabe (1971) as rescaling limits of Galton-Watson processes with immigration (GWI-processes); see also Aliev and Shchurenkov (1982) and Li (2006, 2011). Let $b\in \mbb{R}$ and $c\ge 0$ be given constants. Let $m(dz)$ be a Radon measure on $(0,\infty)$ satisfying $\int_0^\infty (z\land z^2)m(dz)< \infty$. Suppose that $\{W(t): t\ge 0\}$ is a Brownian motion, $\{\eta(t): t\ge 0\}$ is an increasing Lévy process with $\eta(0)=0$ and $\tilde{M}(ds,dz,du)$ is a compensated Poisson random measure on $(0,\infty)^3$ with intensity $dsm(dz)du$. We assume those three noses are independent of each other. By Theorems 5.1 and 5.2 in Dawson and Li (2006), there is a unique positive strong solution to \[eq1.2\] X(t) =X(0) - b\_0\^t X(s)ds + \_0\^tdW(s) + \_0\^t\_0\^\_0\^[X(s-)]{} z (ds,dz,du) + (t). It was shown in Dawson and Li (2006) that the solution $\{X(t): t\ge 0\}$ is a CBI-processes; see also Fu and Li (2010). The process $\{\eta(t): t\ge 0\}$ describes the inputs of the immigrants. Here and in the sequel, we understand $\int_a^b = \int_{(a,b]}$ and $\int_a^\infty = \int_{(a,\infty)}$ for any $a\le b\in \mbb{R}$.
A class of continuous-state branching processes with catastrophe was introduced by Bansaye et al. (2013), who pointed out that the processes can be identified as continuous-state branching processes in random environment (CBRE-processes) with the environment given by a Lévy process with bounded variation. Those authors gave a criticality classification of their CBRE-processes according to the long time behavior of the environmental Lévy process. They also characterized the Laplace exponent of the processes using a backward ordinary differential equation involving the environment process. For stable branching CBRE-processes, Bansaye et al. (2013) calculated explicitly the survival probability and characterized its decay rate in the critical and subcritical cases. In addition, they showed some interesting applications of their results to a cell infection model. The results of Bansaye et al. (2013) were extended in Palau and Pardo (2015a) to the case where the environment was given by a Brownian motion with drift.
The CBRE-processes studied in Bansaye et al. (2013) can be generalized to continuous-state branching processes with immigration in random environment (CBIRE-processes). Let $\{L(t): t\ge 0\}$ be a Lévy process with no jump less than $-1$ and assume it is independent of the three noises in (\[eq1.2\]). It is natural to define a CBIRE-process $\{Y(t): t\ge 0\}$ by the stochastic equation \[eq1.3\] Y(t) =Y(0) - b\_0\^t Y(s)ds + \_0\^tdW(s) + (t) + \_0\^t\_0\^\_0\^[Y(s-)]{} z (ds,dz,du) + \_0\^t Y(s-)dL(s). Here the impact of the environment is represented by the Lévy process $\{L(t): t\ge 0\}$. The existence and uniqueness of the positive strong solution to the above equation follow from the result of Dawson and Li (2012); see Section 2 for the details. In the very recent work of Palau and Pardo (2015b), a further generalization of the CBIRE-process was introduced by considering a competition mechanism. The authors gave a direct construction of their model by arguments similar to those in Dawson and Li (2012) and proved some results on the long time behavior of the process.
The purpose of this paper is to study the basic structures of the CBRE- and CBIRE-processes. In Section 2, we introduce two random cumulant semigroups, which are important tools in the study of those processes. The semigroups are defined in terms of a backward stochastic equations driven by a Lévy process. The existence of them follows from a general result in Li (2011) on Dawson-Watanabe superprocesses. In Section 3, we construct the CBRE-process by applying a theorem in Dawson and Li (2012) on stochastic equations driven by time-space noises. Then we give characterizations of the quenched and annealed transition probabilities of the CBRE-process. In Section 4, we show the CBRE-process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey’s condition. In that case, we give a characterization of the extinction probabilities by the solution of a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. Some of the results are extended in Section 5 to CBIRE-processes. In addition, we give a necessary and sufficient condition for the ergodicity of subcritical CBIRE-processes. Most of the results here are obtained or presented using stochastic equations driven by Lévy processes, which are more elegant than those in the classical discrete setting.
**Acknowledgements.** We thank Professors Yueyun Hu and Zhan Shi for helpful discussions on branching processes in random environments.
Random cumulant semigroups
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In this section, we introduce some random cumulant semigroups, which generalize that of a classical CB-process. Those semigroups are important tools in the study of the CBRE-process. Let $I\subset \mbb{R}$ be an interval and $\zeta = \{\zeta(t): t\in I\}$ a càdlàg function. Let $\phi$ be a *branching mechanism* given by \[eq2.1\] (z) = bz + cz\^2 + \_0\^(e\^[-uz]{}-1+uz) m(du), z0. where $b\in \mbb{R}$ and $c\ge 0$ are constants and $(z\land z^2)m(dz)$ is a finite measure on $(0,\infty)$. From Theorem 6.10 in Li (2011) it follows that, for any $t\in I$ and $\lambda\ge 0$, there is a unique positive solution $r\mapsto u^\zeta_{r,t}(\lambda)$ to the integral evolution equation \[eq2.2\] u\^\_[r,t]{}() = - \_r\^t e\^[(s)]{}(e\^[-(s)]{}u\^\_[s,t]{}())ds, rI(-, t\]. Moreover, there is an inhomogeneous transition semigroup $(P^\zeta_{r,t})_{t\ge r\in I}$ on $[0,\infty)$ defined by \[eq2.3\] \_[\[0,)]{} e\^[-y]{} P\^\_[r,t]{}(x,dy) = e\^[-xu\^\_[r,t]{}()]{}, 0. By a simple transformation, we can define another inhomogeneous transition semigroup $(Q^\zeta_{r,t})_{t\ge r\in I}$ on $[0,\infty)$ by \[eq2.4\] \_[\[0,)]{} e\^[-y]{} Q\^\_[r,t]{}(x,dy) = e\^[-xv\^\_[r,t]{}()]{}, 0, where \[eq2.5\] v\^\_[r,t]{}() = e\^[-(r)]{} u\^\_[r,t]{}(e\^[(t)]{}). The uniqueness of the solution to (\[eq2.2\]) implies that \[eq2.6\] u\^\_[r,t]{}() = u\^\_[r,s]{}u\^\_[s,t]{}(), 0, tsrI. There is a similar relation for $(v^\zeta_{r,t})_{t\ge r\in I}$. By Lebesgue’s theorem one can see $r\mapsto u^\zeta_{r,t}(\lambda)$ is also the unique positive continuous solution to the differential equation \[eq2.7\] u\^\_[r,t]{}() = e\^[(r)]{}(e\^[-(r)]{}u\^\_[r,t]{}()), rI(-,t\] with terminal condition $u^\zeta_{t,t}(\lambda) = \lambda$.
\[th2.1\] Let $(P^\zeta_{r,t})_{t\ge r\in I}$ be defined by (\[eq2.3\]). Then for any $x\ge 0$ and $t\ge r\in I$ we have \_[\[0,)]{} y P\^\_[r,t]{}(x,dy) = xe\^[-b(t-r)]{}.
By differentiating both sides of (\[eq2.2\]) and solving the resulted integral equation we obtain $(d/d\lambda)u^\zeta_{r,t}(0+) = e^{-b(t-r)}$. Then we get the desired equality by differentiating both sides of (\[eq2.3\]).
\[th2.2\] If $b\ge 0$, then $t\mapsto u^\zeta_{r,t}(\lambda)$ is decreasing on $I\cap [r,\infty)$ and $r\mapsto u^\zeta_{r,t}(\lambda)$ is increasing on $I\cap (-\infty, t]$.
From (\[eq2.3\]) we see that $\lambda\mapsto u^\zeta_{r,t}(\lambda)$ is increasing. Since $b\ge 0$, we have $\phi(z)\ge 0$ for every $z\ge 0$. Then (\[eq2.2\]) implies $u^\zeta_{r,t}(\lambda)\le \lambda$. By (\[eq2.6\]) we see $u^\zeta_{r,t}(\lambda)\le u^\zeta_{s,t}(\lambda)$ and $u^\zeta_{r,t}(\lambda)\le u^\zeta_{r,s}(\lambda)$ for $r\le s\le t\in I$.
When the function $\zeta$ is degenerate ($\zeta(t) = 0$ for all $t\in I$), both $(u^\zeta_{r,t})_{t\ge r\in I}$ and $(v^\zeta_{r,t})_{t\ge r\in I}$ reduce to the cumulant semigroup of a classical CB-process with branching mechanism $\phi$; see, e.g., Chapter 3 of Li (2011). In the general case, we may think of $(u^\zeta_{r,t})_{t\ge r\in I}$ as an *inhomogeneous cumulant semigroup* determined by the *time-dependent branching mechanism* $(s,z)\mapsto e^{\zeta(s)} \phi(e^{-\zeta(s)}z)$. The idea of the proof of Theorem 6.10 in Li (2011) is to reduce the construction of an inhomogeneous cumulant semigroup to that of a homogeneous one by some time-space processes. The transformation from $(u^\zeta_{r,t})_{t\ge r\in I}$ to $(v^\zeta_{r,t})_{t\ge r\in I}$ is a time-dependent variation of the one used in the proof of Theorem 6.1 in Li (2011).
We next consider some randomization of the inhomogeneous cumulant semigroups defined above. Let $(\Omega, \mcr{F},\mcr{F}_t,\mbf{P})$ be a filtered probability space satisfying the usual hypotheses. Let $a\in \mbb{R}$ and $\sigma\geq 0$ be given constants and $(1\wedge z^2)\nu(dz)$ a finite measure on $(0,\infty)$. Suppose that $\{B(t): t\ge 0\}$ is an $(\mcr{F}_t)$-Brownian motion and $N(ds,dz)$ is an $(\mcr{F}_t)$-Poisson random measure on $(0,\infty)\times \mbb{R}$ with intensity $ds\nu(dz)$. Let $\{\xi(t): t\ge 0\}$ be a $(\mcr{F}_t)$-Lévy process with the following Lévy-Itô decomposition: \[eq2.8\] (t) = (0) + at + B(t) + \_0\^t \_[\[-1,1\]]{} z (ds,dz) + \_0\^t \_[\[-1,1\]\^c]{} z N(ds,dz), where $[-1,1]^c = \mbb{R}\setminus [-1,1]$. Let $u^\xi_{r,t}(\lambda)$ and $v^\xi_{r,t}(\lambda)$ be defined by (\[eq2.2\]) and (\[eq2.5\]) with $\zeta=\xi$. From (\[eq2.2\]) we see that $r\mapsto v^\xi_{r,t}(\lambda)$ is the unique positive solution to \[eq2.9\] v\^\_[r,t]{}() = e\^[(t)-(r)]{}- \_r\^t e\^[(s)-(r)]{}(v\^\_[s,t]{}()) ds, 0rt. Let $\{L(t): t\ge 0\}$ be the $(\mcr{F}_t)$-Lévy process with Lévy-Itô decomposition: \[eq2.10\] L(t) =L(0) + t + B(t) + \_0\^t \_[\[-1,1\]]{} (e\^z-1)(ds,dz) + \_0\^t \_[\[-1,1\]\^c]{} (e\^z-1) N(ds,dz), where $L(0)=\xi(0)$ and \[eq2.11\] = a + + \_[\[-1,1\]]{} (e\^z-1-z) (dz). Clearly, the two processes $\{\xi(t): t\ge 0\}$ and $\{L(t): t\ge 0\}$ generate the same filtration. From (\[eq2.9\]) we see that the left-continuous process $\{v^\xi_{t-s,t}(\lambda): 0\le s\le t\}$ is progressively measurable with respect to the filtration generated by the Lévy process $\{L_t(s) := L(t-)-L((t-s)-): 0\le s\le t\}$.
\[th2.3\] For any $t\ge 0$ and $\lambda\ge 0$, the process $\{v^\xi_{r,t}(\lambda): 0\le r\le t\}$ is the pathwise unique positive solution to \[eq2.12\] v\^\_[r,t]{}() = - \_r\^t (v\^\_[s,t]{}())ds + \_r\^t v\^\_[s,t]{}()L(), 0rt, where the backward stochastic integral is defined by \_r\^t v\^\_[s,t]{}()L() = \_0\^[(t-r)-]{} v\^\_[t-s,t]{}()L\_t(ds).
Let $\xi_t(r) = \xi(t-) - \xi((t-r)-)$ and $B_t(r) = B(t) - B(t-r)$ for $0\le r\le t$. Let $N_t(ds,dz)$ be the Poisson random measure defined by N\_t(\[0,r\]B) = N(\[t-r,t\]B), 0rt, B(). From (\[eq2.8\]) we have \_t(r) = ar + B\_t(r) + \_0\^r\_[\[-1,1\]]{} z \_t(ds,dz) + \_0\^r \_[\[-1,1\]\^c]{} z N\_t(ds,dz). On the other hand, from (\[eq2.9\]) we have $f_t(r) := v^\xi_{(t-r)-,t}(\lambda) = e^{\xi_t(r)} F_t(r)$, where F\_t(r) = - \_0\^r e\^[-\_t(s)]{}(f\_t(s)) ds. By Itô’s formula, f\_t(r) =+ \_0\^r e\^[\_t(s-)]{}F\_t(s) \_t(ds) + \_0\^r e\^[\_t(s)]{} F\_t(s) ds + \_0\^r\_ e\^[\_t(s-)]{} F\_t(s)(e\^z-1-z) N\_t(ds,dz) + \_0\^r e\^[\_t(s)]{} F\_t(ds) =+ \_0\^r f\_t(s-)L\_t(ds) - \_0\^r (f\_t(s))ds. It follows that v\^\_[r,t]{}() =+ \_0\^[r-]{} f\_t(s-)L\_t(ds) - \_0\^r (f\_t(s))ds =+ \_[0]{}\^[(t-r)-]{} v\^\_[t-s,t]{}()L\_t(ds) - \_0\^[t-r]{} (v\^\_[t-s,t]{}())ds =+ \_r\^t v\^\_[s,t]{}()L() - \_r\^t (v\^\_[t-s,t]{}())ds. That proves the existence of the solution to (\[eq2.12\]). Conversely, assuming $r\mapsto v^\xi_{r,t}(\lambda)$ is a solution to (\[eq2.12\]), one can use similar calculations to see it also solves (\[eq2.9\]). Then the pathwise uniqueness for (\[eq2.12\]) is a consequence of that for (\[eq2.9\]).
Construction of CBRE-processes
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Let $(\Omega, \mcr{F},\mcr{F}_t,\mbf{P})$ be a filtered probability space satisfying the usual hypotheses. Let $\{\xi(t): t\ge 0\}$ and $\{L(t): t\ge 0\}$ be $(\mcr{F}_t)$-Lévy processes given as in Section 2. Let $b\in \mbb{R}$ and $c\geq 0$ be constants and $(z\wedge z^2)m(dz)$ a finite measure on $(0,\infty)$. Suppose that $\{W(t): t\ge 0\}$ is another $(\mcr{F}_t)$-Brownian motion and $M(ds,dz,du)$ is an $(\mcr{F}_t)$-Poisson random measure on $(0,\infty)^3$ with intensity $dsm(dz)du$. We assume both of those are independent of the Lévy process $\{L(t): t\ge 0\}$. Given a positive $\mcr{F}_0$-measurable random variable $X(0)$, we consider the following stochastic integral equation: \[eq3.1\] X(t) =X(0) - b\_0\^t X(s)ds + \_0\^tdW(s) + \_0\^t \_0\^\_0\^[X(s-)]{} z (ds,dz,du) + \_0\^t X(s-)dL(s), where $\tilde{M}(ds,dz,du) = M(ds,dz,du) - dsm(dz)du$.
\[th3.1\] There is a unique positive strong solution $\{X(t): t\ge 0\}$ to (\[eq3.1\]).
Let $E = \{1,2\}$ and $U_0 = A_0\cup B_0$, where $A_0 = \{1\}\times (0,\infty)^2$ and $B_0 = \{2\}\times (0,\infty)$. Let $\pi(du) = \delta_1(du) + \delta_2(du)$ for $u\in E$. Then $W(ds,du) := dW(s)\delta_1(du) + dB(s)\delta_2(du)$ is a Gaussian white noise on $(0,\infty)\times E$ with intensity $ds\pi(dz)$. Let $\mu_0(dy,dz,du) = \delta_1(dy)m(dz)du$ for $(y,z,u)\in A_0$ and $\mu_0(dy,dz) = \delta_2(dy)\nu(dz)$ for $(y,z)\in B_0$. Then $N_0(ds,dy,dz,du) := \delta_1(dy)M(ds,dz,du)$ is a Poisson random measure on $(0,\infty)\times A_0$ with intensity $ds\delta_1(dy)m(dz)du$ and $N_0(ds,dy,dz) := \delta_2(dy)N(ds,dz)$ is a Poisson random measure on $(0,\infty)\times B_0$ with intensity $ds\delta_2(dy)\nu(dz)$. Let $b(x) = (\beta-b)x$ for $x\in [0,\infty)$ and $\sigma(x,u) = \sqrt{2cx}1_{\{u=1\}} + \sigma x1_{\{u=2\}}$ for $(x,u)\in [0,\infty)\times E$. Let $g_0(x,y,z,u) = z1_{\{u\le x\}}$ for $(x,y,z,u)\in [0,\infty)\times A_0$ and $g_0(x,y,z) = x(e^z-1)$ for $(x,y,z)\in [0,\infty)\times B_0$. By Theorem 2.5 in Dawson and Li (2012), there is a unique positive strong solution to the stochastic equation X(t) =X(0) + \_0\^t b(X(s))ds + \_0\^t\_[A\_0]{} g\_0(X(s-),y,z,u)\_0(ds,dy,dz,du) + \_0\^t\_[E]{} (X(s),u)W(ds,du) + \_0\^t\_[B\_0]{} g\_0(X(s-),y,u)\_0(ds,dy,du). The above equation can be rewritten into X(t) =X(0) + \_0\^tdW(s) + \_0\^t \_0\^\_0\^[X(s-)]{} z (ds,dz,du) + (-b)\_0\^t X(s)ds + \_0\^t X(s)dB(s) + \_0\^t \_[\[-1,1\]]{} X(s-)(e\^z-1)(ds,dz). Since the process t\_0\^t \_[\[-1,1\]\^c]{} (e\^z-1) N(ds,dz) has at most a finite number of jumps in each bounded time interval, as in the proof of Proposition 2.2 in Fu and Li (2010), one can see that there is also a pathwise unique positive strong solution to \[eq3.2\] X(t) =X(0) + \_0\^tdW(s) + \_0\^t \_0\^\_0\^[X(s-)]{} z (ds,dz,du) + (-b)\_0\^t X(s)ds + \_0\^t X(s)dB(s) + \_0\^t \_[\[-1,1\]]{} X(s-)(e\^z-1) (ds,dz) + \_0\^t \_[\[-1,1\]\^c]{} X(s-)(e\^z-1) N(ds,dz). The above equation is just a reformulation of (\[eq3.1\]). Then we have the result of the theorem.
\[th3.2\] Since the stochastic equations in Section 2 of Dawson and Li (2012) are formulated for Gaussian and Poisson noises in abstract space, they are quite flexible for applications. For example, the proof of Theorem \[th3.1\] given above can be modified to more complex models with extra structures such as immigration, competition and so on.
We call the solution $\{X(t): t\ge 0\}$ to (\[eq3.1\]) a *CBRE-process*, which is a càdlàg strong Markov process. Here the random environment is provided by the Lévy process $\{L(t): t\ge 0\}$. By Itô’s formula one can see $\{X(t): t\ge 0\}$ has strong generator $A$ defined as follows: For $f\in C^2(\mbb{R}_+)$, \[eq3.3\] Af(x) =(-b) xf’(x) + cxf”(x) + x\_0\^m(dz) + x\^2f”(x) + \_[\[-1,1\]]{} \[f(xe\^z) - f(x) - x(e\^z-1)f’(x)\] (dz) + \_[\[-1,1\]\^c]{} \[f(xe\^z)-f(x)\] (dz).
\[th3.3\] Let $Z(t) = X(t)e^{-\xi(t)}$ for $t\ge 0$. Then we have \[eq3.4\] Z(t) =X(0) - b\_0\^t e\^[-(s)]{}X(s) ds + \_0\^t e\^[-(s)]{} dW(s) + \_0\^t\_0\^\_0\^[X(s-)]{} ze\^[-(s-)]{} (ds,dz,du). In particular, the process $\{Z(t): t\ge 0\}$ is a positive local martingale when $b=0$.
Let $f(x,y) = xe^{-y}$. Then $xf'_x(x,y) = -f'_y(x,y) = f(x,y)$ and $-xf''_{xy}(x,y) = f''_{yy}(x,y) = f(x,y)$. Observe that the Poison random measure $N(ds,dz)$ actually does not produce any jump of $t\mapsto Z(t)$. By (\[eq3.1\]) and Itô’s formula, Z(t) =X(0) - b\_0\^t f’\_x(X(s),(s))X(s)ds + \_0\^t f’\_x(X(s),(s)) dW(s) + \_0\^t\_0\^\_0\^[X(s-)]{} f’\_x(X(s-),(s-))z (ds,dz,du) + \_0\^t f’\_x(X(s),(s))X(s) ds + \_0\^t f’\_x(X(s),(s))X(s) dB((s) + \_0\^t\_[\[-1,1\]]{} f’\_x(X(s-),(s-))X(s-)(e\^z-1) (ds,dz) + \_0\^t\_[\[-1,1\]\^c]{} f’\_x(X(s-),(s-))X(s-)(e\^z-1) N(ds,dz) + a\_0\^t f’\_y(X(s),(s))ds + \_0\^t\_[\[-1,1\]]{} f’\_y(X(s-),(s-))z (ds,dz) + \_0\^t f’\_y(X(s),(s)) dB((s) + \_0\^t\_[\[-1,1\]\^c]{} f’\_y(X(s-),(s-))z N(ds,dz) + \^2\_0\^t ds + \_0\^t\_0\^\_0\^[X(s-)]{} \[f(X(s-)+z,(s)) - f(X(s-),(s)) - f’\_x(X(s-),(s))z\] M(ds,dz,du) + \_0\^t\_ \[f(X(s-)e\^z,(s-)+z) - f(X(s-),(s-)) - f’\_x(X(s-),(s-))X(s-)(e\^z-1) - f’\_y(X(s-),(s-))z\] N(ds,dz) =X(0) - b\_0\^t f’\_x(X(s),(s))X(s)ds + \_0\^t f’\_x(X(s),(s)) dW(s) + \_0\^t\_0\^\_0\^[X(s-)]{} f’\_x(X(s-),(s-))z (ds,dz,du) + \_0\^t f’\_x(X(s),(s))X(s) ds + \_0\^t f’\_x(X(s),(s))X(s) dB((s) + a\_0\^t f’\_y(X(s),(s))ds + \_0\^t f’\_y(X(s),(s)) dB((s) + \^2\_0\^t ds - \_0\^tds \_[\[-1,1\]]{}\[f’\_x(X(s),(s))X(s)(e\^z-1) + f’\_y(X(s),(s))z\](dz). By reorganizing the terms on the right-hand side we get the desired equality.
It is easy to see that the two Lévy processes $\{\xi(t): t\ge 0\}$ and $\{L(t): t\ge 0\}$ generate the same $\sigma$-algebra. Let $\mbf{P}^\xi$ denote the quenched law given $\{\xi(t): t\ge 0\}$ or $\{L(t): t\ge 0\}$.
\[th3.4\] Let $(P^\xi_{r,t})_{t\ge r\ge 0}$ and $(Q^\xi_{r,t})_{t\ge r\ge 0}$ be defined by (\[eq2.3\]) and (\[eq2.4\]), respectively, with $\zeta=\xi$. Then for any $\lambda\geq 0$ and $t\geq r\geq 0$ we have \[eq3.5\] \^= {-Z(r)u\^\_[r,t]{}()} = \_[\[0,)]{} e\^[-y]{} P\_[r,t]{}\^(Z(r),dy) and \[eq3.6\] \^ = {-X(r)v\^\_[r,t]{}()} = \_[\[0,)]{} e\^[-y]{} Q\_[r,t]{}\^(X(r),dy).
Fix $\lambda\geq 0$. For $t\ge r\ge 0$, let $H_t(r) = \exp\{-Z(r)u^\xi_{r,t}(\lambda)\}$. By (\[eq2.2\]) and Proposition \[th3.3\], given the environment $\{\xi(t): t\ge 0\}$, we can use Itô formula to see H\_t(t) =H\_t(r) - \_r\^t H\_t(s)Z(s)e\^[(s)]{}(e\^[-(s)]{}u\^\_[s,t]{}())ds - \_r\^t H\_t(s-)u\^\_[s,t]{}()dZ(s) + c\_r\^t H\_t(s)u\^\_[s,t]{}()\^2 e\^[-2(s)]{}X(s)ds + \_r\^t\_0\^\_0\^[X(s-)]{} H\_t(s-) M(ds,dz,du) =H\_t(r) - \_r\^t H\_t(s)X(s)(e\^[-(s)]{}u\^\_[s,t]{}())ds + b\_r\^t H\_t(s)u\^\_[s,t]{}()e\^[-(s)]{}X(s) ds - \_r\^t H\_t(s)u\^\_[s,t]{}()e\^[-(s)]{} dW(s) + c\_r\^t H\_t(s)e\^[-2(s)]{} u\^\_[s,t]{}()\^2 X(s)ds + \_r\^t\_0\^\_0\^[X(s-)]{} H\_t(s-) M(ds,dz,du) =H\_t(r) - \_r\^t H\_t(s-)u\^\_[s,t]{}()e\^[-(s)]{} dW(s) + \_r\^t\_0\^\_0\^[X(s-)]{} H\_t(s-) (ds,dz,du). Since $\{H_t(r): t\ge r\}$ is a bounded process, by taking the conditional expectation in both sides we get $\mbf{P}^\xi[H_t(t)|\mcr{F}_r] = H_t(r)$. That gives (\[eq3.5\]), and as a consequence we get (\[eq3.6\]).
\[th3.5\] If $\mbf{P}[Z(0)] = \mbf{P}[X(0)]< \infty$, then $\{e^{bt}Z(t): t\ge 0\}$ is a martingale.
Let $t\ge r\ge 0$ and let $F$ be a bounded random variable measurable with respect to the $\sigma$-algebra generated by $\mcr{F}_r\cup \sigma(\xi)$. By (\[eq3.5\]) and Proposition \[th2.1\], we have \[Fe\^[bt]{}Z(t)\] = {Fe\^[bt]{}\^} = \[Fe\^[bt]{}\_[\[0,)]{} y P\_[r,t]{}\^(Z(r),dy)\] = \[Fe\^[br]{}Z(r)\]. Then $\{e^{bt}Z(t): t\ge 0\}$ is a martingale.
By Theorem \[th3.4\] we see that $\{Z(t): t\ge 0\}$ and $\{X(t): t\ge 0\}$ are actually CB-processes under the quenched law with inhomogeneous cumulant semigroups $(u^\xi_{r,t})_{t\ge r\ge 0}$ and $(v^\xi_{r,t})_{t\ge r\ge 0}$, respectively. The next theorem gives a characterization of the transition semigroup of $\{X(t): t\ge 0\}$ under the annealed law.
\[th3.6\] The Markov process $\{X(t): t\ge 0\}$ has Feller transition semigroup $(Q_t)_{t\ge 0}$ defined by \[eq3.7\] \_[\[0,)]{} e\^[-y]{} Q\_t(x,dy) = \[e\^[-xv\^\_[0,t]{}()]{}\], 0.
By (\[eq2.4\]) one can see that (\[eq3.7\]) defines a probability kernel $Q_t(x,dy)$. In view of (\[eq3.6\]), for any bounded $\mcr{F}_r$-measurable random variable $F$ we have \[Fe\^[-X(t)]{}\] = \[F\^(e\^[-X(t)]{}|\_r)\] = \[F{-X(r)v\^\_[r,t]{}()}\]. The pathwise uniqueness of the solution to (\[eq2.12\]) implies that the random variable $v^\xi_{r,t}(\lambda)$ is measurable with respect to the $\sigma$-algebra generated by $\{L(s)-L(t): r\le s\le t\}$ and is identically distributed with $v^\xi_{0,t-r}(\lambda)$. It follows that \[Fe\^[-X(t)]{}\] = \[F\_[\[0,)]{} e\^[-y]{} Q\_[t-r]{}(X(r),dy)\]. Then $\{X(t): t\ge 0\}$ has transition semigroup $(Q_t)_{t\ge 0}$. The Feller property is immediate by (\[eq3.7\]).
Under the annealed law, the process $\{Z(t): t\ge 0\}$ usually does not satisfy the Markov property, but $\{(\xi(t),Z(t)): t\ge 0\}$ is a two-dimensional Markov process. Let $\mcr{F}_\infty = \sigma(\cup_{t\ge 0}\mcr{F}_t)$. By Corollary \[th3.5\], there is a probability measure $\tilde{\mbf{P}}$ on $(\Omega, \mcr{F}_\infty)$ so that $\tilde{\mbf{P}}(F) = \mbf{P}[Fe^{bt}Z(t)]$ for each bounded $\mcr{F}_t$-measurable random variable $F$. Let $\tilde{\mbf{P}}^\xi$ denote the conditional law under $\tilde{\mbf{P}}$ given the environment $\{\xi(t): t\ge 0\}$.
\[th3.7\] For any $\lambda\geq 0$ and $t\geq r\geq 0$, we have \[eq3.8\] \^ = {-X(r)v\^\_[r,t]{}() - \_r\^t\_0’(v\^\_[s,t]{}()) ds}, where $\phi_0'(z) = \phi'(z)-b$.
From (\[eq2.4\]) one can see that $\lambda\mapsto v^\xi_{r,t}(\lambda)$ is infinitely differentiable in $(0,\infty)$. By differentiating both sides of (\[eq2.2\]) we obtain u\^\_[r,t]{}() = 1 - \_r\^t ’(e\^[-(s)]{}u\^\_[s,t]{}()) u\^\_[s,t]{}()ds. Then we can solve the equation to get u\^\_[r,t]{}() = {-\_r\^t ’(e\^[-(s)]{}u\^\_[s,t]{}()) ds}. Let $F$ be a bounded random variable measurable with respect to the $\sigma$-algebra generated by $\mcr{F}_r\cup \sigma(\xi)$. From (\[eq3.5\]) it follows that \[Fe\^[-Z(t)]{}\] = \[F\^(e\^[-Z(t)]{}|\_r)\] = \[F{-Z(r)u\^\_[r,t]{}()}\]. By differentiating both sides in $\lambda>0$ we have \[Fe\^[-Z(t)]{}Z(t)\] = , and hence \[Fe\^[-X(t)]{}Z(t)\] = . It follows that \[Fe\^[-X(t)]{}\] = . Then we have (\[eq3.8\]). The extension of the equality to $\lambda\ge 0$ is immediate.
By Theorem \[th3.7\] one can show as in the proof of Theorem \[th3.6\] that $\{X(t): t\ge 0\}$ is a Markov process under $\tilde{\mbf{P}}$ with Feller transition semigroup $(\tilde{Q}_t)_{t\ge 0}$ defined by \[eq3.9\] \_[\[0,)]{} e\^[-y]{} \_t(x,dy) = , 0. This is a special case of a larger class of transition semigroups to be given in Section 5.
Survival and extinction probabilities
=====================================
In this section, we assume $X(0)=x> 0$ is a deterministic constant for simplicity. Let $\mbf{P}$ or $\mbf{P}_x$ denote the annealed law and $\mbf{P}^\xi$ or $\mbf{P}^\xi_x$ the quenched law given $\{\xi(t): t\ge 0\}$. Let $\tau_0 = \inf\{t\ge 0: X(t)=Z(t)=0\}$ denote the *extinction time* of the CBRE-process. From (\[eq2.4\]) one can see that $v^\xi_{0,t}(\lambda)$ is increasing in $\lambda\ge 0$. For $t>0$ let $\bar{v}^\xi_{0,t} := \lim_{\lambda\to \infty} v^\xi_{0,t}(\lambda) \in[0,\infty]$. Then \[eq5.1\] |[u]{}\^\_[0,t]{} := \_u\^\_[0,t]{}() = \_v\^\_[0,t]{}(e\^[-(t)]{}) = |[v]{}\^\_[0,t]{}. By (\[eq3.5\]) and (\[eq3.6\]) we have the following characterizations of the extinction probabilities: \[eq5.2\] \^\_x(\_0t) = \^\_x(Z(t)=0) = \^\_x(X(t)=0) = e\^[-x|[u]{}\^\_[0,t]{}]{} = e\^[-x|[v]{}\^\_[0,t]{}]{} and \[eq5.3\] \_x(\_0t) = \_x(Z(t)=0) = \_x(X(t)=0) = (e\^[-x|[u]{}\^\_[0,t]{}]{}) = (e\^[-x|[v]{}\^\_[0,t]{}]{}). We say the branching mechanism $\phi$ satisfies *Grey’s condition* if \[eq5.4\] \_1\^(z)\^[-1]{}dz< .
\[th5.1\] The following statements are equivalent:
$\phi$ satisfies Grey’s condition;
$\mbf{P}_x({Z(t)=0}) = \mbf{P}_x({X(t)=0})> 0$ for some and hence all $t> 0$;
$\mbf{P}(\bar{u}^\xi_{0,t}< \infty) = \mbf{P}(\bar{v}^\xi_{0,t}< \infty)> 0$ for some and hence all $t> 0$;
$\mbf{P}(\bar{u}^\xi_{0,t}< \infty) = \mbf{P}(\bar{v}^\xi_{0,t}< \infty) = 1$ for some and hence all $t> 0$.
From (\[eq5.3\]) we see (2)$\Leftrightarrow$(3)$\Leftarrow$(4). Then we only need to show (3)$\Rightarrow$(1)$\Rightarrow$(4). From (\[eq2.7\]) we have \[eq5.5\] t = \_0\^t du\^\_[s,t]{}(). Suppose that $\mbf{P}(\bar{v}^\xi_{0,t}< \infty) = \mbf{P}(\bar{u}^\xi_{0,t}< \infty)> 0$ for some $t> 0$. Choose the constants $0< M_1< M_2< \infty$ so that the event $A := \{\bar{u}^\xi_{0,t}< \infty\}\cap \{M_1\leq e^{\xi(s)}\leq M_2$ for $s\in [0,t]\}$ has strictly positive probability. Since $z\mapsto \phi(z)$ is increasing, by (\[eq5.5\]) we have on $A$ that t \_0\^t du\^\_[s,t]{}() = \_[M\_1\^[-1]{}u\^\_[0,t]{}()]{}\^[M\_1\^[-1]{}]{} dz. By letting $\lambda\to \infty$ we have on $A$ that \_[M\_2|[u]{}\^\_[0,t]{}]{}\^ dzt. Then (\[eq5.4\]) holds. That proves (3)$\Rightarrow$(1). Now suppose that Grey’s condition (\[eq5.4\]) is satisfied. Fix any $t>0$. Choose sufficiently large $n\ge 1$ so that the event $\Omega_n := \{1/n\leq e^{\xi(s)}\leq n$ for $s\in [0,t]\}$ has strictly positive probability. By (\[eq5.5\]), on the event $\Omega_n$ we have t \_0\^t du\^\_[s,t]{}() = n\^2\_[n\^[-1]{}u\^\_[0,t]{}()]{}\^[n\^[-1]{}]{} dz, which implies n\^2\_[n\^[-1]{}|[u]{}\^\_[0,t]{}]{}\^ dzt. It follows that $\bar{u}^\xi_{0,t} = \bar{v}^\xi_{0,t}< \infty$ on $\Omega_n$. Since $\mbf{P}(\cup_{n=1}^\infty \Omega_n) = 1$, we have $\mbf{P}(\bar{v}^\xi_{0,t}< \infty) = \mbf{P}(\bar{u}^\xi_{0,t}< \infty) = 1$.
\[th5.2\] Under Grey’s condition, for any $t>0$, the function $r\mapsto u(r) := \bar{u}^\xi_{r,t} = \bar{v}^\xi_{r,t}$ on $[0,t)$ is the minimal positive continuous solution to \[eq5.6\] u(r) = e\^[(r)]{}(e\^[-(r)]{}u(r)), r(0,t) with terminal condition $u(t-) = \infty$.
For any $t>s>r>0$ we have $\bar{u}^\xi_{r,t} = \lim_{\lambda\to \infty} u^\xi_{r,s}(u^\xi_{s,t}(\lambda)) = u^\xi_{r,s}(\bar{u}^\xi_{s,t})$. From (\[eq2.7\]) we see the differential equation in (\[eq5.6\]) is satisfied first for a.e. $r\in (0,s)$ and then for a.e. $r\in (0,t)$. Since $\bar{u}^\xi_{t-,t}\ge u^\xi_{t-,t}(\lambda) = \lambda$ for any $\lambda\ge0$, we have the terminal property $\bar{u}^\xi_{t-,t} = \infty$. Now suppose that $r\mapsto w(r)$ is another positive continuous solution to (\[eq5.6\]). By the uniqueness of the solution to (\[eq2.7\]) we have $w(r) = u_{r,s}^\xi(w(s))$ for $0\le r\le s<t$. For any $\lambda\ge 0$ we can choose $s\in (r,t)$ so that $w(s)\ge \lambda e^{|b|t}$. By Proposition \[th2.1\] and Jensen’s inequality one can see $u^\xi_{s,t}(\lambda)\le \lambda e^{-b(t-s)}\le \lambda e^{|b|t}$. From monotonicity of $\lambda\mapsto u^\xi_{r,s}(\lambda)$ we get $w(r) = u_{r,s}^\xi(w(s))\ge u^\xi_{r,s}(\lambda e^{|b|t})\ge u^\xi_{r,s}(u^\xi_{s,t}(\lambda)) = u^\xi_{r,t}(\lambda)$. Then $w(r)\ge \bar{u}^\xi_{r,t} = \lim_{\lambda\to \infty} u^\xi_{r,t}(\lambda)$.
\[th5.3\] Let $\bar{v}^\xi := \downarrow\lim_{t\to \infty} \bar{v}^\xi_{0,t}\in[0,\infty]$ and $\tau_0 := \inf\{t\geq 0: X(t)=0\}$. Then $$\mbf{P}_x(\tau_0< \infty)
=
\lim_{t\to \infty} \mbf{P}_x(\tau_0\le t)
=
\lim_{t\to \infty} \mbf{P}_x(X(t)=0)
=
\mbf{P}[e^{-x \bar{v}^\xi}].$$ Moreover, we have $\bar{v}^\xi<\infty$ if and only if Grey’s condition (\[eq5.4\]) holds.
From $\mbf{P}^\xi_x(\tau_0\le t) = \mbf{P}^\xi_x({X(t)=0}) = e^{-x\bar{v}^\xi_{0,t}}$ we see $t\mapsto \bar{v}^\xi_{0,t}$ is decreasing, so $\bar{v}^\xi$ is well defined. From (\[eq5.2\]) it follows that \_x(\_0<) = \_[t]{} \_x(\_0t) = \_[t]{} \_x(X(t)=0) = \_[t]{}\[e\^[-x|[v]{}\^\_[0,t]{}]{}\] = \[e\^[-x |[v]{}\^]{}\]. The second statement follows immediately from Theorem \[th5.1\].
\[th5.4\] Suppose that $\liminf_{t\to \infty} (\xi(t)-bt) = -\infty$ and Grey’s condition (\[eq5.4\]) holds. Then \_x(\_0=) = \_[t]{} \_x(\_0> t) = \_[t]{} \_x(X(t)>0) = 0.
Suppose that $p(x) := \mbf{P}_x(\tau_0=\infty) = 1 - \mbf{P}[e^{-x\bar{v}^\xi}]> 0$. Then we have $\mbf{P}(\bar{v}^\xi>0)> 0$, so $x\mapsto p(x) $ is strictly increasing. Under the assumption $\liminf_{t\to \infty} (\xi(t)-bt) = -\infty$, we have $\liminf_{t\to \infty}X(t) = 0$ as observed in Corollary 2 of Bansaye (2013). Then the stopping time $\sigma = \inf\{t>0: \xi(t)<x/2\}$ is a.s. finite. By Theorem \[th5.3\] and the strong Markov property, we have p(x) = \_x\[\_[X()]{}(\_0=)\] = \_x\[\_[x/2]{}(\_0=)\] = 1-\[e\^[-x|[v]{}\^/2]{}\] = p(x/2), which yields a contradiction. Then we must have $p(x) = \mbf{P}_x(\tau_0=\infty)=0$.
\[th5.5\] Under Grey’s condition, the transition semigroup $(Q_t)_{t\geq 0}$ defined by (\[eq3.7\]) has the strong Feller property.
We here need a construction of the CBRE-process for all initial values. Let $W(ds,du)$ be a time-space Gaussian white noise on $(0,\infty)^2$ with intensity $dsdu$. By a modification of the proof of Theorem \[th3.1\], one can see for each $x\ge 0$ there is a unique positive strong solution to the stochastic equation Y\_t(x)=x - b\_0\^t Y\_s(x) ds + \_0\^t\_0\^[Y\_[s-]{}(x)]{}W(ds,du) + \_0\^t\_0\^\_0\^[Y\_[s-]{}(x)]{}z(ds,dz,du)+ \_0\^t Y\_[s-]{}(x)L(ds). Clearly, the process $\{Y_t(x): t\ge 0\}$ is equivalent to the solution $\{X(t): t\ge 0\}$ to (\[eq3.1\]) with $X(0)=x$. As in the proof of Theorem 3.2 in Dawson and Li (2012), one can show that, for $y\ge x\ge 0$ and $t\ge0$ we have $Y_t(y)\ge Y_t(x)$ and $\{Y_t(y)-Y_t(x): t\ge 0\}$ is equivalent to the solution $\{X(t): t\ge 0\}$ to (\[eq3.1\]) with $X(0)=y-x$. Now let $f\in b\mathscr{B}(\mathbb{R}_+)$ satisfy $\|f\|_{\infty}\leq 1$. Let $T(x,y) = \inf\{t\geq 0: Y_t(x) = Y_t(y)\} = \inf\{t\geq 0: Y_t(y) - Y_t(x) = 0\}$. By (\[eq5.2\]) we have |Q\_tf(x)-Q\_tf(y)| \[|f(Y\_t(x))-f(Y\_t(y))|\_[{T(x,y)>t}]{}\] 2(T(x,y)t) = 2\[e\^[-(y-x)|[v]{}\^\_[0,t]{}]{}\]. The right-hand side tends to zero as $|x-y|\to 0$. That proves the strong Feller property of $(Q_t)_{t\geq 0}$.
In view of the result of Corollary \[th5.4\], one may naturally expect a characterization of the decay rate of the survival probability $\mbf{P}_x(\tau_0> t)$ as $t\to \infty$. This problem for the CBRE-diffusion was studied by Böinghoff and Hutzenthaler (2012). More recently, Bansaye et al. (2013) studied the problem for CB-processes with catastrophes, which is actually a CBRE-process with stable branching and finite variation Lévy environment. Palau and Pardo (2015a) studied the same problem for a CBRE-process with stable branching in a random environment given by a Brownian motion with drift. The decay rate of the survival probability for a CBRE-process with stable branching and a general Lévy environment was studied in Li and Xu (2016). The strong Feller property of classical CBI-processes was proved in Li and Ma (2015).
CBIRE-processes
===============
In this section, we discuss the CBIRE-process defined by (\[eq1.3\]). Let $h\ge 0$ be a constant and $(1\land u)n(du)$ a finite measure on $(0,\infty)$. Suppose that $(\Omega, \mcr{F},\mcr{F}_t,\mbf{P})$ is a filtered probability space satisfying the usual hypotheses. Let $\{W(t): t\ge 0\}$, $\{L(t): t\ge 0\}$ and $M(ds,dz,du)$ be as before. In addition, let $\{\eta(t): t\ge 0\}$ be an increasing $(\mcr{F}_t)$-Lévy process with \[eq6.1\] (e\^[-(t)]{}) = e\^[-t()]{}, 0, where \[eq6.2\] () = h+ \_0\^(1-e\^[-u]{})n(du). We assume that all those noises are independent of each other. The construction and basic properties of the CBIRE-process are provided by the following results. We here omit their proofs since they follow by modifications of the arguments in Sections 3 and 4.
\[th6.1\] For any positive $\mcr{F}_0$-measurable random variable $Y(0)$, there is a unique positive strong solution $\{Y(t): t\ge 0\}$ to (\[eq1.3\]).
\[th6.2\] Let $\{Y(t): t\ge 0\}$ be defined by (\[eq1.3\]) and $Z(t) = Y(t)\exp\{-\xi(t)\}$ for $t\ge 0$. Then we have Z(t) =Y(0) - b\_0\^t e\^[-(s)]{}Y(s) ds + \_0\^t e\^[-(s)]{} dW(s) + \_0\^t\_0\^\_0\^[Y(s-)]{} ze\^[-(s-)]{} (ds,dz,du) + \_0\^t e\^[-(s)]{} d(s).
\[th6.3\] Let $\mbf{P}^\xi$ be the conditional law given $\{\xi(t): t\ge 0\}$. Then for any $\lambda\geq 0$ and $t\geq r\geq 0$, we have \^ = {-Y(r)v\^\_[r,t]{}() - \_r\^t(v\^\_[s,t]{}())ds}.
\[th6.4\] The Markov process $\{Y(t): t\ge 0\}$ defined by (\[eq1.3\]) has Feller transition semigroup $(\bar{Q}_t)_{t\ge 0}$ defined by \[eq6.3\] \_[\[0,)]{} e\^[-y]{} |[Q]{}\_t(x,dy) = .
\[th6.5\] Under Grey’s condition, the transition semigroup $(\bar{Q}_t)_{t\geq 0}$ defined by (\[eq6.3\]) has the strong Feller property.
The transition semigroup $(\bar{Q}_t)_{t\ge 0}$ given by (\[eq6.3\]) generalizes the one defined by (\[eq3.9\]). We can give another useful characterization of this semigroup. For this purpose, let us consider an independent copy $\{L'(t): t\ge 0\}$ of the Lévy process $\{L(t): t\ge 0\}$ with $L'(0)=0$. We may assume the process has the following Lévy-Itô decomposition: L’(t) = t + B’(t) + \_0\^t \_[\[-1,1\]]{} (e\^z-1)’(ds,dz) + \_0\^t \_[\[-1,1\]\^c]{} (e\^z-1) N’(ds,dz), where $\{B'(t): t\ge 0\}$ is a Brownian motion and $N'(ds,dz)$ is a Poisson random measure on $(0,\infty)\times \mbb{R}$ with intensity $ds\nu(dz)$. Let $\{\xi'(t): t\ge 0\}$ be the Lévy process defined by ’(t) = at + B’(t) + \_0\^t \_[\[-1,1\]]{} z’(ds,dz) + \_0\^t \_[\[-1,1\]\^c]{} z N’(ds,dz). Set $L(t) = L(0)-L'(-t-)$ and $\xi(t) = \xi(0)-\xi'(-t-)$ for $t<0$. Then $\{L(t): -\infty< t< \infty\}$ and $\{\xi(t): -\infty< t< \infty\}$ are time homogeneous Lévy processes. We can extend (\[eq2.12\]) easily to $r\le t\in \mbb{R}$. In particular, for any $\lambda\ge 0$ there is a unique positive solution $r\mapsto v^\xi_{r,0}(\lambda)$ to v\^\_[r,0]{}() = -\_r\^0 (v\^\_[s,0]{}())ds + \_r\^0 v\^\_[s,0]{}() L(), r0. The result of Theorem \[th2.3\] can be extended to $r\le t\in \mbb{R}$. Then $r\mapsto v^\xi_{r,0}(\lambda)$ is also the unique positive solution to \[eq6.4\] v\^\_[r,0]{}() = e\^[(0)-(r)]{}- \_r\^0 e\^[(s)-(r)]{}(v\^\_[s,0]{}()) ds, r0. It follows that $r\mapsto u^\xi_{r,0}(\lambda) := e^{\xi(r)} v^\xi_{r,0}(\lambda)$ is the unique positive solution to \[eq6.5\] u\^\_[r,0]{}() = - \_r\^0 e\^[(s)]{}(e\^[-(s)]{}u\^\_[s,t]{}())ds, r0. By the time homogeneity of the Lévy process $\{\xi(t): -\infty< t< \infty\}$, we have \[eq6.6\] \_[\[0,)]{} e\^[-y]{} |[Q]{}\_t(x,dy) = . In the subcricital case, a necessary and sufficient condition for the ergodicity of the transition semigroup $(\bar{Q}_t)_{t\geq 0}$ defined by (\[eq6.3\]) is provided by the following theorem:
\[th6.6\] Suppose that $a_1 := \mbf{P}[\xi(1)]< b$. Then there is a probability measure $\mu$ on $[0,\infty)$ so that $\bar{Q}_t(x,\cdot)\to \mu$ weakly as $t\rightarrow \infty$ for every $x\geq 0$ if and only if \_1\^ (u)n(du)< . Under the above condition, we have \[eq6.7\] \_[\[0,)]{} e\^[-y]{} (dy) = .
Under the assumption, we may adjust the parameters in (\[eq1.3\]) so that $b> 0> a_1$. Then $u^\xi_{-t,0}(\lambda)\le e^{-bt}\lambda$ by Gronwall’s inequality, and hence $u^\xi_{-t,0}(\lambda)\to 0$ as $t\to \infty$. In view of (\[eq6.3\]), by applying Theorem 1.20 in Li (2011) and dominated convergence we conclude that $\bar{Q}_t(x,\cdot)$ converges to a probability measure $\mu$ as $t\rightarrow \infty$ for every $x\geq 0$ if and only if a.s. f(,) := \_[-]{}\^0 (v\^\_[s,0]{}()) ds = \_[-]{}\^0(e\^[-(s)]{}u\^\_[s,0]{}())ds< , 0. Clearly, $\mu$ is given by (\[eq6.7\]) if the above condition is satisfied. For any $z>0$, define $\tau_\lambda(z) = \sup\{r<0: u^\xi_{r,0}(\lambda)\leq z\}$. By (\[eq6.5\]) and a change of the variable, we have f(,) = \_[-]{}\^0 e\^[-(s)]{}du\^\_[s,0]{}() = \_0\^ e\^[-(\_(z))]{}dz. Since $\xi(t)\rightarrow \infty$ as $t\to -\infty$, we have a.s. $M := \sup_{s\le 0} e^{-\xi(s)}< \infty$. It is simple to see that $\phi(z) = bz + o(z)$ as $z\rightarrow 0$. Then $f(\xi,\lambda)< \infty$ if and only if \[eq6.8\] =\_0\^ n(du)\_0\^[u]{} dy< . For all $u>0$ we have $1-\exp\{-e^{-\xi(\tau_\lambda(y/u))}y\}\le My$. It follows that \_[(0,1\]]{} n(du)\_0\^[u]{} dy M\_[(0,1\]]{} u n(du)< . For $u>1$ we have $1 - \exp\{-e^{-\xi(\tau_\lambda(y/u))}y\}\to 1$ as $y\to \infty$. Then (\[eq6.8\]) holds if and only if \_[(1,)]{} n(du)\_0\^[u]{} dy = \_[(1,)]{} (u)n(du)< . That implies the desired result.
\[th6.7\] Suppose that $a_1 := \mbf{P}[\xi(1)]< b$. Let $(\tilde{Q}_t)_{t\ge0}$ be the transition semigroup defined by (\[eq3.9\]). Then there is a probability measure $\mu$ on $[0,\infty)$ so that $\tilde{Q}_t(x,\cdot)\to \mu$ weakly as $t\rightarrow \infty$ for every $x\geq 0$ if and only if \_1\^ u(u)m(du)< . Under the above condition, we have \_[\[0,)]{} e\^[-y]{} (dy) = .
One can see that (\[eq3.9\]) is the special form of (\[eq6.3\]) with $\psi = \phi_0'$. Then we get the results by Theorem \[th6.6\].
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[^1]: Supported by NSFC (No.11131003, No. 11371061 and No.11531001).
[^2]: Corresponding author.
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abstract: 'The population dynamics of bright and dark excitons confined in (In,Ga)As/GaAs quantum dots have been studied as function of magnetic field by two-color pump-probe spectroscopy at cryogenic temperatures. The dark excitons are stable on a few nanoseconds time scale unless the magnetic field induces a resonance with a bright exciton state. At these resonances quasi-elastic spin flips of either electron or hole occur which are initiated by hyperfine interaction with the lattice nuclei. From the relative strength of these resonances the hole-nuclei interaction is estimated to be six times weaker than the one of the electron.'
author:
- 'H. Kurtze$^1$, D.R. Yakovlev$^1$, D. Reuter$^2$, A.D. Wieck$^2$, and M. Bayer$^1$'
title: 'Hyperfine interaction of electron and hole spin with the nuclei in (In,Ga)As/GaAs quantum dots'
---
The spin dynamics of electrons and holes in quantum dots (QDs) has attracted considerable interest recently as it strongly differs from the dynamics observed in higher dimensional systems.[@Dyakonov] At cryogenic temperatures the interaction with the lattice nuclei has been revealed as limiting factor for the spin coherence in QDs while all other interactions become inefficient.[@Merkulov2002; @Khaetskii2002; @Kavokin2004; @Coish2004; @Golovach2004; @Petta2005; @Braun2005] Typically the hyperfine interaction is described by a Fermi-like contact Hamiltonian. Therefore it has been believed that only the electrons in the conduction band with s-wave Bloch functions are subject to this interaction. In contrast, the interaction of holes with their p-wave function character has been thought to be negligible, potentially leading to comparable or even longer coherence times than those of the electrons, despite of strong spin-orbit interaction in the valence band.[@OpticalOrientation; @Laurent2005; @BulaevLoss0507; @Tsitsishvili2003] This has been doubted recently in theoretical works, which show that the dipole-dipole interaction between the hole spin and the nuclei spins may be as strong as the electron hyperfine interaction.[@Coish2008; @Eble2009]
Very lately indications to that end have been found in experiment,[@Eble2009] but still the strength of this dipole-dipole interaction is under debate. It would be beneficial to have access to the nuclear interaction strength of electron and hole spins on the same sample under the same experimental conditions. This is typically not the case for different samples which have been prepared with n- or p-doping or whose charge occupation is changed by gates, as in these cases dot structure and carrier distribution, respectively, may vary.
Here we provide such a measurement by studying the spin relaxation from dark to bright ground state exciton in (In,Ga)As/GaAs quantum dots. The spin relaxation is weak unless a dark exciton state comes into resonance with a bright state, as in this case quasi-elastic scattering with the nuclei can occur. Two resonances are observed which can be attributed to an electron and a hole spin flip with the nuclei, respectively. This confirms that the hole spin hyperfine interaction is important, even though it is found to be about six times weaker than for the electron.
The experiments were performed by a two-color pump-probe technique using two synchronized, independently wavelength-tunable Ti:sapphire lasers. The lasers emit linearly polarized 1.5 ps pulses at a repetition rate of 75.6 MHz with a temporal jitter between the two pulse trains well below 1 ps. We used one laser as a pump, exciting carriers in the GaAs barrier, whereas the other probe laser was used to test the populations in the QD ground state. The temporal delay between pump and probe was adjusted by using a micrometer-precise mechanical delay line. The resulting time-resolved differential transmission signal (TRDT) is detected by a pair of balanced Si-photodiodes connected to a lock-in amplifier, by which we take the difference between the probe beam sent through the sample with and without pump action.
The sample was gown by molecular beam epitaxy and contains 10 layers of nominally undoped (In,Ga)As/GaAs QDs, separated from each other by 100-nm-wide barriers. The structure was exposed to postgrowth thermal annealing to shift the QD emission into the sensitivity range of Si detectors. From studies of QDs with similar processing parameters, we estimate the dot dimensions to be 5-6 nm height along growth direction ${\bf z}$ and approximately 30 nm diameter normal to ${\bf z}$.
In the experiment, the pump excitation density into the GaAs barrier at 1.55 eV photon energy was $I_0 =$10 W/cm$^2$. The probe density was chosen to be ten times weaker $I_0/10$ at an energy matching the QD ground state transition (e.g., 1.37 eV at $T=$10 K). The experiments were performed in magnetic fields $B
\leq 7$ T applied either in longitudinal Faraday configuration (parallel to sample growth direction and optical axis ${\bf z}$) or in transversal Voigt configuration (perpendicular to ${\bf
z}$). The fields were generated by an optical split-coil magnetocryostat in whose sample chamber the temperature was varied down to 5 K.
![(color online) (a) Pump induced time-integrated photoluminescence spectra recorded for excitation powers $I_0$ (solid) and 100$I_0$ (dotted) at $T=$10K in zero magnetic field. (b) Time-resolved DT trace for same conditions as in (a), log scale, $I_0$. (c) Contour plot of DT traces vs applied Voigt magnetic field, $T=$10 K, $I_0$.[]{data-label="fig:A1"}](spinrelaxfig1.EPS){width="\linewidth"}
The pump-excited QD-photoluminescence (PL) for $T=$10 K is shown in Fig. \[fig:A1\] (a). The solid line corresponds to excitation density $I_0$, whereas the dotted line was taken at 100$I_0$. At high densities, the PL spectrum shows several emission features reflecting the QD confined level structure. At the weak pump densities, at which also the TRDT traces were recorded, mostly ground state emission is detected, confirming that the average exciton occupation per QD is below $\sim$2 per excitation cycle.
A ground-state exciton is formed by an electron with spin $S_{e,z}=\pm$1/2 along the optical axis, and a hole with angular momentum $J_{h,z}=\pm3/2$, assuming a pure heavy-hole character (see below). The exchange interaction couples electron and hole spin, resulting in total angular momenta $M=J_{h,z}+S_{e,z}=\pm1$ and $\pm2$, corresponding to bright and dark excitons, respectively. These states are split from each other by the exchange interaction $\delta_0$. Earlier studies on similar QDs gave $\delta_0\sim$ 100 $\mu$eV.[@Yugova]
Fig. \[fig:A1\] (b) shows a typical DT transient at $T=$10 K. The non-resonant pump excitation at $t = 0$ excites carriers which quickly relax towards the dot ground state leading to a fast rise of the DT signal on a ps time scale. The subsequent time evolution shows two components decaying on two different time scales. The first component shows a fast drop with a 0.5 ns time constant, because of which we attribute it to bright excitons, as the same time is observed for the emission decay in time-resolved PL (not shown). The slow component decays on as time of about 8 ns, so that a fraction of this population is still present when the next pump pulse hits the sample. Therefore we associate this population with dark excitons, which are efficiently formed for excitation into GaAs where the photoexcited carriers and in particular the hole may undergo fast spin flips while relaxing towards the QD ground states. In our ensemble we therefore observe QDs with bright and dark exciton occupation.
As has been well established, the typical energy scale for the exciton fine structure given by the exchange interaction between electron and hole and the Zeeman interaction of these carriers in magnetic field is on the order of 100 $\mu$eV (see also above), which is considerably smaller than the thermal energy in experiment at $T=$10 K, equal to about a meV. Therefore the system is comparatively hot, i.e. spin-orbit interaction mediated carrier spin flips involving phonons may be considered as efficient.[@Tsitsishvili2003] However, the experiment reveals dark exciton lifetimes in the ns-range, showing inefficient scattering with phonons due to the significant mismatch of the fine structure energies and the energy of acoustic phonons with a wavelength comparable to the QD size.
The dark exciton assignment is confirmed by experiments in transverse magnetic field shown in Fig. \[fig:A1\] (c). In this configuration the carrier spins precess about the field, so that dark excitons are periodically converted into bright states and vice versa. In a quantum mechanical picture, the Zeeman interaction breaks the rotational symmetry about the growth direction, leading to a mixing of the exciton states which are bright and dark at $B=$0, so that the bright exciton oscillator strength is distributed among the four exciton states and all of them become optically active. This should lead to a strong reduction of the dark exciton lifetime, as confirmed by Fig. \[fig:A1\] (c). Shown are contour plots of the DT transients (along the vertical scale) as function of the applied Voigt field up to 7 T in steps of 0.2 T.
The dark exciton decay continuously decreases with increasing magnetic field, and for field strengths beyond $\sim$5 T has vanished completely. The upper scale of Fig. \[fig:A1\] (c) gives the ratio of the bright exciton radiative decay time of 0.5 ns to the electron spin precession time which is proportional to $B$. The dark exciton background has in effect vanished when the electron perform about 20 revolutions about $B$ within the decay time.
![Fine structure splitting in Faraday geometry for bright and dark exciton states. The arrows represent the exciton spin configuration where electron spin $S_{e,z}= +1/2$ and $-1/2$ are symbolized by the thin arrow pointing up and down, respectively. The thick arrow gives the analogous orientations of the hole spin $J_{h,z}=\pm 3/2$.[]{data-label="fig:A2"}](spinrelaxfig2.EPS){width="\linewidth"}
Here we are interested in the evolution of the dark exciton background when a Faraday field is applied. The exciton fine structure is described then by the effective spin Hamiltonian, following the notations in Ref. : $${\cal H}_X = \mu_B\left(g_{e,z}S_{e,z}+\frac{g_{h,z}}{3}J_{h,z}\right)B -
\frac{4}{3} \delta_0 S_{e,z} J_{h,z},$$ where $\mu_B$ is the Bohr magneton and the $g_{e,z}$ and $g_{h,z}$ are the electron and hole $g$-factors along ${\bf z}$. We also neglect the coupling of electron spin and hole spin normal to ${\bf z}$, as the resulting splittings of the bright and dark excitons are negligibly small for these QDs. Very similar QDs were investigated recently by pump-probe Faraday rotation spectroscopy, to determine the parameters of the exciton fine structure Hamiltonian with high accuracy. Besides the already used $\delta_0=100\pm10$ $\mu$eV, electron and hole $g$-factors of $g_{e,z}=-0.61$ and $g_{h,z}=-0.45$ were obtained.[@Yugova]
These parameters were used as input for calculating the exciton fine structure splitting in a Faraday magnetic field as shown in Fig. \[fig:A2\]. This field configuration does not break the rotational symmetry so that the exciton angular momentum $M$ remains a good quantum number throughout the whole scanned field range. The $B$-linear splitting of the bright and dark excitons leads to crossings in the magnetic field dispersion: The ${\left| -2 \right>}$-exciton, consisting of spin down hole and electron (represented by $\Downarrow$ and $\downarrow$, respectively) crosses with the ${\left| -1 \right>}$-exciton ($\Downarrow\uparrow$) around 3 T. Further, it also crosses with the ${\left| +1 \right>}$-exciton ($\Uparrow\downarrow$) at approximately 4 T.
![(color online) (a) Contour plot of DT traces for different Faraday magnetic field strengths; $T=$10 K, $I_0$. (b) Dots: DT values from (a), averaged over a negative delay time interval before pump arrival vs magnetic field. Solid line is a fit to data by two Gaussians for each resonance, individual Gaussians are shown by dashed lines. The values outside of resonance have been set to zero by baseline subtraction. The inset shows accordant values for $T=$5 K.[]{data-label="fig:A3"}](spinrelaxfig3.EPS){width="\linewidth"}
Figure \[fig:A3\] (a) shows DT transients as function of the magnetic field, applied in Faraday-configuration. At low fields $<$1 T the signal shows the two component behavior already discussed in relation with Fig. \[fig:A1\]. After pump action the fast bright population decay is followed by the significantly slower dark exciton decay. Above 1 T, however, the slow decay becomes drastically shortened with increasing field up to 3 T, while for even higher fields the decay time increases again reaching times almost as long as at zero field. Apparently the decay dynamics of the dark excitons shows a resonance around 3 T: at $B=$0 the estimated dark exciton decay time is 8 ns which is reduced to $\sim$5 ns in resonance and then increases again to $\sim$8 ns. The behavior of the bright exciton decay time is reversed with the dark exciton as it increases from $\sim$0.5 ns at $B=$0 to $\sim$0.6 ns at $B=$3 T and then drops again to 0.5 ns for high fields. Note that some long lived background remains even at 3 T because of the occupation of the ${\left| +2 \right>}$ dark exciton.
The resonance exactly occurs at the field strength at which the crossing of the ${\left| -2 \right>}$ exciton with the ${\left| -1 \right>}$ exciton occurs. However, the resonance behavior is not symmetric with respect to the resonance field but features an asymmetry towards higher $B$. This asymmetry may indicate that also the resonance of the ${\left| -2 \right>}$ exciton with the ${\left| +1 \right>}$ exciton leads to a shortening of the dark exciton lifetime. In combination with the reversed behavior of the magnetic field dependence of the decay of the two exciton components, the observation of field resonant shortening of the dark exciton background can be traced to exchange processes with the bright exciton states, for which spin-flips are required.
To analyse this in more detail, Fig. \[fig:A3\] (b) shows the mean value of the DT traces recorded at negative delay $t<$0 before pump pulse action. Each DT trace was normalized by its maximum; the resulting dataset was baseline substracted. By doing so, values below zero indicate a field induced reduction of the dark exciton population. Besides the main resonance at 2.9 T, clearly a shoulder is observed towards higher magnetic fields, supporting that indeed two field resonances occur. To determine the second resonance field, we have fitted the data with a superposition of two Gaussians with the same halfwidth, resulting to 1.56 T. The Gaussian form is justified as the broadening of the resonances is inhomogeneity related. The data can be well described by this fit as shown by the solid red line. The two dashed curves give the individual resonances. The field position of 4.7 T for the weak resonance is in accord with the crossing point of the ${\left| -2 \right>}$ and ${\left| +1 \right>}$ excitons in Fig. \[fig:A2\]. The difference between the resonance fields of $\sim$0.7 T can be easily achieved in the magnetic field dependent fine structure splitting by varying the $g$-factors by less than 0.1 which is reasonable within the sample imminent variations. Additionally, these values show no significant variations within cryogenic low temperatures, as demonstrated by the inset in Fig. \[fig:A3\] (b) taken at $T=$5 K.
Let us consider more in detail what spin flip scattering events are involved in the resonances. The dominant resonance involves the ${\left| -2 \right>}$ and ${\left| -1 \right>}$ excitons, which can be transferred into each other by an electron flip. We have already discussed that spin-orbit induced flips are generally negligible due to the lack of phonon states which in addition have zero density of states in the resonance case with zero energy difference between two states.[@phononscattering] The quasi-elastic flip can therefore be initiated only by the hyperfine interaction of the electron spin ${\bf S}_e$ with the nuclei ${\bf I}_i$, described by: $$\begin{aligned}
{\cal H}_e = \sum_i A_i \left| \psi \left( {\bf r}_{ei} \right) \right|^2 {\bf
S}_e {\bf I}_i, \label{eqn0}\end{aligned}$$ where the sum goes over all nuclei in the QD electron localization volume. The interaction strength of the electron with the nucleus is determined by the hyperfine constant $A_i$ specific for each nuclear species in the dot and the electron density $\left| \psi \left( {\bf r}_{ei} \right) \right|^2$ at the nuclear site ${\bf r}_{ei}$.
The second weaker resonance involves the ${\left| -2 \right>}$ and ${\left| +1 \right>}$ excitons, which can be transferred into each other by a hole spin flip only. Due to the quasi-elastic scattering in the resonance again this flip-process can be initiated by the nuclei only, demonstrating the importance of the hole-nuclei interaction. The only relevant mechanism in this case would be a dipole-dipole interaction which, however, cannot convert the $\pm3/2$ heavy-hole states into each other due to the mismatch of angular momentum exchange. On the other hand, we know that in the studied QDs the in-plane hole $g$-factor differs considerably from zero ($g_{h,\perp} = 0.15$),[@Yugova] showing that the hole ground state contains significant admixture of light-hole states with $J_{h,z}=\pm1/2$. Due to this admixture, the hole-nuclei flip-flops become possible. The underlying Hamiltonian is: $${\cal H}_h=\sum_{i} C_i \left| \Psi \left( {\bf r}_{hi} \right) \right|^2
\left( c_i \left( I_x^i J_{h,x} + I_y^i J_{h,y} \right) + I_{z}^i J_{h,z}
\right),$$ where the sum goes again over all dot nuclei, the $\Psi$ is the hole envelope wave function, and the $C_i$ are the corresponding interaction constants. The $c_i$ measure the light-hole content of the valence band ground state. Any flip-flop processes can only be initiated by the first two terms on the right-hand side. Thus, if $c_i=0$ for a pure heavy-hole state, such processes would be suppressed.
Despite of recent observations that this dipolar coupling indeed plays an important role[@Eble2009] for the hole spin dynamics, its strength relative to the electron hyperfine interaction is still under heavy debate. From our resonances we obtain an estimate of this relative strength within the same sample: By comparing the enclosed areas of the two fit curves, we estimate that the hole-nuclear interaction is about six times weaker than that of the electrons, but still considerable, contrasted with the previous claims that it can be neglected.
We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG 1549/10-1).
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abstract: 'Generalized Yang-Mills theories are constructed, that can use fields other than vector as gauge fields. Their geometric interpretation is studied. An application to the Glashow-Weinberg-Salam model is briefly review, and some related mathematical and physical considerations are made.'
author:
- |
M. Chaves\
Escuela de Física\
Universidad de Costa Rica\
San José, Costa Rica
title: |
Some mathematical considerations about\
generalized Yang-Mills theories
---
\#1
Introduction
============
Yang-Mills theories have been tremendously useful in high energy physics, where they have served to successfully model the electroweak and strong interactions. The fundamental idea of a Yang-Mills theory is that its mathematical expression must be invariant under a local compact Lie group of transformations. By local here it is meant that the group element varies with the point in Minkowski space that is being considered. In a typical physical theory the quantum fields often appear differentiated with respect to space or time. This means that, when the field transforms under the local Lie group, the differentiation operator is going to act on the transforming group element as well as on the field itself, so there is no invariance of this term under the Lie group. Invariance is reestablished substituting the differentiation operators by covariant derivatives. A covariant derivative is the sum of the differential operator and the Yang-Mills field. These fields are required to transform by adding terms that precisely cancel the extra terms brought in by the differentiation operators. Such a transformation is called a gauge transformation and the theory is left unmodified by it. Fields that perform this kind of service are generically called gauge fields.
Since the differentiation of a scalar field $\partial _\mu \varphi $, $%
\partial _\mu \equiv \partial /\partial x_\mu $, $\mu =0,1,2,3$, results in a four-vector, the gauge fields have also been taken to be four-vectors. In this article we are going to review[@max-herb] how it is possible to generalize the concept of a covariant derivative in a Yang-Mills theory, so that fields other than vector can be used as gauge fields.[@others] The idea of putting the leptons in a triplet and using the graded group $SU(2/1)$ goes back to Ne’eman and Fairlie.[@old] The Lie group $SU(3)$ almost has the right quantum numbers to embed the Glashow-Weinberg-Salam (GWS) Model in it, but not quite. The right group seems to be $U(3)$ in a special representation. An extra boson appears, but it automatically uncouples from the rest of the particles.[@max-herb] However, the emphasis of this paper is on the mathematical aspect, rather than on high energy applications.
The kinetic energy of a vector field transforming spinorially
=============================================================
The quantum electrodynamics Lagrangian is $${\cal L}_{QED}=\bar{\psi}iD\!\!\!\!/\psi -\frac 14F^{\mu \nu }F_{\mu \nu }
\label{QED}$$ where $D_\mu \equiv \partial _\mu +ieA_\mu $ and $F_{\mu \nu }\equiv
\partial _\mu A_\nu -\partial _\nu A_\mu =-ie^{-1}[D_\mu ,D_\nu ]$, and where we are using a metric $\eta _{\mu \nu }=\limfunc{diag}(1,-1,-1,-1).$ This Lagrangian is invariant under a local transformation group based on the $U(1)$ Lie group. If $U=e^{-i\alpha (x)}$ is an element of this group, then the electrically charged fermion field transforms as $\psi \rightarrow U\psi
.$ The vector field is required to obey the gauge transformation law $$A_\mu \rightarrow A_\mu +e^{-1}(\partial _\mu \alpha )\,{,} \label{vec_1}$$ so that, recalling the definition of $D_\mu $, the covariant derivative must transform as $$D_\mu \rightarrow UD_\mu U^{-1}\,{.} \label{derivative}$$
In the previous equation the derivative that is part of the covariant derivative is acting indefinitely to the right. We call such operators [*unrestrained,*]{} while the ones that act only on the immediately following object, such as the partial in (\[vec\_1\]), we call [*restrained*]{}, and use a parenthesis to emphasize that the action of the differentiation operator does not extend any further[*.*]{} We do not find it admissible to have unrestrained operators in a Lagrangian, because, first, they are not gauge invariant, and, second, what they could physically or mathematically mean is not clear. While the operators $D_\mu $ and $D_\mu D_\nu $ are unrestrained, the operator $[D_\mu ,D_\nu ]$ is restrained, and it is entirely appropriate that the kinetic energy of a vector boson can be constructed using this commutator. The way the commutator becomes restrained is as follows: $$\lbrack D_\mu ,D_\nu ]f=\partial _\mu A_\nu f-A_\nu \partial _\mu f-\partial
_\nu A_\mu f+A_\mu \partial _\nu f=(\partial _\mu A_\nu )f-(\partial _\nu
A_\mu )f, \label{Leibnitz}$$ where $f=f(x)$ is some differentiable function and it is seen how four unrestrained operators result in two restrained ones, thanks to Leibnitz’ rule.
Let $S$ be an element of the spinor representation of the Lorentz group, so that, if $\psi $ is a spinor, then it transforms as $\psi \rightarrow S\psi
. $ Then, due to the homomorphism that exists between the vector and spinor representations of the Lorentz group, we have that $A\!\!\!/\rightarrow
SA\!\!\!/S^{-1}$. It was this homomorphism that allowed Dirac to write a spinorial equation that included the vector electromagnetic field. The first step in our road to a generalization of the covariant derivative will be to rewrite the Lagrangian (\[QED\]) using the spinorial representation. To this effect we have the following
[**Theorem.** ]{}[*Let* ]{}$D_\mu =\partial _\mu +B_\mu $[*, where* ]{}$B_\mu $[* is some vector field. Then:* ]{} $$(\partial _\mu B_\nu -\partial _\nu B_\mu )(\partial ^\mu B^\nu -\partial
^\nu B^\mu )=\frac 18\limfunc{Tr}^2D\!\!\!\!/^{\,2}-\frac 12%
\limfunc{Tr}D\!\!\!\!/^{\,4}, \label{teo_1}$$ [*where the traces are to be taken over the Dirac matrices.*]{}
[*Proof.*]{} Notice the partials on the left of this identity are restrained, the ones on the right are not. To prove the theorem it is convenient to use the following trick, which makes the algebra manageable, in this and in more complicated cases to follow. Consider the differentiable operator $O\equiv \partial ^2+2B\cdot \partial +B^2$. Notice that it does not contain any contractions with Dirac matrices, so that $\limfunc{Tr}O=4O,$ $\limfunc{Tr}O\left( \partial \!\!\!/B\!\!\!\!/\right) =4O\left( \partial
\cdot B\right) $, etc. It is not difficult to see then that $%
D\!\!\!\!/=O+\left( \partial \!\!\!/B\!\!\!\!/\right) $, where the slashed partial is acting only on the succeeding slashed field. The trick is to use this form of $D\!\!\!\!/^{\,2}$ in the traces on the right of (\[teo\_1\]). With it one obtains $$\begin{aligned}
\frac 18\limfunc{Tr}^2\left[ O+\left( \partial
\!\!\!/B\!\!\!\!/\right) \right] -\frac 12\limfunc{Tr}\left[ O+\left(
\partial \!\!\!/B\!\!\!\!/\right) \right] ^2 &=&2\left( \partial \cdot
B\right) ^2-\frac 12\limfunc{Tr}\left[ \left( \partial
\!\!\!/B\!\!\!\!/\right) \left( \partial \!\!\!/B\!\!\!\!/\right) \right]
\nonumber \\
&=&(\partial _\mu B_\nu -\partial _\nu B_\mu )(\partial ^\mu B^\nu -\partial
^\nu B^\mu ) \label{proof}\end{aligned}$$ as we wished to demonstrate. The motivation for the additional trace term is the same as for taking the commutator of the covariant derivatives: to ensure that the differential operators be restrained.$\Box$
With the aid of the Theorem we can rewrite the QED Lagrangian in the form $${\cal L}_{QED}=\bar{\psi}iD\!\!\!\!/\psi +e^{-2}\left( \frac 1{32}\limfunc{Tr%
}^2D\!\!\!\!/^{\,2}-\frac 18\limfunc{Tr}D\!\!\!\!/^{\,4}\right)
\,{,} \label{QED'}$$ whose Lorentz invariance can be easily proven using $D\!\!\!\!/\rightarrow
SD\!\!\!\!/S^{-1}$ and the cyclic properties of the trace. As an example of the invariance, observe that $\limfunc{Tr}D\!\!\!\!/^{\,2}\rightarrow
\limfunc{Tr}SD\!\!\!\!/S^{-1}SD\!\!\!\!/S^{-1}=\limfunc{Tr}D\!\!\!\!/^{\,2}$.
A scalar field functioning as a gauge field
===========================================
We are going to construct an example of a theory that employs a scalar instead of a vector boson to maintain gauge invariance. To keep things simple we use $U(1)$ as our Lie group, as in the previous section, and so, again, (\[derivative\]) must hold. This time, however, we take the covariant derivative to be in the spinorial representation, as in (\[QED’\]), so it becomes possible to define it to be: $$D_\varphi =\partial \!\!\!/-e\gamma ^5\varphi \,{.} \label{der_phi}$$ We now require the gauge field to transform as $\gamma ^5\varphi \rightarrow
\gamma ^5\varphi -ie^{-1}\partial \!\!\!/\alpha $. These equations immediately assure us that $D_\varphi \rightarrow UD_\varphi U^{-1},$ and so Lagrangian $${\cal L}_\varphi =\bar{\psi}iD_\varphi \psi +e^{-2}\left( \frac 1{32}%
\limfunc{Tr}^2D_\varphi {}^2-\frac 18\limfunc{Tr}D_\varphi
{}^4\right) \label{lag_phi1}$$ is gauge invariant. The trace terms in this Lagrangian can be simplified algebraically and the Lagrangian written in the more traditional form $${\cal L}_\varphi =\bar{\psi}i\partial \!\!\!/\psi -e\bar{\psi}i\gamma
^5\varphi \psi +\frac 12(\partial _\mu \varphi )(\partial ^\mu \varphi )
\label{lag_phi2}$$ after a bit of algebra. This last calculation is similar to the one done last section, but with the $\gamma ^5$ taking the place of the $\gamma ^\mu $’s of that previous calculation.
Non-abelian Yang-Mills theory with mixed gauge fields
=====================================================
Consider a Lagrangian that transforms under a non-abelian local Lie group that has $N$ generators. The fermion or matter sector of the non-abelian Lagrangian has the form $\bar{\psi}iD\!\!\!\!/\psi $, where $D_\mu $ is a covariant derivative chosen to maintain gauge invariance. This term is invariant under the transformation $\psi \rightarrow U\psi $, where $U=U(x)$ is an element of the fundamental representation of the group. The covariant derivative is $D_\mu =\partial _\mu +A_\mu $, where $A_\mu =igA_\mu ^a(x)T^a$ is an element of the Lie algebra and $g$ is a coupling constant. We are assuming here that the set of matrices $\{T^a\}$ is a representation of the groups generators. Gauge invariance of the matter term is assured if $$A_\mu \rightarrow UA_\mu U^{-1}-(\partial _\mu U)U^{-1}\,{,}
\label{vec_2}$$ or, what is the same, $$A\!\!\!/\rightarrow UA\!\!\!/U^{-1}-(\partial \!\!\!/U)U^{-1} \label{vec_3}$$ We have already seen how scalar fields can function as gauge fields. Our aim in this section is to construct a non-abelian theory that uses both scalar and vector gauge fields. We proceed as follows. For every generator in the Lie group we choose one gauge field, it does not matter whether vector or scalar. As an example, suppose there are $N$ generators in the Lie group; we choose the first $N_V$ to be associated with an equal number of vector gauge fields and the last $N_S$ to be associated with an equal number of scalar gauge fields. Naturally $N_V+N_S=N$. Now we construct a covariant derivative $D$ by taking each one of the generators and multiplying it by one of its associated gauge fields and summing them together. The result is $$D\equiv \partial \!\!\!/+A\!\!\!/+\Phi \label{der_na}$$ where $$\begin{array}{ll}
A\!\!\!/=\gamma ^\mu A_\mu =ig\gamma ^\mu A_\mu ^aT^a, & a=1,\ldots ,N_V, \\
\Phi =\gamma ^5\varphi =-g\gamma ^5\varphi ^bT^b, & b=N_V+1,\ldots ,N.
\end{array}$$
Notice the difference between $A_\mu $ and $A_\mu ^a$, and between $\varphi $ and $\varphi ^b$. We take the gauge transformation for these fields to be $$A\!\!\!/+\Phi \rightarrow U(A\!\!\!/+\Phi )U^{-1}-(\partial \!\!\!/U)U^{-1},
\label{vec_4}$$ from which one can conclude that $D\rightarrow UDU^{-1}.$ The following Lagrangian is constructed based on the requirements that it should contain only matter fields and covariant derivatives, and that it possess both Lorentz and gauge invariance: $${\cal L}_{NA}=\bar{\psi}iD\psi +\frac 1{2g^2}\widetilde{\limfunc{Tr}}\left(
\frac 18\limfunc{Tr}^2D^{\,2}-\frac 12\limfunc{Tr}D^{\,4}\right)
\label{lag_na}$$ where the trace with the tilde is over the Lie group matrices and the one without it is over matrices of the spinorial representation of the Lorentz group. The additional factor of 1/2 that the traces of (\[lag\_na\]) have with respect to (\[teo\_1\]) comes from normalization $\widetilde{\limfunc{%
Tr}}T^aT^b=\frac 12\delta _{ab}$, the usual one in non-abelian gauge theories.
Although we have constructed this non-abelian Lagrangian based only on the requirements just mentioned, its expansion into component fields results in expressions that are traditional in Yang-Mills theories. The reader who wishes to make the expansion herself can substitute (\[der\_na\]) in (\[lag\_na\]), keeping in mind the derivatives are unrestrained, and aim first for the intermediate result $$\begin{aligned}
\frac 1{16}\limfunc{Tr}^2D^{\,2}-\frac 14\limfunc{Tr}D^{\,4}
&=&\left( (\partial \cdot A)+A^2\right) ^2-\limfunc{Tr}\left( (\partial
\!\!\!/A\!\!\!/)+A\!\!\!/A\!\!\!/\right) ^2 \nonumber \\
&&-\frac 14\limfunc{Tr}\left( (\partial \!\!\!/\Phi )+\{A\!\!\!/,\Phi
\}\right) ^2, \label{inter}\end{aligned}$$ where the curly brackets denote an anticommutator. (We recommend to use here the trick explained in section 2.) Notice in this expression that the differentiation operators are restrained, and that the two different types of gauge fields appear only in an anticommutator. The $\gamma ^5$ in the scalar boson term of the generalized covariant derivative $D$ ensures both that the partials become restrained and that these anticommutators become commutators once the properties of the Clifford matrices are taken into account. Substituting (\[der\_na\]) in (\[inter\]) and in the matter term of (\[lag\_na\]) we obtain the non-abelian Lagrangian in expanded form: $$\begin{aligned}
{\cal L}_{NA} &=&\bar{\psi}i(\partial \!\!\!/+A\!\!\!/)\psi -g\bar{\psi}%
i\gamma ^5\varphi ^bT^b\psi +\frac 1{2g^2}\widetilde{\limfunc{Tr}}\left(
\partial _\mu A_\nu -\partial _\nu A_\mu +[A_\mu ,A_\nu ]\right) ^2
\nonumber \\
&&+\frac 1{g^2}\widetilde{\limfunc{Tr}}\left( \partial _\mu \varphi +[A_\mu
,\varphi ]\right) ^2. \label{lag_na'}\end{aligned}$$ The reader will recognize familiar structures: the first term on the right looks like the usual matter term of a gauge theory, the second like a Yukawa term, the third like the kinetic energy of vector bosons in a Yang-Mills theory and the fourth like the gauge-invariant kinetic energy of scalar bosons in the non-abelian adjoint representation. It is also interesting to observe that, if in the last term we set the vector bosons equal to zero, then this term simply becomes $\sum_{b,\mu }\frac 12\partial _\mu \varphi
^b\partial ^\mu \varphi ^b$, the kinetic energy of the scalar bosons. We have constructed a generic non-abelian gauge theory with gauge fields that can be either scalar or vector.
Applying these ideas to the GWS Model
=====================================
These ideas were applied[@max-herb] to the GWS Model of high energy physics. The Higgs fields of the GWS Model were used, along with the usual vector bosons, to construct a generalized covariant derivative. The original intention was to use $SU(3)$ as the gauge Lie group, because it generates quantum numbers for the particles that are very close to the experimental ones. Eventually it was noticed that the complete leptonic and bosonic sectors of the GWS Model could be written in the form of Lagrangian (\[lag\_na\]) using the group $U(3).$
The covariant derivative $D$ contains the gauge vector bosons of $U(1)\times
SU(2),$ the scalar Higgs bosons, and a new scalar boson. The triplet $\psi
=(\nu _L,e_L,e_R)$ contains the leptons and is transformed by the fundamental representation of $U(3)$. All quantum numbers are correctly predicted, and an extra scalar boson, but it automatically decouples from the rest of the model and is thus unobservable except through its gravitational effects. The representation of the group generators is not the usual one, but instead, a special one where the generators are obtained as a linear combination of the usual ones.
The quarks have not been included so far into the scheme. There is one term of the pertinent sector of the GWS Model that is not predicted by this generalized derivative model, and it is the potential of the Higgs field $%
V\left( \varphi \right) $ that could cause the spontaneous symmetry breaking.
A generalized curvature
=======================
In a Yang-Mills theory, be it abelian or not, the terms with physical content must be gauge invariant and not contain unrestrained derivatives. For example, if $D_\mu $ is the covariant derivative of a non-abelian theory, the usual expression for the kinetic energy of the gauge vector bosons is $\widetilde{\limfunc{Tr}}[D_\mu ,D_\nu ][D^\mu ,D^\nu ],$ which satisfies both conditions. But even the expression $F_{\mu \nu }=[D_\mu
,D_\nu ]$ by itself does not have any unrestrained derivatives, while $%
\limfunc{Tr}F_{\mu \nu }$ has the additional property of being gauge invariant. This quantity is the curvature in a principal vector bundle. The question arises if similar results as those for $F_{\mu \nu }$ hold also for a theory with a generalized covariant derivative. The answer to this question is in the affirmative. We proceed now to define a quantity which we shall call the generalized curvature $F$. Let $D$ be the generalized covariant derivative, as given in (\[der\_na\]); then: $$F\equiv {\bf 1}\frac 14\limfunc{Tr}D^{\,2}-D^{\,2}, \label{curvature}$$ or else, in terms of the derivative’s constituents, $$F={\bf 1}\partial \cdot A-(\partial \!\!\!/A\!\!\!/)+{\bf 1}A\cdot
A-A\!\!\!/A\!\!\!/-\partial \!\!\!/\Phi -\{A\!\!\!/,\Phi \},
\label{curvature'}$$ where ${\bf 1}$ is a $4\times 4$ unit matrix, so that $\limfunc{Tr}{\bf 1=}%
4. $ It can be seen that there are no unrestrained derivatives, and clearly $%
\limfunc{Tr}F$ is gauge invariant.
In the case of a Yang-Mills theory the vector boson kinetic energy can be written exclusively in terms of the curvature $F_{\mu \nu },$ the commutator of the covariant derivative with itself. In the generalized case we study here, the kinetic energy can also be written in terms of the quantity $F$ defined above, a quantity, that, as previous examples of the curvature, is quadratic in $D$ and restrained. The relation between these two quantities is $$\frac 18\limfunc{Tr}^2D^{\,2}-\frac 12\limfunc{Tr}D^{\,4}=-\frac 12%
\limfunc{Tr}F^2, \label{curvature2}$$ that uses only traces for the Dirac matrices.
The curvature in terms of the covariant derivative
==================================================
Let us review the curvature concept using Riemannian geometry as example. It is well-known that geodesic deviation and parallel transport around an infinitesimally small closed curve in a Riemann manifold are two aspects of the same construction, and that trivial results do not occur in each case only for curved manifolds.[@Weinberg] Consider thus a four dimensional Riemann manifold with metric $g_{\mu \nu }$ and a vector $B^\mu .$ The parallel transport of a vector $B^\beta $ around a closed curve $C$ is given by $$\Delta B^\beta =\oint_C\Gamma ^\beta {}_{\nu \sigma }B^\sigma {}\frac{dx^\nu
}{ds}ds, \label{path}$$ where $s$ is a parametrization of the curve and $\Gamma ^\beta {}_{\nu
\sigma }$ is the connection in a Riemann space, the Christoffel symbol. Let now $C$ be a small parallelogram made up of two short vectors $x^\mu $ and $%
y^\mu ,$ so that its area tensor is $\Delta S^{\mu \nu }=\frac 12x^{(\mu
}y^{\nu )}.$ The integral can be performed taking the vector field to be constant along the sides of the parallelogram, and expressing the values of the Christoffel symbol and the vector field at the curve as the first two terms of an expansion about, say, the center of the parallelogram. Thus, if the values of those quantities at a point in one of the sides of the parallelogram are $\tilde{\Gamma}^\alpha {}_{\beta \delta }$ and $\tilde{B}%
^\beta ,$ then, for one of the sides, $\tilde{\Gamma}^\alpha {}_{\beta
\delta }=\Gamma ^\alpha {}_{\beta \delta }+\Gamma ^\alpha {}_{\beta \delta
,\sigma }\frac 12x^\sigma $ and $B^\beta +(B^\beta {}_{,\sigma }+\Gamma
^\beta {}_{\sigma \tau }B^\tau )\frac 12x^\sigma ,$ where the quantities without tilde take their values at the center of the parallelogram. The contribution of this side is: $$y^\delta \tilde{\Gamma}^\alpha {}_{\beta \delta }\tilde{B}^\beta
|_1=y^\delta (\Gamma ^\alpha {}_{\beta \delta }+\Gamma ^\alpha {}_{\beta
\delta ,\sigma }x^\sigma /2)[B^\beta +(B^\beta {}_{,\sigma }+\Gamma ^\beta
{}_{\sigma \tau }B^\tau )d^\sigma /2] \label{side1}$$ The contribution of this and its opposite side is: $$y^\delta \tilde{\Gamma}^\alpha {}_{\beta \delta }\tilde{B}^\beta
|_{1+3}=y^\delta dx^\sigma [\Gamma ^\alpha {}_{\beta \delta }B^\beta
{}_{,\sigma }+\Gamma ^\alpha {}_{\beta \delta }\Gamma ^\beta {}_{\sigma \tau
}B^\tau +\Gamma ^\alpha {}_{\beta \delta ,\sigma }B^\beta ]. \label{side34}$$ Summing over the four sides one obtains $$\Delta B^\alpha =-R^\alpha {}_{\beta \gamma \delta }B^\beta x^\gamma
y^\delta =-R^\alpha {}_{\beta \gamma \delta }B^\beta \Delta S^{\gamma \delta
}, \label{deficit}$$ where $R^\alpha {}_{\beta \gamma \delta }=\Gamma ^\alpha {}_{\beta \gamma
,\delta }-\Gamma ^\alpha {}_{\beta \delta ,\gamma }+\Gamma ^\alpha
{}_{\delta \tau }\Gamma ^\tau {}_{\beta \gamma }-\Gamma ^\alpha {}_{\gamma
\tau }\Gamma ^\tau {}_{\beta \delta }$ is the Riemann tensor. The positive contributions to the tensor come from two opposite sides of the parallelogram, and the negative from the other two sides, resulting in structure of commutators that can be seen to arise from a commutator of the covariant derivative with itself.
In the case of a Yang-Mills theory, the vector boson kinetic energy can be written exclusively in terms of the curvature $F_{\mu \nu },$ the commutator of the covariant derivative with itself. The parallel transport of a Yang-Mills field, in its role as a connection, around a closed path[@Rosner] results in the curvature, the Yang-Mills connection squared. For two short vectors $x^\mu $ and $y^\mu $ that form a parallelogram parallel transport results, as can be shown similarly to the way it was done in last paragraph for the Christoffel symbol, in $$\oint A_\mu dx^\mu =F_{\mu \nu }x^\mu y^\nu , \label{YMcurvature}$$ which gives a motivation for the interpretation of $F_{\mu \nu }$ as a curvature.
In the generalized case we have been studying in this paper the kinetic energy can also be written in terms of a curvature, precisely the quantity $%
F $ already defined, which was quadratic in $D$ and restrained. The equality does not need the taking of the Lie group trace, and one simply has $$\frac 18\limfunc{Tr}^2D^{\,2}-\frac 12\limfunc{Tr}D^{\,4}=-\frac 12%
\limfunc{Tr}F^2. \label{curvature4}$$ This equality can be shown to be true substituting $F$ as given by (\[curvature\]) on the right in the equation above. Thus the kinetic energy goes as the square of the curvature in the generalized case, too.
Studying the geometric background
=================================
Let us study the possible geometric interpretations of the formalism. Let then $x^\mu $ and $y^\nu $ be two small vectors that form a parallelogram, as before. To calculate the parallel transport, the dot product between vectors has to be done taking a trace over the Dirac matrices, but otherwise the procedure is the same as before. Using the generalized covariant derivative and the contracted forms $x\!\!\!/$ and $y\!\!\!/$ for the two vectors, and taking the small parallelogram as a path to perform the parallel transport integral, one obtains $$\frac 14\limfunc{Tr}DDx\!\!\!/y\!\!\!/=F_{\mu \nu }x^\mu y^\nu ,
\label{partial}$$ where $F_{\mu \nu }$ is constructed as usual with the vector fields that make up the covariant derivative. While this result is reasonable, it is unsatisfactory in that the scalar fields are left as useless bystanders.
One possible attempt to use the formalism we have developed to its fullest, is adding other terms to $x\!\!\!/$ and $y\!\!\!/.$ Let then $x^5$ and $y^5$ be two numbers and define $$d_x\equiv x\!\!\!/+x^5\gamma ^5\quad{\rm and}\quad d_y\equiv y\!\!\!/+y^5\gamma ^5.
\label{5vectors}$$ It is possible now to generalize the parallel transport integral, and substitute (\[5vectors\]) in the trace expression of (\[curvature4\]). The result is: $$\begin{aligned}
\frac 14\limfunc{Tr}DDx\!\!\!/y\!\!\!/ &=&\left( \partial _{[\mu }A_{\nu
]}+[A_\mu ,A_\nu ]\right) x^\mu y^\nu +\left( \partial _\mu \varphi +[A_\mu
,\varphi ]\right) \left( y^5x^\mu -x^5y^\mu \right) \nonumber \\
&&+\partial _5A_\mu (x^5y^\mu -x^\mu y^5). \label{curvature5}\end{aligned}$$ The geometric interpretation of this equation is straightforward: we are dealing with a five dimensional manifold. The metric is Tr$d_xd_y=4(x\cdot
y+x^5y^5).$
The previous result is interesting from a mathematical point of view, but it does not seem to be very enlightening when it comes to an understanding of the geometric meaning of the formalism as applied to the GWS Model. The reason is that, once we have promoted the linearly independent term $%
x^5\gamma ^5$ in (\[5vectors\]) to represent a new dimension, the covariant derivative (\[der\_na\]) has to include an extra term, a derivative with respect to the new dimension, and become $D\equiv \partial
\!\!\!/+\partial _5\gamma ^5+A\!\!\!/+\Phi .$ This could, in principle, add several new terms to the generalized derivative, and we would not have the GWS Model anymore. As it is, all the new terms vanish except one; but one term alone is bad enough, and we conclude that this five dimensional model cannot represent the GWS Model. For the record we present curvature due to the generalized derivative with the extra dimension: $$F=\partial \cdot A-(\partial \!\!\!/A\!\!\!/)+A\cdot
A-A\!\!\!/A\!\!\!/-(\partial \!\!\!/\Phi )-\{A\!\!\!/,\Phi \}-(\partial
\!\!\!/_5A\!\!\!/);$$ the sole remaining new term is the last one.
Final remarks
=============
We have reviewed the construction of a generalized Yang-Mills theory that uses fields other than vector as gauge fields. We have shown a possible geometric interpretation that requires an additional length parameter, but it requires, to be consistent, an additional term in the covariant derivative with respect to the new dimension. The results are interesting from a mathematical point of view, but not very helpful if one is trying to understand the geometry of the formalism as applied to the GWS Model, since this dimension is not observed.
In this model some of the generators of the Lie group are vectorial, while other are scalar. From a mathematical point of view it make no difference which ones are which, but if one is aiming to reproduce the GWS Model, one is has no free choice in this matter. This is interesting because it relates the symmetries of the base manifold to the internal symmetry space of the particles, the Lie group. One would expect that eventually an association of this kind should follow from first principles.
[9]{} See M. Chaves and H. Morales, Mod. Phys. Lett. [**13A**]{} (1998) 2021, and references therein. E. Gabrielli, Phys. Lett. [**B258**]{} (1991) 151,and D. G. C. McKeon, Int. J. Mod. Phys. [**A9**]{} (1994) 5359. Y. Ne’eman, [*Phys. Lett.*]{} [**B81**]{}, (1978) 190; D. B. Fairlie, [*Phys. Lett.*]{} [**B82**]{}, (1979) 97.
S. Weinberg, [*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,*]{} (John Willey & Sons, Inc., N. Y 1973) p. 133. J. L. Rosner, [*Proc. of 1987 Theor. Advanced Study Institute,*]{} (World Scientific, Singapore 1987).
|
---
abstract: 'We note that the Fubini theorem may be used to prove that an $L^1$ function is determined by its Fourier coefficients.'
address: |
Department of Mathematics\
University of Florida\
Gainesville FL 32611 USA
author:
- 'P.L. Robinson'
title: '$\mathbf L^1$ completeness for Fourier series'
---
One of the most fundamental results in the theory of Fourier series is the uniqueness theorem asserting that an $L^1$ function is essentially determined by its Fourier coefficients. Explicitly, the Fourier coefficients of $f \in L^1 [ - \pi, \pi ]$ are defined by $$a_n := \frac{1}{\pi} \int_{- \pi}^{\pi} f(t) \cos n t \: {\rm d} t \; \; \; \; (n \geqslant 0)$$ $$b_n := \frac{1}{\pi} \int_{- \pi}^{\pi} f(t) \sin n t \; {\rm d} t \; \; \; \; (n > 0)$$ and the fundamental uniqueness theorem to which we refer states that if each of these Fourier coefficients is zero then the function $f$ itself is zero almost everywhere.
By now, this result has been established in many ways. One of the more elegant approaches involves the Cesàro means $\sigma_N = \frac{1}{N + 1} (s_0 + \cdots + s_N)$ of the partial sums $$s_N (t) = \frac{1}{2} a_0 + \sum_{n = 1}^{N} (a_n \cos n t + b_n \sin n t )$$ of the Fourier series of $f$. In 1904, Fejér proved that if $f$ is a continuous function then $\sigma_N {\rightarrow}f$ pointwise and indeed uniformly; in 1905, Lebesgue proved that if $f$ is an $L^1$ function then $\sigma_N {\rightarrow}f$ almost everywhere. The uniqueness theorem is an immediate consequence.
Our sole concern here is with another early approach to the proof of the uniqueness theorem due to Lebesgue. The details of this approach may be found in the classics \[7\] (pages 11-12) and \[4\] (pages 18-19); they may also be found in more recent texts such as \[1\] (pages 55-57), \[3\] (pages 40-41) and \[2\] (pages 226-228). This approach starts with the case in which $f$ is continuous and here employs an auxiliary sequence $(T_n)$ of trigonometric polynomials: \[1\] and \[3\] follow Zygmund in their choice of $T_n$; \[2\] follows Hardy and Rogosinski. All five texts are in substantial agreement on the continuation of the proof, in which $f$ is an $L^1$ function with vanishing Fourier coefficients and the continuous case is applied to a specific indefinite integral $F$ of $f$: integration by parts shows that the Fourier coefficients of the (absolutely) continuous function $F$ also vanish, whence $F$ is zero and therefore $f = 0$ almost everywhere.
In at least the three more recent texts, this reduction of the $L^1$ case to the continuous case rests clearly and firmly on the relationship between differentiation and integration in the Lebesgue theory. Times change. These relatively sophisticated aspects of the Lebesgue theory are now perhaps more likely to be covered in a second course, a first course being perhaps more likely to include the Fubini theorem pertaining to repeated integrals. For this reason, we suggest the following alternative proof of the uniqueness theorem (rather, of the reduction from the $L^1$ case to the continuous case).
If the Fourier coefficients of $f \in L^1 [- \pi, \pi]$ all vanish then $f = 0$ almost everywhere.
As was discussed above, we shall assume the truth of the corresponding statement for a continuous function.
Let $f \in L^1 [- \pi, \pi]$ but temporarily make no assumption regarding the vanishing of its Fourier coefficients. Introduce the continuous function $F$ defined by $$F(t) = \int_0^t f(u) \; {\rm d} u - \frac{1}{2} a_0 t$$ (an indefinite integral of $f$ if $a_0$ vanishes) and note that $F( - \pi) = F( \pi )$ by direct calculation. We claim that the Fourier coefficients of $F$ satisfy $$A_n := \frac{1}{\pi} \int_{- \pi}^{\pi} F(t) \cos n t \: {\rm d} t = -\frac{1}{n} b_n \; \; \; \; (n > 0)$$ and $$B_n := \frac{1}{\pi} \int_{- \pi}^{\pi} F(t) \sin n t \; {\rm d} t = \frac{1}{n} a_n \; \; \; \; (n > 0).$$ To justify this, we apply the Fubini theorem. The function $(t, u) \mapsto f(u) \cos n t$ is integrable over the square $[ - \pi, \pi ] \times [ - \pi, \pi ]$: inverting the order of integration over the indicated triangular subsets, $$\begin{aligned}
\int_0^{\pi} \Big\{ \int_0^t f(u) \; {\rm d} u \Big\} \cos n t \; {\rm d} t & = & \int_0^{\pi} f(u) \; \Big\{ \int_u^{\pi} \cos n t \; {\rm d} t \Big\} \; {\rm d} u \\ & = & \int_0^{\pi} f(u) \Big[ \frac{\sin n t}{n} \Big]_u^{\pi} {\rm d} u\end{aligned}$$ so that $$\int_0^{\pi} \Big\{ \int_0^t f(u) \; {\rm d} u \Big\} \cos n t \; {\rm d} t = - \frac{1}{n} \int_0^{\pi} f(u) \sin n u \; {\rm d} u$$ and similarly $$\int_{-\pi}^0 \Big\{ \int_0^t f(u) \; {\rm d} u \Big\} \cos n t \; {\rm d} t = - \frac{1}{n} \int_{-\pi}^0 f(u) \sin n u \; {\rm d} u;$$ since $$\int_{-\pi}^{\pi} \frac{1}{2} a_0 t \cos n t \; {\rm d} t = 0$$ it follows upon summation that $$A_n = - \frac{1}{n} b_n$$ as claimed. The calculation to establish that $$B_n = \frac{1}{n} a_n$$ is similar, being complicated only by the fact that the integral $$\int_{-\pi}^{\pi} \frac{1}{2} a_0 t \sin n t \; {\rm d} t = (-1)^{n - 1} \frac{\pi}{n} a_0$$ is precisely cancelled by a corresponding term in the integral $$\int_{-\pi}^{\pi} \Big\{ \int_0^t f(u) \; {\rm d} u \Big\} \sin n t \; {\rm d} t$$ on account of the definition $$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f.$$ The function $F$ thus has Fourier coefficients $A_0$ (upon which we need not elaborate here), $A_n = - b_n/n$ and $B_n = a_n/n$ (for $n > 0$).
Now let us assume that the Fourier coefficients of $f$ vanish. It follows that the [*continuous*]{} function $F - \frac{1}{2} A_0$ has vanishing Fourier coefficients and is therefore identically zero; as $F(0) = 0$ it follows that $A_0 = 0$ and therefore that $F$ is identically zero. Thus $f$ has vanishing integral over each interval with $0$ as an endpoint and hence over each interval; so $f$ is zero almost everywhere. Of course, the last conclusion here may be drawn without reference to the relationship between integrals and derivatives.
Incidentally, if we prefer to follow the classics in proving the uniqueness theorem via partial integration, it is still possible to bypass the more sophisticated aspects of Lebesgue theory that relate integration and differentiation: we need only use a version of integration by parts that does not involve differentiation explicitly; such a version appears as Proposition 2 on page 103 of \[6\] and as the first theorem in section 26 on pages 54-55 of the classic \[5\]. On this point it is interesting to note that, in their proofs of the uniqueness theorem, neither Zygmund nor Hardy and Rogosinski seem to specify any particular version of integration by parts.
Finally, it is perhaps worth mentioning that Hardy and Rogosinski \[4\] actually indicate several proofs of the uniqueness theorem: the original proof to which ours is an alternative, at page 18; at page 31, a variant based on termwise integration of Fourier series (our alternative line of argument being so arranged as to allow for its use here, too); at page 43, a variant making use of the fact that indefinite integrals are functions of bounded variation; at page 68, the elegant proof via Cesàro means.
[R]{}[EFERENCES]{}
\[1\] N. Bary, [*A Treatise on Trigonometric Series*]{}, Volume 1, Pergamon Press (1964).
\[2\] S.B. Chae, [*Lebesgue Integration*]{}, Second Edition, Springer-Verlag (1995).
\[3\] R.E. Edwards, [*Fourier Series - A Modern Introduction*]{}, Volume 1, Revised Edition, Springer-Verlag (1979).
\[4\] G.H. Hardy and W.W. Rogosinski, [*Fourier Series*]{}, Cambridge University Press (1944); Dover Publications (1999).
\[5\] F. Riesz and B. Sz.-Nagy, [*Functional Analysis*]{}, Frederick Ungar Publishing Company (1955); Dover Publications (1990).
\[6\] A.J. Weir, [*Lebesgue Integration and Measure*]{}, Cambridge University Press (1973).
\[7\] A. Zygmund, [*Trigonometric Series*]{}, Volume 1, Second Edition, Cambridge University Press (1959).
|
---
abstract: 'We begin with a description of spacetime by a 4-dimensional cubic lattice ${{\mathcal S}}$. It follows from this framework that the the speed of light is the only nonzero instantaneous speed for a particle. The dual space ${\widehat{{{\mathcal S}}}}$ corresponds to a cubic lattice of energy-momentum. This description implies that there is a discrete set of possible particle masses. We then define discrete scalar quantum fields on ${{\mathcal S}}$. These fields are employed to define interaction Hamiltonians and scattering operators. Although the scattering operator $S$ cannot be computed exactly, approximations are possible. Whether $S$ is unitary is an unsolved problem. Besides the definitions of these operators, our main assumption is conservation of energy-momentum for a scattering process. This article concludes with various examples of perturbation approximations. These include simplified versions of electron-electron and electron-proton scattering as well as simple decay processes. We also define scattering cross-sections, decay rates and lifetimes within this formalism.'
author:
- |
S. Gudder\
Department of Mathematics\
University of Denver\
Denver, Colorado 80208, U.S.A.\
[email protected]
title: |
DISCRETE SCALAR\
QUANTUM FIELD THEORY
---
Introduction
============
It has long been recognized that standard quantum field theory suffers from a plague of singularities and infinities. The author would even go so far as to say that in its usual form, a quantum field does not exist! By this we mean that it does not have a mathematically well-defined definition. The problems are compounded by employing these nonexistent quantum fields in rather complicated ways to define interaction Hamiltonians and scattering operators. Perturbation methods are then used to compute numbers that frequently agree with experiment. However, to obtain these numbers, renormalization techniques and infinity cancellations have to be employed [@ps95; @vel94]. Attempts have been made to develop a rigorous mathematically well-define quantum field theory [@sw64]. These attempts have been successful for free quantum fields, but they have not adequately included interacting fields with nontrivial scattering.
It seems to us that the best way to overcome these difficulties is to postulate that spacetime is discrete [@gud68; @hei30; @rus54]. Although such a granular structure for spacetime has not been experimentally observed, with increasingly accurate instruments it may become evident in the future, perhaps in an indirect manner. The granular structure may manifest itself in terms of elementary lengths and times at Planck scales of about $10^{-33}$cm. and $10^{-43}$sec., respectively. The simplest such framework would be a rigid cubic 4-dimensional lattice. The author has previously studied a tetrahedral lattice that appears to have certain advantages [@gud16] but for simplicity and ease of computation, we shall only consider the cubic lattice ${{\mathcal S}}$ here.
It follows from this description that the speed of light is the only nonzero instantaneous speed for a particle. We observe slower speeds because we actually measure average speeds of objects. The dual space ${\widehat{{{\mathcal S}}}}$ corresponds to a 4-dimensional cubic lattice of energy-momentum. This description implies that there is a discrete set of possible particle masses, which we then determine.
We next define discrete scalar quantum fields on ${{\mathcal S}}$. We show that these fields exist mathematically and derive some of their properties. We mention one can also study discrete vector quantum fields that involve spin, but for simplicity we only consider the spin zero case here. Various fields are combined to form interaction Hamiltonians. We postulate that the scattering operator satisfies a “second quantization” Schrödinger equation and derive its form. Besides the definition of these operators, our main assumption is conservation of energy-momentum for scattering processes. Due to this conservation law, we show that a scattering process usually possesses only a finite number of possible outgoing states. These can then be summed over to find scattering amplitudes and probabilities.
The article concludes with various examples of perturbation approximations. These include simplified versions of electron-electron and electron-proton scattering as well as simple decay processes. We also define scattering cross-sections, decay rates and lifetimes within this formalism. We compute these quantities in simple cases. Finally, some speculations are made about the theory’s ability to predict the existence of dark energy and dark matter.
Discrete Spacetime and Energy-Momentum
======================================
Our basic assumption is that spacetime is discrete and has the structure of a 4-dimensional cubic lattice ${{\mathcal S}}$. We regard ${{\mathcal S}}$ as a framework or scaffolding in which the vertices of the lattice (or network) represent tiny cells of Planck scale that may or may not be occupied by a particle. The edges between vertices represent space directions in which particles can propagate at a given time. Let $${{\mathbb Z}}={\left\{0,\pm 1,\pm 2,\ldots\right\}}$$ be the integers and ${{\mathbb Z}}^+={\left\{0,1,2,\ldots\right\}}$ be the nonnegative integers. We have that ${{\mathcal S}}={{\mathbb Z}}^+\times{{\mathbb Z}}^3$ where ${{\mathbb Z}}^+$ represents discrete time and ${{\mathbb Z}}^3$ represents discrete 3-space. If $x=(x_0,x_1,x_2,x_3)\in{{\mathcal S}}$, we write $x=(x_0,{\mathbf{x}})$ where $x_0\in{{\mathbb Z}}^+$ is time and ${\mathbf{x}}\in{{\mathbb Z}}^3$ is a 3-space point. We have that ${{\mathcal S}}$ is a module in the sense that ${{\mathcal S}}$ is closed under addition and multiplication by elements of ${{\mathbb Z}}^+$. The vectors $$\begin{aligned}
d&=(1,{\mathbf{0}})=(1,0,0,0),\enspace e=(0,{\mathbf{e}})=(0,1,0,0)\\
f&=(0,{\mathbf{f}})=(0,0,1,0),\enspace g=(0,{\mathbf{g}})=(0,0,0,1)\end{aligned}$$ form a basis for ${{\mathcal S}}$ and every $x\in{{\mathcal S}}$ has the unique form $$x=nd+me+pf+qg$$ $n\in{{\mathbb Z}}^+$, $m,p,q\in{{\mathbb Z}}$.
We equip ${{\mathcal S}}$ with the Minkowski distance $${\left|\left|x\right|\right|}_4^2=x_0^2-{\left|\left|{\mathbf{x}}\right|\right|}_3^2=x_0^2-x_1^2-x_2^2-x_3^2$$ As usual $x,y\in{{\mathcal S}}$ are *time-like separated* if ${\left|\left|x-y\right|\right|}_4^2\ge 0$ and *space-like separated* if ${\left|\left|x-y\right|\right|}_4<0$. For $x\in{{\mathcal S}}$, the set $${{\mathcal C}}^+(x)={\left\{y\in{{\mathcal S}}\colon y_0\ge x_0,{\left|\left|y-x\right|\right|}_4^2\ge 0\right\}}$$ is the *future light cone at* $x$. Of course, we are using units in which the speed of light is 1. With our interpretation of the structure of ${{\mathcal S}}$, it seems natural that a particle at ${\mathbf{x}}$ can either stay at ${\mathbf{x}}$ or move in one of the six space directions to a point ${\mathbf{y}}$ in one time unit, where ${\mathbf{y}}={\mathbf{x}}\pm{\mathbf{e}}, {\mathbf{x}}\pm{\mathbf{f}},{\mathbf{x}}\pm{\mathbf{g}}$. In this way, the only nonzero instantaneous speed of a particle is the speed of light 1. This gives a primitive reason why the speed of light is the upper limit for all signal speeds. The reason that we observe slower speeds is that we usually measure average speeds and not instantaneous ones. If a particle moves from ${\mathbf{x}}$ to ${\mathbf{y}}$, its *average speed* is ${\overline{v}}={\left|\left|{\mathbf{y}}-{\mathbf{x}}\right|\right|}_3/t$ where $t$ is the elapsed proper time; that is, the time measured along the particle trajectory. For example, suppose a particle initially at ${\mathbf{x}}$, stays at ${\mathbf{x}}$ for one time unit and then moves to ${\mathbf{y}}={\mathbf{x}}+{\mathbf{e}}$ in the next time unit. Then the proper time $t=2$ and the average speed is ${\overline{v}}={\left|\left|{\mathbf{e}}\right|\right|}_3/2=1/2$. As another example, suppose a particle initially at ${\mathbf{0}}$ moves to ${\mathbf{e}}$ at the first time unit and then to ${\mathbf{e}}+{\mathbf{f}}$ at the second time unit. The proper time is again $t=2$ and ${\overline{v}}={\left|\left|{\mathbf{e}}+{\mathbf{f}}\right|\right|}_3/2=1/\sqrt{2}$. We conclude that photons must travel in straight lines along coordinate axes.
The dual of ${{\mathcal S}}$ is denoted by ${\widehat{{{\mathcal S}}}}$. We regard ${\widehat{{{\mathcal S}}}}$ as having the identical structure as ${{\mathcal S}}$ and that ${\widehat{{{\mathcal S}}}}$ again has basis $d,e,f,g$. The only difference is that we denote elements of ${\widehat{{{\mathcal S}}}}$ by $$p=(p_0,{\mathbf{p}})=(p_0,p_1,p_2,p_3)$$ and interpret $p$ as the energy-momentum vector for a particle. In fact, we sometimes even call $p\in{\widehat{{{\mathcal S}}}}$ a particle. Moreover, we only consider the forward light cone $${{\mathcal C}}^+(0)={\left\{p\in{\widehat{{{\mathcal S}}}}\colon{\left|\left|p\right|\right|}_4\ge 0\right\}}$$ in ${\widehat{{{\mathcal S}}}}$. For a particle $p\in{\widehat{{{\mathcal S}}}}$, we call $p_0\ge 0$ the *total energy*, ${\left|\left|{\mathbf{p}}\right|\right|}_3\ge 0$ the *kinetic energy* and $m={\left|\left|p\right|\right|}_4\ge 0$ the *mass* of $p$. The integers $p_1,p_2,p_3$ are *momentum components*. Since $$\label{eq21}
m^2={\left|\left|p\right|\right|}_4^2=p_0^2-{\left|\left|{\mathbf{p}}\right|\right|}_3^2$$ we conclude that Einstein’s energy formula $p_0=\sqrt{m^2+{\left|\left|{\mathbf{p}}\right|\right|}_3^2}$ holds.
It is clear that any nonnegative integer can be a total energy. However, the square of a kinetic energy must be the sum of three squares ${\left|\left|{\mathbf{p}}\right|\right|}_3^2=p_1^2+p_2^2+p_3^2$. A number theory theorem [@vde87] says that $n\in{{\mathbb Z}}^+$ is a sum of three squares if and only if $n$ does not have the form $4^k(8s+7)$, $k,s\in{{\mathbb Z}}^+$. We conclude that $7,15,23,28,\ldots$ are eliminated. Thus, ${\left|\left|{\mathbf{p}}\right|\right|}_3=\sqrt{n}$, $n\in{{\mathbb Z}}^+$ is a kinetic energy if and only if $n\ne 7,15,23,28,\ldots\,$. It follows that every nonnegative number $\sqrt{n}$, $n\in{{\mathbb Z}}^+$ is a possible mass and vice versa, although it may not appear for certain total energy $p_0\ge\sqrt{n}$. Therefore, this theory predicts that there are a countable number of admissible particle masses. In contrast to the speeds considered previously, we also have the concept of a *geometric speed* for $p\in{\widehat{{{\mathcal S}}}}$ given by $$v=\frac{{\left|\left|{\mathbf{p}}\right|\right|}_3}{p_0}=\frac{\sqrt{p_0^2-m^2}}{p_0}=\sqrt{1-\frac{m^2}{p_0^2}}$$ whenever $p_0\ne 0$. Of course, $0\le v\le 1$ and $v=1$ if and only if $m=0$. Moreover, $v=0$ if and only if ${\left|\left|{\mathbf{p}}\right|\right|}=0$.
Table 1 lists kinetic energy values, the corresponding momentum vectors and the number of such vectors. Notice, as previously observed, that $\sqrt{7}$, $\sqrt{15}$, $\sqrt{23}$ are missing.
We can apply Table 1 to find the possible particle masses $m$ corresponding to total energy $p_0$ as well as the number of particle kinetic energies. In this work, we assume that $p_0\ne 0$.
-5pc
=-3pc
[c|c|c]{} Kinentic&&\
Energy&Momentum Vectors&Number\
1&$\pm{\mathbf{e}},\ \pm{\mathbf{f}},\ \pm{\mathbf{g}}$&6\
$\sqrt{2}$&$\pm{\mathbf{e}}\pm{\mathbf{f}},\ \pm{\mathbf{e}}\pm{\mathbf{g}},\ \pm{\mathbf{f}}\pm{\mathbf{g}}$&12\
$\sqrt{3}$&$\pm{\mathbf{e}}\pm{\mathbf{f}}\pm{\mathbf{g}}$&8\
2&$\pm 2{\mathbf{e}},\ \pm 2{\mathbf{f}},\ \pm 2{\mathbf{g}}$&6\
$\sqrt{5}$&$\pm 2{\mathbf{e}}\pm{\mathbf{f}},\ \pm 2{\mathbf{e}}\pm{\mathbf{g}},\ \pm 2{\mathbf{f}}\pm{\mathbf{e}},\ \pm 2{\mathbf{f}}\pm{\mathbf{g}},\ \pm 2{\mathbf{g}}\pm{\mathbf{e}},\ \pm 2{\mathbf{g}}\pm{\mathbf{f}}$&24\
$\sqrt{6}$&$\pm 2{\mathbf{e}}\pm{\mathbf{f}}\pm{\mathbf{g}},\ \pm2{\mathbf{f}}\pm{\mathbf{e}}\pm{\mathbf{g}},\ \pm 2{\mathbf{g}}\pm{\mathbf{e}}\pm{\mathbf{f}}$&24\
$\sqrt{8}$&$\pm 2{\mathbf{e}}\pm 2{\mathbf{f}},\ \pm 2{\mathbf{e}}\pm 2{\mathbf{g}},\ \pm 2{\mathbf{f}}\pm 2{\mathbf{g}}$&12\
3&$\pm 3{\mathbf{e}},\ \pm 3{\mathbf{f}},\ \pm 3{\mathbf{g}},\ \pm 2{\mathbf{e}}\pm 2{\mathbf{f}}\pm{\mathbf{g}},\ \pm 2{\mathbf{e}}\pm 2{\mathbf{g}}\pm{\mathbf{f}},\ \pm 2{\mathbf{f}}\pm 2{\mathbf{g}}\pm{\mathbf{e}}$&30\
$\sqrt{10}$&$\pm 3{\mathbf{e}}\pm{\mathbf{f}},\ \pm 3{\mathbf{e}}\pm{\mathbf{g}},\ \pm 3{\mathbf{f}}\pm{\mathbf{e}},\ \pm 3{\mathbf{f}}\pm{\mathbf{g}},\ \pm 3{\mathbf{g}}\pm{\mathbf{e}},\ \pm 3{\mathbf{g}}\pm{\mathbf{f}}$&24\
$\sqrt{11}$&$\pm 3{\mathbf{e}}\pm{\mathbf{f}}\pm{\mathbf{g}},\ \pm 3{\mathbf{f}}\pm{\mathbf{e}}\pm{\mathbf{g}},\ \pm 3{\mathbf{g}}\pm{\mathbf{e}}\pm{\mathbf{f}}$&24\
$\sqrt{12}$&$\pm 2{\mathbf{e}}\pm 2{\mathbf{f}}\pm 2{\mathbf{g}}$&8\
$\sqrt{13}$&$\pm 3{\mathbf{e}}\pm 2{\mathbf{f}},\ \pm 3{\mathbf{e}}\pm 2{\mathbf{g}},\ \pm 3{\mathbf{f}}\pm 2{\mathbf{e}},\ \pm 3{\mathbf{f}}\pm 2{\mathbf{g}},\ \pm 3{\mathbf{g}}\pm 2{\mathbf{e}},\ \pm 3{\mathbf{g}}\pm 2{\mathbf{f}}$&24\
$\sqrt{14}$&$\pm 3{\mathbf{e}}\pm 2{\mathbf{f}}\pm{\mathbf{g}},\ \pm 3{\mathbf{e}}\pm 2{\mathbf{g}}\pm{\mathbf{f}},\ \pm 3{\mathbf{f}}\pm 2{\mathbf{e}}\pm{\mathbf{g}},\ \pm 3{\mathbf{f}}\pm 2{\mathbf{g}}\pm{\mathbf{e}}$,&\
&$\pm 3{\mathbf{g}}\pm 2{\mathbf{e}}\pm{\mathbf{f}},\ \pm 3{\mathbf{g}}\pm 2{\mathbf{f}}\pm{\mathbf{e}}$&48\
4&$\pm 4{\mathbf{e}},\ \pm 4{\mathbf{f}},\ \pm 4{\mathbf{g}}$&6\
$\sqrt{17}$&$\pm 4{\mathbf{e}}\pm{\mathbf{f}},\ \pm 4{\mathbf{e}}\pm{\mathbf{g}},\ \pm 4{\mathbf{f}}\pm{\mathbf{e}},\ \pm 4{\mathbf{f}}\pm{\mathbf{g}},\ \pm 4{\mathbf{g}}\pm{\mathbf{e}},\ \pm 4{\mathbf{g}}\pm{\mathbf{f}}$,&\
&$\pm 3{\mathbf{e}}\pm 2{\mathbf{f}}\pm 2{\mathbf{g}},\ \pm 3{\mathbf{f}}\pm 2{\mathbf{e}}\pm 2{\mathbf{g}},\ \pm 3{\mathbf{g}}\pm 2{\mathbf{e}}\pm 2{\mathbf{f}}$&48\
$\sqrt{18}$&$\pm 4{\mathbf{e}}\pm{\mathbf{f}}\pm{\mathbf{g}},\ \pm 4{\mathbf{f}}\pm{\mathbf{e}}\pm{\mathbf{g}},\ \pm 4{\mathbf{g}}\pm{\mathbf{e}}\pm{\mathbf{f}},\ \pm 3{\mathbf{e}}\pm 3{\mathbf{f}},\ \pm 3{\mathbf{e}}\pm 3{\mathbf{g}},\ \pm 3{\mathbf{f}}\pm 3{\mathbf{g}}$&36\
$\sqrt{19}$&$\pm 3{\mathbf{e}}\pm 3{\mathbf{f}}\pm{\mathbf{g}},\ \pm 3{\mathbf{e}}\pm 3{\mathbf{g}}\pm{\mathbf{f}},\ \pm 3{\mathbf{f}}\pm 3{\mathbf{g}}\pm{\mathbf{e}}$&24\
$\sqrt{20}$&$\pm 4{\mathbf{e}}\pm 2{\mathbf{f}},\ \pm 4{\mathbf{e}}\pm 2{\mathbf{g}},\ \pm 4{\mathbf{f}}\pm 2{\mathbf{e}},\pm 4{\mathbf{f}}\pm 2{\mathbf{g}},\ \pm 4{\mathbf{g}}\pm 2{\mathbf{e}},\ \pm 4{\mathbf{g}}\pm 2{\mathbf{f}}$&24\
$\sqrt{21}$&$\pm 4{\mathbf{e}}\pm 2{\mathbf{f}}\pm{\mathbf{g}},\ \pm 4{\mathbf{e}}\pm 2{\mathbf{g}}\pm{\mathbf{f}},\ \pm 4{\mathbf{f}}\pm 2{\mathbf{e}}\pm{\mathbf{g}},\ \pm 4{\mathbf{f}}\pm 2{\mathbf{g}}\pm{\mathbf{e}}$,&\
&$\pm 4{\mathbf{g}}\pm 2{\mathbf{e}}\pm{\mathbf{f}},\ \pm 4{\mathbf{g}}\pm 2{\mathbf{f}}\pm{\mathbf{e}}$&48\
$\sqrt{22}$&$\pm 3{\mathbf{e}}\pm 3{\mathbf{f}}\pm 2{\mathbf{g}},\ \pm 3{\mathbf{e}}\pm 3{\mathbf{g}}\pm 2{\mathbf{f}},\ \pm 3{\mathbf{f}}\pm 3{\mathbf{g}}\pm 2{\mathbf{e}}$&24\
$\sqrt{24}$&$\pm 4{\mathbf{e}}\pm 2{\mathbf{f}}\pm 2{\mathbf{g}},\ \pm 4{\mathbf{f}}\pm 2{\mathbf{e}}\pm 2{\mathbf{g}},\ \pm 4{\mathbf{g}}\pm 2{\mathbf{f}}\pm 2{\mathbf{e}}$&24\
5&$\pm 5{\mathbf{e}},\ \pm 5{\mathbf{f}},\ \pm 5{\mathbf{g}},\ \pm 4{\mathbf{e}}\pm 3{\mathbf{f}},\ \pm 4{\mathbf{e}}\pm 3{\mathbf{g}},\ \pm 4{\mathbf{f}}\pm 3{\mathbf{e}}$,&\
&$\pm 4{\mathbf{f}}\pm 3{\mathbf{g}},\ \pm 4{\mathbf{g}}\pm3{\mathbf{e}},\ \pm 4{\mathbf{g}}\pm 3{\mathbf{f}}$&30\
$\vdots$&&\
6&$\pm 6{\mathbf{e}},\ \pm 6{\mathbf{f}},\ \pm 6{\mathbf{g}},\ \pm 4{\mathbf{e}}\pm 4{\mathbf{f}}\pm 2{\mathbf{g}},\ \pm 4{\mathbf{e}}\pm 4{\mathbf{g}}\pm 2{\mathbf{f}},\ \pm 4{\mathbf{f}}\pm 4{\mathbf{g}}\pm 2{\mathbf{e}}$&30\
\
Total Energy
-------------- --- --- --- --- ---- --- --- --- --- ---- --- --- ---- ---- ---- ---- -- --
mass${}^2$ 1 0 4 3 2 1 0 9 8 7 6 5 4 3 1 0
number 1 6 1 6 12 8 6 1 6 12 8 6 24 24 12 30
Total Energy
-------------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- --- ---- ---- --- -- -- --
mass${}^2$ 16 15 14 13 12 11 10 8 7 6 5 4 3 2 0
number 1 6 12 8 6 24 24 12 30 24 24 8 24 48 6
=-5pc
[c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c]{} Total Energy&\
mass${}^2$&25&24&23&22&21&20&19&17&16&15&14&13&12&11&9&8&7&6&5&4&3&1&0\
number&1&6&12&8&6&24&24&12&30&24&24&8&24&48&6&48&36&24&24&48&24&24&30\
\
We classify the mass zero particles (photons) into two types, the one dimensional photons (e.g. ${\mathbf{e}}, 2{\mathbf{f}},\ldots$) that we call *light photons* (pun intended) and the 2-dimensional and 3-dimensional photons (e.g. $4{\mathbf{e}}+3{\mathbf{f}}, 2{\mathbf{e}}+2{\mathbf{f}}+{\mathbf{g}},\ldots$) that we call *dark photons*. Both of these types have geometric speed 1. However, the light photons propagate along coordinate axes so we can consider their average speed to be 1 while the dark photons do not move directly along coordinate axes so we can consider their average speed to be less than 1. We speculate that these latter particles correspond to dark energy. In a similar way the positive mass particles are either 1, 2 or 3-dimensional and we speculate that the first two types correspond to matter and the last type corresponds to dark matter.
Free Quantum Fields
===================
In this section, we study free discrete quantum fields. Since we are considering scalar fields, a particle is essentially determined by its mass. We assume that we are describing a physical system that contains particles of a finite number of various types. For illustrative purposes, suppose there are two types of particles under consideration which we call $p$-particles and $q$-particles with masses $m$ and $M$, respectively, $m\ne M$. The sets $$\begin{aligned}
\Gamma _m&={\left\{p\in{\widehat{{{\mathcal S}}}}\colon{\left|\left|p\right|\right|}_4=m\right\}}\\
\Gamma _M&={\left\{q\in{\widehat{{{\mathcal S}}}}\colon{\left|\left|q\right|\right|}_4=M\right\}}\end{aligned}$$ are called the mass *hyperboloids* for the particles. To describe these particles quantum mechanically, we construct a complex Hilbert space $K$. Technically speaking, $K$ is a symmetric Fock space, but the details are not important here. All we need to know is that $K$ exists and has a very descriptive orthonormal basis of the form $${{\left|p_1p_2\ldots p_nq_1q_2\ldots q_s\right>}}$$ that represents the quantum state in which there are $n$ $p$-particles and $s$ $q$-particles where $p_i,q_j\in{\widehat{{{\mathcal S}}}}$, $i=1,\ldots ,n$, $j=1,\ldots ,s$. The order of the $p$’s and $q$’s is immaterial and different states are mutually orthogonal unit vectors. For example, ${{\left|p_1p_2\right>}}={{\left|p_2p_1\right>}}$ and ${{\left\langlep_1p_2\mid p_1q_1q_2\right\rangle}}=0$. The *vacuum state* in which there are no particles present is the unit vector ${{\left|0\right>}}$. The one-particle states of type $p$ are ${{\left|p\right>}}$, $p\in\Gamma _m$, the two-particle states of the type $p$ are ${{\left|p_1p_2\right>}}$, etc.
For $p\in\Gamma _m$, we define the *annihilation operator* $a(p)$ on $K$ by $a(p){{\left|0\right>}}=0$ where 0 is the zero vector and $$a(p){{\left|pp\ldots pp_1\ldots p_nq_1\ldots q_s\right>}}=\sqrt{n}\,{{\left|pp\ldots pp_1\ldots p_nq_1\ldots q_s\right>}}$$ where $n$ $p$’s appear in the vector on the left side and $n-1$ $p$’s appear in the vector on the right side. We interpret $a(p)$ as the operator that annihilates a particle with energy-momentum $p$. For example, $a(p){{\left|p\right>}}={{\left|0\right>}}$, $a(p){{\left|ppq\right>}}=\sqrt{2}\,{{\left|pq\right>}}$ and $$a(q){{\left|p_1p_2qqq\right>}}=\sqrt{3}\,{{\left|p_1p_2qq\right>}}$$ The adjoint $a(p)^*$ of $a(p)$ is the operator on $K$ defined by $$a(p)^*{{\left|pp\ldots pp_1\ldots p_nq_1\ldots q_s\right>}}=\sqrt{n+1}\,{{\left|pp\ldots pp_1\ldots p_nq_1\ldots q_s\right>}}$$ where $n$ $p$’s appear in the vector on the left side and $n+1$ $p$’s appear in the vector on the right side. For example $a(p)^*{{\left|0\right>}}={{\left|p\right>}}$ and $a(p)^*{{\left|p\right>}}=\sqrt{2}\,{{\left|pp\right>}}$. We interpret $a(p)^*$ as the operator that creates a particle with energy-momentum $p$. To show that $a(p)^*$ is indeed the adjoint of $a(p)$, suppose that $${{\left|\alpha\right>}}={{\left|pp\ldots pq_1\ldots q_s\right>}}$$ has $n$ $p$’s. Then the inner product of $a(p)^*{{\left|\alpha\right>}}$ with any other basis vector ${{\left|\beta\right>}}$ is zero unless $\beta$ has the same form as ${{\left|\alpha\right>}}$ except now there are $n+1$ $p$’s. In this case we have $${{\left<\beta\right|}}a(p)^*{{\left|\alpha\right>}}=\sqrt{n+1}\,{{\left\langle\beta\mid\beta\right\rangle}}={{\left<\alpha\right|}}a(p){{\left|\beta\right>}}$$ which is the defining relationship for the adjoint.
The operators $a(p)$ and $a(p)^*$ have the characteristic property that their commutator ${\left[a(p),a(p)^*\right]}=I$. We illustrate this for two cases which should convince the reader that this holds. If ${{\left|p_1\ldots p_nq_1\ldots q_s\right>}}$ contains no $p$’s, then $$\begin{aligned}
a(p)a(p)^*{{\left|p_1\ldots p_nq_1\ldots q_s\right>}}&=a(p){{\left|pp_1\ldots p_nq_1\ldots q_s\right>}}\\
&={{\left|p_1\ldots p_nq_1\ldots q_s\right>}}\end{aligned}$$ and $a(p)^*a(p){{\left|p_1\ldots p_nq_1\ldots q_s\right>}}=0$. Next, if ${{\left|\alpha\right>}}$ and ${{\left|\beta\right>}}$ are the vectors in the previous paragraph, then $$a(p)a(p)^*{{\left|\alpha\right>}}=\sqrt{n+1}\,a(p){{\left|\beta\right>}}=(n+1){{\left|\alpha\right>}}$$ Since $a(p)^*a(p){{\left|\alpha\right>}}=n{{\left|\alpha\right>}}$ we have that ${\left[a(p),a(p)^*\right]}{{\left|\alpha\right>}}={{\left|\alpha\right>}}$.
One reason that $a(p)$ and $a(p)^*$ are important is that they can be employed to describe physically relevant operators. For example the *free total energy operator* for $p$-particles is defined by $$P_0=\sum _{p\in\Gamma _m}p_0a(p)^*a(p)$$ The eigenvectors of $P_0$ have the form ${{\left|\alpha\right>}}={{\left|p_1p_2\ldots p_n\right>}}$, $p_j\in\Gamma _m$, $j=1,\ldots ,n$, with eigenvalues the total energy $$E_\alpha =(p_1)_0+\cdots +(p_n)_0$$ In a similar way, we define the *free momentum operators* $$P_j=\sum _{p\in\Gamma _m}p_ja(p)^*a(p)$$ for $j=1,2,3$. We also define the *number operator* $$N=\sum _{p\in\Gamma _m}a(p)^*a(p)$$ Then $N{{\left|\alpha\right>}}=n{{\left|\alpha\right>}}$ where $n$ is the number of $p$-particles in the basis vector ${{\left|\alpha\right>}}$. Of course, all these operators are self-adjoint.
The most important operators in this work are the *free quantum fields* $$\label{eq31}
\phi (x)=\sum _{p\in\Gamma _m}\frac{1}{p_0}{\left[a(p)e^{i\pi px/2}+a(p)^*e^{-i\pi px/2}\right]}$$ where $x\in{{\mathcal S}}$ and $px$ is the indefinite inner product $$px=p_0x_0-p_1x_1-p_2x_2-p_3x_3$$ The self-adjoint operator $\phi (x)$ is an observable representing a quantum field for a particle of mass $m$ at the spacetime point $x$. In standard quantum field theory, the summation in is replaced by an integral over the mass hyperboloid and it is questionable whether such an integral exists. (It is not even clear whether the underlying Fock space $K$ exists.) However, in the discrete case, the summation in does exist. Although this holds in general, we shall illustrate it for a simple, but important case. Denoting the mass hyperboloid for light photons by $\Gamma '_0$ we have that $$\phi (0){{\left|0\right>}}=\sum _{p\in\Gamma '_0}\frac{1}{p_0}\,{{\left|p\right>}}$$ Now $\sigma (0){{\left|0\right>}}\in K$ with $${\left|\left|\phi (0){{\left|0\right>}}\right|\right|}^2=6\sum _{n=1}^\infty\frac{1}{n^2}=\pi ^2$$ Hence, ${\left|\left|\phi (0){{\left|0\right>}}\right|\right|}=\pi$.
Let $\phi (x)$ be a general free quantum field given by . The next result gives an expression for the commutator ${\left[\phi (x),\phi (y)\right]}$.
\[thm31\] The commutator $${\left[\phi (x),\phi (y)\right]}=2i\sum _{p\in\Gamma _m}\frac{1}{p_0^2}\,\sin{\left[\pi p_0(x_0-y_0)/2\right]}\cos{\left[{\mathbf{p}}{\mathrel{\mathlarger\cdot}}({\mathbf{x}}-{\mathbf{y}})/2\right]}I$$
We have that $$\begin{aligned}
\phi (x)\phi (y)&=\sum _{p\in\Gamma _m}\frac{1}{p_0}\,{\left[a(p)e^{i\pi px/2}+a(p)^*e^{-i\pi px/2}\right]}\\
&\quad{\mathrel{\mathlarger\cdot}}\sum _{q\in\Gamma _m}\frac{1}{q_0}{\left[a(q)e^{i\pi qy/2}+a(q)^*e^{-i\pi qy/2}\right]}\\
&=\sum _{p,q\in\Gamma _m}\frac{1}{p_0q_0}\left[a(p)a(q)e^{i\pi (px+qy)/2}+a(p)a(q)^*e^{i\pi (px-qy)/2}\right.\\
&\quad\left. +a(p)^*a(q)e^{i\pi(qy-px)/2}+a(p)^*a(q)^*e^{-i\pi (px+qy)/2}\right]\end{aligned}$$ Similarly, $$\begin{aligned}
\phi (y)\phi (x)&=\sum _{p,q\in\Gamma _m}\frac{1}{p_0q_0}\,\left[a(q)a(p)e^{i\pi (px+qy)/2}+a(q)a(p)^*e^{i\pi (qy-px)/2}\right.\\
&\quad\left. +a(q)a(p)^*e^{i\pi (px-qy)/2}+a(q)^*a(p)^*e^{-i\pi (px+qy)/2}\right]\end{aligned}$$ Since ${\left[a(p),a(q)\right]}={\left[a(p)^*,a(q)^*\right]}=0$ and ${\left[a(p),a(q)^*\right]}=\delta _{q,p}I$, we have that $$\begin{aligned}
{\left[\phi (x),\phi (y)\right]}&=\sum _{p\in\Gamma _m}\frac{1}{p_0^2}\,{\left\{{\left[a(p),a(p)^*\right]}e^{i\pi p(x-y)/2}-{\left[a(p),a(p)^*\right]}e^{-i\pi p(x-y)/2}\right\}}I\\
&=2i\sum _{p\in\Gamma _m}\frac{1}{p_0^2}\,\sin{\left[\pi p(x-y)/2\right]}I\\
&=2i\sum _{p\in\Gamma _m}\frac{1}{p_0^2}\,\sin{\left[\pi p_0(x_0-y_0)/2-\pi{\mathbf{p}}{\mathrel{\mathlarger\cdot}}({\mathbf{x}}-{\mathbf{y}})/2\right]}I\\
&=2i\sum _{p\in\Gamma _m}\frac{1}{p_0^2}\,\left[\sin\pi p_0(x_0-y_0)/2\cos\pi{\mathbf{p}}\cdot ({\mathbf{x}}-{\mathbf{y}})/2\right.\\
&\hskip 6pc\left. -\cos\pi p_0(x_0-y_0)/2\sin\pi{\mathbf{p}}{\mathrel{\mathlarger\cdot}}({\mathbf{x}}-{\mathbf{y}})/2\right]I\end{aligned}$$ If $(p_0,{\mathbf{p}})\in\Gamma _m$, then $(p_0,-{\mathbf{p}})\in\Gamma _m$ and the result follows because $\sin$ is an odd function.
The next result is called the *equal-time commutation relation*.
\[cor32\] If $x_0=y_0$, then ${\left[\phi (x),\phi (y)\right]}=0$.
*Quantum locality* says that if ${\left|\left|x-y\right|\right|}_4^2<0$, then ${\left[\phi (x),\phi (y)\right]}=0$. That is, if $x$ and $y$ are space-like separated, then a field measurement at $x$ cannot affect a field measurement at $y$. If $x_0=y_0$ and $x\ne y$ then ${\left|\left|x-y\right|\right|}_4^2<0$ and by Corollary \[cor32\], quantum locality holds. However, in contrast to standard quantum field theory [@vel94], quantum locality does not hold, in general. We show this with the following counterexample. Suppose we have light photons and let $x=(1,2{\mathbf{e}})$, $y=0$ so that ${\left|\left|x-y\right|\right|}_4^2=-3<0$. Applying Theorem \[thm31\] we have that $$\begin{aligned}
{\left[\phi (x),\phi (y)\right]}&=2i\sum _{p\in\Gamma '_0}\frac{1}{p_0^2}\,\sin (\pi p_0/2)\cos\pi{\mathbf{p}}{\mathrel{\mathlarger\cdot}}{\mathbf{e}}I\\
&=4i\sum _{n=1}^\infty\frac{1}{n^2}\,\sin (\pi n/2)\cos\pi nI=4i\sum _{n=1}^\infty\frac{(-1)^n}{n^2}\,\sin\pi n/2I\\
&=-4i\sum _{n=1}^\infty\,\frac{1}{(2n-1)^2}\,\sin (2n-1)\pi /2I=4i\sum _{n=1}^\infty\frac{(-1)^n}{(2n-1)^2}\,I\ne 0\end{aligned}$$
Interacting Quantum Fields
==========================
The important part of quantum fields occurs when we have interactions because then we obtain nontrivial scattering. This section considers interaction Hamiltonians and scattering operators. These are relevant because most of modern theoretical and experimental physics involves some kind of scattering. We shall eventually construct some examples of interaction Hamiltonians, but for now let $H(x_0)$ be self-adjoint operators on $K$ that describe an interaction, where $x_0=0,1,2,\ldots$, represents time. The corresponding *scattering operators* $S(x_0)$ satisfy a “second quantization” equation $$\label{eq41}
\nabla _{x_0}S(x_0)=iH(x_0)S(x_0)$$ Of course, is a generalization of Schrödinger’s equation and for this work $\nabla _{x_0}$ is the difference operator $$\nabla _{x_0}S(x_0)=S(x_0+1)-S(x_0)$$ Starting with the initial condition $S(0)=I$, we obtain from that $$S(1)=I+iH(0)$$ Continuing, we conclude that $$\begin{aligned}
\label{eq42}
S(2)&=S(1)+iH(1)S(1)={\left[I+iH(1)\right]}S(1)={\left[I+iH(1)\right]}{\left[I+iH(0)\right]}\notag\\
S(3)&=S(2)+iH(2)S(2)={\left[I+iH(2)\right]}S(2)={\left[I+iH(2)\right]}{\left[I+iH(1)\right]}{\left[I+iH(0)\right]}\notag\\
\vdots&\notag\\
S(n)&={\left[I+iH(n-1)\right]}{\left[I+iH(n-2)\right]}\cdots{\left[I+iH(1)\right]}{\left[I+iH(0)\right]}\end{aligned}$$
In general, the operators $H(j)$ do not commute so the order in must be retained. This is called a *time ordered product* of operators. Most of the complication in quantum field theory results from trying to solve . In fact, it appears that solving exactly is intractable in general and all we can accomplish is to solve approximately using “perturbation” techniques. For $n\ne 0$, $S(n)$ is not unitary, in general. However, presumably the *limiting scattering operator* $S=\lim\limits _{n\infty}S(n)$ should be unitary in order to preserve probability. Of course, this depends on the interaction Hamiltonian $H(n)$. The author does not know reasonable conditions on $H(n)$ that ensure the unitarity of $S$.
If we multiply out, we obtain the useful form $$\begin{aligned}
\label{eq43}
S(n)&=I+i\sum _{j=0}^{n-1}H(j)+i^2\sum _{j_2<j_1}^{n-1}H(j_1)H(j_2)+i^3\sum _{j_3<j_2<j_1}^{n-1}H(j_1)H(j_2)H(j_3)\notag\\
&\qquad +\cdots +i_nH(n-1)H(n-2)\cdots H(0)\end{aligned}$$ Equation gives a quantum inclusion-exclusion principle with the interaction Hamiltonian again time ordered. The interaction Hamiltonian $H(x_0)$ is usually constructed from an *interaction Hamiltonian density* consisting of self-adjoint operators on $K$ denoted by ${{\mathcal H}}(x)$, $x\in{{\mathcal S}}$. Letting $$V(x_0)={\left|{\left\{{\mathbf{x}}\in{{\mathbb Z}}^3\colon{\left|\left|{\mathbf{x}}\right|\right|}_3\le x_0\right\}}\right|}$$ be the cardinality of the set in brackets (called the *space volume* at $x_0$) we define $$\label{eq44}
H(x_0)=\frac{1}{V(x_0)}\sum{\left\{{{\mathcal H}}(x_0,{\mathbf{x}})\colon{\left|\left|{\mathbf{x}}\right|\right|}_3\le x_0\right\}}$$ The first few terms of are: $H(0)={{\mathcal H}}(0)$ $$\begin{aligned}
H(1)&=\tfrac{1}{7}{\left[{{\mathcal H}}(1,{\mathbf{0}})+{{\mathcal H}}(1,{\mathbf{e}})+{{\mathcal H}}(1,-{\mathbf{e}})+{{\mathcal H}}(1,{\mathbf{f}})+{{\mathcal H}}(1,-{\mathbf{f}})+H(1,{\mathbf{g}})+H(1,-{\mathbf{g}})\right]}\\
H(2)&=\tfrac{1}{33}{\left[{{\mathcal H}}(2,{\mathbf{0}})+{{\mathcal H}}(2,{\mathbf{e}})+\cdots +{{\mathcal H}}(2,2{\mathbf{g}})+{{\mathcal H}}(2,-2{\mathbf{g}})\right]}\end{aligned}$$
We now illustrate this theory with a simplified example that contains the essential elements that can be generalized to more complicated and realistic situations. We consider the scattering of two electrons with mass $m$ and initial energy-momentum $p,q\in{\widehat{{{\mathcal S}}}}$, $p\ne q$. We assume that the electrons interact by exchanging light photons and arrive with final energy-momentum $p',q'\in{\widehat{{{\mathcal S}}}}$. As usual in this article we are neglecting spin so we are really describing spin-zero particles like neutral pions. We say that the *input state* is ${{\left|p,q\right>}}\in K$ and the *output state* is ${{\left|p'q'\right>}}\in K$. Let $\phi (x)$, $\sigma (x)$ be the quantum fields for the electrons and light photons, respectively $$\begin{aligned}
\phi (x)&=\sum _{p\in\Gamma _m}\frac{1}{p_0}{\left[a(p)e^{i\pi px/2}+a(p)^*e^{-i\pi px/2}\right]}\\
\sigma (x)&=\sum _{k\in\Gamma '_0}\frac{1}{k_0}{\left[a(k)e^{i\pi kx/2}+a(k)^*e^{-i\pi kx/2}\right]}\end{aligned}$$
The interaction Hamiltonian density ${{\mathcal H}}(x)$ is frequently constructed by interacting the fields $\phi$ and $\sigma$. A linear combination of $\phi$ and $\sigma$ will not produce any scattering so we take a simple nonlinear combination of the form [@vel94] $$\label{eq45}
{{\mathcal H}}(x)=g\phi (x)^2\sigma (x)$$ where $g$ is called the *coupling constant*. Ideally, we can now find the scattering operator $S$ and compute the *scattering amplitude* ${{\left<p'q'\right|}}S{{\left|pq\right>}}$. The probability of the interaction becomes $$P{\left({{\left|pq\right>}}\to{{\left|p'q'\right>}}\right)}={\left|{{\left<p'q'\right|}}S{{\left|pq\right>}}\right|}^2$$ Unfortunately, we cannot find $S$ exactly so we must be content with computing the lower level perturbation terms, ${{\left<p'q'\right|}}S(n){{\left|pq\right>}}$, $n=0,1,2,\ldots\,$. (We shall stop at $n=2$. The furthest anyone usually can go is $n=5$.) As we shall see, these terms contain a considerable amount of symmetry which could imply conservation of energy-momentum. However, we have not proved this so we shall postulate the conservation law $$\label{eq46}
p'+q'=p+q$$ As one would expect, gives some simplifications.
Applying with $n=2$ we have that $$\begin{aligned}
\label{eq47}
{{\left<p'q'\right|}}S(2){{\left|pq\right>}}&={{\left\langlep'q'\mid pq\right\rangle}}+i{{\left<p'q'\right|}}H(0){{\left|pq\right>}}\notag\\
&\qquad +i{{\left<p'q'\right|}}H(1){{\left|pq\right>}}-{{\left<p'q'\right|}}H(1)H(0){{\left|pq\right>}}\end{aligned}$$
As is usually done, we shall assume that ${{\left|p'q'\right>}}\ne{{\left|pq\right>}}$ [@ps95; @vel94]. We have three reasons for this assumption. One is that ${{\left|pq\right>}}\to{{\left|pq\right>}}$ is not of interest because there is no scattering in this case. Another is that this case is unlikely so it would have small probability. Finally, as we shall later show, this case is very hard to compute. Because of this assumption, the first term in vanishes. The second and third terms contain one $H(n)$ and hence only one $\sigma$-field. This applied to a state ${{\left|pq\right>}}$ that does not have a $\sigma$-particle (light photon) gives 0 for the $a(k)$ part or a state containing a $\sigma$-particle for the $a(k)^*$ part. The inner product of such a state with the state ${{\left|p'q'\right>}}$ containing no $\sigma$-particle is 0. Similarly, any product of an odd number of $H(n)$ gives zero between states without $\sigma$-particles.
We conclude that reduces to $$\label{eq48}
{{\left<p'q'\right|}}S(2){{\left|pq\right>}}=-{{\left<p'q'\right|}}H(1)H(0){{\left|pq\right>}}$$ If we continue to the next order of perturbation, we obtain $$\begin{aligned}
\label{eq49}
{{\left<p'q'\right|}}S(3){{\left|pq\right>}}&=-{{\left<p'q'\right|}}H(1)H(0){{\left|pq\right>}}-{{\left<p'q'\right|}}H(2)H(0){{\left|p\right>}}\notag\\
&\qquad -{{\left<p'q'\right|}}H(2)H(1){{\left|pq\right>}}\end{aligned}$$ As shown earlier, the next term in does not appear in . With some more work we could compute instead of , but to save space we shall only consider . The operator in has the form $$\begin{aligned}
\label{eq410}
H(1)H(0)&=\frac{g^2}{7}\sum _{x,y}\phi (x)^2\sigma (x)\phi (y)^2\sigma (y)\notag\\
&=\frac{g^2}{7}\sum _{x,y}\phi (x)^2\phi (y)^2\sigma (x)\sigma (y)\end{aligned}$$ In order to get a nonzero inner product in , we must have that $H(1)H(0){{\left|pq\right>}}=\alpha{{\left|pq\right>}}$ for some $\alpha\in{{\mathbb C}}$, $\alpha\ne 0$, which we write ${{\left|pq\right>}}\to{{\left|p'q'\right>}}$. As far as $\sigma (x)\sigma (y)$ is concerned, we only have the possibility $a(k)a(k)^*$, $k\in\Gamma '_0$ which corresponds to an exchange of photons. For $\phi (x)^2\phi (y)^2$ we have six possibilities. These can be described by six Feynman diagrams, but we shall employ a symbolic notation.
We could first annihilate $p$ and $q$ using $\phi (y)^2$ and then create $p'$ and $q'$ using $\phi (x)^2$. Another possibility is to first annihilate $p$ and create $p'$ using $\phi (y)^2$ and then annihilate $q$ and create $q'$ using $\phi (x)^2$. The six cases can be described symbolically as follows, with the first two cases given above. [(Case 1)${{\left|pq\right>}}\to{{\left|0\right>}}\to{{\left|p'q'\right>}}$ (Case 2)${{\left|pq\right>}}\to{{\left|p'q\right>}}\to{{\left|p'q'\right>}}$ (Case 3)${{\left|pq\right>}}\to{{\left|q'q\right>}}\to{{\left|p'q'\right>}}$ (Case 4)${{\left|pq\right>}}\to{{\left|pp'\right>}}\to{{\left|p'q'\right>}}$ (Case 5)${{\left|pq\right>}}\to{{\left|pq'\right>}}\to{{\left|p'q'\right>}}$ (Case 6)${{\left|pq\right>}}\to{{\left|pqp'q'\right>}}\to{{\left|p'q'\right>}}$ ]{}
Note that although we have conservation of energy-momentum for the input and output states of ${{\left|pq\right>}}\to{{\left|p'q'\right>}}$, we do not conserve energy-momentum at intermediate times in the interaction. For example, in Case 1 we do not have $p+q=0$. During the interaction, energy-momentum is annihilated and created by the quantum fields.
We now explain specifically why we assume that ${{\left|p'q'\right>}}\ne{{\left|pq\right>}}$. In the situation in which the input and output states are both ${{\left|pq\right>}}$ we have cases like $$\begin{aligned}
{{\left|pq\right>}}&\to{{\left|p'q\right>}}\to{{\left|pq\right>}}\\
{{\left|pq\right>}}&\to{{\left|pq'\right>}}\to{{\left|pq\right>}}\\
{{\left|pq\right>}}&\to{{\left|pqp'q'\right>}}\to{{\left|pq\right>}}\end{aligned}$$ But now $p',q'$ are arbitrary elements of $\Gamma _m$ so we would have to sum over these infinite number of elements which is quite difficult.
The next lemma shows that for a fixed input state ${{\left|pq\right>}}$, there are only a finite number of output states ${{\left|p'q'\right>}}$ satisfying ${{\left|pq\right>}}\to{{\left|p'q'\right>}}$. In this way we can sum probabilities to get various alternatives.
\[lem41\] There exist only finitely many ${{\left|p'q'\right>}}$ such that ${{\left|pq\right>}}\to{{\left|p'q'\right>}}$.
If ${{\left|pq\right>}}\to{{\left|p'q'\right>}}$, then by $p'+q'=p+q$. Hence, $$p'_0,q'_0\le p'_0+q'_0=p_0+q_0$$ We conclude that there are only a finite number of possible values for $p'_0,q'_0$. Since for $p'\in\Gamma _m$ we have that ${\left|\left|{\mathbf{p}}'\right|\right|}_3^2=(p'_0)^2-m^2$, there are only a finite number of possible ${\mathbf{p}}'$ and hence only a finite number of possible $p'$.
Notice that Lemma \[lem41\] is general and does not depend on a particular mass $m$. If ${{\left|p'q'\right>}}\ne{{\left|pq\right>}}$, then we have seen that there are only six possible cases for each of the finite number of scattering situations. We thus have only a finite number of terms to compute.
Scattering Examples
===================
For our first example, we consider the simplified electron scattering process discussed in Section 4. The interaction Hamiltonian density is given by and we want to compute the first order perturbation scattering amplitude . Writing in detail, we have that $$\begin{aligned}
H(1)H(0)&=\frac{1}{7}\,{\left[{{\mathcal H}}(1,{\mathbf{0}})+{{\mathcal H}}(1,{\mathbf{e}})+\cdots +{{\mathcal H}}(1,-{\mathbf{g}})\right]}{{\mathcal H}}(0)\\
&=\frac{g^2}{7}\left[\phi (1,{\mathbf{0}})^2\sigma (1,{\mathbf{0}})+\phi (1,{\mathbf{e}})^2\sigma (1,{\mathbf{e}})\right.\\
&\hskip 3pc\left. +\cdots +\phi (1,-{\mathbf{g}})^2\sigma (1,-{\mathbf{g}})\right]\phi (0)^2\sigma (0)\\
&=\frac{g^2}{7}\left[\phi (1,{\mathbf{0}})^2\phi (0)^2\sigma (1,{\mathbf{0}})\sigma (0)+\phi (1,{\mathbf{e}})^2\phi (0)^2\sigma (1,{\mathbf{e}})\sigma (0)\right.\\
&\hskip 3pc\left. +\cdots +\phi (1,-{\mathbf{g}})^2\phi (0)^2\sigma (1,-{\mathbf{g}})\sigma (0)\right]\end{aligned}$$ Taking the photon field $\sigma$ first, we have that $$\begin{aligned}
\sigma (1,{\mathbf{0}})\sigma (0)&=\sum _{k\in\Gamma '_0}\frac{1}{k_0}\,{\left[a(k)e^{i\pi k_0/2}+a(k)^*e^{-i\pi k_0/2}\right]}\\
&\hskip 3pc{\mathrel{\mathlarger\cdot}}\sum _{k'\in\Gamma '_0}\frac{1}{k'_0}{\left[a(k')+a(k')^*\right]}\end{aligned}$$ By our argument in Section 4, $\sigma (1,{\mathbf{0}})\sigma (0)$ contributes the coefficient $$c_1=6\sum _{n=1}^\infty\frac{1}{n^2}\,e^{i\pi n/2}$$ The next term is $$\begin{aligned}
\sigma (1,{\mathbf{e}})\sigma (0)&=\sum _{k\in\Gamma '_0}\frac{1}{k_0}\,{\left[a(k)e^{i\pi (k_0-k_1)/2}+a(k)^*e^{-i\pi (k_0-k_1)/2}\right]}\\
&\hskip3pc{\mathrel{\mathlarger\cdot}}\sum _{k'\in\Gamma '_0}\frac{1}{k'_0}\,{\left[a(k')+a(k')^*\right]}\end{aligned}$$ Since $k_1=\pm k_0$, this term contributes the coefficient $$\begin{aligned}
c_2&=\sum _{k\in\Gamma '_0}\frac{1}{k_0^2}\,e^{i\pi (k_0-k_1)/2}=\sum _{n=1}^\infty\frac{1}{n_2}\,(1+e^{i\pi n}+4e^{i\pi n/2})\\
&=\frac{\pi ^2}{12}\,+4\sum _{n=1}^\infty\frac{1}{n^2}\,e^{i\pi n/2}\end{aligned}$$ Since the other terms contribute this same coefficient, the nonzero part of $H(1)H(0)$ in becomes $$\begin{aligned}
\label{eq51}
H(1,0)&=\frac{g^2}{7}\left[c_1\phi (1,{\mathbf{0}})^2\phi (0)^2+c_2\phi (1,{\mathbf{e}})^2\phi (0)^2\right.\notag\\
&\hskip 3pc\left. +\cdots +c_2\phi (1,-{\mathbf{g}})^2\phi (0)^2\right]\end{aligned}$$
We now consider the terms of $H(1,0)$ in . We have that $$\begin{aligned}
\label{eq52}
\phi (1,{\mathbf{0}})^2\phi (0)&={\left\{\sum _{p\in\Gamma _m}\frac{1}{p_0}\,{\left[a(p)e^{i\pi p_0/2}+a(p)^*e^{-i\pi p_0/2}\right]}\right\}}^2\notag\\
&\hskip 3pc{\mathrel{\mathlarger\cdot}}{\left\{\sum _{p\in\Gamma _m}\frac{1}{p_0}\,{\left[a(p)+a(p)^*\right]}\right\}}\end{aligned}$$ For the six cases, this term contributes the following coefficients where$\beta = 1/p_0q_0p'_0q'_0$.
[(Case 1)$\beta e^{-i\pi (p'_0+q'_0)/2}$ (Case 2)$\beta e^{i\pi (q_0-q'_0)/2}$ (Case 3)$\beta e^{i\pi (q_0-p'_0)/2}$ (Case 4)$\beta e^{i\pi (p_0-q'_0)/2}$ (Case 5)$\beta e^{i\pi (p_0-p'_0)/2}$ (Case 6)$\beta e^{i\pi (p_0+q_0)/2}$ ]{}
Adding these six coefficients and applying conservation of energy-momentum gives the following contribution of $$d_1=2\beta{\left[\cos\frac{\pi}{2}\,(p_0+q_0)+\cos\frac{\pi}{2}\,(q_0-q'_0)+\cos\frac{\pi}{2}\,(q_0-p'_0)\right]}$$
The next term in is $$\begin{aligned}
\phi (1,{\mathbf{e}})^2\phi (0)^2&={\left\{\sum _{p\in\Gamma _m}\frac{1}{p_0}\,{\left[a(p)e^{i\pi (p_0-p_1)/2}+a(p)^*e^{-i\pi (p_0-p_1)/2}\right]}\right\}}^2\notag\\
&\hskip 3pc{\mathrel{\mathlarger\cdot}}{\left\{\sum _{p\in\Gamma _m}\frac{1}{p_0}\,{\left[a(p)+a(p)^*\right]}\right\}}^2\end{aligned}$$ Again, we have the six cases:
[(Case 1)$\beta e^{-i\pi (p'_0+q'_0-p'_1-q'_1)/2}$ (Case 2)$\beta e^{i\pi (q_0-q'_0-q_1+q'_1)/2}$ (Case 3)$\beta e^{i\pi (q_0-p'_0-q_1+p'_1)/2}$ (Case 4)$\beta e^{i\pi (p_0-q'_0-p_1+q'_1)/2}$ (Case 5)$\beta e^{i\pi (p_0-p'_0-p_1+p'_1)/2}$ (Case 6)$\beta e^{i\pi (p_0+q_0-p_1-q_1)/2}$ ]{}
The sum of these coefficients gives: $$\begin{aligned}
d_2&=2\beta\left[\cos\frac{\pi}{2}\,(p_0+q_0-p_1-q_1)+\cos\frac{\pi}{2}\,(q_0-q'_0-q_1+q'_1)\right.\\
&\hskip 3pc\left. +\cos\frac{\pi}{2}\,(q_0-p'_0-q_1+p'_1)\right]\end{aligned}$$
The next term in has the form $\phi (1,-{\mathbf{e}})^2\phi (0)^2$ and the sum of the corresponding coefficients $d_3$ will be the same as $d_2$ except the $p_1,q_1,p'_1,q'_1$ terms will be the negatives of those in $d_3$. We then have $$\begin{aligned}
d_2+d_3&=4\beta\left[\cos\frac{\pi}{2}\,(p_0+q_0)\cos\frac{\pi}{2}\,(p_1+q_1)\right.\\
&\left. +\cos\frac{\pi}{2}\,(q_0-q'_0)\cos\frac{\pi}{2}\,(-q_1+q'_1)+\cos\frac{\pi}{2}\,(q_0-p'_0)\cos\frac{\pi}{2}\,(-q_1+p'_1)\right]\end{aligned}$$ The other terms will be similar, so adding all these terms gives $$\begin{aligned}
&{{\left<p'q'\right|}}S(2){{\left|pq\right>}}=\frac{-g^2}{7}\,{\left[c_1d_1+c_2(d_2+d_3+d_4+d_5+d_6)\right]}\\
&=\frac{-2g^2}{7}\,\beta\left\{\cos\frac{\pi}{2}\,(p_0+q_0){\left[c_1+2c_2{\left(\cos\frac{\pi}{2}\,(p_1+q_1)+\cos\frac{\pi}{2}\,(p_2+q_2)+\cos\frac{\pi}{2}\,(p_3+q_3)\right)}\right]}\right.\\
\noalign{\medskip}
&\quad +\cos\frac{\pi}{2}\,(q_0-q'_0){\left[c_1+2c_2{\left(\cos\frac{\pi}{2}\,(q'_1-q_1)+\cos\frac{\pi}{2}\,(q'_2-q_2)+\cos\frac{\pi}{2}\,(q'_3-q_3)\right)}\right]}\\
\noalign{\medskip}
&\left.\quad +\cos\frac{\pi}{2}\,(q_0-p'_0){\left[c_1+2c_2{\left(\cos\frac{\pi}{2}\,(p'_1-q_1)+\cos\frac{\pi}{2}\,(p'_2-q_2)+\cos\frac{\pi}{2}\,(p'_3-q_3)\right)}\right]}\right\}\\\end{aligned}$$
We now consider a specific example. Let $p=(2,1,1,1)$, $q=(3,-2,-2,0)$, $p'=(2,-1,1,-1)$ and $q'=(3,0,-2,2)$. In this case, the mass $m=1$ and we have conservation of energy-momentum because $$\begin{aligned}
p+q&=p'+q'=(5,-1,-1,1)\\
\intertext{Moreover,}
\beta&=\frac{1}{p_0q_0p'_0q'_0}=\frac{1}{36}\end{aligned}$$ Applying our previous formula, the amplitude becomes $${{\left<p'q'\right|}}S(2){{\left|pq\right>}}=\frac{-g^2}{126}\,{\left[c_1+2c_2(-1+1-1)\right]}=\frac{g^2}{126}\,(2c_2-c_1)$$ To the first order perturbation we have that $$P{\left({{\left|pq\right>}}\to{{\left|p'q'\right>}}\right)}={\left|{{\left<p'q'\right|}}S(2){{\left|pq\right>}}\right|}^2=\frac{g^4}{(126)^2}\,(2c_2-c_1)^2$$ Since $g$ is unknown, this does not tell us anything. However, we can use this calculation to compare probabilities. Let $p,q$ be as before and let $p''=(2,1,-1,-1)$, $q''=(3,-2,0,2)$. We again obtain $${{\left<p''q''\right|}}S(2){{\left|pq\right>}}=\frac{g^2}{126}\,(2c_2-c_1)$$ which is not surprising due to the symmetry of the situation. It does however, exhibit some consistency. To illustrate a more interesting comparison, suppose the initial state is $p=(3,2,2,0)$, $q=(3,-2,-2,0)$ and the final state is $p'=(3,0,2,2)$, $q'=(3,0,-2,-2)$. We then obtain $${{\left<p'q'\right|}}S(2){{\left|pq\right>}}=\frac{g^2}{126}\,(8c_2-c_1)$$ which is quite different from what we obtained before.
Our second example is a simplified version of electron-proton scattering due to the exchange of photons. As before, we have the electron and photon fields $\phi (x)$, $\sigma (x)$, but now we include a proton field of mass $M$ $$\psi (x)=\sum _{p\in\Gamma _M}{\left[a(p)e^{i\pi px/2}+a(p)^*e^{-i\pi px/2}\right]}$$ Let the interaction Hamiltonian density be ${{\mathcal H}}=g\phi ^2\sigma -g\psi ^2\sigma$ where the negative sign is because the electron and proton have opposite charge. Let $p\in\Gamma _m$ be an initial electron, $q\in\Gamma _M$ be as initial proton and we consider the scattering process ${{\left|pq\right>}}\to{{\left|p'q'\right>}}$ where ${{\left|p'q'\right>}}\ne{{\left|pq\right>}}$ is the final state. As before, we study the first perturbation term where $H(1)H(0)$ now has the form $$\begin{aligned}
H(1)H(0)&=\frac{g^2}{7}\,\sum _{x,y}{\left[\phi (x)^2\sigma (x)-\psi (x)^2\sigma (x)\right]}{\left[\phi (y)^2\sigma (y)-\psi (y)^2\sigma (y)\right]}\\
&=\frac{g^2}{7}\,\sum _{x,y}{\left[\phi (x)^2\phi (y)^2-\phi (x)^2\psi (y)^2-\psi (x)^2\phi (y)^2+\psi (x)^2\psi (y)^2\right]}\sigma (x)\sigma (x)\end{aligned}$$ To map ${{\left|pq\right>}}$ to ${{\left|p'q'\right>}}$, we again have the term $a(k)a(k)^*$ for $k\in\Gamma '_0$ and we obtain the same two constants $c_1$ and $c_2$. For the electron-proton terms we have only two cases:
[(Case 1’)${{\left|pq\right>}}\to{{\left|pq'\right>}}\to{{\left|p'q'\right>}}$ (Case 2’)${{\left|pq\right>}}\to{{\left|p'q\right>}}\to{{\left|p'q'\right>}}$ ]{}
As in , the nonzero part of $H(1)H(0)$ is $$\begin{aligned}
H(1,0)&=\frac{-g^2}{7}\,\left[c_1\phi (1,{\mathbf{0}})^2\psi (0)^2+c_1\psi (1,{\mathbf{0}})^2\phi (0)^2+c_2\phi (1,{\mathbf{e}})^2\psi (0)^2\right.\\
&\quad\left. +c_2\psi (1,{\mathbf{e}})^2\phi (0)^2+\cdots +c_2\phi (1,-{\mathbf{g}})^2\psi (0)^2+c_2\psi (1,-{\mathbf{g}})^2\phi (0)^2\right]\end{aligned}$$ The first term on the right side gives $$\begin{aligned}
\phi (1,{\mathbf{0}})^2\psi (0)^2&={\left\{\sum _{p\in\Gamma _m}\frac{1}{p_0}\,{\left[a(p)e^{i\pi p_0/2}+a(p)^*e^{-i\pi p_0/2}\right]}\right\}}^2\\
&\qquad{\mathrel{\mathlarger\cdot}}{\left\{\sum _{p\in\Gamma _M}\frac{1}{q_0}\,{\left[a(q)+a(q)^*\right]}\right\}}^2\end{aligned}$$ This applies to Case 1’ and contributes the coefficient $$\beta e^{i\pi (p_0-p'_0)/2}$$ In a similar way, the second term on the right side applies to Case 2’ and contributes the coefficient $$\beta e^{i\pi (q_0-q'_0)/2}$$ The third term contributes $$\beta e^{i\pi (p_0-p'_0)/2}e^{i\pi (p'_1-p_1)/2}$$ while the fourth term contributes $$\beta e^{i\pi(q_0-q'_0)/2}e^{i\pi (q'_1-q_1)/2}$$ The other terms are similar. Adding all these terms gives $$\begin{aligned}
{{\left<p'q'\right|}}S(2){{\left|pq\right>}}&=\frac{-g^2\beta}{7}\,\left\{e^{i\pi (p_0-p'_0)/2}{\left[c_1+2c_2\sum _{j=1}^3\cos\frac{\pi}{2}\,(p_j-p'_j)\right]}\right.\\
&\qquad\left. +e^{i\pi (q_0-q'_0)/2}{\left[c_1+2c_2\sum _{j=1}^3\cos\frac{\pi}{2}(q_j-q'_j)\right]}\right\}\end{aligned}$$ Applying conservation of energy-momentum, this becomes $$\label{eq53}
{{\left<p'q'\right|}}S(2){{\left|pq\right>}}=\frac{-2g^2\beta}{7}\,{\left\{\cos\frac{\pi}{2}\,(p_0-p'_0){\left[c_1+2c_2\sum _{j=1}^3\cos\frac{\pi}{2}\,(p_j-p'_j)\right]}\right\}}$$
Notice that is independent of $q$ and $q'$. This may not be true for higher order perturbations such as $S(3)$.
By Lemma \[lem41\], we know that there are only a finite number of possible final states ${{\left|p^jq^j\right>}}$ for the initial state ${{\left|pq\right>}}$. As usual, we assume that ${\mathop{Prob}}{\left({{\left|pq\right>}}\to{{\left|pq\right>}}\right)}$ is small. The *flux* is the number of particles, per unit surface area, per unit time and this is given by the geometric speed $$\frac{{\left|\left|{\mathbf{p}}^j\right|\right|}_3}{p_0^j}+\frac{{\left|\left|{\mathbf{q}}^j\right|\right|}_3}{q_0^j}$$ The *cross section* $\sigma$ is defined to be the probability, per unit time, for unit flux summed over the final states. Since the total time, in our case, is 2, we have that $$\begin{aligned}
\label{eq54}
\sigma&=\frac{1}{2}\,\sum _j{\left[\frac{{\left|\left|{\mathbf{p}}^j\right|\right|}_3}{p_0^j}+\frac{{\left|\left|{\mathbf{q}}^j\right|\right|}_3}{q_0^j}\right]}{\mathop{Prob}}{\left({{\left|pq\right>}}\to{{\left|p^jq^j\right>}}\right)}\notag\\
&=\frac{1}{2}\,\sum _j{\left[\frac{{\left|\left|{\mathbf{p}}^j\right|\right|}_3}{p_0^j}+\frac{{\left|\left|{\mathbf{q}}^j\right|\right|}_3}{q_0^j}\right]}{\left|{{\left<p^jq^j\right|}}S(2){{\left|pq\right>}}\right|}^2\end{aligned}$$
We now restrict our attention to cases of the form $p=(p_0,0,0,p_3)$, $q=(q_0,0,0,0)$. Thus, electrons are moving along the $z$-axis and the proton is initially stationary. Furthermore, we make the approximation that $m=0$ so that $p_3=p_0$ and the electron acts like a photon. This a good approximation because the electron is much less massive than the proton (about 1 to 2000). To be specific, consider the simple example $p=(5,0,0,5)$, $q=(5,0,0,0)$. It is easy to check that conservation of energy-momentum implies that the only possible outgoing state ${{\left|p'q'\right>}}\ne{{\left|pq\right>}}$ is given by $p'=(3,2,2,1)$, $q'=(7,-2,-2,4)$. Substituting these values into gives $${{\left<p'q'\right|}}S(2){{\left|pq\right>}}=\frac{2g^2\beta}{7}\,(c_1+2c_2)$$ Applying , the cross section becomes $$\sigma =\frac{2}{49}\,g^4\beta ^2(c_1+2c_2)^2$$
Finally, we briefly consider particle decay rates and lifetimes. The decay probability per unit time is the *decay rate* and the inverse of the decay rate is the *lifetime*. Suppose we have $\phi$-particles and $\sigma$-particles, but now we assume that the $\sigma$-particle is more massive than two $\phi$-particles so the $\sigma$ decays into two $\phi$’s. We again consider an interaction Hamiltonian density ${{\mathcal H}}=g\phi ^2\sigma$. Let $k$ the initial energy-momentum of the $\sigma$ and $p,q$ the final energy-momentum of the two $\phi$-particles. At the lower order perturbation, we have only one case for the decay, namely ${{\left|k\right>}}\to{{\left|pq\right>}}$. The last term of the scattering operator $$S(2)=I+i{\left[H(0)+H(1)\right]}-H(1)H(0)$$ gives ${{\left<pq\right|}}H(1)H(0){{\left|k\right>}}=0$ and we have $$\begin{aligned}
\label{eq55}
{{\left<pq\right|}}S(2){{\left|k\right>}}&=i{\left[{{\left<pq\right|}}H(0){{\left|k\right>}}+{{\left<pq\right|}}H(1){{\left|k\right>}}\right]}\notag\\
&=ig\biggl\{{{\left<pq\right|}}\phi (0)^2\sigma (0){{\left|k\right>}}+\frac{1}{7}\left[{{\left<pq\right|}}\phi (1,{\mathbf{0}})^2\sigma (1,{\mathbf{0}}){{\left|k\right>}}\right.\biggr.\notag\\
&\quad\biggl.\left. +{{\left<pq\right|}}\phi (1,{\mathbf{e}})^2\sigma (1,{\mathbf{e}}){{\left|k\right>}}+\cdots +{{\left<pq\right|}}\phi (1,-{\mathbf{g}})^2\sigma (1,-{\mathbf{g}}){{\left|k\right>}}\right]\biggr\}\notag\\
&=\frac{ig}{k_0p_0q_0}\,\biggl\{1+\frac{1}{7}\,e^{-i\pi (p_0+q_0)/2}\left[ e^{i\pi k_0/2}+e^{i\pi (k_0-k_1)/2}e^{i\pi (p_1+q_1)/2}\right.\biggr.\notag\\
&\hskip 4pc\biggl.\left. +\cdots +e^{i\pi (k_0-k_3)/2}e^{i\pi (p_3+q_3)/2}\right]\biggr\}\end{aligned}$$
Now suppose we are in the $\sigma$ rest system so that $k=k_0$ and ${\mathbf{k}}=0$. By conservation of energy-momentum, $p_0+q_0=k_0$ and ${\mathbf{p}}+{\mathbf{q}}={\mathbf{0}}$. Then ${\mathbf{p}}=-{\mathbf{q}}$ so $q_0=p_0$ and $k_0=2p_0$. Hence, simplifies to $$\label{eq56}
{{\left<pq\right|}}S(2){{\left|k\right>}}=\frac{8ig}{k_0^3}$$ We can now use for the first approximation to decay rates and lifetimes.
[99]{} S. Gudder, Elementary length topologies in physics, *SIAM J. Appl. Math.* **16**, 1011–1019 (1968). S. Gudder, Discrete quantum gravity and quantum field theory, arXiv: gr-qc 1603.03471v1 (2016). W. Heisenberg, *The Physical Principles of Quantum Mechanics*, University of Chicago Press, Chicago (1930). M. Peskin and D. Schroeder, *An Introduction to Quantum Field Theory*, Addison-Wesely, Reading, Mass. (1995). B. Russell, *The Analysis of Matter*, Dover, New York (1954). R. Streater and A. Wightmann, *PCT, Spin and Statistics and all that*, Benjamin, New York (1964). C. Vanden Eynden, *Elementary Number Theory*, McGraw Hill, Boston, Mass. (1987). M. Veltman, *Diagrammatica*, Cambridge University Press, Cambridge (1994).
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---
abstract: |
A geodesic current on a free group $F$ is an $F$-invariant measure on the set $\partial^2 F$ of pairs of distinct points of the boundary $\partial F$. The main aim of this paper is to introduce and study the notion of *geometric entropy* $h_T(\mu)$ for a geodesic current $\mu$ on a free group $F$ with respect to a point $T$ in the Outer Space $cv(F)$, $T$ thus being an $\mathbb R$-tree equipped with a minimal free and discrete isometric action of $F$. The geometric entropy $h_T(\mu)$ measures the slowest exponential decay rate of the values of $\mu$ on cylinder sets in $T$, with respect to the $T$-length of the segment defining such a cylinder.
We obtain an explicit formula for $h_{T'}(\mu_T)$, where $T,T'\in
cv(F)$ are arbitrary points and where $\mu_T$ is the Patterson-Sullivan current corresponding to $T$, in terms of the volume entropy of $T$ and the extremal distortion of distances in $T$ with respect to distances in $T'$.
We conclude that for $T\in CV(F)$ (where $CV(F)\subseteq cv(F)$ is the projectivized Outer space consisting of all elements of $cv(F)$ with co-volume $1$) and for a Patterson-Sullivan current $\mu_T$ corresponding to $T$, the function $CV(F)\to \mathbb R$ mapping $T'$ to $h_{T'}(\mu_T)$, achieves a strict global maximum at $T'=T$.
We also show that for any $T\in cv(F)$ and any geodesic current $\mu$ on $F$, we have $h_T(\mu)\le h(T)$, where $h(T)$ is the volume entropy of $T$, and the equality is realized when $\mu=\mu_T$. For points $T\in cv(F)$ with simplicial metric (where all edges have length one), we relate the geometric entropy of a current and the measure-theoretic entropy.
address:
- 'Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA http://www.math.uiuc.edu/\~kapovich/'
- 'Section de mathématiques, Université de Genève, 2-4, rue du Lièvre, c.p. 64, 1211 Genève, Switzerland http://www.unige.ch/ tatiana/'
author:
- Ilya Kapovich
- Tatiana Nagnibeda
title: Geometric entropy of geodesic currents on free groups
---
[^1]
Introduction {#intro}
============
In [@CV] Culler and Vogtmann introduced a free group analogue of the Teichmüller space of a hyperbolic surface now known as Culler-Vogtmann’s *Outer space*. The Outer space proved to be a fundamental object in the study of the outer automorphism group of a free group and of individual outer automorphisms.
Let $F$ be a free group of finite rank $k\ge 2$. The *nonprojectivized Outer space $cv(F)$* consists of all minimal free and discrete isometric actions of $F$ on $\mathbb R$-trees. Two trees in $cv(F)$ are considered equal if there exists an $F$-equivariant isometry between them. Note that for every $T\in cv(F)$ the action of $F$ on $T$ is cocompact. There are several topologies on $cv(F)$ that are all known to coincide [@Pau89]: the equivariant Gromov-Hausdorff convergence topology, the point-wise translation length function convergence topology, and the weak $CW$-topology (see Section \[section:cv\] below for more details). There is a natural continuous left action of $Out(F)$ on $cv(F)$ that corresponds to pre-composing an action of $F$ on $T$ with the inverse of an automorphism of $F$. One often works with the projectivized version $CV(F)$ of $cv(F)$, called the *Outer space*, which consists of all $T\in cv(F)$ such that the quotient graph $T/F$ has volume $1$. The space $CV(F)$ is a closed $Out(F)$-invariant subset of $cv(F)$.
A geodesic current is, in the context of negative curvature, a measure-theoretic generalization of the notion of a free homotopy class of a closed curve on a surface and of the notion of a conjugacy class in a group. Let $\partial F$ be the hyperbolic boundary of $F$ and let $\partial^2 F$ be the set of all pairs $(\xi,\zeta)\in \partial F\times\partial F$ such that $\xi\ne \zeta$. There is a natural left translation action of $F$ on $\partial F$ and hence on $\partial^2 F$. A *geodesic current* on $F$ is a positive, finite on compact subsets, Borel measure on $\partial^2 F$ that is $F$-invariant. (One sometimes also requires currents to be invariant with respect to the “flip” map $\partial^2 F\to \partial^2 F$, $(\xi,\zeta)\mapsto (\zeta,\xi)$, but we do not impose this restriction in this paper). The space $Curr(F)$ of all geodesic currents on $F$ is locally compact and comes equipped with a natural continuous action of $Out(F)$ by linear transformations.
The study of geodesic currents in the context of hyperbolic surfaces was initiated by Bonahon [@Bo86; @Bo88]. Bonahon extended the notion of a geometric intersection number between two (free homotopy classes of) closed curves on a hyperbolic surface to a symmetric and mapping-class-group invariant notion of an intersection number between two geodesic currents. He also showed that the Liouville embedding of the Teichmüller space into the space of projectivized geodesic currents extends to a topological embedding of Thurston’s compactification of the Teichmüller space. The study of geodesic currents also proved useful in the context of free groups (see, for example, [@Ma; @Ka; @Ka1; @Ka2; @KL1; @KL2; @KL3; @CHL; @Fra]). Thus in [@Ka; @Ka1] Kapovich constructed a canonical Bonahon-type $Out(F)$-invariant continuous “intersection form” $I:cv(F)\times Curr(F)\to\mathbb R$. In a recent paper [@KL2] Kapovich and Lustig extended this intersection form to the “boundary” of $cv(F)$ and constructed its continuous $Out(F)$-invariant extension $I:\overline{cv}\times Curr(F)\to\mathbb R$. Here $\overline{cv}(F)$ is the closure of $cv(F)$ in the equivariant Gromov-Hausdorff (or the length function) topology. It is known that $\overline{cv}(F)$ consists precisely of all the minimal *very small* isometric actions of $F$ on $\mathbb R$-trees. The projectivization of $\overline{cv}(F)$ gives the *Thurston compactification* $\overline{CV}(F)=CV(F)\cup \partial CV(F)$ of the Outer space $CV(F)$. Motivated by Bonahon’s result, in [@KN] Kapovich and Nagnibeda constructed the *Patterson-Sullivan map* $CV(F)\to\mathbb P Curr(F)$ and proved that this map is an $Out(F)$-equivariant continuous embedding (here $\mathbb P Curr(F)$ is the space of *projectivized geodesic currents* on $F$). Since $\mathbb P Curr(F)$ is compact, the closure of the image of $CV(F)$ under this map gives a compactification of $CV(F)$. However, unlike in the case of hyperbolic surfaces, this compactification is not the same as Thurston’s compactification $\overline{CV}(F)$ of $CV(F)$. Kapovich and Lustig [@KL1] proved moreover that there does not exist a continuous $Out(F)$-equivariant map $\partial CV(F)\to \mathbb PCurr(F)$.
Let $T\in cv(F)$. Note that $T$ is a proper Gromov-hyperbolic geodesic metric space. Denote by $\partial T$ the hyperbolic boundary of $T$ and by $\partial^2 T$ the set of all pairs $(\xi,\zeta)\in \partial^2
T$ such that $\xi\ne \zeta$. Thus for any $(\xi,\zeta)\in \partial^2
T$ there exists a unique bi-infinite (non-parameterized) oriented geodesic line $[\xi,\zeta]\subseteq T$ in $T$ from $\xi$ to $\zeta$. We think of $[\xi,\zeta]\subseteq T$ as the image of an isometric embedding from $\mathbb R$ to $T$, with the correct choice of an orientation on $[\xi,\zeta]$.
Since $F$ acts discretely, isometrically and co-compactly on $T$, the orbit map (for any basepoint in $T$) defines a quasi-isometry $q_T: F\to T$ (where $F$ is taken with any word metric) and hence a canonical $F$-equivariant homeomorphism $\partial q_T: F\to \partial
T$. In turn, $\partial q_T$ defines an $F$-equivariant homeomorphism $\partial^2 q_T: \partial^2F\to\partial^2 T$.
We will use the homeomorphisms $\partial q_T$ and $\partial^2 q_T$ to identify $\partial F$ with $\partial T$ and, similarly, $\partial^2 F$ with $\partial^2 T$. We will often suppress this explicit identification.
The *volume entropy* $h=h(T)$ is defined as $$h(T)=\lim_{R\to\infty}\frac{\log \#\{g\in F: d_T(x_0,gx_0)\le R \}}{R},$$ where $x_0\in T$ is a basepoint. It is well known ([@Coor]) that the limit always exists and does not depend on the choice of a basepoint $x_0\in
T$. It also coincides with the *critical exponent of the Poincaré series* : $$\Pi_{x_0}(s):=\sum_{g\in F} e^{-s\ d_T(x_0,gx_0)}\ ,$$ namely, $\Pi_{x_0}(s)$ converges for all $s>h$ and diverges for all $s<h$. Moreover, for every $x_0\in T$, as $s\to h+$, any weak limit $\nu$ of the probability measures $$\frac{1}{\Pi_{x_0}(s)} \sum_{g\in F} e^{-s\ d(x_0,gx_0)} {\rm Dirac}(gx_0).$$ is a measure supported on $\partial T$. The measure-class of $\nu$ is uniquely determined and does not depend on the choice of $x_0$ or on the choice of a weak limit. Any such $\nu$ is called *a Patterson-Sullivan measure* on $\partial T=\partial F$ corresponding to $T\in cv(F)$.
Furman proved in [@Fur], in a wider context of hyperbolic groups, that there exists a unique, up to a scalar multiple, $F$-invariant and flip-invariant nonzero locally finite measure $\mu_T$ on $\partial^2 T$ in the measure class of $\nu\times \nu$. Such a measure $\mu_T$ is called a *Patterson-Sullivan current* for $T\in cv(F)$. Since $\mu_T$ is unique up to a scalar multiple, its projective class $[\mu_T]$ is called *the projective Patterson-Sullivan current* corresponding to $T\in cv(F)$. Moreover, Furman’s results imply that for $T\in cv(F)$ the projective Patterson-Sullivan current corresponding to $T$ depends only on the projective class $[T]$ of $T$, and thus allow to define the Patterson-Sullivan map $CV(F)\to\mathbb P Curr(F) ; [T]\mapsto\mu_T$. We refer the reader to [@KN] for a more detailed discussion.
Let $T\in cv(F)$. Let $x,y\in T, x\ne y$, and $[x,y]$ denote the unique simplicial geodesic between $x$ and $y$ in $T$. Denote $$\begin{gathered}
Cyl_{[x,y]}^T=Cyl_{[x,y]}:= \{(\zeta_1,\zeta_2)\in \partial^2 F: [x,y]\subseteq
[\partial q_T(\zeta_1),\partial q_T(\zeta_2)] \\ \text{ and the orientations on $[x,y]$ and on
$[\zeta_1,\zeta_2]$ agree}\}\end{gathered}$$ the *two-sided cylinder set corresponding to $[x,y]$*.
For a fixed $T\in cv(F)$, any current $\mu\in Curr(F)$ is uniquely determined by its values on the cylinder sets $Cyl_{[x,y]}\subseteq
\partial^2 F$, where $[x,y]$ varies over all nondegenerate geodesic segments in $T$. Note that since $\mu$ is $F$-invariant, the value $\mu(Cyl_{[x,y]})$ depends only on $\mu$ and the path which is the image of $[x,y]$ in the quotient graph $T/F$. The “weights” $\mu(Cyl_{[x,y]})$ tend to $0$ as $d_T(x,y)\to\infty$, and in many interesting cases, as for example in that of Patterson-Sullivan currents, this convergence is exponential. We introduce the notion of *geometric entropy* $h_T(\mu)$ of $\mu$ with respect to $T$ to measure the slowest exponential rate of decay of the weights $\mu(Cyl_{[x,y]})$ as $d_T(x,y)$ tends to infinity. More precisely (see Definition \[defn:ge\] below): $$h_T(\mu):=\liminf_{d_T(x,y)\to\infty} \frac{-\log \mu (Cyl_{[x,y]})
}{d_T(x,y)}.$$
We first establish some basic properties of geometric entropy in Section \[sect:ge\]. In particular $h_T(\mu)=h_T(c\mu)$ for any $c>0$, $\mu\in Curr(F)$, so that $h_T(\mu)$ depends only on the projective class of $\mu$. We note that for a fixed $\mu\in Curr(F)$ the function $E_\mu:cv(F)\to \mathbb R$, $T\mapsto h_T(\mu)$ is continuous (Proposition \[prop:cont\]). On the other hand, for any $T\in cv(F)$, the function $h_T: Curr(F)\to \mathbb R$, $\mu\mapsto h_T(\mu)$ is highly discontinuous. Indeed, there is a dense subset in $Curr(F)$ consisting of so-called “rational” currents (see Definition 5.1 in [@Ka1]), whose geometric entropy is zero. On the other hand there are many currents with positive geometric entropy.
We obtain an explicit formula for the geometric entropy of a Patterson-Sullivan current $\mu_T$ of $T\in cv(F)$ with respect to an arbitrary $T'\in cv(F)$. The geometric entropy of $\mu_T$ with respect to $T$ coincides with the volume entropy $h(T)$. We then solve two types of extremal problems regarding maximal values of the geometric entropy with either the tree or the current arguments fixed. Our main results are the following.
\[A\] (Corollary \[cor:ps\] and Theorem \[thm:comp1\]). Let $T\in cv(F)$ and let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current corresponding to $T$. Let $h(T)$ be the volume entropy of $T$. Then
1. $h_T(\mu_T)=h(T)$;
2. for any $T'\in cv(F)$ $$h_{T'}(\mu_T)= h(T)\inf_{g\in F\setminus\{1\}}
\frac{||g||_T}{||g||_{T'}}=\frac{h(T)}{\displaystyle \sup_{g\in F\setminus\{1\}} \frac{||g||_{T'}}{||g||_T}} .$$
Here, for $f\in F$ and $T\in cv(F)$, $||f||_T:=\inf_{x\in T} d_T(x,fx)$ is the *translation length* of $f$.
It is known [@Wh; @Ka; @Ka1] that $$\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}=\min_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}},\quad \sup_{f\in F\setminus\{1\}}
\frac{||f||_T}{||f||_{T'}}=\max_{f\in F\setminus\{1\}} \frac{||f||_T}{||f||_{T'}}$$ and, moreover, one can algorithmically find $g,f\in F$ realizing the above equalities in a finite subset of $F\setminus\{1\}$ depending only on $T$. It follows that the geometric entropy as function of $T'$ admits a continuous and strictly positive extension to $\overline{cv}(F)$.
The extremal distortions of the trees $T$ and $T'$ with respect to each other which appear in Theorem \[A\] are key ingredients in the recent construction by Francaviglia and Martino [@FM] of asymmetric metrics on the Outer space. Their construction is inspired by Thurston’s work on mutual extremal stretching factors (extremal Lipshitz constants) of two points in the Teichmüller space [@Thurston].
We further use Theorem \[A\] to compute extremal values of $h_{T'}(\mu_T)$ as function of $T'\in CV(F)$ and show that this function achieves its strict maximum at $T$.
\[B\] (Corollaries \[cor:tw\] and \[cor:inf\]). Let $T,T'\in CV(F)$ be such that $T\ne T'$. Let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current corresponding to $T$ and let $h(T)$ be the volume entropy of $T$. Then
1. for any $T'\in CV(F)$ such that $T'\ne T$ $$h_{T'}(\mu_T)<h_T(\mu_T)=h(T);$$
2. we have $$\inf_{T'\in CV(F)}h_{T'}(\mu_T)=0.$$
We then proceed to study the geometric entropy as a function of $\mu\in Curr(F)$. Although it is highly discontinuous, we compute its maximal value. Given a current $\mu\in Curr(F)$, we consider a family of measures $\{\mu_x\}_{x\in T}$ on $\partial F$ defined by their values on [*one-sided cylinder subsets*]{} of $\partial F$: $$Cyl_{[x,y]}^x := \{\xi\in \partial F: \text{ the geodesic ray }
[x,\partial_T(\xi)] \text{ in $T$ begins with } [x,y]\}\subseteq \partial F,$$ $$\mu_x (Cyl_{[x,y]}^x) := \mu (Cyl_{[x,y]}) .$$ If $\mu\in Curr(F)$, $\mu\ne 0$ then there is $x\in T$ such that $\mu_x\ne 0$ (it is enough to take a segment $[x,y])$ such that $\mu
(Cyl_{[x,y]})\ne 0$). Note however, that the action of $F$ on the set of vertices of $T$ is not necessarily transitive and it may happen that $\mu\ne 0$ but for some vertex $x$ of $T$ we have $\mu_x=0$.
If $\mu_T$ is a Patterson-Sullivan current corresponding to $T$ then $\mu_x$ is a Patterson-Sullivan measure on $\partial F$ corresponding to $T$ (see [@KN]).
\[C\] (Theorem \[thm:vol\] and Corollary \[cor:sharp\].) Let $T\in cv(F)$ and let $h=h(T)$ be the volume entropy of $T$.
1. Let $\mu\in Curr(F)$, $\mu\ne 0$, and let $x\in T$ be such that $\mu_x\ne 0$. Then $$h_T(\mu)\le {\mathbf {HD}} _{\partial T}(\mu_x)\le h_T(\mu_T)=h(T) ,$$ where ${\mathbf {HD}} _{\partial T}(\mu_x)$ is the Hausdorff dimension of $\mu_x$ with respect to $\partial T$ with the metric $d_x$ (definitions are recalled in the beginning of Section \[sect:hd\]).
2. For $T\in cv(F)$ denote by $[T]$ the *projective class* of $T$ that is, the set of all $cT\in cv(F)$ where $c>0$. If $T'\in cv(F)$ is such that $[T']\ne [T]$, then $$h_T(\mu_{T'})< h(T).$$
Part (1) of Theorem \[C\] implies that $$h(T)=h_T(\mu_T)=\max_{\mu\in Curr(F)-\{0\}} h_T(\mu).$$ As was observed in [@KKS], if $T_A\in cv(F)$ is the Cayley graph of $F$ with respect to a free basis $A$ and if $T\in cv(F)$ is arbitrary, then $${\bf HD}_{\partial T}(m_A)=\frac{\log(2k-1)}{\lambda_A(T)}$$ where $k\ge 2$ is the rank of $F$, where $m_A$ is the “uniform” measure on $\partial F$ corresponding to $A$, and where $\lambda_A(T)$ is the “generic stretching factor” of $T$ with respect to $A$. That is, for an element $w_n\in F$ obtained by a simple non-backtracking random walk on $T_A$ of length $n$, we have $||w_n||_T/||w_n||_A\to \lambda_A(T)$ as $n\to\infty$. Note that in this case $h(T_A)=\log(2k-1)$ and $m_A=(\mu_{T_A})_x$ for $x$ being the vertex of the Cayley graph $T_A$ of $F$ corresponding to $1\in F$. Also, obviously $\lambda_A(T)\le \sup_{g\in F\setminus\{1\}}\frac{||g||_T}{||g||_A}$. Thus part (2) of Theorem \[C\] agrees with these observations since it says that $$h_{T}(\mu_{T_A})=\frac{\log(2k-1)}{\sup_{g\in F\setminus\{1\}}\frac{||g||_T}{||g||_A}}\le {\bf HD}_{\partial T}(m_A)=
\frac{\log(2k-1)}{\lambda_A(T)}.$$
These observations also suggest that if $T_0\in cv(F)$ is arbitrary (not necessarily corresponding to a free basis) and if $T\in cv(F)$, one can define the “generic stretching factor” of $T$ with respect to $T_0$ as $\lambda_{T_0}(T):=
\frac{h(T_0)}{{\bf HD}_{\partial T}(\mu_0)}$ where $\mu_0$ is a Patterson-Sullivan measure on $\partial F$ corresponding to $T_0$.
Combining Theorem \[A\] and Theorem \[C\] we obtain:
\[D\] Let $T,T'\in cv(F)$.
1. $$\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}\le \frac{h(T')}{h(T)}\le \sup_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}.$$
2. Suppose that $[T]\ne [T']$. Then $$\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}< \frac{h(T')}{h(T)}< \sup_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}.$$
Here $[T]$ denotes the projective class of a tree $T\in cv(F)$. Part (2) of Corollary \[D\] implies that if $[T]\ne [T']$ and $h(T)=h(T')$ then $$\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}< 1<\sup_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}.$$ Thus there exist $g,f\in F\setminus\{1\}$ such that $||g||_T<||g||_{T'}$ and $||f||_T>||f||_{T'}$. This statement provides an analogue of a theorem of Tad White [@Wh] who proved a similar result for $CV(F)$, that is for the situation where points in $cv(F)$ are normalized by co-volume. The above inequality is an analogue of White’s result for the situation where we normalize points of $cv(F)$ by volume entropy.
In Theorem \[thm:support\] we also bound the geometric entropy $h_T(\mu)$ by the exponential growth rate of the support of $\mu$ (appropriately defined) and observe that currents with support of subexponential growth always have geometric entropy equal to zero.
Suppose that $T\in cv(F)$ is a simplicial tree with all edges of length one, so that we can think of $T$ as $\widetilde\Gamma$ for the finite graph $\Gamma=T/F$ with the standard simplicial metric, and without degree-one vertices. There is a natural shift map $\sigma:\Omega(\Gamma)\to\Omega(\Gamma)$ on the space $\Omega(\Gamma)$ of all semi-infinite reduced edge-paths in $\Gamma$, corresponding to erasing the first edge of a path. The pair $(\Omega(\Gamma),\sigma)$ is an irreducible subshift of finite type. For every finite reduced edge-path $v$ in $\Gamma$ there is a natural cylinder set $Cyl_v\subseteq \Omega(\Gamma)$ consisting of all semi-infinite paths $\gamma\in \Omega(\Gamma)$ that have $v$ as an initial segment. We can think of $\Omega(\Gamma)$ as an analogue of the unit tangent bundle for $\Gamma$. There is also a natural affine correspondence (see [@Ka1] for more details) between the space of geodesic currents $Curr(F)$ and the space $\mathcal M(\Gamma)$ of finite shift-invariant measures on the space $\Omega(\Gamma)$. For $\mu\in Curr(F)$ the corresponding shift-invariant measure $\widehat\mu\in \mathcal M(\Gamma)$ is defined by the condition $\widehat\mu(Cyl_v):=
\mu(Cyl_{[x,y]})$ where $v$ is an arbitrary finite reduced edge path in $\Gamma$ and where $[x,y]$ is a lift of $v$ to $T$. In this setting, for any $\mu\in Curr(F)$, we relate the geometric entropy $h_T(\mu)$ and the measure-theoretic entropy of $\widehat\mu$ (normalized to be a probability measure). We refer the reader to [@Kitchens] for a more detailed discussion regarding measure-theoretic entropy (also known as metric entropy or Kolmogorov-Sinai entropy) of shift-invariant measures on irreducible subshifts of finite type. In particular, it is known that for such subshifts the measure-theoretic entropy of a shift-invariant probability measure never exceeds the topological entropy of the shift and that the equality is realized by a unique shift-invariant probability measure called the *measure of maximal entropy*. For a shift-invariant probability measure $\nu$ on $\Omega(\Gamma)$ we denote its measure-theoretic entropy by $\hbar(\nu)$.
\[E\] (Theorem \[thm:ks\] and Corollary \[cor:ks\].) Let $T\in cv(F)$ be a simplicial tree with simplicial metric and let $\Gamma=T/F$ be the quotient graph (thus all edges in $T$ and $\Gamma$ have length one, and $\Gamma$ is a finite connected graph without degree-one vertices). Let $\mu\in Curr(F)$, $\mu\ne 0$, and let $\widehat\mu\in \mathcal M(\Gamma)$ be the corresponding shift-invariant measure on $\Omega(\Gamma)$, normalized to be a probability measure. Similarly, let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current for $T$ and let $\widehat\mu_T\in \mathcal M(\Gamma)$ be the corresponding shift-invariant measure on $\Omega(\Gamma)$, normalized to have total mass one. Then we have
1. $h_T(\mu)\le \hbar(\widehat\mu)\le h_{topol}(\Omega(\Gamma),\sigma)=h(T)=\hbar(\widehat\mu_T) ;$
2. $h_T(\mu)=h(T)$ if and only if there is $c>0$ such that $c\mu=\mu_T$, that is, $[\mu]=[\mu_T]$ in $\mathbb PCurr(F)$.
(In particular $\widehat\mu_T$ is the measure of maximal entropy for $(\Omega(\Gamma),\sigma)$). Note that Theorem \[E\] implies part (2) of Corollary \[C\] for the case where $T\in cv(F)$ has simplicial metric (all edges have length one).
We believe that a version of Theorem \[E\] should hold for an arbitrary $T\in cv(F)$ and not just for a tree with simplicial metric, that is, for the case where $\Gamma=T/F$ is an arbitrary finite metric graph without degree-one vertices. Indeed, in this general case there is still a natural correspondence between $Curr(F)$ and the space of all $\mathbb R_{\ge 0}$-invariant measures on the space $\Omega_\mathbb R(\Gamma)$ of all locally isometric embeddings $\gamma:[0,\infty)\to \Gamma$, where $\gamma(0)$ is not required to be a vertex. (The space $\Omega_\mathbb R(\Gamma)$ comes equipped with a natural $\mathbb R_{\ge 0}$-action). However, in the case of a non-simplicial metric on $\Gamma$ this $\mathbb R_{\ge 0}$ action does not nicely match the “combinatorial” shift $\sigma$ from Theorem \[E\], which creates unpleasant technical problems with the argument. Proving an analogue of Theorem \[E\] for an arbitrary $T\in cv(F)$ requires first developing basic background machinery and formalism for analyzing $\mathbb R_{\ge 0}$-invariant measures on $\Omega_\mathbb R(\Gamma)$ and matching this information with the combinatorial description of geodesic currents used in this paper. We postpone such analysis till a later date.
Let us note however, in view of the above discussion, that one can directly define an analogue of the notion of [*geometric entropy for any shift-invariant*]{} (or even not necessarily invariant) [*measure on an irreducible subshift of finite type*]{}, and study this notion in its own right. Thus let $A$ be a finite alphabet, let $A^\omega$ be the full shift (the set of all semi-infinite words in $A$), let $\sigma:A^\omega\to A^\omega$ be the standard shift map and let $\Omega\subseteq A^\omega$ be a subshift of finite type. For any finite word $v$ that occurs as an initial segment of some element of $\Omega$ we define $Cyl_v$ to be the set of all $\gamma\in \Omega$ that begin with $v$. If $\mu$ is a finite positive Borel measure on $\Omega$, we can define its *geometric entropy* as: $$h_{geom}(\mu)=\liminf_{|v|\to\infty} \frac{-\log \mu (Cyl_v)
}{|v|},$$ where $|v|$ is the ordinary combinatorial length of the word $v$. Similar to the results above, one can show that $h_{geom}(\mu)\le h_{topol}(\Omega)$ and that if $\mu$ is a shift invariant probability measure then $h_{geom}(\mu)\le \hbar(\mu)\le h_{topol}(\Omega)$. Also, similarly to Theorem \[C\], one can show that $h_{geom}(\mu)\le {\bf HD}_d(\mu)$ with respect to restriction $d$ to $\Omega$ of the standard metric from $A^\omega$. In this paper we concentrate on the setting of geodesic currents since it is more invariant and allows us to avoid choosing $\Omega$, (which would correspond to fixing a particular simplicial tree $T\in cv(F)$).
The authors thank Chris Connell and Seonhee Lim for useful conversations. We also thank the referee for helpful comments.
The Culler-Vogtmann Outer Space {#section:cv}
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The Culler-Vogtmann outer space, introduced by Culler and Vogtmann in a seminal paper [@CV], is a free group analogue of the Teichmüller space of a closed surface of negative Euler characteristic. We refer the reader to the original paper [@CV] and to a survey paper [@Vog] for a detailed discussion of the basic facts listed in this section and for the further references.
Let $F$ be a finitely generated free group of rank $k\ge 2$.
The *non-projectivized outer space* $cv(F)$ consists of all minimal free and discrete isometric actions of $F$ on $\mathbb
R$-trees. Two such trees are considered equal in $cv(F)$ if there exists an $F$-equivariant isometry between them. The space $cv(F)$ is endowed with the equivariant Gromov-Hausdorff convergence topology.
It turns out that every $T\in cv(F)$ is uniquely determined by its *translation length function* $\ell_T: F\to \mathbb R$, where for every $g\in F$ $$\ell_T(g)=||g||_T=\min_{x\in T} d_T(x,gx)$$ is the *translation length* of $g$. Note that $\ell_T(g)=\ell_T(hgh^{-1})$ for every $g,h\in F$. Thus $\ell_T$ can be thought of as a function on the set of conjugacy classes in $F$. The space $cv(F)$ comes equipped with a natural left $Out(F)$-action by homeomorphisms. At the length function level, if $\phi\in Out(F)$, $T\in cv(F)$ and $g\in F$ we have $$\ell_{\phi T}([g])=\ell_T(\phi^{-1} [g]).$$ It is known that the equivariant Gromov-Hausdorff topology on $cv(F)$ coincides with the pointwise convergence topology at the level of length functions. Thus for $T_n,T\in cv(F)$ we have $\lim_{n\to\infty}
T_n=T$ if and only if for every $g\in F$ we have $\lim_{n\to\infty} \ell_{T_n}(g)=\ell_T(g)$.
Points of $cv(F)$ have a more explicit combinatorial description as “marked metric graph structures” on $F$:
Let $\Gamma$ be a finite connected graph without degree-one and degree-two vertices. A *metric graph structure $\mathcal L$* on $\Gamma$ is a function $\mathcal L: E\Gamma\to \mathbb R$ such that for every $e\in E\Gamma$ we have $$\mathcal L(e)=\mathcal L(e^{-1})>0.$$
More generally, define a *semi-metric graph structure $\mathcal L$* on $\Gamma$ to be a function $\mathcal L: E\Gamma\to \mathbb R$ such that for every $e\in E\Gamma$ we have $$\mathcal L(e)=\mathcal L(e^{-1})\ge 0.$$ A semi-metric graph structure $\mathcal L$ on $\Gamma$ is *nondegenerate* if there exists a subforest $Z$ in $\Gamma$ such that $\mathcal L(e)>0$ for every $e\in E\Gamma-EZ$.
For a semi-metric graph structure $\mathcal L$ on $\Gamma$ define the *volume* of $\mathcal L$ as $$vol(\mathcal L)=\sum_{e\in E^+\Gamma} \mathcal L(e)$$ where $E\Gamma=E^+\Gamma\sqcup E^-\Gamma$ is any orientation on $\Gamma$.
If $\mathcal L$ is a semi-metric graph structure on $\Gamma$ and $v=e_1,\dots, e_n$ is an edge-path in $\Gamma$, we denote $$\mathcal L(v)=\sum_{i=1}^n \mathcal L(e_i)$$ and call $\mathcal L(v)$ the $\mathcal L$-*length* of $v$.
For a (finite or infinite) graph $\Delta$, denote by $V\Delta$ the set of all vertices of $\Delta$, and denote by $E\Delta$ the set of all oriented edges of $\Delta$. Combinatorially, we use Serre’s convention regarding graphs. Namely, $\Delta$ comes equipped with three functions: $o:E\Delta\to V\Delta$; $t:E\Delta\to V\Delta$; and ${}^{-1}:E\Delta\to E\Delta$, such that $e^{-1}\ne e$, $(e^{-1})^{-1}=e$ for every $e\in E\Delta$, $o(e)=t(e^{-1})$, and $t(e)=o(e^{-1})$ for every $e\in E\Delta$. The edge $e^{-1}$ is called the *inverse* of $e$. An *orientation* on $\Delta$ is a partition $E\Delta=E^+\Delta\sqcup E^-\Delta$, where for every $e\in E\Delta$ one of the edges $e, e^{-1}$ belongs to $E^+\Delta$ and the other edge belongs to $E^-\Delta$.
An *edge-path* $\gamma$ in $\Delta$ is a sequence of oriented edges which connects a vertex $o(\gamma)$ (origin) with a vertex $t(\gamma)$ (terminus). A path is called *reduced* if it does not contain a back-tracking, that is a path of the form $ee^{-1}$, where $e\in E\Delta$.
If $\gamma=e_1,\dots, e_n$ is an edge-path in $\Delta$, where $e_i\in E\Delta$, we call $n$ the *simplicial length* of $\gamma$ and denote it by $|\gamma|$.
We denote by $\mathcal P(\Delta)$ the set of all finite reduced edge-paths in $\Delta$. For a vertex $x\in V\Delta$, we denote by $\mathcal P_x(\Delta)$ the collection of all $\gamma\in \mathcal P(\Delta)$ that begin with $x$.
Let $\Gamma$ be a finite connected graph without degree-one and degree-two vertices such that $\pi_1(\Gamma)\cong F$. Let $\alpha:F\to \pi_1(\Gamma,p)$ be an isomorphism, where $p$ is a vertex of $\Gamma$. We call such $\alpha$ a *simplicial chart* or a *marking* for $F$.
\[defn:mmgs\] Let $F$ be a free group of finite rank $k\ge 2$. A *marked (semi-)metric graph structure* on $F$ is a pair $(\alpha, \mathcal L$), where $\alpha: F\to \pi_1(\Gamma,p)$ is a simplicial chart for $F$ and $\mathcal L$ is a (semi-)metric structure on $\Gamma$. (A marked semi-metric graph structure $(\alpha, \mathcal L)$ is *non-degenerate* if $\mathcal L$ is nondegenerate.)
\[conv:mgs\] Let $(\alpha, \mathcal L)$ be a marked metric graph structure on $F$. Then $(\alpha, \mathcal L)$ defines a point $T\in cv(F)$ as follows. Topologically, let $T=\widetilde
\Gamma$, with an action of $F$ on $T$ via $\alpha$. We lift the metric structure $\mathcal L$ from $\Gamma$ to $T$ by giving every edge in $T$ the same length as that of its projection in $\Gamma$. This makes $T$ into an $\mathbb R$-tree equipped with a minimal free and discrete isometric action of $F$. Thus $T\in cv(F)$ and in this situation we will sometimes use the notation $T=(\alpha, \mathcal L)\in cv(F)$. Note that $T/F=\Gamma$. Moreover, it is not hard to see that every point of $cv(F)$ arises in this fashion and that $CV(F)$ is exactly the set of all those $T=(\alpha, \mathcal L)\in cv(F)$ where $(\alpha,
\mathcal L)$ is a marked metric graph structure on $F$ with $vol(\mathcal L)=1$.
Note also that any nondegenerate marked semi-metric graph structure on $F$ also defines a point in $cv(F)$ by first contracting the edges of $\mathcal L$-length $0$, and then proceeding as above.
\[defn:ec\] Let $\alpha: F\to \pi_1(\Gamma,p)$ be a simplicial chart for $F$. Fix an orientation $E\Gamma=E^+\Gamma\cup E^-\Gamma$ on $\Gamma$ and let $E^+\Gamma=\{e_1,\dots, e_m\}$, where $m=\#E^+\Gamma$.
Let $V_\alpha\subseteq cv(F)$ be the set of all $T=(\alpha, \mathcal L)$ where $\mathcal L$ is a nondegenerate semi-metric structure on $\Gamma$. Let $U_\alpha$ be the set of all $T=(\alpha, \mathcal L)$ where $\mathcal L$ is a metric structure on $\Gamma$. Thus $U_\alpha\subseteq V_\alpha$. We call $V_\alpha$ the *elementary chart* corresponding to $\alpha$ and we call $U_\alpha$ the *elementary open chart* corresponding to $\alpha$.
There is a natural map $\lambda_\alpha: V_\alpha\to \mathbb R^m$ defined as $\lambda_\alpha (\alpha,\mathcal L)=(\mathcal
L(e_1),\dots, \mathcal L(e_m))$. It is known that $\lambda_\alpha: V_\alpha\to \mathbb R^m$ is injective and is a homeomorphism onto its image. In particular, $\lambda_\alpha(U_\alpha)$ is the positive open cone in $\mathbb R^m$, that is, $\lambda_\alpha(U_\alpha)$ consists of all points in $\mathbb R^m$ all of whose coordinates are positive. Therefore $U_\alpha$ is homeomorphic to an open cone in $\mathbb R^m$.
The space $cv(F)$ is the union of open cones $U_\alpha$ taken over all simplicial charts $\alpha$ on $F$. Moreover, every point $T\in cv(F)$ belongs to only finitely many of the elementary charts $V_\alpha$. It is also known that the standard topology on $cv(F)$ coincides with the weakest topology for which all the maps $\lambda_\alpha^{-1}: \lambda_\alpha(V_\alpha)\to V_\alpha$ are continuous.
Let $\alpha:F\to \pi_1(\Gamma,p)$ be a simplicial chart, let $T=\widetilde \Gamma$ and let $j:T\to \Gamma$ denote the covering. It is easy to see that for $\mu_n, \mu\in Curr(F)$ we have $\displaystyle\lim_{n\to\infty}\mu_n=\mu$ if and only if $\displaystyle\lim_{n\to\infty}\mu_n(Cyl_{\gamma})=\mu(Cyl_{\gamma})$ for every $\gamma \in \mathcal P(T)$. Moreover, for $\mu,\mu'\in
Curr(F)$ we have $\mu=\mu'$ if and only if $\mu(Cyl_{\gamma})=\mu'(Cyl_{\gamma})$ for every $\gamma\in \mathcal
P(T)$.
Note that for any $f\in F$ and $\gamma\in \mathcal P(T)$ we have $f
Cyl_{\gamma}=Cyl_{f \gamma}$. Since geodesic currents are, by definition, $F$-invariant, for a geodesic current $\mu$ and for $\gamma\in \mathcal P(T)$ the value $\mu(Cyl_{\gamma})$ only depends on the label $j(\gamma)\in \mathcal P(\Gamma)$ of $\gamma$.
\[not:sp\] For this reason, for any reduced edge-path $v\in \mathcal P(\Gamma)$, we denote by $\langle v,\mu\rangle_\alpha$ the value $\mu(Cyl_{\gamma})$ where $\gamma\in \mathcal P(T)$ is any reduced edge-path with label $v$.
We finish this Section with the following basic lemma, needed later, which shows that for $T,T'\in cv(F)$ extremal distortions of the translation length functions for $T$ and $T'$ give the optimal stretching constants for $F$-equivariant quasi-isometries between $T$ and $T'$.
\[lem:stretch\] Let $T,T'\in cv(F)$ and let $\phi:T\to T'$ be an $F$-equivariant quasi-isometry. Then the following hold:
1. $$\lambda_1:=\inf_{g\in F\setminus\{1\}}
\frac{||g||_{T'}}{||g||_T}=\liminf_{d_T(x,y)\to\infty} \frac{d_{T'}(\phi(x),\phi(y))}{d_T(x,y)}.$$
2. $$\lambda_2:=\sup_{g\in F\setminus\{1\}}
\frac{||g||_{T'}}{||g||_T}=\limsup_{d_T(x,y)\to\infty} \frac{d_{T'}(\phi(x),\phi(y))}{d_T(x,y)}.$$
3. There is $C>0$ such that for any $x,y\in T$ we have $$\lambda_1 d_T(x,y)-C\le d_{T'}(\phi(x),\phi(y))\le \lambda_2 d_T(x,y)+C .$$
Let $x_0\in T$ be a base-point and let $x_0'=\phi(x_0)\in T'$. Thus $\phi(gx_0)=gx_0'$ for all $g\in F$.
Let us show Part (2). For any $g\in F\setminus\{1\}$ we have $$\lim_{n\to\infty} \frac{d_T(x_0,g^nx_0)}{n}=||g||_T, \quad \lim_{n\to\infty} \frac{d_{T'}(x_0',g^nx_0')}{n}=||g||_{T'}$$ and hence $$\lim_{n\to\infty} \frac{d_{T'}(x_0',g^nx_0')}{d_T(x_0,g^nx_0)}=\frac{||g||_{T'}}{||g||_T}.$$ Then $\frac{||g||_{T'}}{||g||_T}\le \limsup_{d_T(x,y)\to\infty}
\frac{d_{T'}(\phi(x),\phi(y))}{d_T(x,y)}$ and so
$$\sup_{g\in F\setminus\{1\}}
\frac{||g||_{T'}}{||g||_T}\le \limsup_{d_T(x,y)\to\infty} \frac{d_{T'}(\phi(x),\phi(y))}{d_T(x,y)}.$$
Suppose now that $d_T(p_n,q_n)\to\infty$ and $$\lim_{n\to\infty} \frac{d_{T'}(\phi(p_n),\phi(q_n))}{d_T(p_n,q_n)}= \limsup_{d_T(x,y)\to\infty} \frac{d_{T'}(\phi(x),\phi(y))}{d_T(x,y)}.$$
There is a constant $M=M(T')\ge 1$ such that there are some $g_n,h_n\in F$ with $d_T(p_n,g_nx_0),
d_T(q_n,h_nx_0)\le M$ and such that the geodesic $[g_nx_0,h_nx_0]$ projects to a closed cyclically reduced path in $T/F$. That is, the points $g_nx_0,h_nx_0$ belong to the axis of the element $h_ng_n^{-1}$ and $d_T(g_nx_0,h_nx_0)=||h_ng_n^{-1}||_T$. Translating $[p_n,q_n]$ by $g_n^{-1}$ we may assume that $g_n=1$ for every $n\ge 1$. Thus $d(x_0,h_nx_0)=||h_n||_T$ and $d_T(p_n,x_0), d_T(q_n,h_nx_0)\le M$. Hence $| d_T(p_n,q_n)-||h_n||_T|\le 2M$. Note that $x_0$ and $h_nx_0$ belong to the axis of $h_n$ in $T$.
Denote $p_n'=\phi(p_n)$, $q_n'=\phi(q_n)$. Since $\phi$ is an $F$-equivariant quasi-isometry, the $\phi$-image of the axis of $h_n$ in $T$ is an $h_n$-invariant quasigeodesic in $T'$ which is at a bounded Hausdorff distance from an $h_n$-invariant geodesic in $T'$, that is, from the axis of $h_n$ in $T'$. Hence there is some constant $C\ge 1$ such that $| d_{T'}(p_n',q_n')-||h_n||_{T'}|\le C$.
Therefore $$\lim_{n\to\infty} \frac{d_{T'}(\phi(p_n),\phi(q_n))}{d_T(p_n,q_n)}\le
\limsup_{n\to\infty} \frac{||h_n||_T+M}{||h_n||_{T'}-C}\le
\sup_{g\in F\setminus\{1\}}
\frac{||g||_{T'}}{||g||_T}.$$ Hence $$\sup_{g\in F\setminus\{1\}}
\frac{||g||_{T'}}{||g||_T}=\limsup_{d_T(x,y)\to\infty} \frac{d_{T'}(\phi(x),\phi(y))}{d_T(x,y)},$$ as required.
Part (1) is established using a similar argument to part (2) and we omit the details.
For part (3), let $x,y\in T$ be arbitrary and let $x'=\phi(x)$, $y'=\phi(y)$. As in the proof of (2), there exist elements $g,h\in F$ such that $d_T(x,gx_0),
d_T(y,hx_0)\le M$ and such that the geodesic $[gx_0,hx_0]$ projects to a closed cyclically reduced path in $T/F$. By exactly the same argument as above we deduce $$\begin{gathered}
d_{T'}(x',y')\le ||h||_{T'}+C\le \lambda_2||h||_T+C\le \\
\lambda_2 (d_T(x,y)+2M)+C=\lambda_2 d_T(x,y)+(2M\lambda+C),\end{gathered}$$ as required. The proof that $d_{T'}(x',y')\ge\lambda_1 d_T(x,y)-(2M\lambda+C)$, is similar, and we omit the details.
Geometric entropy of a geodesic current {#sect:ge}
=======================================
\[defn:ge\] Let $\mu\in Curr(F)$ and let $T\in cv(F)$. Define the *geometric entropy of $\mu$ with respect to $T$* as $$h_T(\mu):=\liminf_{\overset{d_T(x,y)\to\infty}{x,y\in T}} \frac{-\log \mu (Cyl_{[x,y]})
}{d_T(x,y)}.$$
If $\mu (Cyl_{[x,y]})=0$, we interpret $\log 0=-\infty$. Thus for $\mu=0$ and any $T\in cv(F)$ we have $h_T(\mu)=\infty$. For any $\mu\in Curr(F)$, $\mu\ne 0$ and any $T\in cv(F)$ we have $0\le h_T(\mu)<\infty$. Informally, $h_{T}(\mu)$ measures the slowest exponential decay rate (with respect to $d_T(x,y)$) of the “weights” $\mu (Cyl_{[x,y]})$ as $d_T(x,y)\to\infty$. Thus, taking into account that $T$ is a discrete $\mathbb R$-tree, we see that if $h_{T}(\mu)>s> 0$ then there exists $C>0$ such that for every $x,y\in T$ with $d_T(x,y)\ge 1$ we have $$\mu (Cyl_{[x,y]}) \le C\ \exp(-s\, d_T(x,y)).$$
The following is a more combinatorial interpretation of the geometric entropy that follows immediately from unraveling the definitions (we use notation \[not:sp\]).
Let $\mu\in Curr(F)$, $\mu\ne 0$ and let $T\in cv(F)$ be determined by the pair $(\alpha, \mathcal L)$, where $\alpha: F\to\pi_1(\Gamma,p)$ is an isomorphism and $\mathcal L$ is a metric graph structure on $\Gamma$. Then
1. $$h_T(\mu):=\liminf_{\overset{|v|\to\infty}{v\in \mathcal P(\Gamma)}} \frac{-\log \langle v, \mu\rangle_\alpha}{\mathcal L(v)}=\liminf_{\overset{\mathcal L(v)\to\infty}{v\in \mathcal P(\Gamma)}} \frac{-\log \langle v, \mu\rangle_\alpha
}{\mathcal L(v)} ;$$
2. if $h_{T}(\mu)>s\ge 0$ then there exists $C>0$ such that, for every $v\in \mathcal P(\Gamma)$ $$\langle v,\mu\rangle_\alpha \le C\ \exp(-s \mathcal L(v)).$$
We already noted that if $\mu=0$, we get $h_T(\mu)=\infty$ for any $T\in cv(F)$. On the other hand, there is a family of so-called rational currents $\{\eta_g\}_ {g\in F\setminus\{1\}}$, such that for any $T\in cv(F)$, $h_T(\eta_g)=0$.
The current $\eta_g$ is defined as follows: for every Borel subset $S\subseteq\partial F^2$, $\eta_g(S)$ is equal to the number of $F$-translates of the bi-infinite geodesic $(g^{-\infty},g^{\infty})$ that belong to $S$.
The following statement is an immediate corollary of the definition of geometric entropy:
Let $\mu\in Curr(F), \mu\ne 0$ and let $T\in cv(F)$.
1. For any $c>0$ we have $h_T(\mu)=h_T(c\mu)$.
2. For any $c>0$ we have $h_{cT}(\mu)= \frac{1}{c}h_T(\mu)$.
Thus $h_T(\mu)$ depends only on the projective class $[\mu]\in \mathbb
PCurr(F)$ of $\mu$.
\[lem:qi\] Let $T,T'\in cv(F)$. Let $\phi: T\to T'$ be an $F$-equivariant quasi-isometry. There exists an integer $M=M(\phi)\ge 1$ with the following property.
Let $x_1,x_2\in T$ and let $x_1'=\phi(x_1), x_2'=\phi(x_2)$. Let $y_1,y_2\in [x_1',x_2']$ be such that $d_{T'}(x_1',y_1)=d_{T'}(x_2',y_2)=M$. Then $$\phi(Cyl_{[x_1,x_2]})\subseteq Cyl_{[y_1,y_2]}.$$
This statement is a straightforward consequence of the “Bounded Cancellation Lemma” [@Coo] for quasi-isometries between Gromov-hyperbolic spaces. Indeed, let $[x,y]\subseteq T$ be a geodesic segment in $T$ and let $\gamma$ be a geodesic ray in $T$ with initial point $y$, such that the path $[x,y]\gamma$ is a geodesic (that is, there is no cancellation between $[x,y]$ and $\gamma$. There is a unique geodesic ray $\gamma'$ in $T'$ starting at $\phi(y)$ such that $\gamma'$ is at a finite Hausdorff distance from $\phi(\gamma)$. The “Bounded Cancellation Lemma” implies that the cancellation between $[\phi(x),\phi(y)]$ and $\gamma'$ is bounded by some constant $M\ge 1$ depending only on the quasi-isometry $\phi$. It is not hard to check that this constant $M$ satisfies the requirements of the proposition and we leave the details to the reader.
\[prop:qi1\] Let $T,T'\in cv(F)$. Let $\phi: T\to T'$ be an $F$-equivariant quasi-isometry. There exists an integer $M_1=M_1(\phi)\ge 1$ with the following property.
Let $x_1,x_2\in T$ and let $x_1'=\phi(x_1), x_2'=\phi(x_2)$.
Then there exist points $p_1,p_2, q_1,q_2\in T'$ such that $[p_1,p_2]=[x_1',x_2']\cap [q_1,q_2]$, such that $d(q_i,x_i')\le M_1$ and such that $$Cyl_{[q_1,q_2]}\subseteq \phi(Cyl_{[x_1,x_2]}).$$
Let $\psi: T'\to T$ be an $F$-equivariant quasi-isometry which is a quasi-inverse of $\phi$. Let $M=M(\psi)>0$ be the constant provided by Lemma \[lem:qi\] for $\psi$.
Choose $\xi,\zeta\in \partial T$ such that the bi-infinite geodesic $[x_1,x_2]\subseteq [\xi,\zeta]$. Then $x_1'=\phi(x_1)$ and $x_2'=\phi(x_2)$ are at distance at most $C_1=C_1(\phi)$ from $[\phi(\xi), \phi(\zeta)]$. Thus $[\phi(\xi), \phi(\zeta)]\cap [x_1',x_2']=[p_1,p_2]$, where $d(x_i',p_i)\le C_1$.
Recall that $\psi$ is a quasi-isometry that is a quasi-inverse of $\phi$, so that $\psi(\phi(\xi))=\xi$ and $\psi(\phi(\zeta))=\zeta$. Note also that $[x_1,x_2]\subseteq [\xi,\zeta]=[\psi(\phi(\xi)),\psi(\phi(\zeta))]$ and that $[p_1,p_2]\subseteq
[\phi(\xi),\phi(\zeta)]$. Therefore there is some $C_2=C_2(\phi)>0$ such that if $q_1\in [\phi(\xi), p_1]$, $q_2\in [p_2, \phi(\zeta)]$ are such that $d(q_i,p_i)\ge C_2$ then $[x_1,x_2]\subseteq [\psi(q_1),\psi(q_2)]$ and $d(x_i,\psi(q_i))\ge M=M(\psi)$.
Choose $q_1\in [\phi(\xi), p_1]$, $q_2\in [p_2, \phi(\zeta)]$ such that $d(q_i,p_i)=C_2$. Thus $[x_1,x_2]\subseteq
[\psi(q_1),\psi(q_2)]$ and $d(x_i,\psi(q_i))\ge M$. Note that $d(x_i,\psi(q_i))\le C_3=C_3(\psi)$ since $\psi$ is a quasi-isometry and $d(x_i',q_i)\le C_2+C_1$.
Thus $[x_1,x_2]\subseteq [\psi(q_1),\psi(q_2)]$ and $d(x_i,\psi(q_i))\ge M=M(\psi)$. Then by Lemma \[lem:qi\], applied to $\psi$, we have $$\psi(Cyl_{[q_1,q_2]})\subseteq Cyl_{[x_1,x_2]}.$$ Applying $\phi$, we obtain $$Cyl_{[q_1,q_2]}\subseteq \phi(Cyl_{[x_1,x_2]}).$$ Note that by construction $[p_1,p_2]=[x_1',x_2']\cap [q_1,q_2]$. Moreover, $d(q_i,x_i')\le C_1+C_2$. Thus all the requirements of the proposition are satisfied, which completes the proof.
\[cor:0\] Let $\mu\in Curr(F)$ and let $T,T'\in cv(F)$. Then $$h_{T}(\mu)>0 \iff h_{T'}(\mu)>0.$$
Suppose that $h_{T'}(\mu)>0$. Hence there exists $s>0$, $C>0$ such that for every $x,y\in T'$, $x\ne y$, we have $$\mu(Cyl_{[x,y]}) \le C \exp( -s\, d_{T'}(x,y)).$$ Let $\phi:T\to T'$ be a $F$-equivariant $(\lambda,\lambda)$-quasi-isometry, where $\lambda\ge 1$ (i.e. a quasi-isometry with both the multiplicative and the additive constants equal to
$\lambda$). Let $M,C>0$ be provided by Lemma \[lem:qi\]. Let $x_1,x_2\in
T$ be such that $N:=d_{T}(x_1,x_2)>20\lambda^2M$. Let $x_i'=\phi(x_i)\in
T'$, $i=1,2$. Thus $d_{T'}(x_1',x_2')\ge N/\lambda-\lambda\ge N/{2\lambda}$. Let $y_1,y_2\in [x_1',x_2']$ be such that $d_{T'}(x_1',y_1)=d_{X'}(x_2',y_2)=M$. Thus $$d_{T'}(y_1,y_2)\ge \frac{N}{2\lambda}-2M\ge \frac{N}{3\lambda}=\frac{d_T(x_1,x_2)}{3\lambda}.$$
By Lemma \[lem:qi\] we have: $$\phi(Cyl_{[x_1,x_2]})\subseteq Cyl_{[y_1,y_2]}$$
Recall that under the identifications $\partial q_T:\partial T\rightarrow \partial
F$ and $\partial q_{T'}:\partial T'\rightarrow\partial F$, for any $S\subseteq \partial T$, $\partial q_T(S) = \partial q_{T'}(\phi(S))\subseteq\partial F$. Therefore $$\begin{gathered}
\mu(Cyl_{[x_1,x_2]})\le \mu(Cyl_{[y_1,y_2]})\le C \exp( -s
d_{T'}(y_1,y_2))\le\\
\le C \exp(-s\frac{d_T(x_1,x_2)}{3\lambda}).\end{gathered}$$ This implies that $h_T(\mu)\ge \frac{s}{3\lambda}>0$, as required.
Corollary \[cor:0\] implies that the following notion is well-defined. We say that a current $\mu\in Curr(F)$ has *exponential decay* or *decays exponentially fast* if for some (equivalently, for any) $T\in cv(F)$ we have $h_T(\mu)>0$. Similarly, we say that $\mu$ has *subexponential decay* if for some (equivalently, for any) $T\in cv(F)$ we have $h_T(\mu)=0$.
Recall that for a graph $\Gamma$ we denote by $\mathcal P(\Gamma)$ the set of all nontrivial reduced finite edge-paths in $\Gamma$.
\[prop:cont\] Let $\mu\in Curr(F), \mu\ne 0$. Then the function $$E_\mu: cv(F)\to \mathbb R, \quad T\mapsto h_T(\mu)$$ is continuous.
It suffices to check that for every simplicial chart $\alpha:F\to
\pi_1(\Gamma,p)$ on $F$ the restriction of $E_\mu$ to the elementary chart $V_\alpha\subseteq cv(F)$ is continuous.
Let $E\Gamma=E^+\Gamma\sqcup E^-\Gamma$ be an orientation on $\Gamma$, let $m=\#E^+\Gamma$ and let $E^+\Gamma=\{e_1,\dots, e_m\}$. Recall that $V_\alpha$ consists of all points of the form $(\alpha,
\mathcal L)$, where $\mathcal L$ is a nondegenerate semi-metric structure on $\Gamma$.
Let $T=(\alpha, \mathcal L)\in U_{\alpha}$. Let $\epsilon>0$ be arbitrary. Choose $\epsilon_1>0$ such that $\epsilon_1 h_T(\mu)<\epsilon$. Then there exists a neighborhood $\Omega$ of $T$ in $V_\alpha$ with the following property. For any $v\in \mathcal P(\Gamma)$ and for any $T'=(\alpha, \mathcal L')\in \Omega$ we have $$(1-\epsilon_1) \mathcal L'(v)\le \mathcal L(v)\le (1+\epsilon_1) \mathcal L'(v).$$ Therefore $$\frac{-\log \langle v, \mu\rangle_\alpha}{\mathcal L(v)}(1-\epsilon_1)\le \frac{-\log \langle v, \mu\rangle_\alpha}{\mathcal L'(v)}\le \frac{-\log \langle v, \mu\rangle_\alpha}{\mathcal L(v)}(1+\epsilon_1).$$ It follows that $$h_T(\mu)(1-\epsilon_1)\le h_{T'}(\mu)\le h_T(\mu)(1+\epsilon_1)$$ and hence $$|E_\mu(T)-E_\mu(T')|=|h_{T'}(\mu)-h_T(\mu)|\le h_\mu(T)\epsilon_1\le \epsilon.$$
Thus $E_\mu$ is continuous at the point $T$ and, since $T\in V_\alpha$ was arbitrary, $E_\mu$ is continuous on $V_\alpha$, as required.
\[notation:support\] Let $\alpha: F\to \pi_1(\Gamma)$ be a simplicial chart and let $\mu\in Curr(F)$ be a geodesic current. Define $$supp_\alpha(\mu):=\{v\in \mathcal P(\Gamma): \langle v,\mu\rangle_\alpha>0\}.$$ We refer to $supp_\alpha(\mu)$ as the *support of $\mu$ with respect to $\alpha$*.
Let $\mathcal L$ be a metric graph structure on $\Gamma$ and let $T_{(\alpha, \mathcal L)}\in cv(F)$ correspond to the universal cover of $\Gamma$ with the edge-length lifted from $\mathcal L$. Define $$\rho(T_{(\alpha,\mathcal L)},\mu):=\liminf_{R\to\infty} \frac{\log\beta(R)}{R}$$ where $\beta(R)=\#\{v\in supp_\alpha(\mu): \mathcal L(v)\le R\}$. Thus $\rho(T_{(\alpha,\mathcal L)})$ measures the exponential growth rate of the support $supp_\alpha(\mu)$ with respect to the metric structure $\mathcal L$.
Let now $\mu\in Curr(F)$, $\mu\ne 0$ and let $T\in cv(F)$ be arbitrary. There exists an (essentially unique) marked metric graph structure $(\alpha,\mathcal L)$ such that $T=T_{(\alpha,\mathcal L)}$. Put $\rho(T,\mu):=\rho(T_{(\alpha,\mathcal L)})$.
It is not hard to show that if $\rho(T_0,\mu)=0$ for some $T_0\in cv(F)$ then $\rho(T,\mu)=0$ for every $T\in cv(F)$.
\[thm:support\] Let $T\in cv(F)$ and $\mu\in Curr(F)$, $\mu\ne 0$. Then $$h_T(\mu)\le \rho(T,\mu)$$
We may assume that $T=T_{(\alpha,\mathcal L)}$ for some marked metric graph structure $(\alpha:F\to\pi_1(\Gamma),\mathcal L)$ on $F$. If $h_T(\mu)=0$, there is nothing to prove. Suppose now that $h_T(\mu)>0$. Let $s>0$ be arbitrary such that $s<h_T(\mu)$. Thus there exists $n_0\ge 1$ such that for any $p,q\in T$ with $d_T(p,q)\ge n_0$ we have $\mu(Cyl_{[p,q]})\le \exp(-sd_T(p,q))$.
Since $\mu\ne 0$, there exists an edge $e$ of $\Gamma$ such that $\langle e,\mu\rangle_\alpha>0$, that is $e\in supp_\alpha(\mu)$. Let $[x,y]\subseteq T$ be a lift of $e$ to $T$. Thus $\mu(Cyl_{[x,y]})=\langle e,\mu\rangle_\alpha>0$. For any integer $n\ge n_0$ let $[x,z_1],\dots [x,z_m]$ be all geodesic segments of length $n$ in $T$ that contain $[x,y]$ as an initial segment and such that $\mu(Cyl_{[x,p_i]})>0$, where $p_i=z_i$ if $z_i$ is a vertex of $T$, whereas if $z_i$ is an interior point of an edge, $p_i$ is the endpoint of that edge further away from $x$ than $z_i$. Then $\mu(Cyl_{[x,p_i]})\le \exp(-sn)$. Moreover, the cylinders $Cyl_{[x,p_1]},\dots, Cyl_{[x,p_m]}$ are disjoint and $\mu(Cyl_{[x,y]})=\sum_{i=1}^m \mu(Cyl_{[x,p_i]})$. Let $v_i\in \mathcal P(\Gamma)$ be the path in $\Gamma$ which labels the segment $[x,p_i]$ in $T$. Note that by construction $n\le \mathcal L(v_i)\le n+c$, where $c>0$ is the length of the longest edge in $(\Gamma,\mathcal L)$. Thus, again, by construction, $m\le \beta(n+c)$. We have $$0<\langle e,\mu\rangle_\alpha=\sum_{i=1}^m \langle v_i,\mu\rangle_\alpha\le m \exp(-sn)\le \beta(n+c)\exp(-sn).$$ It follows that $\rho(T,\mu)\ge s$ since if $\rho(T,\mu)<s$ then $\beta(n+c)\exp(-sn)\to 0$ as $n\to\infty$, contradicting the fact that $\langle e,\mu\rangle_\alpha>0$. Thus for every $0<s<h_T(\mu)$ we have $\rho(T,\mu)\ge s$. Therefore $\rho(T,\mu)\ge h_T(\mu)$, as required.
Suppose $T=T_{(\alpha,\mathcal L)}$ where for every edge $e$ of $\Gamma$ we have $\mathcal L(e)=1$. Let $\Omega(\Gamma,\mu)$ be the set of all semi-infinite reduced edge paths $\gamma=e_1,\dots, e_n, \dots $ in $\Gamma$ such that for every finite subpath $v$ of $\gamma$ we have $v\in supp_\alpha(\mu)$. We can think of $\Omega(\Gamma,\mu)$ as a subset of the set $A^\omega$ of all semi-infinite words in the alphabet $A$ consisting of all oriented edges of $\Gamma$. The subset $\Omega(\Gamma,\mu)\subseteq A^\omega$ is clearly invariant under the shift map $\sigma: A^\omega\to A^\omega$ which erases the first letter of a semi-infinite word from $A^\omega$. Thus $\Omega(\Gamma,\mu)$ is a subshift (not necessarily of finite type) of the full shift $(A^\omega,\sigma)$. It is easy to see from the definitions that in this case $\rho(T_{(\alpha,\mathcal L)})$ is equal to the topological entropy $h_{topol}(\Omega(\Gamma,\mu))$ of the subshift $\Omega(\Gamma,\mu)$ of the set $A^\omega$. In this situation Theorem \[thm:support\] says that $h_T(\mu)\le h_{topol}(\Omega(\Gamma,\mu))$.
As we already observed, it is not hard to show that if $\rho(T_0,\mu)=0$ for some $T_0\in cv(F)$ then $\rho(T,\mu)=0$ for every $T\in cv(F)$. In this case Theorem \[thm:support\] implies $h_T(\mu)=0$ for every $T\in cv(F)$.
Tame currents
=============
\[defn:tameT\] Let $\mu\in Curr(F)$ and $T\in cv(F)$. We say that $\mu$ is *tame with respect to $T$* or *$T$-tame* if for every $M\ge 1$ there is $C=C(M)\ge 1$ such that whenever $a_1,b_1,a_2,b_2\in T$ satisfy $d(a_1,a_2)\le M,
d(b_1,b_2)\le M$, $a_1\ne b_1$, $a_2\ne b_2$ then $$\frac{1}{C}\mu ( Cyl_{[a_2,b_2]}) \le \mu ( Cyl_{[a_1,b_1]})\le C \mu ( Cyl_{[a_2,b_2]}).$$ We call $C=C(M)$ the *$T$-tameness constant* corresponding to $M$.
\[lem:wtame\] Let $T\in cv(F)$ and $\mu\in Curr(F)$. Suppose that there is some $N\ge 1$ such that for every $M\ge N$ there exists $D=D(M)\ge 1$ such that whenever $[x,y]\subseteq [a,b]$ where $x\ne y$ and $d_T(a,x)=d_T(y,b)=M$ then $$\mu ( Cyl_{[x,y]})\le D \mu ( Cyl_{[a,b]}).$$ Then $\mu$ is $T$-tame.
Suppose that for every $M\ge N$ there is $D=D(M)\ge 1$ as in Lemma \[lem:wtame\].
We need to prove that $\mu$ is $T$-tame. It is easy to see that it suffices to verify the conditions of Definition \[defn:tameT\] for all sufficiently large $M$.
Let $M\ge N$ be arbitrary. Suppose now that $[x,y]\subseteq [a,b]$ with $d_T(a,x)=d_T(y,b)\le M$ and $x\ne y$. Choose a geodesic segment $[a',b']$ in $T$ such that $[a,b]\subseteq [a',b']$ and such that $d_T(a',x)=d_T(b',y)=M$. Then $Cyl_{[a',b']}\subseteq Cyl_{[a,b]}\subseteq Cyl_{[x,y]}$. Hence by assumption on $D=D(M)$ we have $$\mu(Cyl_{[a,b]})\le \mu(Cyl_{[x,y]})\le D \mu(Cyl_{[a',b']})\le D \mu(Cyl_{[a,b]})$$ so that $$\mu(Cyl_{[a,b]})\le \mu(Cyl_{[x,y]})\le D \mu(Cyl_{[a,b]}).\tag{\dag}$$ Thus $(\dag)$ holds whenever $[x,y]\subseteq [a,b]$ with $d_T(a,x)=d_T(y,b)\le M$.
Suppose now that $M\ge N$ and $a_1,b_1,a_2,b_2\in T$ satisfy $d(a_1,a_2)\le M,
d(b_1,b_2)\le M$. We may assume that $d(a_1,b_1)\ge 3M$ since otherwise the requirements of Definition \[defn:tameT\] are easily satisfied. (Indeed, if $d(a_1,b_1)\le 3M$, then the points $a_1,a_2,b_1,b_2$ lie in an $F$-translated of a fixed closed ball of radius $6M$, which is a finite subtree of $T$). Then $[a_1,b_1]\cap [a_2,b_2]$ is a non-degenerate geodesic segment. Put $[x,y]=[a_1,b_1]\cap [a_2,b_2]$.
Then $$d(x,a_1), d(x,a_2), d(y,b_1),d(y,b_2)\le M$$ and $[x,y]\subseteq [a_1,b_1]$, $[x,y]\subseteq [a_2,b_2]$. Then by $(\dag)$ we have $$\mu(Cyl_{[a_1,b_1]})\le \mu(Cyl_{[x,y]})\le D \mu(Cyl_{[a_2,b_2]})$$ and $$\mu(Cyl_{[a_2,b_2]})\le \mu(Cyl_{[x,y]})\le D \mu(Cyl_{[a_1,b_1]}).$$ Therefore $\mu$ is $T$-tame with as required.
\[prop:tame\] Let $\mu\in Curr(F)$ and let $T,T'\in cv(F)$. Then $\mu$ is tame with respect to $T$ if and only if $\mu$ is tame with respect to $T'$.
Proposition \[prop:tame\] implies that the following notion is well-defined and does not depend on the choice of $T\in cv(F)$:
Let $\mu\in Curr(F)$. We say that $\mu$ is *tame* if for some (equivalently, for any) $T\in cv(F)$ the current $\mu$ is $T$-tame.
Suppose that $\mu$ is tame with respect to $T'$.
Let $x\in T$ and $x'\in T'$ be arbitrary vertices. Let $\phi:T\to T'$ be an $F$-equivariant $(\lambda,\lambda)$-quasi-isometry such that $\phi(x)=x'$. Let $M=M(\phi)\ge
1$ be the constant provided by Lemma \[lem:qi\].
We need to prove that $\mu$ is tame with respect to $T$. By Lemma \[lem:wtame\] it suffices to show that the conditions of Lemma \[lem:wtame\] hold for $\mu$.
Let $M_0\ge 1$ be sufficiently big (to be specified later) and suppose $s,t,a,b\in T$ are such that $[s,t]\subseteq [a,b]$ with $d_T(s,a)=d_T(t,b)=M_0$.
Let $s',t'\in [\phi(a),\phi(b)]$ be such that $$\begin{gathered}
d_{T'}(\phi(s),s')=d_{T'}(\phi(s), [\phi(a),\phi(b)]) \quad \text{ and }\\
d_{T'}(\phi(t),t')=d_{T'}(\phi(t), [\phi(a),\phi(b)]).\end{gathered}$$
Since $\phi$ is a quasi-isometry and $T,T'$ are Gromov-hyperbolic, we have $$d_{T'}(\phi(s),s'), d_{T'}(\phi(t),t')\le C_1,$$ where $C_1=C_1(\phi)>0$ is some constant. Note that $[\phi(a),\phi(b)]\cap [\phi(s),\phi(t)]=[s',t']$. We may assume that $M_0$ was chosen big enough so that $$d_{T'}(\phi(a),s'), d_{T'}(\phi(b),t')\ge M_1=M_1(\psi)$$ where $M_1=M_1(\psi)$ is the constant provided by Proposition \[prop:qi1\].
Proposition \[prop:qi1\] implies that there exist $p_1,p_2,q_1,q_2\in T'$ such that $[p_1,p_2]=[q_1,q_2]\cap [\phi(a),\phi(b)]$ and such that $$Cyl_{[q_1,q_2]}\subseteq \phi (Cyl_{[a,b]})$$ and such that $d_{T'}(q_1,\phi(a)), d_{T'}(q_2,\phi(b))\le M_1$. Thus $d_{T'}(\phi(a),s'), d_{T'}(\phi(b),t')\ge M_1=M_1(\psi)$ implies that $[s',t']\subseteq [p_1,p_2]$.
Let $s'',t''\in [s',t']\subseteq [\phi(a),\phi(b)]$ be such that $d_{T'}(s',s'')=d_{T'}(t',t'')=M$. Thus $s'',t''\in [\phi(s),\phi(t)]$ and $$M\le d_{T'}(\phi(s),s'')\le M+C_1,\quad M\le d_{T'}(\phi(t),t'')\le M+C_1.$$ Then by Lemma \[lem:qi\] $$\phi(Cyl_{[s,t]})\subseteq Cyl_{[s'',t'']}$$ and hence $$\mu(Cyl_{[s,t]} )\le \mu(Cyl_{[s'',t'']}).$$
Note that since $\phi$ is a $(\lambda,\lambda)$-quasi-isometry and $d_{T}(a,s)=M_0$, $d_{T}(b,t)=M_0$ then $$d_{T'}(\phi(a),\phi(s)), d_{T'}(\phi(b),\phi(t))\le \lambda M_0+\lambda.$$ Since $d_{T'}(\phi(s),s'), d_{T'}(\phi(t),t')\le C_1$, it follows that $$d_{T'}(\phi(a),s'), d_{T'}(\phi(b),t')\le \lambda M_0+\lambda+C_1.$$ Since $d_{T'}(s',s'')=d_{T'}(t',t'')=M$, we have $$d(\phi(a),s''), d(\phi(b),t'')\le M+\lambda M_0+\lambda+C_1.$$ Since $p_1\in [\phi(a),s'']$, $p_2\in [t'',\phi(b)]$ and $d_{T'}(q_1,\phi(a)), d_{T'}(q_2,\phi(b))\le M_1$, we get $$d_{T'}(s'',q_1), d_{T'}(t'',q_2)\le M_1+M+\lambda M_0+\lambda+C_1.$$
Put $M_2=M_1+M+\lambda M_0+\lambda+C_1$. Since $\mu$ is $T'$-tame, $$\mu ( Cyl_{[s'',t'']})\le C_2 \mu( Cyl_{[q_1,q_2]}),$$ where $C_2=C(M_2)$ is the $T'$-tameness constant for $\mu$ corresponding to $M_2$.
Recall that $Cyl_{[q_1,q_2]}\subseteq \phi (Cyl_{[a,b]})$ and therefore $$\mu( Cyl_{[q_1,q_2]})\le \mu (Cyl_{[a,b]}).$$ Thus we have $$\begin{gathered}
\mu(Cyl_{[s,t]})\le \mu(Cyl_{[s'',t'']})\le C_2 \mu( Cyl_{[q_1,q_2]})\le\\
\le C_2 \mu (Cyl_{[a,b]}).\end{gathered}$$ Hence by Lemma \[lem:wtame\] $\mu$ is $T$-tame, as required.
The geometric entropy function on the Outer Space
=================================================
\[thm:lowerbound\] Let $T\in cv(F)$ and let $\mu\in Curr(F)$ be a tame current. Then for any $T'\in cv(F)$ we have: $$h_{T'}(\mu)\ge h_T(\mu)\inf_{f\in F\setminus\{1\}}\frac{||f||_T}{||f||_{T'}}.$$
Put $h=h_T(\mu)$.
Let $x\in T$ and $x'\in T'$ be arbitrary vertices. Let $\phi: T'\to T$ be an $F$-equivariant quasi-isometry such that $\phi(x')=x$. Thus $\phi(gx')=gx$ for every $g\in F$. Let $M\ge 1$ be provided by Lemma \[lem:qi\].
Let $a_n,b_n\in T'$ be such that $\lim_{n\to\infty}
d_{T'}(a_n,b_n)=\infty$ and such that $$h_{T'}(\mu)=\lim_{n\to\infty} \frac{-\log \mu(Cyl_{[a_n,b_n]}) }{d_{T'}(a_n,b_n)}.$$
Note that there exists some constant $M'=M'(T')\ge 1$ (it can be taken equal to $vol(\mathcal L')$), such that for any finite reduced edge-path $v$ in $T'/F$ there exists a cyclically reduced closed edge-path $\widehat v$ in $T'/F$ containing $v$ or contained in $v$ (as a subpath) and such that $|\mathcal L'(v)-\mathcal L'(\widehat v)|\le M'$.
Then there exists $h_n,g_n\in F$ such that $d_{T'}(a_n,h_nx')\le M'$, $d_{T'}(b_n,g_nx')\le M'$ and such that $[h_nx',g_nx']$ projects to a closed cyclically reduced path in $T'/F$.
After translating $[a_n,b_n]$ by $h_n^{-1}$, we may assume that $h_n=1$. Thus $[x',g_nx']$ is contained in the axis of $g_n$ and $d_{T'}(x',g_nx')=||g_n||_{T'}$. Note that $\displaystyle\lim_{n\to\infty}d_{T'}(a_n,b_n)=\infty$ implies $\displaystyle\lim_{n\to\infty}||g_n||_{T'}=\infty$.
Since $\mu$ is tame, there exists a constant $C_1\geq 1$ such that $$\mu(Cyl_{[a_n,b_n]})\le C_1 \mu(Cyl_{[x',g_nx']}).$$ Therefore $$\frac{-\log \mu(Cyl_{[a_n,b_n]})}{d_{T'}(a_n,b_n)}\ge \frac{-\log
\mu(Cyl_{[x',g_nx']})-\log C_1}{d_{T'}(x',g_nx')+2M'}.$$ and $$h_{T'}(\mu)\ge \liminf_{n\to\infty} \frac{-\log
\mu(Cyl_{[x',g_nx']})}{d_{T'}(x',g_nx')}.$$
Note that $\phi(x')=x$ and $\phi(g_nx')=g_nx$. Moreover, since $d_{T'}(x',g_nx')=||g_n||_{T'}$ and $\phi$ is a quasi-isometry, there is a constant $C_2>0$ independent of $n$ such that for every $n\ge 1$ $$\left| d_T(x,g_nx)-||g_n||_T \right|\le C_2.$$ Also, we have $\lim_{n\to\infty} ||g_n||_T=\infty$. Let $[y_n,z_n]\subseteq [x,g_nx]$ be such that $d_T(x,y_n)=d_T(g_nx,z_n)=M$, where $M$ is a constant provided by Lemma \[lem:qi\], with $$\phi \left(Cyl_{[x',g_nx']}\right)\subseteq Cyl_{[y_n,z_n]}$$ and hence $$\mu ( Cyl_{[x',g_nx']}) \le \mu( Cyl_{[y_n,z_n]})\le C_3 \mu( Cyl_{[x,g_nx]}),$$ where the constant $C_3\geq 1$ exists and the last inequality holds since $\mu$ is tame.
Thus $$\begin{gathered}
\frac{-\log \mu ( Cyl_{[x',g_nx']})}{d_{T'}(x',g_nx')}\ge
\frac{-\log \mu ( Cyl_{[x,g_nx]})-\log C_2}{d_{T'}(x',g_nx')}=\\
\frac{-\log \mu ( Cyl_{[x,g_nx]})-\log
C_2}{d_T(x,g_nx)}\cdot \frac{d_T(x,g_nx)}{d_{T'}(x',g_nx')}=\\
\frac{-\log \mu ( Cyl_{[x,g_nx]})-\log C_2}{d_T(x,g_nx)}\cdot
\frac{d_T(x,g_nx)}{||g_n||_{T'}}\ge \\
\frac{-\log \mu ( Cyl_{[x,g_nx]})-\log C_2}{d_T(x,g_nx)}\cdot \frac{||g_n||_T-C_3}{||g_n||_{T'}}\end{gathered}$$
Therefore $$\begin{gathered}
h_{T'}(\mu)\ge \liminf_{n\to\infty} \frac{-\log
\mu(Cyl_{[x',g_nx']})}{d_{T'}(x',g_nx')}\ge \\
\liminf_{n\to\infty} \frac{-\log \mu ( Cyl_{[x,g_nx]})-\log
C_2}{d_T(x,g_nx)}\cdot \frac{||g_n||_T-C_3}{||g_n||_{T'}}\ge \\
\ge h_T(\mu)\liminf_{n\to\infty}\frac{||g_n||_T-C_3}{||g_n||_{T'}}
=h_T(\mu)\liminf_{n\to\infty}\frac{||g_n||_T}{||g_n||_{T'}}\ge \\
\ge h_T(\mu)\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}},\end{gathered}$$ as required.
The following statement is an immediate corollary of the explicit formula for the Patterson-Sullivan current in the Outer Space context obtained by Kapovich and Nagnibeda ([@KN], see Proposition 5.3):
\[prop:KN\] Let $T\in cv(F)$ and let $h=h(T)$ be the critical exponent of $T$. Let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current corresponding to $T$. Then there exist constants $C_1>C_2>0$ such that for any distinct vertices $x$ and $y$ of $T$ we have $$C_2 \exp(-h\, d_T(x,y))\le \mu_T(Cyl_{[x,y]} )\le C_1 \exp(-h\, d_T(x,y)).$$
Together with the definitions of geometric entropy and of tameness, Proposition \[prop:KN\] immediately implies:
\[cor:ps\] Let $T\in cv(F)$ and let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current corresponding to $T$. Let $h=h(T)$ be the critical exponent of $T$.
Then $\mu_T$ is tame and $h_T(\mu_T)=h(T)$.
\[prop:comp\] Let $T\in cv(F)$ and let $h=h(T)$ be the critical exponent of $T$. Let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current corresponding to $T$.
Let $T'\in cv(F)$. Then for any $g\in F\setminus\{1\}$ we have $$h_{T'}(\mu_T)\le h\frac{||g||_T}{||g||_{T'}}$$ and therefore $$h_{T'}(\mu_T)\le h\inf_{f\in F\setminus\{1\}}\frac{||f||_T}{||f||_{T'}}.$$
By Proposition \[prop:KN\] there exists $C>0$ such that for any distinct vertices $x,y\in T$ we have $$\mu(Cyl_{[x,y]})\ge C \exp( -h d_T(x,y)).$$
Note that for any $x\in T, x'\in T'$ we have $$\begin{gathered}
||g||_T=\lim_{n\to\infty} \frac{d_T(x,g^nx)}{n}, \quad ||g||_{T'}=\lim_{n\to\infty} \frac{d_{T'}(x',g^nx')}{n}\end{gathered}$$ and hence $$\lim_{n\to\infty} \frac{d_T(x,g^nx)}{d_{T'}(x',g^nx')}=\frac{||g||_T}{||g||_{T'}}.$$
Let $x\in T$ and $x'\in T'$ be arbitrary vertices. Let $\phi: T\to T'$ be an $F$-equivariant quasi-isometry such that $\phi(x)=x'$. Thus $\phi(g^nx)=g^n x'$ for every $n\in \mathbb Z$. Let $M\ge 1$ be provided by Lemma \[lem:qi\]. For $n\to\infty$ let $[y_n,z_n]\subseteq [x',g^n x']$ be such that $d(x',y_n)=d(z_n, g^nx')=M$. Then by Lemma \[lem:qi\] we have $$\phi(Cyl_{[x,g^nx]})\subseteq Cyl_{[y_n,z_n]}.$$ Hence $$\mu(Cyl_{[y_n,z_n]})\ge \mu(Cyl_{[x,g^nx]})\ge C \exp(-h d_T(x,g^nx))$$ and so $$\frac{-\log \big( \mu(Cyl_{[y_n,z_n]}) \big)}{d_{T'}(y_n,z_n)}\le \frac{h d_T(x,g^nx) -\log C}{d_{T'}(y_n,z_n)}\le
\frac{h d_T(x,g^nx) -\log C}{d_{T'}(x',g^nx')-2M}.$$ Hence $$h_{T'}(\mu)\le \liminf_{n\to\infty} \frac{h d_T(x,g^nx) -\log C}{d_{T'}(x',g^nx')-2M}=h\frac{||g||_T}{||g||_{T'}},$$ as required.
Since Patterson-Sullivan currents are tame, Proposition \[prop:comp\] and Theorem \[thm:lowerbound\] imply
\[thm:comp1\] Let $T\in cv(F)$ and let $h=h(T)$ be the critical exponent of $T$. Let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current for $T$. Let $T'\in cv(F)$. Then $$h_{T'}(\mu_T)= h(T)\inf_{f\in F\setminus\{1\}}\frac{||f||_T}{||f||_{T'}}.$$
\[cor:tw\] Let $T\in CV(F)$ and let $h=h(T)$ be the critical exponent of $T$. Let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current for $T$. Let $T'\in CV(F)$ be such that $T'\ne T$. Then $$h_{T'}(\mu_T)<h_T(\mu_T)=h(T).$$ Thus $$h(T)=h_T(\mu_T)=\max_{T'\in CV(T)} h_{T'}(\mu_T)$$ and this maximum is strict.
A result of Tad White [@Wh] implies that, when $T'\ne T$, there exists a nontrivial $g\in F$ such that $||g||_T<||g||_{T'}$. Therefore by Theorem \[thm:comp1\] $$h_{T'}(\mu_T)= h(T)\frac{||g||_T}{||g||_{T'}}<h(T) .$$
\[cor:inf\] Let $T\in CV(F)$ and let $\mu_T\in Curr(F)$ be a Patterson-Sullivan current for $T$. Then $$\inf_{T'\in CV(F)} h_{T'}(\mu_T)=0.$$
Recall that $F$ is a free group of rank $k\ge 2$. Let $A$ be a free basis of $F$ and let $T_A$ be the Cayley graph of $F$ with respect to $A$, where every edge has length $1/k$. Thus $T_A\in CV(F)$. Let $a\in A$. There exists a sequence $\phi_n\in Out(F)$ such that $\lim_{n\to\infty} ||\phi_n a||_A=\infty$ and hence $\lim_{n\to\infty} ||\phi_n a||_{T_A}=\frac{1}{k}\lim_{n\to\infty} ||\phi_n a||_A=\infty$. Put $T_n=\phi_n^{-1} T_A$. Thus $T_n\in CV(F)$ and $$||a||_{T_n}=||a||_{\phi_n^{-1} T_A}=||\phi_n a||_{T_A}\longrightarrow_{n\to\infty}\infty.$$ Therefore by Theorem \[thm:comp1\] we have $$h_{T_n}(\mu)\le h(T)\frac{||a||_T}{||a||_{T_n}}\longrightarrow_{n\to\infty} 0.$$ Hence $$\inf_{T'\in CV(F)} h_{T'}(\mu)=0,$$ as required.
The maximal geometric entropy problem for a fixed tree {#sect:hd}
======================================================
Recall that, as observed in Introduction, the function $h_T(\cdot): Curr(F)\to \mathbb R, \mu\mapsto h_T(\mu)$ is not continuous. Nevertheless, it turns out that it is possible to find the maximal value of $h_T(\cdot)$ on $Curr(F)-\{0\}$.
Recall that if $(X,d)$ is a metric space and $\nu$ is a positive measure on $X$, then the *Hausdorff dimension ${\mathbf {HD}} _X(\nu)$ of $\nu$ with respect to $X$* is defined as $${\mathbf {HD}} _X(\nu):=\inf\{ {\mathbf {HD}} (S): S\subseteq X\text{ such that } \nu(X-S)=0\}.$$ Thus ${\mathbf {HD}} _X(\nu)$ is the smallest Hausdorff dimension of a subset of $X$ of full $\nu$-measure. Note that this obviously implies that ${\mathbf {HD}} _X(\nu)\le {\mathbf {HD}} (X)$.
Let $T\in cv(F)$. Recall that if $x\in T$, $\xi,\zeta\in \partial T$, we denote by $(\xi|\zeta)_x$ the distance $d_T(x,y)$ where $[x,\xi]\cap
[x,\zeta]=[x,y]$. Let $x\in T$ be a base-point. The boundary $\partial T$ is metrized by setting $d_x(\xi,\zeta)=\exp(- (\xi|\zeta)_x )$ for $\xi,\zeta\in \partial T$. It is well-known (see [@Coor]) that ${\mathbf {HD}} (\partial T, d_x)=h(T)$.
Recall that, as explained in the Introduction, given a current $\mu\in
Curr(F)$ and $T\in cv(F)$, we introduce a family of measures $\{\mu_x\}_{x\in T}$ on $\partial F$ defined by their values on all the one-sided cylinder subsets of $\partial F$: $$Cyl_{[x,y]}^x := \{\xi\in \partial F: \text{ the geodesic ray }
[x,\partial_T(\xi)] \text{ in $T$ begins with } [x,y]\}\subseteq \partial F,$$ $$\mu_x (Cyl_{[x,y]}^x) := \mu (Cyl_{[x,y]}) .$$ It is not hard to see that if $\mu\ne 0$ then there exists $x\in T$ such that $\mu_x\ne 0$.
\[thm:vol\] Let $\mu\in Curr(F)$, $\mu\ne 0$, let $T\in cv(F)$, and let $x\in T$ be such that $\mu_x\ne 0$. Then $$h_T(\mu)\le {\mathbf {HD}} _{\partial T}(\mu_x)\le h(T).$$
As observed by Kaimanovich in [@Kaim98] (see formula (1.3.3)), the following formula holds for the Hausdorff dimension of a measure $\nu$ on $\partial T$ endowed with the metric $d_x$ as above. $${\mathbf {HD}} _{\partial T}(\nu)={\rm ess\ sup}_{\xi\in \partial T} \liminf_{k\to\infty} \frac{-\log \nu
\left(B_x(\xi,k)\right)}{k}\tag{$\ddag$}$$
Here $B_x(\xi,k)$ is the set of all $\zeta\in \partial T$ such that $(\xi|\zeta)_x\ge k$, that is $B_x(\xi,k)=Cyl_{[x,y_k]}^x$ where $[x,y_k]$ is the initial segment of $[x,\xi]$ of length $k$. The essential supremum in $(\ddag)$ is taken with respect to $\nu$.
Applied to $\mu_x$, formula $(\ddag)$ yields:
$$\begin{gathered}
h_T(\mu)=\liminf_{d_T(y,z)\to\infty}\frac{-\log \mu (Cyl_{[y,z]})}{d_T(y,z)}\le \\
\le \liminf_{\overset{z\in T}{d_T(x,z)\to\infty}}\frac{-\log \mu (Cyl_{[x,z]})}{d_T(x,z)}\le \\
\le {\rm ess\ sup}_{\xi\in \partial T} \liminf_{\overset{z\in T}{d_T(x,z)\to\infty}} \frac{-\log \mu_x(Cyl_{[x,z]}^x)}{d_T(x,z)}=\\
={\mathbf {HD}} _{\partial T} (\mu_x)\le {\mathbf {HD}} (\partial T)=h(T).\end{gathered}$$
\[prop:proj\] Let $T,T'\in cv(F)$ be such that $h:=h(T)=h(T')$. Let $\mu_T$ be a Patterson-Sullivan current corresponding to $T$ and suppose that $h_{T'}(\mu_T)=h$. Then $T$ and $T'$ are in the same projective class.
Let $\mu_{T'}\in Curr(F)$ be a Patterson-Sullivan current corresponding to $T'$. We will first show that $\mu_T$ is absolutely continuous with respect to $\mu_{T'}$. Let $\phi:T\to T'$ be an $F$-equivariant $(\lambda,\lambda)$-quasi-isometry, where $\lambda\ge 1$. By Proposition \[prop:KN\] there is a constant $C\ge 1$ such that for any $x,y\in T$ with $d_T(x,y)\ge 1$ $$\frac{1}{C} \exp( -h d_T(x,y))\le \mu_T Cyl_{[x,y]} \le C \exp( -h d_T(x,y)).$$
Let $x',y'\in \phi(T')$ be arbitrary such that $d_{T'}(x',y')\ge \lambda^2+\lambda$. Let $x,y\in T$ be such that $x'=\phi(x)$, $y'=\phi(y)$. Since $\mu_T$ is tame, Lemma \[lem:qi\] and Proposition \[prop:qi1\] imply that there is some constant $C_1\ge 1$ such that $$\frac{1}{C_1} \mu_T (Cyl_{[x',y']}) \le \mu_T (Cyl_{[x,y]})\le C_1 \mu_T (Cyl_{[x',y']}).$$ Thus $$\mu_T (Cyl_{[x',y']})\ge \frac{1}{C_1}\mu_T (Cyl_{[x,y]})\ge \frac{1}{C_1C} \exp( -h d_T(x,y)).$$
On the other hand, since by assumption $h_{T'}(\mu_T)=h$, it follows that for any $\epsilon>0$ there exists $C_2=C_2(\epsilon)\ge 1$ such that $$\mu_T (Cyl_{[x',y']})\le C_2 \exp( -(h-\epsilon) d_{T'}(x',y')).$$
Thus $$\frac{1}{C_1C} \exp( -h d_T(x,y)) \le C_2 \exp( -(h-\epsilon) d_{T'}(x',y')).$$ Hence $$h d_T(x,y)\ge (h-\epsilon) d_{T'}(x',y')-\log(C_2C_1C).$$ and so $$\frac{h}{h-\epsilon} \ge \limsup_{d_T(x,y)\to\infty} \frac{d_{T'}(x',y')}{d_T(x,y)}.$$ Since $\epsilon>0$ was arbitrary, it follows that $$\limsup_{d_T(x,y)\to\infty} \frac{d_{T'}(x',y')}{d_T(x,y)}\le 1.$$ Therefore $$\sup_{g\in F\setminus\{1\}} \frac{||g||_{T'}}{||g||_T}\le 1.$$ By Lemma \[lem:stretch\] this implies that there is a constant $C_3\ge 1$ such that $$d_T(x',y')\le d_{T}(x,y)+C_3.$$ Then $$\begin{gathered}
\mu_T(Cyl_{[x',y']})\le \frac{1}{C_1C} \exp( -h d_T(x,y))\le \\
\frac{1}{C_1C} \exp( -h (d_{T'}(x',y')
-C_3))=\frac{\exp(hC_3)}{C_1C}\exp(-h d_{T'}(x',y'))\le\\
\frac{C'\exp(hC_3)}{C_1C}\mu_{T'}(Cyl_{[x',y']}).\end{gathered}$$ The above inequality holds for any $x',y'\in \phi(T)$ with $d_{T'}(x',y')\ge \lambda^2+\lambda$. Since $\phi$ is a quasi-isometry and $\mu_T$ is tame, it follows that there exists a constant $C'\ge 1$ such that for any $x',y'\in T'$ with $d_{T'}(x',y')\ge 1$ we have $$\mu_T(Cyl_{[x',y']})\le C'\mu_{T'}(Cyl_{[x',y']}).$$ Hence $\mu_T$ is absolutely continuous with respect to $\mu_{T'}$. A result of Furman [@Fur] now implies that the translation length functions $||.||_T$ and $||.||_{T'}$ are scalar multiples of each other, as required.
\[cor:sharp\] Let $T_1,T_2\in cv(F)$ be such two elements that do not lie in the same projective class. Let $\mu_{T_2}$ be a Patterson-Sullivan current for $T_2$. Then $h_{T_1}(\mu_{T_2})< h(T_1)$.
After replacing $T_2$ by a scalar multiple of $T_2$ we may assume that $h(T_1)=h(T_2)$. Note that the projective Patterson-Sullivan current depends only on the projective class of an element of $cv(F)$, so that this replacement does not change $\mu_{T_2}$. Now the statement of the corollary follows immediately from Theorem \[thm:vol\] and Proposition \[prop:proj\].
Let $T,T'\in cv(F)$. Then
1. $$\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}\le \frac{h(T')}{h(T)}\le \sup_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}.$$
2. Suppose that $T$ and $T'$ are not in the same projective class. Then $$\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}< \frac{h(T')}{h(T)}< \sup_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}.$$
$ $
\(1) Let $\mu_T$ be a Patterson-Sullivan current corresponding to $T$. By Theorem \[thm:comp1\] and Theorem \[thm:vol\] we have $$h(T)\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}=h_{T'}(\mu_T)\le h(T')$$ and hence $$\inf_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}\le \frac{h(T')}{h(T)}.$$
A symmetric argument shows that $$\inf_{g\in F\setminus\{1\}} \frac{||g||_{T'}}{||g||_{T}}\le \frac{h(T)}{h(T')}.$$ Clearly, $$\inf_{g\in F\setminus\{1\}} \frac{||g||_{T'}}{||g||_{T}}=\frac{1}{\displaystyle\sup_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}}$$ and hence $$\sup_{g\in F\setminus\{1\}} \frac{||g||_T}{||g||_{T'}}\ge \frac{h(T')}{h(T)},$$ as required.
The proof of part (2) is exactly the same, but Corollary \[cor:sharp\] implies that all the inequalities involved are now strict.
Relation to measure-theoretic entropy
=====================================
In this section we relate the geometric entropy of a current $\mu\in Curr(F)$ with respect to $T=\widetilde \Gamma$, where $\Gamma$ is a finite graph with simplicial metric (every edge has length one), to the measure-theoretic entropy of the corresponding shift-invariant measure $\widehat\mu$ on the (appropriately defined) geodesic flow space of the graph $\Gamma$.
\[defn:Omega\] Let $\alpha:F\to \pi_1(\Gamma, p)$ be a simplicial chart. Let $T=\widetilde \Gamma$. We endow both $\Gamma$ and $T$ with simplicial metrics by giving each edge length one. Thus $T\in cv(F)$. Define $\Omega(\Gamma)$ to be the set of all semi-infinite reduced edge-paths $$\gamma=e_1,e_2,\dots, e_n,\dots$$ in $\Gamma$. Note that $\Omega(\Gamma)$ is naturally identified with the disjoint union of $\#V\Gamma$ copies of $\partial T$, corresponding to the $\#V\Gamma$ different possibilities of the initial vertex of $\gamma\in \Omega(\Gamma)$. We topologize $\Omega(\Gamma)$ accordingly. Thus, topologically, $\Omega(\Gamma)$ is a disjoint union of $\#V\Gamma$ copies of the Cantor set.
There is a natural shift transformation $\sigma:\Omega(\Gamma)\to\Omega(\Gamma)$ defined as $$\sigma(e_1,e_2,e_3,\dots)=e_2,e_3,\dots$$ for every $\gamma=e_1,e_2,e_3,\dots \in \Omega(\Gamma)$. Then $\sigma$ is a continuous transformation and the pair $(\Omega(\Gamma),\sigma)$ is easily seen to be an irreducible subshift of finite type, where the alphabet is the set of oriented edges of $\Gamma$.
The space $\Omega(\Gamma)$ has a natural set of cylinder sets which generate its Borel sigma-algebra: for every nontrivial reduced edge-path $v$ in $\Gamma$ let $Cyl_v$ be the set of all $\gamma\in \Omega(\Gamma)$ that have $v$ as an initial segment. It is easy to see that there is a natural affine isomorphism between the space of geodesic currents $Curr(F)$ and the space $\mathcal M(\Gamma)$ of all positive $\sigma$-invariant Borel measures on $\Omega(\Gamma)$ of finite total mass. For a current $\mu\in Curr(F)$ the corresponding measure $\widehat\mu\in \mathcal M(\Gamma)$ is defined by the following condition. For every nontrivial reduced edge-path $v$ in $\Gamma$, let $[x,y]\subseteq T$ be any lift of $v$ to $T$. Then $$\widehat\mu(Cyl_v)=\mu(Cyl_{[x,y]}).$$
Suppose now that $\mu\in Curr(F)$ is normalized so that the corresponding measure $\widehat\mu\in \Omega(\Gamma)$ is a probability measure (recall that multiplication by a positive scalar does not change the geometric entropy). For the probability measure $\widehat\mu$ one can consider its classical *measure-theoretic* entropy $\hbar(\widehat \mu)$ with respect to the shift $\sigma$, also known as the *Kolmogorov-Sinai entropy* or *metric entropy*, see e.g. [@Kitchens].
\[thm:ks\] Let $\alpha:F\to \pi_1(\Gamma, p)$ be a simplicial chart. Let $T=\widetilde \Gamma$ and endow both $\Gamma$ and $T$ with simplicial metric. Let $(\Omega(\Gamma),\sigma)$ be as in Definition \[defn:Omega\]. Let $\mu\in Curr(F)$ be such that the corresponding measure $\widehat\mu\in \mathcal M(\Gamma)$ is a probability measure. Then $$h_{T}(\mu)\le \hbar(\widehat\mu)\le h_{topol}(\sigma)=h(T).$$
The fact that $h_{topol}(\sigma)=h(T)$ is a straightforward exercise which easily follows from the definitions of both quantities. The fact that $\hbar(\widehat\mu)\le h_{topol}(\sigma)$ is also completely general for subshifts of finite type (see e.g. Ch 6. of [@Kitchens]). If $h_T(\mu)=0$ then the inequality $h_T(\mu)\le \hbar(\widehat\mu)$ is obvious. Suppose now that $h_T(\mu)>0$.
Note that $$\Omega(\Gamma)=\sqcup_{e\in E\Gamma} Cyl_e$$ is a generating partition with respect to $\sigma$. Therefore the measure-theoretic entropy of $\widehat\mu$ can be computed using this partition and is easily seen to be $$\hbar(\widehat\mu)=\lim_{n\to\infty} \sum_{|v|=n}-\frac{\widehat\mu(Cyl_v)\log\widehat\mu(Cyl_v)}{n}.$$ Suppose now that $0<s<h_T(\mu)$. Then there exists $n_0\ge 1$ such that for any geodesic edge-path $[x,y]$ in $T$ of length $n\ge n_0$ we have $$\mu(Cyl_{[x,y]})\le \exp(-sn).$$ Hence for any reduced edge-path $v$ in $\Gamma$ of length $n\ge n_0$ we have $\widehat\mu(Cyl_v)\le \exp(-sn)$ and $-\log\widehat\mu(Cyl_v)\ge sn$. Therefore from the above formula for $\hbar(\widehat\mu)$ we get $$\hbar(\widehat\mu)\ge \lim_{n\to\infty}\sum_{|v|=n}\frac{\widehat\mu(Cyl_v) sn}{n}=s,$$ since for every $n\ge 1$ we have $\Omega(\Gamma)=\sqcup_{|v|=n} Cyl_v$ and so $\sum_{|v|=n}\widehat\mu(Cyl_v)=1$. Thus we see that for every $0<s<h_T(\mu)$ we have $\hbar(\widehat\mu)\ge s$. Therefore $\hbar(\widehat\mu)\ge h_T(\mu)$ as claimed.
It is also well-known (again see, for example, Ch. 6 of [@Kitchens]) that for irreducible subshifts of finite type such as $(\Omega(\Gamma),\sigma)$ in Theorem \[thm:ks\], there exists a unique probability measure $\mu_\Omega\in \mathcal M(\Gamma)$ such that $\hbar(\mu_\Gamma)=h_{topol}(\sigma)=h(T)$. The measure $\mu_\Omega$ is known as the *measure of maximal entropy*. We already know that for the Patterson-Sullivan current $\mu_T$ we have $h_T(\mu_T)=h(T)$. Thus, if $\mu_T$ is normalized so that the corresponding measure $\widehat\mu_T\in \mathcal M(\Gamma)$ is a probability measure, then by Theorem \[thm:ks\] we have $h_T(\mu_T)=\hbar(\widehat\mu_T)=h(T)$. Hence $\widehat\mu_T$ is the unique measure of maximal geometric entropy $\mu_\Omega$.
\[cor:ks\] Let $\alpha$, $\Gamma$ and $T$ be as in Theorem \[thm:ks\]. Let $\mu_T\in Curr(F)$ be the Patterson-Sullivan current for $T$ normalized so that the corresponding measure $\widehat\mu_T\in \mathcal M(\Gamma)$ is a probability measure. Then
1. $\hbar(\widehat\mu_T)=h(T)$, so that $\widehat\mu_T=\mu_\Omega$, the unique probability measure of maximal entropy;
2. for $\mu\in Curr(F)$ we have $h_T(\mu)=h(T)$ if and only if $\mu$ is proportional to $\mu_T$.
Theorem \[thm:ks\] and Corollary \[cor:ks\] yield Theorem \[E\] from the Introduction.
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[^1]: The first author was supported by the NSF grants DMS-0603921 and DMS-0904200. Both authors acknowledge the support of the Swiss National Foundation for Scientific Research. The first author also acknowledges the support of the Max-Planck-Institut für Mathematik, Bonn and the activity “Dynamical Numbers” organized there by Sergiy Kolyada.
|
---
abstract: 'This is a latest survey article on embeddings of multibranched surfaces into 3-manifolds.'
address: 'Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan'
author:
- Makoto Ozawa
title: 'Multibranched surfaces in 3-manifolds'
---
[^1]
Throughout this article, we will work in the piecewise linear category. All topological spaces are assumed to be second countable and Hausdorff. It is a fundamental problem that for two topological spaces $X$ and $Y$,
1. Can $X$ be embedded in $Y$?
2. If $X$ can be embedded in $Y$, then
1. When are two embeddings of $X$ in $Y$ equivalent?
2. How can $X$ be embedded in $Y$?
In this article, we consider the case that $X$ is a multibranched surface and $Y$ is a closed orientable 3-manifold.
We say that a 2-dimensional CW complex is a [*multibranched surface*]{} if we remove all points whose open neighborhoods are homeomorphic to the 2-dimensional Euclidean space, then we obtain a 1-dimensional complex which is homeomorphic to a disjoint union of some simple closed curves.
Multibranched surfaces naturally arise in several areas:
- Polycontinuous patterns|a mathematical model of microphase-separated structures made by block copolymers ([@HCO], [@HS], [@FCKHS])
- 2-stratifolds | as spines of closed 3-manifolds ([@GGH], [@GGH16], [@GGH17])
- Trisections, multisections | as an analogue of Heegaard splittings ([@GK], [@Ko], [@RT])
- Essential surfaces | as non-meridional essential surfaces in link exteriors ([@EM], [@EO]), essential surfaces in handlebody-knot exteriors ([@KO2]) and Dehn surgery ([@EM], [@GL])
In Section 1, we define several terms related with multibranched surfaces. In Section 2, we describe several backgrounds of multibranched surfaces. In Section 3, we study embeddings of multibranched surfaces into closed orientable 3-manifolds. In Section 4, we consider on multibranched surfaces which cannot be embedded into the 3-sphere.
Definitions
===========
Definition
----------
Let $\Bbb{R}^2_+$ be the closed upper half-plane $\{(x_1,x_2)\in \Bbb{R}^2 \mid x_2\ge 0\}$. The [*multibranched Euclidean plane*]{}, denoted by $\Bbb{R}^2_i$ $(i\ge 1)$, is the quotient space obtained from $i$ copies of $\Bbb{R}^2_+$ by identifying with their boundaries $\partial \Bbb{R}^2_+=\{(x_1,x_2)\in\Bbb{R}^2\mid x_2=0\}$ via the identity map. See Figure \[model\] for the multibranched Euclidean plane $\Bbb{R}^2_5$.
![The multibranched Euclidean plane $\Bbb{R}^2_5$[]{data-label="model"}](model-crop.pdf){width="0.4\linewidth"}
A second countable Hausdorff space $X$ is called a [*multibranched surface*]{} if $X$ contains a disjoint union of simple closed curves $l_1,\ldots, l_n$ satisfying the following:
1. For each point $x\in l_1\cup \cdots \cup l_n$, there exist an open neighborhood $U$ of $x$ and a positive integer $i$ such that $U$ is homeomorphic to $\Bbb{R}^2_i$.
2. For each point $x\in X-(l_1\cup\cdots\cup l_n)$, there exists an open neighborhood $U$ of $x$ such that $U$ is homeomorphic to $\Bbb{R}^2$.
Construction
------------
To construct a compact multibranched surface, we prepare a closed $1$-dimensional manifold $B$ (corresponding to $l_1,\ldots, l_n$), a compact $2$-dimensional manifold $S$ (corresponding to the closure of each component of $X-(l_1\cup\cdots\cup l_n)$) and a map $\phi: \partial S \to B$ satisfying that for every connected component $c$ of $\partial S$, the restriction $ \phi|_c : c \to \phi(c) $ is a covering map. Then a multibranched surface $X$ can be constructed from the triple $(B, S; \phi)$ as the quotient space $X=B \cup_\phi S$. A connected component of $B$, $S$ or $\partial S$ is said to be a [*branch*]{}, [*sector*]{} or [*prebranch*]{} respectively. The set consisting of all branches or sectors is denoted by $\mathcal{B}(X)$ or $\mathcal{S}(X)$ respectively.
Degree, oriented degree, regularity
-----------------------------------
For a prebranch $c$ of a multibranched surface $X$, the covering degree of $\phi|_c:c\to \phi(c)$ is called the [*degree*]{} of $c$ and denoted by $d(c)$. We give an orientation for each branch and each prebranch $c$ of $X$ (In the case that every sector $s$ is orientable, the orientation of $c$ is induced by that of $s$). The [*oriented degree*]{} of a prebranch $c$ of $X$ is defined as follows: if the covering map $\phi|_c:c\to \phi(c)$ is orientation preserving, the [*oriented degree*]{} $od(c)$ of $c$ is defined by $od(c)=d(c)$ and if it is orientation reversing, the oriented degree is defined by $od(c)=-d(c)$.
A prebranch $c$ of $X$ is said to be [*attached*]{} to a branch $l$ if $\phi(c)=l$. We denote by $\mathcal{A}(l)$ the set consisting of all prebranches which are attached to a branch $l$ and the number of elements of $\mathcal{A}(l)$ is called the [*index*]{} of $l$ and denoted by $i(l)$.
A multibranched surface $X$ is [*regular*]{} if for every branch $l$ and every prebranch $c,c'\in\mathcal{A}(l)$, $d(c)=d(c')$. Let $X$ be a regular multibranched surface. Since each pair of prebranches $c,c'\in\mathcal{A}(l)$ has same degree, the [*degree*]{} of a branch $l$ is well-defined as $d(l)=d(c)=d(c')$.
Graph representation
--------------------
Let $X$ be a compact multibranched surface obtained from $(B, S; \phi)$ such that all components of $S$ are orientable and have non-empty boundary (Hereafter, we assume such conditions on multibranched surfaces unless otherwise stated). The multibranched surface $X=B \cup_\phi S$ has a graph representation ([@EO]) as follows. Let $G=(V_S\cup V_B,E)$ be a bipertite graph such that $|V_S|=|\mathcal{S}(X)|$ and $|V_B|=|\mathcal{B}(X)|$. For each sector $s\in\mathcal{S}(X)$, we assign a vertex $v(s)\in V_S$ with a label $g(s)$. For each branch $l\in\mathcal{B}(X)$, we assign a vertex $v(l)\in V_B$. For a prebranch $c\subset \partial s$, we assign an edge $e\in E$ with a label $od(c)$ connecting $v(s)$ and $v(l)$ where $c\in\mathcal{A}(l)$. Almost similar concept to this graph representation has been defined in [@GGH16].
\[crosscap\] A closed non-orientable surface of crosscap number $h$ can be regarded as a multibranched surface $X$ with $h$ branches $B=l_1\cup \cdots \cup l_h$ and a planar surface $S$ with $h$ boundary components, such that $od(c)=2$ for any prebranch $c\subset \partial S$. Then $X$ has a graph representation $G$ as shown in Figure \[non-orientable\].
------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------
![A multibranched surface $X$ and its graph representation $G$[]{data-label="non-orientable"}](non-orientable-crop.pdf "fig:"){width=".3\linewidth"} ![A multibranched surface $X$ and its graph representation $G$[]{data-label="non-orientable"}](non-orientable_g-crop.pdf "fig:"){width=".27\linewidth"}
$X$ $G$
------------------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------
Incident matrix
---------------
For a sector $s\in\mathcal{S}(X)$ and a branch $l\in\mathcal{B}(X)$, we define the [*algebraic degree*]{} $ad_s(l)$ as follows. $$\displaystyle ad_s(l) = \sum_{c\in \mathcal{A}(l) \cap \partial s} od(c)$$ Then we define the [*incident matrix*]{} $M_X=(a_{ij})$ $(i=1,\ldots,n; j=1,\ldots,m)$ by $$a_{ij} = ad_{s_j} (l_i),$$ where $\mathcal{B}(X)=\{l_1,\ldots,l_n\}$ and $\mathcal{S}(X)=\{s_1,\ldots,s_m\}$.
First homology group
--------------------
For a branch $l$ and a sector $s$ of a regular multibranched surface $X$, we define $d(l;s)=\sum_{c \subset \partial s} od(c)$, where $c$ is a prebranch attached to $l$. The multibranched surface obtained by the removing an open disk from each sector is denoted by $\dot{X}$.
\[homology\] Let $X$ be a regular multibranched surface with $\mathcal{B}(X)=\left\{ l_1, \ldots, l_n \right\}$, $\mathcal{S}(X)=\left\{ s_1, \ldots, s_m \right\}$. Then, $$H_1(X) = \left[ l_1, \ldots, l_n : \sum_{k=1}^{n}d(l_k;s_1)l_k, \ldots, \sum_{k=1}^{n}d(l_k;s_m)l_k \right] \oplus \mathbb{Z}^{r'(X)}$$ where $r'(X)=rank H_1(\dot{X})-n$.
Therefore, the torsion subgroup of $H_1(X)$ can be calculated from the incident matrix $M_X$.
Let $X$ be a multibranched surface which has a graph representation as shown in Figure \[X\_2\], where $n=4$, $g_i=0$ and all wrapping number is 1. In [@MO Example 4.2], the first homology group is calculated by using Theorem \[homology\] as $H_1(X)=(\Bbb{Z}/3\Bbb{Z})\oplus \Bbb{Z}^4$.
As we shall see later, the incident matrix of $X$ is: $$M_{X}=\left( \begin{array}{cccc}
0 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 \\
1 & 1 & 0 & 1 \\
1 & 1 & 1 & 0 \\
\end{array} \right)$$ This matrix is equivalent to $(3)$ as follows. $$\left( \begin{array}{cccc}
0 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 \\
1 & 1 & 0 & 1 \\
1 & 1 & 1 & 0 \\
\end{array} \right)
\sim
\left( \begin{array}{cccc}
3 & 3 & 3 & 3 \\
1 & 0 & 1 & 1 \\
1 & 1 & 0 & 1 \\
1 & 1 & 1 & 0 \\
\end{array} \right)
\sim
\left( \begin{array}{cccc}
3 & 0 & 0 & 0 \\
1 & -1 & 0 & 0 \\
1 & 0 & -1 & 0 \\
1 & 0 & 0 & -1 \\
\end{array} \right)$$ $$\sim
\left( \begin{array}{cccc}
3 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{array} \right)
\sim
\left( \begin{array}{cccc}
3 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right)
\sim
\left( \begin{array}{c}
3 \\
\end{array} \right)$$ This shows that the torsion subgroup of $H_1(X)$ is $\Bbb{Z}/3\Bbb{Z}$.
On the other hand, a natural presentation for the fundamental group of a 2-stratifold was given in [@GGH17]. Thus, we can also obtain the first homology group via abelianization.
Circular permutation system and slope system
--------------------------------------------
A [*permutation*]{} of a set $A$ is a bijection from the additive group $\mathbb{Z} /n \mathbb{Z}$ into $A$. Two permutations $\sigma$ and $\sigma'$ of $A$ are [*equivalent*]{} if there is an element $k \in \mathbb{Z} /n \mathbb{Z}$ such that $\sigma'(x) = \sigma (x+k)$ ($x \in \mathbb{Z} /n \mathbb{Z}$). An equivalent class of a permutation of $A$ is a [*circular permutation*]{}.
For a regular multibranched surface $X$, we define the “circular permutation system” and “slope system” of $X$ as follows. A circular permutation of $\mathcal{A}(l)$ is called a [*circular permutation*]{} on a branch $l$. A collection $\mathcal{P}=\{ \mathcal{P}_l \}_{l \in \mathcal{L}(X)}$ is called a [*circular permutation system*]{} of $X$ if $\mathcal{P}_l$ is a circular permutation on $l$. For a branch $l$, a rational number $p/q$ with $q=d(l)$ is called a [*slope*]{} of $l$. A collection $\{ \mathcal{S}_l \}_{l \in \mathcal{L}(X)}$ is called a [*slope system*]{} of $X$ if $\mathcal{S}_l$ is a slope of $l$.
Neighborhoods
-------------
Let $X=B \cup_\phi S$ be a regular multibranched surface and let $\mathcal{P}=\{ \mathcal{P}_l \}_{l \in \mathcal{L}(X)}$ and $\mathcal{S}=\{ \mathcal{S}_l \}_{ l \in \mathcal{L}(X) }$ be a permutation system and a slope system of $X$ respectively. We will construct a compact and orientable $3$-manifold, which is uniquely determined up to homeomorphism by a pair of $\mathcal{P}$ and $\mathcal{S}$, by the following procedure.
First, for each branch $l\in\mathcal{B}(X)$ and each sector $s\in\mathcal{S}(X)$, we take a solid torus $l \times D^2$, where $D^2$ is a disk and take a product $s \times [-1, 1]$. If $s$ is non-orientable, then we take a twisted $I$-bundle $s \tilde{\times} [-1, 1]$ over $s$. We give orientations for these $3$-manifolds.
Next, we glue them together depending on the permutation system $\mathcal{P}$ and the slope system $\mathcal{S}$, where we assign the slope $\mathcal{S}_l$ of $l$ to the isotopy class of a loop $k_l$ in $\partial ( l \times D^2)$, by an orientation reversing map $\Phi: \partial S \times [-1, 1] \to \partial (B \times D^2)$ satisfying that for every branch $l$ and every prebranch $c$ with $\phi(c)=l$, the restriction $\Phi |_{c \times [-1, 1]} : c \times [-1, 1] \to N \left(k_l; \partial \left( l \times D^2 \right) \right)$ is a homeomorphism.
Then, we uniquely obtain a compact and orientable $3$-manifold with boundary, denoted by $N(X; \mathcal{P}, \mathcal{S})$. The $3$-manifold $N(X; \mathcal{P}, \mathcal{S})$ is called the [*neighborhood*]{} of $X$ with respect to $\mathcal{P}$ and $\mathcal{S}$. The set consisting of all neighborhoods of $X$ is denoted by $\mathcal{N}(X)$.
Background
==========
Graphs
------
A graph $G$ can be regarded as a 1-dimensional CW complex, where a vertex and an edge correspond to a 0-cell and 1-cell respectively, and vertices of an edge specify the attaching map for a 1-cell to 0-cells. This structure can be extended to 2-dimensional objects as in Subsection \[construction\], that is, we extend vertices, edges, the attaching map to a closed $1$-dimensional manifold $B$ (branch), a compact $2$-dimensional manifold without closed component $S$ (sector), a covering map $\phi: \partial S \to B$ respectively. Then a multibranched surface $X$ can be obtained as the quotient space $X=B\cup_{\phi}S$.
Kuratowski ([@K]) proved that a graph $G$ as a 1-dimensional CW complex cannot be embedded into $\mathbb{R}^2$ if and only if $G$ contains the complete graph $K_5$ or the complete bipartite graph $K_{3,3}$ as a subspace. At the present time, this result is stated that $G$ cannot be embedded into $\mathbb{R}^2$ if and only if $G$ has $K_5$ or $K_{3,3}$ as a minor. Robertson–Seymour ([@RS]) showed that for any minor closed property $P$, the set of minor minimal graphs which do not have $P$ is finite. This motivates us to consider the following problem: Characterize all “minor minimal” multibranched surfaces which cannot be embedded in $\mathbb{R}^3$ (Problem \[obstruction\_set\]). Since all closed non-orientable surfaces are minor minimal multibranched surfaces, the set of “minor minimal” multibranched surfaces which cannot be embedded in $\mathbb{R}^3$ is infinite. We will describe the details in Section \[forbidden\].
2-dimensional complexes
-----------------------
A 2-dimensional CW complex is a multibranched surface if we remove all points whose open neighborhoods are homeomorphic to $\mathbb{R}^2$, then we obtain a 1-dimensional complex which is homeomorphic to a disjoint union of some simple closed curves. Thus the set of multibranched surfaces is a subset of the set of 2-dimensional CW complexes.
Embeddings of 2-complexes into manifolds are widely studied in [@HM].
Matousek–Sedgwick–Tancer–Wagner ([@MSTW]) showed that there is an algorithm that, given a 2-dimensional simplicial complex $K$, decides whether $K$ can be embedded (piecewise linearly, or equivalently, topologically) in $\mathbb{R}^3$.
Carmesin ([@C], [@C2], [@C3], [@C4], [@C5]) proved that a locally 3-connected simply connected 2-dimensional simplicial complex has a topological embedding into 3-space if and only if it has no space minor from a finite explicit list $\mathcal{Z}$ of obstructions.
Essential surfaces {#essential}
------------------
The embedding of multibranched surfaces in the 3-sphere $S^3$ is closely related to the existence of essential surfaces in link exteriors. Let $L$ be a link in $S^3$ and $F$ be an essential surface properly embedded in the exterior $E(L)$ of $L$ whose boundary is non-meridional. By shrinking the regular neighborhood $N(L)$ into $L$ and extending $F$ along it, we obtain an essential multibranched surface $X$ embedded in $S^3$, where we say that a multibranched surface $X$ with branches $B$ and sectors $S$ embedded in $S^3$ is [*essential*]{} if $S\cap E(B)$ is essential, namely, incompressible, boundary-incompressible and not boundary-parallel in $E(B)$. Conversely, let $X$ be an essential multibranched surface with branches $B$ and sectors $S$ embedded in $S^3$. Then $B$ is a link in $S^3$ and $S\cap E(B)$ is an essential surface properly embedded in $E(B)$ whose boundary is non-meridional. Therefore, the set of all pairs of a link in $S^3$ and an essential surface properly embedded in the exterior of it whose boundary is non-meridional is equal to the set of all essential multibranched surfaces embedded in $S^3$.
Fundamental problem
-------------------
The Menger–Nöbeling theorem ([[@E Theorem 1.11.4.]]{}) shows that any finite $2$-dimensional CW complex can be embedded in $\mathbb{R}^{5}$. Furthermore, any multibranched surface can be embedded in $\mathbb{R}^4$ ([@MO Proposition 2.3]). More generally, any finite 2-dimensional simplicial complex whose intrinsic 1-skeleton is a proper subset of $K_7$ embeds in $\mathbb{R}^4$ ([@G]).
If for a branch $l$, there exist prebranches $c, c'\in \mathcal{A}(l)$ such that $d(c)\ne d(c')$, then the multibranched surface cannot be embedded in any 3-manifold. The converse also holds, namely we have shown that a multibranched surface can be embedded in some closed orientable 3-manifold if and only if the multibranched surface is regular ([@RBS Corollary 2.4], [@MO Proposition 2.7]).
We remark that any 3-manifold can be embedded in $\Bbb{R}^5$ ([@W]). Thus, we obtain the next diagram on the embedability of multibranched surfaces (Figure \[embedding\]).
![The embeddability of multibranched surfaces[]{data-label="embedding"}](embedding-crop.pdf){width=".85\linewidth"}
The following problems are fundamental for embeddings of multibranched surfaces.
\[3-manifold\] For a regular multibranched surface $X$, find a most simple closed orientable 3-manifold $M$ in which $X$ can be embedded. Moreover, determine the minimal Heegaard genus of such 3-manifold $M$.
\[3-sphere\] For a regular multibranched surface $X$, determine whether or not $X$ can be embedded in the 3-sphere $S^3$.
We consider Problem \[3-manifold\] in Section \[embedding\], and Problem \[3-sphere\] in Section \[forbidden\].
Embeddings into 3-manifolds {#embedding}
===========================
Genus
-----
For a closed orientable 3-manifold $M$, the Heegaard genus is a fundamental index. The [*Heegaard genus*]{} $g(M)$ of $M$ is defined as the minimal genus of a closed orientable surface embedded in $M$ which separates $M$ into two orientable handlebodies.
For an orientable compact 3-manifold $N$ with boundary, the minimal Heegaard genus of closed orientable 3-manifolds in which $N$ can be embedded is denoted by $eg(N)$ and called the [*embeddable genus*]{} of $N$. We remark that $eg(N)\le g(N)$ ([@MO Proposition 3.1]), where $g(N)$ denotes the minimal genus of Heegaard splittings of $N$ in a sense of Casson–Gordon ([@CG]).
For a regular multibranched surface $X$, we define the [*minimum genus*]{} $\min g(X)$ and [*maximum genus*]{} $\max g(X)$ respectively as follows. $$\min g(X)=\min \{ eg(N) \mid N \in \mathcal{N}(X) \}$$ $$\max g(X)=\max \{ eg(N) \mid N \in \mathcal{N}(X) \}$$
Upper bounds
------------
There are following inequalities which give upper bounds for the minimum genus and the maximum genus. The next theorem states originally that $\min g(X) \le |\mathcal{B}(X)| + |\mathcal{S}(X)|$, but its proof is still effective for $\max g(X)$ and implies the latter half.
For a regular multibranched surface $X$, the next inequality holds. $$\max g(X) \le |\mathcal{B}(X)| + |\mathcal{S}(X)|$$ Moreover, if the wrapping number of each branch is 1, then $$\max g(X) \le |\mathcal{S}(X)|.$$
In the proof of [[@MO Theorem 3.5]]{}, it is shown that $X$ can be embedded in a connected sum of $|\mathcal{B}(X)|$ lens spaces and $|\mathcal{S}(X)|$ $S^2\times S^1$’s. Yuya Koda asked me whether any closed orientable 3-manifold contains a minimal genus embedding of some multibranched surfaces.
The next theorem follows from two cited results and gives an estimate for the embeddable genus of a neighborhood of a regular multibranched surface.
For a regular multibranched surface $X$ and any neighborhood $N \in \mathcal{N}(X)$, the next inequality holds. $$rank H_1(X) -g(\partial N)\le eg(N) \le rank H_1(G_N) + g(\partial N),$$ where $G_N$ denotes the abstract dual graph of $N$ and $g(\partial N)$ denotes the sum of genera of all components of $\partial N$.
Lower bounds
------------
The following lower bounds for the minimum genus and the maximum genus are known.
\[lower\] For a regular multibranched surface $X$, the next inequalities hold. $$\begin{aligned}
\min g(X) &\ge& rank H_1(X) - \max_{N \in \mathcal{N}(X)} g(\partial N)\\
\max g(X) &\ge& rank H_1(X) - \min_{N \in \mathcal{N}(X)} g(\partial N)\end{aligned}$$
Graph product $G \times S^1$
----------------------------
For a graph $G$, we obtain a regular multibranched surface by taking a product with $S^1$. We consider the genus of a regular multibranched surface which forms $G\times S^1$, and by using Theorem \[lower\], we obtain the following theorem which is an interplay of the genus of a graph $G$ and the genus of a multibranched surface $G\times S^1$.
The [*minimum genus*]{} $\min g(G)$ of a graph $G$ is defined as the minimal genus of closed orientable surfaces in which $G$ can be embedded. The [*maximum genus*]{} $\max g(G)$ of a graph $G$ is defined as the maximal genus of closed orientable surfaces in which $G$ can be embedded and the complement of $G$ consists of open disks. It is remarkable that Xuong and Nebeský determined the maximum genus of a graph by a completely combinatorial formula ([[@X Theorem 3]]{}, [[@N81 Theorem 2]]{}).
\[product\] For a graph $G$, the next equalities hold. $$\begin{aligned}
\min g(G\times S^1) &=& 2 \min g(G)\\
\max g(G\times S^1) &=& 2 \max g(G)\end{aligned}$$
In [[@T Corollary 1.2]]{}, it was shown that the minimal number $\dim H_1(M ;F)$ for a closed orientable 3-manifolds $M$ containing $G\times S^1$ equals to $2\min g(G)$, where $F=\Bbb{Z}_p$ or $\Bbb{Q}$. It is well-known that $g(M)\ge \dim H_1(M ;F)$. Hence the inequality $\min g(G\times S^1) \ge 2 \min g(G)$ in Theorem 3.5 (3.3) holds.
Spine of closed 3-manifolds
---------------------------
A multibranched surface $X$ is called a [*2-stratifold*]{} if every prebranch $c$ of $X$ satisfies $d(c)>2$. Gómez-Larrañaga–González-Acuña–Heil studied 2-stratifolds from a view point of 3-manifold groups. They asked the following questions.
Which 3-manifolds $M$ have fundamental groups isomorphic to the fundamental group of a 2-stratifold?
Which closed 3-manifolds $M$ have spines that are 2-stratifolds?
Recall that a subpolyhedron $P$ of a 3-manifold $M$ is a spine of $M$, if $M-{Int}(B^3)$ collapses to $P$, where $B^3$ is a 3-ball in $M$. An equivalent definition is that $M-P$ is homeomorphic to an open 3-ball. They completely answered these questions.
Let $M$ be a closed 3-manifold and $X_G$ be a 2-stratifold. If $\pi_1(M)\cong \pi_1(X_G)$, then $\pi_1(M)$ is a free product of groups, where each factor is cyclic or $\mathbb{Z}\times \mathbb{Z}_2$.
A closed 3-manifold $M$ has a 2-stratifold as a spine if and only if $M$ is a connected sum of lens spaces, $S^2$-bundles over $S^1$, and $P^2\times S^1$’s.
Neighborhood equivalence
------------------------
In this subsection, we assume that a multibranched surface is regular, does not have disk sectors, and the degree is greater than 2 for each branch. Let $A$ be an annulus sector of $X$ whose boundary consists of two branches, where at least one branch has the wrapping number $1$, or a Möbius-band sector whose boundary has the wrapping number $1$. An [*IX-move*]{} along $A$ is an operation shrinking $A$ into the core circle, and an [*XI-move*]{} is a reverse operation of an IX-move.
If two multibranched surfaces $X,X'$ embedded in a 3-manifold $M$ are related by a IX-moves or XI-moves, then the regular neighborhoods $N(X), N(X')$ are isotopic in $M$. The following theorem states that the converse holds.
\[IH\] Let $X,\ X'$ be multibranched surfaces embedded in an orientable $3$-manifold $M$. If $N(X)$ is isotopic to $N(X')$ in $M$, then $X$ is transformed into $X'$ by a finite sequence of IX-moves, XI-moves and isotopies.
For more broad class, Matveev–Piergallini theorem is known: Two simple polyhedra embedded in a 3-manifold has isotopic neighborhoods if and only if they are connected by a sequence of [*$2 \leftrightarrow 3$ moves*]{}, [*$0 \leftrightarrow 2$ moves*]{} moves and isotopy ([@M88], [@P88]).
Neighborhood partial order {#NE}
--------------------------
Let $X$ be an essential multibranched surface embedded in a closed orientable 3-manifold $M$. We say that a sector $s$ is [*excess*]{} if it is boundary-parallel in $M-int N(X-s)$. A multibranched surface $X$ is said to be [*efficient*]{} if every sector is not excess.
In this subsection, we restrict multibranched surfaces to the set $\mathcal{X}$ of all connected compact multibranched surfaces $X$ embedded in a closed orientable 3-manifold $M$ satisfying the following conditions: $X$ is maximally spread (that is, applied XI-moves to $X$ as much as possible), essential and efficient in $M$, has no open disk sector, no branch of degree 1 or 2.
Under the influence of Theorem \[IH\], we define an equivalence relation on $\mathcal{X}$ as follows. Two multibranched surfaces $X$ and $X'$ in $\mathcal{X}$ are [*neighborhood equivalent*]{}, denoted by $X\overset{\mathrm{N}}\sim X'$, if $X$ is transformed into $X'$ by a finite sequence of IH-moves. Moreover, we define a binary relation $\le$ over $\mathcal{X}$ as follows.
\[relation\] For $X,\, Y\in \mathcal{X}$, we denote $X \le Y$ if
1. there exists an isotopy of $Y$ in $M$ so that $Y\subset N(X)$ and $B_Y\subset N(B_X)$, and
2. there exists no essential annulus in $N(X)-Y$.
For equivalence classes $[X], [Y]\in \mathcal{X}/\overset{\mathrm{N}}\sim $, we define a [*neighborhood partial order*]{} $\preceq$ over $\mathcal{X}/\overset{\mathrm{N}}\sim $ so that $[X]\preceq [Y]$ if $X\le Y$.
\[partial\] The relation $\preceq$ is well-defined on $\mathcal{X}/\overset{\mathrm{N}}\sim $ and $(\mathcal{X}/\overset{\mathrm{N}}\sim ; \preceq)$ is a partially ordered set.
We say that $B_X$ is [*toroidal*]{} if there exists an essential torus $T$ in the exterior $E(B_X)$ of $B_X$ in $M$, that is, $T$ is incompressible in $E(B_X)$ and $T$ is not parallel to a torus in $\partial E(B_X)$. We say that $E_X$ is [*cylindrical*]{} if there exists an essential annulus $A$ with $A\cap X =A\cap E_X=\partial A$, that is, $A$ is incompressible and $A$ is parallel to neither an annulus in $E_X$ nor an annulus in $\partial E(B_X)$.
\[sufficient\] For equivalence classes $[X], [Y]\in \mathcal{X}/\overset{\mathrm{N}}\sim $, if $[X]\preceq [Y]$ and $[X]\ne [Y]$, then either $B_Y$ is toroidal or $S_Y$ is cylindrical.
Theorem \[sufficient\] provides a sufficient condition for an equivalent class $[X]\in \mathcal{X}/\overset{\mathrm{N}}\sim $ to be minimal with respect to the partial order of $(\mathcal{X}/\overset{\mathrm{N}}\sim ; \preceq)$, that is, if $B_X$ is atoroidal and $E_X$ is acylindrical, then $[X]$ is minimal.
Essential decomposition | Eudave-Muñoz knots type
-------------------------------------------------
Let $X$ be a multibranched surface embedded in the 3-sphere $S^3$, which decomposes $S^3$ into regions $V_1,\ldots, V_n$. If $X$ is essential, then we call this decomposition $S^3=V_1\cup\cdots\cup V_n$ an [*essential decomposition*]{}. As explained in Subsection \[essential\], a link with an essential surface of non-meridional boundary slope gives an essential decomposition.
In this subsection, we recall Eudave-Muñoz knots ([@EM]) in the language of multibranched surface. Let $X$ be a multibranched surface which has a twice punctured torus as a sector $s$ and a single branch $l$, where one prebranch $c$ has $od(c)=2$ and another prebranch $c'$ has $od(c')=-2$.
Suppose that $X$ is embedded in $S^3$ so that it is essential and two regions of $S^3-X$ are genus two handlebodies, say $H$ and $W$. Then, by combining [@EM] with [@GL], the branch $l$ forms an Eudave-Muñoz knot. From the point of view that any essential embedding restricts the knot type of the branch, this phenomenon is especial in low-dimensional geometric topology.
Eudave-Muñoz knots appears in the last piece of the classification of essential annuli in the exterior of genus two handlebody-knots in $S^3$ ([@KO2]). We take a regular neighborhood $N(l)$ and denote two handlebodies $S^3-N(l)-s$ by $H$ and $W$ again. See Figure \[EM\] for the configulation. Put $A=N(l)\cap W$. Then, $H$ is a genus two handlebody-knots with an essential annulus $A$ of Type 4 in [@KO2].
![$(1,2,2;2)$-trisection coming from Eudave-Muñoz knots[]{data-label="EM"}](em-crop.pdf){width=".25\linewidth"}
This configulation as in Figure \[EM\] also provides a nice example of trisection. Let $X'$ be a multibranched surface which has two branches $b\cup b'=N(l)\cap H\cap W$ and three sectors $s_1=H\cap W$, $s_2=N(l)\cap H$ and $s_3=N(l)\cap W$. Then, $X'$ gives an essential decomposition $S^3=N(l)\cup H\cup W$, where the triple of genera of three handlebodies is $(1,2,2)$ and the number of branches is $2$. Thus this gives a $(1,2,2;2)$-trisection of $S^3$. Moreover, it is shown in [[@Ko Proposition 4.7.1]]{} that this trisection is not stabilizations of any other trisection.
Efficient embedding | Universal bounds
--------------------------------------
Recall the relation between essential surfaces in link exteriors and essential multibranched surfaces in Subsection \[essential\], and the definition of efficient embedding in Subsection \[NE\]. Suppose that $X$ is an essential and efficient multibranched surface embedded in a 3-manifold. Then, we have a link and essential surfaces in the link exterior, and moreover any pair of essential surfaces is not mutually parallel.
Let $X$ be a multibranched surface which has a single branch and $n$ once punctured tori each of which has an oriented degree 1. Suppose that $X$ is embedded in $S^3$ so that it is essential and efficient and the branch forms a hyperbolic knot. Then, we have a hyperbolic knot bounding $n$ genus 1 Seifert surfaces which are not mutually parallel. Tsutsumi first showed that the number $n$ is at most 7 ([@Tsu]). After that, Eudave-Muñoz – Ramírez-Losada – Valdez-Sánchez showed that $n$ is at most 6 and provided an example of such embedding of $X$ for $n=5$ ([@ERV]). Finally, Valdez-Sánchez showed that $n$ is at most 5 ([@V]) and therefore this bound is optimal.
This phenomenon is also especial in low-dimensional geometric topology. In general, contrary to the above, there is no upper bound. Tsutsumi showed that for any positive integer $n$, there is a genus one hyperbolic knot in $S^3$ which bounds mutually non-parallel incompressible Seifert surfaces $S, F_1,\ldots, F_n$ where $S$ is genus one and $F_i$ is genus two ([@Tsu Theorem 5.5]).
Forbidden minors for $S^3$ {#forbidden}
==========================
Minor
-----
In this subsection, we allow the degree $deg(B_i)$ of a branch $B_i$ to be $1$ or $2$.
We denote by $\mathcal{M}$ the set of all regular multibranched surfaces (modulo homeomorphism). For $X$, $Y\in \mathcal{M}$, we write $X < Y$ if $X$ is obtained by removing a sector of $Y$, or $X$ is obtained from $Y$ by an IX-move. We define an equivalence relation $\sim$ on $\mathcal{M}$ as follows: if $X < Y$ and $Y < X$, then $X \sim Y$. We define a partial order $\prec$ on $\mathcal{M}/\sim$ as follows. Let $X$, $Y \in \mathcal{M}$. We denote $[X] \prec [Y]$ if there exists a finite sequence $X_1, \ldots, X_n \in \mathcal{M}$ such that $X_1 \sim X$, $X_n \sim Y$ and $X_1 < \cdots < X_n$.
Obstruction set
---------------
A multibranched surface class $[X]$ is called a [*minor*]{} of another multibranched surface class $[Y]$ if $[X] \prec [Y]$. In particular, $[X]$ is called a [*proper minor*]{} of $[Y]$ if $[X] \prec [Y]$ and $[Y] \not= [X]$. A subset $\mathcal{P}$ of $\mathcal{M}/\sim$ is said to be [*minor closed*]{} if for any [$[X] \in \mathcal{P}$, every minor of $[X]$ belongs to $\mathcal{P}$]{}. For a minor closed set $\mathcal{P}$, we define the [*obstruction set*]{} $\Omega(\mathcal{P})$ by all elements $[X] \in \mathcal{M}/\sim $ such that $[X] \not \in \mathcal{P}$ and every proper minor of $[X]$ belongs to $\mathcal{P}$.
The set of multibranched surfaces embeddable into $S^3$, denoted by $\mathcal{P}_{S^3}$, is minor closed. As a 2-dimensional version of Kuratowski’s and Wagner’s theorems, we consider the next problem.
\[obstruction\_set\] Characterize the obstruction set $\Omega(\mathcal{P}_{S^3})$.
We summarize all known results on $\Omega(\mathcal{P}_{S^3})$ at the present moment. As we shall see later, (2) and (3) in Theorem \[obstruction\] are infinite family of multibranched surfaces.
\[obstruction\] The following multibranched surfaces belong to $\Omega(\mathcal{P}_{S^3})$.
1. $K_5\times S^1$ and $K_{3,3}\times S^1$ $($[@S]$)$
2. $X_1$, $X_2$, $X_3$ $($[@EMO]$)$
3. $X_g(p_1,\ldots,p_n)$ $($[@MO]$)$
\(1) Since any proper minor of $K_5$ and $K_{3,3}$ is planar, any proper minor of $K_5\times S^1$ and $K_{3,3}\times S^1$ can be embedded in $D^2\times S^1\subset S^3$.
\(2) We say that a multibranched surface $X$ is [*critical*]{} for $S^3$ if $X$ cannot be embedded in $S^3$ and for any $x\in X$, $X-x$ can be embedded in $S^3$. It is shown in [@EMO] that $X_1$, $X_2$, $X_3$ are critical for $S^3$.
\(3) Since $X_g(p_1,\ldots,p_n)$ has a single sector, the minimality for $\mathcal{P}_{S^3}$ naturally holds.
Theorem \[obstruction\] (1) was proved in [@S Theorem 1]. It also follows Theorem \[product\] and Kuratowski’s and Wagner’s theorem.
The families of multibranched surfaces $X_1,\ X_2,\ X_3$ in Theorem \[obstruction\] (2) are given as follows.
Let $X_1$ be a multibranched surface which is obtained from a single sector of genus $g$, $n$ boundary components and a single branch by a covering map with degree $\epsilon_i$ on each prebranch. See Figure \[X\_1\] for a graph representation. We assume that $\epsilon_i=\pm p$ for the regularity of $X_1$. Then the incident matrix is $M_{X_1}=\big(\sum_{i=1}^n \epsilon_i\big)$. If $\big|\sum_{i=1}^n \epsilon_i\big|>1$, then $H_1(X_1)$ has a torsion and $X_1$ cannot be embedded in $S^3$. Conversely, if $\big|\sum_{i=1}^n \epsilon_i\big|\le 1$, then by [@EMO Theorem 3.2], $X_1$ can be embedded in $S^3$. Hence $X_1\in \Omega(\mathcal{P}_{S^3})$ if and only if $\big|\sum_{i=1}^n \epsilon_i\big|>1$.
![A graph representation of $X_1$[]{data-label="X_1"}](x1-crop.pdf){width=".2\linewidth"}
Let $X_2$ be a multibranched surface which has a graph representation as shown in Figure \[X\_2\], where $n\ge 3$, all wrapping number is 1 (we omit the labels on edges). Then by [@EMO Theorem 3.3], $X_2\in \Omega(\mathcal{P}_{S^3})$.
![A graph representation of $X_2$[]{data-label="X_2"}](x2-crop.pdf){width=".4\linewidth"}
The incident matrix of $X_2$ is: $$M_{X_2}=\left( \begin{array}{cccccc}
0 & 1 & \cdots & \cdots & 1 \\
1 & 0 & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & 1 \\
1 & \cdots & \cdots & 1 & 0
\end{array} \right)$$ Since $det(M_{X_2})=(-1)^{n+1}(n-1)$ and $n\ge 3$, $X_2$ has a torsion.
Let $X_3$ be a multibranched surface which has a graph representation as shown in Figure \[X\_3\], where $n\ge 2,\ k_i\ge 1,\ k_1k_2k_3\cdots k_n\ge3$, all wrapping number is 1 unless otherwise specified. Then by [@EMO Theorem 3.7], $X_3\in \Omega(\mathcal{P}_{S^3})$.
![A graph representation of $X_3$[]{data-label="X_3"}](x3-crop.pdf){width=".45\linewidth"}
The incident matrix of $X_3$ is: $$M_{X_3}=\left( \begin{array}{cccccc}
k_{1} & 0 & \cdots & \cdots & 0 & -1 \\
-1 & k_{2} & 0 & \ddots & \ddots & 0\\
0 & -1 & k_3 & \ddots & \ddots & \vdots\\
\vdots & 0 & -1 & \ddots & \ddots & 0\\
\vdots & \ddots & \ddots & \ddots & k_{n-1} & 0\\
0 & \cdots & \cdots & 0 & -1 & k_n
\end{array} \right)$$ Since $det(M_{X_3})=k_1k_2k_3\cdots k_n -1 \geq 2$, $X_3$ has a torsion.
The multibranched surface $X_g(p_1,\ldots,p_n)$ in Theorem \[obstruction\] (3) was first presented in [@MO Example 4.3]. Let $X_g(p_1,\ldots,p_n)$ be a multibranched surface which has a graph representation as shown in Figure \[X\_g\], where $n\ge 1$, $p=gcd\{ p_1, \ldots, p_n\}>1$. As we have seen in Example \[crosscap\], a closed non-orientable surface of crosscap number $n$ is homeomorphic to $X_0(2,\ldots,2)$.
![A graph representation of $X_g(p_1,\ldots,p_n)$[]{data-label="X_g"}](xg-crop.pdf){width=".3\linewidth"}
As shown in [@MO Example 4.3], we have $H_1(X_g(p_1,\ldots,p_n))=\left( \mathbb{Z}/p \mathbb{Z} \right) \oplus \mathbb{Z}^{2g+n-1}$. Hence, $X_g(p_1,\ldots,p_n)$ cannot be embedded in $S^3$ since $p>1$.
Beyond torsion {#beyond}
--------------
In the previous subsection, we conclude that some multibranched surfaces cannot be embedded in $S^3$ in virtue of torsion part of the first homology group. Then, the following inverse problem naturally arises.
\[inverse\] If $p=1$, then can $X_g(p_1,\ldots,p_n)$ be embedded in $S^3$?
The following theorem gives a partial answer to Problem \[inverse\].
If $p=1$, then $X_g(p_1, p_2, p_3)$ can be embedded in $S^3$ for a sufficiently large $g$.
But, how can we say about Problem \[inverse\] when $g=0$? This is related with a main thema in [@EO]. In [@EO], we characterized non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. This implies that we can derive a formula for the triple $p_1, p_2, p_3$ ([[@EO Proposition 1.4]]{}). For hyperbolic links, we conjectured the following.
\[cabling\] There does not exist an essential n-punctured sphere with non-meridional, non-integral boundary slope in a hyperbolic link exterior in the 3-sphere.
It can be checked that the triple $(5, 7, 18)$ does not satisfy the formula in [[@EO Proposition 1.4]]{}. Therefore, assuming Conjecture \[cabling\], we conclude that $X_0(5, 7, 18)$ cannot be embedded in $S^3$.
On the other hand, if we allow embeddings in 3-manifolds other than $S^3$, then Problem \[inverse\] holds. We use a result in [@GA] that a compact $3$-manifold $M$ with connected boundary can be embedded in a homology $3$-sphere if and only if $H_1(M)$ is free and $H_2(M)=0$. Since for a unique neighborhood $N \in \mathcal{N}(X_g(p_1,\ldots,p_n))$, $H_1(N)$ is free and $H_2(N)=0$ when $p=1$, we have the following.
If $p=1$, then $X_g(p_1,\ldots,p_n)$ can be embedded in a homology $3$-sphere.
The prospects for multibranched surfaces
========================================
The author would like to close this survey article by stating the following prospects.
Firstly, it is important to characterize essential and efficient decompositions of $S^3$, where we say that a decomposition $S^3=V_1\cup \cdots \cup V_n$ by a multibranched surface $X$ is [*efficient*]{} if $X$ is efficient. It can be applied to Polycontinuous patterns, Trisections, Essential surfaces as stated in Introduction.
Secondly, it is a fundamental problem to characterize the obstruction set $\Omega(\mathcal{P}_{S^3})$. This problem has a difficulty as stated in Subsection \[beyond\], but it has also an interest in Conjecture \[cabling\].
[**Acknowledgements.**]{} The author would like to thank to Fico González-Acuña and Arkadiy Skopenkov for informimg me related results.
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[^1]: The author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 17K05262) and (B) (No. 16H03928), The Ministry of Education, Culture, Sports, Science and Technology, Japan
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abstract: 'The advances in Charge Coupled Devices in one hand and the high resolution measurements of holographic technique on the other hand, we have adopted the method of digital real-time holographic interferometry for the diagnostics of high density plasma. The measured values of plasma electron density agree with the measurements from other techniques.'
author:
- Joseph Thomas Andrews
- Kingshuk Bose
title: Plasma diagnostics using digital holographic interferometry
---
In recent years, digital holographic interferometry and shadowgraphy are emerging as a versatile tool for non destructive testing and diagnostics purposes in materials engineering [@recent]. Similar techniques are also adopted for diagnosing plasma and to measure various plasma parameters using pulsed lasers. However a plasma diagnostics method that can reliably and rapidly measure fluctuations with both spatial and time resolution can lead to increased understanding and eventual control of turbulent transport. Additionally, reliable diagnostics that measure the plasma profiles are critical to moving towards an actual prototype magnetic fusion reactor. Conventional methods for plasma diagnostics include [@paul] Langmuir probe [@Dude], microwave, laser interferometry [@Weber] and Thompson scattering [@Weber; @Game], Out of these, laser interferometry has the advantage as it is versatile and gives accurate results. Since the advent of optical lasers, interferometry has been widely used to study different types of plasmas [@LPP]. We have made an attempt to successfully adopt the digital holographic interferometry for plasma diagnostics. To authors knowledge, this is the first time digital holographic interferometry is employed for plasma discharge tubes in real-time. This open up the way for the diagnosis of arc plasma, DC discharge plasma, etc. The advantages of this technique over other conventional spectroscopic methods are its high resolution measurements of the order of few tens of nanometer, real-time measurements and cost effectiveness. Further this methods could give realistic parameters of plasma like electron density and electron temperature [@recent].
The present technique provides information in the form of two-dimensional maps of the electron density, without the need of extensive modeling. In a laser interferometry various types of interferometer can be used such as Michelson interferometer, Mach-Zhender interferometer (MZI) etc. We employed a Mach-Zhender interferometer. Chord integrated phase information is utilized to obtain the radial profile of electron density by using Abel inversion technique with appropriate software code written using MathCAD$\texttrademark$ and LabVIEW$\texttrademark$.
We adopt the fact that the geometrical and optical lengths traveled by light through plasma are different, since the refractive index of plasma is proportional to the density of free electrons. When a laser beam passes through a cylindrical tube containing circular symmetric plasma density $(n_e$), the light suffers a change in optical path length. Light passing through different chords of the tube as shown in Fig. 1, are phase shifted by different amount. The change in phase is estimated from [@book; @Lisi]
{width="6cm"}
$$\begin{aligned}
\Delta \phi(y_i) &=&\Delta k \cdot x_{i} \nonumber \\
&=&2\left({ 1- \frac{\lambda_{0}e^{2}\Delta n_{e}(y_{i})}{4\pi c^{2}m_{e}\epsilon_{0}}}\right) \sqrt{R^{2}-y_{i}^{2}},\end{aligned}$$
where $y_i$ is the horizontal distance of $i$th chord from origin, $\lambda_{0}$, $e$, $\Delta n_{e}(y_i)$, $c$, $m_e$ and $\epsilon_0$ are the wavelength of laser light, charge of an electron, change in electron plasma density at $i$th chord, speed of light, mass of electron and absolute permittivity, respectively. Free electrons produce negative phase shift because the effective index of refraction is less than 1. Hence, the change in refractive index will be less than 1, but positive.
The experimental setup is shown in Fig. 1. A 30 mW polarized He-Ne laser (Melles Griot) having central wavelength at 632.8 nm passes through a spatial filter assembly and a beam expander (BE). A cylindrical lens is used to convert the circular symmetric beam into a line beam. The line helps us to measure the optical properties along a single cross section of the plasma tube only. Also, the plasma tube, if mounted on a linear stage may be useful to scan the plasma tube completely. The expanded beam is divided using a 50:50 non-polarizing cubic beam splitter (BS1). A part of the beam is allowed to incident on a plane mirror (hereafter, we call it reference beam) while the other part is reflected by another mirror (object beam). In this experimental setup,
{width="8cm"}
we have used high pressure mercury lamps (Phillips) as a plasma source. Two identical lamps are used in both the arms of the Mach-Zhender setup. One lamp acts as a compensator while the other lamp is connected to power source. Both the beams interfere inside a non-polarizing cubic beam splitter (BS2). Interference fringes are monitored using a charge coupled device (CCD) (Apogee, Model- LISAA-M). A large area convex lens assembly (Pentax) is used to collect the beam into the CCD. The whole system is mounted on a vibration isolation table (Melles Griot). Interference fringes are viewed from one arm of BS2 using a screen. In the absence of any plasma, the interference fringes (or chord integrated intensity) recorded in the CCD can be expressed as [@Born] $$I(y_i)= I_1+I_2+ 2 \sqrt{I_{1}I_{2}}\cos( \phi ). \label{I0}$$ Here, $I_1$ and $I_2$ are the intensity of the laser in the two arms of the interferometer while $\phi(t)$ is the phase difference between the two beams. Since, the two beams are passing through a path which is identical but intentionally at a small angle so as to obtain straight line fringes. In the presence of plasma eq. (\[I0\]), may be rewritten as $$I(y_i, t)= I_1+I_2+ 2 \sqrt{I_{1}I_{2}}\cos[ \phi(t)+\Delta\phi(y_i,t)]. \label{IP}$$
Simulated results obtained using eq. (\[IP\]) and experimental observation of chord integrated intensity are shown as contour plots in Figs. 3a and 3b, respectively. Due to the limitation of the experimental setup we could obtain only a partial image from the CCD. A typical interference signal obtained using CCD is shown as Fig. 3c. The fringes continuously obtained with time are displayed in Fig. 3b. While obtaining the simulated results, the values of constants are obtained from the present experimental conditions. The experimental data is obtained from CCD, while the simulated results are generated from a software code written using LabVIEW. For times $t = 0$ to 50 sec no current is applied to PT1. Since no temporal change in phase is occurring, the fringe pattern remains almost same as evident from Figs. 3a and 3b. At $t = 50$sec, constant current is applied to the plasma source. In the presence of plasma the interference pattern are modulated
{width="8cm"}
with time. The number of fringes increases due to large change in plasma density, but saturates after the build-up (rising) time of plasma. The unknown parameter $\Delta n_e(y_i)$ for different chords is obtained after performing many iterations of the simulated results for minimum standard deviation. Best profile of $\Delta n_e(y_i)$ which matches gives minimum standard deviation with the experimental profile is used for calculation.
The measurement of phase change indirectly gives the magnitude of electron density. Figure 4, shows the temporal change in plasma density. When the plasma tube is switched ON, the electron density raises rapidly and saturates. However, it cools slowly after switching it OFF.
{width="8cm"}
{width="8cm"}
The chord integrated phase changes with respect to axial distance ($y_i$) are converted radial distance by using Abel inversion method and are shown in Figure 5. The axis on the left shows the numerical values of phase change while the axis on the right side is the estimated values of change in electron density after Abel inversion. These values agree with the recent observations using spectroscopic method [@Nima].
To conclude, we used the method of holographic interferometry to estimate the plasma electron density. The measured value agrees with the standard values of high pressure mercury plasma density. The present work has the potential of measuring, imaging the plasma density variations in real-time. This method can also be used for obtaining the 3D profile of electron density in a DC plasma.
DST & AICTE, New Delhi. The authors also thank Professors P. K. Sen and P. K. Barhai for Discussions.
[9]{}
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abstract: 'Cellular solids usually possess random microstructures that may contain a characteristic length scale, such as the cell size. This gives rise to size dependent mechanical properties where large systems behave differently from small systems. Furthermore, these structures are often irregular, which not only affects the size dependent behavior but also leads to significant property variations among different microstructure realizations. The computational model for cellular microstructures is based on networks of Timoshenko beams. It is a computationally efficient approach allowing to obtain statistically representative averages from computing large numbers of realizations. For detailed analysis of the underlying deformation mechanisms an energetically consistent continuization method was developed which links the forces and displacements of discrete beam networks to equivalent spatially continuous stress and strain fields. This method is not only useful for evaluation and visualization purposes but also allows to perform ensemble averages of, e.g., continuous stress patterns – an analysis approach which is highly beneficial for comparisons and statistical analysis of microstructures with respect to different degrees of structural disorder.'
address:
- 'Institute of Materials Simulation (WW8), Friedrich-Alexander University Erlangen-Nürnberg (FAU) , Dr.-Mack-Strasse 77, 90762 Fürth, Germany'
- 'Chair of Micromechanical Materials Modelling (MiMM), Institute of Mechanics and Fluid Dynamics, Technische Universität Bergakademie Freiberg, Lampadiusstrasse 4, 09599 Freiberg, Germany'
- 'Department of Engineering and Mechanics, Southwest Jiaotong University, Chengdu, P.R. China'
author:
- Stefan Liebenstein
- Stefan Sandfeld
- Michael Zaiser
bibliography:
- 'bibfile.bib'
title: 'Size and Disorder Effects in Elasticity of Cellular Structures: From Discrete Models to Continuum Representations'
---
Cellular Solids ,Size Effects ,Homogenization ,Microstructure ,Timoshenko Beam ,Cosserat ,Disorder
Introduction
============
Cellular solids are a class of lightweight materials which exhibit mass specific mechanical properties superior to their bulk counterparts. Their properties are strongly influenced by geometrical characteristics of the cellular microstructure[^1]. Cellular solids are commonly encountered in complex structures in nature, as for example in cancellous bone, cork or snow. But also man made devices for industrial applications benefit from exploiting microstructure design principles that are analogous to such naturally occurring materials. Typical examples are (metallic) foams, which may either possess a random microstructure or consist of highly regular, periodic arrangements of unit cells (such as e.g. in honeycomb structures). In general, one can distinguish between open- and closed-cellular foams ([@gibson1999]): whereas the microstructure of the former consists of a network of connected struts, the latter consists of thin, quasi two-dimensional walls. These porous, network-like microstructures result in low density, high mechanical energy absorption and good weight specific bending stiffness as compared to bulk materials. The details of the microstructure are controlled by the manufacturing process. This can be a natural growth process for biological materials or technological processes such as extrusion, casting, foaming, or additive manufacturing. Often these processes result in microstructures with significant fluctuations in local materials properties and/or geometry, hence microstructural randomness is an important factor influencing the mechanical performance of most solid foams. Furthermore, the characteristic length scales of the cellular microstructures (cell size or bond length) do in general not scale in proportion with the system size, giving rise to mechanical size effects. First experiments for studying elastic size effects were conducted by [@lakes1983; @lakes1986] who investigated quasi-static bending and torsion of beam-like specimens of polymeric foams. The investigation showed that, for comparable density, bigger beams exhibit lower torsion and bending stiffness compared to smaller beams. Other results by [@brezny1990] for reticulated vitreous carbon, [@choi1990] for cortical bone and [@anderson1994] for closed-cell polymethacrylimide foams indicated an opposite behavior where smaller systems were found to be softer than larger ones. Similar findings – smaller systems are elastically weaker – were reported by [@andrews2001] in uniaxial compression of open and closed cell aluminium foams and by [@bastawros2000] and [@jeon2005] for closed cell aluminium foams. To the knowledge of the authors, stiffness size effects under shear loading were rarely investigated with the exception of [@rakow2004; @rakow2005] who reported softer behavior when cutting out and shear testing sub-regions of decreasing size from a larger specimen. However, they stated that this is not a general result, as they also observed small samples which were stiffer – an observation they attributed to an increasing scatter in materials properties with decreasing sample size.
It should be noted that besides elastic size effects, i.e. size dependency of elastic stiffness, plasticity size effects (size dependency of yield or tensile strength) are a major subject of research (e.g. [@andrews1999], [@onck2001], [@kesler2002], [@chen2002], [@jeon2005]). The mechanisms which lead to these size effects are fundamentally different – unlike yield or failure, elasticity is hardly ever governed by weakest links – and we we do not investigate them in this work. Another important topic of research is the dependency of mechanical properties on density ([@silva1995], [@gibson1999], [@andrews1999], [@zhu2000], [@roberts2001]). The results are also important with regard to size effects or effects of randomness as one must ensure that, when comparing data from different specimens, systematic differences in density do not influence the results.
Detailed experimental characterization and analysis of the structure-property relations of – possibly highly disordered – cellular solids is non-trivial for a number of reasons: First, the geometrical details of the microstructure are usually not a priori known and must be obtained e.g. from 3D computer tomography (see e.g. [@jang2008]), ideally even in-situ under load. Obtaining stresses, strains or energies/energy densities is even more difficult and can usually only be done through post-processing of CT data or by means of digital image correlation of specimen surface images (see e.g. [@bart-smith1998]). Another important point is that a statistically meaningful investigation of disordered systems requires a sufficient number of samples, representing different realizations of the disordered microstructure, to be prepared and tested. For highly disordered microstructures this may require investigation of hundreds of samples, which is very time consuming to do in experiments.
Modelling and simulation approaches, on the other hand, suffer from time restrictions to a much lesser extent and may therefore offer the possibility to analyze in detail the interplay between different microstructure characteristics and the resulting mechanical response. For this purpose, a number of different simulation approaches can be used. The most accurate, but also computationally most expensive approach is to fully resolve the microstructure in all geometrical detail and then to simulate the mechanical response in three dimensions. Often, it is feasible to idealize cellular microstructures as networks of shells or beams. The significant reduction in computational cost for generating and meshing such structures as well as for their simulation makes this approach particularly attractive. This idea to represent a material with a network of beams or as a lattice goes back to [@hrennikoff1941]. Because of its simplicity and numerical efficiently the method became popular for studying fracture and elasticity of continua and especially concrete [@thorpe1990; @roux1985; @schlangen1997; @herrmann1989]. In the context of cellular solids it has been applied for example by [@onck2001] for the analysis of size effects in honeycomb structures, by [@roberts2001] for the study of density dependent elastic properties in 3D closed-cellular solids and by [@zhu2000] for investigating the influence of density and microstructural irregularity on the elastic properties of 2D structures. For a review of lattice/beam models in micromechanics see [@ostoja2002lattice; @ostoja2007microstructural]. An efficient simulation setup is particularly important for statistical analysis of random structures and investigation of the associated size effects, where a large number of simulations with different realizations of a disordered microstructure may need to be performed.
An alternative to fully resolved models of the material microstructure consists in using approaches where the microstructure is averaged out and replaced by a continuous effective medium. Continuum models which contain an instrinsic length scale and therefore can represent size effects are for example so-called micromorphic continua developed by Eringen and coworkers ( [@eringen1964], [@eringen1966], [@eringen1999]) which were used for example by [@dillard2006], [@diebels2002] [@tekoglu2005] in the context of metallic foams. The major benefit of continuum models is the reduced computational cost which allows to simulate complex geometries and loading conditions. Drawbacks consist in reduced accuracy, but even more in the requirement to formulate appropriate constitutive laws, to identify the corresponding parameters which for fully micromorphic theories may be numerous, and to define appropriate boundary conditions for the higher-order stress and strain variables.
In this paper we study the size-dependent elastic behavior of cellular structures by fully resolved simulations. Influences of boundary conditions and microstructural randomness are investigated and statistically analyzed. For detailed investigation of local deformation patterns we introduce a new, energetically consistent continuization scheme which allows us to represent the deformation state of a discrete network-like microstructure of beams in terms of spatially continuous stress and strain fields. We start in by introducing the rules we use for generating the discrete microstructures, and defining the corresponding micro-scale material properties as well as the boundary conditions in our simulated deformation tests. In we present the derivation of the continuization scheme and its verification. Results for macroscopic system response as well as micro-scale deformation patterns are shown in . Finally we discuss the results in and link some aspects of the system-scale deformation behavior to statistical features of the underlying micro-scale deformation patterns before we conclude this work in .
Model description {#sec:model}
=================
We consider two-dimensional systems and envisage the cellular microstructures as network-like arrangements of struts. The aspect ratio between strut length $l$ and thickness $w$ is assumed sufficiently large (i.e. $l/w \geq \approx 5$) such that we may approximate the microstructure as a network of interconnected, quasi-one-dimensional beams. Creating such networks is a well-defined and computationally inexpensive process. In FEM simulations the beams can be described by one-dimensional elements of which only the positions and connectivity of the end points need to be specified. This results in a very significant reduction in degrees of freedom as opposed to solid two- or three-dimensional elements. We note that in real materials struts often show a thickening towards junctions. To take this into account and improve the model it would be possible to use beam elements with varying cross-section as presented e.g. by [@auricchio2015]. Within the present work, however, we consider beam elements of constant cross-section which can be envisaged as Timoshenko beams. This implies that we make the following simplifying assumptions (see e.g. [@zienkiewicz2005b]): (i) each beam has constant cross section; (ii) the dimensions of the cross-section are small compared to the beam length; (iii) the cross-sections remain planar and their shape and size does not change under loading, but they do not necessarily stay perpendicular to the beam axis. Displacements along the beam axis are approximated by shape functions with a linear Ansatz. To prevent shear locking a cubic Ansatz is chosen for beam deflection and cross-section rotation. An overview of the governing equations of the Timoshenko beam can be found in \[appendix:eq\_timoshenko\_beam\].
Microstructure generation
-------------------------
The microstructure of real open-cellular foams is often not regular and exhibits random variations which result from the manufacturing process. One of the most common methods to computationally generate random cellular geometries is the Voronoi tessellation ([@zhu2000; @gibson1999]). As discussed by @boots1982 [@vanderburg1997] it reasonably represents a foaming process under the assumptions that
1. all nuclei appear simultaneously,
2. the nuclei remain fixed in position throughout the growth process,
3. each nucleus grows isotropically, i.e., at the same rate in all directions,
4. the growth stops for each cell whenever it comes into contact with a neighboring cell.
Although the tessellation as well as the whole procedure for the microstructure generation is easily generalized to any dimension we focus here on the 2D case. In a first step spatially distributed seed points are generated in the investigated domain. Then to each seed ${\cal S}$ a cell is assigned which we define as the set of all points whose Euclidean distance to ${\cal S}$ is smaller than that to any other seed. These so-called Voronoi cells are polygons (in 3D: convex polyhedra) which tessellate the plane. The 2D Voronoi tessellation can then be used to generate a cellular microstructure by placing beams on the cell boundaries. The distribution of the Voronoi seeds is a determining factor for the resulting microstructure. As two limit cases we have regular seeding (seeds are placed on a periodic lattice), and a completely random distribution of seed points according to a Poisson point process. A typical 2D example for structures obtained from regular seeding are hexagonal grids (honeycombs) which result from seeds being placed on a triangular lattice.
The emphasis of this work is on systems with tunable degree of randomness. [@zhu2000] proposed a method for creating 2D tesselations, which was used for example by [@onck2001] and [@tekoglu2005]. They start with an initial random seed, followed by generating a sequence of new random seeds, each of which is only accepted if it has a minimal distance $\delta_{\mrm{min}}$ to the initial seeds. The minimal distance acts as an order parameter which controls the randomness of the system. For $\delta_{\mrm{min}}=0$ the seeds are distributed independently, corresponding to a Poisson random process. The other limit is a densest packing of identical cells. For 2D this corresponds to a tessellation with honeycombs of equal edge length ([@hales2001]), in which case $\delta_{\mrm{min}}$ corresponds to the distance $\Delta p$ between the sites of a triangular lattice. However, the numerical generation of highly ordered systems ($\delta_{\mrm{min}} >0.8 \Delta p$) from a random seeding procedure is exceedingly unlikely for simple reasons of (information) entropy. Therefore we use here a different approach (see e.g. [@vanderburg1997; @garboczi1987]) for generating Voronoi tessellations with tunable degree of randomness.
In this approach seeds are first located on a triangular lattice with lattice constant $\Delta p$. The position $\tens{p}$ of each seed point is then perturbed by a stochastic variable $\delta \tens{p}$. Therefore the full range between fully regular and completely random systems can be generated with the same computational effort by simply varying the statistical rules for choosing $\delta \tens{p}$. For the direction of the perturbation an isotropic orientation distribution is used, whereas the distribution of perturbation distances $|\delta \tens{p}|$ is assumed exponential, $$\begin{aligned}
f\lb \frac{|\delta \tens{p}|}{\Delta p},\beta \rb = \frac{1}{\beta} \exp \lb -\frac{1}{\beta}\frac{|\delta \tens{p}|}{\Delta p} \rb,\end{aligned}$$ where the disorder parameter $\beta>0$ defines both the mean value and the standard deviation of the distribution. For $\beta \gg 1$ the final position of the seeds is exclusively governed by the stochastic process, which implies that the seeds are again distributed according to a Poisson point process. The influence of the parameter $\beta$ on the microstructure can be seen in .
In our deformation simulations we consider specimens of rectangular shape. The triangular lattice used in the first step of our seeding procedure is oriented in such a manner that one side of its unit cell aligns with the $y$ axis, which in graphical representations runs across a vertical specimen boundary. The resulting Voronoi tessellation is the dual of the seeding and an arrangement of regular honeycombs, which cell sides are aligned with the $x$ axis (cf. ). To compare systems with different $\beta$ it must be ensured that the areal density of seeds is maintained. This might be relevant in situations where the perturbation results in seeds being displaced outside the investigated domain, leading to a reduction in seed density near the specimen boundaries. Therefore we use periodic boundary conditions to map such seeds back into the specimen domain. After seeding we determine the associated Voronoi cells and place beams on the cell boundaries. If a cell boundary intersects the sample boundary, the beam is terminated at the intersection point.
Microstructural material properties {#sec:microproperties}
-----------------------------------
The mechanical response of the systems depends on the morphology of the microstructure but also on the properties of the beams. In the following we assume that all $N_\mrm{B}$ beams of varying lengths $l_i$ have the same in-plane width $w$ and out of plane thickness $t$. Thus the total volume covered by the beams is $$\begin{aligned}
V_{\mrm B} = t w \sum_{i=1}^{N_\mrm{B}} l_i.\end{aligned}$$ Macroscopically the system has the same thickness $t$ as the beams, its width is denoted by $W$ and its height by $H$. Thus the relative density is given by $$\begin{aligned}
\label{eq:density_scaling}
\rho_{\mrm{rel}} = \frac{V_{\mrm B}}{ t W H} = \frac{w}{W H} \sum_{i=1}^{N_\mrm{B}} l_i.\end{aligned}$$
Since we do not study density dependent behaviour, the relative density must be the same for all systems. We ensure a relative density of $\rho_{\mrm{rel}} = 0.1$ by propotional scaling of $W$ and $H$. The beams are assumed to have linear elastic material properties with Young’s modulus $E_{\mrm{B}} = 0.1 \text{ GPa}$ and Poisson’s ratio $\nu_{\mrm{B}}= 0.3$. The cross-section is assumed quadratic with $t = w = 0.05 \Delta p$ and a beam shear coefficient as proposed by [@cowper1966] of $\kappa = (10+ 10\nu_{\mrm{B}} ) /(12+11 \nu_{\mrm{B}}) \approx 0.85$ (cf. Appendix A).
Macroscopic boundary conditions {#sec:BC}
-------------------------------
Our investigation considers two distinct loading conditions – uniaxial compression and simple shear – which are imposed in a displacement driven manner. In accordance to experiments ([@andrews2001], [@tekoglu2011]) frictionless grip for compressive loading and stick grip for simple shear loading is chosen. Under compressive loading the beams are not clamped at the boundaries, allowing for free rotation (characterized by the rotation vector $\tens{\phi} = \phi \tens{e}_z$), cf. . The simple shear loading is defined by no beam rotations at the top and bottom boundaries and free rotations at the sides. Prescribed nodal displacements $\bar{u}$ are applied to all nodes on the top surface of the discrete beam system. $\bar{u}$ is chosen such that an isotropic continuum system would have an engineering strain of $| \varepsilon^* |=0.05$. This imposed strain is on the upper limit or outside of the elastic regime for real materials, especially if we consider that local strains might be even higher. However, the chosen elastic model scales linearly and thus is representative for any proportional elastic loading, i.e., values for lower strains can be obtained by simple multiplication with the strain ratio. An overview of the applied boundary conditions is given in Table \[tb:Dirichlet BC\].
Loading Bottom ($y=0$) Top ($y=H$) Left ($x=0$) Right ($x=w$)
-------------- ------------------------ ------------------------------- -------------- --------------- --
Compression $\bar{u}_y=0$ $\bar{u}_y= H / 0.05$ free free
Simple Shear $ \bar{u}_x=0, \phi=0$ $ \bar{u}_x= H/ 0.05 ,\phi=0$ free free
: Boundary conditions for the two investigated loading conditions.[]{data-label="tb:Dirichlet BC"}
Studying size effects requires comparison of systems which are geometrically similar, i.e have the same width to height ratio. Furthermore it is necessary that the relative density of the system remains the same (cf. ). A scaling which allows for comparison with closed honeycombs () is given by the height and width values $$\begin{aligned}
\label{eq:scaling1}
H(i) &= \Delta p (4 + 3 i)\qquad \text{and}\qquad
W(j)= \frac{ 2}{\sqrt{3}} \Delta p(4 + 3 j),\end{aligned}$$ with $i,j \in \mathbb{N}$. Throughout the following investigation we shall assume, unless otherwise stated, that $i=j$, corresponding to a ratio $W/H = 2/\sqrt{3} =: R_0$.
Depending on the way boundaries are located, regular honeycombs may differ very substantially from systems with a small degree of disorder. In a regular honeycomb, the position of beam endpoints may exactly coincide with the system boundaries such that all boundary cells are closed (). In slightly perturbed structures, on the other hand, depending on the positions of the seeds at the boundary the corresponding cells may either be closed or (partially) open (). This could lead to the unphysical result that even an infinitesimal perturbation of a regular honeycomb might produce a significant reduction in stiffness. To mitigate this problem we choose the position of the boundaries with respect to the seeds at random. Our regular reference microstructure has a periodicity of $\Delta p$ in $y$-direction and of $(1+\frac{ 2}{\sqrt{3}} )\Delta p$ in $x$-direction. To define our system boundaries we chose random cut-outs where the system boundaries are shifted with respect to the underlying seed lattice by random amounts $$\begin{aligned}
\Delta y = 0.5 \Delta p \,R_{\mrm h}(-1,1) \qquad \text{and} \qquad \Delta x = (0.5+\frac{ 1}{\sqrt{3}} )\Delta p \, R_{\mrm w}(-1,1),\end{aligned}$$ where $R_{\mrm w}(-1,1), R_{\mrm h}(-1,1)$ are random numbers which are uniformly distributed between $-1$ and $1$. It is important to note that different cut-out positions may lead to slightly different total beam lengths. This in turn causes us to re-scale the system size such that a relative density of $\rho_\mrm{rel}=10\%$ is always preserved.
Macroscopic system response {#sec:sys_response}
---------------------------
For comparing the macroscopic mechanical response of different systems we investigate global stiffness parameters of the system. To distinguish between macroscopic properties (e.g. the average engineering strain of the system) and microstructure properties (e.g. axial beam strains) all macroscopic properties are in the following denoted by the superscript ${}^*$. The averaged, macroscopic displacements at the top and bottom specimen boundaries are given by $$\begin{aligned}
\tens{u}^{* \mrm T} = \tens{u}^*(y=H) = \frac{1}{N_{\mrm T}} \sum_{\alpha=1}^{N_{\mrm T}} \tens{u}^\alpha (y=H),\\
\tens{u}^{* \mrm B} = \tens{u}^*(y=0) = \frac{1}{N_{\mrm B}} \sum_{\alpha=1}^{N_{\mrm B}} \tens{u}^\alpha (y=0),\end{aligned}$$ where $N_{\mrm T}$ and $N_{\mrm B}$ are the numbers of surface nodes of the beam network at the top and bottom boundaries, respectively, and $\tens{u}^\alpha(y)$ denotes the displacement vector of surface node number $\alpha$ at position $y$ (cf. ). The macroscopic axial and shear strains are defined as $$\begin{aligned}
\varepsilon_{yy}^* = \frac{|u_y^{* \mrm T}- u_y^{* \mrm B}|}{H} \qquad \text{and}\qquad
\varepsilon_{xy}^* = \frac{|u_x^{* \mrm T}- u_x^{* \mrm B}|}{H}.\end{aligned}$$ The average stiffnesses $C^*_\mrm{c,s}$ for uniaxial compression and simple shear are evaluated by calculating the sums of all load-induced reaction forces $R^\alpha_x$ and $R^\alpha_y$ in $x$ and $y$ direction, and dividing them by the loaded (macroscopic) area $A_{\mrm S} = W t$ and the respective averaged strain components $\varepsilon^*_{yy}, \varepsilon^*_{xy}$: $$\begin{aligned}
\label{eq:macro_stiffness}
C^*_{\mrm c} &= \frac{1}{A_{\mrm S} \varepsilon^*_{yy}} \sum_{\alpha=1}^{N_\mrm{R}} R^\alpha_{y} \qquad \text{for uniaxial compression} \\
C^*_{\mrm s} &= \frac{1}{A_{\mrm S} \varepsilon^*_{xy}} \sum_{\alpha=1}^{N_\mrm{R}} R^\alpha_{x} \qquad \text{for simple shear}.\end{aligned}$$ Note that the stiffnesses $C^*_\mrm{c,s}$ are not intrinsic material properties as they depend on the macroscopic geometry of the sample (cf. ).
Energetically consistent discrete-to-continuum transition {#sec:homogenization}
=========================================================
The mechanical response of our model systems arises from deformation of the beam network representing the microstructure. In this network, forces and displacements are evaluated only at the nodes of the beam elements. In different realizations of the same random microstructure these nodes are located at different points in space. If we want to average over different realizations of the same random microstructure, to make comparisons between different microstructures, or to analyze sample-scale deformation patterns, we need to perform a transition from displacements and forces located at discrete nodal points to a representation of the deformation state in terms of spatially continuous stress and strain fields. The same is true if we want to compare results from our discrete microstructure model to those derived from (generalized) continuum frameworks.
A continuum representation of the local mechanical state of the system should be based on the available data (nodal forces and displacements) and should “guess” as little information as possible by interpolation. To this end the system domain is decomposed into smaller sub-domains in which discrete and continuous formulations are matched locally. We require that these so-called control volumes are (i) in force and moment equilibrium, (ii) can be defined for regular and random microstructures, and (iii) allow for a formulation of the deformation state in terms of stress and strain that is energetically consistent with the discrete system. Schemes which use fixed rectangular subdomains and then compute stresses from the interface forces are microstructure independent but will in general produce artificial oscillations of the stress fields on the scale of the control volumes. A simple example can be seen in where a section from a regular honeycomb system is shown with a homogenization subdomain. Assume that the system is homogeneously compressed in the horizontal direction as shown in the figure. It can be seen that force equilibrium of the given control volume of size $t \times L_x \times L_y$ requires the stress gradient $\Delta \sigma_{xx}/\Delta x = - F_B/(t L_x L_y)$ in $x$ direction to be balanced by an equal but opposing shear stress gradient $\Delta \sigma_{xy}/\Delta y = F_B/(t L_x L_y)$ in the perpendicular direction. This stress gradient oscillates: it will occur with the same magnitude but opposite sign in the adjacent control volumes. Since the stress state is actually homogeneous – all cells of the honeycomb structure are loaded in an identical manner – such oscillating stresses must be envisaged as artifacts of the microstructure-independent control volume.
![Section from the middle of a regular honeycomb system. In blue a microstructure independent control volume which results in artificial stress oscillations. In red: possible triangulations as proposed by [@tekoglu2011].[]{data-label="fig:hom_schemes"}](./Figures/hom_schemes.png){width="30.00000%"}
A cell-wise procedure, e.g. by dividing each Voronoi cell individually into triangles as proposed by [@tekoglu2008; @tekoglu2011] and shown in works well for strain mappings, but is not suited for stress continuization since it is not clear how to evaluate without ambiguity force couples acting on the control volumes from the forces acting on the junction points.
To overcome these limitations we propose a novel approach which allows to compute stresses and strains in an energetically consistent manner while using control volumes that reflect the local microstructure morphology and thus minimize artefacts. Furthermore, this method is suitable for regular as well as for random microstructures. The idea is inspired by a very similar approach commonly used with discrete element methods (see for example [@christoffersen1981], [@mehrabadi1982], [@bagi1996]). The main idea is that energy consistency during transition from a discrete network to an equivalent continuum representation is guaranteed by using the principle of virtual work: For any given control volume $V_\mrm{c}$, the sum of the internal virtual work $\delta W_\mrm{b}^\mrm{int}$ of all $N_\mrm{b}$ beams therein must be the same as the internal virtual work $\delta W_\mrm{c}^\mrm{int}$ done by the continuum, $$\begin{aligned}
\delta W_\mrm{c}^\mrm{int} =\sum_{\alpha=1}^{N_\mrm{b}} \delta W_\mrm{b}^{\mrm{int}, \alpha} \quad \text{in } V_\mrm{c}.\end{aligned}$$ As the systems are in static equilibrium the external virtual work is equal to the internal virtual work so that $$\begin{aligned}
\label{eq:beam_continuum_balance}
\delta W_\mrm{c}^\mrm{int} =\sum_{\alpha=1}^{N_\mrm{b}} \delta W_\mrm{b}^{\mrm{ext}, \alpha} \quad \text{in } V_\mrm{c}.\end{aligned}$$
Energy of the Timoshenko Beam {#sec:beamenergy}
-----------------------------
For derivation of the continuum stresses we first consider a single discretized Timoshenko beam in 2D as used in our finite element simulation, cf. and \[appendix:eq\_timoshenko\_beam\].
![Kinematics (left) and forces (right) in 2D for the Timoshenko beam. For better visualization the rotations are largely exaggerated.[]{data-label="fig:beam_kinematics"}](./Figures/beam_bending){width="95.00000%"}
The beam is defined by the end points $P$ and $Q$ with the corresponding beam forces $\tens{F}$ and torques $\tens{M}$. The external virtual work $\delta W_\mrm{b}^\mrm{ext}$ done by the virtual displacements $\delta \tens{w}$ and the virtual rotations $\delta \tens{\phi}$ is given by $$\begin{aligned}
\label{eq:timoshenko_beam}
\delta W_\mrm{b}^{\mrm{ext}} = \tens{F}^\mrm{P} \cdot \delta \tens{w}^\mrm{P} + \tens{F}^\mrm{Q}\cdot \delta \tens{w}^\mrm{Q} + \tens{M}^\mrm{P}\cdot \delta \tens{\phi}^\mrm{P}+ \tens{M}^\mrm{Q}\cdot \delta \tens{\phi}^\mrm{Q},\end{aligned}$$ where $\cdot$ denotes a single contraction between two vectors or tensors. The so-called beam vector connects $P$ and $Q$: $$\begin{aligned}
\label{eq:beamvec}
\tens{l} = \tens{x}^\mrm{P}-\tens{x}^\mrm{Q}.\end{aligned}$$ Furthermore we define the virtual separation $\delta \tens{\Delta}^w$ as the difference of the virtual displacements $$\begin{aligned}
\label{eq:rel_disp}
\delta \tens{\Delta}^w = \delta \tens{w}^\mrm{P} - \delta \tens{w}^\mrm{Q}.\end{aligned}$$ Because the beam is in static equilibrium it follows with the cross product $\times$ for forces and moments: $$\begin{aligned}
\label{eq:beamforces}
\tens{F} & \defeq \tens{F}^\mrm{P} = -\tens{F}^\mrm{Q},\\
\tens{M} & \defeq \tens{M}^\mrm{P} = - \tens{M}^\mrm{Q} - \tens{l} \times \tens{F}^\mrm{Q} = - \tens{M}^\mrm{Q}+\tens{l} \times \tens{F}.\end{aligned}$$ Therefore we obtain the expression for the virtual work from as $$\begin{aligned}
\begin{split}
\delta W_\mrm{b}^\mrm{ext} &= \tens{F}\cdot \delta \tens{\Delta}^w + \tens{M}\cdot \delta \tens{\phi}^\mrm{P} + \lb \tens{l} \times \tens{F} - \tens{M} \rb\cdot \delta \tens{\phi}^\mrm{Q}.
\end{split}\end{aligned}$$
Derivation of energy equivalent stress fields
---------------------------------------------
The energy equivalence must hold for all local control volumes. To allow for a meaningful computation of forces in conjunction with ensuring equilibrium of forces and moments the boundaries of these control volumes must be chosen such that they intersect with at least three beams that are connected to the same node of the beam network. For the construction of polygonal control volumes we propose the following method:
1. Pick an arbitrary junction point which has three (or more) connecting beams as base.
2. Choose the midpoints between the base point and the connected junctions as corners of the polygons. This assures that all beams are covered by the polygons.
3. Choose the center points of the Voronoi tessellation as additional corner points to ensure a tessellation of the whole domain.
This procedure is carried out for all junction points leading to full coverage of the system by a tessellation of irregular polygons (compare ).
![Two dimensional microstructure with center points of the Voronoi tessellation a control volumes (shown in grey).[]{data-label="fig:tessellation"}](./Figures/single_control_volume){width="50.00000%"}
When deriving energy consistent stress variables for these control volumes, an additional difficulty arises from the fact that the beams - and thus the control volumes - may not only deform but also rotate, which may change the elastic energy. Such behaviour can not be captured in classical continuum formulations because material points do not have rotational degrees of freedom. The simplest extension of classical continuum elasticity which allows for an energy consistent formulation by accounting for rotation effects are the so-called Cosserat or micropolar continua (see for example [@eringen1999] which we choose as our framework. Introducing the Levi-Civita tensor $$\begin{aligned}
\tens{\epsilon} = \epsilon_{ijk} \tens{e}_i \otimes \tens{e}_j \otimes \tens{e}_k =
\begin{cases}
+1 & \text{if } (i,j,k) \text{ is } (1,2,3), (2,3,1) \text{ or } (3,1,2), \\
-1 & \text{if } (i,j,k) \text{ is } (3,2,1), (1,3,2) \text{ or } (2,1,3), \\
\;\;\,0 & \text{if }i=j \text{ or } j=k \text{ or } k=i
\end{cases} \end{aligned}$$ and denoting the double contraction of two tensors by $:$, the static balance equations for a Cosserat continuum without body forces reads $$\begin{aligned}
\label{eq:linear_momentum}
\div{\tens{\sigma}+\tens{S}} = \tens{0},\\
\label{eq:angular_momentum}
\div{\tens{\mu}} - \tens{S}: \tens{\epsilon}= \tens{0}.\end{aligned}$$ The additional couple stress $\tens{\mu}$ balances the angular momentum introduced by the skew-symmetric stress tensor $\tens{S} = -\tens{S} \T$, whereas the stress tensor $\tens{\sigma}$ is symmetric. As a consequence the total stress $\tens{\sigma}+\tens{S}$, which corresponds to the Cauchy stress is, unlike in classical continuum theories, no longer symmetric. Neumann boundary conditions now depend on the traction vector $\tens{t}$ and the vectorial higher order couple traction $ \tens{m}$ acting on the surface: $$\begin{aligned}
\label{eq:traction_bc}
(\tens{\sigma}+\tens{S})\T \cdot \tens{n} &= \tens{t}, \\
\tens{\mu}\T \cdot \tens{n} &= \tens{m},\end{aligned}$$ where $\tens{n}$ is the normal of the surface $S_{\mrm c}$ of the control volume $V_{\mrm c}$. With the (continuum) displacements $\tens{u}$ and micro rotations $\tens{\omega}$ the corresponding work conjugated strain measures are $$\begin{aligned}
\tens{\varepsilon} & = \mathrm{sym} \lb\nabla \tens{u}\rb = \frac{1}{2} ( \nabla \tens{u} + \nabla \tens{u} \T), \\
\tens{\varepsilon}^{\mrm R} &= \mathrm{skw} \lb\nabla \tens{u}\rb + \tens{\epsilon}\cdot\tens{\omega}= \frac{1}{2} ( \nabla \tens{u} - \nabla \tens{u} \T)+ \tens{\epsilon}\cdot\tens{\omega}.\end{aligned}$$ The virtual work of the continuum system follows as $$\begin{aligned}
\delta W_{\mrm c} &= \int_{S_{\mrm c}} \tens{t}\cdot \delta \tens{u} \dif A + \int_{S_{\mrm c}} \tens{m}\cdot \delta \tens{\omega} \dif A - \int_{V_{\mrm c}} \lb \tens{\sigma} : \delta \tens{\varepsilon} + \tens{S} : \delta\tens{\varepsilon}^{\mrm R} + \tens{\mu}: \nabla \delta \tens{\omega} \rb \dif V=0,\end{aligned}$$ which defines the internal virtual work as $$\begin{aligned}
\delta W_{\mrm c}^\mrm{int} &= \int_{V_{\mrm c}} \lb\lb \tens{\sigma} + \tens{S} \rb : \nabla \delta \tens{u} - \tens{S}: \lb\tens{\epsilon}\cdot\delta \tens{\omega} \rb + \tens{\mu}: \nabla \delta \tens{\omega} \rb \dif V.
\label{eq:virt_work}\end{aligned}$$ The position and shape of the control volumes is defined by the positions of the beams. A change of the beam end positions $Q^\alpha$ results in a strain and a rotation around the junction point $\tens{x}^\mrm{P}$. For regular systems the centroid of the control volume $\tens{x}^\mrm{c} = \frac{1}{V_{\mrm c}}\int_{V_{\mrm c}} \tens{x} \dif V$ coincides with this junction point. Irregularity, which is governed by the perturbation $\delta \tens{p}$, leads to a mismatch between the two points. As the continuous system rotates around $\tens{x}^\mrm{c}$ in irregular control volumes the center of rotation is not the same for the continuous and the discrete system. To circumvent this problem we notice, that the two points are often very close to each other, so that we approximate the centroid by $$\begin{aligned}
\tens{x}^\mrm{c} \approx \tens{x}^\mrm{P},
\label{eq:centroid_pos}\end{aligned}$$ which can be envisaged as a zeroth-order expansion of the centroid position with respect to $\delta \tens{p}$. The truncated expansion is exact for regular systems and ensures that (i) the continuous and the discrete system have the same center of rotation and (ii) a pure rigid-body motion of the system, i.e. $$\begin{aligned}
\mathrm{sym}\lb {\nabla \tens{u}}\rb = \tens{0},\qquad \mathrm{skw}\lb {\nabla \tens{u}}\rb = \tens{\epsilon}\cdot\tens{\omega}\end{aligned}$$ does not exhibit any energy.
We assume that stresses as well as couple stresses are constant in each polygonal control volume. This implies that the corresponding virtual displacement and rotation fields can be written as linear mappings $$\begin{aligned}
\label{eq:linear_ansatz}
\delta \tens{u} &= \tens{\Psi} \cdot \tens{x} + \tens{q},\\\
\label{eq:rotation_ansatz}
\delta \tens{\omega} & = \tens{\Phi}\cdot \tens{x} + \tens{r},\end{aligned}$$ with the constant mapping tensors $\tens{\Psi}$ and $\tens{\Phi}$ and the constant vectors $\tens{q}$ and $\tens{r}$. Note that the constant stress/strain assumption is made for the stress/strain field of each control volume separately. The global stress distribution is therefore piecewise constant. The virtual displacement fields for the junction nodes are then given in terms of the respective position vectors $\tens{x}^\mrm{P}$ and $\tens{x}^\mrm{Q}$ by $$\begin{aligned}
\label{eq:P_disp}
\delta \tens{w}^\mrm{P} = \tens{\Psi} \cdot \tens{x}^\mrm{P} + \tens{q},
\quad&\delta \tens{w}^\mrm{Q} = \tens{\Psi} \cdot \tens{x}^\mrm{Q} + \tens{q},\\
\label{eq:P_rot}
\delta \tens{\phi}^\mrm{P} = \tens{\Phi}\cdot \tens{x}^\mrm{P} + \tens{r},
\quad&\delta \tens{\phi}^\mrm{Q} = \tens{\Phi}\cdot \tens{x}^\mrm{Q} + \tens{r}.\end{aligned}$$ Because the virtual micro rotations and the virtual displacements can be considered separately we start by deriving the expressions for the virtual displacements, $$\begin{aligned}
\label{eq:virtual_force_equilibrium}
\int_{V_{\mrm c}} \lb \tens{\sigma} + \tens{S} \rb : \nabla \delta \tens{u} \dif V = \sum_{\alpha=1}^{N_\mrm{b}} \tens{F}^\alpha\cdot \delta \tens{\Delta}^\alpha.\end{aligned}$$ With , and we can write as $$\begin{aligned}
\int_{V_{\mrm c}} \lb \tens{\sigma} +\tens{S} \rb: \tens{\Psi} \dif V = \sum_{\alpha=1}^{N_\mrm{b}} \tens{F}^\alpha\cdot \lb \tens{\Psi}\cdot\tens{l}^\alpha \rb.\end{aligned}$$ Since $\tens{\Psi}$ is arbitrary and constant we obtain $$\begin{aligned}
\int_{V_{\mrm c}}\lb \tens{\sigma} +\tens{S} \rb \dif V = \sum_{\alpha=1}^{N_\mrm{b}} \tens{F}^\alpha \otimes \tens{l}^\alpha
\label{eq:stress_force_relation}\end{aligned}$$ for the stress in the volume. From this the average stress in the control volume evaluates as $$\begin{aligned}
\langle \tens{\sigma} +\tens{S} \rangle_\mrm{c} = \frac{1}{V_{\mrm c}} \int_{V_{\mrm c}} \lb \tens{\sigma} +\tens{S} \rb \dif V, \end{aligned}$$ and we can write as $$\begin{aligned}
\langle \tens{\sigma} +\tens{S} \rangle_\mrm{c} = \frac{1}{V_{\mrm c}} \sum_{\alpha=1}^{N_\mrm{b}}\tens{F}^\alpha \otimes \tens{l}^\alpha.\end{aligned}$$ Because $\tens{\sigma}$ is symmetric and $\tens{S}$ skew-symmetric their averages can be considered separately, such that $$\begin{aligned}
\langle \tens{\sigma} \rangle_\mrm{c} &= \mathrm{sym} \lb\frac{1}{V_{\mrm c}} \sum_{\alpha=1}^{N_\mrm{b}}\tens{F}^\alpha \otimes \tens{l}^\alpha\rb, &
\langle \tens{S} \rangle_\mrm{c} &= \mathrm{skw} \lb\frac{1}{V_{\mrm c}} \sum_{\alpha=1}^{N_\mrm{b}}\tens{F}^\alpha \otimes \tens{l}^\alpha\rb.\end{aligned}$$ The average of a quantity $(\cdot)$ for the whole system can then be approximated by $$\begin{aligned}
\langle \cdot \rangle_\mrm{S} \approx \frac{1}{V_{\mrm S}} \sum_{k=1}^{N_{\mrm c}} V_{\mrm c}^k \langle \cdot \rangle_\mrm{c}^k,\end{aligned}$$ where $V_{\mrm S} = \sum_{k=1}^{N_{\mrm v}} V_c^k$ is the system volume and $ N_{\mrm c}$ the number of control volumes in the system. Thus, the system-averaged stress can be approximated by $$\begin{aligned}
\langle \tens{\sigma} +\tens{S} \rangle_\mrm{S} = \frac{1}{V_{\mrm S}} \sum_{k=1}^{N_{\mrm c}} V_{\mrm c}^k \langle\tens{\sigma} +\tens{S}\rangle_\mrm{c}^k.\end{aligned}$$ Secondly the virtual work equivalence for the virtual rotations is investigated $$\begin{aligned}
\label{eq:virt_rot}
\sum_{\alpha=1}^{N_\mathrm{b}} \tens{M}^\alpha \cdot (\delta\tens{\phi}^\mrm{\alpha,P}-\delta\tens{\phi}^\mrm{\alpha,Q} )+ \lb \tens{l}^\alpha \times \tens{F}^\alpha \rb\cdot \delta\tens{\phi}^\mrm{\alpha,Q} = \int_{V_{\mrm c}} \lb \tens{\mu}: \nabla \delta \tens{\omega} - \tens{S}: \lb\tens{\epsilon}\cdot\delta \tens{\omega} \rb \rb \dif V.\end{aligned}$$ With , , this can be written as $$\begin{aligned}
\sum_{\alpha=1}^{N_\mathrm{b}} \tens{M}^\alpha \cdot \lb \tens{\Phi}\cdot\tens{l}^\alpha \rb
+ \lb \tens{l}^\alpha \times \tens{F}^\alpha \rb\cdot \lb \tens{\Phi}\cdot\tens{x}^\mrm{\alpha,Q} + \tens{r} \rb = \int_{V_{\mrm c}} \lb \tens{\mu}: \tens{\Phi} - \tens{S}: \tens{\epsilon}\cdot\lb \tens{\Phi}\cdot\tens{x} + \tens{r} \rb \rb \dif V.
\label{eq:virt_rot2}\end{aligned}$$ By using Eq. , the stress-force relation and the fact that the stresses and $\tens{\Phi}$ are constant in the control volume, the second term of the right hand side of can be approximated by $$\begin{aligned}
\begin{split}
-\int_{V_{\mrm c}} \tens{S}: \tens{\epsilon}\cdot\lb \tens{\Phi}\cdot\tens{x} + \tens{r} \rb \dif V &= -\lb \langle \tens{S} \rangle_\mrm{c} : \tens{\epsilon} \rb \cdot\lb \tens{\Phi} \cdot \tens{x}^\mrm{P} + \tens{r} \rb \\
&= \sum_{\alpha=1}^{N_\mathrm{b}} \lb \tens{l}^\alpha \times \tens{F}^\mrm{\alpha} \rb \cdot \lb \tens{\Phi} \cdot \tens{x}^\mrm{P} + \tens{r} \rb.
\end{split}
\label{eq:rhs_virt_rot2}\end{aligned}$$ With that can be simplified to $$\begin{aligned}
\sum_{\alpha=1}^{N_\mathrm{b}} \tens{M}^\alpha \cdot \lb \tens{\Phi}\cdot\tens{l}^\alpha \rb
+ \lb \tens{l}^\alpha \times \tens{F}^\alpha \rb\cdot \lb \tens{\Phi}\cdot\tens{x}^\mrm{\alpha,Q} + \tens{r} \rb &= \int_{V_{\mrm c}}\tens{\mu}: \tens{\Phi} \dif V+\sum_{\alpha=1}^{N_\mathrm{b}} \lb \tens{l}^\alpha \times \tens{F}^\mrm{\alpha} \rb \cdot \lb \tens{\Phi} \cdot \tens{x}^\mrm{P} + \tens{r} \rb,\\\sum_{\alpha=1}^{N_\mathrm{b}} \tens{M}^\alpha \cdot \lb \tens{\Phi}\cdot\tens{l}^\alpha \rb
- \lb \tens{l}^\alpha \times \tens{F}^\alpha \rb\cdot \lb \tens{\Phi}\cdot\tens{l}^\mrm{\alpha} \rb &= \int_{V_{\mrm c}}\tens{\mu}: \tens{\Phi} \dif V .\end{aligned}$$ Again the linear mapping $\tens{\Phi}$ is an arbitrary constant tensor so that we obtain for the averaged couple stress $$\begin{aligned}
\langle \tens{\mu} \rangle_\mrm{c} = \frac{1}{V_{\mrm c}} \int_{V_{\mrm c}} \tens{\mu} \dif V
&=\frac{1}{V_{\mrm c}} \sum_{\alpha=1}^{N_\mathrm{b}} \tens{M}^\alpha \otimes \tens{l}^\alpha - \lb \tens{l}^\alpha \times \tens{F}^\mrm{\alpha} \rb \otimes \tens{l}^\alpha \\
&= -\frac{1}{V_{\mrm c}} \sum_{\alpha=1}^{N_\mathrm{b}} \tens{M}^\mathrm{\alpha,Q} \otimes \tens{l}^\alpha.\end{aligned}$$ The system average is computed in the same way as for the stresses $$\begin{aligned}
\langle \tens{\mu} \rangle_\mrm{S} = \frac{1}{V_{\mrm S}} \sum_{k=1}^{N_{\mrm c}} V_{\mrm c}^k \langle \tens{\mu} \rangle_\mrm{c}^k.\end{aligned}$$
Derivation of equivalent strain fields
--------------------------------------
In our derivation we impose the following requirements on the continuous strain and the associated displacement fields:
- discrete and continuous displacements at the positions of the beam nodes are the same
- strain fields are constant over control volumes and piecewise constant over the sample domain
- stress and strain computation is done within identical control volumes
Linear approximations for the spatial dependency of the displacement fields are a natural choice, since they lead to a piecewise constant strain field. For a control volume of arbitrary shape, the averaged strain tensor can be evaluated as $$\begin{aligned}
\langle \tens{e} \rangle_\mrm{c} = \langle \nabla \tens{u} \rangle_{\mrm{c}} = \frac{1}{V_\mrm{c}} \int_{V_\mrm{c}} \nabla \tens{u} \dif V = \frac{1}{V_\mrm{c}} \int_{S_\mrm{c}} \tens{u} \otimes \tens{n} \dif A \end{aligned}$$ where $\tens{n}$ is the outward pointing normal to the control volume surface $S_{\mrm{c}}$. If the control volume is a $N_\mrm{c}$-sided polygon and the dependency of $\tens{u}$ on the spatial coordinates is linear, we find $$\begin{aligned}
\langle \tens{e} \rangle_{\mrm{c}} = \frac{1}{V_\mrm{c}} \sum_{k=1}^{N_\mrm{c}} \left(\frac{\tens{u}^{k} + \tens{u}^{k+1}}{2} \otimes \tens{n}^{(k,k+1)}\right) \tens{b}^{(k,k+1)}.
\label{eq:strainavs}\end{aligned}$$ Here $\tens{u}_k$ is the displacement of corner node number $k$ in a clockwise enumeration, $\tens{b}^{(k,k+1)}$ is the length of the polygon side connecting nodes $k$ and $k+1$, and $\tens{n}^{(k,k+1)}$ the outward pointing normal vector to this side. Note, that similar results have been obtained by [@bagi1996] for an assembly of granules. Applying Eq. to the control volumes, the displacements of the corners of the control volume are only partially known: While the displacements of the corners that are located at the midpoint of a beam are explicitly known, they are unknown for the corners located at the cell midpoints. Owing to the assumption of constant strain/stress any sub-volume of the control volume must have the same (constant) strain field as the whole control volume. Therefore we approximate the control volume strain by computing the strain from the beam midpoints for which the displacements are known. For the calculation of the continuum rotation gradient $\langle \nabla \tens{\omega} \rangle_\mrm{c}$ we follow the same argument for beam rotations (instead of beam displacements). To consistently compute the averaged rotation $\langle \tens{\omega} \rangle_\mrm{c}$ a linear interpolation for the polygon centroid from the same known beam midpoints is used. With the computed averaged quantities the averaged strain measures can be expressed as $$\begin{aligned}
\langle \tens{\varepsilon} \rangle_\mrm{c} &= \frac{1}{2} \lb \langle \tens{e} \rangle_{\mrm{c}} + \langle \tens{e} \rangle_{\mrm{c}}\T \rb, \\
\langle \tens{\varepsilon}^\mrm{R} \rangle_\mrm{c} &= \frac{1}{2} \lb \langle \tens{e} \rangle_{\mrm{c}} - \langle \tens{e} \rangle_{\mrm{c}}\T \rb + \tens{\epsilon}\cdot \langle \tens{\omega} \rangle_\mrm{c}.\end{aligned}$$
Verification of the continuization method
-----------------------------------------
As a first verification of the continuization procedure we test for the energy consistency by comparing for each control volume the continuum strain energy $W_\mrm{c}$ with the energy stored in the beams $W_\mrm{b,c}$. In analogy to the elastic energy of a a discrete beam is given by $$\begin{aligned}
W_\mrm{b} = %\underbrace{
\frac{1}{2} \left(\tens{F}^\mrm{P} \cdot\tens{w}^\mrm{P} + \tens{F}^\mrm{Q} \cdot\tens{w}^\mrm{Q} %\right)
%}_{W_\mrm{b,d}} \underbrace{\frac{1}{2} \left(
+\tens{M}^\mrm{P}\cdot\tens{\phi}^\mrm{P}+\tens{M}^\mrm{Q}\cdot \tens{\phi}^\mrm{Q} \right)
%}_{W_\mrm{b,d}}\end{aligned}$$ with forces $ \tens{F}^\mrm{P}, \tens{F}^\mrm{Q}$, moments $\tens{M}^\mrm{P}, \tens{M}^\mrm{Q} $, displacements $\tens{w}^\mrm{P}, \tens{w}^\mrm{Q}$ and rotations $\tens{\phi}^\mrm{P}, \tens{\phi}^\mrm{Q}$ given at the beam end points P and Q. Note, that P corresponds to the junction point and Q to the midpoint between two junctions. The total beam energy is then given by the sum of energies of all beams in the control volume $V_\mrm{c}$: $$\begin{aligned}
W_\mrm{b,c} = \sum_{\alpha=1}^{N_\mrm{b}} W_\mrm{b}^\alpha.\end{aligned}$$ Because stresses and strains are by construction constant in the control volumes, the continuum strain energy can be computed as $$\begin{aligned}
\label{eq:strain_energy}
W_\mrm{c} = %\underbrace{\frac{1}{2}
\frac{1}{2} \lb \langle\tens{\sigma}\rangle_\mrm{c} : \langle \tens{\varepsilon} \rangle_\mrm{c} + \langle \tens{S}\rangle_\mrm{c} :\langle \tens{\varepsilon}^\mrm{R} \rangle_\mrm{c}
%}_{W_\mrm{c,d}}
%+ \underbrace{
+ \langle \tens{\mu} \rangle_\mrm{c} : \langle \nabla \tens{\omega} \rangle_\mrm{c}\rb.
%}_{W_\mrm{c,r}} .\end{aligned}$$ As a measure for the error the relative energy residual $r_\mrm{W} = {| W_\mrm{c}-W_\mrm{b,c} | }/{W_\mrm{b,c}}$ is used, which is shown in for three structures of increasing randomness. For the regular structure $r_\mrm{W}$ is almost equally distributed, with the trend of smaller errors towards less displaced beams. Increasing randomness leads to an increase in local errors as the volumes become more distorted and the centroid approximation, , is violated. Distortion of the control volumes alone, however, does not necessarily give rise to larger errors, because the violation of the centroid approximation becomes relevant only for rotation dominated energies. Nonetheless, even for the highly irregular system the maximum error is smaller than $1\%$ and the weighted average of the global error is for all systems less than $0.1\% $ (compare the highly distorted control volumes in , of which only some show a large error). We conclude that the chosen approximations are sufficiently accurate and able to represent rotational degrees of freedoms as opposed to a classical Cauchy continuum (see \[appendix:cauchy formulation\]).
As an other verification of the continuization method we analyze the local stresses and strains. As reference system we chose a regular system under uniaxial compression (cf. ). We observe that both stresses and strains show a similar inhomogeneous distribution. The normal stresses $\langle \sigma_{yy}\rangle_\mrm{c}$ are dominating, with a weighted average of $ \langle \sigma_{yy}\rangle_\mrm{S} = 6.6\cdot 10^3$ Pa which is close to the value of $\sigma_{yy}^* = 6.86\cdot 10^3$ Pa obtained by analysis of the reaction forces. Furthermore one can see that boundary effects are present near those boundaries where the system is constrained. Near these boundaries we observe checker-board patterns of shear stress and shear strain which are artifacts of the continuization method. The averaged strain $\langle e_{yy}\rangle_\mrm{S}$ is in good agreement with the imposed engineering strain $e^*=0.05$. The lateral compression is slightly smaller than the axial dilatation, which results in an averaged 2D Poisson ratio of $$\begin{aligned}
\left \langle \nu \right \rangle_\mrm{S} = - \left \langle \frac{e_{yy}}{e_{xx}} \right \rangle_\mrm{S} \approx 0.97
\end{aligned}$$ which is close to the theoretical value for regular honeycombs, cf. Eq. in .
To conclude, our continuization method is not only energy consistent with the discrete system, but also gives comparable results on the macroscopic scale for stresses and strains. Discretization/continuization artifacts such as artificial stress or strain oscillations are largely absent in the bulk of the sample but may be present in the immediate vicinity of constrained boundaries.
[![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_stresses_19-0_xx "fig:"){width=".235\textwidth"}]{} [![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_stresses_19-0_xy "fig:"){width=".235\textwidth"}]{} [![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_stresses_19-0_yx "fig:"){width=".235\textwidth"}]{} [![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_stresses_19-0_yy "fig:"){width=".235\textwidth"} ]{}
[![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_strains_19-0_xx "fig:"){width=".235\textwidth"}]{} [![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_strains_19-0_xy "fig:"){width=".235\textwidth"}]{} [![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_strains_19-0_yx "fig:"){width=".235\textwidth"}]{} [![Stress and strain tensor components for a regular honeycomb structure ($\beta=0,\, H=19 \Delta p$) with closed boundary cells under uniaxial compression. Note that only the left half of the system is shown as it is symmetric with respect to its central axis. \[fig:reg\_stress\_strain\]](./Figures/stress_plots/00_uniaxial_strains_19-0_yy "fig:"){width=".235\textwidth"} ]{}
Results: Size and disorder-dependent elasticity of foam microstructures {#sec:results}
=======================================================================
In the following we study the behavior of regular and random microstructures constructed as discussed in under two loading conditions, simple shear and uniaxial compression, described in and Table \[tb:Dirichlet BC\]. As mentioned in the simulations are performed with the aspect ratio $W/H=R_0=2/\sqrt{3}$ if not stated otherwise. The deformation of the systems is shown for a regular ($\beta=0$) and one realization of a random $(\beta=0.3)$ system in .
[0.245]{} [![Undeformed (in grey) and deformed structures for uniaxial compression along the $y$ direction and for simple shear loading; disorder parameters $\beta=0$ and $\beta=0.3$; the system height is for all systems $H=7 \Delta p$. The colorscale represents the local displacement in $y$ direction for compression and in $x$ direction for shear loading.[]{data-label="fig:disps"}](./Figures/microstructures/07_uniaxial_uy_deformed_00 "fig:"){width="100.00000%"}]{}
[0.245]{} [![Undeformed (in grey) and deformed structures for uniaxial compression along the $y$ direction and for simple shear loading; disorder parameters $\beta=0$ and $\beta=0.3$; the system height is for all systems $H=7 \Delta p$. The colorscale represents the local displacement in $y$ direction for compression and in $x$ direction for shear loading.[]{data-label="fig:disps"}](./Figures/microstructures/07_uniaxial_uy_deformed_03 "fig:"){width="100.00000%"}]{}
[0.245]{} [![Undeformed (in grey) and deformed structures for uniaxial compression along the $y$ direction and for simple shear loading; disorder parameters $\beta=0$ and $\beta=0.3$; the system height is for all systems $H=7 \Delta p$. The colorscale represents the local displacement in $y$ direction for compression and in $x$ direction for shear loading.[]{data-label="fig:disps"}](./Figures/microstructures/07_simple_shear_ux_deformed_00 "fig:"){width="100.00000%"}]{}
[0.245]{} [![Undeformed (in grey) and deformed structures for uniaxial compression along the $y$ direction and for simple shear loading; disorder parameters $\beta=0$ and $\beta=0.3$; the system height is for all systems $H=7 \Delta p$. The colorscale represents the local displacement in $y$ direction for compression and in $x$ direction for shear loading.[]{data-label="fig:disps"}](./Figures/microstructures/07_simple_shear_ux_deformed_03 "fig:"){width="100.00000%"}]{}
It can be seen that in the regular system the beam and junction displacements show an almost continuous, linear distribution. The irregular system on the other hand exhibits a more inhomogeneous distribution of the displacements, corresponding to inhomogeneities in the local strain fields.
Macroscopic system response {#sec:glob_results}
---------------------------
The macroscopic system response (cf. ) is influenced by aspect ratio, system size, irregularity of the microstructure, and configuration of the microstructure at the boundaries. In what follows we compare the relevance of these parameters for systems of different size. Regular honeycomb structures are chosen as reference systems since their deformation properties can be considered isotropic on the macroscopic scale ([@silva1995], [@gibson1999]). Thus, the stiffnesses can be interpreted as a geometry dependent Young’s Modulus $E^*$ for uniaxial compression and shear modulus $G^*$ for pure shear. As an analytical benchmark the solutions of [@gibson1999] are chosen. These solutions take beam bending, axial and shear deformation into account and give for an infinite system and the material properties $\rho_{\mrm{rel}}=0.1, E_{\mrm{B}}=1\cdot 10^8\mrm \, \mrm{Pa}, $ and $\nu_{\mrm{B}}=0.3$ as used in our simulations the theoretical system parameters $$\begin{aligned}
\label{eq:theo_vals}
E^*_{\mrm T} &\approx 1.44 \cdot 10^5 \, \mrm{Pa}, & \nu^*_{\mrm T} &\approx 0.971,& G^*_{\mrm T} &\approx 3.65 \cdot 10^4 \, \mrm{Pa}. \end{aligned}$$ In the following, the values of the macroscopic response parameters $C^*_{\mrm c,s}$ are always normalized by the theoretical Young’s modulus $ E^*_{\mrm T} $ for compression and the theoretical shear modulus $G^*_{\mrm T} $ for shear.
#### Influence of microstructure configuration at the boundary $(\beta=0)$
As described in varying random cut-outs have a large influence on the size dependency of the mechanical response. This is primarily important for regular or slightly perturbed systems whereas this effect cancels out for systems with a large degree of randomness since the boundaries average over many different cell configurations. reveals that the stiffness values of different cut-outs from a regular honeycomb structure exhibit significant scatter, which decreases from about 30 % for the smallest to 3 % for the largest investigated systems, and which is expected to reach zero in the infinite-system limit. Note, that for each investigated system size the realizations slightly vary in size as a result of the scaling procedure (see ). Thus, the average stiffness is obtained by first sorting all cut-outs into bins w.r.t. their original, unscaled size. Then all values corresponding to each size/bin are averaged. Smaller systems are on average weaker than larger ones for both loading conditions. With increasing system size the stiffness approaches asymptotically the bulk value. For uniaxial compression this limit is close to the theoretical Young’s modulus $E^*_{\mrm T}$, whereas for simple shear loading the asymptotic parameter $C^*_{\mrm s}$ remains significantly below the theoretical shear modulus $G^*_{\mrm T}$. This is expected since simple shear loading does not lead to a uniaxial macroscopic stress and strain state. In fact, the asymptotic value of the scaled parameter $C^*_{\mrm s}$ approached in our simulation matches well the result obtained for an isotropic material with the same aspect ratio, calculated with the theoretical bulk values of the elastic moduli. For compressive loading one can see that the size effects are much smaller for the stiffest structures compared to the weaker ones. However, this effect is less pronounced for shear loading.
[![Normalized system responses evaluated for multiple cut-outs from a regular honeycomb structure ($\beta=0$). Each dot represents one simulation, and the average of all cut-outs is represented as the black line.[]{data-label="fig:subrealization"}](./Figures/system_plots/uniaxial_subrealizations_00 "fig:")]{}
#### Influence of system aspect ratio on elastic response $(\beta=0)$
Our second investigation concerns the effect of aspect ratio $R=W/H$ on the mechanical response. For this analysis, we vary for each system height the system width in such a manner that the resulting aspect ratios are multiples of the aspect ratio of a single regular honeycomb $R_0=2/\sqrt{3} \approx 1.155$. Results are shown in . Again, one observes a clear “smaller is weaker” trend. In compression, the stiffness for each height increases with increasing $W$ but this increase soon saturates and even for very wide systems a strong dependency of stiffness on system height remains. In simple shear, by contrast, the size effect becomes smaller for wider systems and eventually inverts to a “smaller is stiffer” behavior. Moreover, for very wide and high systems the shear stiffness approaches the shear modulus of a bulk system, $C^*_{\mrm s} \to G^*_{\mrm T}$, as stress multi-axialities at the specimen corners and stress-free regions near the specimen side boundaries become asymptotically irrelevant. Again the behavior of large systems is consistent with that of an isotropic reference material without microstructure.
#### Influence of microstructure randomness on system response $(\beta\neq 0)$
To determine statistically representative stiffness parameters for microstructurally disordered systems, averages over different statistical realizations of any given microstructure must be considered. In general, bigger system sizes reduce the influence of specific microstructural features as they contain a wider spectrum of local configurations, leading to reduced fluctuations of the system-scale stiffness. Accordingly, we average over 200 different realizations for systems with $H \leq 16 \Delta p$ and 100 different realizations for $H > 16 \Delta p $. For each realization of a disordered microstructure again 20 cut-outs are taken. The resulting average response coefficients are shown for different values of $\beta$ in as functions of system size.
The overall picture is complex, and for interpreting the observations it is important to note that the elastic moduli of bulk systems actually [*increase*]{} with increasing disorder as pointed out by [@zhu2001]. Accordingly, for large systems the macroscopic response coefficients in our simulations increase with increasing $\beta$. At the same time, also irregular systems exhibit size effects though this size-dependent behavior depends on the loading mode. It is a pronounced “smaller is weaker” behavior in compression where disorder exacerbates the weakening effect. As a consequence, in compression for the smallest systems the dependency of stiffness on disorder is reversed (the systems with the largest $\beta$ are weakest). In simple shear, on the other hand, the size dependency of stiffness is much weaker. Here, a complex picture emerges where the size effects invert for increasing degree of disorder. Again this needs to be interpreted in view of the corresponding bulk behaviour: Under shear loading, the individual beams of a honeycomb structure deform primarily by bending. As investigated by [@hyun2002], microstructures which allow for more stretch/compression dominated beam deformations, such as triangular or so-called Kagome structures, are significantly stiffer than regular honeycombs. Irregularities affect the morphology of the local networks by introducing polygons with reduced number of edges, or by shortening some edges to an extent that the cells resemble such polygons.
#### Influence of the spatial distribution of microstructural irregularities
To investigate possible influences of the spatial distribution of disorder, we analyze microstructures where the regular honeycomb pattern has been perturbed by displacement of seeds only in parts of the domain. Specifically, we investigate what happens if perturbations are localized near the sample boundaries or conversely in the center. Starting from a regular seeding, $20 \%$ of the seeds are perturbed with a perturbation factor of $\beta=0.1$. Four different perturbation patterns are considered: (i) vertical bands at the left and right ($0<x_{\cal S}<0.1 W$ or $0.9 W < x_{\cal S} < W $), (ii) a vertical band in the center of the system ($0.4 W < x_{\cal S} < 0.6 W$), (iii) horizontal bands at the top and bottom ($0<y_{\cal S}<0.1 H$ or $0.9 H <y_{\cal S}< H $) and (iv) a horizontal band in the center of the system ($0.4 H <y_{\cal S}< 0.6 H$), where $x_{\cal S}, y_{\cal S}$ are the coordinates of the seeds ${\cal S}$ of the Voronoi tessellation. For each case 100 different random perturbations with 20 random cut-outs are taken. As an example, shows two realizations with vertical perturbation bands for a system of size $H=19 \Delta p$. The resulting system responses, together with that of a regular system as reference, are depicted in . One can see that weakest are microstructures where perturbations are located along the free boundaries of the sample. A vertical perturbation band in the center increases slightly the stiffness in simple shear whereas the perturbations at the boundaries reduce the macroscopic stiffness. This finding indicates that softening associated with microstructure perturbations may be more pronounced near free surfaces.
Microstructural analysis
------------------------
We now turn to analyzing random microstructures in terms of stress and strain patterns. To quantify these patterns we use the continuization method presented in . In order to obtain an averaged stress pattern from different cut-outs and microstructure realizations a common discretization is needed. To that end the stress and strain values are mapped by nearest-neighbor interpolation onto a regular grid which is the same for each system size and regularity factor. By choosing the grid spacing such that there are at least 2–3 grid points in each Voronoi cell, the error introduced by interpolation remains small and local deformation patterns are preserved. In a second step an ensemble average over all $N_{\mrm S}$ microstructure realizations, which themselves are averaged over a set of $N_\mrm{CO}$ random cut-outs, is computed. Because the continuization method is not suitable for the outermost beams and may produce artifacts near boundaries, a layer of width $a=\Delta p/ \sqrt{3}$ at the top and bottom boundaries, and of width $r = \Delta p/2$ at the left and right boundary, is excluded from the analysis. As an example, average stresses for systems of size $H =19 \Delta p$ and disorder parameters $\beta = 0$ and $\beta=0.3$ are shown in . It can be seen that despite the fact that the investigated systems are not continuous, the average response resembles a continuum stress distribution. For uniaxial compression the regular system shows an almost homogeneous stress distribution. The outermost layers near the free boundary (left and right edge) experience about 2% lower than average stresses, whereas in the corners slightly higher stresses are observed. The irregular system shows higher fluctuations in the local stress response with a slight preference for regions of enhanced or reduced stress to mutually align in direction of the loaded stress axis. The results for simple shear show a characteristic deformation pattern with very low stress values near the free boundary, which is again similar to the findings for a homogeneous continuum. In contrast to the compressive case the fluctuations of the random system are without any preferred directionality.
While the inhomogeneous stress distribution in simple shear is mainly dictated by the boundary conditions, for uniaxial compression we would expect a completely homogeneous distribution of stress in a system without microstructure. Conversely, any stress inhomogeneities observed under compressive loading can be directly related to microstructure effects. We thus study the stress distributions emerging under compression by determining, for systems of different size, stress profiles parallel and perpendicular to the compression axis. To this end we perform row- and column-wise averages of the axial stresses: $$\begin{aligned}
\overline{\sigma}_\mrm{row} &= \frac{1}{N_\mrm{S} N_\mrm{CO} } \sum^{N_\mrm{S}} \sum^{N_\mrm{CO}} \frac{1}{W-2 a} \int_a^{W-a} \langle \sigma_{yy} \rangle_\mrm{c}\dif x, \\
\overline{\sigma}_\mrm{col} &= \frac{1}{N_\mrm{S} N_\mrm{CO} }\sum^{N_\mrm{S}} \sum^{N_\mrm{CO}} \frac{1}{H-2 r} \int_r^{H-r} \langle \sigma_{yy} \rangle_\mrm{c} \dif y.\end{aligned}$$ The resulting horizontal and vertical stress profiles are shown in . It can be seen that the averaged stresses for bigger systems are higher and saturate towards a limit value, which is in line with the results of the global stiffness response. Profiles along the stress axis show an approximately constant stress level, whereas profiles perpendicular to the stress axis show a reduced stress level near the free surfaces and a stress plateau in the central region of the specimen. For smaller sizes this plateau occupies a smaller fraction of the specimen cross-section and the difference in stress level between the boundary and the central region is more pronounced.
[.48]{} [![Averaged stress profiles for uniaxial compression of regular honeycomb structures of different sizes for the []{data-label="fig:stress_profiles"}](./Figures/stress_plots/syy_row_avg_uniaxial "fig:")]{}
[.48]{} [![Averaged stress profiles for uniaxial compression of regular honeycomb structures of different sizes for the []{data-label="fig:stress_profiles"}](./Figures/stress_plots/syy_col_avg_uniaxial "fig:")]{}
Discussion {#sec:discussion}
==========
The elastic behavior of small samples of open-cellular solids (i.e. $H<100 \Delta p$) is, in addition to a density dependency, strongly affected by size effects and microstructure irregularities. The size-dependent behavior arises from a complex interplay of bulk and surface effects. Since the surface intersects some of the load-carrying beams, it is natural to assume that a near-surface region with a width of about one cell is less stiff than the bulk, and that this might lead to reduced stiffness of smaller samples where surface effects are more relevant. This is consistent with the observation in which suggests that, even in compression, stresses are reduced in a near-surface region which in smaller samples occupies a larger volume fraction. However, the same figure also demonstrates that the main reason for the stiffening that occurs with increasing system size is not a reduced importance of the softer boundary layer but rather an increased average stiffness in the [*sample interior*]{}.
In our investigation the observed “smaller is weaker” behaviour was studied for systems with non-periodic boundary conditions only. The results are comparable to those obtained in numerical studies by [@diebels2002] for simple shear and [@tekoglu2011] for uniaxial compression. Studies which impose periodic boundary conditions at the sides (at $x=0, x=W$) show the same qualitative behaviour in compression, but a “smaller is stiffer” behaviour for simple shear ([@tekoglu2011] and [@tekoglu2005]). We observe this kind of response for systems with an aspect ratio of $H/W>2 R_0$ as seen in . The explanation is that the clamped boundary locally increases the stiffness whereas the free surface decreases it (see Appendix B). With increasing system width the stiffening effect becomes more and more dominant and thus results in an “smaller is stiffer” size effect for wide and periodic systems.
The system response of locally perturbed regular structures significantly differs depending on where the perturbations of the structure are located (cf. ): for both loading cases a perturbation along the free boundaries of the system results in a macroscopically smaller stiffness compared to a perturbation in the center of the specimen or a perturbation following the constrained boundaries. This observation is also part of the explanation why, under compression, random systems are stiffer compared to regular systems. The surface weakening, which dominates in small samples, competes with bulk strengthening by a stiffer microstructure which results in the observed behavior: small irregular systems are weaker than regular ones, but large irregular systems are stiffer. To better understand the stiffening effect of disorder we study the stiffness variations observed between different specific realizations of a given random microstructures (cf. ). Thus we ask: Among different microstructures constructed according to the same statistical rule, what makes some stiffer and others weaker? In compression, we find that stiffness differences correlate with a patterning of the stress distribution as shown in for $H=19 \Delta p$ and $\beta=0.3$. It can be seen that in the stiffest system the beams are oriented such that they form vertical columns which carry most of the load. The weak system shows fewer and less extended beam chains.
In order to quantify this observation we study the spatial correlation structure of the stress field in terms of the dominant stress tensor component (axial stress for compression, shear stress for simple shear): $$\begin{aligned}
\sigma= \begin{cases}
\sigma_{yy} & \text{for compressive loading},\\
\sigma_{xy} + S_{xy} & \text{for shear loading}.
\end{cases}\end{aligned}$$ As a quantitative characteristic of the spatial correlation structure we use the stress auto-correlation $$\begin{aligned}
C_{\sigma} (\tens{l}) = \left\langle \left( \sigma(\tens{r}) - \langle \sigma \rangle_\mrm{S} \right) \left( \sigma (\tens{r}+\tens{l}) - \langle \sigma \rangle_\mrm{S} \right) \right\rangle_\mrm{S}, \end{aligned}$$ where $\tens{l} = [l \cos \phi, l \sin \phi]$ is the vector between two generic points $\tens{r}, \tens{r}+\tens{l}$, where $\phi$ is the angle with respect to the x-axis ([@Sandfeld2014_JStatMech]). To obtain good statistics, averages of ten simulations (the ten stiffest, ten weakest, and ten average microstructures) are used for the correlation analysis. For both loading cases it can be seen in for $\beta=0.3$ and $H=19 \Delta p$ that stiffer systems show stronger and more extended stress auto-correlations. This allows us to quantify different microstructures based on their correlation functions to determine (macroscopically) stiffer or weaker systems. In addition it shows that for uniaxial compression the correlations are anisotropic and stronger in direction of the stress axis. This relates to the observation of force chains which, in analogy to granular systems, may be supposed to carry most of the load. Stiffer systems show stronger correlations, i.e. more pronounced force chains, and a less homogeneous stress distribution. By looking at the correlations along the vertical and horizontal axis (cf. ) one can see that the horizontal correlation has a dip at $l \approx \Delta p$. This means that the width of the force chains approximately corresponds to the width of one cell.
Also for simple shear it can be observed that stiffer systems show a higher degree of correlation in their stress patterns. However, in comparison to compression the correlation is more isotropic. Furthermore the mechanisms governing local stiffness in this case are more complex and cannot simply be reduced to axial load transmission. Nevertheless a general conclusion can be drawn for both axial and shear deformation: More heterogeneous and more correlated stress patterns imply stiffer samples.
[![Stress auto-correlation functions for uniaxial compression and simple shear; averages over the ten weakest, ten intermediate and ten stiffest systems. System size $H=19 \Delta p$, disorder parameter $\beta=0.3$. For better comparison correlations are normalized by the maximum value of the stiffest system.[]{data-label="fig:correlation_analysis"}](./Figures/correlation_plots/avg_corr "fig:"){width="1.\textwidth"}]{}
[![Stress auto-correlation functions for uniaxial compression and simple shear; averages over the ten weakest, ten intermediate and ten stiffest systems. System size $H=19 \Delta p$, disorder parameter $\beta=0.3$. For better comparison correlations are normalized by the maximum value of the stiffest system.[]{data-label="fig:correlation_analysis"}](./Figures/correlation_plots/syy_avg_corr_weak_colorbar "fig:"){width="1.\textwidth"}]{}
Conclusion {#sec:conclusion}
==========
With the increasing distribution of advanced processes, i.e., additive manufacturing, cellular structures become more and more important because of their superior weight specific properties. This demands a better understanding of the interplay between microstructural length scale, microstructure irregularity, and system size, and their effect on the macroscopic mechanical system response. These questions were studied in this work for randomly generated microstructures with different dimensions. We statistically analyzed large numbers of simulations, varying both morphological parameters (degree of randomness of the microstructure) and geometrical parameters (sample size and aspect ratio). It was found for simple shear and uniaxial compression that most investigated systems show a characteristic “smaller is weaker” behavior. However, the response under simple shear inverts to a “smaller is stiffer” behavior for wide systems as a result of the increasing influence of the boundary layer. This observation might be of interest for future (experimental) studies, as measured system responses cannot easily be interpreted as reflecting geometry independent material properties. From an engineering point of view this size dependent behavior is of great interest, especially for structures which dimensions are less than $\approx 50$ times the averaged cell size. It is important to note that this is not about the absolute size but rather about the ratio between system and cells, so that the studied size effects can appear in many natural or engineered structures.
To better understand deformation on the microstructure level, a continuization scheme was developed. The method provides a continuous representation of stresses and strains of beam networks and therefore allows to visualize local deformation patterns and to average them such as to allow for comparison with continuum models. It thus gives the possibility to statistically analyze and optimize microstructures. By analyzing both ordered and disordered microstructures in terms of the associated continuous representations of stress and strain, we could establish that stiffer microstructures exhibit both increased stress fluctuations and stronger spatial correlations of these fluctuations in the form of long stress transmission chains. These chains may be a quite general feature of disordered systems, see the recent work of @Laubie2017_PRL on disordered porous materials, and correlations in stress transmission chains may be relevant features in the run-up to fracture [@Laubie2017_PRL; @Zaiser2017_P]. A deeper investigation of this issue that goes beyond linear elasticity may be an important topic of future studies.
Beyond the scope of the present investigation, our continuization scheme may be a useful tool for further model development since it allows to directly compare results from discrete microstructure models with those deriving from continuum theories. This capability is of particular interest in view of higher order continuum theories which are also able to model size effects, but suffer from the problem that (higher order) constitutive parameters and boundary conditions are not straightforward to determine. Unlike most top-down attempts, where one takes the (size-dependent) macroscopic system response of the beam model and then tries to fit the parameters of the higher order continuum [see e.g. @diebels2002; @tekoglu2005; @mora2007; @Liebenstein2014_ProcApplMath] our approach allows to directly take the information of the microstructure into account. We showed this in an accompanying paper by [@liebenstein2017] where we identified constitutive parameters of a Cosserat continuum.
Acknowledgements {#acknowledgements .unnumbered}
================
S.L. and M.Z. acknowledge funding by DFG under Grant no. 1 Za 171-9/1. M.Z. also acknowledges support by the Chinese government under the Program for the Introduction of Renowned Overseas Professors (MS2016XNJT044).
Timoshenko Beam equations {#appendix:eq_timoshenko_beam}
=========================
The governing equations of the Timoshenko beam can be found in the standard continuum mechanics literature, e.g. [@zienkiewicz2005b]. The total displacements of the beam are a superposition of displacements $w_i$ along the beam axis (local coordinate $x_1$) and displacements caused by rotations $\phi_i$ of the beam cross-sections, $$\begin{aligned}
\tens{u} = \begin{bmatrix}
w_1(x_1) - x_3 \phi_2(x_1) +x_2 \phi_3(x_1) \\
w_2(x_1) - x_3 \phi_1(x_1) \\
w_3(x_1) + x_2 \phi_1(x_1)
\end{bmatrix}.\end{aligned}$$ Note, that unlike in the Euler-Bernoulli beam theory $\phi_i \neq \tod{w_i}{x_1}$ and the rotation is considered an independent degree of freedom. Disregarding body forces and dynamics the balance equations are given by $\tod{F_i}{x_1} = 0$ ($i=1,2,3$) and $$\begin{aligned}
&\dod{M_1}{x_1} = 0,&
&\dod{M_2}{x_1} - F_3 = 0,&
&\dod{M_3}{x_1} + F_2 = 0,
\end{aligned}$$ where $F_i$ are the beam force components and $M_i$ the bending moments acting around the axis $\tens{e}_i$. Besides the geometrical information regarding the cross-section area $A$ and the area moments of inertia $$\begin{aligned}
I_1 &= \int_A x_2^2 + x_3^2 \dif A,&
I_2 &= \int_A x_3^2 \dif A, &
I_3 &= \int_A x_2^2 \dif A,
\end{aligned}$$ we define the following constitutive parameters for an isotropic linearly-elastic beam: The Young’s Modulus $E$, shear modulus $G$ and correction factors $k_i$ of the corresponding directions which depend on cross-section shape. The forces and moments can then be expressed in terms of the displacements as $$\begin{aligned}
F_1 &= E A \dod{w_1}{x_1}, & F_2 &= k_2 G A \lb \dod{w_2}{x_1} - \phi_3\rb, & F_3 &= k_3 G A \lb \dod{w_3}{x_1}+\phi_2 \rb, \\
M_1 &= k_1 G I_1 \dod{\phi_1}{x_1}, & M_2 &= E I_2 \dod{\phi_2}{x_1}, & M_3 &= E I_3 \dod{\phi_3}{x_1}.
\end{aligned}$$ For a planar model as considered in the main paper these relations simplify since only the force and displacement components $F_{1,2}$ and $w_{1,2}$ need to be considered. The only rotation component that is relevant in the planar case is the angle $\phi_3 =: \phi$ and the associated moment $M_3 =: M$. Accordingly, the only shear correction factor that needs to be considered is $k_2 =: \kappa$.
Simple shear stress profiles
============================
To better understand the size effect change for increasing aspect ratios under simple shear stress profile are analyzed. In analogy to the analysis for compression, the row and column-wise averages for simple shear loading are $$\begin{aligned}
\overline{\sigma}_\mrm{row} &= \frac{1}{N_\mrm{S} N_\mrm{CO} } \sum^{N_\mrm{S}} \sum^{N_\mrm{CO}} \frac{1}{W-2 a} \int_a^{W-a} \langle \sigma_{xy}+S_{xy} \rangle_\mrm{c}\dif x, \\
\overline{\sigma}_\mrm{col} &= \frac{1}{N_\mrm{S} N_\mrm{CO} }\sum^{N_\mrm{S}} \sum^{N_\mrm{CO}} \frac{1}{H-2 r} \int_r^{H-r} \langle \sigma_{xy}+S_{xy} \rangle_\mrm{c} \dif y.\end{aligned}$$ It can be seen in that two competing effects exist. A stiffening at the constrained surfaces (top and bottom) and a softening at the free surfaces (left and right). Smaller system sizes show a significantly stiffer behavior at the constrained surfaces as well as in the bulk, which explains the observed “smaller is stiffer” behavior.
[.49]{} [![Averaged stress profiles for simple shear of regular honeycomb structures of different sizes with aspect ratio $R=10 R_0$.[]{data-label="fig:stress_profiles_shear"}](./Figures/stress_plots/sxy_row_avg_constrained_shear "fig:")]{}
[.49]{} [![Averaged stress profiles for simple shear of regular honeycomb structures of different sizes with aspect ratio $R=10 R_0$.[]{data-label="fig:stress_profiles_shear"}](./Figures/stress_plots/sxy_col_avg_constrained_shear "fig:")]{}
Comparison of elastic strain energies {#appendix:cauchy formulation}
=====================================
Classical continuum formulations do not take the additional micro rotation into account. As a result they are not suitable for homogenization of discrete networks on the meso-scale (the scale of one control volume) but also on the macro-scale (the scale of the system). To illustrate this point we compare the classical Cauchy strain energy $$\begin{aligned}
W_\mrm{c}^\mrm{ca} = \frac{1}{2} \langle\tens{\sigma} \rangle_\mrm{c} : \langle \tens{\varepsilon} \rangle_\mrm{c}\end{aligned}$$ to our formulation . Again as a measure a relative energy residual $$\begin{aligned}
r_\mrm{W}^\mrm{ca} &= \frac{| W_\mrm{c}^\mrm{ca}-W_\mrm{b,c} | }{W_\mrm{b,c}}\end{aligned}$$ is chosen. In the two energy residuals $r_\mrm{W}^\mrm{ca},r_\mrm{W}$ are compared for a regular and a irregular system under simple shear loading. For the simple shear case it can be seen that the maximum error for the irregular system in the Cosserat formulation, is in the range of $2\%$, whereas the overall error, similar to the case of compressive loading, is about $0.1\%$. In the Cauchy formulation which neglects rotations, by contrast, maximum errors are of the order of 1 whereas the average error is of the order of 0.25, rendering such formulations practically useless. As the energy residuals obtained from the Cosserat formulation are about two orders of magnitude smaller than those obtained from the Cauchy formulation we conclude that, for the structures at hand, Cosserat models out-perform classical Cauchy formulations of stress and strain by a very considerable margin.
[^1]: In the following, we denote by ’microstructure’ internal sub-structures of a material which are characterized by length scales well below the size of a specimen or device, the geometry of which defines the ’macrostructure’.
|
---
abstract: 'Two pseudo-Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a complete list of possible values for the degree of mobility of Riemannian and Lorentzian Einstein metrics on simply connected manifolds, and describe all possible dimensions of the space of essential projective vector fields.'
address: 'Institute of Mathematics, Friedrich-Schiller-Universität Jena, 07737 Germany.'
author:
- 'Vladimir S. Matveev and Stefan Rosemann'
nocite: '[@*]'
title: The degree of mobility of Einstein metrics
---
Introduction
============
The aim of this article is to study Einstein metrics (i.e., such that the Ricci curvature is proportional to the metric) of Riemannian and Lorentzian signature in the realm of projective geometry.
Recall that two pseudo-Riemannian metrics $g$ and $\bar g$ on a manifold $M$ are called *projectively equivalent*[^1] if their unparametrized geodesics coincide. Clearly, any constant multiple of $g$ is projectively equivalent to $g$. A generic metric does not admit other examples of projectively equivalent metrics, see [@MatGenRel]. If two metrics $g,\bar g$ are *affinely equivalent*, that is, if their Levi-Civita connections coincide, then they are also projectively equivalent. Affinely equivalent metrics are well-understood at least in Riemannian [@deRham; @ei] and Lorentzian signature [@Sol; @Petrov], see also Lemma \[lem:decomp\] below. The case of arbitrary signature is much more complicated, see [@Sol] or the more recent article [@Boubel] for a local description of all such metrics.
The theory of projectively equivalent metrics has a long and rich history – we refer to the introductions of [@KioMatEinstein; @MatHyper] or to survey [@mikes] for more details, and focus on Einstein metrics in what follows.
Einstein metrics are very natural objects in projective geometry. For instance, as shown in [@KioMatEinstein], the property of a metric $g$ to be Einstein is projectively invariant in the following sense: any metric that projectively equivalent and not affinely equivalent to an Einstein metric is also Einstein. A more educated point of view on the whole subject is the following: a projective geometry, given by a class of projectively equivalent connections (not necessarily Levi-Civita connections), is an example of a parabolic geometry, a special case of a Cartan geometry, see the monographs [@CapBook; @Sharpe]. As shown in [@EastMat], the metrics with Levi-Civita connection contained in the given projective class are in one-one correspondence to solutions of a certain overdetermined system of partial differential equations. This system is a so-called first Bernstein-Gelfand-Gelfand equation [@CalBGG; @CapBGG] and, as shown in [@CapGover], Einstein metrics correspond to a special class of solutions called normal.
The *degree of mobility $D(g)$* of a pseudo-Riemannian metric $g$ is the dimension of the space of $g$-symmetric solutions of the PDE . As we explain in Section \[sec:basic\], nondegenerate solutions of are in one-to-one correspondence with the metrics projectively equivalent to $g$. Hence, intuitively, $D(g)$ is the dimension of the space of metrics projectively equivalent to $g$.
We have $D(g)=1$ for a generic metric $g$ and $D(g)\geq 2$ if $g$ admits a projectively equivalent metric that is nonproportional to $g$. As our main result, we determine all possible values for the degree of mobility $D(g)$ of Riemannian and Lorentzian Einstein metrics, locally or on simply connected[^2] manifolds. Let us denote by “$[\alpha]$” the integer part of a real number $\alpha$.
\[thm:main\] Let $(M,g)$ be a simply connected Riemannian or Lorentzian Einstein manifold of dimension $n\geq 3$. Suppose $g$ admits a projectively equivalent but not affinely equivalent metric.
Then, the degree of mobility $D(g)$ is one of the numbers $\geq 2$ from the following list:
- $\frac{k(k+1)}{2}+l$, where $n\geq 5$, $0\leq k\leq n-4$ and $1\leq l\leq [\frac{n+1-k}{5}]$ for $g$ Riemannian and Lorentzian.
- $\frac{k(k+1)}{2}+l$, where $n\geq 5$, $k=n-3\mbox{ mod }5$, $2\leq k\leq n-3$ and $l=[\frac{n+2-k}{5}]$ for $g$ Lorentzian.
- $\frac{(n+1)(n+2)}{2}$.
Conversely, for $n\geq 3$ and each number $D\geq 2$ from this list, there exist simply connected $n$-dimensional Riemannian resp. Lorentzian Einstein manifolds admitting projectively equivalent but not affinely equivalent metrics and such that $D$ is the degree of mobility $D(g)$.
![ Degree of mobility $D(g)$ from Theorem \[thm:main\] for $3\leq \mathrm{dim}\,M\le 15$. The triangles denote the additional values for Lorentz signature.[]{data-label="1"}](degreeRiemLor){width=".7\textwidth"}
In Theorem \[thm:main\], the degree of mobility is at least $2$ since we assumed that $g$ admits a metric $\bar g$ projectively equivalent to $g$ but not affinely equivalent to it. Suppose this assumption is dropped, that is, let us assume all metrics projectively equivalent to $g$ are affinely equivalent to it. In this case the complete list of possible values of the degree of mobility of $g$ can be easily obtained by combining Lemma \[lem:decomp\] below with methods similar to the ones used in Section \[sec:Bnonzero\] and Section \[sec:Bzeromuzero\]. It is $$\{k(k+1)/2+l:0\leq k\leq n-2,1\leq l\leq [(n-k)/2]\}\cup\{n(n+1)/2\}$$ if $g$ is Einstein with nonzero scalar curvature and $$\{k(k+1)/2+l:0\leq k\leq n-4,1\leq l\leq [(n-k)/4]\}\cup\{n(n+1)/2\}$$ if $g$ is Ricci flat.
It is well-known, see e.g. [@Sinjukov p.134], that if $D(g) $ is equal to its maximal value $(n+1)(n+2)/2$, then $g$ has constant sectional curvature. Conversely, this value is attained on simply connected manifolds of constant sectional curvature. In view of this, the case $n= 3$ in Theorem \[thm:main\] is trivial, since a $3$-dimensional Einstein metric has constant sectional curvature and its degree of mobility takes the maximum value $D(g)=10$.
For $4$-dimensional Einstein metrics, we obtain the following statement as an immediate consequence of Theorem \[thm:main\] (compare also Figure \[1\]):
\[cor:main\] Let $(M,g)$ be a $4$-dimensional Riemannian or Lorentzian Einstein manifold. Suppose $\bar g$ is projectively equivalent to $g$ but not affinely equivalent. Then, $g$ has constant sectional curvature.
Corollary \[cor:main\] was known before, see [@KioMatEinstein Theorem 2] (or, alternatively, [@Hall2]), and it is actually true for metrics of arbitrary signature. However, our methods for proving Theorem \[thm:main\] and Corollary \[cor:main\] are different from that used in [@Hall2; @KioMatEinstein] (although we will rely on some statements from [@KioMatEinstein]). A special case of Corollary \[cor:main\] was also considered in [@Petrov] where it was proven that $4$-dimensional Ricci flat nonflat metrics cannot be projectively equivalent unless they are affinely equivalent. This result was generalized to Einstein metrics of arbitrary scalar curvature in [@Hall1]. Note that by [@KioMatEinstein Theorem 1], the statement of Corollary \[cor:main\] survives for arbitrary dimension under the assumption that both metrics are geodesically complete.
Projective equivalence of Lorentzian Einstein metrics, in particular, the problem we have investigated, was actively studied in general relativity, see the classical references [@eiGR; @eiBook; @Weyl1] and the more recent articles [@Hall1; @Hall2; @MatGenRel]. The motivation to study this problem is based on the description of trajectories of freely falling particles in vacuum as unparametrized geodesics of a Lorentzian Einstein metric. The initial question, studied in [@Ehlers; @Petrov; @Weyl1], is whether and under what conditions one can reconstruct the spacetime metric by only observing freely falling particles. We study the ‘freedom’ of such a reconstruction: the number of parameters is given by Theorem \[thm:main\].
We see from Theorem \[thm:main\] that the list for the values of the degree of mobility for Riemannian Einstein metrics is strictly smaller than the list for Lorentzian Einstein metrics. This difference starts in dimension five: for a $5$-dimensional Riemannian Einstein metric $g$ we have $D(g)=1,2$ or $g$ has constant sectional curvature (i.e., $D(g)=21$). However, according to Theorem \[thm:main\], there exist $5$-dimensional Lorentzian Einstein metrics having $D(g)=4$. For instance, consider
\[ex:counterex1\] The nonconstant curvature metric $$g={\mathrm{d}}t^2+e^{2t}({\mathrm{d}}x_0\odot {\mathrm{d}}x_1+e^{x_2}\mathrm{sin}(x_3){\mathrm{d}}x_1^2+{\mathrm{d}}x_2^2+{\mathrm{d}}x_3^2)$$ on $M={\mathbb{R}}^5$ (with coordinates $(t,x_0,x_1,x_2,x_3)$) is Einstein with scalar curvature $20$ and has signature $(1,4)$. In addition to $g$, the following symmetric $(0,2)$-tensors are solutions of equation : $$L_1=e^{2t}{\mathrm{d}}t^2,\,\,\,L_2=e^{2t}(x_1{\mathrm{d}}t+{\mathrm{d}}x_1)^2,\,\,\,L_3=e^{2t}{\mathrm{d}}t\odot (x_1{\mathrm{d}}t+{\mathrm{d}}x_1).$$
Without the assumption that the metric is Einstein, an analogue of Theorem \[thm:main\] is [@FedMat Theorem 1]. Obviously, the values obtained in Theorem \[thm:main\] are contained in the list of [@FedMat Theorem 1], but our list is of course thinner: not every value from [@FedMat Theorem 1] can be realized as the degree of mobility of an Einstein metric. We suggest to compare Figure \[1\] above with [@FedMat Fig. 1].
Note also that most experts (including us) expected that the list for the values of the degree of mobility should not depend on the signature. This is true (at least when comparing Riemannian and Lorentzian signature) if we do consider general metrics (not necessarily Einstein), see [@FedMat Theorem 1]. As stated in Theorem \[thm:main\], it is not true when we consider Einstein metrics, see also Example \[ex:counterex1\] above.
Note that if the manifold is closed, the list of possible values for the degree of mobility is much shorter. Indeed, by [@KioMatEinstein; @MatMoun], a metric that is projectively equivalent to an Einstein metric of nonconstant sectional curvature on a closed manifold is affinely equivalent to it.
Application: the dimension of the space of essential projective vector fields
-----------------------------------------------------------------------------
Let $(M,g)$ be a pseudo-Riemannian manifold. A diffeomorphism $f:M\rightarrow M$ is called a *projective transformation* if it maps unparametrized geodesics to unparametrized geodesics or, equivalently, if $f^{*}g$ is projectively equivalent to $g$. The isometries of $g$ are clearly projective transformations. A projective transformation is called *essential* if it is not an isometry of the metric.
A vector field $v$ on $(M,g)$ is called *projective* if its local flow consists of projective transformations. A projective vector field is called *essential* if it is not a Killing vector field.
Let $\mathfrak{p}(g)$ and $\mathfrak{i}(g)$ denote the vector spaces (in fact, Lie algebras) of projective and Killing vector fields respectively. The quotient $\mathfrak{p}(g)/\mathfrak{i}(g)$ will be referred to as the *space of essential projective vector fields*. In the generic case, see Remark \[rem:essprojvf\] below, this space can be naturally identified with a subspace (thought, not a subalgebra) of $\mathfrak{p}(g)$.
We determine all possible values for the dimension of the space of essential projective vector fields of a Riemannian or Lorentzian Einstein metric:
\[thm:proj\] Let $(M,g)$ be a simply connected Riemannian or Lorentzian Einstein manifold of dimension $n\geq 3$ which admits a metric that is projectively equivalent but not affinely equivalent to $g$. Then, the possible values for the dimension of the space of essential projective vector fields are given by the numbers $\geq 1$ from the following list:
- $\frac{k(k+1)}{2}+l-1$, where $n\geq 5$, $0\leq k\leq n-4$ and $1\leq l\leq [\frac{n+1-k}{5}]$ for $g$ Riemannian and Lorentzian.
- $\frac{k(k+1)}{2}+l-1$, where $n\geq 5$, $k=n-3\mbox{ mod }5$, $2\leq k\leq n-3$ and $l=[\frac{n+2-k}{5}]$ for $g$ Lorentzian.
- $\frac{(n+1)(n+2)}{2}-1$.
Conversely, for $n\geq 3$ and each number $\geq 1$ from this list, there exists a $n$-dimensional simply connected Riemannian resp. Lorentzian Einstein metric admitting a projectively equivalent but not affinely equivalent metric and for which this number is the dimension of the space of essential projective vector fields.
Comparing the list from Theorem \[thm:proj\] with that in Theorem \[thm:main\], we see that the possible values for $\mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)$ are given by the values for the degree of mobility $D(g)$ subtracted by $1$. Indeed, in the generic case, the number of essential projective vector fields of an Einstein metric is $D(g)-1$. Moreover, if in addition to our assumptions the metric is Riemannian or the scalar curvature is not zero, then there exists a natural linear mapping with $1$-dimensional kernel from the set of solutions of to the space $ \mathfrak{p}(g)/\mathfrak{i}(g)$, see Section \[sec21\] below. There exist though Einstein metrics of Lorentzian signature such that $\mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)< D(g)-1$.
By Theorem \[thm:proj\], any Einstein metric of Riemannian or Lorentzian signature admitting a nonaffinely equivalent projectively equivalent metric also admits an essential projective vector field. The next theorem shows that the assumption on signature is not essential.
\[thm:proj2\] Let $g$ be an Einstein metric of arbitrary signature on a simply connected manifold of dimension $n\geq 3$. If there exists a metric that is projectively equivalent but not affinely equivalent to $g$, there exists at least one essential projective vector field for $g$.
Examples show that the assumption that the metric is Einstein is essential for Theorem \[thm:proj2\].
As we already recalled above, an Einstein metric of arbitrary signature and of nonconstant sectional curvature on a closed manifold does not admit projectively but not affinely equivalent metrics. Therefore, on a closed Einstein manifold of nonconstant sectional curvature every projective transformation is an affine transformation and, hence, every projective vector field is an affine vector field. Actually, in the Riemannian case we do not need the assumption that the metric is Einstein in the latter statement, see [@MatLichOb Corollary 1].
Similar results were also obtained in the case the manifold is not necessarily closed but under the additional assumption that the metric $g$ and a projectively equivalent but not affinely equivalent metric $\bar g$ are complete. By [@KioMatEinstein Theorem 1], projective but not affine equivalence of two complete metrics (of arbitrary signature) one of which is Einstein implies that both metrics have constant sectional curvature. This implies that complete Einstein metrics do not admit complete projective but not affine vector fields. Again in the Riemannian case we do not need the assumption that the metric is Einstein in the latter statement, see [@MatLichOb Theorem 1].
Note that the result of Theorem \[thm:proj\] has a predecessor: in [@FedMat Theorem 3] the possible dimensions of the space of essential projective vector fields have been determined for a general Riemannian or Lorentzian metric. As before the list of values we have obtained in the Einstein case is shorter than the list of values obtained in [@FedMat Theorem 3].
Organisation of the article
---------------------------
In Section \[sec:basic\], we recall basic facts from the theory of projectively equivalent metrics.
The remaining sections deal with the proofs of the Theorems \[thm:main\], \[thm:proj\] and \[thm:proj2\]. As mentioned above, the case of general (= not necessarily Einstein) metrics was solved in [@FedMat]. We extensively use and therefore quote necessary results from [@FedMat] in the paper and indicate the places when the additional condition that the metric is Einstein becomes important.
The proof of Theorem \[thm:main\] will be given in Section \[sec:proofmain\]. It is divided into several parts and a rough discription of how we proceed can be found in Section \[sec:scheme\].
The proof of Theorem \[thm:proj\] and that of Theorem \[thm:proj2\] will be given in Section \[sec:proofproj\].
Basic formulas {#sec:basic}
==============
Let $g,\bar g$ be two pseudo-Riemannian metrics on an $n$-dimensional manifold $M$. We define a symmetric nondegenerate $(0,2)$-tensor $L$ by $$\begin{aligned}
L=L(g,\bar g)=\Big|\frac{\mathrm{det}\,\bar g}{\mathrm{det}\, g}\Big|^{\frac{1}{n+1}}g\bar g^{-1}g.\label{eq:defL}\end{aligned}$$ In the formula above, we view $g,\bar g:TM\rightarrow T^* M$ naturally as bundle isomorphisms and identify $(0,2)$-tensors with endomorphism $TM\rightarrow T^* M$ via $L(X)(Y)=L(X,Y)$ for $X,Y\in TM$. In tensor notation, reads $$L_{ij}=\Big|\frac{\mathrm{det}\,\bar g}{\mathrm{det}\, g}\Big|^{\frac{1}{n+1}}g_{ik}\bar g^{kl}g_{lj},$$ where $\bar g^{ik}\bar g_{kj}=\delta^i_j$. It is a fundamental fact, see [@Sinj], that $g$ and $\bar g$ are projectively equivalent, if and only if the tensor $L$ from is a solution to the following PDE $$\begin{aligned}
\nabla_X L=X^\flat\odot \Lambda,\,\,\,X\in TM,\label{eq:main}\end{aligned}$$ where $\Lambda$ is a certain $1$-form, $\nabla$ denotes the Levi-Civita connection of $g$, $\alpha\odot\beta=\alpha\otimes\beta+\beta\otimes\alpha$ for $1$-forms $\alpha,\beta$ and $X^\flat=g(X,.)$ denotes the metric dual w.r.t. $g$.
Throughout the article, when it is clear which metric is used, we will denote by $X^\flat\in T^*M$ the metric dual of a vector $X\in TM$ and by $\alpha^\sharp\in TM$ the metric dual of a $1$-form $\alpha\in T^*M$. Similarly, for a $(0,2)$-tensor $L$ we let $L^\sharp$ denote the corresponding $(1,1)$-tensor defined by $g(L^\sharp.,.)=L$.
Taking a trace in using $g$ shows that $$\Lambda={\mathrm{d}}\lambda\mbox{, where }\lambda=\frac{1}{2}\mathrm{trace}(L^\sharp).$$ Thus, is in fact a linear PDE of first order on symmetric $(0,2)$-tensors $L$. As stated above, the nondegenerate symmetric solutions of correspond via to metrics projectively equivalent to $g$. In fact, if $L$ is such a solution then $\bar g=(\mathrm{det}\,L^\sharp)^{-1}g((L^\sharp)^{-1}.,.)$ is projectively equivalent to $g$. Since $g$ is always a solution of (corresponding to the fact that $g$ is projectively equivalent to itself), we can (locally) make any symmetric solution of nondegenerate by adding a suitable multiple of $g$. In this sense the linear space of symmetric solutions of corresponds to the space of metrics being projectively equivalent to $g$.
Let $(M,g)$ be a pseudo-Riemannian manifold. We denote by ${\mathcal{A}}(g)$ the linear space of symmetric solutions of . The *degree of mobility $D(g)$* of $g$ is the dimension of ${\mathcal{A}}(g)$.
In view of the above correspondence we will often consider a pair $g,L$, where $L\in {\mathcal{A}}(g)$, instead of a pair $g,\bar g$ of projectively equivalent metrics.
As stated in the introduction, affinely equivalent metrics (i.e. metrics having the same Levi-Civita connections) are projectively equivalent. Obviously, two metrics $g,\bar g$ are affinely equivalent if and only if the tensor $L=L(g,\bar g)$ from is parallel (w.r.t. the Levi-Civita connection of one of the metrics). In view of , this is equivalent to the property that $\Lambda$ from is identically zero. Combining these, we obtain the following wellknown statement:
\[lem:affine\] Let $g,\bar g$ be projectively equivalent pseudo-Riemannian metrics on a manifold $M$ and let $L=L(g,\bar g)\in {\mathcal{A}}(g)$ be given by . Then, $g,\bar g$ are affinely equivalent if and only if $L$ is $g$-parallel if and only if the $1$-form $\Lambda$ corresponding to $L$ is identically zero.
Of fundamental importance for our goals is the following
[@KioMatEinstein]\[thm:extsys\] Let $(M,g)$ be a connected pseudo-Riemannian Einstein manifold of dimension $n\geq 3$ such that at least one $L\in {\mathcal{A}}(g)$ is nonparallel. Let $$B=-\frac{\mathrm{Scal}}{n(n-1)},$$ where $\mathrm{Scal}$ denotes the scalar curvature of $g$.
Then, for every $L\in {\mathcal{A}}(g)$ with corresponding $1$-form $\Lambda$, there exists a function $\mu$ such that $(L,\Lambda,\mu)$ satisfies $$\begin{aligned}
\nabla_X L=X^\flat\odot \Lambda,\,\,\,\nabla\Lambda=\mu g +BL,\,\,\,\nabla\mu=2B\Lambda.
\label{eq:extsys}\end{aligned}$$
Theorem \[thm:extsys\] follows from [@KioMatEinstein Corollary 1 and 2]. As shown in [@KioMat], under the assumption $D(g)\geq 3$, the statement is actually true for any metric (not necessarily Einstein) and a certain constant $B$ (which is not necessarily equal to $-\mathrm{Scal}/n(n-1)$ in this case).
Proof of Theorem \[thm:main\] {#sec:proofmain}
=============================
Scheme of the proof {#sec:scheme}
-------------------
By Theorem \[thm:extsys\], under the assumptions of Theorem \[thm:main\], the degree of mobility $D(g)$ equals the dimension of the space of solutions of the system . The proof of Theorem \[thm:main\] is different for $B=-\mathrm{Scal}/n(n-1)=0$ and for $B\ne 0$.
Consider first the case $B\neq 0$. By scaling the metric $g$ we may assume that $B=-1$. The key observation is that for $B=-1$ the solutions of the system correspond to parallel symmetric $(0,2)$-tensors on the metric cone $(\hat M:= \mathbb{R}_{>0}\times M,\hat g:= {\mathrm{d}}r^2+ r^2 g )$ over $(M,g)$. Depending on the sign of the initial $B$ and on the signature of the metric $g$, the metric cone $(\hat M,\hat g)$ has signature $(0,n+1)$, $(1,n)$, $(n,1)$, or $(n-1,2)$. The space of parallel tensors for cone metrics of these signatures has been described in [@FedMat]. The assumption that the initial metric is Einstein is equivalent to the condition that the cone metric is Ricci-flat. Combining the description of parallel tensors with the Ricci-flat condition, we obtain the list of possible values for $D(g)$.
Consider now the case when $B =0$ but assume that at least one solution of has $\mu\neq 0$. This case is treated in Section \[sec:Bzero\]. We show the local existence of an Einstein metric $\bar g$ of the same signature as $g$ and projectively equivalent to $g$ such that the corresponding constant $\bar B$ for $\bar g$ is nonzero. This allows to reduce the problem to the already solved one.
The remaining case, considered in Section \[sec:Bzeromuzero\], is when $B =0$ and $\mu = 0$ for all solutions of . In this case additional work is necessary, but also here the problem reduces to determining the dimension of the space of parallel symmetric $(0,2)$-tensors (although, this time, we consider such tensors for $g$ and not for the cone metric $\hat g$). We can locally describe all such metrics and the Einstein condition poses additional restrictions on the possible values of the degree of mobility.
Finally, in Section \[sec:realization\] we complete the proof of Theorem \[thm:main\] by showing that actually each number $D$ from the list in the theorem can be realized as the degree of mobility of a certain Lorentzian resp. Riemannian Einstein metric. This is done by going in the opposite direction of the procedure explained in Section \[sec:Bnonzero\]: we construct a Ricci flat cone such that the space of parallel symmetric $(0,2)$-tensor fields has dimension equal to $D$.
The case of nonzero scalar curvature {#sec:Bnonzero}
------------------------------------
The goal of this section is to prove
\[prop:Bnonzero\] Let $(M,g)$ be a simply connected Riemannian or Lorentzian Einstein manifold of dimension $n\geq 3$ with nonzero scalar curvature such that $\Lambda\neq 0$ for at least one solution of the system .
Then, the degree of mobility $D(g)$ is given by one of the values in the list of Theorem \[thm:main\].
We will go along the same line of ideas as in [@FedMat Section 4]. We will start working with a general Riemannian or Lorentzian metric $g$ and implement the condition that $g$ is Einstein at the corresponding places. Since the constant $B:=-\mathrm{Scal}/n(n-1)$ in is nonzero, we can consider the metric $-Bg$ instead of $g$ and for simplicity, we denote this new metric by the same symbol $g$. Because we have rescaled the metric, the system is now satisfied for a new constant $B=-1$, that is, for every $L\in {\mathcal{A}}(g)$ with corresponding $1$-form $\Lambda$, we find a function $\mu$ such that $(A,\Lambda,\mu)$ satisfies $$\begin{aligned}
\nabla_X L=X^\flat\odot \Lambda,\,\,\,\nabla\Lambda=\mu g -L,\,\,\,\nabla\mu=-2\Lambda.
\label{eq:extsys-1}\end{aligned}$$ Note that since the new metric $g$ and the original metric are proportional to each other, they have the same degree of mobility.
Note also that since the initial metric was assumed to be Riemannian or Lorentzian the signature of the new metric $g$ is now $(0,n)$, $(1,n-1)$, $(n,0)$ or $(n-1,1)$, depending on the sign of the scaling constant $B$.
For further use let us recall the following statement which can be found for example in [@MatMoun Proposition 3.1] or [@FedMat Theorem 8] and can be verified by a direct calculation.
\[lem:isom\] There is an isomorphism between the space of solutions of on a pseudo-Riemannian manifold $(M,g)$ and the space of parallel symmetric $(0,2)$-tensors on the metric cone $(\hat M={\mathbb{R}}_{>0}\times M,\,\,\,\hat g={\mathrm{d}}r^2+r^2 g)$ over $(M,g)$.
Since the manifold $(M,g)$ in our case has signature $(0,n)$, $(1,n-1)$, $(n,0)$ or $(n-1,1)$, the signature of the metric $\hat g$ is $(0,n+1)$, $(1,n)$, $(n,1)$ or $(n-1,2)$.
By Lemma \[lem:isom\], in order to determine the possible values of the degree of mobility $D(g)$ of $g$, it is sufficient to calculate the possible dimensions of the space of parallel symmetric $(0,2)$-tensors for the cone metric $\hat g$.
The description of such tensors has been obtained in [@FedMat Theorem 5]. Since we will come back to this result later on, we summarize it in
\[lem:decomp\] Let $(M,g)$ be a simply connected $n$-dimensional pseudo-Riemannian manifold. Assume one of the following:
1. $g$ has signature $(0,n)$ or $(1,n-1)$.
2. $g$ is a metric cone of signature $(n-2,2)$.
Consider the maximal holonomy decomposition $$\begin{aligned}
T M=V_0\oplus V_1\oplus...\oplus V_l\label{eq:decomp}\end{aligned}$$ of the tangent bundle $TM$ into mutually orthogonal subbundles invariant w.r.t. the holonomy group $H( g)$ of $ g$. More precisely, $V_0$ is flat in the sense that $H( g)$ acts trivially on it and $V_1,...,V_l$ are indecomposable, i.e., do not admit an invariant nondegenerate subbundle. Let $g_i$ denote the restriction of $g$ to $V_i$ for $i=0,...,l$. If $\tau_1,...,\tau_k$ is a basis for the space of parallel $1$-forms for $ g$, then any parallel symmetric $(0,2)$-tensor can be written as $$\begin{aligned}
\sum_{i,j=1}^k c_{ij}\tau_i\otimes \tau_j+\sum_{i=1}^l c_i g_i\label{eq:parallel}\end{aligned}$$ for constants $c_{ij}=c_{ji}$ and $c_i$.
The statement of Lemma \[lem:decomp\] is classical for positive definite $g$ [@ei] and for Lorentzian signature [@eiparallel; @Sol]. The description of parallel symmetric $(0,2)$-tensors for metric cones of signature $(n-2,2)$ is given by [@FedMat Theorem 5]. If the metric is not a cone the description of such tensors for metrics of arbitrary signature is in general much more complicated, see [@Boubel].
Formula shows that the dimension of parallel symmetric $(0,2)$-tensors for $\hat g$ and, hence, the degree of mobility $D(g)$ of $g$, is given by $$\begin{aligned}
D(g)=\frac{k(k+1)}{2}+l,\label{eq:degree}\end{aligned}$$ where $k$ is the number of linearly independent parallel vector fields for $\hat g$ and $l$ the number of indecomposable components in the holonomy decomposition of $(\hat M,\hat g)$. To prove the first direction of Theorem \[thm:main\] under the assumption $B\neq 0$, it therefore suffices to determine the range of the integers $k,l$ in . We start listing some known facts concering curvature properties of the metric cone.
\[lem:curvfacts\] Let $(\hat M,\hat g)$ be the metric cone over an $n$-dimensional pseudo-Riemannian manifold $(M,g)$. Then, the following statements hold:
1. $\hat g$ is flat if and only if $g$ has constant sectional curvature equal to $1$.
2. $\hat g$ is Ricci flat if and only if $g$ is Einstein with scalar curvature $n(n-1)$.
The statements follow from the usual formulas relating the curvatures of $\hat g$ and $g$, see for instance [@ACGL equation (3.2)].
Since in our case the given Einstein metric $g$ has $B=-1$, we have $\mathrm{Scal}(g)=n(n-1)$ and therefore $\hat g$ is Ricci flat.
The so-called *cone vector field* $\xi=r\partial_r$ on $\hat M$ satisfies $$\begin{aligned}
\hat \nabla \xi={\mathrm{Id}}.\label{eq:conevf}\end{aligned}$$ This is straight-forward to see (using the formulas for the Levi-Civita connection $\hat\nabla$ of $\hat g$, see for instance [@ACGL equation (3.1)]) and is wellknown, see [@FedMat Lemma 1]. A manifold $(\hat M,\hat g)$ admitting a vector field $\xi$ satisfying will be called a *local cone* in what follows. The name is justified in
\[lem:localcone\] Let $(\hat M,\hat g,\xi)$ be a local cone of dimension $n+1$. Then, $\xi$ is nonvanishing on a dense and open subset and in a neighbourhood of each point of this subset $(\hat M,\hat g,\xi)$ takes the form $$\hat M={\mathbb{R}}_{>0}\times M,\,\,\,\hat g=\varepsilon {\mathrm{d}}r^2+r^2 g,\,\,\,\xi=r\partial_r$$ where $(M,g)$ is a certain $n$-dimensional pseudo-Riemannian manifold and $\varepsilon=\mathrm{sgn}(\hat g(\xi,\xi))$. That is, locally in a neighbourhood of almost every point, $(\hat M,\hat g)$ is a metric cone, up to multiplication by $-1$, over a certain pseudo-Riemannian manifold.
The statement and its proof are standard, see [@FedMat Lemma 1 and Remark 2] (the role of the positive function $v$ used in this reference is played by $\frac{1}{2}\hat g(\xi,\xi)$ for $\hat g(\xi,\xi)>0$).
We will need a dimensional estimate for nonflat Ricci flat local cones.
\[lem:dimRicciflat\] Let $(\hat M,\hat g,\xi)$ be a Ricci flat local cone.
1. If $\hat g$ is nonflat, then $\mathrm{dim}\,\hat M\geq 5$.
2. If $\hat g$ is nonflat and $u$ is a nonzero parallel null vector field for $g$, then $\mathrm{dim}\,\hat M\geq 6$.
$(1)$ follows immediately from Lemma \[lem:curvfacts\]: locally, in a neighborhood of almost every point, $(\hat M, \hat g)$ is a cone over an Einstein manifold $(M,g)$ of dimension $n$ (where $\mathrm{dim}\,\hat M=n+1$) with scalar curvature $\mathrm{Scal}(g)=n(n-1)$. If $n+1= 4$, $g$ is a $3$-dimensional Einstein metric and therefore has constant sectional curvature equal to $1$. This, in turn, implies $\hat g$ is flat.
$(2)$ Let $u$ be a nonzero parallel null vector field for $\hat g$. Suppose $\hat g(u,\xi)=0$ on some open subset $U$. Taking the derivative of this equation and using , we obtain $\hat g (u,.)=0$ on $U$, hence, $u=0$ on $U$, a contradiction. On the other hand, suppose $\xi=fu$ on some open subset $U$ for a smooth function $f:U\rightarrow {\mathbb{R}}$. Again, taking the covariant derivative of this equation and using , we obtain ${\mathrm{Id}}={\mathrm{d}}f\otimes u$ which is clearly a contradiction (since the endomorphism on the right-hand side has rank $1$). We obtain that at every point $p$ of an open and dense subset of $\hat M$, $\xi$ and $u$ are linearly independent (see also [@FedMat Lemma 3]) and $\hat g(u,\xi)(p)\neq 0$. Then, $\hat g$ is nondegenerate on $\mathrm{span}\{\xi(p),u(p)\}$. If $\hat M\leq 5$, the statement that $\hat R(p)=0$ now reduces to the statement that Ricci flat curvature operators in dimensions $\leq 3$ are flat.
The following example shows that the existence of two linearly independent parallel vector fields on a Ricci flat cone $(\hat M,\hat g)$ of dimension $6$ does in general not imply that $\hat g$ is flat:
\[ex:counterex\] The cone metric over the metric from Example \[ex:counterex1\], given by $$\begin{aligned}
\hat g={\mathrm{d}}r^2+r^2[-{\mathrm{d}}t^2+e^{2t}({\mathrm{d}}x_0\odot {\mathrm{d}}x_1+e^{x_2}\mathrm{sin}(x_3){\mathrm{d}}x_1^2-{\mathrm{d}}x_2^2-{\mathrm{d}}x_3^2)],
\label{eq:metriccounterex}\end{aligned}$$ has signature $(4,2)$ and is indecomposable nonflat and Ricci flat. It admits two linearly independent parallel vector fields $$\begin{aligned}
v_1=e^t(\partial_r-\frac{1}{r}\partial_t),\,\,\,
v_2=x_1e^t\partial_r+\frac{1}{r}\left(-x_1 e^t\partial_t+e^{-t}\partial_{x_0}\right)\label{eq:vfcounterex}\end{aligned}$$ such that $\mathrm{span}\{v_1,v_2\}$ is totally isotropic.
Example \[ex:counterex\] is a special case of the following general description (which can be obtained in a straight-forward way by applying, for instance, results of [@BolMat]): any cone $(\hat M={\mathbb{R}}_{>0}\times M,\hat g={\mathrm{d}}r^2+r^2 g)$ with nonzero parallel null vector field $v$, is locally of the form $$\hat M={\mathbb{R}}_{>0}\times {\mathbb{R}}\times N,\,\,\,\hat g={\mathrm{d}}r^2+r^2(-{\mathrm{d}}t^2+e^{2t}h),\,\,\,v=e^t(\partial_r-\frac{1}{r}\partial_t),$$ where $(N,h)$ is a certain pseudo-Riemannian manifold. We have that $\hat g$ is Ricci flat (resp. flat) if and only if $h$ is Ricci flat (resp. flat). If $V$ is another parallel vector field for $\hat g$, we obtain $$V=\left(F e^{t}-\frac{C}{2}e^{-t}\right)\partial_r+\frac{1}{r}\left(-\left(F e^t+\frac{C}{2}e^{-t}\right)
\partial_t+e^{-t} \mathrm{grad}_h F\right)$$ for a certain constant $C$ and a function $F$ on $N$ satisfying $$\nabla^h\nabla^h F=C h,$$ where $\nabla^h$ denotes the Levi-Civita connection of $h$. Since $\hat g(V,V)=-2CF+h(\mathrm{grad}_h F,\mathrm{grad}_h F)$ and $\hat g(v,V)=-C$, we see that $V$ is null and perpendicular to $v$ if and only if $\mathrm{grad}_h F$ is a parallel null vector field on $N$. To construct Example \[ex:counterex\], it remains to find an example of a nonflat Ricci flat Lorentz manifold admitting a nonzero parallel gradient null vector field. Such metrics are described by Walker coordinates [@eiparallel; @Walker].
As explained above, the maximal value $D(g)=(n+1)(n+2)/2$ for the degree of mobility is attained if and only if $g$ has constant sectional curvature, i.e., if and only if $\hat g$ is flat. Thus, in order to seek for the submaximal values of $D(g)$, we may assume that $\hat g$ is nonflat, i.e. $l\geq 1$ in the decomposition . Thus, $(\hat M,\hat g)$ is a Ricci flat but nonflat cone with $k$ parallel vector fields. Let $\hat p\in \hat M$ be a point and denote by $M_i$ the integral leaf containing $\hat p$ of the distribution $V_i$. Then, $(\hat M,\hat g)$ is locally the direct product $$\hat M=M_0\times M_1\times...\times M_l,\,\,\,\hat g=g_0+g_1+...+g_l$$ and, since $\hat g$ is Ricci flat, each of the metrics $g_1,...,g_l$ is Ricci flat as well ($g_0$ is the flat metric by construction). We recall
[@FedMat Lemma 4 and Lemma 5]\[lem:product\] Let $(\hat M,\hat g)=(M_1,g_1)\times (M_2,g_2)$ be a product of pseudo-Riemannian manifolds $(M_i,g_i)$, $i=1,2$. Then, $(M,g)$ is a local cone if and only if both $(M_1,g_1)$ and $(M_2,g_2)$ are local cones. The cone vector fields $\xi$ of $(\hat M,\hat g)$, $\xi_1$ of $(M_1,g_1)$ and $\xi_2$ of $(M_2,g_2)$ are related by $\xi=\xi_1+\xi_2$.
Let $\xi=\xi_1+\xi_2$ be the orthogonal decomposition of the cone vector field $\xi$ of $(\hat M,\hat g)$ w.r.t. the decomposition $T\hat M=TM_1\oplus TM_2$. For $X_1\in TM_1,X_2\in TM_2$, we obtain $X_i=\hat \nabla_{X_i}\xi=\hat\nabla_{X_i}\xi_1+\hat\nabla_{X_i}\xi_2$. Since $\hat \nabla_{X_i} \xi_1\in TM_1$ and $\hat \nabla_{X_i} \xi_2\in TM_2$, we obtain $\hat \nabla_{X_1}\xi_2=\hat \nabla_{X_2}\xi_1=0$. Hence, $\xi_1$, $\xi_2$ are vector fields on $M_1$ resp. $M_2$ and $\nabla^i\xi_i={\mathrm{Id}}_{TM_i}$, $i=1,2$. Thus, $\xi_1,\xi_2$ are cone vector fields for $(M_1,g_1)$ resp. $(M_2,g_2)$.
Conversely, if $\xi_i$ is a cone vector field for $(M_i,g_i)$, $i=1,2$, then, clearly, $\xi=\xi_1+\xi_2$ is a cone vector field for $(\hat M,\hat g)$.
From Lemma \[lem:product\] we conclude that each $(M_i,g_i)$, $i=1,...,l$, is a nonflat Ricci flat local cone which is indecomposable by construction.
Before we determine the range of the integer $l$ in the formula for the degree of mobility $D(g)$, we introduce some notation. For $i=1,...,l$ let $k_i$ denote the dimension of the space $\mathrm{Par}_i$ of parallel vector fields for $\hat g$ which take values in $V_i$. Obviously, when restricted to the integral leaf $M_i$, each vector field in $\mathrm{Par}_i$ is a parallel vector field on $M_i$ for the metric $g_i$. Since $V_i$ is indecomposable, any linear combination of vector fields in $\mathrm{Par}_i$ must be a null vector, that is, at each point, the values of the vector fields in $\mathrm{Par}_i$ span a totally isotropic subspace of the tangent space. Since the only possible signatures of $\hat g$ are $(0,n+1)$, $(1,n)$, $(n,1)$ or $(n-1,2)$, we therefore have $0\leq k_1+...+k_l\leq 2$. Moreover, since by definition, $k$ is the number of parallel vector fields for $\hat g$, we have $k=\mathrm{dim}\,V_0+k_1+...+k_l$.
To determine the range of $l$, we consider two different case:
*Case 1: Suppose $0\leq k_1+...+k_l\leq 1$.* Note that this is the only case which occurs when the initial metric $g$ is Riemannian (where “initial” means before multiplication with $B\neq 0$) – in this case $\hat g$ cannot have signature $(n-1,2)$ and therefore $k_i<2$ for all $i=1,...,l$. Applying Lemma \[lem:dimRicciflat\], we obtain $\mathrm{dim}\,V_i=\mathrm{dim}\,M_i\geq k_i+5$ for $i=1,...,l$ and therefore $$n+1=\mathrm{dim}\,V_0+\mathrm{dim}\,V_1+...+\mathrm{dim}\,V_l\geq \mathrm{dim}\,V_0+k_1+...+k_l+5l=k+5l.$$ Hence, $1\leq l\leq [\frac{n+1-k}{5}]$. Since there is at least one indecomposable component in the decomposition and this component is at least $5$-dimensional, we obtain $0\leq k\leq \mathrm{dim}\,\hat M-5=n-4$. In particular, this completes the proof of Proposition \[prop:Bnonzero\] in case that $g$ is positive definite.
*Case 2: Suppose $k_1=2$ for the component $(M_1,g_1)$.* In this case $\hat g$ necessarily has signature $(n-1,2)$ and therefore also $g_1$ has signature $(\mathrm{dim}\,V_1-2,2)$. Consequently, the remaining components $g_0,g_2,...,g_l$ are negative definite. In particular, we have $k_i=0$ for $i=2,...,l$ and Lemma \[lem:dimRicciflat\] implies $\mathrm{dim}\,V_i\geq 5$ for $i=2,...,l$. From Example \[ex:counterex\] we have learned that $V_1$ is at least $6$ dimensional. Using this, we obtain $$n+1=\mathrm{dim}\,V_0+\mathrm{dim}\,V_1+...+\mathrm{dim}\,V_l\geq \mathrm{dim}\,V_0+6+5(l-1)=k-1+5l.$$ Hence, $1\leq l\leq [\frac{n+2-k}{5}]$. Since $0\leq \mathrm{dim}\,V_0\leq \mathrm{dim}\,\hat M-6=n-5$ and $k=\mathrm{dim}\,V_0+2$, we obtain $2\leq k\leq n-3$. Comparing this with the first case above, the additional values for $D(g)$ appearing in the second case occur for any $k$ in $2\leq k\leq n-3$ satisfying $k=n-3\mbox{ mod }5$ and for $l= [\frac{n+2-k}{5}]$. This completes the proof of Proposition \[prop:Bnonzero\].
The case when the scalar curvature is zero and $\mu\neq 0$ for at least one solution of {#sec:Bzero}
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In this section, we prove the first direction of Theorem \[thm:main\] for a simply connected Riemannian or Lorentzian Einstein manifold $(M,g)$ such that at least one solution $(L,\Lambda,\mu)$ of with $B=0$ has $\mu\neq 0$.
We reduce the proof locally to Proposition \[prop:Bnonzero\] by applying the following lemmas:
[@FedMat Lemma 11]\[lem:changeofmetric\] Let $(M,g)$ be a pseudo-Riemannian manifold. Assume one of the following:
1. $g$ is Riemannian and at least one solution $(L,\Lambda,\mu)$ of with $B=0$ has $\Lambda\neq 0$.
2. $g$ is Lorentzian and at least one solution $(L,\Lambda,\mu)$ of with $B=0$ has $\mu\neq 0$.
Then, on each open subset with compact closure, there exists a metric $\bar g$ of the same signature as $g$ which is projectively equivalent to $g$ and such that the constant $\bar B$ for the system corresponding to $\bar g$ is nonzero.
\[rem:changeofmetric\] Actually, [@FedMat Lemma 11] only contains the statement for Lorentzian signature. However, under the assumption of $(1)$, one can always construct a solution to such that $\mu\neq 0$ and then the proof of [@FedMat Lemma 11] applies. Indeed, let $(L,\Lambda,0)$ be a solution of (with $B=0$) such that $\Lambda\neq 0$. Let $\lambda$ be a function such that $\Lambda={\mathrm{d}}\lambda$. It is easy to check that the $1$-form $\tilde\Lambda=L(\Lambda^\sharp.,.)-\lambda\Lambda$ satisfies $\nabla \tilde\Lambda=\tilde\mu g$ for the nonzero constant $\tilde \mu=|\Lambda|^2$. Then, $(\frac{1}{\tilde \mu}\tilde \Lambda\odot \tilde \Lambda,\tilde \Lambda,\tilde \mu)$ is a solution to . This construction is in general not possible for Lorentzian metrics, see Section \[sec:Bzeromuzero\].
[@KioMatEinstein Lemma 3 and Corollary 5]\[lem:projequivEinstein\] Let $(M,g)$ be a connected pseudo-Riemannian Einstein manifold and let $\bar g$ be projectively equivalent to $g$ but not affinely equivalent. Then, also $\bar g$ is an Einstein metric.
Clearly, all projectively equivalent metrics have the same degree of mobility. Then, by Lemma \[lem:changeofmetric\], Lemma \[lem:projequivEinstein\] and Proposition \[prop:Bnonzero\], the degree of mobility of the restriction $g|_{U}$ of $g$ to any open simply connected subset $U$ with compact closure is given by one of the values in the list of Theorem \[thm:main\].
The extension “local $\rightarrow$ global” follows now directly from [@FedMat Lemma 12]. Alternatively, we may apply [@MatRos Lemma 10] which is a consequence of the Ambrose-Singer theorem [@AmbSing]:
[@MatRos]\[lem:AmbSing\] Let $\pi:E\rightarrow M$ be a vector bundle with connection $\nabla^E$ over a simply connected $n$-dimensional manifold $M$. Denote by $D(E,\nabla^E)$ the dimension of the space of parallel sections and $E|_{U}$ the restriction of $E$ to an open subset $U$ of $M$.
Let $I$ be a subset of integers. Then, if $D(E|_U,\nabla^E)\in I$ for any ball $U$ (that is, $U$ is homeomorphic to a ball in ${\mathbb{R}}^n$ and has compact closure), then also $D(E,\nabla^E)\in I$.
To explain how to apply Lemma \[lem:AmbSing\] in this situation, it suffices to note that ${\mathcal{A}}(g)$ is isomorphic to the space of sections of a certain vector bundle, parallel w.r.t. a certain connection (see [@EastMat Theorem 3.1]).
In our case the situation is more explicit: ${\mathcal{A}}(g)$ is isomorphic to the space of solutions of the system which can be viewed as the space of sections of the vector bundle $E=S^2T^*M\oplus T^*M\oplus {\mathbb{R}}$ (where the fiber $S^2 T_p^* M$ of $S^2T^*M$ over a point $p\in M$ consists of the symmetric $(0,2)$-tensors on $T_p M$) which are parallel w.r.t. the connection $\nabla^E$ defined by $$\nabla^E_X\left(\begin{array}{c}L\\\Lambda\\\mu\end{array}\right)
=\left(\begin{array}{c}\nabla_X L-X^\flat\odot \Lambda\\
\nabla_X\Lambda-\mu X^\flat-B L(X,.)\\
\nabla_X\mu-2B\Lambda(X)\end{array}\right).$$
This completes the proof of the first direction of Theorem \[thm:main\] under the additional assumption that $B=0$ in but at least one solution has $\mu\neq 0$.
The case when the scalar curvature is zero and all solution of have $\mu=0$ {#sec:Bzeromuzero}
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The goal of this section is to prove
\[prop:Bzeromuzero\] Let $(M,g)$ be a simply connected Ricci flat Lorentzian manifold such that $\mu= 0$ for all solutions $(L,\Lambda,\mu)$ of but $\Lambda\neq 0$ for at least one solution. Then, $D(g)$ is given by $$D(g)=k(k+1)/2 +l,$$ where $1\leq k\leq n-4$ and $2\leq l\leq [\frac{n+1-k}{5}]$.
\[rem:Bzeromuzero\] As explained in Remark \[rem:changeofmetric\], the case $B=-\mathrm{Scal}/n(n-1)=0$ in and $\mu=0$ for all solutions cannot happen if $g$ is Riemannian. This section and Proposition \[prop:Bzeromuzero\] are therefore exclusive for the case of Lorentzian signature.
We proceed in the same way as in [@FedMat Section 6.2] and implement the condition that $g$ is Einstein at the corresponding places.
[@FedMat Lemma 13]\[lem:lambdanull\] Let $(M,g)$ be a simply connected Lorentzian manifold such that all solutions of with $B=0$ have $\mu=0$ and at least one solution $(L,\Lambda,0)$ has $\Lambda\neq 0$. Then, $\Lambda$ is parallel and orthogonal to any other parallel $1$-form. In particular, $|\Lambda|=0$, i.e., $\Lambda$ is a null.
Using Lemma \[lem:lambdanull\], it is straight-forward to show that any other $\bar L\in {\mathcal{A}}(g)$ can be written as $$\bar L=cL+L'$$ for a constant $c$ and a parallel symmetric $(0,2)$-tensor $L'$. Thus, $$\begin{aligned}
D(g)=1+\mathrm{dim}\,\mathrm{Par}^{0,2}(g),\label{eq:degree2}\end{aligned}$$ where $\mathrm{Par}^{0,2}(g)$ denotes the space of parallel symmetric $(0,2)$-tensors for $g$. To find the possible values of $D(g)$ we therefore have to find the possible values of $\mathrm{dim}\,\mathrm{Par}^{0,2}(g)$. To do so, we use a maximal holonomy decomposition $TM=\bigoplus_{i=0}^l V_i$ of $TM$ as in into mutually orthogonal holonomy invariant subbundles. The difference to the procedure in Section \[sec:Bnonzero\] is now that $(M,g)$ itself is not a cone and also the integral leafs $M_i$ corresponding to the parallel distributions $V_i$ do in general not carry the structure of a local cone (although, this is still the case for some components $V_i$ in as we shall explain below). We know by Lemma \[lem:decomp\] that every parallel symmetric $(0,2)$-tensor takes the form , hence, $$\begin{aligned}
\mathrm{dim}\,\mathrm{Par}^{0,2}(g)=\frac{k(k+1)}{2}+l.\label{eq:dimPar}\end{aligned}$$ It remains to determine the range of the integers $k,l$. Since $g$ has Lorentzian signature, precisely one of the metrics $g_0,...,g_l$ (we use the notation of Lemma \[lem:decomp\], that is, $g_i$ is the restriction of $g$ to $V_i$) has Lorentzian signature. The flat metric $g_0$ is Riemannian, otherwise, by irreducibility of $V_1,...,V_l$, the parallel null vector field $\Lambda^\sharp$ must take values in $V_0$. However, since by Lemma \[lem:lambdanull\], $\Lambda^\sharp$ is orthogonal to any parallel vector field, this implies that $g_0$ is degenerate which is a contradiction. Therefore, up to rearranging components, we can suppose that $g_1$ is Lorentzian and $\Lambda^\sharp$ takes values in the subbundle $V_1$. It follows that the dimension of the space of parallel vector fields for $g$ is $$\begin{aligned}
k=\mathrm{dim}\,V_0+1.\label{eq:k}\end{aligned}$$ Since $g$ is Ricci flat, each of the components $(M_1,g_1),...,(M_l,g_l)$ is Ricci flat ($(M_0,g_0)$ is flat by definition). The next step in [@FedMat] is to show that the Riemannian manifolds $(M_2,g_2),...,(M_l,g_l)$ each carry the structure of a local cone. Then, since each $(M_i,g_i)$ for $i\geq 2$ is an irreducible nonflat Ricci flat local cone, Lemma \[lem:dimRicciflat\] implies $$\begin{aligned}
\mathrm{dim}\,V_i\geq 5\mbox{ for }i=2,...,l.\label{eq:dimVi}\end{aligned}$$ It remains to establish a lower bound for the dimension of $V_1$. As shown in [@FedMat] the restriction $L_1$ of $L$ to the manifold $(M_1,g_1)$ is contained in ${\mathcal{A}}(g_1)$ with corresponding $1$-form $\Lambda$ and $(L_1,\Lambda,0)$ satisfies for $g_1$ and constant $B=0$. Also any other solution to for $g_1$ has $\mu=0$. In [@FedMat formula (62)] metrics with such properties have been described locally. We summarize this description and other facts (see [@FedMat Lemma 14, 15 and Corollary 2]) in
\[lem:localclass\] Let $(N,h)$ be a Lorentzian manifold such that all solutions $(L,\Lambda,\mu)$ of the system for $h$ with $B=0$ have $\mu=0$ and let $(L,\Lambda,0)$ be a solution with $\Lambda$ not identically zero. Let $\lambda=\frac{1}{2}\mathrm{trace}\,L^\sharp$ such that $\mathrm{grad}\,\lambda$ coincides with the parallel null vector field $\Lambda^\sharp$. Then we have the following
1. The metric $h$ takes the form $$\begin{aligned}
h=h_0+(\lambda+C-\rho_1)^2 h_1+...+(\lambda+C-\rho_m)^2 h_m\label{eq:doublywarped}\end{aligned}$$ in a neighbourhood of almost every point. Here $(N_0,h_0)$ is a $2$-dimensional Lorentzian manifold such that $\Lambda$ is contained in $TN_0$, $(N_1,h_1),...,(N_m,h_m)$ are Riemannian manifolds where we have $m\geq 2$, and $C$ and $\rho_i$ are certain constants
2. W.r.t. the decomposition $TN=TN_0\oplus...\oplus TN_m$, $L^\sharp$ has block-diagonal form, i.e., $L^\sharp(TN_i)\subseteq TN_i$. Moreover, $L^\sharp|_{TN_i}=\rho_i\mathrm{Id}_{TN_i}$ for $i=1,...,m$ and $L^\sharp|_{TN_0}$ is conjugate to a $2$-dimensional Jordan block with eigenvalue $\lambda+C$ and corresponding eigenvector $\Lambda^\sharp$.
3. If $(N,h)$ is indecomposable, then $\mathrm{dim}\,N_i\geq 2$ for $i=1,...,m$.
Using indecomposability of $(M_1,g_1)$, the last statement of the lemma together with $m\geq 2$ shows $\mathrm{dim}\,V_1\geq 6$. However, since $g_1$ is Ricci flat, we obtain a sharper lower bound as we will show next.
\[lem:curvature\] For $i=0,1,...,m$ let $(N_i,h_i)$ be a pseudo-Riemannian manifold. Consider the product $N=N_0\times N_1\times ...\times N_m$ with metric given by $$h=h_0+f_1^2 h_2+...+f_m^2 h_m.$$ Suppose the nowhere vanishing functions $f_1,...,f_m$ on $M_0$ are of the form $f_i=\lambda+c_i$ for constants $c_i$ and a function $\lambda$ such that $\mathrm{grad}\,\lambda$ is parallel and null.
Let $R$ and $\mathrm{Ric}$ be the curvature tensor resp. Ricci tensor of $h$. Let $X_i,Y_i$ denote vector fields on $N_i$ and let $R^i$ and $\mathrm{Ric}^i$ denote the curvature tensor resp. Ricci tensor of $h_i$ for $i=0,1,...,m$. Then, $$\begin{aligned}
R(X_i,Y_i)=R^i(X_i,Y_i),\,\,\,R(X_j,X_k)=0
\label{eq:curvature}\end{aligned}$$ and $$\begin{aligned}
\mathrm{Ric}(X_i,Y_i)=\mathrm{Ric}^i(X_i,Y_i),\,\,\,\mathrm{Ric}(X_j,X_k)=0
\label{eq:Riccicurvature}\end{aligned}$$ for $i,j,k=0,...,m$, $i\neq j$.
Let $\nabla$ resp. $\nabla^i$ denote the Levi-Civita connection of $h$ resp. $h_i$. Using the Koszul formula $$2h(\nabla_X Y,Z)=Xh(Y,Z)+Yh(X,Z)-Zh(X,Y)$$ $$-h(X,[Y,Z])-h(Y,[X,Z])+h(Z,[X,Y]).$$ and the expression for $h$, we derive the following formulas, relating the Levi-Civita connections $\nabla$ and $\nabla^i$: $$\begin{aligned}
\begin{array}{c}
\nabla_{X_0}Y_0=\nabla^0_{X_0}Y_0,\vspace{1mm}\\
\nabla_{X_0}X_i=\nabla_{X_i}X_0=\frac{{\mathrm{d}}f_i(X_0)}{f_i}X_i\mbox{ for }i=1,...,m,\vspace{1mm}\\
\nabla_{X_i}Y_i=\nabla^i_{X_i}Y_i-h(X_i,Y_i)\frac{\mathrm{grad}\, f_i}{f_i}\mbox{ for }i=1,...,m,\vspace{1mm}\\
\nabla_{X_i}X_j=0\mbox{ for }i=1,...,m, \,\,\,i\neq j.
\end{array}\label{eq:LC}\end{aligned}$$ Evaluating the curvature tensor $R(X,Y)Z=\nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z-\nabla_{[X,Y]}Z$ on the vector fields of various types and using that $h(\mathrm{grad}\,f_i,\mathrm{grad}\,f_j)=|\mathrm{grad}\,\lambda|^2=0$, a straight-forward calculation shows that holds and the formulas follow immediately.
Let us use that the component $(M_1,g_1)$ of $(M,g)$ is Ricci flat. Formula in Lemma \[lem:curvature\] shows that all components $h_0,h_1,...,h_m$ of $g_1=h$ in are Ricci flat. Since $3$-dimensional Ricci flat manifolds are flat and, by construction, $g_1$ is nonflat, formula shows that at least one of the Ricci flat components $h_i$, $i\geq 1$, of $g_1$ in is nonflat and therefore must have dimension $\geq 4$. Since there are at least two components $N_1,N_2$ and $N_0$ is $2$-dimensional, we obtain $\mathrm{dim}\,V_1\geq 8$. We claim that this estimate is still too coarse and that instead we actually have $$\begin{aligned}
\mathrm{dim}\,V_1\geq 10.\label{eq:dimV1}\end{aligned}$$ By indecomposability of $(M_1,g_1)$ this follows from
\[lem:parallelvf\] Let $(N,h)$ be a simply connected Lorentzian manifold such that all solutions of the system for $h$ with $B=0$ have $\mu=0$ and let $(L,\Lambda,0)$ be a solution with $\Lambda$ not identically zero. Suppose the metric $h_m$ in the local expression from Lemma \[lem:localclass\] is flat. Let $r$ be the dimension of $N_m$, or equivalently, the multiplicity of the constant eigenvalue $\rho_m$ of $L^\sharp$.
Then, there exist $r$ parallel vector fields $W_1,...,W_r$ on $N$ such that $W_1,...,W_r,\Lambda$ are linearly independent.
Before proving the lemma, we complete the proof of Proposition \[prop:Bzeromuzero\]. By and the estimates and , we have $$n=\mathrm{dim}\,V_0+\mathrm{dim}\,V_1+\mathrm{dim}\,V_2+...+\mathrm{dim}\,V_l\geq \mathrm{dim}\,V_0+5l+5.$$ Taking into account that $k=\mathrm{dim}\,V_0+1$, this yields $1\leq l\leq [\frac{n+1-k}{5}]-1$. From and , we obtain $D(g)=k(k+1)/2 +l'$ and we have shown that $l'=l+1$ is in the range $2\leq l'\leq [\frac{n+1-k}{5}]$. Finally, the estimate shows $0\leq \mathrm{dim}\,V_0\leq n-5$, hence, $1\leq k\leq n-4$. This proves Proposition \[prop:Bzeromuzero\].
Actually the statement is a generalization of [@FedMat Lemma 15(2)] and we will proceed along the same line of arguments to give a proof of it. We work in the local picture described by Lemma \[lem:localclass\] above. Let $u$ be a function on $N_m$ such that ${\mathrm{d}}u$ is parallel and $|{\mathrm{d}}u|_m=1$, where $|\,.\,|_m$ denotes the length of a vector w.r.t. $h_m$. Consider the vector field $U$ on $N$ such that $h(U,X)=u(X)$ for all $X\in TN$. Then, $$\begin{aligned}
h(U,U)=\frac{1}{(\lambda+C-\rho_m)^2}.\label{eq:normU}\end{aligned}$$ Note also that $\nabla U$ is a $h$-symmetric $(1,1)$-tensor on $TN$ and since $U$ takes values in $TN_m$, we have $(L^\sharp-\rho_m{\mathrm{Id}})(U)=0$ (see Lemma \[lem:localclass\](2)). Taking the covariant derivative of this equation in the direction of a vector $X\in TN$, inserting to replace derivatives of $L^\sharp$ and using $h(\Lambda,U)=0$, we obtain $$\begin{aligned}
(L^\sharp-\rho_m{\mathrm{Id}})\nabla_X U=-h(U,X)\Lambda^\sharp.\label{eq:eigvec}\end{aligned}$$ Contracting this with $Y\in TN$ such that $(L^\sharp-\rho{\mathrm{Id}})Y=0$ and using symmetries of $\nabla U$, we obtain $$(\rho-\rho_m)h(\nabla_Y U,X)=-h(U,X)\Lambda(Y).$$ Recall from Lemma \[lem:localclass\](2) that $L^\sharp(\Lambda^\sharp)=(\lambda+C)\Lambda^\sharp$. Then we have $$\begin{aligned}
\nabla_Y U=0\mbox{ for }Y\in TN_i,\,\,\,i=1,...,m-1,\mbox{ and }Y=\Lambda^\sharp.\label{eq:direc1}\end{aligned}$$ Now let $\tilde \Lambda\in TN_0$ be a vector such that $L(\tilde \Lambda)=(\lambda+C)\tilde \Lambda+\Lambda^\sharp$ (recall that by Lemma \[lem:localclass\](2), $L^\sharp|_{TN_0}$ is a Jordan block). Contracting with $\tilde \Lambda$, a straight-forward calculation yields $$\begin{aligned}
\nabla_{\tilde \Lambda} U=-\frac{\Lambda(\tilde\Lambda)}{\lambda+C-\rho_m}U.\label{eq:direc2}\end{aligned}$$
To finally determine $\nabla U$ on a basis of $TN$, let $V$ be another vector tangent to $N_m$. Since $U=h^{-1}{\mathrm{d}}u=\frac{1}{f_m^2}h_m^{-1}{\mathrm{d}}u$ (where $f_m=\lambda+C-\rho_m$), we have that $f_m^2 U$ is a parallel vector field on $N_m$ and using , we calculate $$2f_m V(f_m)U+f_m^2\nabla_V U=-f_m h(V, U)\Lambda.$$ Hence, since $f_m=\lambda+C-\rho_m$ and $V(f_m)=0$, we obtain $$\begin{aligned}
\nabla_V U=-\frac{1}{\lambda+C-\rho_m}h(V, U)\Lambda^\sharp.\label{eq:direc3}\end{aligned}$$ Now consider the vector field $$W=(\lambda+C-\rho_m)U+u\Lambda^\sharp.$$ By definition, $W$ and $\Lambda^\sharp$ are linearly independent. Using $\Lambda^\sharp=\mathrm{grad}\,\lambda$, $|\Lambda|=0$, $\nabla\Lambda^\sharp=0$ and the formulas , and , it is an easy calculation to show that the covariant derivative of $W$ vanishes in all possible directions, hence, $W$ is parallel and linearly independent of $\Lambda^\sharp$. However, we have defined such a $W$ only in a neighbourhood of almost every point of $N$. Actually, what we have shown above is the existence of parallel vector fields $W_1,...,W_r$, where $r=\mathrm{dim}\,N_m$, defined in a neighbourhood of almost every point, such that $\Lambda^\sharp,W_1,...,W_r$ are linearly independent. To see this, we use that $h_m$ is flat and choose a basis of parallel $1$-forms ${\mathrm{d}}u_1,...,{\mathrm{d}}u_r$ of $N_0$ such that $|{\mathrm{d}}u_i|_m=1$ for $i=1,...,r$. As shown above, the vector fields $W_i=(\lambda+C-\rho_m)U_i+u_i\Lambda^\sharp$, $i=1,...,r$, where $U_i=h^{-1}{\mathrm{d}}u_i$, will satisfy the claim.
Thus, we have defined a distribution $\tilde D=\mathrm{span}\{\Lambda,W_1,...,W_r\}$ of rank $r+1$ on a dense and open subset of $N$. We claim $\tilde D$ extends to a smooth distribution $D$ on the whole $N$. Let $E_i(p)$, $i=1,...,m$, denote the generalized eigenspace of $L$ at $p\in N$ corresponding to the constant eigenvalue $\rho_i$. Then we define $$D_p=\{X\in T_p N:X\perp E_i,\,\,\,i=1,...,m-1,\,\,\,X\perp \Lambda^\sharp\}$$ in points $p\in N$ where $(\lambda+C)(p)\neq \rho_i$, $i=1,...,m-1$, and $$D_p={\mathbb{R}}\cdot \Lambda^\sharp(p)\oplus E_m(p)$$ for $(\lambda+C)(p)\neq \rho_m$. Then, $D=\bigsqcup_{p\in N}D_p$ is a smooth distribution of rank $r+1$ which coincides with the parallel and flat distribution $\tilde D$ on a dense and open subset. Then, $D$ is a parallel and flat subbundle of $TN$. This finishes the proof of the lemma.
Realization of the values of the degree of mobility {#sec:realization}
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In this section, we show that for each $n\geq 3$, the values from Theorem \[thm:main\] can be realized as the degree of mobility of an $n$-dimensional Riemannian resp. Lorentzian Einstein metric which admits a projectively equivalent metric that is not affinely equivalent. This will complete the proof of Theorem \[thm:main\]. We may suppose that $n\geq 5$ since the values of Theorem \[thm:main\] for $n=3,4$ are realized by the simply connected spaces of constant sectional curvature.
We will proceed by constructing a Ricci flat local cone $(\hat M,\hat g)$ of suitable signature and of dimension $n+1$ such that the space of parallel symmetric $(0,2)$-tensors of $\hat g$ has dimension $k(k+1)/2+l$ , where the range of integers $k,l$ is as in Theorem \[thm:main\]. Once such a manifold is constructed, we have by Lemma \[lem:curvfacts\] and Lemma \[lem:localcone\] that $(\hat M,\hat g)$ is (locally) the metric cone over a $n$-dimensional Einstein manifold and, in view of Lemma \[lem:isom\], the degree of mobility of $(M,g)$ is given by $k(k+1)/2+l$. Moreover, as can be seen directly from the second and third equations in , any $L\in \mathcal{A}(g)$ that is parallel (that is, we have $\Lambda=0$ for the corresponding vector field) is necessarily proportional to the identity. In particular, $(M,g)$ admits a metric projectively equivalent to $g$ and not affinely equivalent to it.
The Ricci flat cone $(\hat M,\hat g)$ will be constructed by taking a direct product of cones. It is therefore useful to note the following: for any dimension $d+1\geq 5$, there is a Ricci flat nonflat indecomposable cone of any signature $(r,s+1)$ (where $d=r+s$). By Lemma \[lem:curvfacts\], such a cone is obtained by taking the metric cone over a generic $d$-dimensional Einstein metric of scalar curvature $d(d-1)$ and signature $(r,s)$.
We will consider two different cases corresponding respectively to the values from the list of Theorem \[thm:main\] attained by Riemannian *and* Lorentzian Einstein metrics and to the special values only obtained by Lorentzian Einstein metrics.
*1. Case: Let $0\leq k\leq n-4$ and $1\leq l\leq [\frac{n+1-k}{5}]$*. Let $M_0={\mathbb{R}}^k$ with standard flat euclidean metric $g_0$. Clearly, $(M_0,g_0)$ is a cone over the $k-1$-dimensional sphere with standard metric. Since $l\leq [(n+1-k)/5]$, there exist numbers $d_1,...,d_l$ such that $d_i\geq 5$ for $i=1,...,l$ and $d_1+...+d_l=n+1-k$. For each $i=1,...,l$, we take $d_i$-dimensional nonflat Ricci flat indecomposable cones $(M_i,g_i)$ such that $g_1,...,g_{l-1}$ are positive definite. If we want $g$ to be Riemannian, we also let $g_l$ be positive definite. If we want $g$ to be Lorentzian, we let $g_l$ be the metric cone over a Lorentzian Einstein metric. Then, the direct product $$(\hat M,\hat g)=(M_0,g_0)\times ( M_1, g_1)\times ...\times ( M_l, g_l)$$ has Lorentzian signature and the space of parallel symmetric $(0,2)$-tensors has dimension $k(k+1)/2+l$. By Lemma \[lem:isom\], Lemma \[lem:curvfacts\] and Lemma \[lem:localcone\], $(\hat M,\hat g)$ is (locally) the metric cone over a $n$-dimensional Einstein manifold $(M,g)$ with degree of mobility $D(g)=k(k+1)/2+l$.
*2. Case: Let $2\leq k\leq n-3$, $k=n-3$ mod $5$ and $l= [\frac{n+2-k}{5}]$*. We let $M_0={\mathbb{R}}^{k-2}$ with standard flat euclidean metric $g_0$. Since $l-1= [\frac{n-3-k}{5}]$, we find numbers $d_1,...,d_{l-1}\geq 5$ such that $d_1+...+d_{l-1}=n-3-k$. Let $(M_i,g_i)$, $i=1,...,l-1$, be $d_i$-dimensional nonflat Ricci flat indecomposable cones of Riemannian signature. Let $(M_l,g_l)$ be the $6$-dimensional cone of signature $(4,2)$ from Example \[ex:counterex\]. Consider the $n+1$-dimensional manifold $$(\hat M,\hat g)=(M_0,-g_0)\times ( M_1, -g_1)\times ...\times ( M_{l-1}, -g_{l-1})\times ( M_l, g_l).$$ of signature $(n-1,2)$. By construction, it has the property that the space of parallel symmetric $(0,2)$-tensors has dimension $k(k+1)/2+l$. For $i=0,...,l$ let us write $(M_i,g_i)$ in the form $M_i={\mathbb{R}}_{>0}\times N_i$ and $g_i={\mathrm{d}}r_i^2+r_i^2 h_i$. We consider the subset $\hat M^0=\{-r_0^2-r_1^2-...-r_{l-1}^2+r_l^2>0\}\subseteq \hat M$ of points where the cone vector field $\xi=\sum_{i=0}^l \xi_i$ of $(\hat M,\hat g)$ ($\xi_i=r_i\partial_{r_i}$ denoting the cone vector fields for $g_i$) has the property that $\hat g(\xi,\xi)>0$. As above, we have that, locally, in a neighborhood of almost every point of $\hat M^0$, $\hat g$ is the metric cone over an Einstein metric $g$ of signature $(n-1,1)$ such that $D(g)=k(k+1)/2+l$.
Proof of Theorem \[thm:proj\] {#sec:proofproj}
=============================
In this section, we give the proof of Theorems \[thm:proj\] and \[thm:proj2\]. Let $(M,g)$ be an $n$-dimensional pseudo-Riemannian manifold and let $v$ be a projective vector field for $g$. It is straight-forward to show that the symmetric $(0,2)$-tensor $$\begin{aligned}
\varphi(v):=\mathcal{L}_v g-\frac{1}{n+1}\mathrm{trace}(\mathcal{L}_v g)^\sharp\label{eq:phiv}\end{aligned}$$ is a solution of , hence, we have a linear mapping $\varphi:\mathfrak{p}(g)\rightarrow \mathcal{A}(g)$, where $\mathfrak{p}(g)$ denotes the Lie algebra of projective vector fields. Using , one easily concludes (see [@FedMat Lemma 16]) that $\varphi(v)$ is proportional to the metric $g$, if and only if $v$ is a homothety (that is, $\mathcal{L}_v g=cg$ for some constant $c$). Then, denoting by $\mathfrak{h}(g)$ the Lie algebra of homotheties of $g$, we obtain an induced linear injection of quotient spaces $$\begin{aligned}
\varphi:\mathfrak{p}(g)/\mathfrak{h}(g)\rightarrow \mathcal{A}(g)/{\mathbb{R}}\cdot g,\label{eq:injective}\end{aligned}$$ in particular, $$\begin{aligned}
\mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{h}(g)\right)\leq D(g)-1.\label{eq:ineq}\end{aligned}$$
Let $g$ be an Einstein metric and assume moreover, that there exists a nonparallel $L\in {\mathcal{A}}(g)$. By Theorem , the degree of mobility $D(g)$ of $g$ equals the dimension of the space of solutions of . As in the proof of Theorem \[thm:main\], we have to consider different cases according to value of the scalar curvature of $g$.
The case of nonzero scalar curvature and the realization part of Theorem \[thm:proj\] {#sec21}
-------------------------------------------------------------------------------------
Let us prove Theorem \[thm:proj\] under the assumption that the scalar curvature of $g$ is nonzero (see [@FedMat Section 8.3] for details): using that the constant $B=-\mathrm{Scal}/n(n-1)$ in is nonzero, one shows that any homothety for $g$ is actually a Killing vector field, hence, $\mathfrak{h}(g)$ coincides with $\mathfrak{i}(g)$, the Lie algebra of Killing vector fields of $g$. Using the equations from the system it is straight-forward to show that the injective mapping $\varphi$ in is actually an isomorphism, hence, $\mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)= D(g)-1$. Applying Theorem \[thm:main\] to obtain the values for $D(g)$, we obtain the corresponding values for the dimension of the space $\mathfrak{p}(g)/\mathfrak{i}(g)$ of essential projective vector fields from Theorem \[thm:proj\].
The realization part of Theorem \[thm:main\] also shows that each number from the list of Theorem \[thm:proj\] can actually be realized as the dimension of the space of essential projective vector fields for a certain Riemannian resp. Lorentzian Einstein metric. This proves the realization part of Theorem \[thm:proj\].
Let us turn to the prove of Theorem \[thm:proj2\] in case of nonzero scalar curvature. Let $g$ be an Einstein metric of arbitrary signature and with nonzero scalar curvature which admits a projectively equivalent metric that is not affinely equivalent. Let $(L,\Lambda,\mu)$ be a solution of such that $\Lambda\neq 0$. It is wellknown that for $B\neq 0$, $\Lambda^\sharp$ is an essential projective vector field for $g$ which proves Theorem \[thm:proj2\]. For completeness let us show how to verify this fact: we have $$\mathcal{L}_{\Lambda^\sharp}g=2\nabla\Lambda=2\mu g+2BL,$$ hence, $$\mathrm{trace}(\mathcal{L}_{\Lambda^\sharp}g)^\sharp=2n\mu +4B\lambda,$$ where $\lambda=\frac{1}{2}\mathrm{trace}\,L^\sharp$. Since ${\mathrm{d}}\lambda=\Lambda$ and ${\mathrm{d}}\mu=2B\Lambda$, we have that $\mu-2B\lambda$ is equal to a constant. Using this, we obtain $$\begin{aligned}
\mathcal{L}_{\Lambda^\sharp} g-\frac{1}{n+1}\mathrm{trace}(\mathcal{L}_{\Lambda^\sharp} g)^\sharp
=2BL-C g\in {\mathcal{A}}(g),\label{eq:splitting}\end{aligned}$$ where $C$ is a certain constant. This shows that $\Lambda^\sharp$ is an essential projective vector field (compare ) and proves Theorem \[thm:proj2\] for nonzero scalar curvature.
\[rem:essprojvf\] We see from that the mapping $$s:{\mathcal{A}}(g)/{\mathbb{R}}\cdot g\rightarrow \mathfrak{p}(g)/\mathfrak{i}(g),$$ defined by sending $L\in {\mathcal{A}}(g)$ to the corresponding vector field $\frac{1}{2B}\Lambda^\sharp$, is a splitting of the exact sequence $$0\rightarrow \mathfrak{i}(g)\hookrightarrow \mathfrak{p}(g)\overset{\varphi}{\rightarrow }{\mathcal{A}}(g)/{\mathbb{R}}\cdot g,$$ that is $\varphi\circ s={\mathrm{Id}}$. In particular, the space of essential projective vector fields $\mathfrak{p}(g)/\mathfrak{i}(g)$ can be identified with a subspace of $\mathfrak{p}(g)$ (which is not a subalgebra) and each projective vector field for $g$ is of the form $\Lambda+K$, where $K$ is a Killing vector field.
The case of zero scalar curvature and $\mu\neq 0$ for at least one solution of
-------------------------------------------------------------------------------
The proof of Theorem \[thm:proj\] under the assumption that $B=-\mathrm{Scal}/n(n-1)=0$ in the system and at least one solution has $\mu\neq 0$ can be traced back to the case $B\neq 0$ treated in the previous section. We first recall some invariance properties.
\[lem:invariance\] We have $\mathrm{dim}(\mathfrak{p}(g)/\mathfrak{i}(g))=\mathrm{dim}(\mathfrak{p}(\bar g)/\mathfrak{i}(\bar g))$ for any pair of projectively equivalent metrics $g,\bar g$.
By definition of a projective vector field, we have $\mathrm{dim}(\mathfrak{p}(g))=\mathrm{dim}(\mathfrak{p}(\bar g))$ On the other hand, since the defining equation for a Killing vector field is projectively invariant (when we view it as an equation on weighted $1$-forms, see [@East]), we also have $\mathrm{dim}(\mathfrak{i}(g)=\mathrm{dim}(\mathfrak{i}(\bar g))$ and the claim follows.
By Lemma \[lem:changeofmetric\], on each open simply connected subset $U$ of $M$ with compact closure, there exists a metric $\bar g$ having the same signature as $g$ and being projectively equivalent to $g$ such that $\bar B\neq 0$ for the corresponding constant in the system for $\bar g$. By Lemma \[lem:projequivEinstein\], also $\bar g$ is an Einstein metric. It follows from Lemma \[lem:invariance\] and the results of Section \[sec21\] that for each simply connected open subset $U$ with compact closure, $\mathrm{dim}(\mathfrak{p}(g|_U)/\mathfrak{i}(g|_U))$ is given by one of the values from the list of Theorem \[thm:proj\]. However, it is a classical fact that Killing vector fields can be viewed equivalently as parallel sections on a certain vector bundle. The same is true for the projective vector fields of $g$ (since they are the symmetries of the projective geometry determined by the Levi-Civita connection of $g$ [@CapMel; @CapMel2; @Melnick] and general facts about parabolic (projective) geometries assure the existence of a prolongation connection [@Hammerl]). Then, the proof of Theorem \[thm:proj\] under the assumptions $B=0$ but $\mu\neq 0$ for at least one solution of follows from a standard application of the Ambrose-Singer theorem [@AmbSing], see also Lemma \[lem:AmbSing\] and its proof in [@MatRos Lemma 10].
In the same way one proves Theorem \[thm:proj2\] for an Einstein metric of arbitrary signature with vanishing scalar curvature which admits a solution $(L,\Lambda,\mu)$ of such that $\mu\neq 0$: arguing as above (using Lemma \[lem:changeofmetric\] and Lemma \[lem:projequivEinstein\]), the already proven part of Theorem \[thm:proj2\] for nonzero scalar curvature (see Section \[sec21\]) implies that the restriction $g|_U$ of $g$ to any open simply connected subset $U$ with compact closure has $\mathrm{dim}(\mathfrak{p}(g|_U)/\mathfrak{i}(g|_U))\geq 1$, hence, admits an essential projective vector field. A standard application of the Ambrose-Singer theorem yields the desired result for $g$.
The case of zero scalar curvature and $\mu= 0$ for all solutions of
--------------------------------------------------------------------
Let $(M,g)$ be a simply connected Lorentzian manifold such that every solution of the system with $B=0$ has $\mu=0$ and $\Lambda\neq 0$ for at least one solution (recall from Remark \[rem:Bzeromuzero\] that the situation under consideration is exclusive for Lorentzian signature). By [@FedMat Corollary 3], we have that $\mathfrak{p}(g)=\mathfrak{i}(g)$. Thus, $\mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)\leq D(g)-1$ by . It is shown in [@FedMat Section 8.4.2] that we also have $D(g)-2\leq \mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)$, hence $$D(g)-2\leq \mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)\leq D(g)-1.$$ Using Proposition \[prop:Bzeromuzero\], we obtain $$\frac{k(k+1)}{2}+l'-2\leq \mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)\leq \frac{k(k+1)}{2}+l'-1,$$ where $1\leq k\leq n-4$ and $2\leq l'\leq [\frac{n+1-k}{5}]$. Thus, $\mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)=k(k+1)/2+l-1$,where $l=l'$ or $l=l'-1$. Then, $\mathrm{dim}\left(\mathfrak{p}(g)/\mathfrak{i}(g)\right)=k(k+1)/2+l-1$, where $1\leq l\leq [\frac{n+1-k}{5}]$. This proves Theorem \[thm:proj\] under the assumptions $B=0$ and $\mu=0$ for all solutions of .
Finally, let us prove Theorem \[thm:proj2\] for an Einstein metric of arbitrary signature with vanishing scalar curvature such that $\mu=0$ for every solution of but $\Lambda\neq 0$ for at least one solution $(L,\lambda,0)$. Let $\lambda=\frac{1}{2}\mathrm{trace}(L^\sharp)$ such that ${\mathrm{d}}\lambda=\Lambda$. Then, since $\Lambda$ is parallel, $\nabla v^\flat=\Lambda\otimes \Lambda$ for the vector field $v=\lambda\Lambda^\sharp$, hence, $$\mathcal{L}_v g-\frac{1}{n+1}\mathrm{trace}(\mathcal{L}_v g)^\sharp g=2\Lambda\otimes \Lambda-\frac{2g(\Lambda,\Lambda)}{n+1} g.$$ Since $g(\Lambda,\Lambda)$ is a constant, this symmetric $(0,2)$-tensor is clearly contained in ${\mathcal{A}}(g)$. It follows that $v$ is a projective vector field. Moreover, $v$ is essential since it is not an isometry (thought, $v$ is an affine vector field).
**Acknowledgements.** {#acknowledgements. .unnumbered}
---------------------
We thank Deutsche Forschungsgemeinschaft (Research training group 1523 — Quantum and Gravitational Fields) and FSU Jena for partial financial support.
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[^1]: The notions “geodesically equivalent” or “projectively related” are also common.
[^2]: By definition, simply connectedness implies connectedness.
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---
abstract: 'We describe the Einstein Toolkit, a community-driven, freely accessible computational infrastructure intended for use in numerical relativity, relativistic astrophysics, and other applications. The Toolkit, developed by a collaboration involving researchers from multiple institutions around the world, combines a core set of components needed to simulate astrophysical objects such as black holes, compact objects, and collapsing stars, as well as a full suite of analysis tools. The Einstein Toolkit is currently based on the Cactus Framework for high-performance computing and the Carpet adaptive mesh refinement driver. It implements spacetime evolution via the BSSN evolution system and general-relativistic hydrodynamics in a finite-volume discretization. The toolkit is under continuous development and contains many new code components that have been publicly released for the first time and are described in this article. We discuss the motivation behind the release of the toolkit, the philosophy underlying its development, and the goals of the project. A summary of the implemented numerical techniques is included, as are results of numerical test covering a variety of sample astrophysical problems.'
address:
- '$^1$ Center for Computation & Technology, Louisiana State University, Baton Rouge, LA, USA'
- '$^2$ Center for Computational Relativity and Gravitation, School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, USA'
- '$^3$ Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Golm, Germany'
- '$^4$ Center for Relativistic Astrophysics, School of Physics, Georgia Institute of Technology, Atlanta, GA, USA'
- '$^5$ TAPIR, California Institute of Technology, Pasadena, CA, USA'
- '$^6$ Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Japan'
- '$^7$ Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada'
- '$^8$ Department of Physics, University of Guelph, Guelph, ON, Canada'
- '$^9$ Department of Computer Science, Louisiana State University, Baton Rouge, LA, USA'
- '$^{10}$ National Science Foundation, USA'
author:
- 'Frank Löffler $^1$, Joshua Faber $^2$, Eloisa Bentivegna $^3$, Tanja Bode $^4$, Peter Diener $^1$, Roland Haas $^{5,4}$, Ian Hinder $^3$, Bruno C. Mundim $^2$, Christian D. Ott $^{5,1,6}$, Erik Schnetter $^{7,8,1}$, Gabrielle Allen $^{1,9,10}$, Manuela Campanelli $^2$ and Pablo Laguna $^4$'
bibliography:
- 'manifest/einsteintoolkit.bib'
title: 'The Einstein Toolkit: A Community Computational Infrastructure for Relativistic Astrophysics'
---
Introduction
============
Scientific progress in the field of numerical relativity has always been closely tied to the availability and ease-of-use of enabling software and computational infrastructure. This document describes the Einstein Toolkit, which provides such an infrastructure, developed openly and made available freely with grant support from the National Science Foundation.
Now is a particularly exciting time for numerical relativity and relativistic astrophysics, with major advances having been achieved in the study of astrophysical systems containing black holes (BHs) and neutron stars (NSs). The first fully general relativistic (GR) simulations of merging NS-NS binaries were reported in 1999, with further advances over the next few years [@Shibata:1999wm; @Shibata:2002jb; @Shibata:2003ga; @Shibata:2005ss; @Shibata:2006nm]. However, systems containing BHs proved much more difficult to evolve numerically until 2005. That year, computational breakthroughs were made following the development of a generalized harmonic formulation [@Pretorius:2005gq] and then a “moving puncture” approach [@Campanelli:2005dd; @Baker:2005vv] in the BSSN (Baumgarte-Shapiro-Shibata-Nakamura) formalism [@Shibata:1995we; @Baumgarte:1998te] that lead to the first stable long-term evolutions of moving single and multiple BH systems. These results quickly transformed the field which was now able to effectively evolve the Einstein field equations for coalescing BH-BH binaries and other systems containing moving BHs, including merging BH-NS binaries.
These breakthroughs had direct relevance to astrophysics, and enabled exciting new results on recoil velocities from BH-BH mergers (e.g, [@Baker:2006vn; @Campanelli:2007ew; @HolleyBockelmann:2007eh; @Pollney:2007ss; @Lousto:2007db; @Lousto:2008dn] and references therein), post-Newtonian (PN) and numerical waveform comparisons and waveform template generation (e.g., and references therein), comparisons between numerical waveforms [@Baker:2006yw; @Baker:2007fb], determination of the spin of the remnant BH formed in BH-BH mergers (e.g, [@Campanelli:2006uy; @Campanelli:2006fg; @Campanelli:2006fy; @Herrmann:2007ex; @Rezzolla:2007rz; @Berti:2007nw] and references therein), and studies of eccentric BH-BH binaries [@Pretorius:2007jn; @Sperhake:2007gu; @Hinder:2007qu; @Grigsby:2007fu; @Pfeiffer:2007yz; @Stephens:2011as].
Meanwhile, general relativistic magneto-hydrodynamics (GRMHD) on fixed background spacetimes has been successful in multi-dimensional settings since the mid-1990s, focusing on BH accretion processes and relativistic jet production and evolution (see [@Font:2008aa] for a review of the numerical formalism and [@Hawley2009apss] for a review of work on disk and jet models). GRMHD coupled with curvature evolution, on the other hand, which is crucial for modeling large-scale bulk dynamics in compact binary star coalescence or single-star collapse scenarios, has started to produce astrophysically interesting results only in the past $\sim 3-5$ years, enabled primarily by the availability of long-term stable curvature evolution systems as well as improved GRMHD algorithms (see [@Font:2008aa] for a review). In addition to these developments, substantial progress has been made in importing more physically motivated equations of state (EOS), including tabulated versions (e.g., [@Pandharipande:1989hn; @Douchin:2001sv; @Akmal:1998cf]) and temperature-dependent models (e.g., [@Shen:1998by; @Shen:1998gq; @Lattimer:1991nc]). Some codes have also begun to incorporate microphysical effects of neutrino emission and deleptonization [@Sekiguchi:2011zd; @O'Connor:2009vw].
Many of the successful techniques used to evolve BH-BH binaries have proven to be equally applicable to merging NS-NS [@Anderson:2007kz; @Anderson:2008zp; @Baiotti:2008ra; @Baiotti:2009gk; @Baiotti:2010xh; @Baiotti:2011am; @Bernuzzi:2011aq; @Giacomazzo:2009mp; @Giacomazzo:2010bx; @Gold:2011df; @Hotokezaka:2011dh; @Kiuchi:2009jt; @Kiuchi:2010ze; @Liu:2008xy; @Rezzolla:2010fd; @Rezzolla:2011da; @Sekiguchi:2011zd; @Sekiguchi:2011mc; @Thierfelder:2011yi; @Yamamoto:2008js] and BH-NS [@Chawla:2010sw; @Duez:2008rb; @Duez:2009yy; @Etienne:2007jg; @Etienne:2008re; @Foucart:2010eq; @Foucart:2011mz; @Kyutoku:2010zd; @Kyutoku:2011vz; @Lackey:2011vz; @Loffler:2006nu; @Shibata:2006bs; @Shibata:2006ks; @Shibata:2007zm; @Shibata:2009cn; @Shibata:2010zz; @Stephens:2011as; @Yamamoto:2008js] binaries (for reviews, see also [@Faber:2009zz; @Duez:2009yz]), allowing for further investigations into the former and the first full GR simulations of the latter. All recent results use either the general harmonic formalism or the BSSN formalism in the “moving puncture” gauge. Nearly all include some form of adaptive mesh refinement, since unigrid models cannot produce accurate long-term evolutions without requiring exorbitant computational resources, though some BH-NS simulations have been performed with a pseudospectral code [@Duez:2008rb; @Duez:2009yy; @Foucart:2010eq; @Foucart:2011mz]. Many groups’ codes now include GRMHD (used widely for NS-NS mergers, and for BH-NS mergers in [@Chawla:2010sw], and some include microphysical effects as well (e.g., [@Duez:2009yy; @Sekiguchi:2011zd; @Sekiguchi:2011mc]).
In addition to studying binary mergers, numerical relativity is a necessary element for understanding stellar collapse and dynamical instabilities in NSs. GRHD has been used to study, among many other applications, massive stars collapsing to protoneutron stars [@Ott:2006eu; @Ott:2006eh; @Shibata:2004kb], the collapse of rotating, hypermassive NSs to BHs in 2D and 3D (see, e.g., [@Shibata:2006hr; @Shibata:1999yx; @Duez:2005sf; @Duez:2005cj; @Baiotti:2004wn; @Baiotti:2005vi; @Baiotti:2006wn]), and non-axisymmetric instabilities in rapidly rotating polytropic NS models [@Shibata:1999yx; @Baiotti:2006wn; @Manca:2007ca].
Simultaneously with the advances in both our physical understanding of relativistic dynamics and the numerical techniques required to study them, a set of general computational tools and libraries has been developed with the aim of providing a computational core that can enable new science, broaden the community, facilitate collaborative and interdisciplinary research, promote software reuse and take advantage of emerging petascale computers and advanced cyberinfrastructure: the Cactus computational toolkit [@Cactuscode:web]. Although the development of Cactus was driven directly from the numerical relativity community, it was developed in collaboration with computer scientists and other computational fields to facilitate the incorporation of innovations in computational theory and technology.
This success prompted usage of the [Cactus]{} computational toolkit in other areas, such as ocean forecast models [@Djikstra2005] and chemical reaction simulations [@Camarda2001]. At the same time, the growing number of results in numerical relativity increased the need for commonly available utilities such as comparison and analysis tools, typically those specifically designed for astrophysical problems. Including them within the [Cactus]{} computational toolkit was not felt to fit within its rapidly expanding scope. This triggered the creation of the Einstein Toolkit [@EinsteinToolkit:web]. Large parts of the Einstein toolkit presently do make use of the [Cactus]{} toolkit, but this is not an requirement, and other contributions are welcome, encouraged and have been accepted in the past.
Requirements
============
Scientific
----------
While the aforementioned studies collectively represent breakthrough simulations that have significantly advanced the modeling of relativistic astrophysical systems, all simulations are presently missing one or more critical physical ingredients and are lacking the numerical precision to accurately and realistically model the large-scale and small-scale dynamics of their target systems simultaneously.
One of the aims of the Einstein Toolkit is to provide or extend some of these missing ingredients in the course of its development. Over the past three years, routines have been added to the code to allow for a wider range of initial data choices, to allow for multithreading in hydrodynamic evolutions, and to refine the [Carpet]{} adaptive mesh refinement driver. Looking forward, three possible additions to future releases are the inclusion of magnetic fields into the dynamics via an ideal MHD treatment, more physical nuclear matter equations of state (EOSs) including the ability to model finite-temperature effects, and higher-order numerical techniques. All of these are under active development, with MHD and finite-temperature evolution code already available, though not completely documented, within the public toolkit releases, and will be made available once they are thoroughly tested and validated against known results.
Academic and Social
-------------------
A primary concern for research groups is securing reliable funding to support graduate students and postdoctoral researchers. This is easier to achieve if it can be shown that scientific goals can be attacked directly with fewer potential infrastructure problems, one of the goals of the Einstein Toolkit.
While the Einstein Toolkit does have a large group of users, many of them do not directly collaborate on science problems, and some compete. However, many groups agree that sharing the development of the underlying infrastructure is mutually beneficial for every group and the wider community as well. This is achieved by lifting off the research groups’ shoulders much of the otherwise necessary burden of creating such an infrastructure, while at the same time increasing the amount of code review and thus, code quality. In addition, the Einstein Toolkit provides computer scientists an ideal platform to perform state-of-the-art research, which directly benefits research in other areas of science and provides an immediate application of their research.
Design and Strategy
===================
The mechanisms for the development and support of the Einstein Toolkit are designed to be open, transparent and community-driven. The complete source code, documentation and tools included in the Einstein Toolkit are distributed under open-source licenses. The Einstein Toolkit maintains a version control system ([svn.einsteintoolkit.org]{}) with open access that contains software supported by the Einstein Toolkit, the toolkit web pages, and documentation. An open wiki for documentation ([docs.einsteintoollkit.org]{}) has been established where the community can contribute either anonymously or through personal authentication. Almost all discussions about the toolkit take place on an open mail list ([[email protected]]{}). The regular weekly meetings for the Einstein Toolkit are open and the community is invited to participate. Meeting minutes are recorded and publicly available as well. The Einstein Toolkit blog requires users to first request a login, but then allows for posting at will. Any user can post comments to entries already on the blog. The community makes heavy use of an issue tracking system ([trac.einsteintoolkit.org]{}), with submissions also open to the public.
Despite this open design, some actions naturally have to be restricted to a smaller group of maintainers. This is true for administrative tasks like the setup and maintenance of the services themselves, or to avoid large amounts of spam. One of the most important tasks of an Einstein Toolkit maintainer is to review and apply patches sent by users in order to ensure a high software quality level. Every substantial change or addition to the toolkit must be reviewed by another Einstein Toolkit maintainer, and is generally open for discussion on the users mailing list. This convention, though not being technically enforced, works well in practice and promotes active development.
Core Technologies
=================
The Einstein Toolkit modules center around a set of core modules that provide basic functionality to create, deploy and manage a numerical simulation starting with code generation all to way to archiving of simulation results: (i) the [Cactus]{} framework “flesh” provides the underlying infrastructure to build complex simulation codes out of independently developed modules and facilities communication between these modules. (ii) the adaptive mesh refinement driver, [Carpet]{}, is build on top of [Cactus]{} and provides problem independent adaptive mesh refinement support for simulations that need to resolve physics on length scales differing by many orders of magnitude, while relieving the scientist of the need to worry about internal details of the mesh refinement driver. (iii) [Kranc]{}, which generates code in a computer language from a high-level description in Mathematica and (iv) the Simulation Factory, which provides a uniform, high-level interface to common operations, such as submission and restart of jobs, for a large number of compute clusters.
Cactus Framework
----------------
The [Cactus]{} Framework [@Cactuscode:web; @Goodale:2002a; @CactusUsersGuide:web] is an open source, modular, portable programming environment for collaborative HPC computing primarily developed at Louisiana State University, which originated at the Albert Einstein Institute and also has roots at the National Center for Supercomputing Applications (see, e.g., [@Anninos:1995am; @Anninos:1996ai; @Seidel:1999ey] for historical reviews). The [Cactus]{} computational toolkit consists of general modules which provide parallel drivers, coordinates, boundary conditions, interpolators, reduction operators, and efficient I/O in different data formats. Generic interfaces make it possible to use external packages and improved modules which are made immediately available to users.
The structure of the [Cactus]{} framework is completely modular, with only a very small core (the “flesh”) which provides the interfaces between modules both at compile- and run-time. The [Cactus]{} modules (called “thorns”) may (and typically do) specify inter-module dependencies, e.g., to share or extend configuration information, common variables, or runtime parameters. Modules compiled into an executable can remain dormant at run-time. This usage of modules and a common interface between them enables researchers to 1) easily use modules written by others without the need to understand all details of their implementation and 2) write their own modules without the need to change the source code of other parts of a simulation in the (supported) programming language of their choice. The number of active modules within a typical [Cactus]{} simulation ranges from tens to hundreds and often has an extensive set of inter-module dependencies.
The [Cactus]{} Framework was developed originally by the numerical relativity community, and although it is now a general component framework that supports different application domains, its core user group continues to be comprised of numerical relativists. It is not surprising therefore, that one of the science modules provided in the Einstein Toolkit is a set of state of the art modules to simulate binary black hole mergers. All modules to simulate and analyze the data are provided out of the box. This set of modules also provides a way of testing the Einstein Toolkit modules in a production type simulation rather than synthetic test cases. Some of these modules have been developed specifically for the Einstein Toolkit while others are modules used in previous publications and have been contributed to the toolkit. In these cases the Einstein Toolkit provides documentation and best practice guidelines for the contributed modules.
Adaptive Mesh Refinement
------------------------
In [Cactus]{}, infrastructure capabilities such as memory management, parallelization, time evolution, mesh refinement, and I/O are delegated to a set of special *driver* components. This helps separate physics code from infrastructure code; in fact, a typical physics component (implementing, e.g., the Einstein or relativistic MHD equations) does not contain any code or subroutine calls having to do with parallelization, time evolution, or mesh refinement. The information provided in the interface declarations of the individual components allows a highly efficient execution of the combined program.
The Einstein Toolkit offers two drivers, [`PUGH`]{} and [Carpet]{}. [`PUGH`]{} provides domains consisting of a uniform grid with Cartesian topology, and is highly scalable (up to more than 130,000 [@Cactuscode:BlueGene:web].) [Carpet]{} [@Schnetter:2003rb; @Schnetter:2006pg; @CarpetCode:web] provides multi-block methods and adaptive mesh refinement (AMR). Multi-block methods cover the domain with a set of (possibly distorted) blocks that exchange boundary information via techniques such as interpolation or penalty methods.[^1] The AMR capabilities employ the standard Berger-Oliger algorithm [@Berger:1984zza] with subcycling in time.
AMR implies that resolution in the simulation domain is dynamically adapted to the current state of the simulation, i.e., regions that require a higher resolution are covered with blocks with a finer grid (typically by a factor of two); these are called *refined levels*. Finer grids can be also recursively refined. At regular intervals, the resolution requirements in the simulation are re-evaluated, and the grid hierarchy is updated; this step is called *regridding*.
Since a finer grid spacing also requires smaller time steps for hyperbolic problems, the finer grids perform multiple time steps for each coarse grid time step, leading to a recursive time evolution pattern that is typical for Berger-Oliger AMR. If a simulation uses 11 levels, then the resolutions (both in space and time) of the the coarsest and finest levels differ by a factor of $2^{11-1}=1024$. This non-uniform time stepping leads to a certain complexity that is also handled by the [Carpet]{} driver; for example, applying boundary conditions to a fine level requires interpolation in space and time from a coarser level. Outputting the solution at a time in between coarse grid time steps also requires interpolation. These parallel interpolation operations are implemented efficiently in [Carpet]{} and are applied automatically as specified in the execution schedule, i.e. without requiring function calls in user code. Figure \[fig:carpet-details\] describes some details of the Berger-Oliger time stepping algorithm; more details are described in [@Schnetter:2003rb].
![Berger-Oliger time stepping details, showing a coarse and a fine grid, with time advancing upwards. **Left:** Time stepping algorithm. First the coarse grid takes a large time step, then the refined grid takes two smaller steps. The fine grid solution is then injected into the coarse grid where the grids overlap. **Right:** Fine grid boundary conditions. The boundary points of the refined grids are filled via interpolation. This may require interpolation in space and in time.[]{data-label="fig:carpet-details"}](carpet-timestepping "fig:"){width="30.00000%"} ![Berger-Oliger time stepping details, showing a coarse and a fine grid, with time advancing upwards. **Left:** Time stepping algorithm. First the coarse grid takes a large time step, then the refined grid takes two smaller steps. The fine grid solution is then injected into the coarse grid where the grids overlap. **Right:** Fine grid boundary conditions. The boundary points of the refined grids are filled via interpolation. This may require interpolation in space and in time.[]{data-label="fig:carpet-details"}](carpet-interpolation "fig:"){width="30.00000%"}
[Carpet]{} is the main driver used today for [Cactus]{}-based astrophysical simulations. [Carpet]{} offers hybrid MPI/OpenMP parallelization and is used in production on up to several thousand cores [@Reisswig:2010cd; @Lousto:2010ut].
![Results from weak scaling tests evolving the Einstein equations on a mesh refinement grid structure with nine levels. This shows the time required per grid point, where smaller numbers are better (the ideal scaling is a horizontal line). This demonstrates excellent scalability to up to more than 10,000 cores. Including hydrodynamics approximately doubles calculation times without negatively influencing scalability.[]{data-label="fig:weak-scaling"}](results-best){width="85.00000%"}
Figure \[fig:weak-scaling\] shows a weak scaling test of `Carpet`, where `McLachlan` (see section \[sec:Kevol\] below) solves the Einstein equations on a grid structure with nine levels of mesh refinement. This demonstrates excellent scalability up to more than ten thousand cores. In production simulations, smaller and more complex grid structures, serial tasks related to online data analysis and other necessary tasks reduce scalability by up to a factor of ten.
We estimate that in 2010, about 7,000 core years of computing time (45 million core hours) will have been used via [Carpet]{} by more than a dozen research groups world-wide. To date, more than 90 peer-reviewed publications and more than 15 student theses have been based on [Carpet]{} [@CarpetCode:web].
Simulation Factory
------------------
Today’s supercomputers differ significantly in their hardware configuration, available software, directory structure, queuing system, queuing policy, and many other user-visible properties. In addition, the system architectures and user interfaces offered by supercomputers are very different from those offered by laptops or workstations. This makes performing large, three-dimensional time-dependent simulations a complex, time-consuming and difficult task. However, most of these differences are only superficial, and the basic capabilities of supercomputers are very similar; most of the complexity of managing simulations lies in menial tasks that require no physical or numerical insight.
The Simulation Factory [@Thomas:2010aa; @SimFactory:web] offers a set of abstractions for the tasks necessary to set up and successfully complete numerical simulations based on the [Cactus]{} framework. These abstractions hide tedious low-level management operations, capture “best practices” of experienced users, and create a log trail ensuring repeatable and well-documented scientific results. Using these abstractions, most operations are simplified and many types of potentially disastrous user errors are avoided, allowing different supercomputers to be used in a uniform manner.
Using the Simulation Factory, we offer a tutorial for the Einstein Toolkit [@EinsteinToolkit:web] that teaches new users how to download, configure, build, and run full simulations of the coupled Einstein/relativistic hydrodynamics equations on a supercomputer with a few simple commands. Users need no prior knowledge about either the details of the architecture of a supercomputer nor its particular software configuration. The same exact set of SimFactory commands can be used on all other supported supercomputers to run the same simulation there.
The Simulation Factory supports and simplifies three kinds of operations:
1. Remote Access.
: The actual access commands and authentication methods differ between systems, as do the user names that a person has on different systems. Some systems are not directly accessible, and one must log in to a particular “trampoline” server first. The Simulation Factory hides this complexity.
2. Configuring and Building.
: Building [Cactus]{} requires certain software on the system, such as compilers, libraries, and build tools. Many systems offer different versions of these, which may be installed in non-default locations. Finding a working combination that results in efficient code is extremely tedious and requires low-level system experience. The Simulation Factory provides a *machine database* that enables users to store and exchange this information. In many cases, this allows people to begin to use a new machine in a very short time with just a few, simple commands.
3. Submitting and Managing Simulations.
: Many simulations run for days or weeks, requiring frequent checkpointing and job re-submission because of short queue run-time limits. Simple user errors in these menial tasks can potentially destroy weeks of information. The Simulation Factory offers safe commands that encapsulate best practices that prevent many common errors and leave a log trail.
The above features make running simulations on supercomputers much safer and simpler.
Kranc {#sec:kranc}
-----
[`Kranc`]{}[@Husa:2004ip; @Lechner:2004cs; @Kranc:web] is a Mathematica application which converts a high-level continuum description of a PDE into a highly optimized module for [Cactus]{}, suitable for running on anything from a laptop to the world’s largest HPC systems. Many codes contain a large amount of complexity, including expanded tensorial expressions, numerical methods, and the large amount of “glue” code needed for interfacing a modern HPC application with the underlying framework. [`Kranc`]{} absorbs this complexity, allowing the scientist to concentrate on writing only the [`Kranc`]{} script which describes the continuum equations.
This approach brings with it many advantages. With these complicated elements factored out, a scientist can write many different [`Kranc`]{} codes, all taking advantage of the features of [`Kranc`]{} and avoiding unnecessary or trivial but painstaking duplication. The codes might be variations on a theme, perhaps versions which use different sets of variables or formulations of the equations, or they could represent completely different physical systems. The use of a symbolic algebra package, Mathematica, enables high-level optimizations which are not performed by the compiler to be implemented in [`Kranc`]{}.
Any enhancements to [`Kranc`]{} can be automatically applied to all codes which are generated using [`Kranc`]{}. Existing codes have easily benefited from the following features added to [`Kranc`]{} after the codes themselves were written: (i) OpenMP parallelization support, necessary for efficient use of modern multi-core processors; (ii) support for multipatch domains with the Llama [@Pollney:2009yz] code; (iii) automatic generation of vectorized code, where the equations are evaluated simultaneously by the processor for two grid points at the same time; and (iv) common sub-expression elimination, and various other optimization strategies.
Within the Einstein Toolkit, the Einstein evolution thorn [`McLachlan`]{}, as well as the wave extraction thorn [`WeylScal4`]{}, are both generated using [`Kranc`]{}, and hence support all the above features.
Components
==========
The Einstein Toolkit uses the modular [Cactus]{} framework as its underlying infrastructure. A simulation within [Cactus]{} could just use one module, but in practice simulations are often composed from hundreds of components. Some of these modules provide common definitions and conventions (see section \[sec:base\_modules\]). Others provide initial data (see section \[sec:initial\_data\]), which may be evolved using the different evolution methods for vacuum and matter configurations described in sections \[sec:Kevol\] and \[sec:GRHydro\], respectively. The thermodynamic properties of fluids are encoded in equations of state (see section \[sec:eoss\]). Finally, additional quantities which are not directly evolved are often interesting for a detailed analysis of the simulation’s results. Modules providing commonly used analysis methods are described in section \[sec:analysis\].
Base Modules {#sec:base_modules}
------------
Modular designs have proven to be essential for distributed development of complex software systems and require the use of well-defined interfaces. Low-level interoperability within the Einstein Toolkit is provided by the [Cactus]{} infrastructure. One example of this is the usage of one module from within another, e.g., by calling a function within another thorn independent of programming language used for both the calling and called function. Solutions for technical challenges like this can be and are provided by the underlying framework, in this case [Cactus]{}.
However, certain other standards are very hard or impossible to enforce on a technical level. Examples for these include the exact definitions of physical variables, their units, and, on a more technical level, the variable names used for the physical quantities. Even distinct simulation codes typically use very similar scheduling schemes, so conventions describing the behavior of the scheduler can help coordinate the order in which functions in different modules are called.
The Einstein Toolkit provides modules whose sole purpose is to declare commonly used variables and define their meaning and units. These conditions are not strictly enforced, but instead documented for the convenience of the user community. Three of these base modules, [`ADMBase`]{}, [`HydroBase`]{}, and [`TmunuBase`]{}, are described in more detail below.
In the following, we assume that the reader is familiar with the basics of numerical relativity and GR hydrodynamics, including the underlying differential geometry and tensor analysis. Detailed introductions to numerical relativity have recently been given by Alcubierre [@Alcubierre:2008it], Baumgarte & Shapiro [@Baumgarte:2010nu], and Centrella et al. [@Centrella:2010mx]. GR hydrodynamics has been reviewed by Font [@Font:2008aa]. We set $G = c
=1$ throughout this paper, and $M_\odot = 1$ where appropriate.
### ADMBase
The Einstein Toolkit provides code to evolve the Einstein equations $$G^{\mu\nu} = 8 \pi T^{\mu\nu}\,,
\label{eq:einstein}$$ where $G^{\mu\nu}$ is the Einstein tensor, describing the curvature of 4-dimensional spacetime, and $T^{\mu\nu}$ is the stress-energy tensor. Relativistic spacetime evolution methods used within the [Cactus]{} framework employ different formalisms to accomplish this goal, but essentially all are based on the $3+1$ ADM construction [@Arnowitt:1962hi], which makes it the natural choice of a common foundation for exchange data between modules using different formalisms. In the $3+1$ approach, 4-dimensional spacetime is foliated into sequences of spacelike 3-dimensional hypersurfaces (slices) connected by timelike normal vectors. The $3+1$ split introduces 4 gauge degrees of freedom: the lapse function $\alpha$ that describes the advance of proper time with respect to coordinate time for a normal observer[^2] and the shift vector $\beta^i$ that describes how spatial coordinates change from one slice to the next.
According to the ADM formulation, the spacetime metric is assumed to take the form $$ds^2=g_{\mu\nu}dx^\mu dx^\nu\equiv (-\alpha^2+\beta_i\beta^i)dt^2+2\beta_i dt~dx^i+\gamma_{ij} dx^idx^j,\label{eq:adm}$$ where $g_{\mu\nu},~\alpha,~\beta^i$, and $\gamma_{ij}$ are the spacetime 4-metric, lapse function, shift vector, and spatial 3-metric, respectively, and we follow the standard relativistic convention where Latin letters are used to index 3-dimensional spatial quantities and Greek letters to index 4-dimensional spacetime quantities, with the index running from 0 to 3. The remaining dynamical component of the spacetime is contained in the definition of the extrinsic curvature $K_{ij}$, which is defined in terms of the time derivative of the metric after incorporating a Lie derivative with respect to the shift vector: $$K_{ij}\equiv -\frac{1}{2\alpha}(\partial_t-\mathcal{L}_\beta)\gamma_{ij}.$$ The three-metric, extrinsic curvature, lapse function, and shift vector are all declared as variables in the [`ADMBase`]{} module, the latter two together with their first time derivatives. The variables provided by [ADMBase]{} are:
- The 3-metric tensor, $\gamma_{ij}$: [gxx]{}, [gxy]{}, [gxz]{},[gyy]{}, [gyz]{}, [gzz]{}
- The extrinsic curvature tensor, $K_{ij}$: [kxx]{}, [kxy]{}, [kxz]{}, [kyy]{}, [kyz]{}, [kzz]{}
- The lapse function, $\alpha$: [alp]{}
- The shift vector $\beta^i$: [betax]{}, [betay]{}, [betaz]{}
This base module also defines common parameters to manage interaction between different modules. Examples are the type of requested initial data or the used evolution method.
The type of initial data chosen for a simulation is specified by the parameters [initial\_data]{} (3-metric and extrinsic curvature), [initial\_lapse]{}, [initial\_shift]{}. The time derivatives of the gauge variables (the lapse and shift) are set by the parameters [initial\_dtlapse]{} and [initial\_dtshift]{}, respectively. By default, [ADMBase]{} initializes the 3-metric and extrinsic curvature to Minkowski (i.e., $\gamma_{ij}=\delta_{ij}$, the Kronecker delta, and $K_{ij}=0$), the shift to zero, and the lapse to unity. Initial data thorns override these defaults by extending the parameters.
Analogous to specifying initial data, evolution methods are chosen by the parameters [evolution\_method]{} (3-metric and extrinsic curvature), [lapse\_evolution\_method]{}, [shift\_evolution\_method]{}, [dtlapse\_evolution\_method]{} and [dtshift\_evolution\_method]{}. [ADMBase]{} does not evolve the 3-metric or extrinsic curvature, and holds the lapse and shift static. Evolution thorns extend the ranges of these parameters and contain the evolution code.
The variables defined in ADMBase typically are not used for the actual evolution of the curvature. Instead, it is expected that every evolution module converts its internal representation to the form defined in ADMBase after each evolution step. This procedure enables modules which perform analysis on the spacetime variables to use the ADMBase variables without direct dependency on any of the existing curvature evolution methods.
### HydroBase
Similar to [`ADMBase`]{}, the module [`HydroBase`]{} defines a common basis for interactions between modules of a given evolution problem, in this case relativistic hydrodynamics. [`HydroBase`]{} extends the Einstein Toolkit to include an interface within which magnetohydrodynamics may work. [`HydroBase`]{}’s main function is to store variables which are common to most if not all hydrodynamics codes solving the Euler equations, the so-called primitive variables. These are also the variables which are needed to couple to a spacetime solver, and often by analysis thorns as well. As with ADMBase, the usage of a common set of variables by different hydrodynamics codes creates the possibility of sharing parts of the code, e.g., initial data solvers or analysis routines. [`HydroBase`]{} also defines commonly needed parameters and schedule groups for the main functions of a hydrodynamics code.
[`HydroBase`]{} uses a set of conventions known as the Valencia formulation [@Marti:1991wi; @Banyuls:1997zz; @Ibanez:2001:godunov]. In particular, [`HydroBase`]{} defines the primitive variables (see [@Font:2008aa] for details):
- `rho`: rest mass density $\rho$
- `press`: pressure $P$
- `eps`: internal energy density $\epsilon$
- `vel[3]`: contravariant fluid three velocity $v^i$ defined as $$v^i = \frac{u^i}{\alpha u^0} + \frac{\beta^i}{\alpha}$$ in terms of the four-velocity $u^\mu$, lapse, and shift vector .
- `Y_e`: electron fraction $Y_e$
- `temperature`: temperature $T$
- `entropy`: specific entropy per particle $s$
- `Bvec[3]`: contravariant magnetic field vector defined as $$B^i = \frac{1}{\sqrt{4\pi}} n_{\nu} F^{*\nu i}$$ in terms of the dual $F^{*\mu\nu} = \frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ to the Faraday tensor and the unit normal of the foliation of spacetime $n^\mu\equiv \alpha^{-1}[1,-\beta^i]^{\rm T}$.
[`HydroBase`]{} also sets up scheduling blocks that organize the main functions which modules of a hydrodynamics code may need. All of those scheduling blocks are optional, but when used they simplify existing codes and make them more interoperable. [`HydroBase`]{} itself does not schedule routines inside most of the groups that it provides. Currently the scheduling blocks are:
- Initializing the primitive variables
- Converting primitive variables to conservative variables
- Calculating the right hand side (RHS) in the method of lines (MoL)
- Setting and updating an excision mask
- Applying boundary conditions
Through these, the initiation of the primitive variables, methods to recover the conservative variables, and basic atmosphere handling can be implemented in different thorns while allowing a central access point for analysis thorns.
### TmunuBase
In the Einstein Toolkit, we typically choose the stress energy tensor $T^{\mu\nu}$ to be that of an ideal relativistic fluid, $$T^{\mu\nu} = \rho h u^\mu u^\nu - g^{\mu\nu} P\,\,,$$ where $\rho$, $u^\mu$, and $g^{\mu\nu}$ are defined above, and $h = 1 + \epsilon + P/\rho$ is the relativistic specific enthalpy.
The thorn [`TmunuBase`]{} provides grid functions for the stress-energy tensor $T_{\mu\nu}$ as well as schedule groups to manage when $T_{\mu\nu}$ is calculated. In a simulation, many different thorns may contribute to the stress-energy tensor and this thorn allows them to do so without explicit interdependence. The resulting stress-energy tensor can then be used by the spacetime evolution thorn (again without explicit dependence on any matter thorns). When thorn [`MoL`]{} is used for time evolution this provides a high-order spacetime-matter coupling.
The grid functions provided by [`TmunuBase`]{} are:
- The time component $T_{00}$: [eTtt]{}
- The mixed components $T_{0i}$: [eTtx]{}, [eTty]{}, [eTtz]{}
- The spatial components $T_{ij}$: [eTxx]{}, [eTxy]{}, [eTxz]{}, [eTyy]{}, [eTyz]{}, [eTzz]{}
In addition, the grid scalar [stress\_energy\_state]{} has the value 1 if storage is provided for the stress-energy tensor and 0 if not.
Thorn [`ADMCoupling`]{} provides a similar (but older) interface between space-time and matter thorns. However, since it is based on an include file mechanism it is more complicated to use. We recommend all new matter thorns to use [`TmunuBase`]{} instead.
Initial Data {#sec:initial_data}
------------
The Einstein Toolkit contains many modules used to generate initial data for GR simulations, including both vacuum and hydrodynamic configurations. These include modules used primarily for testing of various components, as well as physically motivated configurations that describe, for example, single or binary BHs and/or NSs. Many of the modules are self-contained, consisting of either all the code to generate exact initial solutions or the numerical tools required to construct solutions known semi-analytically. Others, though, require the installation of numerical software packages that are included in the toolkit as external libraries. One example is the [`TwoPunctures`]{} module [@Ansorg:2004ds] — commonly used in numerical relativity to generate BH-BH binary initial data — which makes use of the GNU Scientific Library \[GSL; [@GSL:web; @Galassi:2009]\]. Several modules have also been implemented to read in data files generated by the [Lorene]{} code [@Lorene:web; @Gourgoulhon:2000nn].
Initial data setup is in most cases clearly separated from the evolution that follows. Typically, initial data routines provide the data in terms of the quantities defined in the Base modules (see section \[sec:base\_modules\]), and the evolution modules will later convert these quantities to forms used for the evolution. For example, an initial data module must supply $g_{ij}$, the spatial 3-metric, and $K_{ij}$, the extrinsic curvature. The conversion between the physical metric and extrinsic curvature and conformal versions of these is handled solely within evolution modules, which are responsible for calculating the conformally related three metric $\tilde{\gamma}_{ij}$, the conformal factor $\psi$, the conformal traceless extrinsic curvature $\tilde{A}_{ij}$ and the trace of the extrinsic curvature $K$, as well as initializing the BSSN variable $\tilde{\Gamma}^i$ should that be the evolution formalism chosen (see section \[sec:Kevol\] for definitions of these). Optionally, many initial data modules also supply values for the lapse and shift vector and in some cases their time derivatives. It is important to note, though, that many dynamical calculations run better from initial gauge choices set by ansatz rather than those derived from stationary approximations that are incompatible with the gauge evolution equations. In particular, conformal thin-sandwich initial data for binaries include solutions for the lapse and shift that are frequently replaced by simpler analytical values that lead to more circular orbits under standard “moving puncture” gauge conditions (see, e.g., [@York:1998hy; @Etienne:2007jg] and other works).
We turn our attention next to a brief discussion of the capabilities of the aforementioned modules as well as their implementation.
### Simple Vacuum initial data
Vacuum spacetime tests in which the constraint equations are explicitly violated are provided by [`IDConstraintViolate`]{} and [`Exact`]{}, a set of exact spacetimes in various coordinates including Lorentz-boosted versions of traditional solutions. Vacuum gravitational wave configurations can be obtained by using either [`IDBrillData`]{}, providing a Brill wave spacetime [@Brill:1959zz]; or [`IDLinearWaves`]{}, for a spacetime containing a linear gravitational wave. Single BH configurations include [`IDAnalyticBH`]{} which generates various analytically known BH configurations; as well as [`IDAxibrillBH`]{}, [`IDAxiOddBrillBH`]{}, [`DistortedBHIVP`]{} and [`RotatingDBHIVP`]{}, which introduce perturbations to isolated BHs.
### Hydrodynamics Tests
Initial data to test different parts of hydrodynamics evolution systems are provided by [`GRHydro_InitData`]{}. This module includes several shock tube problems that may be evolved in any of the Cartesian directions or diagonally. All of these have been widely used throughout the field to evaluate a diverse set of solvers [@Marti:1999wi]. Conservative to primitive variable conversion and vice versa are also supported, as are tests to check on the reconstruction of hydrodynamical variables at cell faces (see Sec. \[sec:GRHydro\] for more on this). Along similar lines, the [`Hydro_InitExcision`]{} module sets up masks for different kinds of excised regions, which is convenient for hydrodynamics tests.
### TwoPunctures: Binary Black Holes and extensions {#sec:twopunctures}
A substantial fraction of the published work on the components of the Einstein toolkit involves the evolution of BH-BH binary systems. The most widely used routine to generate initial data for these is the [`TwoPunctures`]{} code (described originally in [@Ansorg:2004ds]) which solves the binary puncture equations for a pair of BHs [@Brandt:1997tf]. To do so, one assumes the extrinsic curvature for each BH corresponds to the Bowen-York form [@Bowen:1980yu]: $$\begin{aligned}
K_{(m)}^{ij}&=&\frac{3}{2r^2}(p_{(m)}^i\hat{N}^j+p_{(m)}^j\hat{N}^i-(\gamma^{ij}-\hat{N}^i\hat{N}^j)p_{(m)}^k\hat{N}_k))\nonumber\\
&&+\frac{3}{r^3}(\varepsilon^{ikl}S^{(m)}_k\hat{N}_l\hat{N}^j+\varepsilon^{jkl}S^{(m)}_k\hat{N}_l\hat{N}^i),\end{aligned}$$ where the sub/superscript $(m)$ refers to the contribution from BH $m=1,2$; the 3-momentum is $p^i$; the BH spin angular momentum is $S_i$; the conformal 3-metric $\gamma^{ij}$ is assumed to be flat, i.e. $\gamma_{ij}=\eta_{ij}$, and $\hat{N}^i=x^i/r$ is the Cartesian normal vector relative to the position of each BH in turn. This solution automatically satisfies the momentum constraint, and the Hamiltonian constraint may be written as an elliptic equation for the conformal factor, defined by the condition $g_{ij}=\psi^4\gamma_{ij}$ or equivalently $\psi\equiv (\det|g_{ij}|)^{1/12}$: $$\Delta \psi+\frac{1}{8}K^{ij}K_{ij}\psi^{-7}=0$$ Decomposing the conformal factor into a singular analytical term and a regular term $u$, such that $$\psi = \frac{m_1}{2r_1}+\frac{m_2}{2r_2}+u\equiv \frac{1}{\Psi}+u$$ where $m_1,~m_2$ and $r_1,~r_2$ are the mass of and distance to each BH, respectively, and $\Psi$ is defined by the equation itself, the Hamiltonian constraint may be written as $$\Delta u +\left[\frac{1}{8}\Psi^7K^{ij}K_{ij}\right](1+\Psi u)^{-7}\label{eq:twopunc_u}$$ subject to the boundary condition $u\rightarrow 1$ as $r\rightarrow\infty$. In Cartesian coordinates, the function $u$ is infinitely differentiable everywhere except the two puncture locations. [`TwoPunctures`]{} resolves this problem by performing a coordinate transformation modeled on confocal elliptical/hyperbolic coordinates. This transforms the spatial domain into a finite cube with the puncture locations mapped to two parallel edges, as can be seen in figure \[fig:TP\_BHNS\_coordinates\]. The coordinate transformation is: $$\begin{aligned}
x&=&b\frac{A^2+1}{A^2-1}\frac{2B}{1+B^2}\nonumber\\
y&=&b\frac{2A}{1-A^2}\frac{1-B^2}{1+B^2}\cos\phi\nonumber\\
z&=&b\frac{2A}{1-A^2}\frac{1-B^2}{1+B^2}\sin\phi\end{aligned}$$ which maps $\mathcal{R}^3$ onto $0\le A\le 1$ (the elliptical quasi-radial coordinate), $-1\le B\le 1$ (the hyperbolic quasi-latitudinal coordinate), and $0\le\phi<2\pi$ (the longitudinal angle). Since $u$ is smooth everywhere in the interior of the remapped domain, expansions into modes in these coordinates are [*spectrally convergent*]{} and thus capable of extremely high accuracy. In practice, the field is expanded into Chebyshev modes in the quasi-radial and quasi-latitudinal coordinates, and into Fourier modes around the axis connecting the two BHs. The elliptic solver uses a stabilized biconjugate gradient method to achieve rapid solutions and to avoid ill-conditioning of the spectral matrix.
![Example of a TwoPunctures coordinate system for BH-NS binary initial data[]{data-label="fig:TP_BHNS_coordinates"}](TwoPunctures_grid_BHNS "fig:"){width="50.00000%"}\
### Lorene-based binary data
The ET contains three routines that can read in publicly available data generated by the [Lorene]{} code [@Lorene:web; @Gourgoulhon:2000nn], though it does not currently include the capability of generating such data from scratch. For a number of reasons, such functionality is not truly required; in particular, [Lorene]{} is a serial code and to call it as an ET initial data generator saves no time. Also, it is not guaranteed to be convergent for an arbitrary set of parameters; thus the initial data routine itself may never finish its iterative steps. Instead, recommended practice is to let Lorene output data into files, and then read those into ET at the beginning of a run.
Lorene uses a multigrid spectral approach to solve the conformal thin-sandwich equations for binary initial configurations [@York:1998hy] and a single-grid spectral method for rotating stars. For binaries, five elliptic equations for the shift, lapse, and conformal factor are written down and the source terms are divided into pieces that are attributed to each of the two objects. Matter source terms are ideal for this split, since they are compactly supported, while extrinsic curvature source terms are spatially extended but with sufficiently rapid falloff at large radii to yield convergent solutions. Around each object, a set of nested spheroidal sub-domains (see figure \[fig:Lorene\_coordinates\]) is constructed to extending through all of space, with the outermost domain incorporating a compactification to allow it to extend to spatial infinity. Within each of the nested sub-domains, fields are decomposed into Chebyshev modes radially and into spherical harmonics in the angular directions, with elliptic equation solving reduced to a matrix problem. The nested sub-domains need not be perfectly spherical, and indeed one may modify the outer boundaries of each to cover any convex shape. For NSs, this allows one to map the surface of a particular sub-domain to the NS surface, minimizing Gibbs error there. For BHs, excision boundary conditions are imposed at the horizon. To read a field solution describing a given quantity onto a [Cactus]{}-based grid, one must incorporate the data from each star’s domains at that point.
![Example of a Lorene multi-domain coordinate system for binary initial data. The outermost, compactified domain extending to spatial infinity is not shown.[]{data-label="fig:Lorene_coordinates"}](Lorene_Grid "fig:"){width="50.00000%"}\
[`Meudon_Bin_BH`]{} can read in BH-BH binary initial data described in [@Grandclement:2001ed], while [`Meudon_Bin_NS`]{} handles binary NS data from [@Gourgoulhon:2000nn]. [`Meudon_Mag_NS`]{} may be used to read in magnetized isolated NS data [@Lorene:web].
### TOVSolver {#sec:TOVSolver}
The [`TOVSolver`]{} routine in the ET solves the standard TOV equations [@Tolman:1939jz; @Oppenheimer:1939ne] expressed using the Schwarzschild (areal) radius $r$ in the interior of a spherically symmetric star in hydrostatic equilibrium: $$\begin{aligned}
\label{eq:TOViso}
\frac{d P}{d r} & = & -(e + P) \frac{m + 4\pi r^3 P}{r(r - 2m)}\nonumber\\
\frac{d m}{d r} & = & 4 \pi r^2 e\nonumber\\
\frac{d \Phi}{d r} & = & \frac{m + 4\pi r^3 P}{r(r -
2m)},\end{aligned}$$ where $e\equiv \rho(1+\epsilon)$ is the energy density of the fluid, including the internal energy contribution, $m$ is the gravitational mass inside a sphere of radius $r$, and $\Phi$ the logarithm of the lapse. The routine also supplies the analytically known solution in the exterior, $$\begin{aligned}
P & = & {\tt TOV\_atmosphere},\nonumber \\
m & = & M, \nonumber\\
\Phi & = &{{\frac{1}{2}}} \log(1-2M / r)
\label{eq:TOVexterior}\end{aligned}$$ where [TOV\_atmosphere]{} is a parameter used to define the density of the ambient atmosphere. Since the isotropic radius $\bar{r}$ is the more commonly preferred choice to initiate dynamical calculations, the code then transforms all variables into isotropic coordinates, integrating the radius conversion formula $$\label{eq:rbar}
\frac{d (\log(\bar{r} / r))}{\partial r} = \frac{r^{1/2} - (r-2m)^{1/2}}{r(r-2m)^{1/2}} \ .$$ subject to the boundary condition that in the exterior, $$\begin{aligned}
\bar{r} &=& {{\frac{1}{2}}}\left(\sqrt{r^2-2Mr}+r -M\right)\nonumber \\
r&=&\bar{r}\left(1+{{\frac{M}{2\bar{r}}}}\right)^2 \ .\end{aligned}$$ handling with some care the potentially singular terms that appear at the origin.
To facilitate the construction of stars in more complicated dynamical configurations, one may also apply a uniform velocity to the NS, though this does not affect the ODE solution nor the resulting density profile.
Spacetime Curvature Evolution {#sec:Kevol}
-----------------------------
The Einstein Toolkit curvature evolution code [`McLachlan`]{} [@Brown:2008sb; @Reisswig:2010cd] is auto-generated from tensor equations via [`Kranc`]{} (Sec. \[sec:kranc\]). It implements the Einstein equations in a $3+1$ split as a Cauchy initial boundary value problem [@York:1979sg]. For this, the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) conformal-tracefree reformulation [@Shibata:1995we; @Baumgarte:1998te; @Alcubierre:2000xu] of the original Arnowitt-Deser-Misner (ADM) formalism [@Arnowitt:1962hi] is employed. [`McLachlan`]{} uses fourth-order accurate finite differencing for the spacetime variables and adds a fifth-order Kreiss-Oliger dissipation term to remove high frequency noise. The evolved variables are the conformal factor $\Phi$, the conformal 3-metric $\tilde{\gamma}_{ij}$, the trace $K$ of the extrinsic curvature, the trace free extrinsic curvature $A_{ij}$ and the conformal connection functions $\tilde{\Gamma}^i$. These are defined in terms of the standard ADM 4-metric $g_{ij}$, 3-metric $\gamma_{ij}$, and extrinsic curvature $K_{ij}$ by:
$$\begin{aligned}
\phi & \equiv & \log \left[ \frac{1}{12} \det \gamma_{ij} \right]\,,
\\
\tilde\gamma_{ij} & \equiv & e^{-4\phi}\; \gamma_{ij}\,,
\\
K & \equiv & g^{ij} K_{ij}\,,
\\
\tilde A_{ij} & \equiv & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K\,,
\right]
\\
\tilde\Gamma^i & \equiv & \tilde\gamma^{jk} \tilde\Gamma^i_{jk} .\end{aligned}$$
The evolution equations are then:
$$\begin{aligned}
\partial_0 \alpha & = & -\alpha^2 f(\alpha, \phi, x^\mu) (K -
K_0(x^\mu))
\\
\partial_0 K & = & -e^{-4\phi} \left[ \tilde{D}^i \tilde{D}_i \alpha
+ 2 \partial_i \phi \cdot \tilde{D}^i \alpha \right] + \alpha
\left( \tilde{A}^{ij} \tilde{A}_{ij} + \frac{1}{3} K^2 \right) - \alpha S
\\
\partial_0 \beta^i & = & \alpha^2 G(\alpha,\phi,x^\mu) B^i
\\
\partial_0 B^i & = & e^{-4\phi} H(\alpha,\phi,x^\mu)
\partial_0\tilde{\Gamma}^i - \eta^i(B^i,\alpha,x^\mu)
\\
\partial_0 \phi & = & -\frac{\alpha}{6}\, K +
\frac{1}{6}\partial_k\beta^k
\\
\partial_0 \tilde{\gamma}_{ij} & = & -2\alpha\tilde{A}_{ij}
+ 2\tilde{\gamma}_{k(i}\partial_{j)}\beta^k
- \frac{2}{3}\tilde{\gamma}_{ij}\partial_k\beta^k
\\
\partial_0 \tilde{A}_{ij} & = & e^{-4\phi}\left[
\alpha\tilde{R}_{ij} + \alpha R^\phi_{ij} - \tilde{D}_i\tilde{D}_j\alpha
+ 4\partial_{(i}\phi\cdot\tilde{D}_{j)}\alpha\right]^{TF}
\nonumber\\
& & {} + \alpha K\tilde{A}_{ij} - 2\alpha\tilde{A}_{ik}\tilde{A}^k_{\; j}
+ 2\tilde{A}_{k(i}\partial_{j)}\beta^k
- \frac{2}{3}\tilde{A}_{ij}\partial_k\beta^k
- \alpha e^{-4\phi} \hat{S}_{ij}
\\
\partial_0\tilde{\Gamma}^i & = &
\tilde{\gamma}^{kl}\partial_k\partial_l\beta^i
+ \frac{1}{3} \tilde{\gamma}^{ij}\partial_j\partial_k\beta^k
+ \partial_k\tilde{\gamma}^{kj} \cdot \partial_j\beta^i
- \frac{2}{3}\partial_k\tilde{\gamma}^{ki} \cdot \partial_j\beta^j
\nonumber\\
& & {} - 2\tilde{A}^{ij}\partial_j\alpha
+ 2\alpha\left[ (m-1)\partial_k\tilde{A}^{ki} - \frac{2m}{3}\tilde{D}^i K
\right. \nonumber \\
& & {} + m(\tilde{\Gamma}^i_{\; kl}\tilde{A}^{kl} +
6\tilde{A}^{ij}\partial_j\phi) \Bigg] - S^i,\end{aligned}$$
with the momentum constraint damping constant set to $m=1$. The stress energy tensor $T_{\mu\nu}$ is incorporated via the projections $$\begin{aligned}
E & \equiv & \frac{1}{\alpha^2} \left( T_{00} - 2 \beta^i T_{0i} +
\beta^i \beta^j T^{ij} \right)
\\
S & \equiv & g^{ij} T_{ij}
\\
S_i & \equiv & - \frac{1}{\alpha} \left( T_{0i} - \beta^j T_{ij} \right) .\end{aligned}$$ We have introduced the notation $\partial_0 = \partial_t -
\beta^j\partial_j$. All quantities with a tilde involve the conformal 3-metric $\tilde{\gamma}_{ij}$, which is used to raise and lower indices. In particular, $\tilde{D}_i$ and $\tilde{\Gamma}^k_{ij}$ refer to the covariant derivative and the Christoffel symbols with respect to $\tilde{\gamma}_{ij}$. The expression $[ \cdots ]^{TF}$ denotes the trace-free part of the expression inside the parentheses, and we define the Ricci tensor contributions as:
$$\begin{aligned}
\tilde{R}_{ij}
&=& -\frac{1}{2} \tilde{\gamma}^{kl}\partial_k\partial_l\tilde{\gamma}_{ij}
+ \tilde{\gamma}_{k(i}\partial_{j)}\tilde{\Gamma}^k
- \tilde{\Gamma}_{(ij)k}\partial_l\tilde{\gamma}^{lk}
+ \tilde{\gamma}^{ls}\left( 2\tilde{\Gamma}^k_{\; l(i}\tilde{\Gamma}_{j)ks}
+ \tilde{\Gamma}^k_{\; is}\tilde{\Gamma}_{klj} \right)
\\
R^\phi_{ij} &=& -2\tilde{D}_i\tilde{D}_j\phi
- 2\tilde{\gamma}_{ij} \tilde{D}^k\tilde{D}_k\phi
+ 4\tilde{D}_i\phi\, \tilde{D}_j\phi
- 4\tilde{\gamma}_{ij}\tilde{D}^k\phi\, \tilde{D}_k\phi .\end{aligned}$$
This is a so-called $\phi$-variant of BSSN. The evolved gauge variables are lapse $\alpha$, shift $\beta^i$, and a quantity $B^i$ related to the time derivative of the shift. The gauge parameters $f$, $G$, $H$, and $\eta$ are determined by our choice of a $1+\log$ [@Alcubierre:2002kk] slicing: $$\begin{aligned}
f(\alpha,\phi,x^\mu) & \equiv & 2/\alpha
\\
K_0(x^\mu) & \equiv & 0\end{aligned}$$ and $\Gamma$-driver shift condition [@Alcubierre:2002kk]: $$\begin{aligned}
G(\alpha,\phi,x^\mu) & \equiv & (3/4)\, \alpha^{-2}
\\
H(\alpha,\phi,x^\mu) & \equiv & \exp\{4\phi\}
\\\label{eq:eta}
\eta(B^i,\alpha,x^\mu) & \equiv & (1/2)\, B^i q(r) .\end{aligned}$$ The expression $q(r)$ attenuates the $\Gamma$-driver depending on the radius as described below.
The $\Gamma$-driver shift condition is symmetry-seeking, driving the shift $\beta^i$ to a state that renders the conformal connection functions $\tilde\Gamma^i$ stationary. Of course, such a stationary state cannot be achieved while the metric is evolving, but in a stationary spacetime the time evolution of the shift $\beta^i$ and thus that of the spatial coordinates $x^i$ will be exponentially damped. This damping time scale is set by the gauge parameter $\eta$ (see ) which has dimension $1/T$ (inverse time). As described in [@Muller:2009jx; @Schnetter:2010cz], this time scale may need to be adapted in different regions of the domain to avoid spurious high-frequency behavior in regions that otherwise evolve only very slowly, e.g., far away from the source.
Here we use the simple damping mechanism described in (12) of [@Schnetter:2010cz], which is defined as: $$\begin{aligned}
\label{eq:varying-simple}
q(r) & \equiv & \left\{
\begin{array}{llll}
1 & \mathrm{for} & r \le R & \textrm{(near the origin)}
\\
R/r & \mathrm{for} & r \ge R & \textrm{(far away)}
\end{array}
\right.\end{aligned}$$ with a constant $R$ defining the transition radius between the interior, where $q\approx1$, and the exterior, where $q$ falls off as $1/r$. A description of how $q$ appears in the gauge parameters may be found in .
### Initial Conditions
Initial conditions for the ADM variables $g_{ij}$, $K_{ij}$, lapse $\alpha$, and shift $\beta^i$ are provided by the initial data routines discussed in Sec. \[sec:initial\_data\]. From these the BSSN quantities are calculated via their definitions, setting $B^i=0$, and using cubic extrapolation for $\tilde\Gamma^i$ at the outer boundary. This extrapolation is necessary since the $\tilde\Gamma^i$ are calculated from derivatives of the metric, and one cannot use centered finite differencing stencils near the outer boundary.
The extrapolation stencils distinguish between points on the faces, edges, and corners of the grid. Points on the faces are extrapolated via stencils perpendicular to that face, while points on the edges and corners are extrapolated with stencils aligned with the average of the normals of the adjoining faces. For example, points on the $(+x,+y)$ edge are extrapolated in the $(1,1,0)$ direction, while points in the $(+x,+y+z)$ corner are extrapolated in the $(1,1,1)$ direction. Since several layers of boundary points have to be filled for higher order schemes (e.g., three layers for a fourth order scheme), one proceeds outwards starting from the innermost layer. Each subsequent layer is then defined via the points in the interior and the previously calculated layers.
### Boundary Conditions {#sec:curv_boundaries}
During time evolution, a Sommerfeld-type radiative boundary condition is applied to all components of the evolved BSSN variables as described in [@Alcubierre:2000xu]. The main feature of this boundary condition is that it assumes approximate spherical symmetry of the solution, while applying the actual boundary condition on the boundary of a cubic grid where the face normals are not aligned with the radial direction. This boundary condition defines the right hand side of the BSSN state vector on the outer boundary, which is then integrated in time as well so that the boundary and interior are calculated with the same order of accuracy.
The main part of the boundary condition assumes that one has an outgoing radial wave with some speed $v_0$: $$\begin{aligned}
X & = & X_0 + \frac{u(r - v_0 t)}{r},\end{aligned}$$ where $X$ is any of the tensor components of evolved variables, $X_0$ the value at infinity, and $u$ a spherically symmetric perturbation. Both $X_0$ and $v_0$ depend on the particular variable and have to be specified. This implies the following differential equation: $$\begin{aligned}
\partial_t X & = & - v^i \partial_i X - v_0\, \frac{X - X_0}{r}\,,\end{aligned}$$ where $v^i = v_0\, x^i/r$. The spatial derivatives $\partial_i$ are evaluated using centered finite differencing where possible, and one-sided finite differencing elsewhere. Second order stencils are used in the current implementation.
In addition to this main part, it is also necessary to account for those parts of the solution that do not behave as a pure wave, e.g., Coulomb type terms caused by infall of the coordinate lines. It is assumed that these parts decay with a certain power $p$ of the radius. This is implemented by considering the radial derivative of the source term above, and extrapolating according to this power-law decay.
Given a source term $(\partial_t X)$, one defines the corrected source term $(\partial_t X)^*$ via $$\begin{aligned}
(\partial_t X)^* & = & (\partial_t X) + \left( \frac{r}{r - n^i
\partial_i r} \right)^p\; n^i \partial_i (\partial_t X)\,,\end{aligned}$$ where $n^i$ is the normal vector of the corresponding boundary face. The spatial derivatives $\partial_i$ are evaluated by comparing neighboring grid points, corresponding to a second-order stencil evaluated in the middle between the two neighboring grid points. Second-order decay is assumed, hence $p=2$.
As with the initial conditions above, this boundary condition is evaluated on several layers of grid points, starting from the innermost layer. Both the extrapolation and radiative boundary condition algorithms are implemented in the publicly available `NewRad` component of the Einstein Toolkit.
This boundary condition is only a coarse approximation of the actual decay behavior of the BSSN state vector, and it does not capture the correct behavior of the evolved variables. However, one finds that this boundary condition leads to stable evolutions if applied sufficiently far from the source. Errors introduced at the boundary (both errors in the geometry and constraint violations) propagate inwards with the speed of light [@Brown:2008sb]. Gauge changes introduced by the boundary condition, which are physically not observable, propagate faster, with a speed up to $\sqrt{2}$ for the gauge conditions used in [`McLachlan`]{}.
Hydrodynamics Evolution {#sec:GRHydro}
-----------------------
Hydrodynamic evolution in the Einstein Toolkit is designed so that it interacts with the metric curvature evolution through a small set of variables, allowing for maximum modularity in implementing, editing, or replacing either evolution scheme.
The primary hydrodynamics evolution routine in the Einstein Toolkit is [`GRHydro`]{}, a code derived from the public [`Whisky`]{} code [@Baiotti:2004wn; @Hawke:2005zw; @Baiotti:2010zf; @Whisky:web] designed primarily by researchers at AEI and their collaborators. It includes a high resolution shock capturing (HRSC) scheme to evolve hydrodynamic quantities, with several different reconstruction methods and Riemann solvers, as we discuss below. In such a scheme, we define a set of “conserved” hydrodynamic variables, defined in terms of the “primitive” physical variables such as mass and internal energy density, pressure, and velocity. Wherever derivatives of hydrodynamic terms appear in the evolution equations for the conserved variables, they are restricted to appear only inside divergence terms (referred to as fluxes) and never in the source terms. By calculating fluxes at cell faces, we may obtain a consistent description of the inter-cell values using reconstruction techniques that account for the fact that hydrodynamic variables are not smooth and may not be finite differenced accurately. All other source terms in the evolution equations may contain only the hydrodynamic variables themselves and the metric variables and derivatives of the latter, since the metric must formally be smooth and thus differentiable using finite differencing techniques. Summarizing these methods briefly, the following stages occur every timestep:
- The primitive variables are “reconstructed” at cell faces using shock-capturing techniques, with total variation diminishing (TVD), piecewise parabolic (PPM), and essentially non-oscillatory (ENO) methods currently implemented.
- A Riemann problem is solved at each cell face using an approximate solver. Currently implemented versions include HLLE (Harten-Lax-van Leer-Einfeldt), Roe, and Marquina solvers.
- The conserved variables are advanced one timestep, and used to recalculate the new values of the primitive variables.
We discuss the GRHD formalism, the stages within a timestep, and the other aspects of the code below, noting that the documentation included in the released version is quite extensive and covers many of these topics in substantially more detail.
### Ideal general relativistic hydrodynamics (GRHD)
The equations of ideal GR hydrodynamics evolved by [`GRHydro`]{} are derived from the local GR conservation laws of mass and energy-momentum: $$\nabla_{\!\mu} J^\mu = 0, \qquad \nabla_{\!\mu} T^{\mu \nu} = 0\,\,,
\label{eq:equations_of_motion_gr}$$ where $ \nabla_{\!\mu} $ denotes the covariant derivative with respect to the 4-metric, and $ J^{\,\mu} = \rho u^{\,\mu} $ is the mass current. The 3-velocity $v^i$ may be calculated in the form $$v^i = \frac{u^i}{W} + \frac{\beta^i}{\alpha}\,\,,
\label{eq:vel}$$ where $W = (1-v^i v_i)^{-1/2}$ is the Lorentz factor. The contravariant 4-velocity is then given by: $$u^0 = \frac{W}{\alpha}\,,\qquad
u^i = W \left( v^i - \frac{\beta^i}{\alpha}\right)\,\,,$$ and the covariant 4-velocity is: $$u_0 = W(v^i \beta_i - \alpha)\,,\qquad
u_i = W v_i\,\,.$$
The [GRHydro]{} evolution scheme is a first-order hyperbolic flux-conservative system for the conserved variables $D$, $S^i$, and $\tau$, which may be defined in terms of the primitive variables $\rho, \epsilon, v^i$, such that: $$\begin{aligned}
D &=& \sqrt{\gamma} \rho W,\label{eq:p2c1}\\
S^i &=& \sqrt{\gamma} \rho h W^{\,2} v^i,\label{eq:p2c2}\\
\tau &=& \sqrt{\gamma} \left(\rho h W^{\,2} - P\right) - D\label{eq:p2c3}\,,\end{aligned}$$ where $ \gamma $ is the determinant of $\gamma_{ij} $. The evolution system then becomes $$\frac{\partial \mathbf{U}}{\partial t} +
\frac{\partial \mathbf{F}^{\,i}}{\partial x^{\,i}} =
\mathbf{S}\,\,,
\label{eq:conservation_equations_gr}$$ with $$\begin{aligned}
\mathbf{U} & = & [D, S_j, \tau], \nonumber\\
\mathbf{F}^{\,i} & = & \alpha
\left[ D \tilde{v}^{\,i}, S_j \tilde{v}^{\,i} + \delta^{\,i}_j P,
\tau \tilde{v}^{\,i} + P v^{\,i} \right]\!, \nonumber \\
\mathbf{S} & = & \alpha
\bigg[ 0, T^{\mu \nu} \left( \frac{\partial g_{\nu j}}{\partial x^{\,\mu}} -
\Gamma^{\,\lambda}_{\mu \nu} g_{\lambda j} \right), \nonumber\\
& &\qquad\alpha \left( T^{\mu 0}
\frac{\partial \ln \alpha}{\partial x^{\,\mu}} -
T^{\mu \nu} \Gamma^{\,0}_{\mu \nu} \right) \bigg]\,.\end{aligned}$$Here, $ \tilde{v}^{\,i} = v^{\,i} - \beta^i / \alpha $ and $
\Gamma^{\,\lambda}_{\mu \nu} $ are the 4-Christoffel symbols. The time integration and coupling with curvature are carried out with the Method of Lines [@Hyman:1976cm]. The expressions for $\mathbf{S}$ are calculated in [`GRHydro`]{} by using the definition of the extrinsic curvature to avoid any time derivatives whatsoever, as discussed in detail in the code’s documentation, following a suggestion by Mark Miller based on experience with the [`GR3D`]{} code.
### Reconstruction techniques
In order to calculate fluxes at cell faces, we first must calculate function values on either side of the face. In practice, reconstructing the primitive variables yields more stable and accurate evolutions than reconstructing the conservatives. In what follows, we assume a Cartesian grid and loop over faces along each direction in turn. We define $q^L_{i+1/2}$ to be the value of a quantity $q$ on the left side of the face between $q_i\equiv q(x_i,y,z)$ and $q_{i+1}\equiv q(x_{i+1},y,z)$, where $x_i$ is the $i$th point in the $x$-direction, and $q^R_{i+1/2}$ the right side of the same face.
For total variation diminishing (TVD) methods, we let: $$q^L_{i+1/2} = q_i+\frac{f(q_i)\Delta x}{2};~~q^R_{i+1/2} = q_{i+1}-\frac{f(q_{i+1})\Delta x}{2}$$ where $f(q_i)$ is a slope-limited gradient function, typically determined by the values of $q_{i+1}-q_i$ and $q_i-q_{i-1}$, with a variety of different forms of the slope limiter available. In practice, all try to accomplish the same task of preserving monotonicity and removing the possibility of spuriously creating local extrema. Implemented methods include minmod, superbee [@Roe:1986cb], and monotonized central [@vanLeer:1977aa].
The piecewise parabolic method (PPM) is a multi-step method based around a quadratic fit to nearby points interpolated to cell faces [@Colella:1982ee], for which $q^L$ and $q^R$ are generally equivalent except near shocks and local extrema. The version implemented in [`GRHydro`]{} includes the steepening and flattening routines described in the original PPM papers, with a simplified dissipation procedure. Essentially non-oscillatory (ENO) methods use a divided differences approach to achieve third-order accuracy via polynomial interpolation [@Harten:1987un; @Shu:1999ho].
Both ENO and PPM yield third-order accuracy for smooth monotonic functions, whereas TVD methods typically yield second-order accurate values. Regardless of the reconstruction scheme chosen, all of these methods reduce to first order near local extrema and shocks.
### Riemann solvers
The Riemann problem involves the solution of the equation $$\partial_t q+\partial_i f^i(q)=0\label{eq:Riemann}$$ at some point $X$ representing a discontinuity between constant states. The exact solution can be quite complicated, involving five different waves with different characteristic speeds for a hydrodynamic problem (8 for GRMHD), so [`GRHydro`]{} implements three different approximate solvers to promote computational efficiency. In each case, the solution takes a self-similar form $q(\xi)$, where $\xi\equiv (x-X)/t$ represents the characteristic speed from the original shock location to the point in question in space and time.
The simplest method implemented is the Harten-Lax-van Leer-Einfeldt solver [@Harten:1983on; @Einfeldt:1988og] (HLL or HLLE, depending on the reference), which uses a two wave approximation to calculate the evolution along the shock front. With $\xi_-$ and $\xi_+$ the most negative and most positive wave speeds present on either side of the interface, the solution $q(\xi)$ is assumed to take the form $$\label{hlle1}
q = \left\{ \begin{array}[c]{r c l} q^L & {\rm if} & \xi
< \xi_- \\ q_* & {\rm if} & \xi_- < \xi < \xi_+ \\
q^R & {\rm if} & \xi > \xi_+, \end{array}\right.$$ with the intermediate state $q_*$ given by $$\label{hlle2}
q_* = \frac{\xi_+ q^R - \xi_- q^L - f(
q^R) + f(q^L)}{\xi_+ - \xi_-}.$$ The numerical flux along the interface takes the form $$\label{hlleflux}
f(q) = \frac{\widehat{\xi}_+f(q^L) -
\widehat{\xi}_-f(q^R) + \widehat{\xi}_+ \widehat{\xi}_-
(q^R - q^L)}{\widehat{\xi}_+ - \widehat{\xi}_-},$$ where $$\label{hlle3}
\widehat{\xi}_- = {\rm min}(0, \xi_-), \quad \widehat{\xi}_+ =
{\rm max}(0, \xi_+).$$ It is these flux terms that are then used to evolve the hydrodynamic quantities.
The Roe solver [@Roe:1981ar] involves linearizing the evolution system for the hydrodynamic evolution, defining the Jacobian matrix $A\equiv \frac{\partial f}{\partial q}$ (see ), and working out the eigenvalues $\lambda^i$ and left and right eigenvectors, $l_i$ and $r^j$, assumed to be orthonormalized so that $l_i\cdot r^j=\delta_i^j$. Defining the characteristic variables $w_i=l_i\cdot q$, the characteristic equation becomes $$\partial_t w+\Lambda \partial_x w=0$$ with $\Lambda$ the diagonal matrix of eigenvalues. Letting $\Delta w_i\equiv w_i^L-w_i^R=l_i\cdot (q^L-q^R)$ represent the differences in the characteristic variables across the interface, the Roe flux is calculated as $$f(q)=\frac{1}{2}\left(f(q^L)+f(q^R)-\sum |\lambda_i| \Delta w_i r^i\right)$$ where the eigenvector appearing in the summed term are evaluated for the approximate Roe average flux $q_{\rm Roe}=\frac{1}{2}(q^L+q^R)$. The Marquina flux routines use a similar approach to the Roe method, but provide a more accurate treatment for supersonic flows (i.e., those for which the characteristic wave with $\xi=0$ is within a rarefaction zone) [@Donat:1996cs; @Aloy:1999ne].
### Conservative to primitive conversion
In order to invert eqs. – , solving for the primitive variables based on the values of the conservative ones, [`GRHydro`]{} uses a 1-dimensional Newton-Raphson approach that solves for a consistent value of the pressure. Defining the (known) undensitized conservative variables $\hat{D}\equiv D/\sqrt{\gamma} =
\rho W$, $\hat{S}^i=S^i/\sqrt{\gamma} = \rho h W^2 v^i$ and $\hat{\tau}\equiv \tau/\sqrt{\gamma} = \rho h W^2-P-\hat{D}$, as well as the auxiliary quantities $Q\equiv \rho h W^2 = \hat{\tau}+\hat{D}+P$ and $\hat{S}^2 = \gamma_{ij}\hat{S}^i\hat{S}^j = (\rho h W)^2(W^2-1)$, the former of which depends on $P$ and the latter of which is known, we find that $\sqrt{Q^2-\hat{S}^2} = \rho h W$ and thus $$\begin{aligned}
\rho&=&\frac{\hat{D}\sqrt{Q^2-\hat{S}^2}}{Q}\\
W&=&\frac{Q}{\sqrt{Q^2-\hat{S}^2}}\\
\epsilon&=&\frac{\sqrt{Q^2-\hat{S}^2}-PW-\hat{D}}{\hat{D}}.\end{aligned}$$ Given the new values of $\rho$ and $\epsilon$, one may then find the residual between the pressure and $P(\rho,\epsilon)$ and perform the Newton-Raphson step, so long as the values of $\frac{\partial P}{\partial \rho}$ and $\frac{\partial P}{\partial\epsilon}$ are known.
### Atmospheres, boundaries, and other code details
[`GRHydro`]{} uses an atmosphere, or extremely-low density floor, to avoid problems involving sound speeds and conservative-to-primitive variable conversion near the edges of matter distributions. The floor density value may be chosen in either absolute ([`rho_abs_min`]{}) or relative ([`rho_rel_min`]{}) terms. The atmosphere is generally assumed to have a specified polytropic EOS, regardless of the EOS describing the rest of the matter within the simulation. Whenever the numerical evolution results in a grid cell where conservative to primitive variable conversion yields negative values of either $\rho$ or $\epsilon$, the cell is reassigned to the atmosphere, with zero velocity.
At present, only flat boundary conditions are supported for hydrodynamic variables, since it is generally recommended that the outer boundaries of the simulation be placed far enough away so that all cells near the edge of the computational domain represent the atmosphere.
[`GRHydro`]{} has the ability to advect a set of passive scalars, referred to as “tracers”, as well as the electron fraction of a fluid, under the assumption that each tracer $X$ follows the conservation law $$\partial_t (DX)+\partial_i(\alpha \tilde{v}^iDX)=0.$$
Equations of State {#sec:eoss}
------------------
An equation of state connecting the primitive state variables is needed to close the system of GR hydrodynamics equations. The module [`EOS_Omni`]{} provides a unified general equation of state (EOS) interface and back-end for simple analytic and complex microphysical EOSs.
The polytropic EOS $$P = K\rho^\Gamma\,\,,$$ where $K$ is the polytropic constant and $\Gamma$ is the adiabatic index, is appropriate for adiabatic (= isentropic) evolution without shocks. When using the polytropic EOS, one does not need to evolve the total fluid energy equation, since the specific internal energy $\epsilon$ is fixed to $$\epsilon = \frac{K\rho^\Gamma}{(\Gamma - 1)\rho}\,.$$ Note that the adiabatic index $\Gamma = d\ln{P}/
d\ln{\rho}$ is related to the frequently used polytropic index $n$ via $n = 1 / (\Gamma - 1)$.
The gamma-law EOS[^3], $$P = (\Gamma - 1) \rho \epsilon\,\,,$$ allows for non-adiabatic flow but still assumes fixed microphysics, which is encapsulated in the constant adiabatic index $\Gamma$. This EOS has been used extensively in simulations of NS-NS and BH-NS mergers.
The hybrid EOS, first introduced by [@Janka:1993da], is a 2-piecewise polytropic with a thermal component designed for the application in simple models of stellar collapse. At densities below nuclear density, a polytropic EOS with $\Gamma = \Gamma_1 \approx 4/3$ is used. To mimic the stiffening of the nuclear EOS at nuclear density, the low-density polytrope is fitted to a second polytrope with $\Gamma = \Gamma_2 \gtrsim 2$. To allow for thermal contributions to the pressure due to shock heating, a gamma-law with $\Gamma = \Gamma_\mathrm{th}$ is used. The full EOS then reads $$\begin{aligned}
P & = & \frac{\Gamma - \Gamma_{\rm th}}{\Gamma - 1}
K \rho_{\rm nuc}^{\Gamma_1 - \Gamma}
\rho^{\Gamma} - \frac{(\Gamma_{\rm th} - 1) (\Gamma - \Gamma_1)}
{(\Gamma_1 - 1) (\Gamma_2 - 1)}
K \rho_{\rm nuc}^{\Gamma_1 - 1} \rho \nonumber \\
& & + (\Gamma_{\rm th} - 1) \rho \epsilon\,.
\label{eq:hybrid_eos}\end{aligned}$$In this, the total specific internal energy $\epsilon$ consists of a polytropic and a thermal contribution. In iron core collapse, the pressure below nuclear density is dominated by the pressure of relativistically degenerate electrons. For this, one sets $K = 4.897 \times 10^{14}$ \[cgs\] in the above. The thermal index $\Gamma_{\rm th}$ is usually set to $1.5$, corresponding to a mixture of relativistic ($\Gamma=4/3$) and non-relativistic ($\Gamma=5/3$) gas. Provided appropriate choices of EOS parameters (e.g., [@Dimmelmeier:2007ui]), the hybrid EOS leads to qualitatively correct collapse and bounce dynamics in stellar collapse.
[`EOS_Omni`]{} also integrates the [`nuc_eos`]{} driver routine, which was first developed for the [`GR1D`]{} code [@O'Connor:2009vw] for tabulated microphysical finite-temperature EOS which assume nuclear statistical equilibrium (NSE). [`nuc_eos`]{} handles EOS tables in [`HDF5`]{} format which contain entries for thermodynamic variables $X = X(\rho,T,Y_e)$, where $T$ is the matter temperature and $Y_e$ is the electron fraction. [`nuc_eos`]{} also supports calls for $X = X(\rho,\epsilon,Y_e)$ and carries out a Newton iteration to find $T(\rho,\epsilon,Y_e)$. For performance reasons, [`nuc_eos`]{} employs simple tri-linear interpolation in thermodynamic quantities and thus requires finely spaced tables to maintain thermodynamic consistency at an acceptable level. EOS tables in the format required by [`nuc_eos`]{} are freely available from [http://stellarcollapse.org]{}.
Analysis {#sec:analysis}
--------
It is often beneficial and sometimes necessary to evaluate analysis quantities during the simulation rather than post-processing variable output. Beyond extracting physics, these quantities are often used as measures of how accurately the simulation is progressing. In the following, we describe the common quantities available through Einstein Toolkit modules, and how different modules approach these quantities with differing assumptions and algorithms. The most common analysis quantities provided fall broadly into several different categories, including horizons, masses and momenta, and gravitational waves. Note that several modules bridge these categories and some fall outside them, including routines to perform constraint monitoring and to compute commonly used derived spacetime quantities. The following discussion is meant as an overview of the most common tools rather than an exhaustive list of the functionality provided by the Einstein Toolkit. In most cases, the analysis modules work on the variables stored in the base modules discussed in Sec. \[sec:base\_modules\], [`ADMBase`]{}, [`TmunuBase`]{}, and [`HydroBase`]{}, to be as portable as possible.
### Horizons
When spacetimes contain a BH, localizing its horizon is necessary for describing time-dependent quasi-local measures such as its mass and spin. The Einstein Toolkit provides two modules — [`AHFinder`]{} and [`AHFinderDirect`]{} — for locating the defined locally on a hypersurface. The module [`EHFinder`]{} is also available to search an evolved spacetime for the globally defined event horizons.
[`EHFinder`]{} [@Diener:2003jc] evolves a null surface backwards in time given an initial guess (e.g., the last apparent horizon) which will, in the vicinity of an event horizon, converge exponentially to its location. This must be done after a simulation has already evolved the initial data forward in time with enough 3D data written out that the full 4-metric can be recovered at each timestep.
In [`EHFinder`]{}, the null surface is represented by a function $f(t,x^i)=0$ which is required to satisfy the null condition $g^{\alpha\beta} \partial_\alpha f \partial_\beta f = 0$. In the standard numerical 3+1 form of the metric, this null condition can be expanded out into an evolution equation for $f$ as $$\partial_t f = \beta^i \partial_i f - \sqrt{\alpha^2 \gamma^{ij}
\partial_i f \partial_j f}$$ where the roots are chosen to describe outgoing null geodesics. The function $f$ is chosen such that it is negative within the initial guess of the horizon and positive outside, initially set to a distance measure from the initial surface guess $f(t_0,x^i)=\sqrt{(x^i-x^i_0)(x_i-x_{i(0)})}-r_0$. There is a numerical problem with the steepening of $\nabla f$ during the evolution, so the function $f$ is regularly re-initialized during the evolution to satisfy $|\nabla f|\simeq1$. This is done by evolving $$\frac{df}{d\lambda} = -\frac{f}{\sqrt{f^2+1}}\left(|\nabla f|-1\right)$$ for some unphysical parameter $\lambda$ until a steady state has been reached. As the isosurface $f=0$ converges exponentially to the event horizon, it is useful to evolve two such null surfaces which bracket the approximate position of the anticipated event horizon to further narrow the region containing the event horizon.
However, event horizons can only be found after the full spacetime has been evolved. It is often useful to know the positions and shapes of any BH on a given hypersurface for purposes such as excision, accretion, and local measures of its mass and spin. The Einstein Toolkit provides several algorithms of varying speed and accuracy to find marginally trapped surfaces, of which the outermost are . All finders make use of the fact that null geodesics have vanishing expansion on an which, in the usual 3+1 quantities, can be written $$\label{eq:ah_theta}
\Theta \equiv \nabla_i n^i + K_{ij} n^i n^j - K = 0$$ where $n^i$ is the unit outgoing normal to the 2-surface.
The module [`AHFinder`]{} provides two algorithms for locating . The minimization algorithm [@Anninos:1996ez] finds the local minimum of $\oint_S (\Theta - \Theta_o )^2 d^2S$ corresponding to a surface of constant expansion $\Theta_o$, with $\Theta_o=0$ corresponding to the For time-symmetric data, the option exists to find instead the minimum of the surface area, which in this case corresponds to an An alternative algorithm provided by [`AHFinder`]{}, the flow algorithm [@Gundlach:1997us], on which [`EHFinder`]{} is also based. Defining a surface as a level set $f(x^i)=r-h(\theta,\phi)=0$, and introducing an unphysical timelike parameter $\lambda$ to parametrize the flow of $h$ towards a solution, can be rewritten $$\partial_\lambda h = - \left( \frac{\alpha}{\ell_\mathrm{max}
(\ell_\mathrm{max}+1)} + \beta \right) \left( 1 -
\frac{\beta}{\alpha} L^2\right)^{-1} \rho \Theta$$ where $\rho$ is a strictly positive weight, $L^2$ is the Laplacian of the 2D metric, and $\alpha$, $\beta$, and $\ell_\mathrm{max}$ are free parameters. Decomposing $h(\theta,\phi)$ onto a basis of spherical harmonics, the coefficients $a_{\ell m}$ evolve iteratively towards a solution as $$a_{\ell m}^{(n+1)} = a_{\ell m}^{(n)} -
\frac{\alpha + \beta \ell_\mathrm{max}
\left(\ell_\mathrm{max}+1\right) }
{\ell_\mathrm{max}\left(\ell_\mathrm{max}+1\right)
\left(1+\beta\ell(\ell+1)/\alpha\right)}
\left(\rho\Theta\right)_{\ell m}^{(n)}$$
The [`AHFinderDirect`]{} module [@Thornburg:2003sf] is a faster alternative to [`AHFinder`]{}. Its approach is to view as an elliptic PDE for $h(\theta,\phi)$ on $S^2$ using standard finite differencing methods. Rewriting in the form $$\Theta \equiv \Theta\left(h,\partial_u h,\partial_{uv}h;
\gamma_{ij},K_{ij},\partial_k \gamma_{ij}\right) = 0\,,$$ the expansion $\Theta$ is evaluated on a trial surface, then iterated using a Newton-Raphson method to solve $\bf{J}\cdot\delta h=-\Theta$, where $\bf{J}$ is the Jacobian matrix. The drawback of this method is that it is not guaranteed to give the outermost marginally trapped surface. In practice however, this limitation can be easily overcome by either a single good initial guess, or multiple less accurate initial guesses.
### Masses and Momenta
Two distinct measures of mass and momenta are available in relativistic spacetimes. First, ADM mass and angular momentum evaluated as either surface integrals at infinity or volume integrals over entire hypersurfaces give a measure of the total energy and angular momentum in the spacetime. The module [`ML_ADMQuantities`]{} of the McLachlan code [@McLachlan:web] uses the latter method, creating gridfunctions containing the integrand of the volume integrals [@Yo:2002bm]: $$\begin{aligned}
M &=& \frac{1}{16\pi} \int_\Omega d^3 x \left[ e^{5 \phi}
\left( 16 \pi E + \tilde{A}_{ij} \tilde{A}^{ij} - \frac23 K^2
\right) - e^\phi \tilde{R} \right] \\
J_i &=& \frac{1}{8 \pi} \varepsilon_{ij}{}^k \int_\Omega d^3 x \left[
e^{6\phi}\left( \tilde{A}^j{}_k + \frac23 x^j \tilde{D}_k K
- \frac12 x^j \tilde{A}_{\ell n} \partial_k \tilde{\gamma}^{\ell n}
+ 8 \pi x^j S_k \right) \right]\end{aligned}$$ on which the user can use the reduction functions provided by [Carpet]{} to perform the volume integral. We note that [`ML_ADMQuantities`]{} inherits directly from the BSSN variables stored in [`McLachlan`]{} rather than strictly from the base modules. As the surface terms required when converting a surface integral to a volume integral are neglected, this procedure assumes that the integrals of $\tilde{D}^i e^\phi$ and $e^{6\phi} \varepsilon_{ij}{}^k x^j \tilde{A}^\ell{}_k$ over the boundaries of the computational domain vanish. The ADM mass and angular momentum can also be calculated from the variables stored in the base modules using the [`Extract`]{} module, as surface integrals [@Bowen:1980yu] $$\begin{aligned}
M &=& - \frac{1}{2\pi} \oint \tilde{D}^i \psi d^2 S_i \\
J_i &=& \frac{1}{16\pi} \varepsilon_{ijk} \oint \left( x^j K^{km}
- x^k K^{jm} \right) d^2 S_m \end{aligned}$$ on a specified spherical surface, preferably one far from the center of the domain since these quantities are only properly defined when calculated at infinity.
There are also the quasi-local measures of mass and angular momentum, from any found during the spacetime. Both [`AHFinderDirect`]{} and [`AHFinder`]{} output the corresponding mass derived from the area of the horizon $m_H =
\sqrt{A/(16\pi)}$.
The module [`QuasiLocalMeasures`]{} implements the calculation of mass and spin multipoles from the isolated and dynamical horizon formalism [@Dreyer:2002mx; @Schnetter:2006yt], as well as a number of other proposed formulæ for quasilocal mass, linear momentum and angular momentum that have been advanced over the years [@Szabados:2004ql]. Even though there are only a few rigorous proofs that establish the properties of these latter quantities, they have been demonstrated to be surprisingly helpful in numerical simulations (see, e.g., [@Lovelace:2009dg]), and are therefore an indispensable tool in numerical relativity. [`QuasiLocalMeasures`]{} takes as input a horizon surface, or any other surface that the user specifies (like a large coordinate sphere) and can calculate useful quantities such as the Weyl or Ricci scalars or the three-volume element of the horizon world tube in addition to physical observables such as mass and momenta.
Finally, the module [`HydroAnalysis`]{} additionally locates the (coordinate) center of mass as well as the point of maximum rest mass density of a matter field.
### Gravitational Waves
One of the main goals of numerical relativity to date is modeling gravitational waveforms that may be used in template generation to help analyze data from the various gravitational wave detectors around the globe. The Einstein Toolkit includes modules for extracting gravitational waves via either the Moncrief formalism of a perturbation on a Schwarzschild background or the calculation of the Weyl scalar $\Psi_4$.
The module [`Extract`]{} uses the Moncrief formalism [@Moncrief:1974am] to extract gauge-invariant wave functions $Q_{\ell m}^\times$ and $Q_{\ell
m}^+$ given spherical surfaces of constant coordinate radius. The spatial metric is expressed as a perturbation on Schwarzschild and expanded into a tensor basis of the Regge-Wheeler harmonics [@Regge:1957td] described by six standard Regge-Wheeler functions $\lbrace c_1^{\times\ell m}, c_2^{\times\ell m},
h_1^{+\ell m}, H_2^{+\ell m},K^{+\ell m}, G^{+ \ell m} \rbrace$. From these basis functions the gauge-invariant quantities: $$\begin{aligned}
Q_{\ell m}^\times &=& \sqrt{\frac{2(\ell+2)!}{(\ell-2)!}}
\left[c_1^{\times\ell m} + \frac12\left( \partial_r - \frac{2}{r}
\right) c_2^{\times\ell m} \right] \frac{S}{r} \\
Q_{\ell m}^+ &=& \frac{1}{\Lambda} \sqrt{ \frac{2(\ell-1)(\ell+2)}
{\ell(\ell+1)}} \Bigg( \ell(\ell+1)S(r^2 \partial_r G^{+ \ell m}
- 2 h_1^{+\ell m} ) \nonumber \\
& & + 2rS(H_2^{+\ell m}-r\partial_rK^{+\ell m})
+ \Lambda r K^{+\ell m} \Bigg)\end{aligned}$$ are calculated, where $S=1-2M/r$ and $\Lambda=(\ell-1)(\ell+2)+6M/r$. These functions then satisfy the wave equations: $$\begin{aligned}
(\partial_t^2-\partial_{r^*}^2)Q_{\ell m}^\times &=&
- S \left[ \frac{\ell(\ell+1)}{r^2}-\frac{6M}{r^3} \right]
Q_{\ell m}^\times \\
(\partial_t^2-\partial_{r^*}^2)Q_{\ell m}^+ &=&
- S \Bigg[ \frac{1}{\Lambda^2} \left( \frac{72M^3}{r^5}-\frac{12M
(\ell-1)(\ell+2)}{r^3}\left(1-\frac{3M}{r}\right) \right)
\nonumber \\
& & +
\frac{\ell(\ell^2-1)(\ell+2)}{r^2\Lambda} \Bigg] Q_{\ell m}^+\end{aligned}$$ where $r^*=r+2M \ln(r/2M-1)$. Since these functions describe the 4-metric as a perturbation on Schwarzschild, the spacetime must be approximately spherically symmetric for the output to be interpreted as first-order gauge-invariant waveforms.
For more general spacetimes, the module [`WeylScal4`]{} calculates the complex Weyl scalar $\Psi_4=C_{\alpha\beta\gamma\delta}\,n^\alpha
\bar{m}^\beta n^\gamma \bar{m}^\delta$, which is a projection of the Weyl tensor onto components of a null tetrad. [`WeylScal4`]{} uses the fiducial tetrad [@Baker:2001sf], written in 3+1 decomposed form as: $$\begin{aligned}
\ell^\mu &=& \frac{1}{\sqrt{2}}\left(u^\mu+\tilde{r}^\mu\right) \\
n^\mu &=& \frac{1}{\sqrt{2}}\left(u^\mu-\tilde{r}^\mu\right) \\
m^\mu &=&\frac{1}{\sqrt{2}}\left(\tilde{\theta}^\mu+i\tilde{\phi}^\mu\right)\end{aligned}$$ where $u^\mu$ is the unit normal to the hypersurface. The spatial vectors $\lbrace \tilde{r}^\mu, \tilde{\theta}^\mu, \tilde{\phi}^\mu
\rbrace$ are created by initializing $\tilde{r}^\mu =
\lbrace0,x^i\rbrace$, $\tilde{\phi}^\mu = \lbrace0,-y,x,0\rbrace$, and $\tilde{\theta}^\mu=\lbrace0,\sqrt{\gamma} \gamma^{ik} \varepsilon_{k\ell
m} \phi^\ell r^m\rbrace$, then orthonormalizing starting with $\tilde{\phi}^i$ and invoking a Gram-Schmidt procedure at each step to ensure the continued orthonormality of this spatial triad.
The Weyl scalar $\Psi_4$ is calculated explicitly in terms of projections of the 3-Riemann tensor onto a null tetrad, such that $$\begin{aligned}
\Psi_4 &=& \mathcal{R}_{ijk\ell} n^i \bar{m}^j n^k \bar{m}^\ell
+ 2 \mathcal{R}_{0jk\ell} \left( n^0 \bar{m}^j n^k \bar{m}^\ell
- \bar{m}^0 n^j n^k \bar{m}^\ell \right) \nonumber \\
&+& \mathcal{R}_{0j0\ell} \left( n^0 \bar{m}^j n^0 \bar{m}^\ell
+ \bar{m}^0 n^j \bar{m}^0 n^\ell - 2n^0 \bar{m}^j \bar{m}^0
n^\ell \right)\,.\end{aligned}$$ For a suitably chosen tetrad, this scalar in the radiation zone is related to the strain of the gravitational waves since $$h = h_+ - i h_\times = - \int_{-\infty}^t dt^\prime
\int_{-\infty}^{t^\prime} \Psi_4 dt^{\prime\prime}\,.$$
While the waveforms generated by [`Extract`]{} are already decomposed on a convenient basis to separate modes, the complex quantity $\Psi_4$ is provided by [`WeylScal4`]{} as a complex grid function. For this quantity, and any other real or complex grid function, the module [`Multipole`]{} interpolates the field $u(t,r,\theta,\phi)$ onto coordinate spheres of given radii and calculates the coefficients $$C^{\ell m} \left(t,r\right) = \int {}_s Y_{\ell m}^* u(t,r,\theta,\phi)
r^2 d\Omega$$ of a projection onto spin-weighted spherical harmonics ${}_s Y_{\ell m}$.
### Object tracking {#sec:object-tracking}
We provide a module ([`PunctureTracker`]{}) for tracking BH positions evolved with moving puncture techniques. It can be used with ([`CarpetTracker`]{}) to have the mesh refinement regions follow the BHs as they move across the grid. The BH position is stored as the centroid of a spherical surface (even though there is no surface) provided by [`SphericalSurface`]{}.
Since the punctures only move due to the shift advection terms in the BSSN equations, the puncture location is evolved very simply as $$\frac{d x^i}{d t} = -\beta^i, \label{eq:puncturetracking}$$ where $x^i$ is the puncture location and $\beta^i$ is the shift. Since the puncture location usually does not coincide with grid points, the shift is interpolated to the location of the puncture. Equation () is implemented with a simple first-order Euler scheme, accurate enough for controlling the location of the mesh refinement hierarchy.
Another class of objects which often needs to be tracked are neutron stars. Here is it usually sufficient to locate the position of the maximum density and adapt AMR resolution in these regions accordingly, coupled with the condition that this location can only move at a specifiable maximum speed.
### Other analysis modules
The remaining analysis capabilities of the Einstein Toolkit span a variety of primarily vacuum-based functions. First, modules are provided to calculate the Hamiltonian and momentum constraints which are used to monitor how well the evolved spacetime satisfies the Einstein field equations. Two modules, [`ADMConstraints`]{} and [`ML_ADMConstraints`]{} provide these quantities. Both calculate these directly from variables stored in the base modules described in Sec. \[sec:base\_modules\], explicitly written as: $$\begin{aligned}
H &=& R - K^i{}_j K^j{}_i + K^2 - 16 \pi E \label{eqn:analysis_hamiltonian_constraint}\\
M_i &=& \nabla_j K_i{}^j - \nabla_i K - 8 \pi S_i\end{aligned}$$ where $S_i=-\frac{1}{\alpha} \left( T_{i0} - \beta^j T_{ij} \right)$. The difference between these modules lies in how they access the stress energy tensor $T_{\mu\nu}$, as the module [`ADMConstraints`]{} uses a deprecated functionality which does not require storage for $T_{\mu\nu}$.
Finally, [`ADMAnalysis`]{} calculates a variety of derived spacetime quantities that are often useful in post-processing such as the determinant of the 3-metric $\det{\gamma}$, the trace of the extrinsic curvature $K$, the 3-Ricci tensor in Cartesian coordinates $\mathcal{R}_{ij}$ and its trace $\mathcal{R}$, as well as the 3-metric and extrinsic curvature converted to spherical coordinates.
Simulation Domain, Symmetries, Boundaries
-----------------------------------------
### Domains and Coordinates.
[Cactus]{} distinguishes between the *physical* domain, which lives in the continuum, and *discrete* domain, which consists of a discrete set of grid points. The physical domain is defined by its coordinate extent and is independent of the numerical resolution; in particular, the boundary of the physical domain has a width of zero (and is thus a set of measure zero). The discrete domain is defined indirectly via a discretization procedure that specifies the number of boundary points, their location with respect to the physical boundary, and either the grid spacing or the number of grid points spanning the domain. This defines the number and location of the grid points in the discrete domain. The discrete domain may have grid points outside of the physical domain, and may have a non-zero boundary width. This mechanism ensures that changes in the numerical resolution do not affect the extent of the physical domain, i.e., that the discrete domains converge to the physical domain in the limit of infinite resolution. The Einstein Toolkit provides the [`CoordBase`]{} thorn that facilitates the definition of the simulation domain independent of the actual evolution thorn used, allowing it to be specified at run time via parameters in the same way that parameters describing the physical system are specified. [`CoordBase`]{} exposes a public runtime interface that allows other thorns to query the domain description in a uniform way. This is used by [`Carpet`]{} to query [`CoordBase`]{} for the discrete grid when creating the hierarchy of grids, automatically ensuring a consistent grid description between the two thorns. Evolution thorns such as [`McLachlan`]{} use the domain information to decide which points are evolved and therefore require the evaluation of the right-hand-side expression, and which ones are set via boundary or symmetry conditions.
### Symmetries and Boundary Conditions.
The Einstein Toolkit includes two thorns, [`Boundary`]{} and [`SymBase`]{}, to provide a generic interface to specify and implement boundary and symmetry conditions. The toolkit includes built-in support for a set of reflecting or rotating symmetry conditions that can be used to reduce the size of the simulation domain. These symmetries include periodicity in any of the coordinate directions (via the [`Periodic`]{} module), reflections across the coordinate planes (via the [`Reflection`]{} module), $90^{\circ}$ and $180^{\circ}$ rotational symmetries (via the [`RotatingSymmetry90`]{} and [`RotatingSymmetry180`]{} modules respectively) about the $z$ axis, and a continuous rotational symmetry (via the [`Cartoon2D`]{} thorn) [@Alcubierre:1999ab]. [`Cartoon2D`]{} allows fully three dimensional codes to be used in axisymmetric problems by evolving a slice in the $y=0$ plane and using the rotational symmetry to populate ghost points off the plane (see Figure \[fig:cartoon-plane\]).
![Grid layout of a simulation using [`Cartoon2D`]{}. The $z$-axis is the axis of rotational symmetry. Image courtesy of Denis Pollney.[]{data-label="fig:cartoon-plane"}](cartoon_plane){width="20.00000%"}
In applying symmetries to populate ghost zones, the transformation properties of tensorial quantities (including tensor densities and non-tensors such as Christoffel symbols) are correctly taken into account, just as they are in the interpolation routines present in [Cactus]{}. Thus, symmetries are handled transparently from the point of view of user modules (see Figure \[fig:faces\] for an illustration).
![Iterative transformation of a point $x$ in quadrant 3 to the corresponding point $x''$ for which there is actual data stored. In this example, two reflection symmetries along the horizontal and vertical axis are present. Notice how the vector components change in transformations $A$ and $B$.[]{data-label="fig:faces"}](faces)
The [`Boundary`]{} thorn serves as a registry for available boundary conditions and provides basic scheduling to enforce all requested boundary conditions at the proper times. It also provides a basic set of boundary conditions to be used by user thorns. The “flat” boundary conditions often used for hydrodynamic variables that approach an atmosphere value fall in this category. More complicated boundary conditions are often implemented as modifications to the evolution equations and are not handled directly by [`Boundary`]{}. Examples are the radiative (Sommerfeld) and extrapolation boundary conditions provided by thorn [`NewRad`]{}.
### Adaptive Mesh Refinement
The Einstein toolkit currently supports feature-based mesh refinement, which is based on extracting quantities such as the locations of BHs or NSs and then constructing a mesh hierarchy (stacks of refined regions) based on the locations, sizes, and speeds of these objects. This allows tracking objects as they move through the domain. One can also add or remove stacks if, for instance, the number of objects changes. Full AMR based on a local error estimate is supported by [`Carpet`]{}, but the Einstein Toolkit does not presently provide a suitable regridding thorn to create such a grid. If initial conditions are constructed outside of [Carpet]{} (which is often the case), then the initial mesh hierarchy has to be defined manually. In order to facilitate the description of the mesh hierarchy the Einstein toolkit provides two regridding modules in the [`CarpetRegrid`]{} and [`CarpetRegrid2`]{} thorns. Both thorns primarily support box-in-box type refined meshes, which are well suited to current binary BH simulations in which the high-resolution regions are centered on the individual BHs. Figure \[fig:bbh-boxes\] shows a typical set of nested boxes during the inspiral phase of a binary BH merger simulation.
![Nested boxes following the individual BHs in binary BH merger simulation (see Section \[sec:bbh-example\]), with the location of the individual BHs found by [`PunctureTracker`]{}. The innermost three of the nine levels of mesh refinement used in this simulation are shown. Notice the use of [`RotatingSymmetry180`]{} to reduce the computational domain.[]{data-label="fig:bbh-boxes"}](bbh-boxes)
[`CarpetRegrid`]{} provides a number of different ways to specify the refined regions, e.g., as a set of boxes centered around the origin or as an explicit list of regions that make up the grid hierarchy. Traditionally, groups using [`CarpetRegrid`]{} have employed auxiliary thorns that are not part of the Einstein Toolkit to create this list of boxes based on information obtained from apparent horizon tracking or other means. [`CarpetRegrid2`]{} provides a user-friendly interface to define sets of nested boxes that follow BHs or other tracked objects. Object coordinates are updated by [`CarpetTracker`]{}, which provides a simple interface to the object trackers [`PunctureTracker`]{} and [`NSTracker`]{} (see section \[sec:object-tracking\]) in order to have the refined region follow the moving objects. [`CarpetRegrid2`]{} contains code to handle the $\pi$-symmetry provided by [`RotatingSymmetry180`]{}, enforcing the symmetry on the resulting grid layout (see Figure \[fig:rot180-grid\]).
![Grid layout created by [`CarpetRegrid2`]{}. In this example we use one ghost point, one boundary point, and two buffer points as well as [`RotatingSymmetry180`]{}. There are two refinement levels present, a coarse one represented by big red circles and a fine one represented by small black circles. The filled black circle is the single point specified by the user. [`CarpetRegrid2`]{} surrounded it with a layer of buffer points, indicated by the cyan filled circles. The open circles are ghost and boundary points which are maintained by [`Carpet`]{}. The presence of the $\pi$-symmetry forces [`CarpetRegrid2`]{} to create the tiny region to the bottom left of the grid. It serves only as a source for the boundary condition.[]{data-label="fig:rot180-grid"}](rot180-grid){width="30.00000%"}
Examples
========
To demonstrate the properties of the code and its capabilities, we have used it to simulate common astrophysical configurations of interest. Given the community-oriented direction of the project, the parameter files required to launch these simulations and a host of others are included and documented in the code releases, along with the data files produced by a representative set of simulation parameters to allow for code validation and confirmation of correct code performance on new platforms and architectures. As part of the internal validation process, nightly builds are checked against a set of benchmarks to ensure that consistent results are generated with the inclusion of all new commits to the code.
The performance of the Toolkit for vacuum configurations is demonstrated through evolutions of single, rotating BHs and the merger of binary black hole configurations (sections \[sec:1bh-example\] and \[sec:bbh-example\], respectively). Linear oscillations about equilibrium for an isolated NS are discussed in section \[sec:tov\_oscillations\], and the collapse of a NS to a BH, including dynamical formation of a horizon, in section \[sec:collapse\_example\]. Finally, to show a less traditional application of the code, we show its ability to perform cosmological simulations by evolving a Kasner spacetime (see section \[sec:cosmology\]).
Spinning BH {#sec:1bh-example}
-----------
As a first example, we perform simulations of a single distorted rotating BH. We use [`TwoPunctures`]{} to set up initial data for a single puncture of mass $M_{\mathrm{bh}}=1$ and dimensionless spin parameter $a = S_{\mathrm{bh}}/M_{\mathrm{bh}}^2 = 0.7$. Evolution of the data is performed by [`McLachlan`]{}, apparent horizon finding by [`AHFinderDirect`]{} and gravitational wave extraction by [`WeylScal4`]{} and [`Multipole`]{}. Additional analysis of the horizons is done by [`QuasiLocalMeasures`]{}. The runs were performed with fixed mesh refinement provided by [`Carpet`]{}, using 8 levels of refinement on a quadrant grid (symmetries provided by [`ReflectionSymmetry`]{} and [`RotatingSymmetry180`]{}). The outer boundaries were placed at $R=256M$. We performed runs at 3 different resolutions: the low resolution was $0.024M (3.072M)$, medium was $0.016M (2.048M)$ and high was $0.012M (1.536M)$, where the numbers refer to the resolution on the finest (coarsest) grid. The runs where performed using the tapering evolution scheme in [`Carpet`]{} to avoid interpolation in time during prolongation. The initial data corresponds to a rotating BH perturbed by a Brill wave and, as such, has a non-zero gravitational wave content. We evolved the BH using 4th-order finite differencing from $T=0M$ until it had settled down to a stationary state at $T=120M$.
Figure \[fig:kerr\_waves\] shows the $\ell =2, m=0$ mode of $r\Psi_4$ extracted at $R=30M$, and its numerical convergence.
![The extracted $\ell =2, m=0$ mode of $\Psi_4$ as function of time from the high resolution run (top plot). The extraction was done at $R=30M$. Shown is both the real (solid black curve) and the imaginary (dashed blue curve) part of the waveform. At the bottom, we show the difference between the medium and low resolution runs (solid black curve), between the high and medium resolution runs (dashed blue curve), and the scaled difference (for 4th order convergence) between the medium and low resolution runs (dotted red curve) for the real part of the $\ell =2, m=0$ waveforms.[]{data-label="fig:kerr_waves"}](waves "fig:"){width="90.00000%"} ![The extracted $\ell =2, m=0$ mode of $\Psi_4$ as function of time from the high resolution run (top plot). The extraction was done at $R=30M$. Shown is both the real (solid black curve) and the imaginary (dashed blue curve) part of the waveform. At the bottom, we show the difference between the medium and low resolution runs (solid black curve), between the high and medium resolution runs (dashed blue curve), and the scaled difference (for 4th order convergence) between the medium and low resolution runs (dotted red curve) for the real part of the $\ell =2, m=0$ waveforms.[]{data-label="fig:kerr_waves"}](waves_conv "fig:"){width="90.00000%"}
In the top plot the black (solid) curve is the real part and the blue (dashed) curve is the imaginary part of $r \Psi_4$ for the high resolution run. Curves from the lower resolution are indistinguishable from the high resolution curve at this scale. In the bottom plot the black (solid) curve shows the absolute value of the difference between the real part of the medium and low resolution waveforms while the blue (dashed) curve shows the absolute value of the difference between the high and medium resolution waveforms in a log-plot. The red (dotted) curve is the same as the blue (dashed) curve, except it is scaled for 4th order convergence. With the resolutions used here this factor is $\left (0.016^4-0.024^4\right )/\left ( 0.012^4-0.016^4\right) \approx 5.94$.
Figure \[fig:kerr\_waves\_l4\] shows similar plots for the $\ell =4, m=0$ mode of $r\Psi_4$, again extracted at $R=30 M$.
![Real part of the extracted $\ell =4, m=0$ mode of $\Psi_4$ as function of time (top plot) for the high (solid black curve), medium (dashed blue curve) and low (dotted red curve) resolution runs. The extraction was done at $R=30M$. The bottom plot shows for the real part of the $\ell =4, m=0$ waveforms the difference between the medium and low resolution runs (solid black curve), the difference between the high and medium resolution runs (dashed blue curve) as well as the scaled (for 4th order convergence) difference between the medium and low resolution runs (dotted red curve).[]{data-label="fig:kerr_waves_l4"}](waves_l4 "fig:"){width="90.00000%"} ![Real part of the extracted $\ell =4, m=0$ mode of $\Psi_4$ as function of time (top plot) for the high (solid black curve), medium (dashed blue curve) and low (dotted red curve) resolution runs. The extraction was done at $R=30M$. The bottom plot shows for the real part of the $\ell =4, m=0$ waveforms the difference between the medium and low resolution runs (solid black curve), the difference between the high and medium resolution runs (dashed blue curve) as well as the scaled (for 4th order convergence) difference between the medium and low resolution runs (dotted red curve).[]{data-label="fig:kerr_waves_l4"}](waves_l4_conv "fig:"){width="90.00000%"}
The top plot in this case shows only the real part of the extracted waveform but for all three resolutions (black solid curve is high, blue dashed curve is medium and red dotted curve is low resolution). Since the amplitude of this mode is almost a factor of 20 smaller than the $\ell =2, m=0$ mode there are actually small differences visible between resolutions in the beginning of the waveform. The bottom plot shows the convergence of the real part of the $\ell =4, m=0$ mode (compare with the bottom plot in Figure \[fig:kerr\_waves\]) and demonstrates that even though the amplitude is much smaller we still obtain close to perfect fourth-order convergence.
In addition to the modes shown in Figure \[fig:kerr\_waves\] and \[fig:kerr\_waves\_l4\] we note that the extracted $\ell =4, m=4$ mode is non-zero due to truncation error, but shows fourth-order convergence to zero with resolution (this mode is not present in the initial data and is not excited during the evolution). Other modes are zero to round-off due to symmetries at all resolutions.
Since there is non-trivial gravitational wave content in the initial data, the mass of the BH changes during its evolution. In figure \[fig:ah\_mass\], we show in the top plot the irreducible mass as calculated by [`AHFinderDirect`]{} as a function of time at the high (black solid curve), medium (blue dashed curve) and low (red dotted curve) resolutions.
![The top plot shows the irreducible mass of the apparent horizon as a function of time at low (black solid curve), medium (blue dashed curve) and high (red dotted curve) resolutions. The inset is a zoom in on the $y$-axis to more clearly show the differences between the resolutions. The bottom plot shows the convergence of the irreducible mass. The black (solid) curve shows the difference between the medium and low resolution results, the blue (dashed) curve shows the difference between the high and medium resolution results. The red (dotted) and green (dash-dotted) show the difference between the high and medium resolutions scaled according to fourth and third-order convergence respectively.[]{data-label="fig:ah_mass"}](ah_mass "fig:"){width="90.00000%"} ![The top plot shows the irreducible mass of the apparent horizon as a function of time at low (black solid curve), medium (blue dashed curve) and high (red dotted curve) resolutions. The inset is a zoom in on the $y$-axis to more clearly show the differences between the resolutions. The bottom plot shows the convergence of the irreducible mass. The black (solid) curve shows the difference between the medium and low resolution results, the blue (dashed) curve shows the difference between the high and medium resolution results. The red (dotted) and green (dash-dotted) show the difference between the high and medium resolutions scaled according to fourth and third-order convergence respectively.[]{data-label="fig:ah_mass"}](ah_mass_conv "fig:"){width="90.00000%"}
The inset shows in more detail the differences between the different resolutions. The irreducible mass increases by about 0.3% during the first $40M$ of evolution and then remains constant (within numerical error) for the remainder of the evolution. The bottom plot shows the convergence of the irreducible mass by the difference between the medium and low resolutions (black solid curve), the difference between the high and medium resolutions (blue dashed curved) as well as the scaled difference between the high and medium resolutions for fourth-order (red dotted curve) and third-order (green dash-dotted curve). The convergence is almost perfectly fourth-order until $T=50M$, then better than fourth-order until $T=60M$, and finally between third-order and fourth-order for the remainder of the evolution. The lack of perfect fourth-order convergence at late times may be attributed to non-convergent errors from the puncture propagating to the horizon location at the lowest resolution.
Finally, in Figure \[fig:ah\_mass\_spin\] we show the total mass (top plot) and the change in the spin, $\Delta S = S(t) - S(t=0)$, as calculated by [`QuasiLocalMeasures`]{}.
![The top plot shows the total mass and the bottom plot shows the change in spin (i.e. $\Delta S=S(t)-S(t=0)$ of the BH as a function of time. In both plots the black (solid) curve is for high, blue (dashed) for medium and red (dotted) for low resolution. In the bottom plot the green (dash-dotted) curve shows the high resolution result scaled for second-order convergence. The agreement with the medium resolution curve shows that the change in spin converges to zero as expected.[]{data-label="fig:ah_mass_spin"}](qlm_mass "fig:"){width="90.00000%"} ![The top plot shows the total mass and the bottom plot shows the change in spin (i.e. $\Delta S=S(t)-S(t=0)$ of the BH as a function of time. In both plots the black (solid) curve is for high, blue (dashed) for medium and red (dotted) for low resolution. In the bottom plot the green (dash-dotted) curve shows the high resolution result scaled for second-order convergence. The agreement with the medium resolution curve shows that the change in spin converges to zero as expected.[]{data-label="fig:ah_mass_spin"}](qlm_spin "fig:"){width="90.00000%"}
In both cases the black (solid) curve is for high, blue (dashed) for medium and red (dotted) for low resolution. Since the spacetime is axisymmetric the gravitational waves cannot radiate angular momentum. Thus any change in the spin must be due to numerical error and $\Delta S$ should converge to zero with increasing resolution. This is clearly shown in the bottom plot of Figure \[fig:ah\_mass\_spin\]; the green (dash-dotted) curve (the high resolution result scaled by a factor of $1.78$ for second-order convergence to the resolution of the medium resolution) and the blue (dashed) curve are on top of each other. Since the [`QuasiLocalMeasures`]{} thorn uses an algorithm which is only second-order accurate overall, this is the expected result. The increase of about 0.22% in the mass of the BH is caused solely by the increase in the irreducible mass.
BH Binary {#sec:bbh-example}
---------
To demonstrate the performance in the code for a current problem of wide scientific interest, we have evolved a non-spinning equal-mass BH binary system. The initial data represent a binary system in a quasi-circular orbit, with an initial separation chosen to be $r=6M$ so we may track the later inspiral, plunge, merger and ring down phases of the binary evolution. Table \[table:BHB\_ID\] provides more details about the initial binary parameters used to generate the initial data. The [`TwoPunctures`]{} module uses these initial parameters to solve , the elliptic Hamiltonian constraint for the regular component of the conformal factor (see section \[sec:twopunctures\]). The spectral solution for this example was determined by using $[n_A,n_B,n_{\phi}]=[28,28,14]$ collocation points, and, along with the Bowen-York analytic solution for the momentum constraints, represents constrained GR initial data $\{\gamma_{ij},K_{ij}\}$. The evolution is performed by the [`McLachlan`]{} module.
\[table:BHB\_ID\]
Configuration $x_1$ $x_2$ $p_x$ $p_y$ $m$ $M_{\rm ADM}$
--------------- ------- ------- ------- --------- --------- ---------------
QC3 3.0 -3.0 0.0 0.13808 0.47656 0.984618
: Initial data parameters for a non-spinning equal mass BH binary. The punctures are located on the $x$-axis at positions $x_1$ and $x_2$, with puncture bare mass parameters $m_1 = m_2 = m$, and momenta $\pm\vec p$.
\
The simulation domain spans the coordinate range $[[x_{\rm min},x_{\rm max}],[y_{\rm min},y_{\rm max}],[z_{\rm min},z_{\rm max}]]
= [[0,120],[-120,120],[0,120]]$, where we have taken advantage of both the equatorial symmetry (implemented by the [`ReflectionSymmetry`]{} module) and the $180\degree$ rotational symmetry around the $z$-axis, which we apply at the $x=0$ plane using the [`RotatingSymmetry180`]{} module. [`Carpet`]{} provides a hierarchy of refined grids centered at each puncture. Here, we used $7$ levels of refinement, where the box edge coordinate lengths are given by $[128,32,16,8,4,2]$ in units of the total binary mass, which is set to unity. Note that overlapping boxes are automatically redefined by [`Carpet`]{} into one unique region before the domain decomposition takes place.
Figure \[fig:tracks\_waveform\] shows the two puncture tracks throughout all phases of the binary evolution, provided by the [`PunctureTracker`]{} module. In the same plot we have recorded the intersection of the apparent horizon $2$-surface with the $z=0$ plane every time interval $t=10M$ during the evolution. A common horizon is first observed at $t=116M$. These apparent horizons were found by the [`AHFinderDirect`]{} module and their radius and location information stored as a $2$-surface with spherical topology by the [`SphericalSurface`]{} module. The irreducible mass and dimensionless spin of the merged BH were calculated by the [`QuasiLocalMeasures`]{} module, and were found to be $0.647 M$ and $-0.243 M^{-2}$, respectively.
Two modules are necessary to perform the waveform extraction. The first one, [`WeylScal4`]{}, calculates the Weyl scalar $\Psi_4$ in term of the metric components and its derivatives; these were computed to be $4$-th order accurate in this example. The second module, [`Multipole`]{}, interpolates the Weyl scalars onto spheres with centers and radii specified by the user, and performs a spherical harmonic multipole mode decomposition. Figure \[fig:tracks\_waveform\] shows the real and imaginary parts of the ($l=2$, $m=2$) mode for $\Psi_4$ extracted on a sphere centered at the origin at $R_{\rm obs} = 60M$. The number of grid points on the sphere was set to be $[n_{\theta},n_{\phi}]=[120,240]$, which yields an angular resolution of $2.6 \times 10^{-2}$ radians, and an error of the same order, since the surface integrals were calculated by midpoint rule – a first order accurate method. In order to evaluate the convergence of the numerical solution, we ran five simulations with different resolutions, and focus our analysis on the convergence of the phase and amplitude of the Weyl scalar $\Psi_4$. The mesh spacings adopted for the coarser grid in the AMR hierarchy for these different runs were $\{h_{\rm low},h_{\rm med},h_{\rm medh},h_{\rm high},h_{\rm higher}\}
=\{2.0M,1.5M,1.25M,1.0M,0.75M\}$, respectively, while the finer grid spacings can be easily found by dividing them by $2^k$ for the $k$th level of mesh refinement.For this example, we set $\{h^f_{\rm low},h^f_{\rm med},h^f_{\rm medh},h^f_{\rm high},h^f_{\rm higher}\}
=\{3.125M,2.344M,1.953M,1.563M,1.172M\}\times 10^{-2}$ for the finest grid in the different AMR hierarchies, respectively. Here, we consider the phase $\phi(t)$ and the amplitude $A(t)$ of the Weyl scalar $\Psi_4$ at $R_{\rm obs}=60M$. In order to take differences between the numerical values at two different grid resolutions, we use an $8$-th order accurate Lagrange operator to interpolate the higher-accuracy finite difference solution into the immediately coarser grid. We have experimented with $4$-th and $6$-th order as well, to evaluate the level of noise these interpolations could potentially introduce, but did not observe any noticeable difference and we report here on results from the higher-order option.
In Figure \[fig:amp\_phs\_convergence\], we show the convergence of the amplitude and phase of the Weyl scalar by plotting the logarithm of the absolute value of the differences between two levels of resolution. The differences clearly converge to zero as the resolution is increased. We also indicate on both plots the time at which the gravitational wave frequency reaches $\omega=0.2/M$. We follow a community standard, agreed to over the course of the NRAR[@NRAR:web] collaboration, that constrains the numerical resolution so that the accumulated phase error is not larger than $0.05$ radians at a gravitational wave frequency of $\omega=0.2/M$. From the plot, we assert that the phase error between the higher and high resolutions and the one between high and medium-high resolutions satisfies this criterion, while the phase error between the medium-high and medium resolutions barely satisfies the criterion; and the one between medium and low resolutions does not. We conclude then that the three highest resolution runs do have sufficient resolution to extract waveforms for use in the construction of analytic waveform templates.
![In the left panel, we plot the tracks corresponding to the evolution of two punctures initially located on the $x$-axis at $x=\pm 3$. The solid blue line represents puncture 1, and the dashed red line puncture 2. The circular dotted green lines are the intersections of the apparent horizons with the $z=0$ plane plotted every $10M$ during the binary evolution. A common horizon appears at $t=116M$. In the right panel, we plot the real (solid blue line) and imaginary (dotted red line) parts of the ($l=2$,$m=2$) mode of the Weyl scalar $\Psi_4$ as extracted at an observer radius of $R_{\rm obs}=60M$.[]{data-label="fig:tracks_waveform"}](tracks "fig:"){width="45.00000%"} ![In the left panel, we plot the tracks corresponding to the evolution of two punctures initially located on the $x$-axis at $x=\pm 3$. The solid blue line represents puncture 1, and the dashed red line puncture 2. The circular dotted green lines are the intersections of the apparent horizons with the $z=0$ plane plotted every $10M$ during the binary evolution. A common horizon appears at $t=116M$. In the right panel, we plot the real (solid blue line) and imaginary (dotted red line) parts of the ($l=2$,$m=2$) mode of the Weyl scalar $\Psi_4$ as extracted at an observer radius of $R_{\rm obs}=60M$.[]{data-label="fig:tracks_waveform"}](mp_psi4_l2_m2_r60 "fig:"){width="45.00000%"}
![Weyl scalar amplitude (left panel) and phase (right panel) convergence. The long dashed red curves represent the difference between the medium and low-resolution runs. The short dashed orange curves show the difference between the medium-high and medium resolution runs. The dotted brown ones, the difference between high and medium-high resolutions, while the solid blue curves represent the difference between the higher and high resolution runs. The dotted vertical green line at $t=154M$ indicates the point during the evolution at which the Weyl scalar frequency reaches $\omega=0.2/M$. Observe that the three highest resolutions accumulate a phase error below the standard of $0.05$ radians required by the NRAR collaboration. []{data-label="fig:amp_phs_convergence"}](amp_convergence_all_8th "fig:"){width="45.00000%"} ![Weyl scalar amplitude (left panel) and phase (right panel) convergence. The long dashed red curves represent the difference between the medium and low-resolution runs. The short dashed orange curves show the difference between the medium-high and medium resolution runs. The dotted brown ones, the difference between high and medium-high resolutions, while the solid blue curves represent the difference between the higher and high resolution runs. The dotted vertical green line at $t=154M$ indicates the point during the evolution at which the Weyl scalar frequency reaches $\omega=0.2/M$. Observe that the three highest resolutions accumulate a phase error below the standard of $0.05$ radians required by the NRAR collaboration. []{data-label="fig:amp_phs_convergence"}](phase_convergence_all_8th "fig:"){width="45.00000%"}
Linear oscillations of TOV stars {#sec:tov_oscillations}
--------------------------------
The examples in the previous subsections did not include the evolution of matter within a relativistic spacetime. One interesting test of a coupled matter-spacetime evolution is to measure the eigenfrequencies of a stable TOV star (see, e.g., [@Gourgoulhon:1991aa; @Romero:1996aa; @Shibata:1998sg; @Font:2001ew; @Shibata:2003iy]). These eigenfrequencies can be compared to values known from linear perturbation theory.
We begin our simulations with a self-gravitating fluid sphere, described by a polytropic equation of state. This one-dimensional solution is obtained by the code described in section \[sec:TOVSolver\], and is interpolated on the three-dimensional, computational evolution grid. This system is then evolved using the BSSN evolution system implemented in [`McLachlan`]{} and the hydrodynamics evolution system implemented in [`GRHydro`]{}.
For the test described here, we set up a stable TOV star described by a polytropic equation of state $p=K\rho^\Gamma$ with $K=100$ and $\Gamma=2$, and an initial central density of $\rho_c=1.28\times10^{-3}$. This model can be taken to represent a non-rotating NS with a mass of $M=1.4\mathrm{M}_\odot$. The computational domain is a cube of length $640\mathrm{M}$ with a base resolution of $2\mathrm{M}$ ($4\mathrm{M}$, $8\mathrm{M}$) in each dimension. Four additional grids refine the region around the star centered at the origin, each doubling the resolution, with sizes of $120\mathrm{M}$, $60\mathrm{M}$, $30\mathrm{M}$ and $15\mathrm{M}$, resulting in a resolution of $0.125\mathrm{M}$ ($0.25\mathrm{M}$, $0.5\mathrm{M}$) across the entire star.
In Figure \[fig:tov\_rho\_max\] we show the evolution of the central density of the star over an evolution time of $1300\mathrm{M}$ ($6.5\mathrm{ms}$). The initial spike is due to the perturbation of the solution resulting from the interpolation onto the evolution grid. The remaining oscillations are mainly due to the interaction of the star and the artificial atmosphere and are present during the whole evolution. Given enough evolution time, the frequencies of these oscillations can be measured with satisfactory accuracy.
\[fig:tov\_rho\_max\] {width="90.00000%"}
In Figure \[fig:tov\_mode\_spectrum\] we show the power spectral density (PSD) of the central density oscillations computed from a full 3D relativistic hydrodynamics simulation, compared to the corresponding frequencies as obtained with perturbative techniques (kindly provided by Kentaro Takami and computed using the method described in [@Yoshida:1999vj]). The PSD was computed using the entire time series of the high-resolution run, by removing the linear trend and averaging over Hanning windows overlapping half the signal length after padding the signal to five times its length. The agreement of the fundamental mode and first three overtone frequencies is clearly visible, but are limited beyond this by the finite numerical resolution. Higher overtones should be measurable with higher resolution, but at substantial computational cost.
\[fig:tov\_mode\_spectrum\] {width="90.00000%"}
Within this test it is also interesting to study the convergence behavior of the coupled curvature and matter evolution code. One of the variables often used for this test is the Hamiltonian constraint violation. This violation vanishes for the continuum problem, but is non-zero and resolution-dependent in discrete simulations. The expected rate of convergence of the hydrodynamics code lies between $1$ and $2$. It cannot be higher than $2$ due to the directional flux-split algorithm which is of second order. Depending on solution itself, the hydrodynamics code is only of first order in particular regions, e.g., at extrema (like the center of the star), or at the stellar surface.
Figure \[fig:tov\_ham\_conv\] shows the order of convergence of the Hamiltonian constraint violation, using the three highest-resolution runs, at the stellar center and a coordinate radius of $r=5\mathrm{M}$ which is about half way between the center and the surface. The observed convergence rate for most of the simulation time lies between $1.4$ and $1.5$ at the center, and between $1.6$ and $2$ at $r=5\mathrm{M}$, consistent with the expected data-dependent convergence order of the underlying hydrodynamics evolution scheme.
\[fig:tov\_ham\_conv\] {width="90.00000%"}
Neutron star collapse {#sec:collapse_example}
---------------------
The previous examples dealt either with preexisting BHs, either single or in a binary, or with a smooth singularity free spacetime, as in the case of the TOV star. The evolution codes in the toolkit are, however, also able to handle the dynamic formation of a singularity, that is follow a neutron star collapse into a BH. As a simple example of this process, we study the collapse of a non-rotating TOV star. We create initial data as in section \[sec:tov\_oscillations\] using $\rho_c=3.154\times10^{-3}$ and $K_{\mathrm{ID}} = 100$, $\Gamma
= 2$, yielding a star model of gravitational mass $1.67\,M_\odot$, that is at the onset of instability. As is common in such situations (e.g., [@Baiotti:2005vi]), we trigger collapse by reducing the pressure support after the initial data have been constructed by lowering the polytropic constant $K_{\mathrm{ID}}$ from its initial value to $K = 0.98 \, K_{\mathrm{ID}} = 98$. To ensure that the pressure-depleted configuration remains a solution of the Einstein constraint equations in the presence of matter, we rescale the rest mass density $\rho$ such that the total energy density $T_{nn}$ does not change: $$\rho' + K (\rho')^2 = \rho + K_{\mathrm{ID}} \rho^2.
\label{eqn:collapse_rho_rescaled}$$ Compared to the initial configuration, this rescaled star possesses a slightly higher central density and lower pressure. This change in $K$ accelerates the onset of collapse that would otherwise rely on being triggered by numerical noise, which would not be guaranteed to converge to a unique solution with increasing resolution. In order to resolve the star as well as to push the outer boundary far enough away (so that the star and the numerical outer boundary are not in causal contact during the simulation) we employ a fixed mesh refinement scheme. The outermost box has a radius of $R_0 = 204.8\,M_\odot$ and a resolution of $3.2\,M_\odot$ ($2.4\,M_\odot$, $1.6\,M_\odot$, $0.6\,M_\odot$ for higher convergence levels). Around the star, centered about the origin, we stack $5$ extra boxes of approximate size $8\times2^\ell\,M_\odot$ for $0 \le \ell \le 4$, where the resolution on each level is twice that of the surrounding level. In order to resolve the large density gradients developing during the collapse, two more levels with radii $4\,M_\odot$ and $2\,M_\odot$ are placed inside the star. We use the PPM reconstruction method and the HLLE Riemann solver to obtain second-order convergent results in smooth regions. Due to the presence of the density maximum at the center of the star and the non-smooth atmosphere at the edge of the star, we expect the observed convergence rate to be somewhat lower than second order, but higher than first order.
\[fig:tov\_collapse\_radii\] {width="90.00000%"}
\[fig:tov\_collapse\_H\_convergence\_at0\] ![Convergence factor for the Hamiltonian constraint violation at the center of the collapsing star. We plot convergence factors computed using a set of 4 runs covering the diameter of the star with $\approx$ 60, 80, 120, and 240 grid points. The units of time on the upper and lower $x$-axes are identical to those of Figure \[fig:tov\_collapse\_radii\]. ](H_convergence_at0 "fig:"){width="90.00000%"}
In Figure \[fig:tov\_collapse\_radii\], we plot the approximate coordinate size of the star as well as the circumferential radius of the apparent horizon that eventually forms in the simulation. The apparent horizon is first found at approximately the time when the star’s coordinate radius approaches its Schwarzschild radius, though one needs to keep in mind that the Schwarzschild radius is a circumferential radius, whereas the meaning of the coordinate radius in our BSSN calculation is likely somewhat different. In Figure \[fig:tov\_collapse\_H\_convergence\_at0\], we display the convergence factor obtained from $$\frac{H_{h_1}-H_{h_2}}{H_{h_2}-H_{h_3}} = \frac{h_1^Q-h_2^Q}{h_2^Q-h_3^Q}\,,
\label{eq:convergence-factor-definition}$$ for the Hamiltonian constraint violation at the center of the collapsing star. Here $H_{h_i}$ is the Hamiltonian constraint violation at the center of the star for a run with resolution $h_i$. Up to the time when the apparent horizon forms, the convergence order is an expected $\approx 1.5$. At later times, the singularity forming at the center of the collapsing star renders a pointwise measurement of the convergence factor at the center impossible.
Cosmology {#sec:cosmology}
---------
The Einstein Toolkit is not only designed to evolve compact-object spacetimes, but also to solve the initial-value problem for spacetimes with radically different topologies and global properties. In this section, we illustrate the evolution of an initial-data set representing a constant-$t$ section of a spacetime from the Gowdy $T^3$ class [@Gowdy:1971jh; @New:1997me]. Models in this class have the line element: $$\label{eq:gowdyT3}
ds^2=\tau^{-1/2}e^{\lambda/2}(-d\tau^2+dz^2)+\tau[e^P(dx+Qdy)^2+e^{-P}dy^2]$$ defined on a 3-torus $-x_0 \leq x \leq x_0$, $-y_0 \leq y \leq y_0$, $-z_0 \leq z \leq z_0$, with the functions $P$, $Q$ and $\lambda$ to be determined by the Einstein equations. For $P=Q=\lambda=0$, a coordinate transformation $t=4/3 \, \tau^{3/4}$ (plus a rescaling of the spatial coordinates) casts the line element into the form: $$\label{eq:kasner}
ds^2=-dt^2+t^{4/3}(dx^2+dy^2)+t^{-2/3}dz^2$$ which represents the familiar Kasner spacetime for a homogeneous but anisotropically expanding universe. In the 3+1 decomposition described above, this reads:
$$\begin{aligned}
\alpha(t) &=& 1 \\
\beta^i(t) &=& 0 \\
\gamma_{ij}(t) &=& {\rm diag}(t^{4/3},t^{4/3},t^{-2/3}) \\
K_{ij}(t) &=& - {\rm diag}(\frac{2}{3} \, t^{4/3},\frac{2}{3} \, t^{4/3},\frac{1}{3} \, t^{-2/3})\end{aligned}$$
In Figure \[fig:kasner\], we show the full evolution of the $t=1$ slice of spacetime , along with the associated error for a sequence of time resolutions.
![Top: the evolution of a vacuum spacetime of the type , with $P=Q=\lambda=0$; the initial data are chosen as $\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$. Bottom: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$; the errors are scaled according to the expectation for fourth-order convergence. \[fig:kasner\]](kasner.pdf "fig:"){width="90.00000%"} ![Top: the evolution of a vacuum spacetime of the type , with $P=Q=\lambda=0$; the initial data are chosen as $\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$. Bottom: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$; the errors are scaled according to the expectation for fourth-order convergence. \[fig:kasner\]](err.pdf "fig:"){width="90.00000%"}
Conclusion and Future Work
==========================
In this article, we described the Einstein Toolkit, a collection of freely available and easy-to-use computational codes for numerical relativity and relativistic astrophysics. The code details and example results present in this article represent the state of the Einstein Toolkit in its release ET\_2011\_05 “Curie,” released on April 21, 2011.
The work presented here is but a snapshot of the Einstein Toolkit’s ongoing development, whose ultimate goal it is to provide an open-source set of robust baseline codes to realistically and reproducibly model the whole spectrum of relativistic astrophysical phenomena including, but not limited to, isolated black holes and neutron stars, binary black hole coalescence in vacuum and gaseous environments, double neutron star and neutron star – black hole mergers, core-collapse supernovae, and gamma-ray bursts.
For this, much future work towards including proper treatments of magnetic fields, more complex equations of state, nuclear reactions, neutrinos, and photons will be necessary and will need to be matched by improvements in infrastructure (e.g., more flexible AMR on general grids) and computing hardware for the required fully coupled 3-D, multi-scale, multi-physics simulations to become reality. These tasks, as well as the others mentioned below, are likely to occupy a great deal of the effort spent developing future versions of the Einstein Toolkit over the next few years.
Without a doubt, collapsing stars and merging BH-NS and NS-NS binaries must be simulated with GRMHD to capture the effects of magnetic fields that in many cases will alter the simulation outcome on a qualitative level and may be the driving mechanisms behind much of the observable EM signature from GRBs (e.g., [@Woosley:2006fn]) and magneto-rotationally exploding core-collapse supernovae (e.g., [@Burrows:2007yx]). To date, all simulations that have taken magnetic fields into account are still limited to the ideal MHD approximation, which assumes perfect conductivity. Non-ideal GRMHD schemes are just becoming available (see, e.g., [@Palenzuela:2008sf; @DelZanna:2007pk]), but have yet to be implemented widely in many branches of numerical relativity.
Most presently published 3D GR(M)HD simulations, with the exception of recent work on massive star collapse (see, e.g., [@Ott:2006eu]) and binary mergers (see, e.g., [@Sekiguchi:2011zd]), relied on simple zero-temperature descriptions of NS stellar structure, with many assuming simple polytropic forms. Such EOSs are computationally efficient, but are not necessarily a good description for matter in relativistic astrophysical systems. The inclusion of finite-temperature EOSs, derived from the microphysical descriptions of high-density matter, will lead to qualitatively different and much more astrophysically reliable results (see, e.g., [@Ott:2006eu]). In addition, most GR(M)HD studies neglect transport of neutrinos and photons and their interactions with matter. Neutrinos in particular play a crucial role in core-collapse supernovae and in the cooling of NS-NS merger remnants, thus they must not be left out when attempting to accurately model such events. Few studies have incorporated neutrino and/or photon transport and interactions in approximate ways (see, e.g., [@Ott:2006eu; @Farris:2008fe; @Sekiguchi:2011zd; @Sekiguchi:2011mc]).
Besides new additions of physics modules, existing techniques require improvement. One example is the need for the gauge invariant extraction of gravitational waves from simulation spacetimes as realized by the Cauchy Characteristic Extraction (CCE) technique recently studied in [@Babiuc:11; @Reisswig:2010cd; @Reisswig:2011a]. The authors of one such CCE code [@Babiuc:11] have agreed to make their work available to the whole community by integrating their CCE routines into the Einstein Toolkit release 2011\_11 “Maxwell,” which will be described elsewhere.
A second much needed improvement of our existing methods is a transition to cell-centered AMR for GR hydrodynamic simulations, which would allow for exact flux conservation across AMR interfaces via a refluxing step that adjusts coarse and/or fine grid fluxes for consistency (e.g., [@Berger:1984zza]). This is also a prerequisite for the constrained transport method [@Toth:00] for ensuring the divergence-free condition for the magnetic field in a future implementation of GRMHD within the Einstein Toolkit. Work towards cell-centered AMR, refluxing, and GRMHD is underway and will be reported in a future publication.
While AMR can increase resolution near regions of interest within the computational domain, it does not increase the convergence order of the underlying numerical methods. Simulations of BHs can easily make use of high-order numerical methods, with eighth-order convergence common at present. However, most GRMHD schemes, though they implement high-resolution shock-capturing methods, are limited to 2nd-order numerical accuracy in the hydrodynamic/MHD sector while performing curvature evolution with 4th-order accuracy or more. Higher order GRMHD schemes are used in fixed-background simulations (e.g., [@Tchekhovskoy:2007zn]), but still await implementation in fully dynamical simulations.
Yet another important goal is to increase the scalability of the [ Carpet]{} AMR infrastructure. As we have shown, good scaling is limited to only a few thousand processes for some of the most widely used simulation scenarios. Work is in progress to eliminate this bottleneck [@Zebrowski:2011bl]. On the other hand, a production simulation is typically composed of a large number of components, and even analysis and I/O routines have to scale well to achieve overall good performance. This is a highly non-trivial problem, since most Einstein Toolkit physics module authors are neither computer scientists nor have they had extensive training in parallel development and profiling techniques. Close collaboration with experts in these topics has been fruitful in the past and will be absolutely necessary for the optimization of Einstein Toolkit codes for execution on the upcoming generation of true petascale supercomputers on which typical compute jobs are expected to be running on 100,000 and more compute cores.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors wish to thank Ed Seidel whose inspiration and vision has driven work towards the Einstein Toolkit over the past 15 years. We are also grateful to the large number of people who contributed to the Einstein Toolkit via ideas, code, documentation, and testing; without these contributions, this toolkit would not exist today.
The Einstein Toolkit is directly supported by the National Science Foundation in the USA under the grant numbers 0903973/0903782/0904015 (CIGR). Related grants contribute directly and indirectly to the success of CIGR, including NSF OCI-0721915, NSF OCI-0725070, NSF OCI-0832606, NSF OCI-0905046, NSF OCI-0941653, NSF AST-0855535, NSF DMS-0820923, NASA 08-ATFP08-0093, EC-FP7 PIRG05-GA-2009-249290 and Deutsche Forschungsgemeinschaft grant SFB/Transregio 7 “Gravitational-Wave Astronomy”. Results presented in this article were obtained through computations on the Louisiana Optical Network Initiative under allocation loni\_cactus05 and loni\_numrel07, as well as on NSF XSEDE under allocations TG-MCA02N014, TG-PHY060027N, TG-PHY100033, at the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the US Department of Energy under contract DE-AC03-76SF00098, at the Leibniz Rechenzentrum of the Max Planck Society, and on Compute Canada resources via project cfz-411-aa.
G. Allen acknowledges that this material is based upon work supported while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
[^1]: Although multi-block methods are supported by [Carpet]{}, the Einstein Toolkit itself does not currently contain any multi-block coordinate systems.
[^2]: A normal observer follows a worldline tangent to the unit normal on the 3-hypersurface.
[^3]: For historic reasons, this EOS is referred to as the “ideal fluid” EOS in [`GRHydro`]{}.
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author:
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bibliography:
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title: Light meson form factors at high $Q^2$ from lattice QCD
---
Introduction {#intro}
============
The electromagnetic form factor of the meson parameterises the deviations from the behaviour of a point-like particle when hit by a photon. By determining the form factor at different values of the square of the 4-momentum transfer, $Q^2$, we can test our knowledge of QCD as a function of $Q^2$. Measurements of $\pi$ and K form factors are key experiments in the new Jefferson Lab $12$ GeV upgrade (experiments E12-06-101 [@JLABE12-06-101] and E12-09-11 [@JLABE12-09-011]). The pion form factor is known experimentally but with sizeable uncertainties up to $Q^2 < 2.45$ GeV$^2$, and the new experiment will extend the $Q^2$ range up to $6$ GeV$^2$.
Lattice QCD calculations have been done at small $Q^2$ (see [@Brandt:2013ffb] for a review) as doing a calculation at small momenta is easier because of deteriorating signal to noise at large momentum. In [@pionradius] we studied the pion form factor close to $Q^2=0$ and determined the charge radius of the pion. The goal of this study is to provide predictions of the form factors at high $Q^2$ ahead of experiments and to test the applicability of asymptotic perturbative QCD (PQCD). Here we use the $\eta_s$ meson, a pseudoscalar meson made of strange quarks, as a “pseudo pion” to see how (and if) the form factor approaches the PQCD value. The strange quark is light ($m_s << \Lambda_{\textrm{QCD}}$) from the PQCD point of view, and the behaviour is expected to be qualitatively similar for $\eta_s$ and $\pi$. The advantage is that strange quarks are computationally cheaper to simulate on the lattice, and the signal to noise ratio is better than for lighter quarks. We are now extending the study to pions and kaons, although the maximum $Q^2$ we can reach in the current calculations is not as high as for the $\eta_s$.
Lattice configurations
======================
We use lattice ensembles generated by the MILC Collaboration, with 3 different lattice spacings (ranging from $0.15$ fm to $0.09$ fm) and different light quark masses to allow a reliable continuum and chiral extrapolation. The Higly Improved Staggered Quark (HISQ) action is used for both valence and sea quarks, with $u/d$, $s$ and $c$ quarks included in the sea. The strange quark mass has been tuned to the physical mass by using the $\eta_s$ mass. The ensembles are listed in Table \[tab:ensembles\].
Set $\beta$ $w_0/a$ $am_l$ $am_s$ $am_c$ $L_s/a \times L_t/a$ $M_{\pi}$ $n_{\textrm{conf}}$ $T/a$
----- --------- --------------- ----------- ---------- --------- ---------------------- ----------- --------------------- ------------
1 $5.8$ $1.1119(10)$ $0.01300$ $0.0650$ $0.838$ $16\times 48$ $300$ MeV $1020$ $9,12,15$
2 $6.0$ $1.3826(11)$ $0.01020$ $0.0509$ $0.635$ $24\times 64$ $300$ MeV $1053$ $12,15,18$
3 $6.0$ $1.4029(9)$ $0.00507$ $0.0507$ $0.628$ $32\times 64$ $220$ MeV $1000$ $12,15,18$
4 $6.3$ $1.9006(20)$ $0.00740$ $0.0370$ $0.440$ $32\times 96$ $310$ MeV $1008$ $15,18,21$
: Lattice ensembles used in this study: Set 1 is ’very coarse’ ($a\sim 0.15$ fm), sets 2 and 3 ’coarse’ ($a\sim 0.12$ fm) and set 4 ’fine’ $a\sim 0.09$ fm. Lattice spacing is set using the Wilson flow parameter $w_0=0.1715(9)$ fm. $am_q$ are the sea quark masses in lattice units and $L_s/a \times L_t/a$ gives the lattice size in spatial and time directions. $M_{\pi}$ and $n_{\textrm{conf}}$ are the pion mass and the number of configurations. The last column gives the time extent of the 3-point correlators. More details of the lattice ensembles can be found in [@PhysRevD.82.074501; @PhysRevD.87.054505].[]{data-label="tab:ensembles"}
Electromagnetic form factors on the lattice
===========================================
The electromagnetic form factor is extracted from the 3-point correlation function depicted in Fig. \[fig:3ptcorr\], where a current $V$ is inserted in one of the meson’s quark propagators. We also need the standard 2-point correlation function of the meson that propagates from time $0$ to time $t$. The 3-point correlation function gives $$\langle P(p_f)|V_{\mu}|P(p_i)\rangle = F_P(Q^2)\cdot (p_f+p_i)_{\mu},$$ where $p_i$ and $p_f$ are the initial and final momenta of the pseudoscalr meson $P$, respectively. We use a 1-link vector current $V_{\mu}$ in the time direction, and the Breit frame $\vec{p}_i=-\vec{p}_f$ to maximise $Q^2$ for a given momentum $pa$. This leads to the simple relation $Q^2=|2\vec{p}_i|^2$. The form factor is normalised by requiring $F_P(0)=1$.
![A 3-point correlation function. The meson, here a kaon, is created at time $t=t_0$, and destroyed at time $t=t_0+T$. A vector current $V$ is inserted at time $t'$, where $t_0 < t' < t_0+T$. We use multiple $T$ values to fully map the oscillating states that are a feature of staggered fermions.[]{data-label="fig:3ptcorr"}](threept.pdf){width="35.00000%"}
To extract the properties of the meson we use multi-exponential fits with Bayesian priors to fit both 2-point and 3-point correlators simultaneously. The fit functions are $$\begin{aligned}
C_{\textrm{2pt}}(\vec{p}) &= \sum_i b_i^2 f(E_i(p),t') + \textrm{o.p.t.}, \nonumber \\
C_{\textrm{3pt}}(\vec{p},-\vec{p}) &= \sum_{i,j} \big[b_i(p) f(E_i(p),t) J_{ij}(Q^2) b_j(p) f(E_j(p),T-t)\big]
+ \textrm{o.p.t.}, \nonumber \\
f(E,t) &= e^{-Et}+e^{-E(L_t-t)},\end{aligned}$$ where $E_i$ is the energy of the state $i$ and $\vec{p}$ is the spatial momentum, and o.p.t. stands for the opposite parity terms. Note that $E_i$ and the amplitudes $b_i$ are common fit parameters for the 2-point and 3-point functions. We are interested in the ground state parameters $E_o$ (the mass of the meson if momentum $p=0$), $b_0$ (which is associated with the decay constant of the meson) and $J_{00}(Q^2)$, but use 6 exponentials to make sure that effects of the excited states are properly included in the error estimates. $J_{00}$ gives the matrix element of the vector current that we need to extract the form factor. More details can be found in [@etasff].
![Mapping the domain of analyticity in $t = q^2$ onto the unit circle in $z$.[]{data-label="fig:Q2toz"}](q2toz.pdf){width="42.00000%"}
To determine the form factor $F$ in the physical continuum limit we must extrapolate in the lattice spacing and $u/d$ quark mass. We first remove the pole in $F_P(Q^2)$ by multiplying the form factor by $P_V(Q^2)$, where $$P^{-1}_{V}(Q^2)=\frac{1}{1+Q^2/M^2_{V}}.
\label{eq:pole}$$ The pole mass $M_{V}$ is the mass of the vector meson that corresponds to the quarks at the current $V$. If the quarks are light quarks the mass is $M_{\rho}$, if the quarks are strange quarks the pole mass is $M_{\phi}$. The product $P_VF$ has reduced $Q^2$-dependence because $P^{-1}_{V}$ is a good match to the form factor at small $Q^2$. We then map the domain of analyticity in $t = q^2$ onto the unit circle in $z$ — see Fig. \[fig:Q2toz\]: $$z(t,t_{\textrm{cut}})=\frac{\sqrt{t_{\textrm{cut}}-t}-\sqrt{t_{\textrm{cut}}}}{\sqrt{t_{\textrm{cut}}-t}
+\sqrt{t_{\textrm{cut}}}}$$ and choose $t_{\textrm{cut}}=4M^2_K$ for the $\eta_s$. Now $|z| < 1$ and we can do a power series expansion in $z$, and use a fit form $$\begin{aligned}
\label{eq:zfit}
&P_{V}F(z,a,m_{\textrm{sea}})=1+\sum_i z^iA_i\bigg [ 1+B_i(a\Lambda)^2+C_i(a\Lambda)^4+D_i\frac{\delta m}{10} \bigg], \\
& \delta m = \sum_{u,d,s}(m_q-m_q^{\textrm{tuned}})/m_s^{\textrm{tuned}},~\Lambda = 1.0~\textrm{GeV}\nonumber.\end{aligned}$$ The terms with $B_i$ and $C_i$ parametrise lattice discretisation effects and the last term takes into account possible mistunings of the sea quark masses.
Results
=======
![Pion, kaon and $\eta_s$ form factors $Q^2F_P$ on the coarse lattice (set 2). The form factors with a strange current are found to be very similar, and so are the form factors with a light current. The spectator quark has only very small effect to the form factor. The dashed lines show the corresponding pole forms $Q^2P^{-1}_{V}(Q^2)$ (equation ) with pole masses $M_{\phi}$ and $M_{\rho}$ respectively.[]{data-label="fig:piKetasFF"}](q2fls.pdf){width="55.00000%"}
Figure \[fig:piKetasFF\] shows results for pion, kaon and $\eta_s$ form factors $Q^2F_P$. Let us start by noting how small the effect of the spectator quark is in the pseudoscalar meson electromagnetic form factor. The pion is made of two light quarks, whereas the $\eta_s$ is made of two strange quarks. The $K$ meson has one strange quark and one light quark, and the current can thus be either light or strange. Fig. \[fig:piKetasFF\] illustrates how the form factors can be grouped according to the flavor of the quarks at the current insertion: the $\eta_s$ and the strange-current $K$ form factors are very similar as are the pion and light-current $K$ form factors. The form factors follow the pole form at small $Q^2$, but peel away from it when the momentum transfer grows larger.
![The $\eta_s$ form factor $Q^2F_{\eta_s}$ as a function of $Q^2$. At small $Q^2$ the form factor follows the pole form (with pole mass $M_{\phi}$) as expected. The discretisation effects are very small. The grey band shows the continuum and chiral extrapolation (equation ). ’PQCD1’ is the asymptotic value from perturbative QCD, and ’PQCD2’ shows the perturbative value with corrections added to the asymptotic PQCD. We plot $Q^2F_{\eta_s}$ rather than $F_{\eta_s}$ to compare to the asymptotic value (eq. \[eq:aPQCD\] multiplied by $Q^2$ gives $8\pi\alpha_s f_P^2$.)[]{data-label="fig:etasFF"}](q2fetas.pdf){width="60.00000%"}
![The perturbative QCD description of a meson electromagnetic form factor (here the pion is used as an example, but the calculation is analogous for the $\eta_s$). $\phi_{\pi}$ is the distribution amplitude and the blue colour marks the high momentum photon and gluon.[]{data-label="fig:highmomfact"}](highmomfact.pdf){width="55.00000%"}
In Figure \[fig:etasFF\] we plot the $\eta_s$ form factor obtained on very coarse, coarse and fine lattices as a function of $Q^2$. We can reach $Q^2\sim 6$ GeV$^2$ on the fine lattice, and the form factor multiplied by $Q^2$ is found to be almost flat in the $Q^2$ range $3$ – $6$ GeV$^2$. This can be compared to the asymptotic value marked with ’PQCD1’. At high $Q^2$ the electromagnetic form factor can be calculated using perturbative QCD, because the process in which the hard photon scatters from the quark or antiquark factorises from the distribution amplitudes which describe the quark-antiquark configuration in the meson, as is illustrated in figure \[fig:highmomfact\] using a pion as an example. The asymptotic value is $$\label{eq:aPQCD}
F_P(Q^2)=\frac{8\pi\alpha_s f_P^2}{Q^2},$$ where $f_P$ is the decay constant of the pseudoscalar meson (pion, kaon, $\eta_s$). The value we obtain for the $\eta_s$ form factor is much higher than the asymptotic value at $Q^2=6$ GeV$^2$. On the other hand, the curve ’PQCD2’ that includes non-asymptotic corrections to the distribution amplitude lies above the $\eta_s$ form factor. More details can be found in [@etasff].
![The pion form factor $Q^2F_{\pi}$ as a function of $Q^2$. The agreement with experimental results at small $Q^2$ is excellent, and peeling away from the pole form (shown as the continuous line) is observed as expected. The results are preliminary as we are pushing to higher $Q^2$, and no continuum extrapolation is done at this time. Also smaller light quark masses have to be included in the study to do a reliable chiral extrapolation: the pion masses used here are $\sim 300$ MeV. The experimental results are from [@NA7pi; @JLAB1; @JLAB2].[]{data-label="fig:q2pi"}](piq2f.pdf){width="55.00000%"}
![The kaon form factor $Q^2F_K$ as a function of $Q^2$. The agreement with experimental results at small $Q^2$ is excellent. The results are preliminary as we are pushing to higher $Q^2$, and no continuum extrapolation is done at this time. Also smaller light quark masses have to be included in the study to do a reliable chiral extrapolation. The experimental results are from [@NA7K].[]{data-label="fig:q2K"}](Kq2f.pdf){width="55.00000%"}
In figures \[fig:q2pi\] and \[fig:q2K\] we show our preliminary results for pion and kaon electromagnetic form factors as a function of $Q^2$. These are the first predictions of the $K^0$ and $K^+$ form factors from lattice QCD ahead of the Jefferson Lab experiment. The $K^0$ and $K^+$ form factors are calculated from the strange and light current $K$ form factors by combining with the electric charges of the quarks: $K^+$ is $u\bar{s}$ and $K^0$ is $d\bar{s}$. Work is underway to go to higher $Q^2$ values and to study the dependence of the pion and kaon form factors on the light quark mass. The light quark masses used at this preliminary stage correspond to pion mass of $\sim 310$ MeV. This has been studied in the case of the $\eta_s$ form factor, where the effect is negligible, but smaller masses are needed to do the chiral extrapolation for the pion and kaon form factors. We plan to include results from physical light quarks in our final analysis. No continuum or chiral extrapolation is presented at this time for the pion and kaon form factors.
Conclusions and outlook
=======================
Our $\eta_s$ form factor results indicate that asymptotic perturbative QCD is not applicable at $Q^2\sim 6$ GeV$^2$ or below — much larger $Q^2$ are needed. Using strange quarks instead of light quarks allows us to get some qualitative knowledge of light pseudoscalar meson form factors (pion and kaon form factors) at high $Q^2$ ahead of the more lengthy calculations required for $K$ and $\pi$. We can also probe higher $Q^2$ values with strange quarks than with light quarks. However, we can already provide first, preliminary predictions of the $K^+$ and $K^0$ form factors ahead of the upcoming Jefferson Lab experiment. The pion form factor is the most challenging. By gathering more statistics and pushing to higher $Q^2$ we will have good theoretical understanding of the form factors in the momentum range that the Jefferson Lab pion and kaon experiments will use.
Acknowledgements
================
We are grateful to the MILC collaboration for the use of their gauge configurations and code. Our calculations were done on the Darwin Supercomputer as part of STFC’s DiRAC facility jointly funded by STFC, BIS and the Universities of Cambridge and Glasgow. This work was funded by a CNPq-Brazil scholarship, the National Science Foundation, the Royal Society, the Science and Technology Facilities Council and the Wolfson Foundation.
[^1]: Speaker,
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---
abstract: 'In this paper, using a criterion given by J. Brough and B. Späth recently, we verify the inductive blockwise Alperin weight condition for the simple groups ${\operatorname{PSp}}_{2n}(q)$ and any odd prime $\ell$ not dividing $q$ under some assumptions concerning the decomposition matrices. *2020 Mathematics Subject Classification:* 20C20, 20C33. *Keyword:* Alperin weight conjecture, inductive condition, projective symplectic groups.'
author:
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Conghui Li\
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title: 'The inductive blockwise Alperin weight condition for ${\operatorname{PSp}}_{2n}(q)$ and odd primes [^1]'
---
Introduction {#sect:intro}
============
An important longstanding global-local conjecture, called the blockwise Alperin weight conjecture, is proposed by Alperin in [@Al87]. The conjecture is stated as follows; see §\[subsec:general\] for the definition of weight.
Let $G$ be a finite group, $\ell$ a prime and $B$ an $\ell$-block of $G$. Denote the set of all $G$-conjugacy classes of $B$-weights by ${\operatorname{Alp}}(B)$, then $|{\operatorname{Alp}}(B)|=|{\operatorname{IBr}}(B)|$.
This conjecture has been verified for numerous groups, such as symmetric groups, many sporadic simple groups and many finite groups of Lie type. Here we just mention the papers of Alperin-Fong [@AF90] and An [@An94], which classify the weights of some classical groups for odd primes.
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Since the general proof of this conjecture seems extremely difficult, an accessible way is to reduce it to simple groups. First, Navarro and Tiep reduced the non-blockwise version of this conjecture to simple groups in [@NT11], and then Späth reduced the blockswise version to simple groups in [@Sp13]. Thus to prove this conjecture, it suffices to verify the inductive blockwise Alperin weight (BAW) condition defined in [@Sp13] for all simple groups. This inductive condition has been verified for many cases. In [@Sp13], Späth gives a verification for finite simple groups of Lie type and the defining characteristic. Malle verifies it in [@Ma14] for simple alternating groups, Suzuki groups and Ree groups. All blocks with cyclic defect groups have been proved to satisfy this inductive condition by Koshitani and Späth in [@KS16a; @KS16b]. Schulte gives a verification for it for simple groups of type $G_2$ and $^3D_4$ in [@Sch16]. A special case of type A is verified by Li-Zhang in [@LZ18; @LZ19]. Feng verifies it for the unipotent blocks of type A in [@Feng19], and together with Z. Li and J. Zhang for unipotent blocks of classical groups and some other cases of classical groups in [@FLZ19]. Some more particular simple groups of small rank are considered in [@SF14; @FLL17a; @BSF19; @LL19].
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The inductive BAW condition is highly complicated, which makes it very challenging to verify it for simple groups of Lie type. Roughly speaking, it consists of two parts: the first requires an equivariant bijection between irreducible Brauer characters and weights; the second is some requirements concerning the Clifford theory for characters under this bijection. Fortunately, Brough and Späth prove a criterion for this inductive condition for simple groups with abelian outer automorphism groups, which makes it possible to consider the simple groups of Lie type B and C; see Theorem \[thm:criterion\] for the statement. We consider the simple group ${\operatorname{PSp}}_{2n}(q)$ and odd primes $\ell\nmid q$ in this paper. First, assume $q$ is odd, in which case, according to the criterion, the main task is to establish the BAW conjecture itself for the conformal symplectic groups.
\[mainthm-1\] Let $p$ be an odd prime, $q=p^f$ and $\ell$ an odd prime different from $p$, then the blockwise Alperin weight conjecture holds for ${\operatorname{CSp}}_{2n}(q)$ and $\ell$.
The proof of the above theorem relies heavily on the results of Fong-Srinivasan in [@FS89] and An in [@An94]. To continue the verification, we prove that a certain bijection between irreducible Brauer characters and weights of conformal symplectic groups satisfies the requirements of part (iii) in Theorem \[thm:criterion\]. The remaining requirements concerning the symplectic groups can be verified easily, using a result of Cabanes and Späth in [@CS17C] and a result in [@Li19] of the author. In the process, to obtain information about irreducible Brauer characters, we need to make some assumptions about the decomposition matrices. Thus the main result for $q$ odd is stated as follows.
\[mainthm-2\] Keep the assumptions in Theorem \[mainthm-1\]. Assume the decomposition matrices of ${\operatorname{Sp}}_{2n}(q)$ and ${\operatorname{CSp}}_{2n}(q)$ with respect to ${\mathcal{E}}({\operatorname{Sp}}_{2n}(q),\ell')$ and ${\mathcal{E}}({\operatorname{CSp}}_{2n}(q),\ell')$ are unitriangular. Then the inductive blockwise Alperin weight condition holds for the simple group ${\operatorname{PSp}}_{2n}(q)$ and $\ell$.
Here, recall that ${\mathcal{E}}(G,\ell') = \bigcup_{s\in G^*_{ss,\ell'}} {\mathcal{E}}(G,(s))$ for a finite group $G$ of Lie type.
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Next, we consider the relatively easy case for $q=2^f$.
\[mainthm-3\] Let $q=2^f$ and $\ell$ be an odd prime, then the blockwise Alperin weight conjecture holds for ${\operatorname{Sp}}_{2n}(2^f)$ and $\ell$. Furthermore, assume $n\geq2$ and $(n,f)\neq(2,1)$ or $(3,1)$, then if the decomposition matrix of ${\operatorname{Sp}}_{2n}(2^f)$ to ${\mathcal{E}}({\operatorname{Sp}}_{2n}(2^f),\ell')$ is unitriangular, the inductive blockwise Alperin weight condition holds for ${\operatorname{Sp}}_{2n}(2^f)$ and $\ell$.
The structure of this paper is as follows. In §\[sec:pre\], we give some notation and preliminaries. Then in sections §\[sec:prepare\], §\[sec:weights\], §\[sec:inductive\], we consider the case when $q$ is odd. First, we consider the Brauer pairs associated to radical subgroups and recall the results about blocks of conformal symplectic groups in §\[sec:prepare\]. In §\[sec:weights\], we classify the weights of conformal symplectic groups and prove Theorem \[mainthm-1\]. In §\[sec:inductive\], we verify the inductive BAW condition for the cases in Theorem \[mainthm-2\]. Finally, in §\[sec:main-3\], we consider the case when $q$ is even.
Preliminaries {#sec:pre}
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{#subsec:general}
For an element $g$ of a finite group $G$, we denote by ${\operatorname{Cl}}_G(g)$ its $G$-conjugacy class. The notation for representations of finite groups in this paper is standard, which can be found for example in [@NT89] except that we use ${\operatorname{Ind}}$ and ${\operatorname{Res}}$ for induction and restriction and use $\chi^0$ for the restriction of an ordinary character $\chi$ of $G$ to the $\ell$-regular elements. We will consider the modular representations with respect to a fixed prime $\ell$, thus we will abbreviate $\ell$-Brauer characters, $\ell$-block, etc. as Brauer characters, blocks, etc.
For a finite group $G$, we denote by ${\operatorname{dz}}(G)$ the set of all irreducible defect zero characters of $G$. For $K \unlhd G$ and $\theta\in{\operatorname{Irr}}(K)$, we set ${\operatorname{dz}}(G \mid \theta) = {\operatorname{dz}}(G) \cap {\operatorname{Irr}}(G \mid \theta)$. For an $\ell$-subgroup $R$ of $G$ and a block $B$ of $G$, ${\operatorname{dz}}(N_G(R)/R,B)$ denotes the set of all characters $\varphi$ of $N_G(R)$ which lift characters in ${\operatorname{dz}}(N_G(R)/R)$ and satisfy ${\operatorname{bl}}(\varphi)^G=B$. Here, ${\operatorname{bl}}(\varphi)$ is the block of $N_G(R)$ containing $\varphi$. Here and often in the sequel, we will identify characters of $N_G(R)/R$ and their lifts to $N_G(R)$.
An $\ell$-weight of $G$ means a pair $(R,\varphi)$ with $\varphi\in{\operatorname{dz}}(N_G(R)/R)$. In this case, $R$ is necessarily an $\ell$-radical subgroup of $G$, *i.e.* $R={\operatorname{O}}_\ell(N_G(R))$, and $\varphi$ is called a weight character. If furthermore $\varphi\in{\operatorname{dz}}(N_G(R)/R,B)$, then $(R,\varphi)$ is called a $B$-weight. Denote the set of all $B$-weights by ${\operatorname{Alp}}^0(B)$ and the set of all $G$-conjugacy classes of $B$-weights by ${\operatorname{Alp}}(B)$. For $(R,\varphi) \in {\operatorname{Alp}}^0(B)$, denote by $\overline{(R,\varphi)}$ the $G$-conjugacy class of $(R,\varphi)$.
Recently, J. Brough and B. Späth gave a new criterion for the inductive condition particularly suitable for simple groups of type B and C. Denote by ${\operatorname{Rad}}(G)$ the set of all radical subgroups of $G$ and by ${\operatorname{Rad}}(G)/{\!\!\sim\!\!}G$ a $G$-transversal of ${\operatorname{Rad}}(G)$.
\[thm:criterion\] Let $S$ be a finite non-abelian simple group and $\ell$ a prime dividing $|S|$. Let $G$ be an $\ell'$-covering group of $S$ and assume there are groups ${\tilde{G}}$, $E$ such that $G \unlhd {\tilde{G}}\times E$. Assume ${\mathcal{B}}\subseteq{\operatorname{Bl}}(G)$ is a ${\tilde{G}}$-stable set satisfying $({\tilde{G}}E)_B\leq({\tilde{G}}E)_{\mathcal{B}}$ for any $B\in{\mathcal{B}}$ and the following hold:
(i) - $G=[{\tilde{G}},{\tilde{G}}]$ and $E$ is abelian,
- $C_{{\tilde{G}}E}(G)=Z({\tilde{G}})$ and ${\tilde{G}}E/Z({\tilde{G}}) \cong {\operatorname{Aut}}(G)$ by the natural map,
- any element of ${\operatorname{IBr}}({\mathcal{B}})$ extends to its stabilizer in ${\tilde{G}}$,
- for any $R \in {\operatorname{Rad}}(G)$, any element of ${\operatorname{dz}}(N_G(R)/R,B)$ with $B\in{\mathcal{B}}$ extends to its stabilizer in $N_{{\tilde{G}}}(R)/R$.
(ii) Let ${\tilde{\mathcal{B}}}={\operatorname{Bl}}({\tilde{G}}\mid{\mathcal{B}})$. There exists an ${\operatorname{IBr}}({\tilde{G}}/G) \rtimes E_{{\mathcal{B}}}$-equivariant bijection $${\tilde{\Omega}}_{{\tilde{\mathcal{B}}}}:\quad {\operatorname{IBr}}({\tilde{\mathcal{B}}}) \to {\operatorname{Alp}}({\tilde{\mathcal{B}}})$$ with ${\tilde{\Omega}}_{{\tilde{\mathcal{B}}}}({\operatorname{IBr}}({\tilde{B}})) = {\operatorname{Alp}}({\tilde{B}})$ for every ${\tilde{B}}\in {\tilde{\mathcal{B}}}$ and ${\operatorname{J}}_G({\tilde{\psi}}) = {\operatorname{J}}_G({\tilde{\Omega}}_{{\tilde{\mathcal{B}}}}({\tilde{\psi}}))$ for every ${\tilde{\psi}}\in {\operatorname{IBr}}({\tilde{\mathcal{B}}})$.
(iii) For every ${\tilde{\chi}}\in{\operatorname{IBr}}({\tilde{G}})$, there exists some $\chi_0\in{\operatorname{IBr}}(G\mid{\tilde{\chi}})$ such that
- $({\tilde{G}}\rtimes E)_{\chi_0}={\tilde{G}}_{\chi_0}\rtimes E_{\chi_0}$,
- $\chi_0$ extends to $G\rtimes E_{\chi_0}$.
(iv) For every $\overline{(R,\psi_0)} \in {\operatorname{Alp}}({\mathcal{B}})$, there exists some $x \in {\tilde{G}}$ with
- $({\tilde{G}}E)_{R,\psi_0}= {\tilde{G}}_{R,\psi_0} (GE^x)_{R,\psi_0}$,
- $\psi_0$ extends to $(G\rtimes E^x)_{R,\psi_0}$.
Then the inductive blockwise Alperin weight condition holds for any $B$ in ${\mathcal{B}}$ with abelian ${\operatorname{Out}}(G)_{{\tilde{G}}\textrm{-orbit of }B}$.
For the definition of ${\operatorname{J}}_G({\tilde{\psi}})$ and ${\operatorname{J}}_G({\tilde{\Omega}}({\tilde{\psi}}))$, see [@BS19]. In the case when $G={\operatorname{Sp}}_{2n}(q)$, ${\tilde{G}}={\operatorname{CSp}}_{2n}(q)$ and $\ell$ is an odd prime, ${\operatorname{J}}_G({\tilde{\psi}}) = {\tilde{G}}= {\operatorname{J}}_G({\tilde{\Omega}}({\tilde{\psi}}))$ is trivially satisfied.
{#sect:gLt}
Since the case ${\operatorname{PSp}}_2(q)={\operatorname{PSL}}_2(q)$ has been considered, we may assume $n\geq2$. We will frequently use the notation in [@FS89] and [@Li19], so we review those which will be used from now on; for more notation, we will refer to the definitions in [@FS89] and [@Li19] when they appear for the first time in the sequel.
Assume $q$ is odd. Let ${\mathbf{V}}$ (${\mathbf{V}}^*$) be the symplectic space of dimension $2n$ (the orthogonal space of dimension $2n+1$) over the algebraic closure ${\overline{\mathbb{F}}}_p$ of the finite field ${\mathbb{F}}_p$ of $p$ elements and ${\mathbf{G}}={\operatorname{Sp}}({\mathbf{V}}),{\tilde{\mathbf{G}}}={\operatorname{CSp}}({\mathbf{V}}),{\mathbf{G}}^*={\operatorname{SO}}({\mathbf{V}}),{\tilde{\mathbf{G}}}^*={\operatorname{D}}_0({\mathbf{V}}^*)$. Denote by $F$ both the Frobenius maps on ${\tilde{\mathbf{G}}}$ and ${\tilde{\mathbf{G}}}^*$ defining an ${\mathbb{F}}_q$-structure and by $G,{\tilde{G}},G^*,{\tilde{G}}^*$ the corresponding groups of fixed points. These finite groups can be viewed as groups on a symplectic space $V$ or an orthogonal space $V^*$ over ${\mathbb{F}}_q$. In particular, $G^* \cong {\operatorname{SO}}_{2n+1}(q)$. The regular embedding $i: {\mathbf{G}}\to {\tilde{\mathbf{G}}}$, its dual $i^*: {\tilde{\mathbf{G}}}^* \to {\mathbf{G}}^*$ and related constructions are as in [@Li19 §2.B]. Let $E={\langleF_p\rangle}$ be the group of field automorphisms, then ${\tilde{G}}\rtimes E$ affords all automorphisms of the finite simple group $S=G/Z(G)$ when $n\geq2$.
When $q$ is a power of $2$ and $n\geq2$, $G$ is simple if $(n,f)\neq(2,1)$ and is its own universal covering group of itself if furthermore $(n,f)\neq(3,1)$, thus we will not need to introduce ${\tilde{\mathbf{G}}},{\tilde{\mathbf{G}}}^*,{\tilde{G}},{\tilde{G}}^*$. In this case, there are isomorphisms of abstract groups ${\mathbf{G}}\cong {\mathbf{G}}^*$ (but not as algebraic groups) and $G \cong G^*$.
Assume $q=p^f$ is a power of an odd prime $p$ and $\ell$ is an odd prime different from $p$. Let ${\mathbb{F}}_q$ be the field of $q$ elements. The sets ${\mathcal{F}}_0,{\mathcal{F}}_1,{\mathcal{F}}_2$ of polynomials are defined as in [@FS89 (1.7)] and ${\mathcal{F}}={\mathcal{F}}_0\cup{\mathcal{F}}_1\cup{\mathcal{F}}_2$. For any $\Gamma\in{\mathcal{F}}$, $\delta_\Gamma$ and $\varepsilon_\Gamma$ are defined as in [@FS89 (1.8),(1.9)]. We will use ${\tilde{s}}$ to denote a semisimple element of ${\tilde{G}}^*$ and use $s=i^*({\tilde{s}})$ to denote the image of ${\tilde{s}}$ under $i^*$, which is different from the convention in [@FS89]. The $G^*$-conjugacy class of $s$ is determined by the multiplicity function $\Gamma\in{\mathcal{F}}\mapsto m_\Gamma(s)$ and the type function $\Gamma\in{\mathcal{F}}\mapsto \eta_\Gamma(s)$; see [@FS89 pp.125–126].
If $q=2^f$, we define ${\mathcal{F}}_0$ as $${\mathcal{F}}_0={\{X-1\}}$$ and let ${\mathcal{F}}_1,{\mathcal{F}}_2$ be defined in the same way as in [@FS89 (1.7)] with odd $q$ replaced by $2^f$. Set then ${\mathcal{F}}={\mathcal{F}}_0\cup{\mathcal{F}}_1\cup{\mathcal{F}}_2$. For any $\Gamma\in{\mathcal{F}}$, $\delta_\Gamma$ and $\varepsilon_\Gamma$ are defined in the same way as in [@FS89 (1.8),(1.9)]. The $G$-conjugacy class of a semisimple element $s$ is determined by the multiplicity function $\Gamma\in{\mathcal{F}}\mapsto m_\Gamma(s)$; for the structure of $C_G(s)$, see [@SF14 Lemma 2.2]. Since $G^* \cong G$, this also gives a parametrization of conjugacy classes of semisimple elements of $G^*$ and a description of their centralizers.
{#subsec:radical}
The radical subgroups of $G$ are given by An in [@An94]. We recall the construction using the twisted basic subgroups introduced in [@Li19].
Recall that $e$ is the multiplicative order of $q^2$ in ${\mathbb{Z}}/\ell{\mathbb{Z}}$ and $\ell$ is said to be linear or unitary if $\ell$ divides $q^2-1$ or $q^e+1$. Set $\varepsilon=1$ or $-1$ when $\ell$ is linear or unitary. Let $a=v(q^{2e}-1)$, where $v_\ell$ is the discrete valuation such that $v(\ell)=1$. A group of symplectic type is a central product $Z_\alpha E_\gamma$ of a cyclic groups $Z_\alpha$ of order $\ell^{a+\alpha}$ and an extraspecial group $E_\gamma$ of order $\ell^{2\gamma+1}$; since only extraspecial groups of exponent $\ell$ occurs in radical subgroups, we always assume the exponent is $\ell$. Let $R_{m,\alpha,\gamma}^0$ be the embedding of $Z_\alpha E_\gamma$ in $G_{m,\alpha,\gamma}^0:={\operatorname{GL}}(m\ell^\gamma,\varepsilon q^{e\ell^\alpha})$ given in [@Li19 §6.A]. Let ${\mathbf{G}}_{m,\alpha,\gamma}={\operatorname{Sp}}(2me\ell^{\alpha+\gamma},{\overline{\mathbb{F}}}_p)$ and $v_{m,\alpha,\gamma}$ be as in [@Li19 §6.A]. Set $G_{m,\alpha,\gamma}^{tw}:={\mathbf{G}}_{m,\alpha,\gamma}^{v_{m,\alpha,\gamma}F}$, then the hyperbolic embedding $\hbar: G_{m,\alpha,\gamma}^0 \to G_{m,\alpha,\gamma}^{tw}$ is defined as in [@Li19 §6.A]. The image of $R_{m,\alpha,\gamma}^0$ under $\hbar$ is denoted by $R_{m,\alpha,\gamma}^{tw}$. Let $g_{m,\alpha,\gamma}$ be an element in ${\mathbf{G}}_{m,\alpha,\gamma}$ such that $g_{m,\alpha,\gamma}^{-1}F(g_{m,\alpha,\gamma}^{-1})=v_{m,\alpha,\gamma}$; see for example [@CE04 Theorem 7.1]. The map $\iota: G_{m,\alpha,\gamma}^{tw} \to G_{m,\alpha,\gamma} := {\mathbf{G}}_{m,\alpha,\gamma}^F$ defined by the conjugation by $g_{m,\alpha,\gamma}$ is an isomorphism. Denote $R_{m,\alpha,\gamma}=\iota(R_{m,\alpha,\gamma}^{tw})$.
Let ${\mathbf{c}}$ be defined as in [@Li19 §6.A], then $R_{m,\alpha,\gamma,{\mathbf{c}}}^{tw}=R_{m,\alpha,\gamma}^{tw}\wr A_{\mathbf{c}}$ is called a twisted basic subgroup, which is a subgroup of $G_{m,\alpha,\gamma,{\mathbf{c}}}^{tw}:={\mathbf{G}}_{m,\alpha,\gamma,{\mathbf{c}}}^{v_{m,\alpha,\gamma,{\mathbf{c}}}F}$, where $v_{m,\alpha,\gamma,{\mathbf{c}}}$ and ${\mathbf{G}}_{m,\alpha,\gamma,{\mathbf{c}}}$ is as in [@Li19 §6.A]. Let $g_{m,\alpha,\gamma,{\mathbf{c}}}=g_{m,\alpha,\gamma}\otimes I_{\ell^{|{\mathbf{c}}|}}$ and the isomorphism (and all similar homomorphisms) induced by conjugation by $g_{m,\alpha,\gamma,{\mathbf{c}}}$ is again denoted by $\iota$. Set $R_{m,\alpha,\gamma,{\mathbf{c}}}=\iota(R_{m,\alpha,\gamma,{\mathbf{c}}}^{tw})$, called a basic subgroup, which is conjugate to the basic subgroup denoted by the same symbol in [@An94]. Obviously, $R_{m,\alpha,\gamma,{\mathbf{c}}}=R_{m,\alpha,\gamma}\wr A_{{\mathbf{c}}}$.
Let $\tau_{m,\alpha,\gamma,{\mathbf{c}}}^{tw}$ be as in [@Li19 (6.3)]. The notation and results on centralizers and normalizers of the basic subgroups are collected in [@Li19 Lemma 6.4, Lemma 6.6]. Set $\tau_{m,\alpha,\gamma,{\mathbf{c}}}=\iota(\tau_{m,\alpha,\gamma,{\mathbf{c}}}^{tw})$.
We remark that all the above constructions also apply to the case when $q$ is even. In fact, since $-1=1$ for even $q$, the structure of normalizers in [@Li19 Lemma 6.6] can be even slightly simplified. To be more specific, $N_{m,\alpha,\gamma}^{tw} = \hbar(N_{m,\alpha,\gamma}^0) \rtimes V_{m,\alpha,\gamma}^{tw}$ while for odd $q$, $\hbar(N_{m,\alpha,\gamma}^0) \cap V_{m,\alpha,\gamma}^{tw} = Z(G_{m,\alpha,\gamma}^{tw})$.
Assume $q$ is odd. By [@An94], all radical subgroups of $G$ are conjugate to subgroups of the form $R_0 \times R_1 \times \cdots \times R_u$, where $R_0$ is the trivial group and $R_i=R_{m_i,\alpha_i,\gamma_i,{\mathbf{c}}_i}$ is a basic subgroup for $i>0$. In fact, this also holds for even $q$; see §\[sec:main-3\]. For such a radical subgroup $R$ of $G$, let $v,g,\iota$ be as in [@Li19 Lemma 6.7] and set ${\tilde{G}}^{tw}={\tilde{\mathbf{G}}}^{vF}$. In the sequel, when we say “consider the twisted groups” or “transfer to twisted groups”, we will always mean transfer the problems to ${\tilde{G}}^{tw}$ using $\iota$. The method to use twisted groups to consider local structures is introduced in [@CS17]. This method has some technical advantages. For example, the diagonal automorphisms and field automorphisms fix the twisted version of radical subgroups (see [@Li19 Lemma 6.8]), which makes the calculation of automorphisms on weights easier. See §\[sec:inductive\] for further applications of the twisted version of radical subgroups.
{#subsec:ano-conju}
The version of basic subgroups given above is convenient when one considers the action of automorphisms on weights (see [@Li19]) and the extension problem. Now, we will give another conjugate of the basic subgroup $R_{m,\alpha,\gamma,{\mathbf{c}}}$, which is convenient when one considers the inclusion of some Brauer pairs in §\[subsec:Brauerpair\].
Note that $R_{m,\alpha}$ is a cyclic subgroup of ${\operatorname{Sp}}_{2me\ell^\alpha}(q)$ of order $\ell^{a+\alpha}$ consisting of the elements of the form $z_{m,\alpha}(\zeta):=(\iota\circ\hbar)(\zeta I_m)$ for some $\zeta\in{\overline{\mathbb{F}}}_p$ with $\zeta^{\ell^{a+\alpha}}=1$. Let $\zeta_\ell$ be a primitive $\ell$-th root of unity, then $R_{m,\alpha,\gamma}$ defined as above is conjugate to the group $R_{m,\alpha,\gamma,1}$ generated by the following elements $$R_{m,\alpha}\otimes I_{\ell^\gamma}, {\operatorname{diag}}\left\{z_{m,\alpha}(1),z_{m,\alpha}(\zeta_\ell),\cdots,z_{m,\alpha}(\zeta_\ell^{\ell-1})\right\}, I_{2me\ell^\alpha}\otimes Y_j^0, j=1,\ldots,\gamma,$$ where $Y_j^0$ is as in [@Li19 §6.A]. Set $R_{m,\alpha,\gamma,{\mathbf{c}},1}=R_{m,\alpha,\gamma,1}\wr A_{\mathbf{c}}$, then it is conjugate to the basic subgroup $R_{m,\alpha,\gamma,{\mathbf{c}}}$ defined above. From now on, we will denote any conjugate of the basic subgroup $R_{m,\alpha,\gamma,{\mathbf{c}}}$ by the same notation; it should be clear which conjugate is being used by the context. Note that the definition of $\tau_{m,\alpha,\gamma,{\mathbf{c}}}$ and the statement of [@Li19 Lemma 6.4] should be adjusted accordingly.
For any non negative integer $\beta$, denote ${\mathbf{c}}_\beta=(1,\dots,1)$ with $\beta$ 1’s and set $D_{m,\alpha,\beta}:=R_{m,\alpha,0,{\mathbf{c}}_\beta}$. In [@FS89], $D_{m,\alpha,\beta}$ is denoted by $R_{m,\alpha,\beta}$, but to avoid any confusion of notation, we introduce this new notation. Then by [@FS89 (5K)], the defect groups of blocks of ${\operatorname{Sp}}_{2n}(q)$ are of the form $D_0 \times D_1 \times\cdots\times D_u$ with $D_0$ the trivial group and $D_i=D_{m_i,\alpha_i,\beta_i}$ for $i=1,\cdots,u$.
{#subsec:partition-symbol}
We now recall some facts about partitions and Lusztig symbols and fix some notation. In this subsection, $e$ is an arbitrary positive integer (not necessarily a prime).
For a given partition $\lambda$ of some natural number $n$, the $e$-core $\lambda_{(e)}$ and the $e$-quotient $\lambda^{(e)}$ of $\lambda$ is uniquely determined; conversely, the partition is determined by its core and quotient; for details, see [@Ol93 §3]. Here, the $e$-core $\lambda_{(e)}$ is again a partition of some natural number $n_0\leq n$, while the $e$-quotient $\lambda^{(e)} = (\lambda_1^{(e)},\ldots,\lambda_e^{(e)})$ is an $e$-tuple (ordered sequence) of partitions. Given an $e$-core $\kappa$ and an $e$-quotient $Q$, the unique partition $\lambda$ determined by $\lambda_{(e)}=\kappa$ and $\lambda^{(e)}=Q$ is denoted as $\lambda=\kappa*Q$.
For Lusztig symbols, defects and ranks, cores and quotients, degenerate Lusztig symbols, etc., see [@Ol93 §5]. *We remark that degenerate symbols are counted twice when one parametrizes characters, blocks and weights for conformal symplectic groups (see [@FS89] and §\[sec:weights\] in this paper), while degenerate symbols are counted only once when one parametrizes characters, blocks and weights for symplectic groups (see [@Li19]).* The two copies of the degenerate symbol $\lambda$ are denoted as $\lambda$ and $\lambda'$. As in [@Li19], the empty Lusztig symbol is viewed as degenerate, thus a non-degenerate symbol is *a fortiori* not empty. But in many ocasions, the empty Lusztig symbol will be distinguished from the non-empty ones in this paper; the reason for this is that for conformal symplectic groups, the non-empty degenerate symbols are counted twice while the empty symbol is only counted once; see for example Table \[tab:block\].
For a given Lusztig symbol $\lambda$ of rank $n$, the $e$-core $\lambda_{(e)}$ and the $e$-quotient $\lambda^{(e)}$ of $\lambda$ is uniquely determined. But conversely, given an $e$-core $\kappa$ and an $e$-quotient $Q$, there may be one or two Lusztig symbols with $\kappa$ and $Q$ as their $e$-core and $e$-quotient respectively; there are two such symbols if and only if both $\kappa$ and $Q$ are non degenerate. For details, see [@Ol93 §5]. Here, the $e$-core $\lambda_{(e)}$ is again a Lusztig symbol of rank $n_0\leq n$, while the $e$-quotient $\lambda^{(e)}$ is an unordered pair of two $e$-tuples of partitions $[\lambda_1,\dots,\lambda_e;\mu_1,\dots,\mu_e]$, which means that $[\lambda_1,\dots,\lambda_e;\mu_1,\dots,\mu_e]$ and $[\mu_1,\dots,\mu_e;\lambda_1,\dots,\lambda_e]$ are identified; see [@Ol93 §5]. An $e$-quotient $[\lambda_1,\dots,\lambda_e;\mu_1,\dots,\mu_e]$ is called degenerate if and only if $(\lambda_1,\dots,\lambda_e)=(\mu_1,\dots,\mu_e)$.
Associated with a given $e$-quotient $Q=[\lambda_1,\dots,\lambda_e;\mu_1,\dots,\mu_e]$, there are one or two ordered sequences of partitions $$(\lambda_1,\dots,\lambda_e,\mu_1,\dots,\mu_e), \quad (\mu_1,\dots,\mu_e,\lambda_1,\dots,\lambda_e).$$ called the ordered quotient(s) with respect to $Q$ and denoted as $Q_0,Q_0'$. Thus $Q_0=Q_0'$ if and only if $Q$ is degenerate.
When $\kappa$ is non degenerate, the number of Lusztig symbols with $\kappa$ and $Q$ as their $e$-core and $e$-quotient respectively is exactly the number of ordered quotient(s) with respect to $Q$. The one or two Lusztig symbols are denoted as $\kappa*Q_0,\kappa*Q_0'$. Equivalently, the (necessarily non degenerate) Lusztig symbols with non degenerate cores are determined uniquely by the cores and the ordered quotients.
When $\kappa$ is degenerate, then for any given quotient $Q$, there is only one Lusztig symbol $\lambda$ with $\kappa$ and $Q$ as their $e$-core and $e$-quotient respectively. When $Q$ is non degenerate, $\lambda$ is non degenerate, and we set $\lambda=\kappa*Q_0=\kappa*Q_0'$. When $Q$ is degenerate (thus we can identify $Q$ with its associated ordered quotient), $\lambda$ is degenerate. Note that $\lambda$ should be counted twice when one considers conformal groups. In this case, the two copies of the degenerate symbols are denoted as $\lambda=\kappa*(Q,0)$ and $\lambda'=\kappa*(Q,1)$.
Note that our convention of notation used to consider characters and weights of conformal symplectic groups is slightly different from those in [@Li19] used to consider characters and weights of symplectic groups.
{#subsec:action-tz}
Assume $q$ is odd. Now, we recall the Jordan decomposition of characters of ${\tilde{G}}$ and consider the action of ${\operatorname{Irr}}({\tilde{G}}/G)$ on ${\operatorname{Irr}}({\tilde{G}})$.
\[thm:Jordan\] Assume ${\tilde{\mathbf{G}}}$, ${\tilde{G}}$, ${\tilde{\mathbf{G}}}^*$, ${\tilde{G}}^*$ are as in §\[sect:gLt\]. There is a bijection for any ${\tilde{s}}\in{\tilde{G}}^*$ between Lusztig series (note that ${\tilde{\mathbf{G}}}$ has connected center): $${\mathcal{J}}_{{\tilde{s}}}: \quad {\mathcal{E}}({\tilde{G}},{\tilde{s}}) \longleftrightarrow {\mathcal{E}}(C_{{\tilde{G}}^*}({\tilde{s}}),1)$$ such that $${\left\langle{\tilde{\chi}},R_{{\tilde{\mathbf{T}}}}^{{\tilde{\mathbf{G}}}}\hat{{\tilde{s}}}\right\rangle}_{{\tilde{G}}} = \varepsilon_{{\tilde{\mathbf{G}}}}\varepsilon_{C_{{\tilde{\mathbf{G}}}^*}({\tilde{s}})} {\left\langle{\mathcal{J}}_{{\tilde{s}}}({\tilde{\chi}}),R_{{\tilde{\mathbf{T}}}^*}^{C_{{\tilde{\mathbf{G}}}^*}({\tilde{s}})}1\right\rangle}_{C_{{\tilde{G}}^*}({\tilde{s}})},$$ where the ${\tilde{G}}$-conjugacy class of $({\tilde{\mathbf{T}}},\hat{{\tilde{s}}})$ corresponds to the ${\tilde{G}}^*$-conjugacy class of $({\tilde{\mathbf{T}}}^*,{\tilde{s}})$ via duality; see [@CE04 Theorem 8.21]. Furthermore ${\tilde{\chi}}\in{\mathcal{E}}({\tilde{G}},{\tilde{s}})$ is uniquely determined by the scalar products ${\left\langle{\tilde{\chi}},R_{{\tilde{\mathbf{T}}}}^{{\tilde{\mathbf{G}}}}\hat{{\tilde{s}}}\right\rangle}_{{\tilde{G}}}$. The set ${\operatorname{Irr}}({\tilde{G}})$ of characters of ${\tilde{G}}$ has a decomposition $${\operatorname{Irr}}({\tilde{G}}) = \bigcup\limits_{{\tilde{s}}} {\mathcal{E}}({\tilde{G}},{\tilde{s}}),$$ where ${\tilde{s}}$ runs over a ${\tilde{G}}^*$-transversal of semisimple elements of ${\tilde{G}}^*$.
Let $s=i^*({\tilde{s}})$, then $C_{{\mathbf{G}}^*}^\circ(s)^F$ is as in [@Li19 §3.A]. We can identify ${\mathcal{E}}(C_{{\tilde{G}}^*}({\tilde{s}}),1)$ with ${\mathcal{E}}(C_{{\mathbf{G}}^*}^\circ(s)^F,1)$, which can be parametrized by $\lambda=\prod_\Gamma\lambda_\Gamma$, where $\lambda_\Gamma$ is a partition of $m_\Gamma(s)$ for $\Gamma\in{\mathcal{F}}_1\cup{\mathcal{F}}_2$ while $\lambda_\Gamma$ is a Lusztig symbol of rank $[\frac{m_\Gamma(s)}{2}]$ for $\Gamma\in{\mathcal{F}}_0$; see [@Car85 §13.8]. Recall that the degenerate symbols in component $X+1$ are counted twice. For $\lambda$ as above, $\lambda'$ is defined as below: $\lambda'_\Gamma=\lambda_\Gamma$ when $\lambda_\Gamma$ is a partition or non degenerate symbol and $(\lambda')_\Gamma = (\lambda_\Gamma)'$ when $\lambda_\Gamma$ is a degenerate symbol. Denote the characters in ${\mathcal{E}}(C_{{\tilde{G}}^*}({\tilde{s}}),1)$ and ${\mathcal{E}}({\tilde{G}},{\tilde{s}})$ corresponding to $\lambda$ by $\chi_\lambda$ and $\chi_{{\tilde{s}},\lambda}$ respectively.
\[rem:action-tz\] The duality induces an isomorphism of abelian groups (see [@CE04 (8.19)]): $$Z({\tilde{G}}^*) \to {\operatorname{Irr}}({\tilde{G}}/G), \quad {\tilde{z}}\mapsto \hat{{\tilde{z}}}.$$ We consider the action of ${\operatorname{Irr}}({\tilde{G}}/G)$ on ${\operatorname{Irr}}({\tilde{G}})$. Note that by [@CE04 (8.20)], $\hat{{\tilde{z}}}\chi_{{\tilde{s}},\lambda} \in {\mathcal{E}}({\tilde{G}},{\tilde{z}}{\tilde{s}})$. Then $$\begin{aligned}
{\left\langle\hat{{\tilde{z}}}\chi_{{\tilde{s}},\lambda},R_{{\tilde{\mathbf{T}}}}^{{\tilde{\mathbf{G}}}}\widehat{{\tilde{z}}{\tilde{s}}}\right\rangle}_{{\tilde{G}}}
&= {\left\langle\chi_{{\tilde{s}},\lambda},\hat{{\tilde{z}}}^{-1}R_{{\tilde{\mathbf{T}}}}^{{\tilde{\mathbf{G}}}}\widehat{{\tilde{z}}{\tilde{s}}}\right\rangle}_{{\tilde{G}}}
= {\left\langle\chi_{{\tilde{s}},\lambda},R_{{\tilde{\mathbf{T}}}}^{{\tilde{\mathbf{G}}}}\hat{{\tilde{z}}}^{-1}\widehat{{\tilde{z}}{\tilde{s}}}\right\rangle}_{{\tilde{G}}}
= {\left\langle\chi_{{\tilde{s}},\lambda},R_{{\tilde{\mathbf{T}}}}^{{\tilde{\mathbf{G}}}}\hat{{\tilde{s}}}\right\rangle}_{{\tilde{G}}} \\
&= \varepsilon_{{\tilde{\mathbf{G}}}}\varepsilon_{C_{{\tilde{\mathbf{G}}}^*}({\tilde{s}})} {\left\langle\chi_\lambda,R_{{\tilde{\mathbf{T}}}^*}^{C_{{\tilde{\mathbf{G}}}^*}({\tilde{s}})}1\right\rangle}_{C_{{\tilde{G}}^*}({\tilde{s}})}.\end{aligned}$$ On the other hand, note that $C_{{\tilde{\mathbf{G}}}^*}({\tilde{z}}{\tilde{s}})=C_{{\tilde{\mathbf{G}}}^*}({\tilde{s}})$, then $${\left\langle\chi_{{\tilde{z}}{\tilde{s}},\lambda},R_{{\tilde{\mathbf{T}}}}^{{\tilde{\mathbf{G}}}}\widehat{{\tilde{z}}{\tilde{s}}}\right\rangle}_{{\tilde{G}}}
= \varepsilon_{{\tilde{\mathbf{G}}}}\varepsilon_{C_{{\tilde{\mathbf{G}}}^*}({\tilde{s}})} {\left\langle\chi_\lambda,R_{{\tilde{\mathbf{T}}}^*}^{C_{{\tilde{\mathbf{G}}}^*}({\tilde{s}})}1\right\rangle}_{C_{{\tilde{G}}^*}({\tilde{s}})}.$$ Thus with the Jordan decomposition of ${\operatorname{Irr}}({\tilde{G}})$ as in Theorem \[thm:Jordan\], $\hat{{\tilde{z}}}\chi_{{\tilde{s}},\lambda} = \chi_{{\tilde{z}}{\tilde{s}},\lambda}$.
#### {#section-3}
Fix a generator ${\tilde{z}}_0$ of $Z({\tilde{G}}^*)$, then ${\tilde{z}}_0^{(q-1)/2}=-e$, where $e$ is the identity element of the special Clifford group ${\operatorname{D}}_0(V^*)$. For each $G^*$-conjugacy class of semisimple elements, fix a representative $s$ and a preimage ${\tilde{s}}$ of $s$ under $i^*$.
(1) If $m_{X+1}(s)=0$, then ${i^*}^{-1}({\operatorname{Cl}}_{G^*}(s))$ consists of $q-1$ conjugacy classes of ${\tilde{G}}^*$, each of which contains exactly one element ${\tilde{z}}{\tilde{s}}$ of ${\mathbb{F}}_q^\times{\tilde{s}}$ by [@FS89 (2D)]. Then $Z({\tilde{G}}^*) \cong {\operatorname{Irr}}({\tilde{G}}/G)$ acts regularly on ${\{ \chi_{{\tilde{z}}{\tilde{s}},\lambda} \mid {\tilde{z}}\in Z({\tilde{G}}^*)\}}$ as follows $$\chi_{{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{z}}_0{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \cdots \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{z}}_0^{q-2}{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{s}},\lambda}.$$
(2) If $m_{X+1}(s)\neq0$, then ${i^*}^{-1}({\operatorname{Cl}}_{G^*}(s))$ consists of $(q-1)/2$ conjugacy classes of ${\tilde{G}}^*$, each of which contains exactly two elements ${\tilde{z}}{\tilde{s}},-{\tilde{z}}{\tilde{s}}$ of ${\mathbb{F}}_q^\times{\tilde{s}}$ by [@FS89 (2D)].
1. If furthermore $\lambda_{X+1}$ is non degenerate, then by [@FS89 (4C)], $\widehat{-e}\chi_{{\tilde{s}},\lambda}=\chi_{{\tilde{s}},\lambda}$. So, with the Jordan decomposition of ${\operatorname{Irr}}({\tilde{G}})$ as in Theorem \[thm:Jordan\], we have $\chi_{-{\tilde{s}},\lambda}=\chi_{{\tilde{s}},\lambda}$. Thus $Z({\tilde{G}}^*) \cong {\operatorname{Irr}}({\tilde{G}}/G)$ acts transitively on ${\{ \chi_{{\tilde{z}}{\tilde{s}},\lambda} \mid {\tilde{z}}\in Z({\tilde{G}}^*)\}}$ with kernel ${\langle-e\rangle}$ as follows $$\chi_{{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{z}}_0{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \cdots \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{z}}_0^{(q-1)/2-1}{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{s}},\lambda}.$$
2. If furthermore $\lambda_{X+1}$ is degenerate, then $\widehat{-e}\chi_{{\tilde{s}},\lambda}=\chi_{{\tilde{s}},\lambda'}$ by [@FS89 (4C)]. Thus with the Jordan decomposition of ${\operatorname{Irr}}({\tilde{G}})$ as in Theorem \[thm:Jordan\], we have $\chi_{-{\tilde{s}},\lambda}=\chi_{{\tilde{s}},\lambda'}$. Then $Z({\tilde{G}}^*) \cong {\operatorname{Irr}}({\tilde{G}}/G)$ acts regularly on ${\{ \chi_{{\tilde{z}}{\tilde{s}},\lambda},\chi_{{\tilde{z}}{\tilde{s}},\lambda'} \mid {\tilde{z}}\in Z({\tilde{G}}^*)\}}$ and we can choose the labels such that $$\chi_{{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \cdots \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{z}}_0^{(q-1)/2-1}{\tilde{s}},\lambda} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{s}},\lambda'} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \cdots \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{z}}_0^{(q-1)/2-1}{\tilde{s}},\lambda'} \xmapsto{\widehat{{\tilde{z}}_0}\cdot} \chi_{{\tilde{s}},\lambda}.$$
Brauer pairs and blocks of conformal groups {#sec:prepare}
===========================================
From now on until the last section, we assume $q$ is odd. Before considering the weights of conformal symplectic groups, we make some preparations, including classification of radical subgroups, some considerations of Brauer pairs and blocks.
{#subsec:Rad-tG}
We begin by dealing with the conjugacy classes of radical subgroups of ${\tilde{G}}$. Denote by $Z({\tilde{G}})_\ell$ the $\ell$-part of the cyclic group $Z({\tilde{G}})$. For any symplectic space $V$, denote by ${\operatorname{I}}_0(V)$ and ${\operatorname{J}}_0(V)$ the symplectic group and the conformal symplectic group on $V$ respectively. Recall that for any radical subgroup $R= R_0 \times R_1 \times\cdots\times R_u$ with $R_i=R_{m_i,\alpha_i,\gamma_i,{\mathbf{c}}_i}$ for $i=1,\ldots,u$, there is a corresponding decomposition of the space $V= V_0\perp V_1\perp\cdots\perp V_u$.
\[prop:Rad(tG)\] For any radical subgroup $R$ of $G$ as above, denote ${\tilde{R}}:=Z({\tilde{G}})_\ell R$.
(1) ${\tilde{N}}:=N_{{\tilde{G}}}({\tilde{R}})=N_{{\tilde{G}}}(R)={\langle\tau,N\rangle}$ with $N:=N_G(R)$ and $\tau=\tau_0\times\tau_1\times\cdots\times\tau_u$, where $\tau_0$ is an element of order $q-1$ generating ${\operatorname{J}}_0(V_0)$ modulo ${\operatorname{I}}_0(V_0)$ and $\tau_i=\tau_{m_i,\alpha_i,\gamma_i,{\mathbf{c}}_i}$ for $i=1,\ldots,u$.
(2) The map $R \mapsto {\tilde{R}}$ defines a bijection between ${\operatorname{Rad}}(G)$ and ${\operatorname{Rad}}({\tilde{G}})$ with inverse ${\tilde{R}}\mapsto G\cap{\tilde{R}}$, which induces a bijection between ${\operatorname{Rad}}(G)/{\!\!\sim\!\!}{G}$ and ${\operatorname{Rad}}({\tilde{G}})/{\!\!\sim\!\!}{{\tilde{G}}}$.
The first assertion follows from [@Li19 Lemma 6.6 (4)]. Let ${\hat{G}}=Z({\tilde{G}})G$. Since $|{\tilde{G}}:{\hat{G}}|=2$ and $\ell$ is odd, ${\operatorname{Rad}}({\tilde{G}})={\operatorname{Rad}}({\hat{G}})$. Now ${\hat{G}}$ is the central product of $Z({\tilde{G}})$ and $G$ over $Z(G)$ and $|Z(G)|=2$, the maps given in (2) are bijections between ${\operatorname{Rad}}(G)$ and ${\operatorname{Rad}}({\hat{G}})$. By the definition, $\tau$ generates ${\tilde{G}}$ modulo $G$. Thus $|{\tilde{G}}:N_{{\tilde{G}}}(R)|=|G:N_G(R)|$ and the given maps induce also a bijection between conjugacy classes.
Many results about the defect groups of blocks of ${\tilde{G}}$ in [@FS89] hold for radical subgroups of ${\tilde{G}}$. In the sequel, when we state such a result for all radical subgroups, we will often refer to [@FS89] for the corresponding result for defect groups. We will also give new proofs for some of these statements using explicit calculations in twisted groups.
Brauer pairs {#subsec:Brauerpair}
------------
Let $R$ be a radical subgroup of $G$ as above. Set $R_+=R_1 \times\cdots\times R_u$, then $R=R_0\times R_+$, $V = V_0 \perp V_+$ and all related constructions can be decomposed correspondingly. Then $C:=C_G(R)= C_0 \times C_+$, where $C_0={\operatorname{I}}_0(V_0)$ and $C_+=C_1 \times\cdots\times C_u$ with $C_i=C_{m_i,\alpha_i,\gamma_i,{\mathbf{c}}_i}$ for $i>0$; for $C_{m_i,\alpha_i,\gamma_i,{\mathbf{c}}_i}$, see the definition before [@Li19 Lemma 6.4]. Let $\tau$ be as before, then by [@Li19 Lemma 6.4], ${\tilde{C}}:=C_{{\tilde{G}}}(R)=C_{{\tilde{G}}}({\tilde{R}})={\langle\tau,C\rangle}$, $[\tau,RC]=1$ and $\tau^{q-1}\in Z(C)$. The following is a generalization of [@FS89 (5J)] to radical subgroups.
\[lem:cen-prod\] Let $R$ be a radical subgroup of $G$, ${\tilde{R}}=Z({\tilde{G}})_\ell R$ and $Z_+$ be a subgroup of $Z(C_+)$ containing $\tau_+^{q-1}$.
(1) ${\tilde{C}}$ is a central product of ${\langle\tau,C_0Z_+\rangle}$ and $C_+$ over $Z_+$.
(2) Assume $Z_+=Z(C_+)$, then $Z({\tilde{G}})_\ell\leq{\langle\tau,C_0Z(C_+)\rangle}$. If $g\in N$ and $g_0,g_+$ are the restriction of $g$ to $V_0,V_+$ respectively, then $[\tau_0,g_0]\in C_0, [\tau_+,g_+]\in Z(C_+)$. In particular, $N$ and thus ${\tilde{N}}$ normalize ${\langle\tau,C_0Z(C_+)\rangle}$ and $C_+$.
This can be proved by the same method as in [@FS89 (5J)], or can be proved with explicit calculation by transferring to the twisted group ${\tilde{G}}^{tw}$ and using the choice of $\tau_{m,\alpha,\gamma,{\mathbf{c}}}^{tw}$ in [@Li19 (6.3)].
\[rem:cen-prod\] As in [@FS89], mainly the following two cases of $Z_+$ are used:
(1) $Z_+={\langle\tau_+^{q-1}\rangle}$, in which case, ${\tilde{C}}$ is viewed as a central product of ${\langle\tau,C_0\rangle}$ and $C_+$ over ${\langle\tau_+^{q-1}\rangle}$.
(2) $Z_+=Z(C_+)$, in which case, we denote ${\tilde{C}}_0={\langle\tau,C_0Z(C_+)\rangle}$. As shown in the above lemma, the advantage of this case is that ${\tilde{N}}$ stabilize this central product decomposition.
Let ${\mathcal{F}}'$ be the subset of polynomials in ${\mathcal{F}}$ whose roots have $\ell'$-order. For any $\Gamma\in{\mathcal{F}}'$, $G_\Gamma,R_\Gamma,C_\Gamma,N_\Gamma,\theta_\Gamma$ are as on [@An94 p.22]. We refer to [@Li19 §6.B] for the constructions for symplectic groups. In particular, $\Gamma$ determines an $N_\Gamma$-conjugacy class in ${\operatorname{dz}}(C_\Gamma/R_\Gamma)$ and $\theta_\Gamma$ is a chosen one in this conjugacy class. Defined as in [@Li19 §6.B] are also $R_{\Gamma,\gamma,{\mathbf{c}}}$, $\theta_{\Gamma,\gamma,{\mathbf{c}}}$, the set ${\mathcal{R}}_{\Gamma,\delta}={\{R_{\Gamma,\delta,1},R_{\Gamma,\delta,2},\ldots\}}$, $\theta_{\Gamma,\delta,i}$, and the Notation 6.10. Recall that $D_{m,\alpha,\beta}$ is defined at the end of §\[subsec:ano-conju\]. Then similarly, we denote $D_{\Gamma,\beta}=D_{m_\Gamma,\alpha_\Gamma,\beta}$. So $D_{\Gamma,\beta}=R_\Gamma\wr A_{{\mathbf{c}}_\beta}$.
Let $\theta\in{\operatorname{dz}}(C/Z(R))$, then $\theta=\theta_0\times\theta_+$, where $\theta_0$ is a character of ${\operatorname{I}}_0(V_0)$ of defect zero, while $\theta_+$ and $R_+$ can be decomposed as follows: $$\label{equ:theta_+R_+}
\theta_+ = \prod_{\Gamma,\delta,i} \theta_{\Gamma,\delta,i}^{t_{\Gamma,\delta,i}},\quad
R_+=\prod_{\Gamma,\delta,i}R_{\Gamma,\delta,i}^{t_{\Gamma,\delta,i}}.
\addtocounter{thm}{1}\tag{\thethm}$$
Recall that ${\tilde{R}}=Z({\tilde{G}})_\ell R$. Then by Lemma \[lem:cen-prod\], the results for canonical characters associated with defect groups in [@FS89 §7] can be generalized to radical subgroups. Specifically, any ${\tilde{\theta}}\in{\operatorname{dz}}({\tilde{C}}/Z({\tilde{R}})\mid\theta_R)$ is of the form $${\tilde{\theta}}={\tilde{\theta}}_0\theta_+,$$ where ${\tilde{\theta}}_0$ is the canonical character of a block of ${\tilde{C}}_0={\langle\tau,C_0Z(C_+)\rangle}$ of the central defect group $Z({\tilde{R}})=Z({\tilde{G}})_\ell Z(R)$, and the linear characters of $Z(C_+)$ induced by ${\tilde{\theta}}_0$ and $\theta_+$ are the same. Note that ${\tilde{\theta}}_0\in{\operatorname{Irr}}({\tilde{C}}_0\mid\theta_0)$. Recall that $$\label{equ:cen-prod-tC-R}
{\tilde{C}}_0 = {\langle\tau,C_0\rangle} \times_{{\langle\tau_+^{q-1}\rangle}} Z(C_+)
\addtocounter{thm}{1}\tag{\thethm}$$ is a central product over ${\langle\tau_+^{q-1}\rangle}$.
Let $D'=D'_0\times D'_+$, $\theta'=\theta'_0\times\theta'_+$ with $D'_0=R_0$, $\theta'_0=\theta_0$ and $$D'_+= \prod_\Gamma R_\Gamma^{m_\Gamma},\quad \theta'_+=\prod_\Gamma\theta_\Gamma^{m_\Gamma},$$ where, $m_\Gamma = \sum_{\delta,i}t_{\Gamma,\delta,i}\ell^\delta$. Let ${\tilde{D}}'=Z({\tilde{G}})_\ell D'$, $C',{\tilde{C}}'$ denote the centralizers of $D',{\tilde{D}}'$ in $G,{\tilde{G}}$ respectively and $C' = C'_0 \times C'_+$ with $C'_0=C_0={\operatorname{I}}_0(V_0)$. Then $\theta' \in {\operatorname{dz}}(C'/Z(D'))$. Assume ${\tilde{\theta}}'={\tilde{\theta}}'_0\theta'_+\in{\operatorname{dz}}({\tilde{C}}'/Z({\tilde{D}}')\mid\theta')$ and $\theta'$ are in the same relation as ${\tilde{\theta}}$ and $\theta$. Note that $$\label{equ:cen-prod-tC-D'}
{\tilde{C}}'_0:= {\langle\tau,C'_0Z(C'_+)\rangle} = {\langle\tau,C'_0\rangle} \times_{{\langle\tau_+^{q-1}\rangle}} Z(C'_+)
\addtocounter{thm}{1}\tag{\thethm}$$ is a central product over ${\langle\tau_+^{q-1}\rangle}$. Since $Z(C_+) \leq Z(C'_+)$, ${\tilde{\theta}}'_0$ can be chosen to be an extension of ${\tilde{\theta}}_0$ by (\[equ:cen-prod-tC-R\]) and (\[equ:cen-prod-tC-D’\]).
Let $D=D_0\times D_+$, $\theta_D=\theta_{D,0}\times\theta_{D,+}$ with $D_0=R_0$, $\theta_{D,0}=\theta_0$ and $$D_+= \prod_{\Gamma,\beta} D_{\Gamma,\beta}^{r_{\Gamma,\beta}},\quad \theta_{D,+}=\prod_{\Gamma,\beta}(\theta_\Gamma\otimes I_{\ell^\beta})^{r_{\Gamma,\beta}},$$ where, $m_\Gamma = \sum_\beta r_{\Gamma,\beta}\ell^\beta$ is the $\ell$-adic decomposition of $m_\Gamma$. Let ${\tilde{D}}=Z({\tilde{G}})_\ell D$, $C_D,{\tilde{C}}_D$ denote the centralizers of $D,{\tilde{D}}$ in $G,{\tilde{G}}$ respectively and $C_D = C_{D,0} \times C_{D,+}$ with $C_{D,0}=C_0={\operatorname{I}}_0(V_0)$. Then $\theta_D \in {\operatorname{dz}}(C_D/Z(D))$. Assume ${\tilde{\theta}}_D={\tilde{\theta}}_{D,0}\theta_{D,+}\in{\operatorname{dz}}({\tilde{C}}_D/Z({\tilde{D}})\mid\theta_D)$ and $\theta_D$ are in the same relation as ${\tilde{\theta}}$ and $\theta$. Similarly as above, $$\label{equ:cen-prod-tC-D}
{\tilde{C}}_{D,0} = {\langle\tau,C_{D,0}Z(C_{D,+})\rangle} = {\langle\tau,C_{D,0}\rangle} \times_{{\langle\tau_+^{q-1}\rangle}} Z(C_{D,+})
\addtocounter{thm}{1}\tag{\thethm}$$ is a central product over ${\langle\tau_+^{q-1}\rangle}$. Since $Z(C_{D,+}) \leq Z(C'_+)$, ${\tilde{\theta}}_{D,0}$ can be chosen to be the restriction of ${\tilde{\theta}}'_0$.
When every basic subgroup in $R$ is chosen to be the conjugate in §\[subsec:ano-conju\], the element $\tau=\iota(\tau^{tw})$ for $R,D',D$ can be chosen to be the same one and we make this choice in the sequel.
\[prop:Brauerpair\] With the above notation,
(1) $(D',\theta') \unlhd (D,\theta_D)$ as Brauer pairs in $G$. $(D,\theta_D)$ is a maximal pair for some block $B$ of $G$ and all maximal pairs are of this form.
(2) $({\tilde{D}}',{\tilde{\theta}}') \unlhd ({\tilde{D}},{\tilde{\theta}}_D)$ as Brauer pairs in ${\tilde{G}}$. $({\tilde{D}},{\tilde{\theta}}_D)$ is a maximal pair for some block ${\tilde{B}}$ of ${\tilde{G}}$ and all maximal pairs are of this form.
(3) ${\tilde{B}}$ covers $B$.
(4) $(R,\theta) \leq (D,\theta_D)$ as Brauer pairs in $G$ and $({\tilde{R}},{\tilde{\theta}}) \leq ({\tilde{D}},{\tilde{\theta}}_D)$ as Brauer pairs in ${\tilde{G}}$.
\(1) and (2) are just [@FS89 (8A)]. For (3), note first that $N_{{\tilde{G}}}({\tilde{D}})=N_{{\tilde{G}}}(D)$. Let $b,{\tilde{b}}$ be the Brauer correspondents of $B,{\tilde{B}}$ respectively, then by the definition, ${\tilde{b}}$ covers $b$. Thus (3) follows from the Harris-Knörr correspondence [@HK85]. To prove $(R,\theta) \leq (D,\theta_D)$ as Brauer pairs in $G$, it suffices to prove that $(R_+,\theta_+) \leq (D_+,\theta_{D,+})$ as Brauer pairs in $G_+={\operatorname{I}}_0(V_+)$. Let $z_+$ be an element in $Z(D_+)$ such that (i) $o(z_+)=\ell$, (ii) $[z_+,V_+]=V_+$, (iii) $z_+$ is primary; see [@FS89 p.178]. Then by the above constructions from $R$ to $D$, $z_+\in Z(R_+)$. So both $R_+C_{G_+}(R_+)$ and $D_+C_{G_+}(D_+)$ are contained in $C_{G_+}(z_+)$, which is isomorphic to a general linear or unitary group when $\ell$ is a linear or unitary prime respectively. By the construction of $D$ and results of [@Brou86], $(R_+,\theta_+) \leq (D_+,\theta_{D,+})$ as Brauer pairs in $C_{G_+}(z_+)$ and so as Brauer pairs in $G_+={\operatorname{I}}_0(V_+)$; see also [@An94 p.18]. Then $({\tilde{R}},{\tilde{\theta}}) \leq ({\tilde{D}},{\tilde{\theta}}_D)$ as Brauer pairs in ${\tilde{G}}$ follows by a similar argument as in [@FS89 (8A)] using the first kind of central product decompositions in Remark \[rem:cen-prod\].
\[cor:weights-into-blocks\] Keep the above notation and assume $({\tilde{R}},{\tilde{\varphi}})$ is a weight such that ${\tilde{\varphi}}\in {\operatorname{Irr}}({\tilde{N}}\mid{\tilde{\theta}})$, then $({\tilde{R}},{\tilde{\varphi}})$ belongs to the block ${\tilde{B}}$ in the above proposition.
{#subsec:block}
Let $R$ be a radical subgroup of $G$ and keep the above constructions and notation. In [@FS89 §11], the authors use the Brauer pair $({\tilde{D}}',{\tilde{\theta}}')$ to give a label $({\tilde{s}},{\mathcal{K}})$ for the block ${\tilde{B}}$. Recall that we use ${\tilde{s}}$ to denote a semisimple element in ${\tilde{G}}^*$ and denote by $s=i^*({\tilde{s}})$ the image of ${\tilde{s}}$ in $G^*$, while in [@FS89], the corresponding notation is $s$ and $\bar{s}$; some other notation used here is also slightly different from the one in [@FS89]. For convenience, we first recall the results as follows.
Since ${\tilde{C}}'$ is a Levi subgroup of ${\tilde{G}}$, ${\tilde{\theta}}'$ is contained in some Lusztig series ${\mathcal{E}}({\tilde{C}}',({\tilde{s}}))$ with ${\tilde{s}}$ a semisimple $\ell'$-element in $({\tilde{C}}')^* \leq {\tilde{G}}^*$. This ${\tilde{s}}$ is the semisimple part of the label $({\tilde{s}},{\mathcal{K}})$.
By [@FS89 (11.2),(11.3)], $$\label{equ:tC'_0}
\begin{aligned}
{\tilde{C}}'_0/{\operatorname{I}}_0(V_0) &\cong {\langle\tau_+,Z(C'_+)\rangle},\\
{\tilde{C}}'_0/Z(C'_+) &\cong {\langle\tau_0,{\operatorname{I}}_0(V_0)\rangle} = {\operatorname{J}}_0(V_0).
\end{aligned}
\addtocounter{thm}{1}\tag{\thethm}$$ Then ${\tilde{\theta}}'$ can be decomposed as ${\tilde{\theta}}'={\tilde{\chi}}'\zeta'\theta'_+$, where ${\tilde{\chi}}'$ is a character of ${\operatorname{J}}_0(V_0)$ and $\zeta'$ is an extension to ${\langle\tau_+,Z(C'_+)\rangle}$ of the linear character $\zeta'_+$ of $Z(C'_+)$ induced by $\theta'_+$ and ${\tilde{\theta}}'_0$. By the Jordan decomposition of characters of ${\operatorname{J}}_0(V_0)$, ${\tilde{\chi}}'={\tilde{\chi}}_{{\tilde{s}}_0,\kappa}^{{\operatorname{J}}_0(V_0)}$ for some semisimple $\ell'$-element ${\tilde{s}}_0$ of ${\operatorname{J}}_0(V_0)^*$ and $\kappa=\prod_{\Gamma\in{\mathcal{F}}'}\kappa_\Gamma$, where $\kappa_\Gamma$ is a Lusztig symbol or partition according to $\Gamma\in{\mathcal{F}}_0$ or $\Gamma\notin{\mathcal{F}}_0$. But since the decomposition ${\tilde{\theta}}'={\tilde{\chi}}'\zeta'\theta'_+$ is not unique, $\kappa$ has different choices. By the process described in [@FS89 §11], a carefully chosen set ${\mathcal{K}}$ of the unipotent labels $\kappa$ appearing in the label of ${\tilde{\chi}}'$ in some decompositions ${\tilde{\theta}}'={\tilde{\chi}}'\zeta'\theta'_+$ is the unipotent label of the block ${\tilde{B}}$. For symplectic groups, $|{\mathcal{K}}|$ is one or two.
In [@FS89 §11, §13], the authors consider both conformal symplectic and conformal orthogonal groups. Many of their complicated considerations are for conformal orthogonal groups. For convenience, we list in Table \[tab:block\] their results about conformal symplectic groups and the results in [@Li19] for symplectic groups.
cases conditions ${\mathcal{B}}$ ${\tilde{\mathcal{B}}}$ ${\operatorname{Irr}}({\tilde{B}})\cap{\mathcal{E}}({\tilde{G}},\ell')$
------- ----------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------- -- --
(I) $m_{X+1}(s)=0$ $B_{s,\kappa}$ $\begin{array}{c} {\tilde{B}}_{{\tilde{s}}{\tilde{z}},{\{\kappa\}}},\\ {\tilde{z}}\in Z({\tilde{G}}^*)_{\ell'}\\ \end{array}$ $\begin{array}{c} \chi_{{\tilde{s}}{\tilde{z}},\lambda},\\ {\{\kappa\}}=\textrm{ core of }\lambda\\ \end{array}$
(II) $\begin{array}{c} \kappa_{X+1}\neq\O\\ \textrm{non-degenerate}\\ 2{\operatorname{rk}}\kappa_{X+1}=m_{X+1}(s)\\ \end{array}$ $\begin{array}{c} B_{s,\kappa,i}, \\ i=0,1 \\ \end{array}$ $\begin{array}{c} {\tilde{B}}_{{\tilde{s}}{\tilde{z}},{\{\kappa\}}},\\ {\tilde{z}}\in Z({\tilde{G}}^*)_{\ell'}\\ \end{array}$ $\begin{array}{c} \chi_{{\tilde{s}}{\tilde{z}},\lambda},\\ {\{\kappa\}}=\textrm{ core of } \lambda\\ \end{array}$
(III) $\begin{array}{c} \kappa_{X+1}\neq\O\\ \textrm{degenerate}\\ 2{\operatorname{rk}}\kappa_{X+1}=m_{X+1}(s)\\ \end{array}$ $B_{s,\kappa}$ $\begin{array}{c} {\tilde{B}}_{{\tilde{s}}{\tilde{z}},{\mathcal{K}}},\\ {\tilde{z}}\in Z({\tilde{G}}^*)_{\ell'}\\ {\mathcal{K}}={\{\kappa\}}\textrm{~or~}{\{\kappa'\}}\\ \end{array}$ $\begin{array}{c} \chi_{{\tilde{s}}{\tilde{z}},\lambda},\\ {\mathcal{K}}=\textrm{ core of } \lambda\\ \end{array}$
(IV) $\begin{array}{c} \kappa_{X+1}=\O\\ m_{X+1}(s)\neq0\\ \end{array}$ $B_{s,\kappa}$ $\begin{array}{c} {\tilde{B}}_{{\tilde{s}}{\tilde{z}},{\{\kappa\}}},\\ {\tilde{z}}\in Z({\tilde{G}}^*)_{\ell'}\\ \end{array}$ $\begin{array}{c} \chi_{{\tilde{s}}{\tilde{z}},\lambda},\\ {\{\kappa\}}=\textrm{ core of } \lambda\\ \end{array}$
(V) $\begin{array}{c} \kappa_{X+1}\neq\O\\ \textrm{non-degenerate}\\ 2{\operatorname{rk}}\kappa_{X+1}<m_{X+1}(s) \end{array}$ $\begin{array}{c} B_{s,\kappa,i}, \\ i=0,1 \\ \end{array}$ $\begin{array}{c} {\tilde{B}}_{{\tilde{s}}{\tilde{z}},{\{\kappa\}}},\\ {\tilde{z}}\in Z({\tilde{G}}^*)_{\ell'}\\ \end{array}$ $\begin{array}{c} \chi_{{\tilde{s}}{\tilde{z}},\lambda},\\ {\{\kappa\}}=\textrm{ core of } \lambda\\ \end{array}$
(VI) $\begin{array}{c} \kappa_{X+1}\neq\O\\ \textrm{degenerate} \\ 2{\operatorname{rk}}\kappa_{X+1}<m_{X+1}(s)\\ \end{array}$ $B_{s,\kappa}$ $\begin{array}{c} {\tilde{B}}_{{\tilde{s}}{\tilde{z}},{\{\kappa,\kappa'\}}},\\ {\tilde{z}}\in Z({\tilde{G}}^*)_{\ell'}\\ \end{array}$ $\begin{array}{c} \chi_{{\tilde{s}}{\tilde{z}},\lambda},\\ {\{\kappa,\kappa'\}}=\textrm{ core of } \lambda\\ \end{array}$
: Blocks of $G$ and ${\tilde{G}}$
\[tab:block\]
\[rem:block\] We give some remarks about the notation in Table \[tab:block\].
(1) ${\mathcal{B}}$ is a ${\tilde{G}}$-conjugacy class of blocks of $G$, i.e. covered by a certain block of ${\tilde{G}}$; ${\tilde{\mathcal{B}}}={\operatorname{Bl}}({\tilde{G}}\mid{\mathcal{B}})$. These sets satisfy the requirements in Theorem \[thm:criterion\].
(2) Recall that for symplectic groups, degenerate Lusztig symbols are counted once. See [@Li19 Theorem 3.2, Theorem 3.7] for notation of characters and blocks for symplectic groups.
(3) For conformal symplectic groups, every degenerate Lusztig symbol is counted twice and $\kappa'$ is defined as in §\[subsec:action-tz\]; see [@FS89 p.132]. For the definition of the core of $\lambda$ in the case of conformal groups, see [@FS89 §9]. The definition of $e_\Gamma$ in [@FS89 §9] is the same as that before [@Li19 Theorem 3.7].
Set $$i{\operatorname{IBr}}({\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}) = {\left\{ \lambda=\prod_\Gamma\lambda_\Gamma ~\middle|~ {\mathcal{K}}\textrm{~is the core of~} \lambda \right\}}.$$ Then by [@Ge93] and [@FS89 §13], the following holds.
\[lem:IBr\] There are bijections $i{\operatorname{IBr}}({\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}) \leftrightarrow {\operatorname{Irr}}({\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}) \cap {\mathcal{E}}({\tilde{G}},{\tilde{s}}) \leftrightarrow {\operatorname{IBr}}({\tilde{B}}_{{\tilde{s}},{\mathcal{K}}})$, denoted by $\lambda \mapsto \chi_{{\tilde{s}},\lambda} \mapsto \phi_{{\tilde{s}},\lambda}$.
The blockwise Alperin weight conjecture for ${\tilde{G}}$ {#sec:weights}
=========================================================
In this section, we classify the weights of ${\tilde{G}}$ and prove the blockswise Alperin weight conjecture for ${\tilde{G}}$.
{#subsec:N(ttheta)}
Let $({\tilde{R}},{\tilde{\varphi}})$ be a weight of ${\tilde{G}}$ belonging to the block ${\tilde{B}}$ with label $({\tilde{s}},{\mathcal{K}})$ as above. Then ${\tilde{\varphi}}$ is of the form ${\tilde{\varphi}}={\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}}{\tilde{\psi}}$, where ${\tilde{\theta}}\in{\operatorname{dz}}({\tilde{C}}/Z({\tilde{R}}))$ with $({\tilde{R}},{\tilde{\theta}})$ a ${\tilde{B}}$-Brauer pair and ${\tilde{\psi}}\in{\operatorname{dz}}({\tilde{N}}({\tilde{\theta}})/{\tilde{R}}\mid{\tilde{\theta}})$. When ${\tilde{\theta}}$ runs over a set of representatives of ${\tilde{N}}$-conjugacy classes of ${\operatorname{dz}}({\tilde{C}}/Z({\tilde{R}}))$, the above construction gives all weight characters associated with ${\tilde{R}}$. Thus we need to consider the ${\tilde{N}}$-conjugacy classes of ${\tilde{\theta}}$ and ${\tilde{N}}({\tilde{\theta}})$.
We first show how a result concerning $R_{\Gamma,\gamma}$ can be derived from [@FS89 (6A)] which deals only with $R_\Gamma$. But we will state and prove it in the twisted group $G_{\Gamma,\gamma}^{tw}$. Here, recall that for all related constructions, we replace the two parameters $m_\Gamma,\alpha_\Gamma$ by $\Gamma$. Let $s_\Gamma$ and $s_\Gamma^*$ be as in [@Li19 §6.B]. Let $s_\Gamma^0, \theta_\Gamma^0 = \pm R_{T_\Gamma^0}^{C_\Gamma^0} \hat{s_\Gamma^0}, \theta_{\Gamma,\gamma}^0 = \theta_\Gamma^0 \otimes I_{\ell^\gamma}, \theta_{\Gamma,\gamma}^{tw}$ be as in [@Li19 §6.C]. Set $\phi_\Gamma^0 = \hat{s_\Gamma^0}$ and denote by $\phi_\Gamma^{tw}$ the induced character on $Z(C_{\Gamma,\gamma}^{tw}) = \hbar(Z(C_\Gamma^0) \otimes I_{\ell^\gamma}) = \hbar({\mathfrak{Z}}_{q^{e\ell^{\alpha_\Gamma}}-\varepsilon}I_{m_\Gamma\ell^\gamma})$, where ${\mathfrak{Z}}_{q^{e\ell^{\alpha_\Gamma}}-\varepsilon}$ is the cyclic subgroup of ${\overline{\mathbb{F}}}_p^\times$ of order $q^{e\ell^{\alpha_\Gamma}}-\varepsilon$.
\[lem:phi-Gamma\] Let $g\in N_{\Gamma,\gamma}^{tw}(\theta_{\Gamma,\gamma}^{tw})$ and $z=[\tau_{\Gamma,\gamma}^{tw},g]$, then $z\in Z(C_{\Gamma,\gamma}^{tw})$ and the following hold:
(1) $\phi_\Gamma^{tw}(z)=\pm1$.
(2) If $\Gamma\neq X+1$, then $\phi_\Gamma^{tw}(z)=1$.
(3) Assume $\Gamma=X+1$, then $\phi_{X+1}^{tw}(z)=1$ or $-1$ according to whether $g$ is a square or a non-square in $N_{\Gamma,\gamma}^{tw}/\hbar(N_{\Gamma,\gamma}^0)$.
Recall from [@Li19 Lemma 6.6] that $N_{\Gamma,\gamma}^{tw} = \hbar(N_{\Gamma,\gamma}^0) V_{\Gamma,\gamma}^{tw}$, where $N_{\Gamma,\gamma}^0=C_{\Gamma,\gamma}^0L_{\Gamma,\gamma}^0$ is a central product and $V_{\Gamma,\gamma}^{tw}$ is as in [@Li19 Lemma 6.6 (2)]. Also, $\tau_{\Gamma,\gamma}^{tw}$ is as [@Li19 (6.3)], then by the definition of $\hbar$ in [@Li19 §6.A], $[\tau_{\Gamma,\gamma}^{tw},N_{\Gamma,\gamma}^{tw}] = [\tau_{\Gamma,\gamma}^{tw},V_{\Gamma,\gamma}^{tw}] \leq Z(C_{\Gamma,\gamma}^{tw})$. By the structure of $\tau_{\Gamma,\gamma}^{tw}$ and $Z(C_{\Gamma,\gamma}^{tw})$, there is no loss to assume $\gamma=0$ and thus this lemma follows from [@FS89 (6A)].
As for defect groups in [@FS89 §7], for any radical subgroup ${\tilde{R}}$, we have ${\tilde{N}}({\tilde{\theta}})={\langle\tau,N({\tilde{\theta}})\rangle}$ and $$N({\tilde{\theta}}) = N({\tilde{\theta}}_0) \cap N(\theta_+), \quad N(\theta_+) \leq N(\zeta_+),$$ where $\zeta_+$ is the linear character of $Z(C_+)$ induced by ${\tilde{\theta}}_0$ and $\theta_+$. Similarly as in [@FS89 (7B)], it is easy to see from Equation (\[equ:theta\_+R\_+\]) that $$N(\theta_+) = I_0(V_0) \times \prod_{\Gamma,\delta,i} N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})\wr{\mathfrak{S}}(t_{\Gamma,\delta,i}).$$ Note that the notation here is slightly different from the one in [@FS89]; in particular, $\gamma$ in [@FS89] and here have different meanings. For $g \in N$, the function $\omega_g$ on ${\tilde{C}}_0$ is defined in the same way as in [@FS89 (7.5)]. Then the conclusions of [@FS89 (7D), (7E)] for defect groups hold for any radical subgroups from Lemma \[lem:phi-Gamma\]. In particular, for any $g \in N(\theta_+)$, $\omega_g$ is a linear character of ${\tilde{C}}_0/C_0Z(C_+)$ of order $1$ or $2$. *In the sequel, when we refer to [@FS89 (7D), (7E)], we mean the generalized results for all radical subgroups.* For conformal symplectic groups, [@FS89 (7F)] can be simplified as follows.
\[lem:N(ttheta)\] Keep the above notation, then $$N({\tilde{\theta}}) = {\{ g \in N(\theta_+) \mid {\tilde{\theta}}_0\omega_g={\tilde{\theta}}_0 \}}$$ and $N({\tilde{\theta}})$ is a normal subgroup of $N(\theta_+)$ of index $1$ or $2$.
The proof is the same as for [@FS89 (7F)], noting that for symplectic groups, $g_0$ in the decomposition $g=g_0g_+$ satisfies $g_0\in{\operatorname{I}}_0(V_0)$ and thus ${^{g_0}{\tilde{\theta}}_0}={\tilde{\theta}}_0$.
\[prop:N(theta+):N(ttheta)\] Keep the above notation and let cases (I)$\sim$(VI) be as in Table \[tab:block\]. $$|N(\theta_+):N({\tilde{\theta}})|=
\left\{\begin{array}{ll}
1, & \textrm{cases (I)$\sim$(III), (V);}\\
2, & \textrm{cases (IV), (VI).}
\end{array}\right.$$
Note that $$\label{equ:tC_0}
\begin{aligned}
{\tilde{C}}_0/{\operatorname{I}}_0(V_0) &\cong {\langle\tau_+,Z(C_+)\rangle},\\
{\tilde{C}}_0/Z(C_+) &\cong {\langle\tau_0,{\operatorname{I}}_0(V_0)\rangle} = {\operatorname{J}}_0(V_0).
\end{aligned}
\addtocounter{thm}{1}\tag{\thethm}$$ Then similarly as on [@FS89 p.169], ${\tilde{\theta}}_0$ can be decomposed as ${\tilde{\theta}}_0 = {\tilde{\chi}}_0\zeta$, where ${\tilde{\chi}}_0$ is a character of ${\operatorname{J}}_0(V_0)$ and $\zeta$ is an extension to ${\langle\tau_+,Z(C_+)\rangle}$ of the linear character $\zeta_+$ induced by $\theta_+$ and ${\tilde{\theta}}_0$. Fix a choice $\zeta$ of the extension, then all possible choices are $\zeta\omega$ with $\omega \in {\operatorname{Irr}}({\operatorname{J}}_0(V_0)/{\operatorname{I}}_0(V_0))$.
\[rem:zeta\] From now on, we fix an extension $\zeta$ of $\zeta_+$ to ${\langle\tau_+,Z(C_+)\rangle}$, then ${\tilde{\chi}}_0$ and ${\tilde{\theta}}_0$ determine each other since $\zeta$ is a linear character.
$s=i^*({\tilde{s}})$ can be decomposed as $s=s_0s_+$, where $s_0\in{\operatorname{I}}_0(V_0)$ and $s_+\in{\operatorname{I}}_0(V_+)$ can be determined by $\theta_0$ and $\theta_+$ respectively; see [@Li19 §6.B]. Since $({\tilde{R}},{\tilde{\theta}})$ is a ${\tilde{B}}$-Brauer pair and ${\tilde{B}}$ has label $({\tilde{s}},{\mathcal{K}})$, then by Proposition \[prop:Brauerpair\] and the results in [@FS89 §11], ${\tilde{\chi}}_0$ can be chosen to be of the form ${\tilde{\chi}}_0 = \chi_{{\tilde{s}}_0,\kappa}^{{\operatorname{J}}_0(V_0)}$, where ${\tilde{s}}_0\in{\operatorname{D}}_0(V_0^*)$ satisfies $s_0=i^*({\tilde{s}}_0)$ and $\kappa\in{\mathcal{K}}$. By Lemma \[lem:N(ttheta)\] and Remark \[rem:zeta\], $N({\tilde{\theta}}) = {\{ g \in N(\theta_+) \mid {\tilde{\chi}}_0={\tilde{\chi}}_0\omega_g \}}$. By [@FS89 (7E)], $\omega_g\neq1$ for some $g \in N(\theta_+)$ if and only if $m_{X+1}(s_+)\neq0$. For cases (I)$\sim$(III), we then have $\omega_g=1$ for all $g \in N(\theta_+)$, thus $|N(\theta_+):N({\tilde{\theta}})|=1$. For cases (IV)$\sim$(VI), $\omega_g\neq1$ for some $g \in N(\theta_+)$, then the assertion follows from Remark \[rem:action-tz\].
Recall that the ${\tilde{N}}$-conjugacy class of ${\tilde{\theta}}$ is the same as the $N$-conjugacy class of ${\tilde{\theta}}$. The following is essentially in the spirit of [@FS89 §11].
\[cor:cc-cano-chars\] Let $({\tilde{R}},{\tilde{\varphi}})$ be a weight of ${\tilde{G}}$ belonging to the block ${\tilde{B}}$ with label $({\tilde{s}},{\mathcal{K}})$, where ${\tilde{\varphi}}$ lies over some ${\tilde{\theta}}\in{\operatorname{dz}}({\tilde{C}}/Z({\tilde{R}}))$ such that $({\tilde{R}},{\tilde{\theta}})$ is a ${\tilde{B}}$-Brauer pair. Let $\theta_+$ be as in (\[equ:theta\_+R\_+\]). Let ${\operatorname{Cl}}_{{\tilde{N}}}({\tilde{\theta}})$ be the ${\tilde{N}}$-conjugacy class of ${\tilde{\theta}}$. Denote ${\operatorname{Cl}}_{N(\theta_+)}({\tilde{\theta}}) = {\{ {\tilde{\theta}}_1\in {\operatorname{Cl}}_{{\tilde{N}}}({\tilde{\theta}}) \mid {\tilde{\theta}}_1={\tilde{\theta}}_{1,0}\theta_{1,+}, \theta_{1,+}=\theta_+ \}}$. Then the set $${\left\{ \kappa \mid {\tilde{\theta}}_1={\tilde{\theta}}_{1,0}\theta_+\in {\operatorname{Cl}}_{N(\theta_+)}({\tilde{\theta}}), {\tilde{\theta}}_{1,0} \textrm{~induces~} \chi_{{\tilde{s}}_0,\kappa}^{{\operatorname{J}}_0(V_0)} \right\}}$$ coincides with ${\mathcal{K}}$.
By the proof of Proposition \[prop:N(theta+):N(ttheta)\] and Table \[tab:block\].
\[rem:cc-cano-chars\] By the remark at the beginning of this subsection, when we construct weight characters, we start with a representative ${\tilde{\theta}}$ of an ${\tilde{N}}$-conjugacy class of ${\operatorname{dz}}({\tilde{C}}/Z({\tilde{R}}))$. We fix a way to choose such a representative. Recall that ${\tilde{\theta}}={\tilde{\theta}}_0\theta_+$ has two parts: the $0$-part ${\tilde{\theta}}_0$ and the $+$-part $\theta_+$. Since $\Gamma$ determines an $N_\Gamma$-conjugacy class in ${\operatorname{dz}}(C_\Gamma/R_\Gamma)$, once we fix $\theta_\Gamma\in{\operatorname{dz}}(C_\Gamma/R_\Gamma)$ for each $\Gamma$, we fix the way to choose $\theta_+$; as in [@FS89 §7], call it normalized. For cases (I)$\sim$(III), (V) and any ${\tilde{N}}$-conjugacy class of ${\operatorname{dz}}({\tilde{C}}/Z({\tilde{R}}))$, by Proposition \[prop:N(theta+):N(ttheta)\], there is only one ${\tilde{\theta}}$ in it whose $+$-part $\theta_+$ is normalized; we take this ${\tilde{\theta}}$ as the representative of this ${\tilde{N}}$-conjugacy class. For cases (IV), (VI) and any ${\tilde{N}}$-conjugacy class of ${\operatorname{dz}}({\tilde{C}}/Z({\tilde{R}}))$, by Proposition \[prop:N(theta+):N(ttheta)\], there are two ${\tilde{\theta}}$ in it whose $+$-part $\theta_+$ is normalized. We can take either of the two choices as the representative of this ${\tilde{N}}$-conjugacy class, but once we fix one, we should always use that one.
\[rem:ts0\] For the proof of Lemma \[lem:(3)of(iii)\], we make a convention related to Remark \[rem:action-tz\] and Remark \[rem:zeta\]. Fixing an extension $\zeta$ as in Remark \[rem:zeta\] gives a decomposition ${\tilde{s}}={\tilde{s}}_0{\tilde{s}}_+$, where ${\tilde{\chi}}_0 = \chi_{{\tilde{s}}_0,\kappa}^{{\operatorname{J}}_0(V_0)}$ as in the proof of Proposition \[prop:N(theta+):N(ttheta)\]. Let $s$, ${\tilde{s}}$ and the labels for characters in ${\operatorname{Irr}}({\tilde{G}})$ be as in Remark \[rem:action-tz\]; but let ${\tilde{z}}_0$ be a generator of $Z({\tilde{G}}^*)_{\ell'}$. We start with ${\tilde{G}}$ of a fixed dimension $2n$, and consider blocks with labels $({\tilde{s}}{\tilde{z}}_0^i,{\mathcal{K}})$, where $i \in {\{0,1,\ldots,(q-1)_{\ell'}-1\}}$ if $m_{X+1}(s)=0$ while $i \in {\{0,1,\ldots,(q-1)_{\ell'}/2-1\}}$ if $m_{X+1}(s)\neq0$. Then we have a fixed symplectic space $V_0$ and the conformal symplectic group ${\operatorname{J}}_0(V_0)$ over $V_0$. Thus we can fix particular $\zeta$’s for all blocks with labels $({\tilde{s}}{\tilde{z}}_0^i,{\mathcal{K}})$ such that the induced characters of ${\operatorname{J}}_0(V_0)$ are $\chi_{{\tilde{s}}_0{\tilde{z}}_0^i,\kappa}^{{\operatorname{J}}_0(V_0)}$’s.
{#subsec:weight}
In this subsection, we label the weights of ${\tilde{G}}$ and prove the blockwise Alperin weight conjecture for ${\tilde{G}}$. We start with a lemma about the construction of characters under some *ad hoc* conditions.
\[lem:cons-char\] Let $K= K_0 \times_Z K_+$ be a central product over $Z= K_0 \cap K_+$. Assume ${\tilde{\theta}}={\tilde{\theta}}_0\theta_+$ is a character of $K$. Denote by $\zeta_+$ the linear character of $Z$ induced by ${\tilde{\theta}}_0$ and $\theta_+$. Let $H=K_0H_+$ be such that $K_0 \cap H_+ =Z$, $K_0 \unlhd H$, $K_+ \unlhd H_+ \unlhd H$. Assume $H_+$ stabilizes ${\tilde{\theta}}_0$ and $\theta_+$, and assume $[K_0,H_+] \leq {\operatorname{Ker}}\zeta_+$. Let $\psi_+ \in {\operatorname{Irr}}(H_+\mid\theta_+)$ and define $\psi:={\tilde{\theta}}_0\psi_+$ by $\psi(k_0h_+)={\tilde{\theta}}_0(k_0)\psi_+(h_+)$ for any $k_0 \in K_0$ and $h_+ \in H_+$. Then $\psi$ is an irreducible character of $H$.
By the assumption, we have ${\operatorname{Irr}}(Z\mid\psi_+) = {\{\zeta_+\}}$. Let $\rho_0: K_0 \to {\operatorname{GL}}(V_0)$, $\rho_+: H_+ \to {\operatorname{GL}}(V_+)$ be representations affording ${\tilde{\theta}}_0, \psi_+$ respectively. Define $\rho: H \to {\operatorname{GL}}(V_0 \otimes V_+), k_0h_+ \mapsto \rho_0(k_0)\otimes\rho_+(h_+)$. We first check that this map is well defined. Assume $k_0h_+=k'_0h'_+$, then $k_0^{-1}k'_0=h_+{h'_+}^{-1} \in Z$ and $$\begin{aligned}
\rho(k'_0h'_+) &= \rho_0(k'_0) \otimes \rho_+(h'_+) = \rho_0 (k_0k_0^{-1}k'_0) \otimes \rho_+(h'_+) \\
&= \rho_0(k_0)\zeta_+(k_0^{-1}k'_0) \otimes \rho_+(h'_+) = \rho_0(k_0) \otimes \zeta_+(h_+{h'_+}^{-1})\rho_+(h'_+) \\
&= \rho_0(k_0) \otimes \rho_+(h_+{h'_+}^{-1}h'_+) = \rho_0(k_0) \otimes \rho_+(h_+) = \rho(k_0h_+).\end{aligned}$$ To show that $\rho$ is multiplicative, note that $k_0h_+k'_0h'_+=k_0k'_0[{k'_0}^{-1},h_+]h_+h'_+$, so $$\begin{aligned}
\rho(k_0h_+k'_0h'_+) &= \rho_0(k_0k'_0[{k'_0}^{-1},h_+]) \otimes \rho_+(h_+h'_+) \\
&= \rho_0(k_0k'_0)\zeta_+([{k'_0}^{-1},h_+]) \otimes \rho_+(h_+h'_+) \\
&= \rho_0(k_0)\rho_0(k'_0) \otimes \rho_+(h_+)\rho_+(h'_+) \\
&= \big(\rho_0(k_0) \otimes \rho_+(h_+)\big) \big(\rho_0(k'_0) \otimes \rho_+(h'_+)\big) \\
&= \rho(k_0h_+)\rho(k'_0h'_+).\end{aligned}$$ Thus $\rho$ is a representation of $H$. To see that $\rho$ is irreducible, it suffices to note that $\rho$ can be lifted to $K_0 \times H_+$ via the surjective homomorphism $K_0 \times H_+ \to H, (k_0,h_+) \mapsto k_0h_+$.
Set ${\mathscr{C}}_{\Gamma,\delta} = \cup_{i} {\operatorname{dz}}(N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})/R_{\Gamma,\delta,i}\mid\theta_{\Gamma,\delta,i})$ and assume ${\mathscr{C}}_{\Gamma,\delta}={\{\psi_{\Gamma,\delta,i,j}\}}$ with $\psi_{\Gamma,\delta,i,j}$ a character of $N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})$. Note that ${\mathscr{C}}_{\Gamma,\delta}$ here is different from but in bijection with that in [@Li19 §6.B]. Here, we define ${\mathscr{C}}_{\Gamma,\delta}$ as above so that we can use Lemma \[lem:cons-char\] to construct ${\operatorname{dz}}({\tilde{N}}({\tilde{\theta}})/{\tilde{R}}\mid {\tilde{\theta}})$ from ${\operatorname{dz}}(N_+(\theta_+)/R_+ \mid \theta_+)$.
Recall that $N:=N_G(R)$ can be decomposed as $N = N_0 \times N_+$. From Equation (\[equ:theta\_+R\_+\]), we have $N_+(\theta_+) = \prod_{\Gamma,\delta,i} N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})\wr{\mathfrak{S}}(t_{\Gamma,\delta,i})$. Any $\psi_+ \in {\operatorname{dz}}(N_+(\theta_+)/R_+ \mid \theta_+)$ can be decomposed as $\psi_+ = \prod_{\Gamma,\delta,i} \psi_{\Gamma,\delta,i}$, where $\psi_{\Gamma,\delta,i}$ is a character of $N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})\wr{\mathfrak{S}}(t_{\Gamma,\delta,i})$. Then by Clifford theory, $\psi_{\Gamma,\delta,i}$ is of the form $$\label{equ:psi}
{\operatorname{Ind}}_{N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})\wr\prod_j{\mathfrak{S}}(t_{\Gamma,\delta,i,j})}^{N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})\wr{\mathfrak{S}}(t_{\Gamma,\delta,i})}
\overline{\prod_j\psi_{\Gamma,\delta,i,j}^{t_{\Gamma,\delta,i,j}}} \cdot \prod_j\phi_{\kappa_{\Gamma,\delta,i,j}},
\addtocounter{thm}{1}\tag{\thethm}$$ where $t_{\Gamma,\delta,i}=\sum_jt_{\Gamma,\delta,i,j}$, $\overline{\prod_j\psi_{\Gamma,\delta,i,j}^{t_{\Gamma,\delta,i,j}}}$ is the canonical extension of $\prod_j\psi_{\Gamma,\delta,i,j}^{t_{\Gamma,\delta,i,j}}\in{\operatorname{Irr}}(N_{\Gamma,\delta,i}^{t_{\Gamma,\delta,i}})$ to $N_{\Gamma,\delta,i}\wr\prod_j{\mathfrak{S}}(t_{\Gamma,\delta,i,j})$ as in the proof of [@Bon99b Proposition 2.3.1], $\kappa_{\Gamma,\delta,i,j}\vdash t_{\Gamma,\delta,i,j}$ without $\ell$-hook and $\phi_{\kappa_{\Gamma,\delta,i,j}}$ is the character of ${\mathfrak{S}}(t_{\Gamma,\delta,i,j})$ corresponding to $\kappa_{\Gamma,\delta,i,j}$. Now, define $K_\Gamma:\cup_\delta{\mathscr{C}}_{\Gamma,\delta}\to\{\ell\textrm{-cores}\}$, $\psi_{\Gamma,\delta,i,j}\mapsto\kappa_{\Gamma,\delta,i,j}$ and $K= \prod_\Gamma K_\Gamma$. We call $K$ the label of $\psi_+$, which will be denoted as $\psi_{+,K}$ from now on.
As before, the block ${\tilde{B}}$ of ${\tilde{G}}$ with label $({\tilde{s}},{\mathcal{K}})$ is denoted by ${\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}$. Denote by $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$ the set of $K$’s satisfying $\sum_{\delta,i,j}\ell^\delta |K_\Gamma(\psi_{\Gamma,\delta,i,j})|=w_\Gamma$ and $m_\Gamma(s)=m_\Gamma(s_0)+\beta_\Gamma e_\Gamma w_\Gamma$, where $\beta_\Gamma=1$ or $2$ according to $\Gamma\notin{\mathcal{F}}_0$ or $\Gamma\in{\mathcal{F}}_0$. Then by the above constructions, $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$ is in bijection with the set $\cup_{{\{({\tilde{R}},{\tilde{\theta}})\in{\tilde{B}}\}}/\sim{\tilde{N}}}{\operatorname{dz}}(N_+(\theta_+)/R_+ \mid \theta_+)$, where $({\tilde{R}},{\tilde{\theta}})$ runs over an ${\tilde{N}}$-transversal of Brauer pairs $({\tilde{R}},{\tilde{\theta}})$ belonging to the block ${\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}$. Denote this bijection as $K \mapsto \psi_{+,K}$. Recall that for any $K \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$, $K'$ is defined as in the paragraph before [@Li19 Proposition 6.25].
\[lem:alpha-(IV)\] Assume we are in case (IV) or (VI) in Table \[tab:block\] and $\alpha\in{\operatorname{Irr}}\left(N_+(\theta_+)/N_+({\tilde{\theta}})\right)$ is the unique non-trivial character, then $\alpha\cdot\psi_{+,K}=\psi_{+,K'}$.
Recall that $N_+(\theta_+) = \prod_{\Gamma,\delta,i} N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})\wr{\mathfrak{S}}(t_{\Gamma,\delta,i})$. By [@FS89 (7E)] and Lemma \[lem:N(ttheta)\], $N_{\Gamma,\delta,i}(\theta_{\Gamma,\delta,i})\wr{\mathfrak{S}}(t_{\Gamma,\delta,i}) \leq N_+({\tilde{\theta}})$ for $\Gamma \neq X+1$. So there is no loss to assume that $N_+(\theta_+) = \prod_{\delta,i} N_{X+1,\delta,i}\wr{\mathfrak{S}}(t_{\delta,i})$; note that $N_{X+1,\delta,i}(\theta_{X+1,\delta,i}) = N_{X+1,\delta,i}$. Since $N_{X+1,\delta,i}\wr{\mathfrak{S}}(t_{\delta,i}) \nleq N_+({\tilde{\theta}})$ for each component, we may assume there is only one component in the radical subgroup $R_+ = R_{X+1,\delta,i}^t$. Thus $N_+(\theta_+) = N_{X+1,\delta,i}\wr{\mathfrak{S}}(t)$ and $$\psi_+ = {\operatorname{Ind}}_{N_{X+1,\delta,i}\wr\prod_j{\mathfrak{S}}(t_j)}^{N_{X+1,\delta,i}\wr{\mathfrak{S}}(t)}
\overline{\prod_j\psi_{X+1,\delta,i,j}^{t_j}} \cdot \prod_j\phi_{\kappa_{X+1,\delta,i,j}}.$$ So $$\alpha\cdot\psi_+ = {\operatorname{Ind}}_{N_{X+1,\delta,i}\wr\prod_j{\mathfrak{S}}(t_j)}^{N_{X+1,\delta,i}\wr{\mathfrak{S}}(t)}
\overline{\prod_j({\operatorname{Res}}^{N_+(\theta_+)}_{N_{X+1,\delta,i}}\alpha\cdot\psi_{X+1,\delta,i,j})^{t_j}} \cdot \prod_j\phi_{\kappa_{X+1,\delta,i,j}}.$$ By [@FS89 (7E)] again, ${\operatorname{Res}}^{N_+(\theta_+)}_{N_{X+1,\delta,i}}\alpha \neq 1$ and $${\operatorname{Res}}^{N_+(\theta_+)}_{N_{X+1,\delta,i}}\alpha\cdot\psi_{X+1,\delta,i,j} = \psi_{X+1,\delta,i,j'},$$ where $\psi_{X+1,\delta,i,j'}$ is as in [@Li19 Lemma 6.24]. Thus $\alpha\cdot\psi_{+,K}=\psi_{+,K'}$ by the definition of $K'$.
Now, set $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}=i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$ for cases (I)$\sim$(III) and (V), and set $$i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}} = {\left\{ {\{K,K'\}} \mid K\neq K' \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0 \right\}} \cup {\left\{ (K,i) \mid K=K' \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0, i\in {\mathbb{Z}}/2{\mathbb{Z}}\right\}}$$ for cases (IV) and (VI).
\[prop:weights\] Assume ${\tilde{B}}$ is a block of ${\tilde{G}}$ with label $({\tilde{s}},{\mathcal{K}})$, then ${\operatorname{Alp}}({\tilde{B}})$ is in bijection with $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}$.
Recall that $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$ is in bijection with $\cup_{{\{({\tilde{R}},{\tilde{\theta}})\in{\tilde{B}}\}}/\sim{\tilde{N}}}{\operatorname{dz}}(N_+(\theta_+)/R_+ \mid \theta_+)$.
\(1) Assume we are in one of the cases (I)$\sim$(III). Then by Proposition \[prop:N(theta+):N(ttheta)\] and its proof, $N_+(\theta_+) = N_+({\tilde{\theta}}_0) \cap N_+(\theta_+) = N_+({\tilde{\theta}})$ and $\omega_g=1$ for any $g \in N_+(\theta_+)$. By the definition of $\omega_g$ ([@FS89 (7.5)]), $[{\tilde{C}}_0,N_+(\theta_+)] \leq {\operatorname{Ker}}\zeta_+$, where $\zeta_+$ is the linear character of $Z(C_+)$ induced by ${\tilde{\theta}}_0$ and $\theta_+$. Then by applying Lemma \[lem:cons-char\] to ${\tilde{C}}={\tilde{C}}_0C_+$ and ${\tilde{N}}({\tilde{\theta}}) = {\tilde{C}}_0N_+(\theta_+)$, $K \mapsto \psi_K := {\tilde{\theta}}\psi_{+,K}$ gives a bijection from $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$ to $\cup_{{\{({\tilde{R}},{\tilde{\theta}})\in{\tilde{B}}\}}/\sim{\tilde{N}}}{\operatorname{dz}}({\tilde{N}}({\tilde{\theta}})/{\tilde{R}}\mid {\tilde{\theta}})$. By Clifford theory, the latter set is in bijection with the set of ${\tilde{B}}$-weights via $\psi_K \mapsto {\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}} \psi_K$ and thus the assertion holds in this case.
\(2) Assume we are in case (V). Then by Proposition \[prop:N(theta+):N(ttheta)\] and its proof, $N_+(\theta_+) = N_+({\tilde{\theta}}_0) \cap N_+(\theta_+) = N_+({\tilde{\theta}})$ but $\omega_g\neq1$ for some $g \in N_+(\theta_+)$. We can not apply Lemma \[lem:cons-char\] directly. Let ${\hat{C}}_0 = {\langle\tau^2,C_0Z(C_+)\rangle}$. As in the proof of Proposition \[prop:N(theta+):N(ttheta)\], ${\tilde{\theta}}_0 = {\tilde{\chi}}_0 \zeta$ with ${\tilde{\chi}}_0 = \chi_{{\tilde{s}}_0,\kappa}^{{\operatorname{J}}_0(V_0)}$, where ${\mathcal{K}}={\{\kappa\}}$. Then ${\operatorname{Res}}^{{\tilde{C}}_0}_{{\hat{C}}_0} {\tilde{\theta}}_0$ has two irreducible constituents since ${\operatorname{Res}}^{{\tilde{C}}_0}_{C_0} {\tilde{\theta}}_0 = {\operatorname{Res}}^{{\operatorname{J}}_0(V_0)}_{{\operatorname{I}}_0(V_0)} {\tilde{\chi}}_0$ has two irreducible constituents by [@Li19 Theorem 3.2]. Let ${\hat{\theta}}_0^{(0)}$ and ${\hat{\theta}}_0^{(1)}$ be the two irreducible constituents of ${\operatorname{Res}}^{{\tilde{C}}_0}_{{\hat{C}}_0} {\tilde{\theta}}_0$. By [@FS89 (7E)], ${\operatorname{Res}}^{{\tilde{C}}_0}_{{\hat{C}}_0}\omega_g=1$, which means $[{\hat{C}}_0,N_+(\theta_+)] \leq {\operatorname{Ker}}\zeta_+$ by definition of $\omega_g$. Then by applying Lemma \[lem:cons-char\] to ${\hat{C}}:={\hat{C}}_0C_+$ and ${\hat{N}}({\tilde{\theta}}) := {\hat{C}}_0N_+(\theta_+)$, there is a character ${\hat{\psi}}_K := {\hat{\theta}}_0^{(0)}\psi_{+,K}\in {\operatorname{dz}}({\hat{N}}({\tilde{\theta}})/{\tilde{R}})$. Then $\psi_K = {\operatorname{Ind}}_{{\hat{N}}({\tilde{\theta}})}^{{\tilde{N}}({\tilde{\theta}})} {\hat{\psi}}_K \in {\operatorname{dz}}({\tilde{N}}({\tilde{\theta}})/{\tilde{R}}\mid {\tilde{\theta}})$. If we start with ${\hat{\theta}}_0^{(1)}$, we get the same $\psi_K$. This gives a bijection from $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$ to $\cup_{{\{({\tilde{R}},{\tilde{\theta}})\in{\tilde{B}}\}}/\sim{\tilde{N}}}{\operatorname{dz}}({\tilde{N}}({\tilde{\theta}})/{\tilde{R}}\mid {\tilde{\theta}})$, and proves the assertion in this case by Clifford theory again.
\(3) Assume we are in case (IV) or (VI). Then $|N_+(\theta_+):N_+({\tilde{\theta}})|=2$ and $\omega_g=1$ for any $g \in N_+({\tilde{\theta}})$, thus $[{\tilde{C}}_0,N_+({\tilde{\theta}})] \leq {\operatorname{Ker}}\zeta_+$. This case is divided into two sub-cases.
(3.1) Assume $K \neq K'$. By Lemma \[lem:alpha-(IV)\] and Clifford theory, $\psi_{+,K}^0:={\operatorname{Res}}^{N_+(\theta_+)}_{N_+({\tilde{\theta}})} \psi_{+,K}$ is irreducible. Applying Lemma \[lem:cons-char\] to ${\tilde{C}}={\tilde{C}}_0C_+$, ${\tilde{N}}({\tilde{\theta}})={\tilde{C}}_0N_+({\tilde{\theta}})$ and $\psi_{+,K}^0$, there is an irreducible character $\psi_K^0 := {\tilde{\theta}}_0\psi_{+,K}^0$ of ${\tilde{N}}({\tilde{\theta}})$. Then ${\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}}\psi_K^0$ gives a weight associated to ${\tilde{R}}$. If we start with $\psi_{+,K'}$, we will get the same weight, since ${\operatorname{Res}}^{N_+(\theta_+)}_{N_+({\tilde{\theta}})} \psi_{+,K} = {\operatorname{Res}}^{N_+(\theta_+)}_{N_+({\tilde{\theta}})} \psi_{+,K'}$. Give this weight the label ${\{K,K'\}}$.
(3.2) Assume $K=K'$. By Lemma \[lem:alpha-(IV)\] and Clifford theory, ${\operatorname{Res}}^{N_+(\theta_+)}_{N_+({\tilde{\theta}})} \psi_{+,K}$ has two irreducible constituents, denoted as $\psi_{+,K}^{(i)}$, $i\in{\mathbb{Z}}/2{\mathbb{Z}}$. Applying Lemma \[lem:cons-char\] to ${\tilde{C}}={\tilde{C}}_0C_+$, ${\tilde{N}}({\tilde{\theta}})={\tilde{C}}_0N_+({\tilde{\theta}})$ and $\psi_{+,K}^{(i)}$, there is an irreducible character $\psi_K^{(i)} := {\tilde{\theta}}_0\psi_{+,K}^{(i)}$ of ${\tilde{N}}({\tilde{\theta}})$. Then ${\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}}\psi_K^{(i)}$ gives a weight associated to ${\tilde{R}}$. Give this weight the label $(K,i)$.
Combining (3.1) and (3.2), we have a bijection from $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}$ to ${\operatorname{Alp}}({\tilde{B}})$, which proves the assertion in this case.
Denote the conjugacy class of weights corresponding by the above Proposition to $K \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}$ for cases (I)$\sim$(III) and (V) by $w_K$ or $w_{{\tilde{s}},{\mathcal{K}},K}$; denote the conjugacy class of weights corresponding to ${\{K,K'\}} \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}$ for cases (IV) and (VI) by $w_{{\{K,K'\}}}$ or $w_{{\tilde{s}},{\mathcal{K}},{\{K,K'\}}}$ and the conjugacy class of weights corresponding to $(K,i) \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}$ for cases (IV) and (VI) by $w_{(K,i)}$ or $w_{{\tilde{s}},{\mathcal{K}},(K,i)}$.
Recall that $|{\mathscr{C}}_{\Gamma,\delta}| = \beta_\Gamma e_\Gamma\ell^\delta$ by [@An94 (4A)]. By [@AF90 (1A)], there is a bijection between $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^0$ and the following set $$i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^{q,0} = {\left\{ Q = \prod_\Gamma Q_\Gamma ~\middle|~
\begin{array}{c} Q_\Gamma=(Q_\Gamma^{(1)},\cdots,Q_\Gamma^{(\beta_\Gamma e_\Gamma)}),\\
Q_\Gamma\textrm{'s are partitions, } \sum_{i=1}^{\beta_\Gamma e_\Gamma} |Q_\Gamma^{(i)}| = w_\Gamma. \end{array}\right\}}$$ For $Q_{X+1} = (\lambda_1,\dots,\lambda_e,\mu_1,\dots,\mu_e)$, define $(Q_{X+1})' = (\mu_1,\dots,\mu_e,\lambda_1,\dots,\lambda_e)$, and define $Q'$ by $Q'_\Gamma = Q_\Gamma$ for $\Gamma \neq X+1$ and $Q'_{X+1} = (Q_{X+1})'$. So if $K$ corresponds to $Q$, $K'$ corresponds to $Q'$. Similarly as before, set $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^q=i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^{q,0}$ for cases (I)$\sim$(III) and (V), and set $$i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^q = {\left\{ {\{Q,Q'\}} \mid Q\neq Q' \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^{q,0} \right\}} \cup {\left\{ (Q,i) \mid Q=Q' \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^{q,0}, i\in {\mathbb{Z}}/2{\mathbb{Z}}\right\}}$$ for cases (IV) and (VI). Then $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^q$ is in bijection with $i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}$ and thus with ${\mathcal{W}}({\tilde{B}}_{{\tilde{s}},{\mathcal{K}}})$. Denote the conjugacy class of weights corresponding to $Q \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^q$ for cases (I)$\sim$(III) and (V) by $w_Q$ or $w_{{\tilde{s}},{\mathcal{K}},Q}$; denote the conjugacy class of weights corresponding to ${\{Q,Q'\}} \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^q$ for cases (IV) and (VI) by $w_{{\{Q,Q'\}}}$ or $w_{{\tilde{s}},{\mathcal{K}},{\{Q,Q'\}}}$ and the conjugacy class of weights corresponding to $(Q,i) \in i{\mathcal{W}}_{{\tilde{s}},{\mathcal{K}}}^q$ for cases (IV) and (VI) by $w_{(Q,i)}$ or $w_{{\tilde{s}},{\mathcal{K}},(Q,i)}$.
#### {#section-4}
Now, Theorem \[mainthm-1\] follows from the following explicit bijection.
\[thm:bijection\] Let ${\tilde{B}}={\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}$ be a block of ${\tilde{G}}$. Keep the notation above and that of Lemma \[lem:IBr\].
(1) Assume we are in one of the cases (I)$\sim$(III) and (V) with ${\mathcal{K}}={\{\kappa\}}$. Then there is a bijection $${\tilde{\Omega}}_{{\tilde{B}}}: \quad {\operatorname{IBr}}({\tilde{B}}_{{\tilde{s}},{\{\kappa\}}}) \longleftrightarrow {\operatorname{Alp}}({\tilde{B}}_{{\tilde{s}},{\{\kappa\}}})$$ satisfying that if ${\tilde{\Omega}}_{{\tilde{B}}}(\phi_{{\tilde{s}},\lambda}) = w_{{\tilde{s}},{\mathcal{K}},Q}$, then $\lambda_\Gamma=\kappa_\Gamma*Q_\Gamma$. Note that in cases (II) or (III), then $\lambda_{X+1}=\kappa_{X+1}$.
(2) Assume we are in case (IV) with ${\mathcal{K}}={\{\kappa\}}$ and $\kappa_{X+1}=\O$ or in case (VI) with ${\mathcal{K}}={\{\kappa,\kappa'\}}$ and $\kappa_{X+1}\neq\O$ degenerate. Then there is a bijection $${\tilde{\Omega}}_{{\tilde{B}}}: \quad {\operatorname{IBr}}({\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}) \longleftrightarrow {\operatorname{Alp}}({\tilde{B}}_{{\tilde{s}},{\mathcal{K}}})$$ satisfying that if ${\tilde{\Omega}}_{{\tilde{B}}}(\phi_{{\tilde{s}},\lambda})=w_{{\tilde{s}},{\mathcal{K}},{\{Q,Q'\}}}$ or $w_{{\tilde{s}},{\mathcal{K}},(Q,i)}$, then
(i) $\lambda_\Gamma=\kappa_\Gamma*Q_\Gamma$ for $\Gamma \neq X+1$;
(ii) $\lambda_{X+1}=\kappa_{X+1}*Q_\Gamma$ if $Q \neq Q'$;
(iii) $\lambda_{X+1}=\kappa_{X+1}*(Q,i)$ if $Q=Q'$.
The theorem follows obviously from the §\[subsec:partition-symbol\].
The inductive condition {#sec:inductive}
=======================
In this section, we verify the inductive condition for the cases in Theorem \[mainthm-2\]. Let $E$ be the set of field automorphisms on ${\tilde{G}}$ and ${\mathcal{B}}$ be as in Table \[tab:block\]. Then $({\tilde{G}}E)_B\leq({\tilde{G}}E)_{\mathcal{B}}$ holds obviously for any $B \in {\tilde{B}}$. We first remark that the part (i) of Theorem \[thm:criterion\] holds. The first and the second requirements are obvious. The rest follows from the fact that ${\tilde{G}}/G$ is cyclic.
{#subsec:pro-bij}
In this subsection, we prove part (ii) of Theorem \[thm:criterion\]. Set ${\tilde{\Omega}}^{{\tilde{G}}} = \cup_{{\tilde{B}}\in{\operatorname{Bl}}({\tilde{G}})} {\tilde{\Omega}}_{{\tilde{B}}}$, where ${\tilde{\Omega}}_{{\tilde{B}}}$ is the bijection in Theorem \[thm:bijection\]
\[lem:equi\] Assume the decomposition matrix with respect to ${\mathcal{E}}({\tilde{G}},\ell')$ is unitriangular. Then the bijection ${\tilde{\Omega}}^{{\tilde{G}}}$ is ${\operatorname{Aut}}(G)$-equivariant.
It suffices to consider a field automorphism $\sigma\in E$.
Let $\chi_{{\tilde{s}},\lambda} \in {\operatorname{Irr}}({\tilde{G}})$, then by [@CS13 Theorem 3.1], $\chi_{{\tilde{s}},\lambda}^\sigma = {\tilde{\chi}}_{\sigma^*({\tilde{s}}),\sigma^*(\lambda)}$, where $\sigma^*$ is the corresponding field automorphism on ${\tilde{G}}^*$ via duality as in [@CS13 Definition 2.1] and $\sigma^*(\lambda)_{\sigma^*(\Gamma)} = \lambda_\Gamma$; see also [@Li19 Theorem 5.2] and the proof of [@Li19 Proposition 5.6]. By the assumption on the decomposition matrices and [@CS13 Lemma 7.5], the labels of irreducible Brauer characters in Lemma \[lem:IBr\] can be chosen to satisfy that $\phi_{{\tilde{s}},\lambda}^\sigma = \phi_{\sigma^*({\tilde{s}}),\sigma^*(\lambda)}$. Let ${\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}$ be a block of ${\tilde{G}}$, then ${\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}^\sigma = {\tilde{B}}_{\sigma^*({\tilde{s}}),\sigma^*({\mathcal{K}})}$, where $\sigma^*({\mathcal{K}})$ is similarly defined.
Let $({\tilde{R}},{\tilde{\varphi}})$ be a ${\tilde{B}}_{{\tilde{s}},{\mathcal{K}}}$-weight; we use the notation in the proof of Proposition \[prop:weights\]. Assume $({\tilde{R}},{\tilde{\varphi}})$ has label $({\tilde{s}},{\mathcal{K}},K)$ or $({\tilde{s}},{\mathcal{K}},{\{K,K'\}})$. Then $({\tilde{R}},\varphi)^\sigma$ is a ${\tilde{B}}_{\sigma^*({\tilde{s}}),\sigma^*({\mathcal{K}})}$-weight. Assume ${\tilde{\varphi}}$ comes from a Brauer pair $({\tilde{R}},{\tilde{\theta}})$ and is constructed from $\psi_{+,K}\in{\operatorname{dz}}(N_+(\theta_+)/R_+ \mid \theta_+)$ as in Proposition \[prop:weights\]. By [@Li19 Proposition 6.20], $\psi_{+,K}^\sigma = \psi_{+,\sigma^*(K)}$ up to $N_+$-conjugate, where $\sigma^*(K)$ is defined similarly as before. Thus $({\tilde{R}},\varphi)^\sigma$ has label $(\sigma^*({\tilde{s}}),\sigma^*({\mathcal{K}}),\sigma^*(K))$ or $(\sigma^*({\tilde{s}}),\sigma^*({\mathcal{K}}),{\{\sigma^*(K),\sigma^*(K)'\}})$. Assume $({\tilde{R}},{\tilde{\varphi}})$ has label $({\tilde{s}},{\mathcal{K}},(K,i))$ with $K=K'$. We claim that $(\psi_{+,K}^{(i)})^\sigma = \psi_{+,\sigma^*(K)}^{(i)}$ up to $N_+$-conjugate, whose proof is left to Lemma \[lem:equi-claim\]. Thus $({\tilde{R}},\varphi)^\sigma$ has label $(\sigma^*({\tilde{s}}),\sigma^*({\mathcal{K}}),(\sigma^*(K),i))$.
Now, it is easy to see that the bijection in Theorem \[thm:bijection\] is equivariant.
\[lem:equi-claim\] With the notation in the proof of the above lemma, $(\psi_{+,K}^{(i)})^\sigma = \psi_{+,\sigma^*(K)}^{(i)}$ up to $N_+$-conjugate.
To avoid constant use of “up to conjugation”, we transfer to twisted groups. As in the proof of Lemma \[lem:alpha-(IV)\], we may assume $R_+^{tw} = (R_{X+1,\gamma,{\mathbf{c}}}^{tw})^t$. Thus $N_+^{tw}(\theta_+^{tw}) = N_{X+1,\gamma,{\mathbf{c}}}^{tw}\wr{\mathfrak{S}}(t)$ and $$\psi_{+,K}^{tw} = {\operatorname{Ind}}_{N_{X+1,\gamma,{\mathbf{c}}}^{tw}\wr\prod_j{\mathfrak{S}}(t_j)}^{N_{X+1,\gamma,{\mathbf{c}}}^{tw}\wr{\mathfrak{S}}(t)}
\overline{\prod_j(\psi_{X+1,j}^{tw})^{t_j}} \cdot \prod_j\phi_{\kappa_{X+1,j}}.$$ Then $K$ is defined as $\psi_{X+1,j} \mapsto \kappa_{X+1,j}$. Since $N_{X+1,\gamma,{\mathbf{c}}}^{tw}/R_{X+1,\gamma,{\mathbf{c}}}^{tw} \cong N_{X+1,\gamma}^{tw}/R_{X+1,\gamma}^{tw} \times N_{{\mathfrak{S}}(\ell^{|{\mathbf{c}}|})}(A_{{\mathbf{c}}})/A_{{\mathbf{c}}}$, we may assume ${\mathbf{c}}={\mathbf{0}}$. By [@Li19 Lemma 6.15], we may assume $\gamma=0$.
Note that $|{\operatorname{dz}}(N_{X+1}/R_{X+1}\mid\theta_{X+1})| = 2e$ and by [@Li19 Lemma 6.23, Lemma 6.24] the characters in ${\operatorname{dz}}(N_{X+1}/R_{X+1}\mid\theta_{X+1})$ can be labelled as $\psi_{X+1,1}, \psi_{X+1,1'}, \cdots, \psi_{X+1,e}, \psi_{X+1,e'}$. Since $K=K'$, $\kappa_{X+1,j}=\kappa_{X+1,j'}$ by the definition of $K'$; in particular, $t_j=t_{j'}$. Thus $$\psi_{+,K}^{tw} = {\operatorname{Ind}}_{N_{X+1}^{tw}\wr\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}^{N_{X+1}^{tw}\wr{\mathfrak{S}}(t)}
\overline{\prod_{j=1}^e(\psi_{X+1,j}^{tw})^{t_j}(\psi_{X+1,j'}^{tw})^{t_j}} \cdot \prod_j\phi_{\kappa_{X+1,j}}\phi_{\kappa_{X+1,j}}.$$ Let $T,V$ be as in the proof of [@Li19 Lemma 6.23], thus $N_{X+1}^{tw}=TV$. Set $\bar{N}_{X+1}^{tw} = N_{X+1}^{tw}/T$ and use the bar convention. Set $B_0 = {\left\{ (n_1,\cdots,n_t) \mid n_i \in N_{X+1}^{tw}, \bar{n}_1 \cdots \bar{n}_t \in \bar{V}^2 \right\}}$. By Lemma \[lem:N(ttheta)\] and [@FS89 (7E)], $N_+({\tilde{\theta}}) = B_0 \rtimes {\mathfrak{S}}(t)$. To prove this lemma, it is equivalent to prove a similar assertion for the following character $${\operatorname{Ind}}_{N_{X+1}^{tw}\wr\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}^{N_{X+1}^{tw}\wr{\mathfrak{S}}(t)}
\overline{\prod_{j=1}^e(\psi_{X+1,j}^{tw})^{t_j}(\psi_{X+1,j'}^{tw})^{t_j}},$$ which, for convenience, is still denoted by $\psi_{+,K}$. Then $${\operatorname{Res}}^{N_+^{tw}(\theta_+)}_{N_+^{tw}({\tilde{\theta}})} \psi_{+,K}^{tw} = {\operatorname{Ind}}_{B_0\rtimes\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}^{B_0\rtimes{\mathfrak{S}}(t)}
{\operatorname{Res}}^{N_{X+1}^{tw}\wr\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}_{B_0\rtimes\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}
\overline{\prod_{j=1}^e(\psi_{X+1,j}^{tw})^{t_j}(\psi_{X+1,j'}^{tw})^{t_j}}.$$ Set $N_{+,0}^{tw}({\tilde{\theta}}) = \left(B_0\rtimes\prod_{j=1}^e{\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j)\right)\rtimes{\mathfrak{S}}(2)$, where the action of ${\mathfrak{S}}(2)$ transposes each pair $(\psi_{X+1,j}^{tw},\psi_{X+1,j'}^{tw})$. Note that $\psi_{X+1,j'}^{tw}=\alpha\psi_{X+1,j}^{tw}$ by the proof of [@Li19 Lemma 6.23], where $\alpha$ is the unique character of $\bar{V}$ of order $2$. Then by the definition of $B_0$, $N_{+,0}^{tw}({\tilde{\theta}})$ fixes $${\operatorname{Res}}^{N_{X+1}^{tw}\wr\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}_{B_0\rtimes\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}
\overline{\prod_{j=1}^e(\psi_{X+1,j}^{tw})^{t_j}(\psi_{X+1,j'}^{tw})^{t_j}}.$$ Thus this character has two extensions to $N_{+,0}^{tw}({\tilde{\theta}})$, denoted $\psi_{+,K,0}^{(0)}$ and $\psi_{+,K,0}^{(1)}$. So ${\operatorname{Ind}}_{N_{+,0}^{tw}({\tilde{\theta}})}^{N_+^{tw}({\tilde{\theta}})}\psi_{+,K,0}^{(i)}$ are the two irreducible constituents of ${\operatorname{Res}}^{N_+^{tw}(\theta_+)}_{N_+^{tw}({\tilde{\theta}})} \psi_{+,K}^{tw}$; we may set $\psi_{+,K}^{(i),tw} = {\operatorname{Ind}}_{N_{+,0}^{tw}({\tilde{\theta}})}^{N_+^{tw}({\tilde{\theta}})}\psi_{+,K,0}^{(i)}$, where $\psi_{+,K}^{(i),tw}$ is the twisted version of $\psi_{+,K}^{(i)}$.
By [@Li19 Lemma 6.17], $\sigma$ fixes $${\operatorname{Res}}^{N_{X+1}^{tw}\wr\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}_{B_0\rtimes\prod_{j=1}^e({\mathfrak{S}}(t_j)\times{\mathfrak{S}}(t_j))}
\overline{\prod_{j=1}^e(\psi_{X+1,j}^{tw})^{t_j}(\psi_{X+1,j'}^{tw})^{t_j}}.$$ Since the above character is a linear character, $\sigma$ fixes both the extensions $\psi_{+,K,0}^{(i)}$, and thus fixes $\psi_{+,K}^{(i),tw}$, which means that $(\psi_{+,K}^{(i)})^\sigma = \psi_{+,\sigma^*(K)}^{(i)}$ up to $N_+$-conjugacy.
Recall that ${\tilde{\mathcal{B}}}= {\operatorname{Bl}}({\tilde{G}}\mid{\mathcal{B}})$. Set ${\operatorname{IBr}}({\tilde{\mathcal{B}}}) = \cup_{{\tilde{B}}\in{\tilde{\mathcal{B}}}} {\operatorname{IBr}}({\tilde{B}})$ and ${\operatorname{Alp}}({\tilde{\mathcal{B}}}) = \cup_{{\tilde{B}}\in{\tilde{\mathcal{B}}}} {\operatorname{Alp}}({\tilde{B}})$. Denote by ${\tilde{\Omega}}_{{\tilde{\mathcal{B}}}}$ the restriction of ${\tilde{\Omega}}^{{\tilde{G}}}$ to ${\operatorname{IBr}}({\tilde{\mathcal{B}}})$. As a corollary of Lemma \[lem:equi\], ${\tilde{\Omega}}_{{\tilde{\mathcal{B}}}}$ is $E_{{\mathcal{B}}}$-equivariant. Since the bijection ${\tilde{\Omega}}^{{\tilde{G}}}$ comes from the blockwise bijections ${\tilde{\Omega}}_{{\tilde{B}}}$, ${\tilde{\Omega}}_{{\tilde{\mathcal{B}}}}({\operatorname{IBr}}({\tilde{B}})) = {\operatorname{Alp}}({\tilde{B}})$ for every ${\tilde{B}}\in {\tilde{\mathcal{B}}}$.
\[lem:(3)of(iii)\] Assume the decomposition matrix with respect to ${\mathcal{E}}({\tilde{G}},\ell')$ is unitriangular. Then ${\tilde{\Omega}}_{{\tilde{\mathcal{B}}}}$ is ${\operatorname{IBr}}({\tilde{G}}/G)$-equivariant.
First, by [@De17 Lemma 2.3] and the assumption on the decomposition matrix, it suffices to prove the same statement for the irreducible ordinary characters corresponding to irreducible Brauer characters in ${\operatorname{IBr}}({\tilde{B}}\mid{\tilde{R}})$. Recall that the action of ${\operatorname{Irr}}({\tilde{G}}/G)$ on ${\operatorname{Irr}}({\tilde{G}})$ is described in Remark \[rem:action-tz\].
We use the construction of weights in Proposition \[prop:weights\] to consider the action of ${\operatorname{IBr}}({\tilde{G}}/G) = {\operatorname{Irr}}({\tilde{G}}/G)_{\ell'}$ on weights. First note that ${\tilde{C}}={\langle\tau,C\rangle}$ and ${\tilde{N}}={\langle\tau,N\rangle}$, thus ${\tilde{C}}/C \cong {\tilde{N}}/N \cong {\tilde{G}}/G$, so we can identify ${\operatorname{Irr}}({\tilde{G}}/G)$ with ${\operatorname{Irr}}({\tilde{C}}/C)$ and ${\operatorname{Irr}}({\tilde{N}}/N)$ by restriction. To simplify the notation, we abbreviate ${\operatorname{Res}}^{{\tilde{G}}}_{{\tilde{N}}}$ and ${\operatorname{Res}}^{{\tilde{G}}}_{{\tilde{C}}}$.
Let ${\tilde{s}},{\tilde{z}}_0,{\tilde{s}}_0$ be as in Remark \[rem:ts0\]. It suffices to consider the action of $\widehat{{\tilde{z}}_0}$.
Assume we are in case (1) in the proof of Proposition \[prop:weights\], then the weight $({\tilde{R}},{\tilde{\varphi}})$ has the label of the form $({\tilde{s}}{\tilde{z}}_0^i,{\{\kappa\}},K)$. We may assume $0\leq i\leq(q-1)_{\ell'}/2-1$ for cases (II) and (III). In these cases, ${\tilde{N}}(\theta_+)={\tilde{N}}({\tilde{\theta}})$. By the proof of Proposition \[prop:weights\], the weight character ${\tilde{\varphi}}$ is of the form ${\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}} {\tilde{\theta}}_0\psi_{+,K}$. Then $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}(\theta_+)}^{{\tilde{N}}} (\widehat{{\tilde{z}}_0}{\tilde{\theta}}_0)\psi_{+,K}$. By Remark \[rem:ts0\], ${\tilde{\theta}}_0$ induces a character $\chi_{{\tilde{s}}_0{\tilde{z}}_0^i,\kappa}$ of ${\operatorname{J}}_0(V_0)$. By Remark \[rem:zeta\], $\widehat{{\tilde{z}}_0}{\tilde{\theta}}_0$ is determined by $\widehat{{\tilde{z}}_0}\chi_{{\tilde{s}}_0{\tilde{z}}_0^i,\kappa}$. Thus by Remark \[rem:action-tz\], $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}$ is the weight character associated with ${\tilde{R}}$ with label
(a) $({\tilde{s}}{\tilde{z}}_0^{i+1},{\{\kappa\}},K)$ for case (I), case (II) or (III) and $i<(q-1)_{\ell'}/2-1$;
(b) $({\tilde{s}},{\{\kappa\}},K)$ for case (II) and $i=(q-1)_{\ell'}/2-1$;
(c) $({\tilde{s}},{\{\kappa'\}},K)$ for case (III) and $i=(q-1)_{\ell'}/2-1$.
Assume we are in case (2) in the proof of Proposition \[prop:weights\] i.e. it is the case (V), then the weight $({\tilde{R}},{\tilde{\varphi}})$ has the label of the form $({\tilde{s}}{\tilde{z}}_0^i,{\{\kappa\}},K)$. We may assume $0\leq i\leq(q-1)_{\ell'}/2-1$. In this case, ${\tilde{N}}(\theta_+)={\tilde{N}}({\tilde{\theta}})$. By the proof of Proposition \[prop:weights\], the weight character ${\tilde{\varphi}}$ is of the form ${\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}} \psi_K$ with $\psi_K = {\operatorname{Ind}}_{{\hat{N}}({\tilde{\theta}})}^{{\tilde{N}}({\tilde{\theta}})} {\hat{\theta}}_0^{(i)}\psi_{+,K}$, $i=0$ or $1$. Thus $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}} \widehat{{\tilde{z}}_0}\psi_K$ and $\widehat{{\tilde{z}}_0}\psi_K$ is determined by $\widehat{{\tilde{z}}_0}{\tilde{\theta}}_0$ by construction. Then by the same arguments as in the above paragraph, $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}$ is the weight character associated with ${\tilde{R}}$ with label $({\tilde{s}}{\tilde{z}}_0^{i+1},{\{\kappa\}},K)$ if $i<(q-1)_{\ell'}/2-1$ and $({\tilde{s}},{\{\kappa\}},K)$ if $i=(q-1)_{\ell'}/2-1$.
It remains to consider the case (3) in the proof of Proposition \[prop:weights\], i.e. (IV) or (VI), in which case, ${\tilde{N}}(\theta_+)\neq{\tilde{N}}({\tilde{\theta}})$.
Assume we are in case (IV), then the weight $({\tilde{R}},{\tilde{\varphi}})$ has the label of the form $({\tilde{s}}{\tilde{z}}_0^i,{\{\kappa\}},{\{K,K'\}})$ or $({\tilde{s}}{\tilde{z}}_0^i,{\{\kappa\}},(K,j))$ with $j\in{\mathbb{Z}}/2{\mathbb{Z}}$. Assume first the label is $({\tilde{s}}{\tilde{z}}_0^i,{\{\kappa\}},{\{K,K'\}})$ with $K \neq K'$. We may assume $0\leq i\leq(q-1)_{\ell'}/2-1$. By the proof of Proposition \[prop:weights\], the weight character ${\tilde{\varphi}}$ is of the form ${\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}} {\tilde{\theta}}_0\psi_{+,K}^0$. Then $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}(\theta_+)}^{{\tilde{N}}} (\widehat{{\tilde{z}}_0}{\tilde{\theta}}_0)\psi_{+,K}^0$. By Remark \[rem:ts0\], ${\tilde{\theta}}_0$ induces a character $\chi_{{\tilde{s}}_0{\tilde{z}}_0^i,\kappa}$ of ${\operatorname{J}}_0(V_0)$. By Remark \[rem:zeta\], $\widehat{{\tilde{z}}_0}{\tilde{\theta}}_0$ is determined by $\widehat{{\tilde{z}}_0}\chi_{{\tilde{s}}_0{\tilde{z}}_0^i,\kappa}$. If $i<(q-1)_{\ell'}/2-1$, by Remark \[rem:action-tz\], $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}$ is the weight character associated with ${\tilde{R}}$ with label $({\tilde{s}}{\tilde{z}}_0^{i+1},{\{\kappa\}},{\{K,K'\}})$. If $i=(q-1)_{\ell'}/2-1$, $\widehat{{\tilde{z}}_0}\chi_{{\tilde{s}}_0{\tilde{z}}_0^i,\kappa}=\chi_{-{\tilde{s}},\kappa}$. By Proposition \[prop:N(theta+):N(ttheta)\] and its proof, there is $g_+ \in N_+(\theta_+)-N_+({\tilde{\theta}})$ such that $\chi_{-{\tilde{s}},\kappa} = \chi_{{\tilde{s}},\kappa}^{g_+}$. By Remark \[rem:cc-cano-chars\], we should always start with ${\tilde{\theta}}_0$ corresponding to $\chi_{{\tilde{s}},\kappa}$ to label the weights. Thus $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}(\theta_+)}^{{\tilde{N}}} {\tilde{\theta}}_0^{g_+}\psi_{+,K}^0 = {\operatorname{Ind}}_{{\tilde{N}}(\theta_+)}^{{\tilde{N}}} {\tilde{\theta}}_0({^{g_+}\psi_{+,K}^0}) = {\operatorname{Ind}}_{{\tilde{N}}(\theta_+)}^{{\tilde{N}}} {\tilde{\theta}}_0\psi_{+,K}^0$, which has label $({\tilde{s}},{\{\kappa\}},{\{K,K'\}})$. In summary, $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}$ is the weight character associated with ${\tilde{R}}$ with label
(a) $({\tilde{s}}{\tilde{z}}_0^{i+1},{\{\kappa\}},{\{K,K'\}})$ if $i<(q-1)_{\ell'}/2-1$;
(b) $({\tilde{s}},{\{\kappa\}},{\{K,K'\}})$ if $i=(q-1)_{\ell'}/2-1$.
Assume then the label is $({\tilde{s}}{\tilde{z}}_0^i,{\{\kappa\}},(K,j))$ with $K=K'$ and $j\in{\mathbb{Z}}/2{\mathbb{Z}}$. We may assume $0\leq i\leq(q-1)_{\ell'}/2-1$. By the proof of Proposition \[prop:weights\], the weight character ${\tilde{\varphi}}$ is of the form ${\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}({\tilde{\theta}})}^{{\tilde{N}}} {\tilde{\theta}}_0\psi_{+,K}^{(j)}$. Then $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}= {\operatorname{Ind}}_{{\tilde{N}}(\theta_+)}^{{\tilde{N}}} (\widehat{{\tilde{z}}_0}{\tilde{\theta}}_0)\psi_{+,K}^{(j)}$. Thus the result can be obtained similarly as above, noting that ${^{g_+}\psi_{+,K}^{(j)}}=\psi_{+,K}^{(j+1)}$. In summary, $\widehat{{\tilde{z}}_0}{\tilde{\varphi}}$ is the weight character associated with ${\tilde{R}}$ with label
(a) $({\tilde{s}}{\tilde{z}}_0^{i+1},{\{\kappa\}},(K,j))$ if $i<(q-1)_{\ell'}/2-1$;
(b) $({\tilde{s}},{\{\kappa\}},(K,j+1))$ if $i=(q-1)_{\ell'}/2-1$.
The result for case (VI) is similar with ${\{\kappa\}}$ replaced by ${\{\kappa,\kappa'\}}$ and can be proved in the same way, noting that when $i=(q-1)_{\ell'}/2-1$, $\chi_{-{\tilde{s}}_0,\kappa}$ should be replaced by $\chi_{{\tilde{s}}_0,\kappa'}$ for case (VI).
Then it is easy to see that the bijection in Theorem \[thm:bijection\] is equivariant under the action of ${\operatorname{Irr}}({\tilde{G}}/G)_{\ell'}$, which proves this lemma.
{#section-5}
In this subsection, we finish the proof of Theorem \[mainthm-2\]. Since ${\operatorname{Out}}(G)$ is abelian, it suffices to verify parts (iii) and (iv) of Theorem \[thm:criterion\], and in fact, the requirements in parts (iii) and (iv) are satisfied for any $\chi\in{\operatorname{IBr}}(G)$ and any $\psi\in{\operatorname{dz}}(N_G(R)/R)$ respectively. This would complete the proof of Theorem \[mainthm-2\].
Assume the decomposition matrix with respect to ${\mathcal{E}}(G,\ell')$ is unitriangular. Then for any $\chi\in{\operatorname{IBr}}(G)$,
(1) $({\tilde{G}}\rtimes E)_\chi={\tilde{G}}_\chi \rtimes E_\chi$,
(2) $\chi$ extends to $G\rtimes E_\chi$.
By [@CS17C Theorem 3.1], $({\tilde{G}}\rtimes E)_\chi={\tilde{G}}_\chi\rtimes E_\chi$ holds for any $\chi\in{\operatorname{Irr}}(G)$. By [@Ge93], ${\mathcal{E}}(G,\ell')$ is a basic set of ${\operatorname{Irr}}(G)$. Then (1) follows from the assumption about the decomposition matrix with respect to ${\mathcal{E}}(G,\ell')$ and [@CS13 Lemma 7.5]. Since $E$ is cyclic, (2) obviously holds.
Let $R = R_0 \times R_1 \times \cdots \times R_u$ be a radical subgroup of $G$, where $R_0$ is the trivial group and $R_i=R_{m_i,\alpha_i,\gamma_i,{\mathbf{c}}_i}$ ($i\geq1$) is a basic subgroup. For any $\psi\in{\operatorname{dz}}(N_G(R)/R)$, there exists some $x \in {\tilde{G}}$ with
(1) $({\tilde{G}}E)_{R,\psi}= {\tilde{G}}_{R,\psi} (GE)_{R,\psi}$,
(2) $\psi$ extends to $(G\rtimes E)_{R,\psi}$.
To prove this lemma, we transfer to the twisted groups as in [@CS17 Proposition 5.3]. Let $v,g,\iota$ be as in [@Li19 Lemma 6.7]. Let ${\tilde{G}}^{tw}={\tilde{\mathbf{G}}}^{vF}$. Then $\iota$ can be extended to a surjective homomorphism $$\iota:\quad {\tilde{G}}^{tw} \rtimes {\hat{E}}\to {\tilde{G}}\rtimes E$$ where ${\hat{E}}={\langle{\hat{F}}_p\rangle}$ is the group of field automorphisms of ${\tilde{G}}^{tw}$. Note that ${\operatorname{Ker}}\iota = {\langlev{\hat{F}}_q\rangle}$, where ${\hat{F}}_q={\hat{F}}_p^f$. Let $R^{tw} = \iota^{-1} (R)$ and $\psi^{tw} = \psi\circ\iota$. By [@Li19 Lemma 6.8], $\sigma(R^{tw})=R^{tw}$ for any $\sigma\in{\hat{E}}$ and $N_{{\tilde{G}}^{tw}{\hat{E}}}(R^{tw}) = N_{{\tilde{G}}^{tw}}(R^{tw}) \rtimes {\hat{E}}$. Then by a similar argument as in [@CS17 Proposition 5.3], it suffices to prove the following
1. $(N_{{\tilde{G}}^{tw}}(R^{tw}) \rtimes {\hat{E}})_{\psi^{tw}} = N_{{\tilde{G}}^{tw}}(R^{tw})_{\psi^{tw}} \rtimes {\hat{E}}_{\psi^{tw}}$,
2. $\psi^{tw}$ can be extended to a character ${\tilde{\psi}}^{tw}$ of $N_{G^{tw}}(R^{tw}) \rtimes {\hat{E}}_{\psi^{tw}}$ with $v{\hat{F}}_q \in {\operatorname{Ker}}{\tilde{\psi}}^{tw}$.
(1’) follows from [@Li19 Proposition 6.20, Proposition 6.25] and their proofs. For (2’), note that $v{\hat{F}}_q$ acts trivially on $G^{tw}$. Thus we can view $\psi^{tw}$ as a character of $N_{G^{tw}}(R^{tw}) \rtimes {\langle{\hat{F}}_q\rangle} = N_{G^{tw}}(R^{tw}) \times {\langlev{\hat{F}}_q\rangle}$ containing $v{\hat{F}}_q$ in the kernel. Now, since $N_{G^{tw}}(R^{tw}) \rtimes {\hat{E}}_{\psi^{tw}} / N_{G^{tw}}(R^{tw}) \rtimes {\langle{\hat{F}}_q\rangle}$ is cyclic, (2’) follows obviously. Of course, (2) also follows directly from the fact that $E$ is cyclic.
When the radical subgroup $R^x$ ($R$ is as in the above lemma) is considered, we need to replace $E$ with $E^x$.
Proof of Theorem \[mainthm-3\] {#sec:main-3}
==============================
In this section, we assume $q=2^f$ and consider the simple group $G={\operatorname{Sp}}_{2n}(2^f)$ with $(n,f)\neq(2,1)$ or $(3,1)$. For this case, our proof is just to say that all the relevant arguments in [@FS89; @An94; @Li19] and the previous sections apply. We list all the statements and point out where to find the relevant proofs.
Any radical subgroup of $G$ is conjugate to a subgroup of the form $R_0 \times R_1 \times \cdots \times R_u$, where $R_0$ is a trivial group and $R_i=R_{m_i,\alpha_i,\gamma_i,{\mathbf{c}}_i}$ is a basic subgroup for $i>0$.
The relevant arguments in [@An94 §1,§2] apply to this case.
In [@SF14 Proposition 3.1], the author considered ${\operatorname{Sp}}_6(q)$ and already noted that the relevant arguments in [@An94 §1, §2] apply when $\ell \mid (q^2-1)$. For radical subgroups $Q^{(3)}$ when $3\neq\ell\mid(q^4+q^2+1)$ and $Q^{(2)}$ when $\ell\mid(q^2+1)$ in [@SF14 Proposition 3.1], we can find the corresponding notation used in §\[subsec:radical\]. Note that $e$ is necessarily odd if $\ell$ is linear. When $3\neq\ell\mid(q^4+q^2+1)$, $e=3$, and $\ell$ is linear if $\ell\mid(q^2+q+1)$ while $\ell$ is unitary if $\ell\mid(q^2-q+1)$. In this case, $Q^{(3)}$ is just $R_1$ defined in §\[subsec:radical\]; here, recall that $R_1$ is the abbreviation of $R_{1,0,0,{\mathbf{0}}}$. When $\ell\mid(q^2+1)$, $e=2$ and $\ell$ is unitary, and $Q^{(2)}$ should be again denoted as $R_1$ using the notation system in §\[subsec:radical\].
Let $D_{m,\alpha,\beta}$ be defined in the same way as in §\[subsec:ano-conju\].
Any defect group of $G$ is of the form $D_0 \times D_1 \times \cdots \times D_u$, where $D_0$ is the trivial group and $D_i=D_{m_i,\alpha_i,\beta_i}$ for $i>0$.
The argument in [@FS89 (5K)] applies.
All the relevant constructions and notation in §\[subsec:Brauerpair\] apply to the case in this section. In particular, let $(R,\varphi)$ be a weight of $G$ and $\varphi$ lies over $\theta\in{\operatorname{dz}}(C/Z(R))$, then we can start with a Brauer pair $(R,\theta)$ and construct $(D',\theta')$ and $(D,\theta_D)$.
$(R,\theta) \leq (D,\theta_D)$ as Brauer pairs in $G$. $(D,\theta_D)$ is a maximal Brauer pair and all maximal Brauer pair of $G$ are of this form.
The argument of Proposition \[prop:Brauerpair\] applies.
With this lemma, we can determine which block the weight $(R,\varphi)$ belongs to as we have done for $q$ odd.
Let $(D,\theta_D)$ be a maximal Brauer pair for the block $B$ of $G$. Let $(s,\kappa)$ be defined in the same way as in [@FS89 §10], then $(s,\kappa)$ is a label of $B$; such labelling gives a bijection as in [@FS89 (10B)]. The dual defect group of $B$ is defined as the image of $D$ under the isomorphism $G \cong G^*$.
Let $B$ be the block of $G$ with label $(s,\kappa)$. Then for any $\chi\in{\operatorname{Irr}}(G)$, $\chi\in{\operatorname{Irr}}(B)$ if and only if $\chi=\chi_{t,\lambda}$ and (1) $t_{\ell'}$ is conjugate to $s$; (2) $t_\ell$ is contained in a dual defect group of $B$; (3) $\kappa$ is the core of $\lambda$.
The arguments of [@FS89 (12A)] apply. Here, note that $\kappa$ is the core of $\lambda$ means $\kappa_\Gamma$ is the $e_\Gamma$-core of $\lambda_\Gamma$.
${\operatorname{IBr}}(B_{s,\kappa})$ can be labelled by $\phi_{s,\lambda}$, where $\kappa$ is the core of $\lambda$.
As an example, see [@Whi95] and [@Whi00] for irreducible ordinary characters in unipotent blocks of ${\operatorname{Sp}}_4(2^f)$ and ${\operatorname{Sp}}_6(2^f)$, and see [@SF13] for irreducible ordinary characters in non-unipotent blocks of ${\operatorname{Sp}}_6(2^f)$.
The parametrization of weights is the same as in [@An94]; see [@Li19 §6.B] for a description of the construction of weights. In the above constructions, the duals of semisimple elements are used; see on [@An94 p.18] for the definition of dual for the case $q$ odd. In the case $q$ even, we define the dual of semisimple elements via the isomorphism $G \cong G^*$. We state the result as follows.
Let $B=B_{s,\kappa}$ be a block of $G$. Assume $s= s_0 \times s_+$ as in [@FS89 §10]. Then $m_\Gamma(s)-m_\Gamma(s_0)=w_\Gamma\beta_\Gamma e_\Gamma$ for some natural number $w_\Gamma$. Set $$i{\mathcal{W}}(B)=\left\{
Q=\prod_\Gamma Q_\Gamma ~\middle|~
\begin{array}{c}
Q_\Gamma=\left(Q_\Gamma^{(1)},Q_\Gamma^{(2)},\dots,Q_\Gamma^{(\beta_\Gamma e_\Gamma)}\right),\\
\textrm{$Q_\Gamma^{(j)}$'s are partitions},\sum\limits_{j=1}^{\beta_\Gamma e_\Gamma} |Q_\Gamma^{(j)}|=w_\Gamma.
\end{array}
\right\}$$ Here $Q_\Gamma$ is an ordered sequence of $\beta_\Gamma e_\Gamma$ partitions. Then there is a bijection between $i{\mathcal{W}}(B)$ and ${\operatorname{Alp}}(B)$.
The conjugacy class of weights in $B_{s,\kappa}$ corresponding to $Q$ will be denoted as $w_{s,\kappa,Q}$.
Let $B=B_{s,\kappa}$ be a block of $G$. Then there is a bijection $${\operatorname{IBr}}(B) \to {\operatorname{Alp}}(B), \quad \phi_{s,\lambda} \mapsto w_{s,\kappa,Q},$$ where $\lambda_\Gamma=\kappa_\Gamma*Q_\Gamma$.
Noting that $\lambda_{X-1}$ and $\kappa_{X-1}$ are both non-degenerate Lusztig symbols, while $\lambda_\Gamma$ and $\kappa_\Gamma$ for $\Gamma\neq X-1$ are partitions, the result follows from §\[subsec:partition-symbol\].
Assume the sub-matrix of the decomposition matrix of ${\operatorname{Sp}}_{2n}(2^f)$ to ${\mathcal{E}}({\operatorname{Sp}}_{2n}(2^f),\ell')$ is unitriangular. Then the above bijection is ${\operatorname{Aut}}(G)$-equivariant.
Since the cases $n=1$ and $n=2$ have been considered in [@Sp13] and [@SF14] respectively, we may assume $n\geq3$, thus it suffices to consider $\sigma\in E$. For $\chi_{s,\lambda}\in{\operatorname{Irr}}(B)$, $\chi_{s,\lambda}^\sigma = \chi_{\sigma^*(s),\sigma^*(\lambda)}$ by [@CS13 Theorem 3.1] and similar arguments in [@Li19 Proposition 5.6]. Then by the assumption on the decomposition matrix, $\phi_{s,\lambda}^\sigma = \phi_{\sigma^*(s),\sigma^*(\lambda)}$. For the action of $\sigma$ on weights, the arguments in [@Li19 §6.C] applies. Then it is easy to see the above bijection is equivariant.
Let ${\tilde{G}}=G$ and ${\mathcal{B}}={\{B\}}$, then the other requirements in Theorem \[thm:criterion\] are obviously satisfied. This completes the proof of Theorem \[mainthm-3\].
Acknowledgements {#acknowledgements .unnumbered}
================
The author is extremely grateful to Julian Brough, Marc Cabanes and Britta Späth for their hospitality during his visit at Wuppertal in May 2019, and especially for the fruitful discussions and keen suggestions, which made this paper possible. The author acknowledges the great working atmosphere during the stay of a month in the summer of 2019 at Sustech International Center for Mathematics, where the main part of this paper was written.
[99]{}
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[^1]: Supported by National Natural Science Foundation of China (No. 11901478 and No. 11631001) and Fundamental Research Funds for the Central Universities (No. 2682019CX48).
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abstract: 'Gibbs partition models are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of the probability function of the two-parameter Poisson-Dirichlet family. Recently those models have been investigated in a Bayesian nonparametric approach to species sampling problems as alternatives to the Dirichlet and the Pitman-Yor process priors. Here we derive marginals of conditional and unconditional multivariate distributions arising from exchangeable Gibbs partitions to obtain explicit formulas for joint falling factorial moments of corresponding conditional and unconditional Gibbs sampling formulas. Our proofs rely on a known result on factorial moments of sum of non independent indicators. We provide an application to a Bayesian nonparametric estimation of the predictive probability to observe a species already observed a certain number of times.'
address:
- |
Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza\
Università degli Studi di Roma La Sapienza\
Via del Castro Laurenziano, 9 00161 Rome, Italy\
\
- 'August 8, 2012'
author:
-
title: Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling
---
Introduction
============
Exchangeable [*Gibbs*]{} partitions ([@gnepit06]) are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of exchangeable partition probability function (EPPF) of the two parameter $(\alpha, \theta)$ Poisson-Dirichlet partition model ([@pit95], [@pityor97]), namely $$\label{2par}
p_{\alpha, \theta}(n_1, \dots, n_k)= \frac{(\theta +\alpha)_{k-1 \uparrow \alpha}}{(\theta +1)_{n-1}} \prod_{j=1}^{k} (1 -\alpha)_{n_j-1},$$ for $\alpha \in (0,1)$, $\theta > -\alpha$, $(n_1, \dots, n_k)$ a composition of $n$, $1 \leq k \leq n$ and $(x)_{y \uparrow \alpha}=x(x+\alpha)\cdots(x+(y-1)\alpha)$ generalized rising factorials. Their EPPF is characterized by the [*Gibbs*]{} product form $$\label{EPPFgibbs}
p_{\alpha, V}(n_1, \dots, n_k)= V_{n,k} \prod_{j=1}^{k} (1 -\alpha)_{n_j-1},$$ for $\alpha \in (-\infty, 1)$, and $V=(V_{n,k})$ weights satisfying the backward recursive relation $V_{n,k}=(n -k\alpha)V_{n+1, k} + V_{n+1, k+1}$, for $V_{1,1}=1$. By Theorem 12 in [@gnepit06] each element of (\[EPPFgibbs\]) arises as a probability mixture of extreme partitions, namely: Fisher’s (1943) partitions ([@fis43]) for $\alpha < 0$, Ewens $(\theta)$ partitions ([@ewe72], [@kin75]) for $\alpha=0$, and Poisson-Kingman conditional partitions driven by the stable subordinator ([@pit03]) for $\alpha \in (0,1)$.
By an application of Eq. (2.6) in [@pit06], given an infinite EPPF in the form (\[EPPFgibbs\]), for each $n \geq 1$ the corresponding joint distribution of the random vector $(N_{1,n} \dots, N_{K_n, n}, K_n)$ of the sizes and number of the blocks in [*size biased order*]{} (i.e. in order of their least elements) is given by $$\label{sizebias2}
\mathbb{P}_{\alpha, V} (N_{1,n}=n_1, \dots, N_{K_n, n}=n_k, K_n=k)=$$ $$=\frac{n!}{n_k(n_k+n_{k-1})\cdots (n_k +\dots+n_1) \prod_{j=1}^k (n_j-1)!} V_{n,k} \prod_{j=1}^k (1 -\alpha)_{n_j -1},$$ where the combinatorial factor accounts for the number of partitions of $[n]$ in which the $j$-th block in order of appearance has $n_j$ elements. When the order of the blocks is irrelevant an alternative, more tractable coding for the joint distribution (\[sizebias2\]) is in [*exchangeable random order*]{} (cfr. Eq. (2.7) in [@pit06]) $$\label{prior}
\mathbb{P}_{\alpha, V}(N_1^{ex}=n_1, \dots, N_{K_n}^{ex}=n_k, K_n=k)= \frac{n!}{\prod_{j=1}^k n_j!}\frac{1}{k!} V_{n,k} \prod_{j=1}^{k} (1 -\alpha)_{n_j-1},$$ that, from now on, we term [*multivariate Gibbs distribution*]{} of parameters $(n, \alpha, V)$. Corresponding [*Gibbs sampling formula*]{}, encoding the partition of $n$ by the vector of the numbers of blocks of different sizes, is obtained by the obvious change of variable in (\[prior\]) and is given by $$\label{sampling}
\mathbb{P}_{\alpha, V}(C_{1,n}=c_1, \dots, C_{n,n}=c_n)={n!}V_{n,k}\prod_{i=1}^n \frac{[(1-\alpha)_{i-1}]^{c_i}}{(i!)^{c_i} c_i!},$$ for $c_i=\sum_{j=1}^k 1\{n_j=i\}$, for $i=1, \dots, n$, $\sum_{i=1}^n i c_i=n$ and $\sum_{i=1}^n c_i=k$. Note that this is the general [*Gibbs*]{} analog of the [*Ewens sampling formula*]{} (cfr. [@ewe72]) $$\label{ewensEPPF}
\mathbb{P}_{\theta}(C_{1,n}=c_1,\dots, C_{n,n}=c_n)= \frac{n! \theta^k}{(\theta)_n} \prod_{i=1}^n \frac{1}{(i)^{c_i} c_i!},$$ encoding by the vector of counts the Dirichlet $(\theta)$ partition model, ([@fer73; @kin75]), whose EPPF is well-known to arise for $\alpha=0$ in (\[2par\]). A comprehensive reference for the study of (\[sampling\]), also called [*component frequency spectrum*]{}, for general combinatorial random structures is [@arr03].\
In this paper we study marginals of (\[prior\]), both conditional and unconditional, in order to derive joint falling factorial moments of corresponding conditional and unconditional sampling formulas. Our main motivation comes from applications in Bayesian nonparametric estimation in species sampling problems. In this setting, given $n$ observations from a population of species with multiplicities of the first $k$ species observed $(n_1, \dots, n_k)$, interest may lie in conditional predictive estimation of quantities related to a further sample of $m$ observations (cfr. e.g. [@lmp07], [@lpw08]), or in conditional estimation of some diversity index of the whole population, (see e.g. [@cer12]). A common [*prior*]{} assumption in the Bayesian nonparametric approach is that the unknown relative abundances $(P_i)_{i\geq 1}$ of the species in the population follow a random discrete distribution belonging to the Gibbs family, i.e. are such that, by Kingman’s correspondence (cfr. [@kin78]), $$\label{kingm}
\sum_{(i_1, \dots, i_k)} \mathbb{E} \left[ \prod_{j=1}^k P_{i_j}^{n_j} \right]= V_{n,k} \prod_{j=1}^k (1-\alpha)_{n_j-1},$$ where $(i_1, \dots, i_k)$ ranges over all ordered $k$-tuples of distinct positive integers. This is equivalent to assume that the theoretically infinite sequence of species [*labels*]{} $(X_i)_{i \geq 1}$ is exchangeable with almost surely discrete [*de Finetti*]{} measure representable as $P(\cdot)=\sum_{i=1}^{\infty} P_i \delta_{Y_i}(\cdot)$, for $(P_i)$ any rearrangement of the ranked frequencies $(P_i^{\downarrow})$ satisfying (\[kingm\]), independent of $(Y_i) \sim$ IID $H(\cdot)$, for $H$ some non atomic probability distribution.
Actually the study of [*conditional Gibbs structures*]{} in this perspective has been initiated in [@lmp07] and [@lpw08] and some results for conditional falling factorial moments of [*components*]{} of (\[sampling\]) are in [@flp12a]. Nevertheless in those papers some confusion arises between [*conditional EPPFs*]{}, and [*conditional multivariate distributions*]{} of the vector of sizes and number of the blocks in exchangeable random order, which heavily affects the complexity of the proofs.
Here, after deriving marginals of conditional and unconditional [*multivariate Gibbs distributions*]{}, we obtain [*joint*]{} falling factorial moments of any order of (\[sampling\]), both conditional and unconditional, and explicit formulas for some distributions of interest generalizing some particular cases obtained in [@flp12a], in a direct way. Our analysis, besides providing a more effective technique for the study of Gibbs sampling formulas, with a view toward Bayesian nonparametric applications, establishes the first systematic study of joint multivariate distributions arising from Gnedin-Pitman’s Gibbs partition models. The paper is organized as follows: in Section 2 we provide marginals of (\[prior\]) and, resorting to a result in [@johkot05] for sum of non independent indicators, derive general formulas for joint falling factorial moments of (\[sampling\]), together with some explicit marginal distributions and their expected values. In Section 3 we derive [*conditional multivariate Gibbs distributions*]{} and their marginals, for sizes and number of [*new*]{} blocks induced by the additional $m$-sample. A complete analysis is performed for [*conditional Gibbs sampling formulas*]{} exploiting the same technique adopted in Section 2. In Section 4 we focus on [*multivariate Pólya-like*]{} distributions arising by the conditional allocation of the additional sample in [*old*]{} blocks. Finally, in Section 5, we provide an application of marginals of multivariate Gibbs distributions to a Bayesian nonparametric estimation of a $m$-step ahead probability to detect at observation $n+m+1$ a species already observed a certain number of times.
Marginals of multivariate Gibbs distributions
=============================================
To obtain the marginal distributions for general Multivariate Gibbs distributions (\[prior\]) it is enough to resort to the definition of generalized [*central*]{} Stirling numbers (cfr. Eq. 1.9 and 1.19 in [@pit06]) (see the Appendix for further details) $$S_{n,k}^{-1, -\alpha}= \frac{n!}{ k!}\sum_{(n_1, \dots, n_k)} \prod_{j=1}^k \frac{(1-\alpha)_{n_j-1}}{n_j!},
%n! \sum_{c_1, \dots, c_n} \prod_{i=1}^n \frac{[(1 -\alpha)_{l-1}]^{c_i}}{(l!)^{c_i} c_i!}$$ where the sum ranges over all $(n_1, \dots, n_k)$ compositions of $n$. From now on we refer to (\[prior\]) omitting the [*ex*]{} power in the notation.
Under a general Gibbs partition model (\[EPPFgibbs\]) of parameters $(\alpha, V)$, for each $n \geq 1$ the $r$-dimensional marginal of (\[prior\]), for $0 \leq k-r \leq n -\sum_{j=1}^r n_j$, is given by $$\label{rmarg}
\mathbb{P} (N_1=n_1, \dots, N_r=n_r, K_n=k)=$$ $$=\frac{n!}{\prod_{j=1}^r n_j!}\prod_{j=1}^r (1 -\alpha)_{n_j-1}
\frac{V_{n,k}}{k!} \sum_{(b_1, \dots, b_{k-r})} \frac{1}{\prod_i b_i!}\prod_{i=1}^{k-r} (1-\alpha)_{b_i-1}$$ for $(b_1, \dots, b_{k-r})$ such that $b_i >0$ $\forall i$ and $\sum_{i} b_i= n-\sum_{j=1}^r n_j$. Multiplying and dividing by $(n-\sum_{j=1}^{r} n_j)!$ and $(k-r)!$ yields $$=\frac{n!}{\prod_{j=1}^r n_j! (n -\sum_{j=1}^r n_j)!}\prod_{j=1}^r (1 -\alpha)_{n_j-1} \frac{V_{n,k}}{k_{[r]}} S_{n-\sum_{j=1}^r n_j, k-r}^{-1, -\alpha},$$ for $(x)_{[n]}= (x)(x-1)\cdots(x-n+1)$.
By a known result in [@gnepit06] for each model (\[EPPFgibbs\]) the number of blocks $K_n$ has distribution $$\mathbb{P}(K_n=k)= V_{n,k}S_{n,k}^{-1, -\alpha},$$ hence, conditioning (\[rmarg\]) on $K_n=k$ yields $$\label{marg_2}
\mathbb{P} (N_1=n_1, \dots, N_r=n_r| K_n=k)=$$ $$\frac{n!}{\prod_{j=1}^r n_j! (n -\sum_{j=1}^{r} n_j)!}\frac{\prod_{j=1}^r (1 -\alpha)_{n_j-1} }{k_{[r]}} \frac{S_{n-\sum_{j=1}^r n_j, k-r}^{-1, -\alpha}}{S_{n,k}^{-1, -\alpha}}$$ independently of the specific Gibbs model. For $r=k$ this is the general Gibbs analog of Eq. (41.8) in [@ewetav95], and for $r=1$, $0 \leq k-1 \leq n-n_1$ and $n_1=1, \dots, n-k+1$ $$\mathbb{P}(N_1=n_1| K_n=k)= {n \choose n_1} \frac{(1-\alpha)_{n_1-1}}{k} \frac{S_{n-n_1, k-1}^{-1, -\alpha, }}{S_{n,k}^{-1, -\alpha}}$$ with expected value $$\mathbb{E}(N_1| K_n=k)= \frac{n}{k} \frac{S_{n-1, k-1}^{-1, -\alpha, -(1-\alpha)}}{S_{n,k}^{-1, -\alpha}},$$ for $S_{n,k}^{-1, -\alpha, \gamma}$ generalized non-central Stirling numbers (see (\[noncentralsti\]) in the Appendix). Marginalizing (\[rmarg\]) with respect to $K_n$ yields $$\mathbb{P} (N_1=n_1, \dots, N_r=n_r)=$$ $$=\frac{n!}{\prod_{j=1}^r n_j! (n -\sum_{j=1}^{r} n_j)!}\prod_{j=1}^r (1 -\alpha)_{n_j-1} \sum_{k-r=0}^{n -\sum_{j=1}^r n_j} \frac{V_{n,k} }{k_{[r]}} S_{n-\sum_{j=1}^r n_j, k-r}^{-1, -\alpha}.$$
Joint factorial moments of Gibbs sampling formulas
--------------------------------------------------
Joint falling factorial moments for the Ewens’ sampling formula (\[ewensEPPF\]) of order $(r_1, \dots, r_n)$, for $r_l$ non negative integers and $n-\sum_l lr_l \geq 0$, are in [@ewetav95] (cfr. Eq. (41.9)) and correspond to $$\mathbb{E}_{\theta} \left[\prod_{l=1}^n (C_{l,n})_{[r_l]}\right]= \frac{n!}{(n -\sum_{l=1}^n lr_l)!} \frac{(\theta)_{n -\sum_{l=1}^n lr_l}}{(\theta)_n} \prod_{l=1}^n \left(\frac{\theta }{l} \right)^{r_l}.$$ Under the same conditions, the generalization to the $(\alpha, \theta)$ Poisson-Dirichlet partition model (\[2par\]) has been obtained is [@yamsib00] and is given by $$\mathbb{E}_{\alpha, \theta} \left[\prod_{l=1}^n (C_{l,n})_{[r_l]}\right]= \frac{n!}{(n -\sum_{l=1}^n lr_l)!} \frac{(\theta + \alpha)_{\sum_l r_l -1 \uparrow \alpha}}{(\theta +1)_{n-1}} \times$$ $$\times \prod_{l=1}^n \left(\frac{(1- \alpha)_{l-1}}{l!}\right)^{r_l} (\theta + \alpha \sum_l r_l)_{n -\sum lr_l}.$$ In the following Proposition we obtain the general result for the Gibbs sampling formula (\[sampling\]) by resorting to a result in [@johkot05], first established in [@dem18] then studied in [@jor67]. See also [@iye49; @iye58].
\[joint\_sampl\]Under a general $(\alpha, V)$ Gibbs partition model, joint falling factorial moments of the vector of counts $(C_{1,n}, \dots, C_{n,n})$ of order $(r_1, \dots, r_n)$ for $\sum_l lr_l \leq n$ are given by $$\label{jointprior}
\mathbb{E} \left[\prod_{l=1}^n (C_{l,n})_{[r_l]}\right]= \frac{n!}{\prod_{l=1}^n (l!)^{r_l}}
\frac{\prod_{l=1}^n \left[(1-\alpha)_{l-1}\right]^{r_l}}{(n -\sum_l l r_l)!} \sum_{k- \sum_l r_l=0}^{n- \sum_l l r_l} V_{n,k} S_{n-\sum_l l r_l, k- \sum_l r_l}^{-1, -\alpha}$$ for $0 \leq k-\sum_{l=1}^n r_l \leq n-\sum_{l=1}^n lr_l$. For $r_l=r \leq \lceil{\frac nl \rceil}$ and $r_j=0$ for every $j \neq l$, the $r$-th falling factorial moment of $C_{l,n}$ results $$\label{factmom_1}
\mathbb{E}\left[(C_{l,n})_{[r]}\right]= \frac{n! [(1-\alpha )_{l-1}]^r}{(l!)^r (n-lr)!} \sum_{k-r=0}^{n-rl}
V_{n,k} S_{n-rl, k-r}^{-1, -\alpha}.$$
For $K_n=k$, let $C_{l,n}= \sum_{j=1}^k 1 (N_{j}=l)$. Then by a result for sum of non independent indicators r.v.s in Johnson & Kotz (2005, Sect. 10.2), or Charalambides (2005, Example 1.12), for $r \leq n$ $$\label{momentr}
\mathbb{E}\left[(C_l)_{[r]}\right]= \mathbb{E}(\sum_{j=1}^k {1\{N_j=l\}})_{[r]}= r! \sum_{(a_1, \dots, a_r)} \mathbb{P}(N_{a_1}=l, \dots, N_{a_r}=l),$$ where the summation is extended over all $r$-combinations $(a_1, \dots, a_r)$ of $\{1, \dots, k\}$. Since in our case the number of blocks $K_n$ is random, and the vector $(N_1, \dots, N_r|K_n=k)$ is exchangeable then, for $l=1, \dots, n$, $$\mathbb{E}\left[(C_l)_{[r]}\right]= \sum_{k-r=0}^{n-rl} \mathbb{E}\left[(C_l|K_n=k)_{[r]}\right] \mathbb{P}(K_n=k)=$$ $$= \sum_{k-r=0}^{n-rl}r! {k \choose r} \mathbb{P}(N_1=l, \dots, N_r=l|K_n=k) \mathbb{P}(K_n=k)=$$ $$=\sum_{k-r=0}^{n-rl} r! {k \choose r} \mathbb{P}(N_1=l, \dots, N_r=l, K_n=k).$$ hence $$\mathbb{E} \left[\prod_{l=1}^n (C_{l,n})_{[r_l]}\right]= \sum_{k- \sum_l r_l=0}^{n -\sum_{l} lr_l} (\prod_{l=1}^n r_l!) \frac{k!}{\prod_l r_l! (k- \sum_l r_l)! }\times$$ $$\label{kalea}
\times \mathbb{P}(N_1=1, \dots, N_{r_1}=1, \dots, N_{\sum_l r_l -r_{n}+1}= n, \dots, N_{\sum_l r_l}= n, K_n=k).$$ Inserting (\[rmarg\]) in (\[kalea\]) the result follows.
Notice that (\[jointprior\]) generalizes the result in [@flp12a] Eq. (11), stated in terms of [*generalized factorial coefficients*]{}, (cfr. Eq. and in the Appendix), which corresponds to (\[factmom\_1\]). Next Proposition generalizes Proposition 2 (Dirichlet case) and Proposition 4 (two parameter Poisson-Dirichlet case) in [@flp12a].
\[dist\_marg\_uncon\] Under a general $(\alpha, V)$ Gibbs partition model, for each $n \geq 1$ the law of $C_{l,n}$, the number of blocks of size $l$, has distribution $$\label{margL}
\mathbb{P}(C_{l,n}=x)=\frac{n! [(1-\alpha)_{l-1}]^x}{x!(l!)^{x}} \sum_{r=0}^{\lceil\frac nl\rceil-x} \frac{(-1)^{r}[(1-\alpha)_{l-1}]^r}{r! (l!)^r (n-rl-lx)!}\times$$ $$%\begin{equation}
\times \sum_{k-r-x=0}^{n-rl-xl} V_{n,k} S_{n-rl-xl, k-r-x}^{-1, -\alpha}$$ for $x=0, \dots, \lceil{n/l}\rceil$, with expected value $$\label{meanL}
\mathbb{E}(C_{l,n})= {n \choose l} (1 -\alpha)_{l-1} \sum_{k-1=0}^{n-l} V_{n,k}S_{n-l, k-1}^{-1, -\alpha}$$ and the distribution of the number of singleton species $C_{1,n}$ follows from (\[margL\]) $$\label{marg10}
\mathbb{P}(C_{1,n}=x)= \frac{n!}{x!} \sum_{r=0}^{n-x} \frac{(-1)^r}{r! (n-r-x)!} \sum_{k-r-x=0}^{n-r-x} V_{n,k} S_{n-r-x, k-r-x}^{-1, -\alpha}.$$ with expected value $$\label{mean_10}
\mathbb{E}(C_{1,n})= n \sum_{k-1=0}^{n-1} V_{n,k} S_{n-1, k-1}^{-1, -\alpha}.$$
(\[margL\]) arises by the known relationship between discrete probability distributions and falling factorial moments (cfr. (\[momprob\]) in the Appendix). (\[meanL\]) follows from (\[factmom\_1\]) for $r=1$. and follow for $l=1$ and generalize (41.10) and (41.11) in [@ewetav95] to the entire Gibbs family.
Conditional multivariate Gibbs distributions
============================================
The study of [*conditional exchangeable random partitons*]{}, i.e. random partitions starting with an initial allocation of the first $n$ natural integers in a certain number $k$ of blocks, has been initiated in [@lmp07] in view of proposing a Bayesian conditional nonparametric estimation of the richness of a population of species under [*priors*]{} on the unknown relative abundances belonging to the Gibbs class. (See also [@gne10], Sect. 7). In [@cer09] it has been shown that the corresponding conditional partition probability function, describing the conditional allocation in [*new*]{} and [*old*]{} blocks of integers $n+1, n+2, \dots$, can be obtained by a multi-step variation of the classical [*Chinese restaurant process*]{} construction (CRP) for exchangeable partitions, (first devised by Dubins and Pitman, see [@pit06] Ch. 3). This variation helps to properly place the Bayesian nonparametric approach to species sampling problems under Gibbs priors into the Gnedin-Pitman’s exchangeable random partitions theoretical framework. Here we recall the multi-step CRP for completeness.
(Cerquetti, 2009) Given an infinite EPPF model (\[EPPFgibbs\]), assume that an unlimited numbers of groups of customers arrive sequentially in a restaurant with an unlimited numbers of circular tables, each capable of sitting an unlimited numbers of customers. Given the placement of the first group of $n$ customers in a ${\bf n}=(n_1, \dots, n_j)$ configuration in $j$ tables, a new [*group*]{} of $m \geq 1$ customers is\
\
a) all seated at the $j$ old tables in configuration ${\bf m}=(m_1,\dots, m_j)$, for $m_i \geq 0$, $\sum_{i=1}^j m_i=m$, with probability $$\label{gibbsallold}
p_{\bf m}({\bf n})=\frac{V_{n+m,j}\prod_{i=1}^j (1-\alpha)_{n_i+m_i-1}}{V_{n,j}\prod_{i=1}^j (1-\alpha)_{n_i -1}}=\frac{V_{n+m,j}}{V_{n,j}} \prod_{i=1}^j (n_i-\alpha)_{m_i},$$ b) all seated at $k$ [*new*]{} tables in configuration ${\bf s}=(s_1, \dots, s_{k})$, for $\sum_{i=1}^{k} s_i =m$, $1 \leq k \leq m$, $s_i \geq 1$, with probability $$\label{gibbsallnew}
p^{\bf s}({\bf n})=\frac{V_{n+m,j+k}\prod_{i=1}^j (1-\alpha)_{n_i -1}\prod_{i=1}^{k} (1-\alpha)_{s_i -1}}{V_{n,j} \prod_{i=1}^j (1-\alpha)_{n_i -1}}=\frac{V_{n+m, j+k}}{V_{n,j}}\prod_{i=1}^{k} (1-\alpha)_{s_i -1},\\$$ c) a subset $s < m$ of the new customers is seated at $k$ [*new*]{} tables in configuration $(s_1,\dots,s_{k})$ and the remaining $m-s$ customers are seated at the [*old*]{} tables in configuration $(m_1,\dots, m_j)$ for $\sum_{i=1}^{j} m_i= m-s$, $1 \leq s \leq m$, $\sum_{i=1}^{k} s_i=s$, $m_i \geq 0$, $s_i \geq 1$ with probability $$%\label{gibbsoldnew}
p_{\bf m}^ {\bf s}({\bf n})=\frac{V_{n+m,j+k} \prod_{i=1}^j (1-\alpha)_{n_i+m_i-1}\prod_{i=1}^{k}(1-\alpha)_{s_i-1}}{V_{n,j}\prod_{i=1}^j (1-\alpha)_{n_i-1}}=$$ which, by the multiplicative property of rising factorials (\[multiplicative\]), simplifies to $$\label{oldenew}
=\frac{V_{n+m,j+k}}{V_{n,j}}\prod_{i=1}^j (n_i-\alpha)_{m_i}\prod_{i=1}^{k}(1-\alpha)_{s_i-1}.$$
Now, as in [@lpw08], given the allocation of the first $n$ integers in $j$ blocks with multiplicities $(n_1, \dots, n_j)$, let $K_m$ be the number of [*new*]{} blocks generated by the additional $m$ integers, $(S_1, \dots, S_{K_m})$ the vector of the sizes of the [*new*]{} blocks in exchangeable random order and $S_{m}=\sum_{i=1}^{K_m} S_{i}$ the total number of [*new*]{} integers in [*new*]{} blocks. To obtain the joint conditional distribution of the vector $(K_m, S_m, S_1, \dots, S_{K_m})$ of the number and multiplicities of new blocks, and total observations in new blocks, it is enough to marginalize (\[oldenew\]) with respect to all $(m_1, \dots, m_j)$ allocations of $m-S_m$ observations in [*old*]{} blocks, and to multiply for the combinatorial coefficient accounting for the number of partitions of $[m]$ providing the same sizes and the same number $k$ of new blocks and the same number $s$ of integers in new blocks. We can hence state the following.
Under a general $(\alpha, V)$ Gibbs partition model the joint conditional distribution of $(S_m, K_m, S_1, \dots, S_{K_m})$, for $S_1, \dots, S_{K_m}$ in [*exchangeable random order*]{}, given the initial allocation of $n$ integers in $j$ blocks, corresponds to $$\mathbb{P} (K_m=k, S_m=s, S_{1}= s_1, \dots, S_{K_m}=s_k| n_1, \dots, n_{j})=$$ $$= \frac{s!}{s_1! \cdots s_k! k!} \frac{V_{n+m, j+k}}{V_{n,j}} {m \choose s} (n -j\alpha)_{m-s} \prod_{i=1}^{k} (1 -\alpha)_{s_i -1}=$$ or alternatively $$\label{newblock}
=\frac{m!}{s_1! \cdots s_k! k! m-s!} \frac{V_{n+m, j+k}}{V_{n,j}} (n -j\alpha)_{m-s} \prod_{i=1}^{k} (1 -\alpha)_{s_i -1}.$$ Moreover, conditioning on $S_m$, by Eq. (11) in [@lpw08], yields $$\label{uffa_2}
\mathbb{P} (K_m= k, S_1=s_1, \dots, s_{K_m}=s_k| K_n=j, S_m=s)=$$ $$=\frac{s!}{s_1! \cdots s_k! k!} \frac{V_{n+m, j+k}}{\sum_{i=0}^{s} V_{n+m, j+i} S_{s, i}^{-1, -\alpha}} \prod_{i=1}^k (1 -\alpha)_{s_i-1},$$ while conditioning on $K_m$, by Eq. (4) in [@lmp07], eliminates the dependency on the specific $(V_{n,k})$ Gibbs model as in $$\label{uffa_3}
\mathbb{P} (S_{1}= s_1, \dots, S_{K_m}=s_k| K_{m}=k, S_{m}=s, K_n=j)=$$ $$= \frac{s!}{s_1! \cdots s_k! k!} \frac {\prod_{i=1}^k (1-\alpha)_{s_i -1}}{S_{s,k}^{-1, -\alpha}}.$$
Further results for the conditional moments of any order of $K_m$ and for the conditional asymptotic distribution of a proper normalization of $K_m$ under $(\alpha, \theta)$ Poisson-Dirichlet partition models are in [@flp09]. A simplified approach to the posterior analysis of the two-parameter model exploiting the [*deletion of classes property*]{} and the Beta-Binomial distribution of $S_m|K_n=j$ is in [@cer11a]. A general result for [*conditional $\alpha$ diversity*]{} for Poisson-Kingman partition models driven by the stable subordinator ([@pit03]) has been obtained in [@cer11b].
Notice that equations (\[newblock\]), (\[uffa\_2\]), and (\[uffa\_3\]) fix corresponding formulas (9), (19) and (34) in [@lpw08] which are missing the combinatorial coefficients. The problem in Lijoi et al. (2008) seems to follow from some confusion between conditional Gibbs EPPFs, as arising from the multistep sequential construction of Proposition 4, and joint conditional distributions of the corresponding random vectors. We stress here that an EPPF provides the probability of a particular partition characterized by a certain allocation in a certain number of blocks with certain multiplicities. This differs from the probability of the random vector of the multiplicities to assume that specific value, which is obtained by summing over all different partitions providing the same multiplicities in the same number of blocks. The results in the following sections show that once the corrected formulas for the joint conditional distribution are properly identified, the derivation of estimators for quantities of interest in Bayesian nonparametric species sampling modeling simply follows by working with joint conditional marginals of (\[condmulti\]).
The first step is to define the conditional analog of (\[prior\]).
Under a general $(\alpha, V)$ Gibbs partition model, the multivariate distribution of the vector $(S_1, \dots, S_{K_m}, K_m)$, which arises by marginalizing (\[newblock\]) with respect to $S_m$, $$\label{condmulti}
\mathbb{P}_{\alpha, V} (S_1=s_1, \dots, S_{K_m}=s_k, K_m=k| {\bf n})=$$ $$= \frac{m!}{s_1! \cdots s_{k}! k! } \frac{V_{n+m, j+k}}{V_{n,j}} \sum_{s=k}^{m} \frac{(n-j\alpha)_{m-s}}{(m-s)!} \prod_{i=1}^k (1 -\alpha)_{s_i -1}$$ for $(s_1, \dots, s_{k}): \sum_j{s_j} \in [k, m]$, $k \in [1, m]$, is termed [*conditional multivariate Gibbs distribution of parameters $(m, \alpha, j, n)$*]{}, for $m \geq 1$, $\alpha \in (-\infty, 1)$ and $j \leq n$.
In the next Proposition, mimicking the technique adopted in the previous section for the unconditional case, we derive marginals of (\[condmulti\]) as the tools to obtain joint conditional falling factorial moments of the [*conditional Gibbs sampling formula*]{}. For $W_{l,m}= \sum_{i=1}^{K_m} 1 \{S_i=l\}$ this is given by the usual change of variable in (\[condmulti\]), hence $$\label{condsampl}
\mathbb{P}(W_{1,m}=w_1, \dots, W_{m,m}=w_m|n_1, \dots, n_j)=$$ $$=\frac{m!V_{n+m, j+k}}{V_{n,j}} \sum_{s=k}^{m} \frac{(n-j\alpha)_{m-s}}{(m-s)!} \prod_{i=1}^s
\frac{[(1-\alpha)_{i-1}]^{w_i}}{(i!)^{w_i} w_i!}.$$ In what follows we will resort to the convolution relation which defines [*non-central*]{} generalized Stirling numbers in terms of [*central*]{} generalized Stirling numbers $$\label{convo_1}
S_{n,k}^{-1, -\alpha, \gamma}= \sum_{s=k}^{n} {n \choose s} S_{s,k}^{-1, -\alpha} (-\gamma)_{n-s},$$ see the Appendix (cfr. (\[convo\])) for further details.
\[marg\_new\_cond\] Under a general $(\alpha, V)$ Gibbs partition model the r-dimensional marginal of (\[condmulti\]), for $(s_1, \dots, s_r): \sum_i s_i \leq s \leq m$ and $0 \leq k-r \leq m -\sum_{i=1}^r s_i$, is given by $$\label{margnew}
\mathbb{P} (S_1=s_1, \dots, S_r=s_r, K_m= k| {\bf n} )=$$ $$= \frac{m! [\prod_{i=1}^r (1-\alpha)_{s_i-1}]}{\prod_{i=1}^r s_i! (m-\sum_{i=1}^r s_i)!} \frac{(k-r)!}{ k!} \frac{V_{n+m, j+k}}{V_{n,j}} S_{m -\sum_{i=1}^r s_i, k- r}^{-1, -\alpha, -(n-j\alpha)}.$$
Multiplying and dividing (\[condmulti\]) by $(s-\sum_{i=1}^r s_i)!$ and $(m-\sum_{i=1}^r s_i)!$ and marginalizing yields $$\mathbb{P} (S_1=s_1, \dots, S_r=s_r, K_m= k| {\bf n} )=\frac{m! [\prod_{i=1}^r (1-\alpha)_{s_i-1}]}{\prod_{i=1}^r s_i! (m-\sum_{i=1}^r s_i)!} \frac{ 1}{ k! } \frac{V_{n+m, j+k}}{V_{n,j}} \times$$ $$\times \sum_{s-\sum_{i=1}^r s_i= k-r}^{m-\sum_{i=1}^r s_i} \frac{(m-\sum_{i=1}^r s_i)! (n-j\alpha)_{m-s}}{(s-\sum_{i=1}^r s_i)! (m-s)!} \sum_{(b_1, \dots, b_{k-r})} \frac{ (s-\sum_{i=1}^r s_i)!}{\prod_i b_i!} \prod_{i=1}^{k-r} (1 -\alpha)_{b_i -1}=$$ further multiplying and dividing by $(k -r)!$ we obtain $$=\frac{m! [\prod_{i=1}^r (1-\alpha)_{s_i-1}]}{\prod_{i=1}^r s_i! (m-\sum_{i=1}^r s_i)!} \frac{(k-r)!}{ k!} \frac{V_{n+m, j+k}}{V_{n,j}} \times$$ $$\times \sum_{s-\sum_{i=1}^r s_i= k-r}^{m-\sum_{i=1}^r s_i} {m-\sum_{i=1}^r s_i \choose s- \sum_{i=1}^r s_i} (n -j\alpha)_{m-s} S_{s-\sum_{i=1}^r s_i, k-r}^{-1, -\alpha},$$ and the result follows by (\[convo\_1\]).
The following Proposition generalizes Theorem 2. in [@flp12a]. We adopt the notation $W_{l,m}^{(n)}$ to indicate components of (\[condsampl\])
\[joint\_cond2\] Under a general $(\alpha, V)$ Gibbs partition model, joint falling factorial moments of order $(r_1, \dots, r_m)$ of the conditional sampling formula (\[condsampl\]) arise by an application of (\[margnew\]) in (\[kalea\]). For $m- \sum_l lr_l \geq 0$ $$\label{jointnew}
\mathbb{E} \left[ \prod_{l=1}^m (W_{l,m}^{(n)})_{(r_l)}\right]=$$ $$= \frac{m! \prod_{l=1}^m \left[(1-\alpha)_{l-1}\right]^{r_l}}{(m- \sum_{l} l r_l)! \prod_l (l!)^{r_l}} \frac{1}{V_{n,j}} \sum_{k - \sum_l r_l =0}^{m- \sum_l l r_l} V_{n+m, j+k} S_{m -\sum_l l r_l, k- \sum_l r_l}^{-1, -\alpha, -(n-j\alpha)}.$$ For $r_l=r \leq \lceil{\frac ml \rceil}$ and $r_j=0$ for $j\neq l$ then $$\label{mmnew}
\mathbb{E}[(W_{l,m}^{(n)})_{[r]}]=\frac{m!}{(m-rl)!} \frac{[(1-\alpha)_{l-1}]^r}{(l!)^r} \frac{1}{V_{n,j}} \sum_{k-r=0}^{m-rl} V_{n+m, j+k} S_{m-rl, k-r}^{-1, -\alpha, -(n- j\alpha)},$$ which agrees with the result in Theorem 2. in [@flp12a] expressed in terms of non central generalized factorial numbers (cfr. (\[coeffsti\]) in the Appendix).
By the analogy between (\[margnew\]) and (\[rmarg\]) the proof moves along the same lines as the proof of Proposition \[joint\_sampl\], exploiting the marginals obtained in Proposition \[marg\_new\_cond\].
Notice the great computational advantage provided by the technique based on marginals of multivariate Gibbs distributions devised in the previous section with respect to the complexity of the approach adopted in [@flp12a]. (\[jointnew\]) immediately follows as the conditional analog of the result obtained in Proposition \[joint\_sampl\] without any need to provide a new proof.
In [@flp12a], (cfr. Propositions 6 and 9), explicit marginals of (\[condsampl\]) have been derived for the Dirichlet $(\theta)$, the $(\alpha, \theta)$ Poisson-Dirichlet and the Gnedin-Fisher $(\gamma)$ ([@gne10]) partition models, . In the next Proposition we obtain the general result for the entire Gibbs family thus providing the conditional analog of Proposition \[dist\_marg\_uncon\].
Under a general $(\alpha, V)$ Gibbs partition model the marginal distribution of (\[condsampl\]) for $x=0, \dots, \lceil m/l \rceil$ corresponds to $$\label{margcondw}
\mathbb{P}(W_{l,m}^{(n)}= x)= \frac{[(1-\alpha)_{l-1}]^x}{x! (l!)^x}\frac{m!}{V_{n,j}}\times$$ $$\times \sum_{r= 0}^{\lceil{\frac ml \rceil}-x}\frac{(-1)^r [(1-\alpha)_{l-1}]^r}{r! (l!)^r (m -rl-xl)!} \sum_{k-r-x=0}^{m-rl-xl} V_{n+m, j+k} S_{m-rl-xl, k-r-x}^{-1, -\alpha, -(n-j\alpha)}.$$ Its expected value, which provides the Bayesian nonparametric estimator under quadratic loss function, for the number of [*new*]{} species represented $l$ times, arises from (\[mmnew\]) for $r=1$ $$\label{mean_cond}
\mathbb{E}(W_{l,m}^{(n)})={m \choose l} \frac{(1-\alpha)_{l-1}}{V_{n,j}} \sum_{k-1=0}^{m-l} V_{n+m, j+k} S_{m-l, k-1}^{-1, -\alpha, -(n-j\alpha)}.$$ The conditional distribution of the number of new singleton species $W_{1,m}^{(n)}$ will be $$\mathbb{P}(W_{1,m}^{(n)}=x)=\frac{m!}{x!}\frac{1}{V_{n,j}} \sum_{r=0}^{m-x} \frac{(-1)^r}{r! (m-r-x)!} \sum_{k-r-x=0}^{m-r-x} V_{n+m, j+k} S_{m-r-x, k-r-x}^{-1, -\alpha, -(n-j\alpha)}$$ and a Bayesian estimator of the number of new singleton species follows from (\[mean\_cond\]) as $$\mathbb{E}(W_{1,m}^{(n)})= \frac{m}{V_{n,j}} \sum_{k-1=0}^{m-1} V_{n+m, j+k} S_{m-1,k-1}^{-1, -\alpha, -(n-j\alpha)}.$$
(\[margcondw\]) arises by an application of (\[momprob\]), (\[mean\_cond\]) follows from (\[mmnew\]) for $r=1$ and corresponds to Eq. (17) in [@flp12a] expressed in terms of generalized non central factorial numbers (cfr. (\[coeffsti\]) in the Appendix).
Multivariate Pólya-Gibbs distributions
======================================
In this Section we focus on the conditional random allocation of the additional $m$ integers in the $j$ [*old*]{} blocks. First we derive the conditional joint distribution of the random vector $(M_{1,m}, \dots, M_{j,m}, S_m)$ of the sizes of the $m-S_m$ observations falling in the $j$ [*old*]{} blocks and of the total number of [*new*]{} observations $S_m$ falling in [*new*]{} blocks. Then, similarly to the previous sections, we move attention to the corresponding vector of counts and its joint falling factorial moments. From (\[oldenew\]), marginalizing with respect to the partitions in new blocks, and multiplying for the combinatorial coefficient accounting for the number of allocations providing the same sizes of [*old*]{} blocks and the same number of total observations in [*new*]{} blocks, we obtain $$\label{jointold}
\mathbb{P}(M_{1,m}=m_1, \dots, M_{j,m}=m_j, S_m=s|{n_1, \dots, n_j})=$$ $$=\frac{m!}{\prod_{i=1}^j m_i! s!} \prod_{i=1}^j (n_i -\alpha)_{m_i}\sum_{k=0}^s \frac{V_{n+m, j+k}}{V_{n,j}} S_{s, k}^{-1, -\alpha},$$ for $m_{i} \geq 0$ for $i=1, \dots, j$ and $\sum_{i=1}^j m_{i}=m -S_m$.
Since the number of old blocks is fixed, (\[jointold\]) may be interpreted as a generalization of [*multivariate Pólya distributions*]{}. If $Q_{V_{n,k}}$ is the conditional law, given $(n_1, \dots, n_j)$, of the vector $(\tilde{P}_{1,n}, \dots, \tilde{P}_{j,n}, R_{j,n})$, for $\tilde{P}_{j,n}= \tilde{P}_j|n_1, \dots, n_j$ the conditional random relative abundance of the $j$-th species to appear, and $R_{j,n}=1 -\sum_{i=1}^j \tilde{P}_{i,n}$, then (\[jointold\]) turns out to be a $Q_{V}$-multinomial mixture that we term [*multivariate Pólya-Gibbs distribution*]{} of parameters $(n_1-\alpha, \dots, n_j- \alpha, V)$. Moreover $Q_{V}$ will be the limit law, for $m \rightarrow \infty$, of the random vector $$\frac{M_{1,m}^{(n)}}{m}, \dots, \frac{M_{j,m}^{(n)}}{m}, \frac{S_m}{m},$$ where $M_{i, m}^{(n)}$ stands for a component of . Notice that for the two-parameter Poisson-Dirichlet $(\alpha, \theta)$ model, by a result in [@pit96], (cfr. Sect. 3.7, Corollary 20), $$(\tilde{P}_{1,n}, \dots, \tilde{P}_{j,n}, R_{j,n})\sim Dir[n_1 -\alpha, \dots, n_j-\alpha, \theta +j\alpha],$$ and substituting $V_{n,k}= (\theta +\alpha)_{k-1 \uparrow \alpha} / (\theta +1)_{n-1}$ in (\[jointold\]) yields $$\label{jointoldPD}
\mathbb{P}_{\alpha, \theta}(M_{1,m}=m_1, \dots, M_{j,m}=m_j, S_m=s|{\bf n})= $$
$$
=\frac{m!}{\prod_{i=1}^j m_i! s!} \frac{\prod_{i=1}^j (n_i -\alpha)_{m_i} (\theta +j\alpha)_{s}}{(n +\theta)_{m}}$$ which is a proper [*multivariate Pólya distribution*]{} of parameters $(m, n_1-\alpha, \dots, n_j-\alpha, \theta+j\alpha)$.
Next Proposition provides the general marginal that we need to obtain joint falling factorial moments of the vector of counts corresponding to .
Under a general $(\alpha, V)$ Gibbs model, the conditional joint marginal distribution of the vector of the sizes $(M_{1,m} \dots, M_{r,m})$ of the additional [*new*]{} observations falling in the first $r$ [*old*]{} blocks corresponds to $$\label{margvecchi}
\mathbb{P}(M_{1,m}=m_1, \dots, M_{r,m}=m_r|n_1, \dots, n_j)=$$ $$=\frac{m! \prod_{i=1}^r (n_i -\alpha)_{m_i} }{\prod_{i=1}^r m_i! (m- \sum_{i=1}^r m_i)!} \sum_{k=0}^{m-\sum_{i=1}^r m_i} \frac{V_{n+m, j+k}}{V_{n,j}} S_{m- \sum_{i=1}^r m_i, k}^{-1, -\alpha, -(n-(j-r)\alpha -\sum_{i=1}^r n_i)}.$$
By (\[jointold\]), the jont marginal of the first $r$ blocks and $S_m$ is easily obtained as $$\mathbb{P}(M_{1,m}=m_1, \dots, M_{r,m}=m_r, S_m=s|{\bf n})=\frac{m!}{\prod_{i=1}^r m_i! (m -s-\sum_{i=1}^r m_i)! s!}\times$$ $$\times \prod_{i=1}^r (n_i -\alpha)_{m_i} (n- j\alpha - \sum_{i=1}^r n_i +r\alpha)_{m-s-\sum_{i=1}^r m_i} \sum_{k=0}^s \frac{V_{n+m, j+k}}{V_{n,j}} S_{s, k}^{-1, -\alpha},$$ marginalizing with respect to $S_m$, and multiplying and dividing by $(m -\sum_{i=1}^r m_i)!$ yields $$\mathbb{P}(M_{1,m}=m_1, \dots, M_{r,m}=m_r|n_1, \dots, n_j)
=\sum_{s=0}^{m -\sum_{i=1}^r m_i} {m - \sum_{i=1}^r m_i \choose s} \times$$ $$\times \frac{m! \prod_{i=1}^r (n_i -\alpha)_{m_i} }{\prod_{i=1}^r m_i! (m- \sum_{i=1}^r m_i)!} (n- j\alpha - \sum_{i=1}^r n_i +r\alpha)_{m-s-\sum_{i=1}^r m_i} \sum_{k=0}^s \frac{V_{n+m, j+k}}{V_{n,j}} S_{s, k}^{-1, -\alpha}=$$ which reduces to $$=\frac{m! \prod_{i=1}^r (n_i -\alpha)_{m_i} }{\prod_{i=1}^r m_i! (m- \sum_{i=1}^r m_i)!} \sum_{k=0}^{m-\sum_{i=1}^r m_i} \frac{V_{n+m, j+k}}{V_{n,j}}\sum_{s=k}^{m-\sum_{i=1}^{r} m_i} {m - \sum_{i=1}^r m_i \choose s} \times$$ $$\times(n- j\alpha - \sum_{i=1}^r n_i +r\alpha)_{m-s-\sum_{i=1}^r m_i}S_{s, k}^{-1, -\alpha},$$ and the result follows by an application of (\[convo\_1\]).
Now let $O_{l,m}^{(n)}=\sum_{i: n_i \leq l} 1\{n_i+M_{i,m}=l| n_1, \dots, n_j\}$, for $l=1, \dots, n+m$, be the number of [*old*]{} blocks of size $l$ after the allocation of the additional $m$-sample, then, to obtain the joint falling factorial moments of any order for the sampling formula of (\[jointold\]) we exploit the multivariate version of the result (\[momentr\]) recalled in the proof of Proposition 2, namely $$\label{johgen}
\mathbb{E}\left[(O_{l,m}^{(n)})_{[r]}\right]= r! \sum_{(\xi_1, \dots, \xi_r)} \mathbb{P} (M_{\xi_1}= l-n_1, \dots, M_{\xi_r}=l-n_r).$$ For $m_i=l-n_i$, (\[margvecchi\]) specializes as $$\label{oldmarg}
\mathbb{P}(M_{1,m}=l-n_1, \dots, M_{r,m}=l-n_r|n_1, \dots, n_j)=$$ $$=\frac{m! \prod_{i=1}^r (n_i -\alpha)_{l -n_i} }{\prod_{i=1}^r (l-n_i)! (m- rl +\sum_{i=1}^r n_i)!}\frac{1}{V_{n,j}} \times$$ $$\times \sum_{k=0}^{m-lr + \sum_{i=1}^r n_i} V_{n+m, j+k} S_{m- lr +\sum_{i=1}^r n_i, k}^{-1, -\alpha, -(n-(j-r)\alpha -\sum_{i=1}^r n_i)},$$ and the one-dimensional marginal of (\[oldmarg\]) corresponds to $$\label{onemargold}
\mathbb{P}(M_{i,m}=l-n_i|{\bf n})= {m \choose {l-n_i}}\frac{ (n_i-\alpha)_{l-n_i}}{V_{n,j}} \sum_{k=0}^{m-l+n_i} V_{n+m, j+k} S_{m-l +n_i,k}^{-1, -\alpha, -(n-j\alpha+\alpha -n_i)}.$$
The following result easily follows from (\[johgen\]) as the analog of Propositions \[joint\_sampl\] \[joint\_cond2\].
Under a general $(\alpha, V)$ Gibbs model, the joint falling factorial moments of the vector of the number of old blocks of different size $(O_{1,m}^{(n)}, \dots, O_{n+m,m}^{(n)})$, after the allocation of the additional $m$-sample, given the initial allocation $n_1, \dots, n_k$ is given by $$\mathbb{E}\left[ (\prod_{l=1}^{n+m}(O_{l,m}^{(n)})_{[r_l]})\right]=$$ $$=\prod_{l=1}^{n+m} r_l! \sum_{({\bf \Xi}_{r_1}, \dots, {\bf \Xi}_{r_{n+m}})} \frac{m! \prod_{l=1}^{n+m} \prod_{i=1}^{r_l} (n_{\xi_i}- \alpha)_{l-n_{\xi_i}}}{\prod_{l=1}^{n+m} \prod_{i=1}^{r_l} (l- n_{\xi_i})! (m -\sum_l lr_l + \sum_l \sum_{i=1}^{r_l} n_{\xi_i})!} \times$$ $$\times\sum_{k=0}^{m -\sum_l l r_l + \sum_l \sum_{i=1}^{r_l} n_{\xi_i}} \frac{V_{n+m, j+k}}{V_{n,j}} S_{m -\sum_l l r_l+ \sum_l \sum_{i=1}^{r_l} n_{\xi_i}, k}^{-1, -\alpha, -(n -(j -\sum_l r_l)\alpha - \sum_l \sum_i n_{\xi_i} )},$$ for ${\bf \Xi}_{r_1}=(\xi_1, \dots, \xi_{r_1}), \dots, {\bf \Xi}_{r_{n+m}}= (\xi_{\sum_l r_l - r_{n+m}}, \dots, \xi_{\sum_{l=1}^{n+m} r_l}$), $\xi_i: n_{\xi_i} \leq l$, and each ${\bf \Xi}_{r_l}$ ranges over all the combinations of $r_l$ elements of $j$. For $r_l=r$ and $r_j=0$ for $j\neq l$, then $$\label{momoldcond}
\mathbb{E}\left[(O_{l,m}^{(n)})_{[r]}\right]= r! \sum_{(\xi_1, \dots, \xi_r)}\frac{m! \prod_{i=1}^r (n_{\xi_i} -\alpha)_{l -n_{\xi_i}} }{\prod_{i=1}^r (l-n_{\xi_i})! (m- rl+ \sum_{i=1}^r n_{\xi_i})!} \times$$ $$\times\sum_{k=0}^{m-lr +\sum_{i=1}^r n_{\xi_i}} \frac{V_{n+m, j+k}}{V_{n,j}} S_{m- lr +\sum_{i=1}^r n_{\xi_i}, k}^{-1, -\alpha, -(n-(j-r)\alpha -\sum_{i=1}^r n_{\xi_i})}$$ for $\xi_i:n_{\xi_i} \leq l$, which agrees with the result in Theorem 1. in [@flp12a].
Next Proposition generalizes the results in Proposition 5 (two-parameter Poisson-Dirichlet case) and Proposition 9 (one parameter Gnedin-Fisher case [@gne10]) in [@flp12a] to the entire $(\alpha, V)$ Gibbs family.
Under a general $(\alpha, V)$ Gibbs model, from (\[momoldcond\]) and (\[momprob\]), the conditional marginal law of $O_{l,m}^{(n)}$ is given by $$\mathbb{P}(O_{l,m}^{(n)}=y)= \sum_{r=0}^{\lceil{\frac{m -rl +\sum_{i=1}^{r+y} n_{\xi_i}}{l} \rceil} -y} \frac{(-1)^r (r+y)!}{y! r!} \frac{1}{V_{n,j}} \times$$ $$\times \sum_{(\xi_1, \dots, \xi_{r+y})} \frac{m!}{\prod_{i=1}^{r+y} (l-n_{\xi_i})! (m -rl-yl+ \sum_{i=1}^{r+y} n_{\xi_i})! }{\prod_{i=1}^{r+y} (n_{\xi_i}- \alpha)_{l-n_{\xi_i}}}\times$$ $$\times \sum_{k=0}^{m- rl-ly+\sum_{i=1}^{r+y} n_{\xi_i}} V_{n+m, j+k} S_{m-lr-ly+\sum_{i=1}^{r+y} n_{\xi_i}, k}^{-1, -\alpha, -(n-(j-r-y)\alpha- \sum_{i=1}^{r+y} n_{\xi_i})}.$$ Its expected value, which plays the role of the Bayesian nonparametric estimator, under quadratic loss function, of the number of old species represented $l$ times, follows from (\[onemargold\]) as $$\label{old_marg}
\mathbb{E}(O_{l,m}^{(n)})=
\mathbb{E}\left({\sum_{i:n_i \leq l} 1(n_i+M_{i,m}=l|n_1, \dots, n_j)}\right)=$$ $$= \sum_{i: n_i \leq l} \mathbb{E}(1(M_{i,m}=l-n_i|n_1, \dots, n_j))= \sum_{i:n_i \leq l} \mathbb{P}(M_{i,m}=l-n_i|n_1, \dots, n_j)=$$ $$=\sum_{i:n_i \leq l} {m \choose {l-n_i}} \frac{(n_i-\alpha)_{l-n_i}}{V_{n,j}} \sum_{k=0}^{m-l+n_i} V_{n+m,j+k} S_{m-l+n_i, k}^{-1, -\alpha, -(n -j\alpha+\alpha-n_i)},$$ or from for $r=1$ and agrees with Eq. (15) in [@flp12a].
Relying on the technique presented in this paper, falling factorial $r$-th moments of $Z_{l,m}^{(n)}$, the total number of [*old*]{} and [*new*]{} blocks of size $l$ after the allocation of the additional $m$-sample, as derived in Th. 3 in [@flp12a] by means of a very complex procedure, may be obtained in a straightforward way by the full conditional joint distribution $$\small{\mathbb{P}(S_1=s_1, \dots, S_k=s_k, S_{m}=s, K_m=k, { M}_{1, m}={m}_{1}, \dots, M_{j,m}=m_{j}| {\bf n})=}$$ $$=\frac{m!}{\prod_{i=1}^k s_i! k! \prod_{i=1}^j m_i! s!} \frac{V_{n+m, j+k}}{V_{n,j}} \prod_{i=1}^k (1 -\alpha)_{s_i-1}\prod_{i=1}^j (n_i-\alpha)_{m_i}.$$ Multiplying for the way to choose $t$ blocks among the [*old*]{} and $r-t$ among the [*new*]{} for every $t$, from (\[mmnew\]) and (\[momoldcond\]) we get $$\mathbb{E}\left[(Z_{l,m}^{(n)})_{[r]}\right]=$$ $$=\sum_{t=0}^r {r \choose t} t! \sum_{(\xi_{i_1}, \dots, \xi_{i_t})}\frac{m! [(1 -\alpha)_{l-1}]^{r-t} \prod_{i=1}^t (n_{\xi_i}-\alpha)_{l-n_{\xi_i}} }{\prod_{i=1}^t (l-n_{\xi_i})! (l!)^{r-t} (m - tl +\sum_{i=1}^t n_{\xi_i} - (r-t)l)!} \times$$ $$\times \sum_{k-r+t=0}^{m - rl +\sum n_{\xi_i}} \frac{V_{n+m, k+j}}{V_{n,j}} S_{m - rl +\sum n_{\xi_i}, k - r+t}^{-1, -\alpha, -(n -j\alpha -\sum n_{\xi_i} + t\alpha)}$$ which agrees with Theorem 3. in [@flp12a].
In the next section we provide one more example of the importance of working with marginals of conditional multivariate Gibbs distributions in the implementation of the Bayesian nonparametric approach to species sampling problems under Gibbs priors.
Bayesian nonparametric estimation of the probability to observe a species of a certain size
===========================================================================================
In species sampling problems, particularly in ecology or genomics, given a basic sample $(n_1, \dots, n_j)$, interest may be in estimating the [*probability*]{} to observe at step $n+m+1$ a species already represented $l$ times both belonging to an [*old*]{} species or to a [*new*]{} species eventually arising in the $m$-additional sample which is still not observed. This is the topic of a recent paper by Favaro [*et al.*]{} (2012b) and can be seen as a generalization of the problem of estimating the [*discovery probability*]{}, i.e. the probability to discover a [*new*]{} species, not represented in the previous $n+m$ observations. A Bayesian nonparametric estimator of the discovery probability under general $(\alpha, V)$ Gibbs partition models has been first derived in [@lmp07].
In this Section we show that working with marginals of conditional Gibbs multivariate distributions greatly simplifies the derivation of the results obtained in [@flp12b], thus providing another example of the importance of the technique proposed in this paper.\
First recall that by sequential construction of exchangeable partitions, the probability to observe an [*old*]{} species observed $l$ times in the basic $n$-sample at observation $n+1$, easily follows by one-step prediction rules for general Gibbs EPPFs (see e.g. [@pit06]). For $c_{l,n}= \sum_{i=1}^j 1\{n_i=l\}$, for $l=1, \dots, n$ then $$p_{l,n}(n_1, \dots, n_j)=c_{l,n}\frac{p(n_1, \dots, l+1, \dots, n_j)}{p(n_1, \dots, l , \dots, n_j)}=c_{l,n} \frac{V_{n+1, j}}{V_{n,j}} (l -\alpha).$$
Given a basic sample $(n_1, \dots, n_j)$, but assuming as in [@flp12b] an intermediate $m$-sample still to be observed, the probability to observe a species represented $l$ times among [*new*]{} species at observation $n+m+1$ will be a random variable, namely $$\label{newelle}
P^{n+m+1}_{new,l}(\alpha, V)=\frac{V_{n+m+1, j+K_m}}{V_{n+m, j +K_m}} (l- \alpha)W_{l, m}^{(n)},$$ for $K_m$ the random number of [*new*]{} species induced by the additional sample and $W_{l,m}^{(n)}$ the random number of species represented $l$ times in the additional sample given the basic sample.
In the following Proposition we show how the Bayesian nonparametric estimator, under quadratic loss function, of (\[newelle\]), (cfr. Theorem 2. in [@flp12b]), may be obtained in few elegant steps.
Under a general $(\alpha, V)$ Gibbs partition model, for $W_{l,m}^{(n)}= \sum_{i=1}^{K_m}1 \{S_i=l|K_n=j\}$ the Bayesian nonparametric estimator of $P^{m+n+1}_{new,l}(\alpha, V)$ is given by $$\label{newdisco}
\mathbb{E}_{(S_1,\dots, S_{K_m}, K_m| K_n=j)} \left( \frac{V_{n+m+1, j+K_m}}{V_{n+m, j +K_m}} (l- \alpha)W_{l, m}^{(n)}\right)=
$$
$$
=(l-\alpha) \sum_{k-1=0}^{m-l} \frac{V_{n+m+1, j+k}}{V_{n,j}} {m \choose l} (1 -\alpha)_{l-1} S_{m-l, k-1}^{-1, -\alpha, -(n-j\alpha)}.$$
Let $f(K_m)= \frac{V_{n+m+1, j+K_m}}{V_{n+m, j+K_m}}$ then, by definition of $W_{l,m}^{(n)}$, $$\mathbb{E}_{(S_1,\dots, S_{K_m}, K_m| K_n=j)} \left( \frac{V_{n+m+1, j+K_m}}{V_{n+m, j +K_m}} (l- \alpha)W_{l, m}^{(n)}\right)=$$ $$= (l- \alpha) \sum_{k=1}^ {m-l+1} \mathbb{E}_{(S_1,\dots, S_{K_m}| K_m=k, K_n=j)} \left(f(k) \sum_{i=1}^{k} 1 \{S_{i}=l|K_n=j\}\right) \times$$ $$\times\mathbb{P}(K_m=k| K_n=j)=$$ $$\label{quasi}
= (l- \alpha) \sum_{k-1=0}^ {m-l} f(k) k \mathbb{P} (S_i=l| K_m=k, K_n=j) \mathbb{P}(K_m=k|K_n=j).$$ Now, specializing (\[margnew\]), $$\mathbb{P}(S_1=l, \dots, S_r=l, K_m=k|K_n=j)=$$ $$= \frac{m!}{m- rl!} \frac{(k-r)!}{(l!)^r} \frac{[(1-\alpha)_{l-1}]^r}{k!} \frac{V_{n+m, j+k}}{V_{n,j}} S_{m-rl, k-r}^{-1, -\alpha, -(n-j\alpha)}$$ and inserting the marginal for $r=1$ in (\[quasi\]), the result follows.
By analogous approach we provide a straightforward derivation for the Bayesian nonparametric estimator for the probability to observe a species represented $l$ times among the [*old*]{} species, namely $$P_{old, l}^{m+n+1}(\alpha, V)=\frac{V_{n+m+1, j+K_m}}{V_{n+m, j+K_m}}(l-\alpha) O_{l,m}^{(n)}.$$
Under a general $(\alpha, V)$ Gibbs partition model, for $O_{l,m}^{(n)}= \sum_{i=1}^{j} 1\{n_i+M_{i,m}=l|n_1, \dots, n_j\}$ then a Bayesian nonparametric estimator under quadratic loss function of $P_{old, l}^{m+n+1}(\alpha, V_{n,j})$ is given by $$\label{discoold}
\mathbb{E}_{(M_{1,m}, \dots, M_{j,m}, K_m|n_1, \dots, n_j)} \left( \frac{V_{n+m+1, j+K_m}}{V_{n,m, j+K_m}}(l-\alpha) O_{l,m}^{(n)}\right)=$$ $$=(l-\alpha) \sum_{\xi=1}^{l} m_\xi {m \choose {l-\xi}} (\xi-\alpha)_{l-\xi} \sum_{k=0}^{m-l+\xi} \frac{V_{n+m+1, j+k}}{V_{n,j}} S_{m-l+\xi, k}^{-1, -\alpha, -(n-j\alpha + \xi -\alpha)}$$
Let $f(K_m)=\frac{V_{n+m+1, j+K_m}}{V_{n,m, j+K_m}}$, then $$\mathbb{E}_{(M_{1,m}, \dots, M_{j,m}, K_m|n_1, \dots, n_j)} \left( \frac{V_{n+m+1, j+K_m}}{V_{n,m, j+K_m}}(l-\alpha) O_{l,m}^{(n)}\right)=$$ $$= (l-\alpha) \sum_{k=0}^{m}f(k) \times$$ $$\times \mathbb{E}_{(M_{1,m}, \dots, M_{j,m}| K_m=k, n_1,\dots, n_j)} \left(\sum_{i=1}^{j} 1 \left\{n_i+M_{i,m}=l | K_m=k, n_1, \dots, n_j\right\} \right)\times$$ $$\times \mathbb{P}(K_m=k|K_n=j),$$ which is equal to $$=(l-\alpha) \sum_{k=0}^m f(k) \sum_{i:n_i \leq l} \mathbb{P} (M_{i,m}= l-n_i, K_m=k |n_1,\dots, n_j)$$ and by (\[onemargold\]) $$=(l-\alpha) \sum_{i:n_i \leq l} {m \choose {l-n_i}} (n_i-\alpha)_{l-n_i} \sum_{k=0}^{m-l+n_i} \frac{V_{n+m+1, j+k}}{V_{n,j}} S_{m-l+n_i, k}^{-1, -\alpha, -(n-j\alpha -n_i +\alpha)}$$ and the result follows.
This Appendix contains some basic facts on rising and falling factorial numbers, partitions and compositions of the natural integers, together with known results and definitions of generalized [*central*]{} and [*non central*]{} Stirling numbers that are exploited in the proofs and derivations all over the paper. The main reference is [@pit06]. Additionally, to facilitate the reading of the results contained in [@lmp07; @lpw08; @flp12a] and [@flp12b], the relationship between central and non central generalized [*factorial*]{} coefficients and generalized [*Stirling*]{} numbers is reported.\
Generalized rising factorials
-----------------------------
For $n=0,1,2,\dots,$ and arbitrary real $x$ and $h$, $(x)_{n\uparrow h}$ denotes the $n$th factorial power of $x$ with increment $h$ (also called generalized [*rising*]{} factorial) $$\label{factdef}
(x)_{n \uparrow h}:= x(x+h)\cdots(x+(n-1)h)=\prod_{i=0}^{n-1}(x+ih)=h^n(x/h)_{n},$$ where $(x)_{n}$ stands for $(x)_{n\uparrow 1}$, and $(x)_{h\uparrow 0}=x^h$, for which the following multiplicative law holds $$\label{multiplicative}
(x)_{n+r \uparrow h}=(x)_{n\uparrow h} (x +n h)_{r \uparrow h}.$$ From e.g. [@nor04] (cfr. eq. 2.41 and 2.45) a binomial formula also holds, namely $$\label{bino}
(x+y)_{n \uparrow h}=\sum_{k=0}^n {n \choose k} (x)_{k \uparrow h} (y)_{n-k \uparrow h},$$ as well as a generalized version of the multinomial theorem, i.e. $$\label{multi}
(\sum_{j=1}^p z_j)_{n \uparrow h}= \sum_{n_j \geq 0, \sum n_j=n} \frac{n!}{n_1!\cdots n_p!} \prod_{j=1}^p (z_j)_{n_j \uparrow h}.$$ For $m_j>0$, for every $j$, and $\sum_j m_j=m$, an application of (\[multiplicative\]) yields $$\label{miaaa}
(z_j)_{n_j +m_j -1}=(z_j)_{m_j-1} (z_j +m_j -1)_{n_j}$$ and by (\[multi\]) $$\sum_{n_j \geq 0, \sum n_j=n} \frac{n!}{n_1!\cdots n_p!} \prod_{j=1}^p (z_j)_{n_j +m_j -1}=\prod_{j=1}^p (z_j)_{m_j -1 }(\sum_{j=1}^p (z_j +m_j -1))_{n}=$$ $$=\prod_{j=1}^p (z_j)_{m_j -1}(m +\sum_{j=1}^p z_j -p)_{n},$$ which makes unnecessary the proof of Lemma 1 in [@lpw08].
Partitions and compositions
---------------------------
A [*partition*]{} of the finite set $[n]=(1,\dots, n)$ into $k$ blocks is an [*unordered*]{} collection of non-empty disjoint sets $\{A_1,\dots, A_k\}$ whose union is $[n]$, where the blocks $A_i$ are assumed to be listed in order of appearance, i.e. in the order of their least elements. The sequence $(|A_1|,\dots, |A_k|)$ of the sizes of blocks, $(n_1, \dots, n_k)$, defines a [*composition*]{} of $n$, i.e. a sequence of positive integers with sum $n$ and $\mathcal{P}_{[n]}^k$ denotes the space of all partitions of $[n]$ with $k$ blocks. From [@pit06] (cfr. eq. (1.9)) the number of ways to partition $[n]$ into $k$ blocks and assign each block a $W$ combinatorial structure such that the number of $W$-structures on a set of $j$ elements is $w_j$, in terms of sum over [*compositions*]{} of $n$ into $k$ parts is given by $$\label{bell}
B_{n,k}(w_\bullet)=\frac{n!}{k!} \sum_{(n_1,\dots, n_k)} \prod_{i=1}^k \frac{w_{n_i}}{n_i!},$$ where $B_{n, k}(w_\bullet)$ is a polynomial in variables $w_1, \dots, w_{n-k+1}$ known as the $(n,k)$th [*partial Bell polynomial*]{}.
Generalized Stirling numbers
----------------------------
(For a comprehensive treatment see [@hsushi98], see also [@pit06] Ex. 1.2.7). For arbitrary distinct reals $\eta$ and $\beta$, these are the connection coefficients $S_{n,k}^{\eta, \beta}$ defined by $$\label{connection}
(x)_{n \downarrow \eta}= \sum_{k=0}^n S_{n,k}^{\eta, \beta} (x)_{k \downarrow \beta}$$ and correspond to $$S_{n,k}^{\eta, \beta}=B_{n,k}((\beta -\eta)_{\bullet -1 \downarrow \eta}),$$ where $(x)_{n \downarrow h}$ are generalized [*falling*]{} factorials and $(x)_{n \downarrow -h}=(x)_{n \uparrow h}$, while $(x)_{n\downarrow 1}=(x)_{[n]}$. Hence for $\eta=-1$, $\beta=-\alpha$, and $\alpha \in (-\infty, 1)$, $S_{n,k} ^{-1, -\alpha}$ is defined by $$\label{unoalpha}
(x)_{n}=\sum_{k=0}^{n} S_{n,k} ^{-1, -\alpha} (x)_{k \uparrow \alpha},$$ and for $w_{n_i}=(1 -\alpha)_{n_i-1}$ and $\alpha \in [0,1)$, equation (\[bell\]) yields $$\label{bellalpha}
B_{n,k}((1-\alpha)_{\bullet-1})=\sum_{\{A_1,\dots, A_k\}\in \mathcal{P}_{[n]}^k }\prod_{i=1}^k (1-\alpha)_{n_i-1}=
$$
$$=\frac{n!}{k!}\sum_{(n_1,\dots,n_k)}\prod_{i=1}^k \frac{(1-\alpha)_{n_i-1}}{n_i!}=S_{n,k}^{-1,-\alpha}.$$ In [@lmp07; @lpw08; @flp12a; @flp12b] the treatment is in term of [*generalized factorial coefficients*]{}, which are the connection coefficients $\mathcal{C}^\alpha_{n,k}$ defined by $$\label{chara}
(\alpha y)_{n}=\sum_{k=0}^{n} \mathcal{C}^\alpha_{n,k}(y)_{k},$$ (cfr. [@cha05]). From (\[factdef\]) and (\[unoalpha\]), if $x=y \alpha$ then $$(y \alpha)_{n}= \sum_{k=0}^{n} S_{n, k}^{-1, -\alpha}(y \alpha)_{k \uparrow \alpha}=\sum_{k=0}^n S_{n,k}^{-1, -\alpha} \alpha^k
(y)_{k},$$ hence $$\label{coeff}
S_{n,k}^{-1, -\alpha}=\frac{\mathcal{C}_{n,k}^\alpha}{\alpha^k}.$$ The representation (37) in [@lpw08], ([@tos39]), also holds for generalized Stirling numbers with the obvious changes (cfr. e.g. [@pit06], Eq. 3.19). Additionally, specializing formula (16) in [@hsushi98], the following convolution relation holds, which defines [*non-central*]{} generalized Stirling numbers $$\label{convo}
S_{n,k}^{-1, -\alpha, \gamma}= \sum_{s=k}^{n} {n \choose s} S_{s,k}^{-1, -\alpha} (-\gamma)_{n-s},$$ and by (\[coeff\]), $$\label{coeffsti}
\mathcal{C}_{n,k}^{\alpha, \gamma}=\alpha^k S_{n,k}^{-1, -\alpha, \gamma}= \sum_{s=k}^{n} {n \choose s} \mathcal{C}_{s,k}^{\alpha} (-\gamma)_{n-s}.$$ Hence the following variation of equation (38) in [@lpw08] defines [*non-central*]{} generalized Stirling numbers as the connection coefficients $S_{n, k}^{-1, -\alpha, \gamma}$ such that $$\label{noncentralsti}
(y \alpha-\gamma)_{n}= \sum_{k=0}^{n} S_{n, k}^{-1, -\alpha, \gamma} \alpha^k
(y)_{k}=\sum_{k=0}^{n} S_{n, k}^{-1, -\alpha, \gamma}
(y\alpha)_{k \uparrow \alpha} .$$
Factorial moments and discrete distributions
--------------------------------------------
(Cfr. e.g. [@johkot05]). Falling factorial moments of a discrete r.v. $X$ provide the distribution function by the relationship $$\label{momprob}
\mathbb{P}(X=x)= \sum_{r \geq 0} \frac{(-1)^r}{x! r!} \mathbb{E}[(X)_{[x+r]}]$$ and standard moments by the definition as connection coefficients of the Stirling numbers of the second kind $S_{r,j}^{0,1}$, (cfr. Eq. (\[connection\])) $$\label{momfact}
\mathbb{E}[(X)^r] = \sum_{j=0}^r S_{r,j}^{0,1} \mathbb{E}[(X)_{[r]}].$$
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---
abstract: 'In this paper, we consider multiple cache-enabled clients connected to multiple servers through an intermediate network. We design several topology-aware coding strategies for such networks. Based on topology richness of the intermediate network, and types of coding operations at internal nodes, we define three classes of networks, namely, dedicated, flexible, and linear networks. For each class, we propose an achievable coding scheme, analyze its coding delay, and also, compare it with an information theoretic lower bound. For flexible networks, we show that our scheme is order-optimal in terms of coding delay and, interestingly, the optimal memory-delay curve is achieved in certain regimes. In general, our results suggest that, in case of networks with multiple servers, type of network topology can be exploited to reduce service delay.'
author:
- |
Seyed Pooya Shariatpanahi$^1$ , Seyed Abolfazl Motahari$^{2,1}$, Babak Hossein Khalaj$^{3,1}$\
1: School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.\
2: Department of Computer Engineering, Sharif University of Technology, Tehran, Iran.\
3: Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran.\
(emails: [email protected], {motahari,khalaj}@sharif.edu)\
title: 'Multi-Server Coded Caching'
---
Introduction {#Sec_Intro}
============
Unprecedented growth in transmit data volumes throughout the networks in recent years demands more efficient use of storage devices while providing high quality of service (QoS) to the users. Currently, large files are stored on servers and users’ requests are stored in queues waiting to get service from them. Naturally, one approach to reduce congestion in such networks is to increase the service rate of such servers. However, this will put additional burden on such nodes. As the cost of storage devices has decreased over the years, another viable option is to provide geographical content replication in the network through use of the so-called low-capacity *caching* nodes. The idea of using such nodes for data replication and providing easier local access to data is already covered in the literature (see for example [@Kangasharju; @Baev_2008; @Dowdy_2012; @Podlipnig; @Borst_2010; @Gitzenis_2013]).
Recently, in their seminal work, Maddah-Ali and Niesen considered a single server network and have shown that through a two-phase cache placement and content delivery strategy, server load can be reduced inversely proportional to the total size of cache introduced in the network. In fact, in the cache placement phase, contents are stored on caches without knowing the actual demands of the users and in content delivery phase the server transmits packets to fulfill the demands. The fact that such *global caching gain* can be achieved in such network is surprising as the demands are not known apriori at the cache placement phase.
Maddah-Ali and Niesen’s cache placement strategy is based on shattering each file into many pieces and only distributing them throughout the caching nodes *without* replication. It should be noted that such approach is in contrast to the conventional local cache placement strategies where a file or a single piece of it is replicated in caches. The astounding feature of their strategy is that transmission of a single packet at content delivery phase can then simultaneously serve several users. Imagine that two pieces of two files are stored at two different caches and each of them requires the piece available at the other. A single packet containing the sum of two packets can be sent to fulfill both users’ demands. They have shown that their strategy is 12-approximation of the optimal strategy.
The network considered in [@Maddah-Ali_Fundamental_2014] is a simple broadcast network where a packet transmitted by the server arrives unaltered at all users. A fundamental problem is to see how network topology affects the optimal coding strategy through both placement and delivery phases.
One of the simplest topologies is the tree network. In [@Maddah-Ali_Decentralized_2014], Maddah-Ali and Niesen proved that their original strategy can be used directly for such a network and what is needed to achieve 12-approximation of the optimal strategy is a simple topology-aware routing strategy at the internal nodes; An internal node routes a packet on its output port if the packet is useful for at least one of the port’s children.
While the topology-aware routing scheme for tree networks is shown to be an order-optimal solution, real-world topologies are much more sophisticated than the simple tree structure. In this paper, we characterize the effect of network topologies on code design and performance analysis of coded caching in a more general setup. In particular, we investigate a multi-server network topology where a set of servers are connected to the clients through an error-free and delay-free intermediate network of nodes (see Fig. \[Fig\_Multi\_Server\_Model\]). We assume that each node in the intermediate network can perform any causal processing on its input data, to generate its outgoing data. This can consist of simple routing or more sophisticated network coding schemes.
The objective considered in [@Maddah-Ali_Fundamental_2014] is minimizing the traffic load imposed to the single server. However, in general, other objectives may be of higher importance when designing network operation strategy. One such key criterion is the *service delay* of the network which is specially critical in content delivery networks (see eg. [@Chen_2002; @Vakali_2003; @Niesen_Delay_2014]). We define the service delay of the network as the total time required to serve any given set of the clients’ requests for a specific strategy. We distinguish between two types of delay, *Network Delay* $T_N$ and *Coding Delay* $T_C$, where the total service delay, $T$, is given by $T=T_N+T_C$. To be more precise, $T_N$ is the time it takes for packets to be routed through the network and arrive at their requesting nodes. Naturally, $T_N$ mainly captures the links and queues delays in the network which are intrinsic characteristics of the network. On the other hand, $T_C$ captures the transmission block length required to serve all the users for a specific coding strategy. In this paper, we focus on the coding delay and design strategies to minimize such delays.
We consider three classes of networks: 1- dedicated networks, 2- flexible networks, and 3- linear networks. These networks are characterized based on the richness of their internal connections, as shown in Fig. \[Fig\_Model\_Example\]. In each class, an important network topology aspect is the number of servers connected to the network, and their points of contact. In dedicated networks, we can dedicate each server to serve a fixed subset of clients, where each server can send a common message to its corresponding subset, interference-free from other servers. Although in dedicated networks the assignment of clients to the servers is fixed, in flexible networks the network topology is rich-enough to let us adapt these assignments during network operation. Finally, in linear networks we assume random linear network coding operations at the internal nodes. Consequently, in linear networks, the network input-output relation is characterized by a random matrix. As we show in this paper, in order to minimize the coding delay, designing the coding strategy for each class should carefully utilize the flexibility of that class. As will be shown subsequently, there exist coding strategies outperforming that of [@Maddah-Ali_Decentralized_2014] for all of the three classes of networks. Interestingly, we obtain an order optimal solution for the flexible networks.
Finally, let us review some notations used in this paper. We use lower case bold-face symbols to represent vectors, and upper case bold-face symbols to represent matrices. For any matrix $\mathbf{A}$, $\mathbf{A}^t$ denotes the transpose of $\mathbf{A}$ and for any vector $\mathbf{a}$, $\mathbf{a}^{\perp}$ shows that the condition $\mathbf{a}.\mathbf{a}^{\perp}=0$ is satisfied. For any two sets $S_1$ and $S_2$, the set $S_1 \backslash S_2$ consists of those elements of $S_1$ not present in $S_2$. Also we define $[K]=\{1,\dots,K\}$ and $\mathbb{N}$ to be the set of integer numbers. Moreover, $\mathbb{F}_q$ shows a finite field with $q$ elements, and $\mathbb{F}_q^{a \times b}$ denotes the set of all $a$-by-$b$ matrices whose elements belong to $\mathbb{F}_q$. Finally, let $x_1,\dots,x_m \in \mathbb{F}_q$, then $L(x_1,\dots,x_m)$ is a random linear combination of $x_1,\dots,x_m$ where the random coefficients are uniformly chosen from $\mathbb{F}_q$.
The rest of the paper is organized as follows. In Section \[Sec\_Model\], we describe the network model and different classes of networks. In Section \[Sec\_Main\_Results\], we review the main results of the paper, present some examples, and discuss their implications. The next two sections, i.e. Sections \[Sec\_Flexible\] and \[Sec\_Linear\], present the details of the coding strategies proposed for flexible and linear networks, respectively. Finally, we conclude the paper in Section \[Sec\_Conclusions\].
Model and Assumptions {#Sec_Model}
=====================
![Network Model. \[Fig\_Multi\_Server\_Model\]](Multi_Server_Model_General.eps){width="40.00000%"}
Consider $L$ servers connected to $K$ users through a network. By network we mean a *Directed Acyclic Graph* (DAG) $\mathcal{G}=(V,E)$, in which the set of vertices $V$ consists of internal nodes, and every edge $e\in E$ on the graph represents an *error-free* and *delay-free* link with capacity of one symbol per channel use. Each server and each user is connected to the network by a single link with capacity of one symbol per channel use. At each channel use each node inside the network sends symbols on its output links based on (deterministic/random) functions of the symbols on its input links, without introducing any delay, where functions corresponding to different output ports need not be the same. Also, we assume that there is no inter-link interference. Data is represented by $m$-bit *symbol*s which are members of a finite field $\mathbb{F}_{q}$, where $q=2^m$.
Consider a library of $N$ files $\{W_1,\dots,W_N\}$ each of $F$ bits is available to all servers. Each user is also assumed to have a cache of size $MF$ bits. During its operation, the network experiences two different traffic conditions, namely *low-peak* and *high-peak* leading to different network transmission costs for the two conditions. Based on the given traffic condition, the network operates in two distinctive phases. The first phase that is performed during low-peak condition is called the *cache content placement* phase at which servers send data to the users without knowing the actual requests of the users. This data is cached at the users with the size constraint of $MF$ bits and is stored to be used in the future. In the second phase that is performed during high-peak network condition, each user requests one of the files (demand $d_k$ of user $k$ denotes requesting file $W_{d_k}$), and according to these requests the servers send proper packets over the network. Subsequently, upon receipt of packets over the network, users try to decode their requested files with the help of their own cache contents. Assuming that the cache placement transmission delay during the low-peak condition puts no constraint on overall network performance, the goal is to design the cache placement strategy such that the service delay at the time of *content delivery* is minimized.
Channel uses in the network are indexed by time slots $t=1,2,\dots$. At time slot $t$, servers transmit symbols $s_1(t),\dots,s_L(t)$ and users receive symbols $r_1(t), \dots, r_K(t)$ without delay. We consider the most general case, i.e., $$\begin{aligned}
\nonumber r_k(t)=f_k(s_1(t),\dots,s_L(t)), k=1,\dots,K,\end{aligned}$$ in which we have assumed that the network is memory-less across time slots. Functions $f_k(.)$ depend on the topology of the network and the local operations of the nodes inside the network.
We define: $$\begin{aligned}
\nonumber
\mathbf{s}(t) \triangleq \left( \begin{array}{c} s_1(t) \\ \vdots \\ s_L(t) \end{array} \right),
\mathbf{r}(t) \triangleq \left( \begin{array}{c} r_1(t) \\ \vdots \\ r_K(t) \end{array} \right), \end{aligned}$$ where $\mathbf{s} \in \mathbb{F}_{q}^{L \times 1}$, and $\mathbf{r} \in \mathbb{F}_{q}^{K \times 1}$.
In the first phase, users store data from the servers without knowing the actual requests. The only concern in the first phase is respecting the memory constraint of each user. However, in the second phase, we focus on the time needed to deliver the requested files to the users. The second phase consists of $T_C$ time slots (channel uses). In other words, during the second phase, servers sequentially transmit $\mathbf{s}(1), \mathbf{s}(2), \dots, \mathbf{s}(T_C)$, and the users receive $\mathbf{r}(1), \mathbf{r}(2), \dots, \mathbf{r}(T_C)$. Consequently, $T_C(d_1,\dots,d_K)$ is the number of times slots required to satisfy demands $d_1,\dots,d_k$. Then, we define the optimum *Coding Delay* as: $$\begin{aligned}
D^*=\min {\max_{d_1,\dots,d_k}{T_C(d_1,\dots,d_K)}}, \end{aligned}$$ where the minimization is over all strategies. In this paper, we are interested in characterizing $D^*$ for a network, given its specific topology.
For a given network topology, the network input-output relation depends on operational design of internal nodes. As we will show, the *richer* the network topology is, the broader the design space will be. Therefore, we consider the following three classes of network topologies:
![Examples for dedicated, flexible, and linear networks.\[Fig\_Model\_Example\]](Model_Example.eps){width="100.00000%"}
- **Dedicated Networks**
In this class of networks, each packet transmitted by a server is routed to a fixed subset of the users. In other words, we can dedicate each server to a fixed subset of users, and this server can send packets to these users, concurrently and without interference to other servers. We assume these subsets to be non-overlapping so that each user is assigned to a single server. Also, we assume we can balance these assignments such that the number of users assigned to a server is almost the same for all servers. If network topology allows us to perform such assignments, we call the network a *Dedicated Network*.
More formally, there exists a coding (in this case just routing would suffice) strategy at the network nodes such that there *exists a partitioning* $\{P_1,\dots,P_L\}$ of $[K']=\{1,2, \dots,K'\}$ where $$\begin{aligned}
\nonumber & &|P_l|=\frac{K'}{L}, l=1,\dots,L
\\ & &\forall k=1,\dots,K, \hspace{2mm} \mathrm{if} \hspace{2mm} k \in P_l, \hspace{2mm} \mathrm{then} \hspace{2mm} f_k(s_1,\dots,s_L)=s_l,\end{aligned}$$ in which $K'$ is the smallest number larger than or equal to $K$ which is divisible by $L$.
Consider Fig. \[Fig\_Model\_Example\]-(a) in which $L=2$ servers are connected to $K=4$ users via a dedicated network. In this example, we have $K'=K$, and it is easy to verify that we can find a routing strategy at intermediate nodes such that: $$\begin{aligned}
\nonumber
&&P_1=\{1, 2\}, P_2=\{3, 4\} \\ \nonumber
&&f_1(s_1,s_2)=f_2(s_1,s_2)=s_1, \\ \nonumber
&&f_3(s_1,s_2)=f_4(s_1,s_2)=s_2.\end{aligned}$$
- **Flexible Networks**
In this class of networks, we assume that there exists a coding (routing) strategy at network nodes such that for *every partitioning* $\{P_1,\dots,P_L\}$ of $[K]=\{1,2, \dots,K\}$ we have: $$\begin{aligned}
\forall k=1,\dots,K, \hspace{2mm} \mathrm{if} \hspace{2mm} k \in P_l, \hspace{2mm} \mathrm{then} \hspace{2mm} f_k(s_1,\dots,s_L)=s_l.\end{aligned}$$
It should be noted that in the dedicated networks, each server was assigned to a *fixed* subset of users, while in flexible networks we can flexibly change these assignments during the data delivery phase. In the example shown in Fig. \[Fig\_Model\_Example\]-(b), we have chosen two sample partitionings, i.e. $P_1=\{1,4\}, P_2=\{2,3\}$ for the top figure, and $P_1=\{2,4\}, P_2=\{1,3\}$ for the bottom figure. It is obvious that every flexible network is a dedicated network, but the converse is not true. Hence, flexible networks are generally *richer* than dedicated networks in terms of their internal connectivity.
- **Linear Networks**
In the aforementioned dedicated and flexible networks, the intermediate nodes should know the topology of the network in order to do a proper routing of their input data onto their output ports. However, in the case of linear networks, we assume that such knowledge is not available at intermediate nodes. Thus, we assume that each node generates a random linear combination of data at its input ports to be transmitted on its output ports. Consequently, the overall transmit and receive vectors of the network are linearly related at each time slot: $$\mathbf{r}(t) = \mathbf{H} \mathbf{s}(t),$$ where $\mathbf{H} \in \mathbb{F}_{q}^{K \times L}$. $\mathbf{H}$ is called the *Network Transfer Matrix* (NTM). Let us define: $$\begin{aligned}
\nonumber \mathbf{X}&\triangleq&[\mathbf{s}(1), \mathbf{s}(2), \dots, \mathbf{s}(T_C)],
\\ \mathbf{Y}&\triangleq&[\mathbf{r}(1), \mathbf{r}(2), \dots, \mathbf{r}(T_C)].\end{aligned}$$ We call matrices $\mathbf{X} \in \mathbb{F}_{q}^{L \times T_C}$ and $\mathbf{Y} \in \mathbb{F}_{q}^{K \times T_C}$, transmit and receive blocks, respectively. Then, transmit and receive blocks are also linearly related: $$\mathbf{Y}=\mathbf{H}\mathbf{X}.$$
In *Linear Networks*, we assume that network topology is *rich-enough* to guarantee that the elements of $\mathbf{H}$ are i.i.d. random variables. Similar to most existing papers employing random linear network coding, we assume large-enough $q=2^m$ to assure that NTM exhibits full rank matrix properties, with high probability [@Yang_2011; @Li_2003]. Also, we assume uniform distribution on the elements of $\mathbf{H}$, which is a proper assumption for large scale networks with many sources of randomness [@Mahdi_2008; @Mahdi_2011; @Silva_2010; @Yang_2010].
Finally, for later reference, define $\mathbf{h}_k$ as $$\begin{aligned}
\label{Eq_Model_Linear_Channel_Vector_Def}
\mathbf{h}_k \triangleq [h_{k,1},\dots,h_{k,L}]^t, k=1,\dots,K.\end{aligned}$$
It should be noted that we assume a static network transfer matrix $\mathbf{H}$, such that it does not change for the duration of $T_C$ time slots. As changes in the network transfer matrix is due to topology changes (e.g. failure of a node), such assumption is valid in most practical scenarios. Fig. \[Fig\_Model\_Example\]-(c) illustrates an example of a linear network in the case of $L=2$ and $K=4$.
Finally, it should be noted that in this paper, we assume $N \geq K$. Such assumption will lead to more clear presentation in the rest of this paper and will also exclude the possibility of using uncoded multi-casting schemes that may trivially be adopted for the case of small number of files. Extending the results to the case $N <K$ is straightforward, and the readers are referred to [@Maddah-Ali_Fundamental_2014].
It should be noted that if a server is connected to the network by a number of links (each of integer capacity) with the total capacity of $t$ symbols per time slot, our model can accommodate this scenario by splitting this server into $t$ separate servers.
The random linear network coding approach at intermediate nodes is also used in other papers such as [@Das_2010], [@Meng_2012], and [@Meng_2013], in the context of uni-casting via interference alignment.
Main Results: Review and Discussion {#Sec_Main_Results}
===================================
![The Super-Server Strategy.\[Fig\_Main\_Result\_1\]](Main_Results_1.eps){width="40.00000%"}
The simplest approach in designing a coding scheme for the multi-server case is to directly transform it to a single-server scenario and use the scheme presented in [@Maddah-Ali_Fundamental_2014]. Such approach can be simply adopted by adding a *Super Server* node and connecting it with edges of infinite capacity to all other servers (see Fig. \[Fig\_Main\_Result\_1\]). As shown in [@Maddah-Ali_Decentralized_2014], we only need to route packets that are transmitted by the super-server to those users that can benefit from receiving them. For tree networks, such approach results in the following simple topology-aware routing scheme: at each interior node, the packets received at the input port benefiting at least one of the descendants of the node, is sent on the corresponding output port. As proved in [@Maddah-Ali_Decentralized_2014], the minimum traffic load imposed on each link, in the scaling sense, can be achieved by such simple routing scheme. Such approach also leads to an order-optimal coding delay for tree networks under our formulation.
One can, however, think of another naive and simple approach to the multi-server problem. We can simply dedicate each server to a subset of users and make it responsible for satisfying the requests of the corresponding subset of users. It is clear that, in order to prevent congestion at a specific server, we should balance out loads of the servers so that each of $L$ servers will be responsible for about $K/L$ users. Consequently, one can easily arrive at the following theorem for the coding delay in dedicated networks:
\[Thm\_dedicated\] The coding delay for a dedicated network is upper bounded by a piecewise-linear curve with corner points $$\label{Eq_Dedicated_Theorem}
D^*(M) \leq \frac{K' \left( 1-\frac{M}{N} \right)}{\min{\left(K',L+K'\frac{M}{N}\right)}} \frac{F}{m},$$ where $\frac{K'M}{LN} \in \mathbb{N}$ should be satisfied, and $K'$ is the smallest number larger than or equal to $K$ which is divisible by $L$.
The proof of Theorem \[Thm\_dedicated\] is straightforward, and thus, we just draw the main sketch here. First, let us review the main concept behind the coded caching scheme for a single server in a broadcast scenario [@Maddah-Ali_Fundamental_2014]. In this case, if we do not have any cache at the users, it is clear that the server should in sequence send all the requested files to the users (considering that the users request different files). This will lead to a total amount of $KF$ bits to be transmitted. Since the server is only able to transmit $m$ bits (a symbol in $\mathbb{F}_q$) at each time slot, the coding delay will be $K\frac{F}{m}$ time slots. By providing cache at the users, the *local caching gain* will reduce the coding delay to $K(1-\frac{M}{N})\frac{F}{m}$. The main result in [@Maddah-Ali_Fundamental_2014] indicates that by exploiting the additional *global caching gain*, the coding delay for $KM/N \in \mathbb{N}$ reduces to: $$\label{Eq_Single_Server_Coding_Delay}
T_C=\frac{K(1-M/N)}{1+KM/N}\frac{F}{m},$$ which is order optimal for this scenario.
As we extend to the multi-server case, let us assume for simplicity that $K$ is divisible by $L$. Splitting the original $L$-server problem with $K$ users into $L$ single-server problems with $\frac{K}{L}$ users is possible in this case. Since the sub-networks may operate in parallel, the delay is further reduced to: $$\begin{aligned}
\nonumber T_C&=&\frac{\frac{K}{L}\left(1-\frac{M}{N}\right)}{1+\frac{K}{L}\frac{M}{N}} \frac{F}{m}\\ \nonumber
&=& \frac{K(1-M/N)}{L+KM/N}\frac{F}{m},\end{aligned}$$ where $KM/LN \in \mathbb{N}$. Since in any scheme we can benefit at most all the $K$ users simultaneously, the total multi-casting gain of any scheme is at most $K$, and the denominator should be compared to $K$ (by the $\min$ operator in the denominator of (\[Eq\_Dedicated\_Theorem\])). Extension to the case where $K$ is not divisible by $L$ can be accomplished by adding virtual users. The following example compares the above two naive approaches:
\[Examp\_Intro\_1\]
Consider the network shown in Fig. \[Fig\_Intro\_Examp1\] for $K=4$ users. We also assume the library contains $N=4$ files, and each user can store $M=2$ files during the cache content placement phase. By adding a super server a tree network is obtained, and in the delivery phase, the scheme in [@Maddah-Ali_Decentralized_2014] suggests to send $$\begin{aligned}
\nonumber R_1&=&\frac{K(1-M/N)}{1+KM/N}F \\ \nonumber
&=&\frac{4\left(1-\frac{2}{4}\right)}{1+\frac{4\times 2}{4}} F \\ \nonumber
&=&\frac{2}{3} F,\end{aligned}$$ bits at the super server’s output. In their scheme, at each node only those packets benefiting the descendants of an output port will be copied on that port. However, in our case each packet benefits $1+\frac{KM}{N}=3$ users, and thus should be copied on both output ports of node $n_1$. This results in: $$\begin{aligned}
\nonumber R_2=R_1, \end{aligned}$$ and since we assumed a capacity of one symbol per time slot for each internal edge, the delay of this scheme is: $$\begin{aligned}
T_C=\frac{R_2}{m} = \frac{2}{3} \frac{F}{m}.\end{aligned}$$
At this stage, the key question is whether it is possible to further reduce the required number of time slots or not? In fact, with a closer look at this network it becomes evident that we can reduce this network to a dedicated network with: $$\begin{aligned}
\nonumber P_1&=&\{1,2\} \\ \nonumber
P_2&=&\{3,4\}.\end{aligned}$$ Therefore, the original problem can be divided into two sub-problems (see Fig. \[Fig\_Intro\_Examp1\]) and each server can address the load of its corresponding sub-network by: $$\begin{aligned}
\nonumber R_3&=&\frac{\frac{K}{L}(1-M/N)}{1+\frac{K}{L}\frac{M}{N}}F \\ \nonumber
&=& \frac{F}{2}.\end{aligned}$$ Since the sub-networks operate in parallel, the delay of this scheme will be $$\begin{aligned}
T_C=\frac{R_3}{m}=\frac{1}{2}\frac{F}{m}\end{aligned}$$ time slots.
![Example \[Examp\_Intro\_1\]. \[Fig\_Intro\_Examp1\]](Intro_Examp1.eps){width="80.00000%"}
The above example shows that although the scheme in [@Maddah-Ali_Decentralized_2014] is order-optimal for tree networks, however, by designing a topology-aware scheme it may be possible to arrive at a better pre-constant factor.
Next, let us consider another class of networks with more flexibility, i.e. Flexible Networks. In such networks, similar to dedicated networks, we can assign a subset of users to each server, and the network allows parallel operation of the servers. However, unlike dedicated networks, such assignment can be changed arbitrarily in subsequent transmissions. Such extra freedom in user assignments allows a significant reduction in the coding delay as shown in the following example.
\[Examp\_Flexible\_1\] For a single server case, the scheme proposed in [@Maddah-Ali_Fundamental_2014] achieves the following delay for $M=1$: $$\begin{aligned}
\nonumber T_C=\frac{K(1-M/N)}{1+KM/N}\frac{F}{m}=\frac{3}{2} \frac{F}{m}.\end{aligned}$$ In order to get a better insight on this result, consider Fig. \[Fig\_Flexible\_Exmp1\]-(a) which shows the cache content placement and the delivery scheme for requests $A,B,C,D$ by users $1,2,3,4$, respectively. In the cache content placement phase, each file is divided into four equal-sized parts and cached as shown in Fig. \[Fig\_Flexible\_Exmp1\]-(a). In the delivery phase, the single server sends the following data in sequence: $$\begin{aligned}
\nonumber A_2+B_1, A_3+C_1, A_4+D_1, B_3+C_2, B_4+D_2, C_4+D_3.\end{aligned}$$ As a result, six transmissions are required while each has the delay $\frac{1}{4}\frac{F}{m}$. Thus, the total delay will be $T_C=\frac{6}{4}\frac{F}{m}=\frac{3}{2}\frac{F}{m}$. In the above scheme, each transmission benefits a pair of users, and is of no value for the other pair.
If we have two servers, by the definition of flexible networks each server is able to transmit a given data to a pair of users simultaneously and interference-free from transmission of the other server. In Fig. \[Fig\_Flexible\_Exmp1\]-(b), transmissions of the left and right servers are colored as blue and red, respectively. Thus, a pair of transmissions in Fig. \[Fig\_Flexible\_Exmp1\]-(a) can be sent simultaneously as shown in Fig. \[Fig\_Flexible\_Exmp1\]-(b), resulting in the achievable pair $(M,T_C)=(1,\frac{3}{4}\frac{F}{m})$. Thus, exploiting the extra flexibility of the network in this example results in the coding delay enhancement, compared with the single-server case.
![Flexible Network Example \[Examp\_Flexible\_1\]. \[Fig\_Flexible\_Exmp1\]](Flexible_Example1.eps){width="80.00000%"}
In dedicated networks, we exploit the network topology to assign a fixed number of users to each server. In this way, a user receives packets only from a certain server and this assignment is fixed during the course of transmission. In flexible networks, however, at different time slots users can be served by different servers where the assignment strategy is fixed for each server. Fig. \[Fig\_Strategy\] shows two servers connected to three users through such flexible network. The blue packets originating from server 1 are intended for one user (which may change at different time slots) and the red packets originating from server 2 are intended for two users (which may change at different time slots). We assign blue packets to be associated with Strategy $1$ and red packets with Strategy $2$. Fig. \[Fig\_Strategy\] shows consequent transmissions in such network where Strategy 1 is associated with server 1 and Strategy 2 with server 2. In general, we associate Strategy $p$ to a packet if it is intended for $p$ users. Now, if we fix a strategy for a server, it means that all the packets transmitted by that server have the same strategy. It is worth mentioning that packets received by a user do not necessarily have the same strategy, since they may have arrived from different servers (see Fig. \[Fig\_Strategy\]).
![Server 1 and blue packets are associated with Strategy 1, and Server 2 and red packets are associated with Strategy 2. \[Fig\_Strategy\]](Strategy_Description.eps){width="80.00000%"}
Consider server $i$ with Strategy $p_i$. Also, we assign a fraction $F_i$ bits of each file to be delivered by Server $i$. In order to employ the scheme in [@Maddah-Ali_Fundamental_2014] for this server, we allocate a memory of size $\bar{M_i}$ bits from all the users to be used only by Server $i$ where $$\begin{aligned}
\nonumber \bar{M_i}=\frac{N}{K}\left(p_i-1\right)F_i.\end{aligned}$$ Therefore, Server $i$ can deliver $F_i$ bits to all the users in $T_C(i)$ time slots where $$\begin{aligned}
\nonumber T_C(i)&=&\frac{K\left(1-\frac{\bar{M_i}/F_i}{N}\right)}{1+\frac{K\bar{M_i}/F_i}{N}}\frac{F_i}{m} \\
&=&\frac{K-p_i+1}{p_i}\frac{F_i}{m}.\end{aligned}$$ We assume that a routing strategy exists where packets from different servers do not interfere with each other. In this case, the total delay is limited by the maximum delay of the servers. Therefore, in order to balance out the servers’ loads, we can simply set: $$\begin{aligned}
\nonumber F_i=\alpha \frac{p_i}{K-p_i+1}F,\end{aligned}$$ where $\alpha$ does not depend on $i$ and satisfies: $$\begin{aligned}
\nonumber \sum_{i=1}^{L}{F_i}=\alpha \sum_{i=1}^{L}{\frac{p_i}{K-p_i+1}}F=F.\end{aligned}$$ Therefore, $$\begin{aligned}
\alpha=1/\sum_{i=1}^{L}{\frac{p_i}{K-p_i+1}}.\end{aligned}$$ Since the total memory is $M$, we have $$\begin{aligned}
\nonumber M=\sum_{i=1}^{L} {\bar{M_i}} /F &=& \frac{N}{KF} \sum_{i=1}^{L}{\left(p_i-1\right)F_i} \\ \nonumber &=& \frac{N}{K} \sum_{i=1}^{L}{\left(p_i-1\right)\alpha \frac{p_i}{K-p_i+1}} \\
&=& \frac{N}{K} \frac{\sum_{i=1}^{L}{\frac{p_i(p_i-1)}{K-p_i+1}}}{\sum_{i=1}^{L}{\frac{p_i}{K-p_i+1}}}.\end{aligned}$$ Hence, $$\begin{aligned}
\nonumber T_C&=&\alpha \frac{F}{m} \\
&=& \frac{\frac{F}{m}}{\sum_{i=1}^{L}{\frac{p_i}{K-p_i+1}}}.\end{aligned}$$
The aforementioned result is based on a strong assumption that a routing strategy exists for parallel and interference-free transmission of the packets. In Section \[Sec\_Flexible\], we show that such a strategy does in fact exist for flexible networks. The preceding discussion is a rough proof of the following Theorem:
\[Thm\_flexible\] Suppose a flexible network with $L$ servers. Then, for all $Q \in \{0,\dots,K-L\}$ the following $(M,T_C)$ pairs (and the straight lines connecting them) are achievable $$\label{Eq_Main_Resuls_Flexile_Th}
(M,T_C)=\left\{\left(\frac{N}{K}\frac{\sum_1^L{\frac{p_i(p_i-1)}{K-p_i+1}}}{\sum_1^L{\frac{p_i}{K-p_i+1}}},\frac{1}{\sum_1^L{\frac{p_i}{K-p_i+1}}}\frac{F}{m}\right) \mathrm{,\hspace{1 mm} for \hspace{1 mm} all \hspace{5 mm}} p_1+\dots+p_{L}=K-Q \mathrm{,\hspace{1 mm} where \hspace{5 mm}} p_i \geq 2 \right\},$$ and thus lead to an upper bound for the optimum coding delay $D^*$.
See Section \[Sec\_Flexible\] for the proof.
In the following example, we present a network in which employing the flexible network strategy results will go beyond earlier results and paves the way for scaling improvement in the coding delay compared with the super-server strategy.
\[Examp\_Intro\_2\] Consider the network depicted in Fig. \[Fig\_Intro\_Examp2\]-(a). In this network, $L$ (an even number) servers are connected to $K=L^2/2$ users via $L$ intermediate nodes where each intermediate node has dedicated links to all the users. We also assume: $$\begin{aligned}
\nonumber \frac{M}{N}=\frac{2}{L^2}\left(\frac{L}{2}-1\right).\end{aligned}$$ In order to use the super-server strategy with the tree approach proposed in [@Maddah-Ali_Decentralized_2014], we need to choose an appropriate tree inside the network. It can be easily verified that the tree illustrated in Fig. \[Fig\_Intro\_Examp2\]-(b) is the best choice. Therefore, $R_1$, the minimum rate of the super-server, is given by $$\begin{aligned}
\nonumber R_1&=&\frac{K(1-M/N)}{1+KM/N} F \\ \nonumber
&=&\frac{\frac{L^2}{2}(1-M/N)}{\frac{L}{2}} F \\ \nonumber
&=& L(1-M/N) F.\end{aligned}$$ The load $R_2$ on each server consists of those packets that are useful for at least a user which is a descendant of that server. We know that each packet benefits a subset of users of size: $$\begin{aligned}
\nonumber 1+\frac{KM}{N}=\frac{L}{2}.\end{aligned}$$ Therefore, the ratio of packets routed on a specific edge to the total number of packets is: $$\begin{aligned}
\nonumber \frac{R_2}{R_1}&=& \frac{\sum_{i=1}^{L/2}{{L/2 \choose i}{{L^2/2-L/2 \choose L/2-i}}}}{{L^2/2 \choose L/2}} \\ \nonumber
&=&
1-\left(1-\frac{L/2}{L^2/2}\right) \left(1-\frac{L/2}{L^2/2-1}\right) \dots \left(1-\frac{L/2}{L^2/2-(L/2-1)}
\right) \\ \nonumber
&\geq& 1- \left(1-\frac{1}{L}\right)^{\frac{L}{2}} \\ \nonumber
&\sim&1-e^{-1/2},\end{aligned}$$ for large $L$. Thus, almost a constant number of packets generated by the server will be routed on each edge. This will result in a delay of:
$$\begin{aligned}
\label{Eq_MainResults_Examp3_Delay_Super}
\nonumber T_C&=& \frac{R_2}{m} \\
&\sim& \left(1-e^{-1/2}\right) L(1-\frac{M}{N}) \frac{F}{m}\end{aligned}$$
time slots.
A closer look at the network topology shows that the network is indeed flexible. Setting $p_i=L/2$ which satisfies $\sum{p_i}=K$ and using memory size $M$ where $$\begin{aligned}
\nonumber M&=&\frac{N}{K}\frac{\sum_1^L{\frac{p_i(p_i-1)}{K-p_i+1}}}{\sum_1^L{\frac{p_i}{K-p_i+1}}} \\ \nonumber
&=& \frac{N}{K} \left(\frac{L}{2}-1\right) \\
&=& \frac{N}{L^2/2} \left(\frac{L}{2}-1\right),\end{aligned}$$ Theorem \[Thm\_flexible\] can be used to achieve the following coding delay: $$\begin{aligned}
\label{Eq_MainResults_Examp3_Delay_Flex}
\nonumber T_C&=&\frac{1}{\sum_1^L{\frac{p_i}{K-p_i+1}}}\frac{F}{m} \\ \nonumber
&=& \frac{F/m}{L \frac{L/2}{L^2/2-L/2+1}} \\
&=&\left(1-\frac{M}{N}\right) \frac{F}{m}.\end{aligned}$$
The above delay in (\[Eq\_MainResults\_Examp3\_Delay\_Flex\]) is not only a scaling improvement compared with the super-server tree-based strategy with delay (\[Eq\_MainResults\_Examp3\_Delay\_Super\]), but also the optimal delay. This is due to the fact that each user can store at most $\frac{M}{N}F$ bits of each file.
![Example \[Examp\_Intro\_2\]. \[Fig\_Intro\_Examp2\]](Intro_Examp2.eps){width="80.00000%"}
The optimality of the preceding coding scheme can be generalized to any flexible network where $K$ is divisible by $L$ as the following theorem states.
\[Thm\_Order\_Optimal\] If $K$ is divisible by $L$, then the upper bound in Theorem \[Thm\_flexible\] is optimal within a multiplicative constant gap.
See Section \[Sec\_Flexible\] for the proof.
For flexible and topologically complex networks, finding a proper routing strategy that achieves the optimal coding delay may not be straightforward. To overcome this difficulty, internal nodes can perform simple random linear network coding which is oblivious to the network’s topology. Although this strategy may not be optimal, it has the advantage of being practical and robust. In this way, the network model reduces to a linear network model and the following theorem provides an achievable coding delay for such networks.
\[Thm\_linear\] The coding delay for a linear network with $L$ servers is upper bounded by a piecewise-linear curve with the corner points $$D^*(M) \leq \frac{K(1-M/N)}{\min(K,L+KM/N)}\frac{F}{m},$$ where $KM/N \in \mathbb{N}$ should be satisfied.
See Section \[Sec\_Linear\] for the proof.
In linear networks, a packet intended for a certain number of users, in general, interferes with all other users. Proper pre-coding schemes can be adopted to reduce interference in such networks. Consequently, simultaneous transmission of multiple packets will further reduce network coding delay. In order to clarify the implications of Theorem \[Thm\_linear\], we present the following example:
![Example \[Examp\_Linear\_2\]: $N=4, K=4$.\[Fig\_4\_4\_Curves\_Together\]](4_4_Curves_Together.eps){width="40.00000%"}
\[Examp\_Linear\_2\] Consider a network with $K=N=4$. Using Theorem \[Thm\_linear\], the coding delay for any $L \in \{1,2,3,4\}$ is given by $$T_C = \frac{4-M}{\min(4,L+M)}\frac{F}{m}.$$ The above delay is plotted in Fig. \[Fig\_4\_4\_Curves\_Together\] for $L \in \{1,2,3,4\}$. The problem for $L=1$ reduces to that of [@Maddah-Ali_Fundamental_2014]. For $L=4$, we obtain a multiplexing gain of $4$ by constructing four parallel interference-free links each from one server to one user (e.g. through Singular Value Decomposition) and the optimal coding delay is achieved. Networks with $L\in \{2,3\}$ are interesting cases where interference management is required to achieve the gain $\min(4,L+M)$ in the denominator. The detail of the coding strategy is rather involved and we delegate it to Appendices B and C.
Flexible Networks: Details {#Sec_Flexible}
==========================
In this section, we present an achievable scheme for the flexible networks leading to the result given in Theorem \[Thm\_flexible\]. We also provide a proof for the optimality result in Theorem \[Thm\_Order\_Optimal\] through cut-set analysis.
For the achievability part, we need to provide the cache content placement and content delivery strategies. Let us start with defining the following parameters: let $Q \in \{0,\dots,K-L\}$ and consider an integer solution of the following equation: $$\begin{aligned}
\nonumber p_1+\dots+p_L+p_{L+1}=K,\end{aligned}$$ where $p_{L+1}=Q$ and $p_i\geq 2, i=1,\dots,L$. We also define $$\begin{aligned}
\nonumber \alpha_i &\triangleq& {K \choose p_i-1}, i=1,\dots,L \\ \nonumber
\gamma_i &\triangleq& \frac{(K-p_i)!p_i!}{p_1!\dots p_{L+1}!}, i=1,\dots,L+1 \\ \nonumber
x &\triangleq& 1/\sum_1^L \alpha_i \gamma_i \\
x_i &\triangleq& \cases{
\gamma_i x &\quad $i=1,\dots,L$ \cr
0 &\quad $i=L+1$ \cr
}.\end{aligned}$$\
**Cache Placement Strategy:** First, split each file $W_n$ into $L$ sub-files $$\begin{aligned}
\nonumber W_n=\left(W_n^i : i=1,\dots,L\right),\end{aligned}$$ where $W_n^i$ is of size $\alpha_i x_i F$. Then, split each sub-file $W_n^i$ into $\alpha_i$ equal-sized mini-files: $$\begin{aligned}
\nonumber W_n^i=\left(W_{n,\tau_i}^i: \tau_i \subseteq [K], |\tau_i|=p_i-1 \right).\end{aligned}$$ Finally, split each mini-file $W_{n,\tau_i}^i$ into $\gamma_i$ equal-sized pico-files of size $xF$ bits: $$\begin{aligned}
\nonumber W_{n,\tau_i}^i=\left( W_{n,\tau_i}^{i,j}: j=1,\dots,\gamma_i \right),\end{aligned}$$ where $\gamma_i$ is an integer number. For each user $k$, we cache pico-file $W_{n,\tau_i}^{i,j}$ if $k \in \tau_i$, for all possible $i,j,n$. Then, the required memory size for each user is: $$\begin{aligned}
\nonumber M&=&\frac{1}{F} N \left( \sum_{i=1}^L{{K-1 \choose p_i-2} \gamma_i x F} \right) \\ \nonumber
&=& N \frac{\sum_{i=1}^L{ {K-1 \choose p_i-2} \gamma_i }}{\sum_{i=1}^L{ {K \choose p_i-1} \gamma_i }} \\
&=& \frac{N}{K} \frac{ \sum_{i=1}^L {\frac{p_i(p_i-1)}{K-p_i+1}} } { \sum_{i=1}^L {\frac{p_i}{K-p_i+1}} },\end{aligned}$$ which is consistent with the assumptions of Theorem \[Thm\_flexible\].\
\
**Content Delivery Strategy:** Define $P_1^i,\dots,P_{{K \choose p_i}}^i$ to be the collection of all $p_i$-subsets of $[K]$ for all $i=1,\dots L+1$. The delivery phase consists of $\frac{K!}{p_1!\dots p_{L+1}!}$ transmit slots. Each transmit slot is in one-to-one correspondence with one $(p_1,\dots,p_{L+1})$-partition of $[K]$. Consider the transmit slot associated with the partition $$\begin{aligned}
\nonumber \left\{ P^1_{\theta_1},\dots,P^{L+1}_{\theta_{L+1}} \right\},\end{aligned}$$ where $\theta_i \in \left\{1,\dots,{K \choose p_i}\right\}$. Then, the server $i$ sends $$\begin{aligned}
\nonumber {+}_{r \in P^i_{\theta_i}}W_{d_r,P^i_{\theta_i} \backslash \{r\}}^{i,N(P^i_{\theta_i})}\end{aligned}$$ to the subset of users $P^i_{\theta_i}$, interference-free from other servers, where the sum is in $\mathbb{F}_q$ and is over all $r \in P^i_{\theta_i}$. Since we have assumed a flexible network, simultaneous transmissions by all servers is feasible. Also, the index $N(P^i_{\theta_i})$ is chosen such that each new transmission consists of a fresh (not transmitted earlier) pico-file. Obviously, the virtual server $L+1$ does not transmit any packet.
Since each pico-file consists of $x\frac{F}{m}$ symbols, at each transmission slot we should send a block of size $L$-by-$x\frac{F}{m}$ by the servers. Also, since this action should be performed for all $\frac{K!}{p_1!\dots p_{L+1}!}$ slots, the delay of this scheme will be: $$\begin{aligned}
\nonumber T_c&=&\frac{K!}{p_1!\dots p_{L+1}!} \times x\frac{F}{m} \\
&=&\frac{1}{\sum_1^L{\frac{p_i}{K-p_i+1}}} \frac{F}{m},\end{aligned}$$ as stated in Theorem \[Thm\_flexible\]. Consequently, if we show that through the aforementioned number of transmit slots all users will be able to recover their requested files, the proof is complete.\
**Correctness Proof:** The main theme of this scheme is to divide each file into $L$ sub-files, and to assign each sub-file to a single server. Then, each server’s task is to deliver the assigned sub-files to the desired users (see Fig. \[Fig\_Flexible\_Proof\]).
Consider server $i$. This server handles sub-files $W_n^i, n \in [N]$ though the following delivery tasks: $$\begin{aligned}
\nonumber & & W_{d_1}^i \hspace{2mm}
\stackrel{\textrm{server i} }{\Longrightarrow} \hspace{2mm} \textrm{User 1} \\ \nonumber
& & W_{d_2}^i \hspace{2mm} \stackrel{\textrm{server i} }{\Longrightarrow} \hspace{2mm} \textrm{User 2} \\ \nonumber
& & \vdots \\ \nonumber
& & W_{d_K}^i \hspace{2mm} \stackrel{\textrm{server i} }{\Longrightarrow} \hspace{2mm} \textrm{User K} \end{aligned}$$ The above formulation leads to a single server problem [@Maddah-Ali_Fundamental_2014] with files of size $F_i= \alpha_i x_i F$ bits. It can be easily verified that the proposed cache placement strategy for each sub-file mimics that of [@Maddah-Ali_Fundamental_2014] for single-server problems. Therefore, if we demonstrate that this server is able to send a common message of size $x_i\frac{F}{m}$ symbols to all $p_i$-subsets of users, then this server can handle this single-server problem successfully. However, in the above scheduling scheme, the server benefits each $p_i$-subset of the users by a common message of size $x\frac{F}{m}$ symbols (a pico-file size), $\gamma_i$ times. Consequently, the total volume of common message that this server is able to send to each $p_i$-subset is $\gamma_i \cdot x\frac{F}{m}=x_i\frac{F}{m}$ symbols.
Since by proper scheduling scheme in flexible networks all servers can perform the same task simultaneously, all requested portions of files will be delivered. It should be noted that the portion of each file assigned to the virtual server is $x_{L+1}=0$. Algorithm 1 presents the pseudo-code of the procedure described above.
![Flexible network file distribution for proof of Theorem \[Thm\_flexible\]. \[Fig\_Flexible\_Proof\]](Felexible_Proof.eps){width="70.00000%"}
\[Alg\_Main\_Flexible\]
$\alpha_i \gets {K \choose p_i-1}$, $i=1,\dots,L$ $\gamma_i \gets ((K-p_i)!p_i!)/(p_1!\dots,p_{L+1}!)$, $i=1,\dots,L+1$ $x \gets 1/(\sum_1^L{\alpha_i \gamma_i})$ $x_i \gets \gamma_i x$, $i=1,\dots,L$ $x_{L+1} \gets 0$ split $W_n$ into $(W_n^i: i=1,\dots,L)$, where $|W_n^i|=\alpha_i x_i$ split $W_n^i$ into $(W_{n,\tau_i}^i: \tau_i \subset [K], |\tau_i|=p_i-1)$ of equal size split $W_{n,\tau_i}^i$ into $(W_{n,\tau_i}^{i,j}: j=1,\dots,\gamma_i)$ of equal size $Z_k \gets (W_{n,\tau_i}^{i,j}: \tau_i \subset [K], |\tau_i|=p_i-1, k \in \tau_i, j=1,\dots,\gamma_i, n \in [N])$\
$N({P}^i_j) \gets 1 $ $\{P^1_{\theta_1},\dots,P^{L+1}_{\theta_{L+1}}\} \gets $ selected partition **transmit** $ \mathbf{X}(\{P^1_{\theta_1},\dots,P^{L+1}_{\theta_{L+1}}\}) = \left[ {\begin{array}{c}
{+}_{r \in P^1_{\theta_1}}W_{d_r,P^1_{\theta_1} \backslash \{r\}}^{1,N(P^1_{\theta_1})} \Rightarrow P^1_{\theta_1} \\
\vdots \\
{+}_{r \in P^L_{\theta_L}} W_{d_r,P^L_{\theta_L} \backslash \{r\}}^{L,N(P^L_{\theta_L})} \Rightarrow P^L_{\theta_L} \\
\end{array} } \right] $
$N(P^i_{\theta_i}) \leftarrow N(P^i_{\theta_i}) +1$
To prove Theorem \[Thm\_Order\_Optimal\], we first state the following lemma:
\[Lem\_Converse\] The coding delay for a general network with $L$ servers is lower bounded by $$\label{Eq_Converse_Theorem}
D^*(M) \geq \max_{s \in \{1,\dots,K\}}\frac{1}{\min(s,L)}\left(s-\frac{s}{\lfloor \frac{N}{s} \rfloor}M\right) \frac{F}{m}.$$
See Appendix A for the proof.
The above lemma can be used to prove optimality of the proposed scheme in some range of parameters. The following corollary states the result.
\[Cor\_Converse\] All $(M-T_C)$ pairs in Theorem \[Thm\_flexible\] corresponding to $Q=0$ are optimal. Thus, the converse line $\left(1-\frac{M}{N}\right)\frac{F}{m}$ is achieved for $M^*\leq M \leq N$, where $$M^*=\min_{p_1+\dots+p_L=K}\frac{N}{K}\frac{\sum_1^L{\frac{p_i(p_i-1)}{K-p_i+1}}}{\sum_1^L{\frac{p_i}{K-p_i+1}}}.$$
Theorem \[Thm\_flexible\] states that all the $(M-T_C)$ pairs in (\[Eq\_Main\_Resuls\_Flexile\_Th\]) are achievable. By some simple calculations one can show that for these achievable pairs we have: $$\begin{aligned}
\label{Eq_Flexible_Converse_Cor_1}
\left(1-\frac{M}{N}\right)\frac{F}{m}=\left(1-\frac{Q}{K}\right) T_C.\end{aligned}$$ Therefore, if we put $Q=0$ in Theorem \[Thm\_flexible\], all the corresponding $(M-T_C)$ pairs satisfy $$\begin{aligned}
\nonumber T_C=\left(1-\frac{M}{N}\right)\frac{F}{m}.\end{aligned}$$ On the other hand, by considering the case of $s=1$ in Lemma \[Lem\_Converse\] we know that the optimal coding delay satisfies: $$\begin{aligned}
\nonumber D^*(M) \geq \left(1-\frac{M}{N}\right)\frac{F}{m},\end{aligned}$$ which is matched to our achievable coding delay . Therefore, by setting $Q=0$ in Theorem \[Thm\_flexible\], for all $$\begin{aligned}
\nonumber M=\frac{N}{K}\frac{\sum_1^L{\frac{p_i(p_i-1)}{K-p_i+1}}}{\sum_1^L{\frac{p_i}{K-p_i+1}}}, p_1+\dots,p_L=K, p_i \geq 2,\end{aligned}$$ the achievable coding delay is optimum. By minimizing the cache size, over all partitionings satisfying $p_1+\dots,p_L=K, p_i \geq 2$, the proof is complete.
There is an interesting intuition behind Eq. (\[Eq\_Flexible\_Converse\_Cor\_1\]). In the proposed scheme for flexible networks, we assigned a subset of $Q$ users to the virtual server, and all the other $K-Q$ users benefited from other servers. Thus, through each transmission, the ratio $\frac{K-Q}{K}$ of users will be real users. This is exactly the coefficient that shows how close is the achieved delay to the optimal curve $(1-M/N)F/m$.
Finally, we are ready to prove Theorem \[Thm\_Order\_Optimal\]. We consider two regimes for cache sizes. First , we let $$\begin{aligned}
\nonumber M^*=\frac{N}{K}\left(\frac{K}{L}-1\right).\end{aligned}$$ In the first regime where $M \geq M^*$, using Theorem \[Thm\_flexible\] with $Q=0$ and $p_1,\dots,p_L=\frac{K}{L}$, we obtain: $$\begin{aligned}
\nonumber T_C = \left(1-\frac{M}{N}\right)\frac{F}{m}.\end{aligned}$$ As Corollary \[Cor\_Converse\] states, for this case the optimal curve is achieved.
For the second regime where $M<M^*$ (such that $KM/N \in \mathbb{N}$), set $$\begin{aligned}
\nonumber & &Q=K-\left(\frac{KM}{N}+1\right)L \\ \nonumber
& & p_1,\dots,p_L=\frac{K-Q}{L} = \left(\frac{KM}{N}+1\right).\end{aligned}$$ Then, we obtain: $$\begin{aligned}
\nonumber T_C = \frac{1}{L}\frac{K(1-M/N)}{1+KM/N}.\end{aligned}$$
On the other hand, from Lemma \[Lem\_Converse\] we have:
$$\begin{aligned}
\nonumber D^* &\geq& \max_{s \in \{1,\dots,K\}}\frac{1}{\min(s,L)}\left(s-\frac{s}{\lfloor \frac{N}{s} \rfloor}M\right) \frac{F}{m} \\ \nonumber
&\geq& \max_{s \in \{1,\dots,K\}}\frac{1}{L}\left(s-\frac{s}{\lfloor \frac{N}{s} \rfloor}M\right) \frac{F}{m} \\ \nonumber
&\geq& \frac{1}{L}\frac{1}{12}\frac{K(1-M/N)}{1+KM/N}, \\
&\geq& \frac{1}{12} T_C,\end{aligned}$$
where the last inequality follows from [@Maddah-Ali_Fundamental_2014]. This concludes the proof of Theorem \[Thm\_Order\_Optimal\].
Linear Networks: Details {#Sec_Linear}
========================
In order to explain the main concepts behind the coding strategy proposed for linear networks, we will first present a simple example:
![Example \[Examp\_Linear\_1\] ($L=2, K=3, N=3$): Lower and upper bounds on the coding delay.\[Fig\_Multi\_Server\_3\_3\_2\_Example\]](Multi_Server_3_3_2_Example.eps){width="50.00000%"}
\[Examp\_Linear\_1\] In this example, we consider a network consisting of $L=2$ servers, $K=3$ users, and a library of $N=3$ files, namely $W_1=A$, $W_2=B$, and $W_3=C$. By definition of linear networks the input-output relation of this network is characterized by a $3$-by-$2$ random matrix $\mathbf{H}$. Lower and upper bounds for the coding delay of this setting are shown in Fig. \[Fig\_Multi\_Server\_3\_3\_2\_Example\]. The lower bound is due to Lemma \[Lem\_Converse\] as follows: $$D^* \geq \max\left(1-\frac{M}{3},\frac{3-3M}{2} \right)\frac{F}{m}.$$ The upper bound is due to Theorem \[Thm\_linear\] except the achievable pair $(M,T_C)=(\frac{1}{3},1)$, which will be discussed later. We have also exploited the fact that the straight line connecting every two achievable points on the $M-T_C$ curve is also achievable, as shown in [@Maddah-Ali_Fundamental_2014]. In order to get a glimpse of the ideas of the coding strategy behind Theorem \[Thm\_linear\], next we discuss the achievable $(M,T_C)$ pair $(1,\frac{2}{3})$. In this case, as we will show, we can benefit both from the local/global caching gain (provided by cache of each user), and the multiplexing gain (provided by multiple servers in the network). The question is how to design an scheme so that we can exploit both gains simultaneously. In what follows we provide the solution:
Suppose that (without loss of generality) in the second phase, the first, second, and third users request files $A$, $B$, and $C$ respectively. Assume that the cache content placement is similar to that of [@Maddah-Ali_Fundamental_2014]: First, divide each file into three equal-sized non-overlapping sub-files: $$\begin{aligned}
\nonumber A&=&[A_1,A_2,A_3] \\ \nonumber
B&=&[B_1,B_2,B_3] \\ \nonumber
C&=&[C_1,C_2,C_3].\end{aligned}$$ Then, put the following contents in the cache of users: $$\begin{aligned}
\nonumber Z_1&=&[A_1,B_1,C_1] \\ \nonumber
Z_2&=&[A_2,B_2,C_2] \\ \nonumber
Z_3&=&[A_3,B_3,C_3].\end{aligned}$$ Let $L(x_1,\dots,x_m)$ be a random linear combination of $x_1,\dots,x_m$ as defined earlier. Consequently, in this strategy, the two servers send the following transmit block: $$\label{Eq_L2_K3_N3_M1_transmit}
\mathbf{X}=[\mathbf{h}_1^{\perp}L_1^1(C_2,B_3)+\mathbf{h}_2^{\perp}L_2^1(A_3,C_1)+\mathbf{h}_3^{\perp}L_3^1(A_2,B_1), \mathbf{h}_1^{\perp}L_1^2(C_2,B_3)+\mathbf{h}_2^{\perp}L_2^2(A_3,C_1)+\mathbf{h}_3^{\perp}L_3^2(A_2,B_1) ].$$ where the random linear combination operator $L(\cdot,\cdot)$ operates on sub-files, in an element-wise manner, and $\mathbf{h}_i^{\perp}$ is an orthogonal vector to $\mathbf{h}_i$ (i.e. $\mathbf{h}_i.\mathbf{h}_i^{\perp}=0$). Let us focus on the first user who will receive: $$\begin{aligned}
\nonumber \mathbf{h}_1.\mathbf{X}&=&[(\mathbf{h}_2^{\perp} . \mathbf{h}_1) L_2^1(A_3,C_1)+(\mathbf{h}_3^{\perp} . \mathbf{h}_1)L_3^1(A_2,B_1), (\mathbf{h}_2^{\perp} . \mathbf{h}_1)L_2^2(A_3,C_1)+(\mathbf{h}_3^{\perp} . \mathbf{h}_1)L_3^2(A_2,B_1) ] \\
&=& [L^1(A_2,A_3,C_1,B_1),L^2(A_2,A_3,B_1,C_1)].\end{aligned}$$ As the first user has already cached $B_1$ and $C_1$ in the first phase, by subtracting the effect of interference terms, the first user can recover: $$\begin{aligned}
\nonumber [L(A_2,A_3),L'(A_2,A_3)],\end{aligned}$$ which consists of two independent (with high probability for large field size $q$) linear combinations of $A_2$ and $A_3$. By solving these independent linear equations, such user can decode $A_2$ and $A_3$, and with the help of $A_1$ cached at the first phase, he can recover the whole requested file $A$. It can easily be verified that other users can also decode their requested files in a similar fashion. The transmit block size indicated in (\[Eq\_L2\_K3\_N3\_M1\_transmit\]) is $2$-by-$\frac{2F}{3m}$, resulting in $T_C=\frac{2F}{3m}$ time slots.
Let us forget about one of the servers for a moment and assume we have just one server. Then, the scheme proposed in [@Maddah-Ali_Fundamental_2014] only benefits two users per transmission through pure global caching gain. Also, in the case of two servers and no cache memory (the aforementioned case of $M=0$), we could design an scheme which benefited only two users through pure multiplexing gain. However, through the proposed strategy, we have designed an scheme which exploited both the global caching and multiplexing gains such that all the three users could take advantage from each transmission.
Finally, let us discuss the achievable pair $(M,T_C)=(\frac{1}{3},1)$, where we need to adopt a different strategy. Assume we divide each of files $A$, $B$ and $C$ into three equal parts and fill the caches as follows: $$\begin{aligned}
\nonumber Z_1&=&[A_1+B_1+C_1] \\ \nonumber
Z_2&=&[A_2+B_2+C_2] \\ \nonumber
Z_3&=&[A_3+B_3+C_3].\end{aligned}$$ Consequently, the servers transmit the following vectors: $$\begin{aligned}
\nonumber \mathbf{X}_1&=&\frac{\mathbf{h}_3^{\perp}}{\mathbf{h}_1.\mathbf{h}_3^{\perp}}B_1 + \frac{\mathbf{h}_2^{\perp}}{\mathbf{h}_1.\mathbf{h}_2^{\perp}}C_1 \\ \nonumber
\mathbf{X}_2&=&\frac{\mathbf{h}_3^{\perp}}{\mathbf{h}_2.\mathbf{h}_3^{\perp}}A_2 + \frac{\mathbf{h}_1^{\perp}}{\mathbf{h}_2.\mathbf{h}_1^{\perp}}C_2 \\
\mathbf{X}_3&=&\frac{\mathbf{h}_2^{\perp}}{\mathbf{h}_3.\mathbf{h}_2^{\perp}}A_3 + \frac{\mathbf{h}_1^{\perp}}{\mathbf{h}_3.\mathbf{h}_1^{\perp}}B_3.\end{aligned}$$ It can be easily verified that the first user receives $A_2$, $A_3$, and $B_1+C_1$. So, with the help of its cache content, it can decode the whole file $A$. Similarly, the other users can decode their requested files. As each block $\mathbf{X}_i$ is a $2$-by-$\frac{F}{3m}$ matrix of symbols, the total delay required to fulfill the users’ demands is $T_C=\frac{F}{m}$ time slots.
Example \[Examp\_Linear\_2\], also, discusses the coding delay for a linear network with four users. The details of the coding strategy of Example \[Examp\_Linear\_2\], which are provided at Appendices B and C, further clarify the basic ideas behind the proposed scheme. However, in the rest of this section, we provide the formal proof of Theorem \[Thm\_linear\].\
\
**Cache Placement Strategy:** The cache content placement phase is identical to [@Maddah-Ali_Fundamental_2014]: Define $t\triangleq MK/N$, and divide each file into ${K \choose t}$ non-overlapping sub-files as[^1]: $$\begin{aligned}
\nonumber W_n=\left(W_{n,\tau}: \tau \subset [K], |\tau|=t\right), n=1,\dots,N,\end{aligned}$$ where each sub-file consists of $F/{K \choose t}$ bits. In the first phase, we store the sub-file $W_{n,\tau}$ in the cache of user $k$ if $k \in \tau$. Therefore, the total amount of cache each user needs for this placement is: $$\begin{aligned}
\nonumber N\frac{F}{{K \choose t}}{K-1 \choose t-1} = MF\end{aligned}$$ bits.
We further divide each sub-file into ${K-t-1 \choose L-1}$ non-overlapping equal-sized mini-files as follows: $$\begin{aligned}
\nonumber W_{n,\tau}=\left(W_{n,\tau}^j: j=1,\dots,{K-t-1 \choose L-1}\right).\end{aligned}$$ Thus, each mini-file consists of $F/\left({K \choose t} {K-t-1 \choose L-1}\right)$ bits.\
\
**Content Delivery Strategy:** Consider an arbitrary $(t+L)$-subset of users denoted by $S$ (i.e. $S \subseteq [K], |S|=t+L$). For this specific subset $S$ denote all $(t+1)$-subsets of $S$ by $T_i, i=1,\dots,{t+L \choose t+1}$ (i.e. $T_i \subseteq S, |T_i|=t+1$). First, we assign a $L$-by-$1$ vector $\mathbf{u}_{S}^{T_i}$ to each $T_i$ such that $$\begin{aligned}
\label{Eq_Vector_Constraints}
\nonumber \mathbf{u}_{S}^{T_i} &\perp& \mathbf{h}_j \hspace{5mm} \mathrm{for \hspace{2mm} all} \hspace{5mm} j\in S \backslash T_i \\
\mathbf{u}_{S}^{T_i} &\not \perp& \mathbf{h}_j \hspace{5mm} \mathrm{for \hspace{2mm} all} \hspace{5mm} j\in T_i.\end{aligned}$$ The following lemma specifies the required field size such that the aforementioned condition is met with high probability:
If the elements of the network transfer matrix $\mathbf{H}$ are uniformly and independently chosen from $\mathbb{F}_q$, then we can find vectors which satisfy (\[Eq\_Vector\_Constraints\]) with high probability if: $$q \gg (t+1) {K \choose t+L}{t+L \choose t+1}.$$
First, since the set $S \backslash T$ has $L-1$ elements, we require $\mathbf{u}_{S}^{T_i}$ to be orthogonal to $L-1$ arbitrary vectors, which is feasible in an $L$ dimensional space of any field size.
Second, the total number of non-orthogonality constraints in (\[Eq\_Vector\_Constraints\]) for all possible subsets $S$ is $(t+1) {K \choose t+L}{t+L \choose t+1}$. On the other hand, it can be easily verified that the probability that two uniformly chosen random vectors in $\mathbb{F}_q$ are orthogonal is $1/q$. Thus, by using the union bound, the probability that at least one non-orthogonality constraint in (\[Eq\_Vector\_Constraints\]) is violated is upper bounded by $$\begin{aligned}
\nonumber \frac{(t+1) {K \choose t+L}{t+L \choose t+1}}{q} \ll 1,\end{aligned}$$ which concludes the proof.
For each $T_i$ define: $$\label{Eq_Linear_Proof_G_T}
G(T_i)=L_{r \in T_i}\left(W_{d_r,T_i \backslash \{r\}}^j\right),$$ where $W_{d_r,T_i \backslash \{r\}}^j$ is a mini-file which is available in the cache of all users in $T_i$, except $r$, and is required by user $r$. Also $L_{r \in T_i}$ represents a random linear combination of the corresponding mini-files for all $r \in T_i$. Note that the index $j$ is chosen such that such mini-files have not been observed in the previous $(t+L)$-subsets. Thus, if we define $N(r,T \backslash \{r\})$ as the index of the next fresh mini-file required by user $r$, which is present in the cache of users $T \backslash \{r\}$, then we can rewrite: $$\label{Eq_Linear_General_GTi_1}
G(T_i)=L_{r \in T_i}\left(W_{d_r,T_i \backslash \{r\}}^{N(r,T_i \backslash \{r\})}\right),$$ Subsequently, we make the following definition for such $(t+L)$-subset $S$: $$\label{Eq_Linear_Proof_X_S}
\mathbf{X}(S)=\sum_{T \subseteq S, |T|=t+1}{\mathbf{u}_{S}^{T}G(T)}.$$ We repeat the above procedure ${t+L-1 \choose t}$ times for the given $(t+L)$-subset $S$ in order to derive different independent versions of $\mathbf{X}_\omega(S), \omega=1,\dots,{t+L-1 \choose t}$. In other words, $\mathbf{X}_\omega(S)$’s only differ in the random coefficients chosen for calculating the linear combinations in (\[Eq\_Linear\_General\_GTi\_1\]), which makes them independent linear combinations of the corresponding mini-files, with high probability. Thus, to distinguish between these different versions notationally we define: $$\label{Eq_Linear_General_GTi_2}
G_\omega(T_i)=L_{r \in T_i}^\omega\left(W_{d_r,T_i \backslash \{r\}}^{N(r,T_i \backslash \{r\})}\right), \mathbf{X}_\omega(S)=\sum_{T \subseteq S, |T|=t+1}{\mathbf{u}_{S}^{T}G_\omega(T)}.$$
Subsequently, for this $(t+L)$-subset $S$, the servers transmit the block $$\left[\mathbf{X}_1(S),\dots,\mathbf{X}_{{t+L-1 \choose t}}(S)\right],$$ and we update $N(r, T \backslash \{r\})$ for those mini-files which have appeared in the linear combinations in (\[Eq\_Linear\_General\_GTi\_1\]). When the above procedure for this specific subset $S$ is completed, we consider another $(t+L)$-subset of users and do the above procedure for that subset, and repeat this process until all $(t+L)$-subsets of $[K]$ have been taken into account.
Next, let us calculate the coding delay of this scheme, after which we prove the correctness of this content delivery strategy. For a fixed $(t+L)$-subset $S$ each $\mathbf{X}_\omega(S)$ is a $L$-by-$\frac{F/m}{{K \choose t} {K-t-1 \choose L-1}}$ block of symbols. Thus, the transmit block for $S$, i.e. $\left[\mathbf{X}_1(S),\dots,\mathbf{X}_{{t+L-1 \choose t}}(S)\right]$, is a $L$-by-$\frac{F/m}{{K \choose t} {K-t-1 \choose L-1}} {t+L-1 \choose t}$ block. Since this transmission should be repeated for all ${K \choose t+L}$ $(t+L)$-subsets of users, the whole transmit block size will be $$\begin{aligned}
\nonumber L \mathrm{-by-} \frac{ {t+L-1 \choose t}}{{K \choose t} {K-t-1 \choose L-1}} {K \choose t+L}\frac{F}{m}= L \mathrm{-by-} \frac{K(1-M/N)}{L+MK/N} \frac{F}{m},\end{aligned}$$ which will result in the coding delay of $$T_C=\frac{K(1-M/N)}{L+MK/N}\frac{F}{m}$$ time slots. Algorithm 2 shows the pseudo-code of the aforementioned procedure for linear networks.\
**Correctness Proof**: Suppose the user $k$, who is interested in acquiring the file $W_{d_k}$. This file is partitioned into two parts: 1- The part already cached in this user at the first phase and constitutes of sub-files: $$\begin{aligned}
\left(W_{d_k,\tau}: \tau \subseteq [K], |\tau|=t, k \in \tau\right).\end{aligned}$$ 2- Those parts which should be delivered to this user through the content delivery strategy, which constitutes of sub-files: $$\begin{aligned}
\left(W_{d_k,\tau}: \tau \subseteq [K], |\tau|=t, k \not \in \tau\right).\end{aligned}$$ Thus, since due to the following Lemma \[Lem\_Linear\_Proof\_2\], the sub-files in the second category are successfully delivered to this user through the content delivery strategy, this user will decode the requested file. Moreover, since this user was arbitrarily chosen, all users will similarly decode their requested files.
Before proving Lemma \[Lem\_Linear\_Proof\_2\] we need another lemma which is proved first:
\[Lem\_Linear\_Proof\_1\] Suppose an arbitrary subset $T \subseteq [K]$ such that $|T|=t+1$, and $k \in T$. Then, through the above content placement and delivery strategy, user $k$ will be able to decode the sub-file $W_{d_k, T \backslash \{k\}}$.
Consider those transmissions which are assigned to the $(t+L)$-subsets which contain $T$. There exist ${K-t-1 \choose L-1}$ of such subsets. Let us focus on one of them, namely $S$. Corresponding to $S$, the following transmit block is sent by the servers: $$\begin{aligned}
\left[\mathbf{X}_1(S),\dots,\mathbf{X}_{{t+L-1 \choose t}}(S)\right],\end{aligned}$$ and subsequently, user $k$ receives: $$\begin{aligned}
\label{Eq_Linear_Proof_Recieve_k_1}
\mathbf{h}_k .\left[\mathbf{X}_1(S),\dots,\mathbf{X}_{{t+L-1 \choose t}}(S)\right].\end{aligned}$$ Let’s focus on $\mathbf{h}_k .\mathbf{X}_1(S)$: $$\begin{aligned}
\label{Eq_Linear_Proof_Recieve_k_2}
\nonumber \mathbf{h}_k .\mathbf{X}_1(S)&\stackrel{(a)}=&\mathbf{h}_k . \sum_{T \subseteq S, |T|=t+1}{\mathbf{u}_{S}^{T}G_1(T)} \\ \nonumber
&\stackrel{(b)}=&\sum_{T \subseteq S, |T|=t+1, k \in T}{\left(\mathbf{h}_k .\mathbf{u}_{S}^{T}\right)G_1(T)} \\
&\stackrel{(c)}=&\sum_{T \subseteq S, |T|=t+1, k \in T}{\left(\mathbf{h}_k .\mathbf{u}_{S}^{T}\right)L_{r \in T}^1(W_{d_r,T \backslash \{r\}}^j)},\end{aligned}$$ where (a) follows from (\[Eq\_Linear\_Proof\_X\_S\]), (b) follows from the fact that $$\begin{aligned}
\mathbf{u}_{S}^{T} &\perp& \mathbf{h}_k \hspace{5mm} \mathrm{for \hspace{2mm} all} \hspace{5mm} k\in S \backslash T,\end{aligned}$$ and (c) is due to (\[Eq\_Linear\_Proof\_G\_T\]). In (\[Eq\_Linear\_Proof\_Recieve\_k\_2\]), user $k$ can extract $W_{d_k,T \backslash \{k\}}^j$ from the linear combination $L_{r \in T}^1(W_{d_r,T \backslash \{r\}}^j)$, since all the other interference terms are present at his cache. Thus, by removing interference terms, user $k$ can carve the following linear combination from (\[Eq\_Linear\_Proof\_Recieve\_k\_2\]): $$\begin{aligned}
\nonumber L_{T \subseteq S, |T|=t+1, k \in T}^1\left(W_{d_k,T \backslash \{k\}}^j\right),\end{aligned}$$ which is a random linear combination of ${t+L-1 \choose t}$ mini-files desired by user $k$. However, since in (\[Eq\_Linear\_Proof\_Recieve\_k\_1\]) user $k$ receives ${t+L-1 \choose t}$ independent random linear combinations of these mini-files, he can recover the whole set of mini-files: $$\begin{aligned}
\nonumber \left(W_{d_k,T \backslash \{k\}}^j: T \subseteq S, |T|=t+1, k \in T\right).\end{aligned}$$ Thus, for the $T$ specified in this lemma, he can recover the mini-file $W_{d_k,T \backslash \{k\}}^j$. Now, since there exist a total of ${K-t-1 \choose L-1}$ $(t+L)$-subsets containing this specific $T$, by considering the transmissions corresponding to each, this user will recover ${K-t-1 \choose L-1}$ *distinct* mini-files of form $W_{d_k,T \backslash \{k\}}^j$. The distinctness is guaranteed by the appropriate updating of the index $N(\cdot,\cdot)$. These mini-files will recover the sub-file $W_{d_k,T \backslash \{k\}}$ and the proof is concluded.
\[Lem\_Linear\_Proof\_2\] Through the above content delivery strategy an arbitrary user $k$ will be able to decode all the sub-files: $$\begin{aligned}
\left(W_{d_k,\tau}: \tau \subseteq [K], |\tau|=t, k \not \in \tau\right).\end{aligned}$$
Consider an arbitrary $\tau \subseteq [K]$ such that $|\tau|=t, k \not \in \tau$. Define $T=\tau \cup \{k\}$. Then, since to Lemma \[Lem\_Linear\_Proof\_1\], user $k$ is able to decode $W_{d_k,\tau}$. Since $\tau$ was chosen arbitrarily, the proof is complete.
\[Alg\_Main\]
$t \gets MK/N$ split $W_n$ into $(W_{n,\tau}: \tau \subset [K], |\tau|=t)$ of equal size split $W_{n,\tau}$ into $(W_{n,\tau}^j: j=1,\dots,{K-t-1 \choose L-1})$ of equal size $Z_k \gets (W_{n,\tau}^j: \tau \subset [K], |\tau|=t, k \in \tau, j=1,\dots,{K-t-1 \choose L-1}, n \in [N])$\
$t \gets MK/N$ $N(r,T\backslash\{r\}) \gets 1 $ Design $\mathbf{u}_{S}^{T}$ such that: for all $j \in S$, $\mathbf{h}_j \perp \mathbf{u}_{S}^{T}$ if $j \not\in T$ and $\mathbf{h}_j \not \perp \mathbf{u}_{S}^{T}$ if $j \in T$
$G_\omega(T) \gets L_{r \in T}^\omega\left( W_{{d_r},T\backslash\{r\}}^{N(r,T\backslash\{r\})}\right)$ $\mathbf{X}_\omega(S) \gets \sum_{T \subseteq S, |T|=t+1} {\mathbf{u}_{S}^{T} G_\omega(T)}$
**transmit** $\mathbf{X}(S)=\left[\mathbf{X}_1(S),\dots,\mathbf{X}_{{t+L-1 \choose t}}(S)\right]$
$N(r,T\backslash\{r\}) \gets N(r,T\backslash\{r\}) + 1$
Conclusions {#Sec_Conclusions}
===========
In this paper, we investigated coded caching in a multi-server network where servers are connected to multiple cache-enabled clients. Based on the topology of the network, we defined three types of networks, namely, dedicated, flexible, and linear networks. In dedicated and flexible networks, we assume that the internal nodes are aware of the network topology, and accordingly route the data. In linear networks, we assume no topology knowledge at internal nodes, and thus, internal nodes perform random linear network coding. We have shown that knowledge of type of network topology plays a key role in design of proper caching mechanisms in such networks. Our results show that all network types can benefit from both caching and multiplexing gains. In fact, in dedicated and linear networks the global caching and multiplexing gains appear in additive form. However, in flexible networks they appear in multiplicative form, leading to an order-optimal solution in terms of coding delay.
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Appendix A: Converse Proof {#Appendix_Converse .unnumbered}
==========================
![Converse Proof. \[Fig\_Converse\_Proof\]](Converse_Proof.eps){width="60.00000%"}
The proof is similar to the cut-set method presented in [@Maddah-Ali_Fundamental_2014]. See Fig. \[Fig\_Converse\_Proof\] and let us concentrate on the first $s$ users. Define $\mathbf{X}_1$ to be the transmit block sent by the servers such that these users, with the help of their cache contents $Z_1,\dots,Z_s$, will be able to decode $W_1,\dots,W_s$. Also, define $\mathbf{X}_2$ to be the block which enables the users to decode $W_{s+1},\dots,W_{2s}$, and continue the same process such that $\mathbf{X}_{\lfloor N/s \rfloor}$ is the block which enables the users to decode $W_{s\lfloor N/s \rfloor -s+1},\dots,W_{s\lfloor N/s \rfloor}$. Also, define $R$ to be the maximum information needed to pass through the two cuts shown in the figure, by each transmit block transmission. Then we will have: $$\begin{aligned}
\nonumber s\lfloor \frac{N}{s} \rfloor F \leq \lfloor \frac{N}{s} \rfloor R+sMF,\end{aligned}$$ which will result in $$\begin{aligned}
\nonumber R \geq \left(s-\frac{s}{ \lfloor \frac{N}{s} \rfloor}M\right)F.\end{aligned}$$ However, we have: $$\begin{aligned}
\nonumber D^*(M) &\geq& \frac{R}{\mathrm{min-cut}} \\ \nonumber
&\geq& \frac{R}{\min(s,L)m} \\
&\geq& \frac{1}{\min(s,L)} \left(s-\frac{s}{ \lfloor \frac{N}{s} \rfloor}M\right)\frac{F}{m}.\end{aligned}$$ Now we can maximize on the free parameter $s$ to arrive at the tightest bound, which concludes the proof.
Appendix B: Details of Example \[Examp\_Linear\_2\] ($L=2,N=4,K=4$) {#Appendix_Two_Servers .unnumbered}
===================================================================
In this appendix, we consider the scenario in Example \[Examp\_Linear\_2\] for the case of two servers. For each memory size $M=0,\dots,4$, we present the scheme which achieves the coding delay as stated in Example \[Examp\_Linear\_2\].
- $M=0$
In this case, we do not have any cache space available at the users. Suppose we divide each file into three equal-sized non-overlapping parts: $$\begin{aligned}
\nonumber A&=&[A^1,A^2,A^3] \\ \nonumber
B&=&[B^1,B^2,B^3] \\ \nonumber
C&=&[C^1,C^2,C^3] \\ \nonumber
D&=&[D^1,D^2,D^3].\end{aligned}$$ Then, the servers transmit the following blocks, in sequence: $$\begin{aligned}
\label{Eq_L2_K4_N4_M0_transmit}
\nonumber \mathbf{X}(\{1,2\})&=& \mathbf{h}_1^{\perp}B^1+\mathbf{h}_2^{\perp}A^1 \\ \nonumber
\mathbf{X}(\{1,3\})&=& \mathbf{h}_1^{\perp}C^1+\mathbf{h}_3^{\perp}A^2 \\ \nonumber
\mathbf{X}(\{1,4\})&=& \mathbf{h}_1^{\perp}D^1+\mathbf{h}_4^{\perp}A^3 \\ \nonumber
\mathbf{X}(\{2,3\})&=& \mathbf{h}_2^{\perp}C^2+\mathbf{h}_3^{\perp}B^2 \\ \nonumber
\mathbf{X}(\{2,4\})&=& \mathbf{h}_2^{\perp}D^2+\mathbf{h}_4^{\perp}B^3 \\
\mathbf{X}(\{3,4\})&=& \mathbf{h}_3^{\perp}D^3+\mathbf{h}_4^{\perp}C^3.\end{aligned}$$ Let’s focus on the first user which receives: $$\begin{aligned}
\nonumber \mathbf{h}_1.\mathbf{X}(\{1,2\})&=& (\mathbf{h}_1.\mathbf{h}_2^{\perp})A^1 \\ \nonumber
\mathbf{h}_1.\mathbf{X}(\{1,3\})&=& (\mathbf{h}_1.\mathbf{h}_3^{\perp})A^2 \\ \nonumber
\mathbf{h}_1.\mathbf{X}(\{1,4\})&=& (\mathbf{h}_1.\mathbf{h}_4^{\perp})A^3. \end{aligned}$$ From the above data, this user can recover the whole file $A$. Similarly, other users can decode their requested files.
The transmission stated in (\[Eq\_L2\_K4\_N4\_M0\_transmit\]) consists of six blocks of size $2$-by-$\frac{F}{3m}$, resulting in a coding delay of $T_C=6 \frac{F}{3m}=2\frac{F}{m}$.
- $M=1$
Consider the cache content placement used in [@Maddah-Ali_Fundamental_2014]: First divide each file into $4$ equal-sized non-overlapping sub-files: $$\begin{aligned}
\nonumber A&=&[A_1,A_2,A_3,A_4] \\ \nonumber
B&=&[B_1,B_2,B_3,B_4] \\ \nonumber
C&=&[C_1,C_2,C_3,C_4] \\ \nonumber
D&=&[D_1,D_2,D_3,D_4],\end{aligned}$$ and then, fill the caches as follows: $$\begin{aligned}
\nonumber Z_1&=&[A_1,B_1,C_1,D_1] \\ \nonumber
Z_2&=&[A_2,B_2,C_2,D_2] \\ \nonumber
Z_3&=&[A_3,B_3,C_3,D_3] \\ \nonumber
Z_4&=&[A_4,B_4,C_4,D_4].\end{aligned}$$ Such placement respects the memory constraint of $M=1$. Also, divide each sub-file into two equal parts of size $\frac{1}{2}\frac{F}{4}=\frac{F}{8}$ bits: $$\begin{aligned}
\nonumber A_i&=&[A_i^1,A_i^2], \\ \nonumber
B_i&=&[B_i^1,B_i^2], \\ \nonumber
C_i&=&[C_i^1,C_i^2], \\ \nonumber
D_i&=&[D_i^1,D_i^2], \end{aligned}$$ where $i=1,2,3,4$. In the second phase, we send the following blocks of size $2$-by-$\frac{F}{4}$ bits: $$\begin{aligned}
\label{Eq_L2_K4_N4_M1_transmit}
\nonumber \mathbf{X}(\{1,2,3\})&=&[\mathbf{h}_1^{\perp} L_{\{2,3\}}^1(B_3^1,C_2^1)+\mathbf{h}_2^{\perp} L_{\{1,3\}}^1(A_3^1,C_1^1)+\mathbf{h}_3^{\perp} L_{\{1,2\}}^1(A_2^1,B_1^1), \\ \nonumber
& &\mathbf{h}_1^{\perp} L_{\{2,3\}}^2(B_3^1,C_2^1)+\mathbf{h}_2^{\perp} L_{\{1,3\}}^2(A_3^1,C_1^1)+\mathbf{h}_3^{\perp} L_{\{1,2\}}^2(A_2^1,B_1^1)] \\ \nonumber
\mathbf{X}(\{1,2,4\})&=&[\mathbf{h}_1^{\perp} L_{\{2,4\}}^1(B_4^1,D_2^1)+\mathbf{h}_2^{\perp} L_{\{1.4\}}^1(A_4^1,D_1^1)+\mathbf{h}_4^{\perp} L_{\{1,2\}}^1(A_2^2,B_1^2), \\ \nonumber
& &\mathbf{h}_1^{\perp} L_{\{2,4\}}^2(B_4^1,D_2^1)+\mathbf{h}_2^{\perp} L_{\{1,4\}}^2(A_4^1,D_1^1)+\mathbf{h}_4^{\perp} L_{\{1,2\}}^2(A_2^2,B_1^2)] \\ \nonumber
\mathbf{X}(\{1,3,4\})&=&[\mathbf{h}_1^{\perp}L_{\{3,4\}}^1(C_4^1,D_3^1)+\mathbf{h}_3^{\perp}L_{\{1,4\}}^1(A_4^2,D_1^2)+\mathbf{h}_4^{\perp}L_{\{1,3\}}^1(A_3^2,C_1^2), \\ \nonumber
& &\mathbf{h}_1^{\perp}L_{\{3,4\}}^2(C_4^1,D_3^1)+\mathbf{h}_3^{\perp}L_{\{1,4\}}^2(A_4^2,D_1^2)+\mathbf{h}_4^{\perp}L_{\{1,3\}}^2(A_3^2,C_1^2)] \\ \nonumber
\mathbf{X}(\{2,3,4\})&=&[\mathbf{h}_2^{\perp}L_{\{3,4\}}^1(C_4^2,D_3^2)+\mathbf{h}_3^{\perp}L_{\{2,4\}}^1(B_4^2,D_2^2)+\mathbf{h}_4^{\perp}L_{\{2,3\}}^1(B_3^2,C_2^2), \\ \nonumber
& &\mathbf{h}_2^{\perp}L_{\{3,4\}}^2(C_4^2,D_3^2)+\mathbf{h}_3^{\perp}L_{\{2,4\}}^2(B_4^2,D_2^2)+\mathbf{h}_4^{\perp}L_{\{2,3\}}^2(B_3^2,C_2^2)]. \\\end{aligned}$$ Let’s focus on the first user. From the above transmissions he recovers: $$\begin{aligned}
\nonumber \mathbf{h}_1 . \mathbf{X}(\{1,2,3\})&=&[(\mathbf{h}_1.\mathbf{h}_2^{\perp}) L_{\{1,3\}}^1(A_3^1,C_1^1)+(\mathbf{h}_1.\mathbf{h}_3^{\perp}) L_{\{1,2\}}^1(A_2^1,B_1^1), \\ \nonumber
& &(\mathbf{h}_1.\mathbf{h}_2^{\perp}) L_{\{1,3\}}^2(A_3^1,C_1^1)+(\mathbf{h}_1.\mathbf{h}_3^{\perp}) L_{\{1,2\}}^2(A_2^1,B_1^1)] \\ \nonumber
&=& [L^1(A_3^1,C_1^1,A_2^1,B_1^1),L^2(A_3^1,C_1^1,A_2^1,B_1^1)] \\ \nonumber
\mathbf{h}_1 . \mathbf{X}(\{1,2,4\})&=&[(\mathbf{h}_1.\mathbf{h}_2^{\perp}) L_{\{1,4\}}^1(A_4^1,D_1^1)+(\mathbf{h}_1.\mathbf{h}_4^{\perp}) L_{\{1,2\}}^1(A_2^2,B_1^2), \\ \nonumber
& &(\mathbf{h}_1.\mathbf{h}_2^{\perp}) L_{\{1,4\}}^2(A_4^1,D_1^1)+(\mathbf{h}_1.\mathbf{h}_4^{\perp}) L_{\{1,2\}}^2(A_2^2,B_1^2)] \\ \nonumber
&=&[L^1(A_4^1,D_1^1,A_2^2,B_1^2),L^2(A_4^1,D_1^1,A_2^2,B_1^2)] \\ \nonumber
\mathbf{h}_1 .\mathbf{X}(\{1,3,4\})&=&[(\mathbf{h}_1 .\mathbf{h}_3^{\perp})L_{\{1,4\}}^1(A_4^2,D_1^2)+(\mathbf{h}_1 .\mathbf{h}_4^{\perp})L_{\{1,3\}}^1(A_3^2,C_1^2), \\ \nonumber
& &(\mathbf{h}_1 .\mathbf{h}_3^{\perp})L_{\{1,4\}}^2(A_4^2,D_1^2)+(\mathbf{h}_1 .\mathbf{h}_4^{\perp})L_{\{1,3\}}^2(A_3^2,C_1^2)] \\ \nonumber
&=& [L^1(A_4^2,D_1^2,A_3^2,C_1^2),L^2(A_4^2,D_1^2,A_3^2,C_1^2)]. \\\end{aligned}$$ (Although user 1 also receives $\mathbf{h}_1 .\mathbf{X}(\{2,3,4\})$, such information is of no value to him.) With the help of its cache contents the first user can eliminate the undesired terms and obtain: $$\begin{aligned}
\nonumber & &[L(A_3^1,A_2^1),L'(A_3^1,A_2^1)] \rightarrow A_3^1,A_2^1 \\ \nonumber
& &[L(A_4^1,A_2^2),L'(A_4^1,A_2^2)] \rightarrow A_4^1,A_2^2 \\ \nonumber
& & [L(A_4^2,A_3^2),L'(A_4^2,A_3^2)] \rightarrow A_4^2,A_3^2.\end{aligned}$$ Since $A_1^1$ and $A_1^2$ is already available in first user’s cache location, he can subsequently recover the whole block $A$. Similarly, all other users can recover their requested files.
The transmission scheme adopted in (\[Eq\_L2\_K4\_N4\_M1\_transmit\]) consists of four $2$-by-$\frac{F}{4m}$ blocks which will result in the coding delay $T_C=4 \frac{F}{4m}=\frac{F}{m}$ time slots.
- $M=2$
Consider the cache content placement used in [@Maddah-Ali_Fundamental_2014]: First divide each file into $6$ equal-sized non-overlapping sub-files: $$\begin{aligned}
\nonumber A&=&[A_1,A_2,A_3,A_4,A_5,A_6] \\ \nonumber
B&=&[B_1,B_2,B_3,B_4,B_5,B_6] \\ \nonumber
C&=&[C_1,C_2,C_3,C_4,C_5,C_6] \\ \nonumber
D&=&[D_1,D_2,D_3,D_4,D_5,D_6],\end{aligned}$$ and then, fill the caches as follows: $$\begin{aligned}
\nonumber Z_1&=&[A_1,A_2,A_3,B_1,B_2,B_3,C_1,C_2,C_3,D_1,D_2,D_3] \\ \nonumber
Z_2&=&[A_1,A_4,A_5,B_1,B_4,B_5,C_1,C_4,C_5,D_1,D_4,D_5] \\ \nonumber
Z_3&=&[A_2,A_4,A_6,B_2,B_4,B_6,C_2,C_4,C_6,D_2,D_4,D_6] \\ \nonumber
Z_4&=&[A_3,A_5,A_6,B_3,B_5,B_6,C_3,C_5,C_6,D_3,D_5,D_6].\end{aligned}$$ In the second phase, we send the following block of symbols of size $2$-by-$\frac{F}{2m}$: $$\begin{aligned}
\label{Eq_L2_K4_N4_M2_transmit}
\nonumber \mathbf{X}=[& &\mathbf{h}_1^{\perp} L_{\{2,3,4\}}^1(B_6,C_5,D_4)+\mathbf{h}_2^{\perp}L_{\{1,3,4\}}^1(A_6,C_3,D_2)+\mathbf{h}_3^{\perp}L_{\{1,2,4\}}^1(A_5,B_3,D_1)+\mathbf{h}_4^{\perp}L_{\{1,2,3\}}^1(A_4,B_2,C_1), \\ \nonumber
& &\mathbf{h}_1^{\perp} L_{\{2,3,4\}}^2(B_6,C_5,D_4)+\mathbf{h}_2^{\perp}L_{\{1,3,4\}}^2(A_6,C_3,D_2)+\mathbf{h}_3^{\perp}L_{\{1,2,4\}}^2(A_5,B_3,D_1)+\mathbf{h}_4^{\perp}L_{\{1,2,3\}}^2(A_4,B_2,C_1),\\ \nonumber
& &\mathbf{h}_1^{\perp} L_{\{2,3,4\}}^3(B_6,C_5,D_4)+\mathbf{h}_2^{\perp}L_{\{1,3,4\}}^3(A_6,C_3,D_2)+\mathbf{h}_3^{\perp}L_{\{1,2,4\}}^3(A_5,B_3,D_1)+\mathbf{h}_4^{\perp}L_{\{1,2,3\}}^3(A_4,B_2,C_1) \hspace{6mm}]. \\\end{aligned}$$ Let’s focus on the first user who receives: $$\begin{aligned}
\nonumber [& &L^1(A_4,A_5,A_6,B_2,B_3,C_1,C_3,D_1,D_2), \\ \nonumber
& &L^2(A_4,A_5,A_6,B_2,B_3,C_1,C_3,D_1,D_2),\\ \nonumber
& &L^3(A_4,A_5,A_6,B_2,B_3,C_1,C_3,D_1,D_2) \hspace{6mm}].\end{aligned}$$ This user also has the unwanted terms $B_2,B_3,C_1,C_3,D_1,D_2$ in his cache, and after removing them from above linear combinations he has three different linear combinations of its required terms $A_4$, $A_5$, and $A_6$. After solving these equations, and with the help of $A_1$, $A_2$, and $A_3$ stored in his cache, he can recover the whole file $A$. Similarly the other users are able to decode their required files.
The transmit block stated in (\[Eq\_L2\_K4\_N4\_M2\_transmit\]) is of size $2$-by-$\frac{F}{2m}$ vector, resulting in $T_C=\frac{1}{2}\frac{F}{m}$ time slots.
- $M=3$
In this case, by the scheme proposed in [@Maddah-Ali_Fundamental_2014], all four users can get useful information through a single transmission from a single server. Thus, we cannot further reduce the delay by activating the other server. Thus, by activating just one server and based on [@Maddah-Ali_Fundamental_2014] a coding delay of $T_C=\frac{1}{4}\frac{F}{m}$ time slots is obtained.
- $M=4$
In the case of $M=4$, all four files can be stored in the cache of each user, and the required delivery delay in the second phase is zero $T_C=0$.
Appendix C: Details of Example \[Examp\_Linear\_2\] ($L=3,N=4,K=4$) {#Appendix_Three_Servers .unnumbered}
===================================================================
In this example, we consider the three server case in Example \[Examp\_Linear\_2\], and for all values of $M=0,\dots,4$ present the schemes that lead to achievable rates.
- $M=0$
In this case, we do not have any cache space available at the user locations. Suppose we divide each file into three equal-sized non-overlapping parts: $$\begin{aligned}
\nonumber A&=&[A^1,A^2,A^3] \\ \nonumber
B&=&[B^1,B^2,B^3] \\ \nonumber
C&=&[C^1,C^2,C^3] \\ \nonumber
D&=&[D^1,D^2,D^3].\end{aligned}$$ The three servers can then send the following $3$-by-$1$ vectors: $$\begin{aligned}
\label{Eq_L3_K4_N4_M0_transmit}
\nonumber \mathbf{X}(\{1,2,3\})&=& \mathbf{u}_{\{1,2,3\}}^{\{1\}}A^1+\mathbf{u}_{\{1,2,3\}}^{\{2\}}B^1+\mathbf{u}_{\{1,2,3\}}^{\{3\}}C^1\\ \nonumber
\mathbf{X}(\{1,2,4\})&=& \mathbf{u}_{\{1,2,4\}}^{\{1\}}A^2+\mathbf{u}_{\{1,2,4\}}^{\{2\}}B^2+\mathbf{u}_{\{1,2,4\}}^{\{4\}}D^1\\ \nonumber
\mathbf{X}(\{1,3,4\})&=& \mathbf{u}_{\{1,3,4\}}^{\{1\}}A^3+\mathbf{u}_{\{1,3,4\}}^{\{3\}}C^2+\mathbf{u}_{\{1,3,4\}}^{\{4\}}D^2\\
\mathbf{X}(\{2,3,4\})&=& \mathbf{u}_{\{2,3,4\}}^{\{2\}}B^3+\mathbf{u}_{\{2,3,4\}}^{\{3\}}C^3+\mathbf{u}_{\{2,3,4\}}^{\{4\}}D^3,\end{aligned}$$ where we require $$\begin{aligned}
\nonumber &&\mathbf{u}_S^T \perp \mathbf{h}_j, \forall \mathbf{h}_j \in S \backslash T \\
&&\mathbf{u}_S^T \not \perp \mathbf{h}_j, \forall \mathbf{h}_j \in T.\end{aligned}$$ In this example, since we have three dimensional transmit vectors (three servers) and $|S \backslash T|=2$, such vectors can be found.
Let’s focus on the first user who receives: $$\begin{aligned}
\nonumber \mathbf{h}_1.\mathbf{X}(\{1,2,3\})&=& \left(\mathbf{h}_1.\mathbf{u}_{\{1,2,3\}}^{\{1\}}\right)A^1+\left(\mathbf{h}_1.\mathbf{u}_{\{1,2,3\}}^{\{2\}}\right)B^1+\left(\mathbf{h}_1.\mathbf{u}_{\{1,2,3\}}^{\{3\}}\right)C^1=\left(\mathbf{h}_1.\mathbf{u}_{\{1,2,3\}}^{\{1\}}\right)A^1\\ \nonumber
\mathbf{h}_1.\mathbf{X}(\{1,2,4\})&=& \left(\mathbf{h}_1.\mathbf{u}_{\{1,2,4\}}^{\{1\}}\right)A^2+\left(\mathbf{h}_1.\mathbf{u}_{\{1,2,4\}}^{\{2\}}\right)B^2+\left(\mathbf{h}_1.\mathbf{u}_{\{1,2,4\}}^{\{4\}}\right)D^1=\left(\mathbf{h}_1.\mathbf{u}_{\{1,2,4\}}^{\{1\}}\right)A^2\\
\mathbf{h}_1.\mathbf{X}(\{1,3,4\})&=& \left(\mathbf{h}_1.\mathbf{u}_{\{1,3,4\}}^{\{1\}}\right)A^3+\left(\mathbf{h}_1.\mathbf{u}_{\{1,3,4\}}^{\{3\}}\right)C^2+\left(\mathbf{h}_1.\mathbf{u}_{\{1,3,4\}}^{\{4\}}\right)D^2=\left(\mathbf{h}_1.\mathbf{u}_{\{1,3,4\}}^{\{1\}}\right)A^3.\end{aligned}$$ The first user can then successfully decode its requested file. Similarly, the other users will also be able to decode their requested files.
The transmission stated in (\[Eq\_L3\_K4\_N4\_M0\_transmit\]) consists of four $3$-by-$\frac{F}{3m}$ blocks, resulting in $T_C=\frac{4F}{3m}$ time slots.
- $M=1$
The cache content placement is the same as [@Maddah-Ali_Fundamental_2014]. Then, the transmit block by the three servers is: $$\label{Eq_L3_K4_N4_M1_transmit}
\mathbf{X}=[\mathbf{X}_1,\mathbf{X}_2, \mathbf{X}_3],$$ where (for $\omega=1,2,3$) $$\begin{aligned}
\nonumber \mathbf{X}_\omega&=&\mathbf{u}_{\{1,2,3,4\}}^{\{1,2\}}L_{\{1,2\}}^\omega(A_2,B_1)+\mathbf{u}_{\{1,2,3,4\}}^{\{1,3\}}L_{\{1,3\}}^\omega(A_3,C_1)+\mathbf{u}_{\{1,2,3,4\}}^{\{1,4\}}L_{\{1,4\}}^\omega(A_4,D_1) \\
&+&\mathbf{u}_{\{1,2,3,4\}}^{\{2,3\}}L_{\{2,3\}}^\omega(B_3,C_2)+\mathbf{u}_{\{1,2,3,4\}}^{\{2,4\}}L_{\{2,4\}}^\omega(B_4,D_2)+\mathbf{u}_{\{1,2,3,4\}}^{\{3,4\}}L_{\{3,4\}}^\omega(C_4,D_3).\end{aligned}$$
Now let’s focus on the first user who receives: $$y_1=\mathbf{h}_1 . \mathbf{X}=[\mathbf{h}_1 .\mathbf{X}_1,\mathbf{h}_1 .\mathbf{X}_2, \mathbf{h}_1 .\mathbf{X}_3].$$ Let’s consider first the term: $$\begin{aligned}
\nonumber \mathbf{h}_1 .\mathbf{X}_1&=&\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{1,2\}}\right)L_{\{1,2\}}^1(A_2,B_1)+\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{1,3\}}\right)L_{\{1,3\}}^1(A_3,C_1)+\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{1,4\}}\right)L_{\{1,4\}}^1(A_4,D_1) \\ \nonumber
&+&\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{2,3\}}\right)L_{\{2,3\}}^1(B_3,C_2)+\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{2,4\}}\right)L_{\{2,4\}}^1(B_4,D_2)+\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{3,4\}}\right)L_{\{3,4\}}^1(C_4,D_3)\\ \nonumber
&=&\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{1,2\}}\right)L_{\{1,2\}}^1(A_2,B_1)+\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{1,3\}}\right)L_{\{1,3\}}^1(A_3,C_1)+\left(\mathbf{h}_1 .\mathbf{u}_{\{1,2,3,4\}}^{\{1,4\}}\right)L_{\{1,4\}}^1(A_4,D_1) \\
&=& L^1(A_2,A_3,A_4,C_1,B_1,D_1).\end{aligned}$$ As this user has cached $B_1,C_1,D_1$ in the first phase, it can remove these terms from this linear combination to obtain $$\begin{aligned}
\nonumber L(A_2,A_3,A_4).\end{aligned}$$ Thus, user $1$ can recover a linear combination of its requested sub-files from $\mathbf{h}_1 .\mathbf{X}_1$. From, $\mathbf{h}_1 .\mathbf{X}_2$ and $\mathbf{h}_1 .\mathbf{X}_3$ he can obtain two other independent linear combinations from which he can recover all three subfiles $A_2,A_3,A_4$. Since he already has $A_1$ in his cache, he can decode the whole $A$ file. Similarly, all the other users can also decode their requested files.
The transmit block stated in (\[Eq\_L3\_K4\_N4\_M1\_transmit\]) consists of one $3$-by-$\frac{3F}{4m}$ vectors, resulting in $T_C=\frac{3}{4}\frac{F}{m}$ time slots.
- $M=2$ In this case, we only activate two of the servers and thus the problem reduces to the case with $L=2,N=4,K=4$ for which we achieved $T_C=\frac{1}{2}\frac{F}{m}$.
- $M=3$ In this case, we only activate one server and thus the problem reduces to [@Maddah-Ali_Fundamental_2014] with $T_C=\frac{1}{4}\frac{F}{m}$.
- $M=4$ In this case we have $T_C=0$.
[^1]: It should be noted that the definition of sub-files and mini-files here differs from that of flexible networks.
|
---
abstract: 'In this work, the out-of-equilibrium dynamics of the Kardar-Parisi-Zhang equation in (1+1) dimensions is studied by means of numerical simulations, focussing on the two-times evolution of an interface in the absence of any disordered environment. This work shows that even in this simple case, a rich aging behavior develops. A multiplicative aging scenario for the two-times roughness of the system is observed, characterized by the same growth exponent as in the stationary regime. The analysis permits the identification of the relevant growing correlation length, accounting for the important scaling variables in the system. The distribution function of the two-times roughness is also computed and described in terms of a generalized scaling relation. These results give good insight into the glassy dynamics of the important case of a non-linear elastic line in a disordered medium.'
address: 'DPMC-MaNEP, Universit[é]{} de Gen[è]{}ve, 24 Quai Ernest Ansermet, 1211 Gen[è]{}ve 4, Switzerland'
author:
- Sebastian Bustingorry
title: 'Aging dynamics of non-linear elastic interfaces: the Kardar-Parisi-Zhang equation'
---
[**Keywords:**]{} Slow dynamics and aging (theory), self–affine roughness (theory), kinetic growth processes (theory)
Introduction {#sec:intro}
============
Due to its relevance on fundamental problems of condensed matter physics, the statics and dynamics of elastic manifolds are a subject of great current interest. In general, elastic manifolds can be related to problems such as domain wall motion in magnetic [@magnetic] or ferroelectric [@ferroelectric] materials, vortex matter in high temperature superconductors [@rusos; @vortex; @us], grain boundary fluctuations in materials science [@grains], interface dynamics in deposition problems [@Barabasi-Stanley], or crack propagation [@cracks]. The main ingredients characterizing these problems might include the elastic energy of the internal degrees of freedom, the quenched disorder, and the interaction between different components of the system. It is known that when quenched disorder is present, these systems presents glassy characteristics such as disorder roughness exponents or non-stationary dynamics. In order to better understand such features as the slow out-of-equilibrium dynamics, it is important to study in the first place the properties of simpler models, i.e. systems without disorder and/or interactions.
It has been shown that even the simple Edwards-Wilkinson equation, a very well studied model of interface dynamics, presents glassy behavior [@Cukupa; @Pleimling; @cuentas_EW]. Since this is an harmonic solvable model, it is possible to obtain from it a lot of information about the out-of-equilibrium dynamics, such as the multiplicative aging scaling of the two-times roughness, the inclusion of finite size equilibration, and the generalization of scaling distribution functions. It is important to understand how these properties, obtained for the simple EW equation, can be generalized to other models as the non-linear Kardar-Parisi-Zhang (KPZ) equation [@KPZ], different discrete models [@Barabasi-Stanley], and even different dimensions. For these cases, however, the universal exponents characterizing static and dynamic properties might change, and then it becomes important to understand how such a change affects the glassy properties.
In particular, the study of the non-linear contributions to the elasticity should allow a better understanding of the glassy properties of more realistic elastic models. In this sense, the study of the out-of-equilibrium dynamics of the pure KPZ equation serve as a first step to rationalize the aging behavior reported in [@Spaniards], where a driven non-linear elastic model with quenched disorder was analyzed. Moreover, it shall also give a first insight into how the general picture obtained for the EW case can be generalized to more complex, and analytically non-solvable, models.
In addition, it is now clear how important fluctuations are to describe the out-of-equilibrium dynamics of complex systems [@Bramwell; @Racz; @Chamon-etal; @Chamon-Cugliandolo]. The thermally induced distribution functions of a given quantity contain more information than the first moments, which resulted in the proposal of using these fluctuations to characterize different universality classes [@Racz; @Aarao]. The roughness distribution of elastic lines was studied in the steady state for various models on the saturation regime [@Aarao; @Marinari; @static-inter], and also in the dynamic growth regime [@Zoltan]. It was also analyzed in the context of depinning of elastic lines in random environments [@Rosso] and in relation to the $1/f$ noise [@gyorgyi]. Its generalization to non-equilibrium situations has also been recently considered [@cuentas_EW; @Bustingorryetal]. Within this context, it seems important to test the scaling of the out-of-equilibrium roughness distribution for the KPZ equation.
Therefore, with the aim of analyzing the glassy properties of non-linear elastic models, the numerical solution of the KPZ equation in (1+1) dimensions is presented in this work. Two-times correlation functions, as the roughness and structure factor, are computed and compared with the EW results. The distribution function of the roughness in the aging regime is also considered. These results allow to generalize those previously obtained for the EW equation by properly including the KPZ universal exponents. The paper is organized as follows. The numerical details of the simulations, together with the out-of-equilibrium protocol, are presented in . The glassy properties for the different two-times quantities are presented and discussed in . Finally, presents the conclusions.
Numerical details {#sec:model}
=================
In the present section the numerical details of the simulation of the out-of-equilibrium dynamics of the KPZ equation are presented. This equation describes, in (1+1) dimensions, the evolution of a one-dimensional non-linear elastic object embedded in a two-dimensional space in the limit of small fluctuations. The non-linear contribution is a first order correction to elasticity, thus allowing the characterization of the line in terms of the univaluated field $x(z,t)$. The KPZ equation in (1+1) dimensions is then [@KPZ] $$\label{eq:kpz}
\frac{\partial x(z,t)}{\partial t} = \nu \frac{\partial^2 x(z,t)}{\partial z^2}+ \lambda \left[ \frac{\partial x(z,t)}{\partial z} \right]^2+\xi(z,t),$$ where the thermal noise $\xi$ is characterized by $$\begin{aligned}
\label{eq:noise}
\langle \xi(z,t) \rangle = 0,\\
\langle \xi(z,t) \xi(z',t') \rangle = 2\,T\, \delta(z-z')
\delta(t-t'),\end{aligned}$$ with $\nu$ the elastic constant, $\lambda$ characterizing the strength of the non-linear term, $T$ the temperature of the thermal bath and $\langle \cdots \rangle$ the average over the white noise $\xi$, [*i.e.*]{} the thermal average.
The KPZ equation is numerically solved using the following finite difference approach: $$\label{eq:disc-kpz}
\fl
x_i(t+t_0)=x_i(t)+t_0 \nu \left[ x_{i+1}(t)-2 x_i(t)+x_{i-1}(t) \right]+
\frac{t_0\,\lambda}{3} \Psi_i^2
+\sqrt{24 T t_0} \; \xi_i(t).$$ Here, the internal dimension is replaced by a one-dimensional lattice of unitary space, which sets the length scales in the following. The noise $\xi_i(t)$ is now a random number uniformly distributed in $[-1/2,1/2)$. The particular choice of the finite difference representation of the non-linear term is given by $$\label{eq:disc-KPZ-term}
\fl
\Psi_i^2=\left[ x_{i+1}(t)-x_i(t)\right]^2 +\left[x_{i+1}(t)-x_i(t)\right]\left[x_i(t)-x_{i-1}(t)\right]+\left[x_i(t)-x_{i-1}(t)\right]^2,$$ which not only gives the correct exponents in the KPZ universality class but also the correct prefactors and the correct fluctuations in the saturation regime [@Lam]. This is important in order to perform a direct comparison with the EW case. In this work, the parameters used to numerically solve the KPZ equation are $L=1024$ (except for the computation of distribution functions where the values $L=256$ and $L=64$ are also used in order to highlight the scaling properties), $t_0=0.01$, which sets the time units, $\nu=1$, and $\lambda=1$. For the thermal average, $N_T=10^3$ noise realizations were considered for the evaluation of two-times correlations. In order to compute the roughness distributions, $N_T=10^5$ thermal noise realizations were used for $L=256$ and $L=64$, while $N_T=10^4$ were used for $L=1024$.
The out-of-equilibrium dynamics of the KPZ equation was analyzed adopting the usual two-times protocol used to study glassy systems [@Leto]. Starting from a flat configuration, the system is first equilibrated at a given initial temperature $T_0$. This means that the system evolves until the stationary state is reached, corresponding to the size-dependent saturation regime of the roughness. Then, the temperature is suddenly changed to the working temperature $T$, and the time is set to zero. Two-times correlation functions are then defined in terms of the waiting time $t_w$ and the elapsed time $\Delta t=t-t_w$. In the present work, the working temperature was set to $T=1$ and different initial temperatures, both smaller and larger than $T$, were studied. In particular, the values $T_0=0$, which corresponds to a perfectly flat initial condition, and $T_0=5$ were used.
Results {#sec:results}
=======
In the following, the results for different two-times correlation quantities are described. Particular attention is given to correlation functions showing a clearly non-trivial out-of-equilibrium generalization of the scaling properties previously found in the EW equation.
Roughness {#sec:roughness}
---------
The one-time roughness, which characterizes the width of the discrete fluctuating line, is commonly defined as [@Barabasi-Stanley] $$\label{eq:w2det}
w^2(t)=\frac{1}{L} \sum_{i=1}^L \left\langle \left[
\delta x_i(t) \right]^2 \right\rangle,$$ where $\delta x_i(t)=x_i(t)-\overline{x(t)}$ and $\overline{x(t)}=L^{-1} \sum_{i=1}^L x_i(t)$ is the instantaneous center of mass position. In general, the time evolution of the roughness can be described through the Family-Vicsek scaling [@Favi], for which the roughness can be written as $w^2(t) \sim L^\zeta f(t/t_x)$, with $t_x \sim L^z$ the saturation time and the scaling function $f(u)\sim u^\beta$ for $u \ll 1$ and $f(u) \sim const.$ for $u \gg 1$. This defines the growth exponent $\beta$, the roughness exponent $\zeta$ and the dynamic exponent $z$, related through the scaling relation $z=\zeta/\beta$. The values of these exponents are $\zeta=1$, $z=2$, and $\beta=1/2$ for the EW universality class, and $\zeta=1$, $z=3/2$, and $\beta=2/3$ for KPZ universality class, both in (1+1) dimensions.
The two-times roughness, which is the out-of-equilibrium generalization of the one-time roughness , is defined as $$\label{eq:w2} w^2(\Delta t,t_w)=\frac{1}{L}\sum_{i=1}^L \left\langle \left[
\delta x_i(t_w + \Delta t) - \delta x_i(t_w) \right]^2 \right\rangle.$$ This quantity measures the relative fluctuation between the line’s configuration at times $t_w$ and $t$. It was recently shown that, for the EW equation with finite size $L$, the two-times roughness ages [@cuentas_EW], now scaling as $w^2 \sim F(t_w/t_x,\Delta t/t_x)$. The aging regime may be understood as the pre-asymptotic non-equilibrium regime prior to finite size equilibration, which takes a time $t_x \sim L^z$. When $t_w \gg t_x$, the stationary solution and the Family-Vicsek scaling are recovered. On the other hand, in the aging regime $t_w \ll t_x$, the scaling function behaves as $$\begin{aligned}
\label{eq:gen-fv}
F\left(\frac{t_w}{t_x},\frac{\Delta t}{t_x}\right) \sim \left\{
\begin{array}{lll}
t_w^\beta f_1\left(\frac{\Delta t}{t_w}\right) & \mbox{for} & \Delta t \ll t_x,
\\
\\
L^\zeta f_2\left(\frac{t_w}{t_x}\right)& \mbox{for} & \Delta t \gg t_x,
\end{array}
\right.\end{aligned}$$ with $\beta=1/2$ and $\zeta=1$ the EW scaling exponents. In the case where $\Delta t \ll t_x$, the roughness scales as $w^2 \sim t_w^\beta f_1(\Delta t/t_w)$, with $f_1(u) \sim c(T) u^\beta$ for $\Delta t \ll t_w$ and $f_1(u) \sim c_0(T_0,T) u^\beta$ for $\Delta t \gg t_w$. This scaling describes a multiplicative aging scenario [^1] for the two-times roughness [@Cukupa; @cuentas_EW; @Pleimling], and effectively corresponds to the infinite size limit, $L \to \infty$. On the other hand, when $\Delta t \gg t_x$, i.e. in the saturation regime, the scaling function $f_2(t_w/t_x)$ behaves as $f_2(u) \sim s_0(T,T_0)$ for $t_w \ll t_x$ and $f_2(u) \sim s(T)$ for $t_w \gg t_x$, thus leading to a $t_w$-dependent saturation value for the roughness which evolves from $s_0(T,T_0)L^\zeta$ to $s(T)L^\zeta$ while $t_w$ increases. The scaling form generalizes the Family-Vicsek scaling to a non-equilibrium situation. It is interesting to test if this generalization is valid for the KPZ equation, using its respective exponents. This point is addressed in the following.
In this work, the two-times roughness $w^2(\Delta t,t_w)$, has been numerically obtained from the evolution of the discrete KPZ equation, and it is shown in . Panels (a) and (b) correspond to initial conditions with $T_0=0<T$ and $T_0=5>T$, respectively. The different curves correspond to different waiting times, as indicated. After an initial time interval where the components of the line fluctuate independently, $w^2 \sim \Delta t$, the line becomes correlated in the longitudinal direction and starts aging. Then, ones the line is correlated, two different time regimes are observed, corresponding to $\Delta t$ larger or smaller than $t_x$, and both showing aging. When the waiting time is larger than the saturation time, aging stops and the stationary situation is reached.
In the growth regime, for each waiting time, the curves jump from the equilibrated asymptote to the non-equilibrated asymptote with increasing $\Delta t$. Both asymptotes are growing as $\Delta t^\beta$, with $\beta =2/3$ the KPZ value. Therefore, a multiplicative aging scenario with the corresponding KPZ growth exponent is found. In the saturation regime the curves jump from the $s_0(T,T_0)L^\zeta$ asymptotic value, corresponding to $t_w=0$, to the value $s(T)L^\zeta$, corresponding to the stationary solution. The values of the numerically obtained prefactors $s_0$ and $s$ are equal, within numerical accuracy, to those analytically found for the EW equation, i.e. $s_0=(T_0+T)/12$ and $s=T/6$. It is well known that the KPZ and EW continuum equations have the same steady state solution for the distribution function of a given configuration in the saturation regime [@Barabasi-Stanley]. Furthermore, it has been also shown that this property holds for the discretized model used here [@Lam]. However, the fact that the $t_w$-dependent saturation values obtained in this work coincide with the ones for the EW equation suggests that the non-stationary solution in the saturation regime may also be the same for both models. shows the $t_w$-dependent saturation value for both initial conditions, and compares it with the EW equation solution [@cuentas_EW]. The asymptotic limits are clearly the same, and a small difference is observed in the crossover region.
These results support the generalized Family-Vicsek scaling, although the growth regime is too short in time to accurately test the $\Delta t/t_w$ scaling. Larger system sizes would be necessary for a good test using the two-times roughness. However, a better way to observe the growing correlation length associated with aging is to focus on the two-times structure factor. This quantity also reflects the scaling properties of the roughness, but a better scaling test is obtained, as described in the next section.
Structure factor {#sec:structure-factor}
----------------
The one-time dynamical structure factor is usually defined as $$\label{eq:Sonet}
S_n(t)=L\langle \left| c_n(t) \right|^2 \rangle,$$ where $c_n(t)$ represents the Fourier modes of a given configuration at time $t$, and where the discreetness of the lattice was already taken into account with the wave vectors given by $k_n=2 \pi n/L$, $n=1,...,L$. This structure factor gives information on how the fluctuations of the modes evolve in time, containing also information on the growing correlation length [@Barabasi-Stanley; @Kolton]. The definition should not be confused with the dynamical structure factor $S^D_n(t)=L\langle c_n(t) c_{-n}(0) \rangle$, which gives information on how a given mode correlates between times $t=0$ and $t>0$. This dynamical structure factor presents non-trivial stretched exponential relaxation characteristics in the KPZ case [@Eytan].
In general, the structure factor scales as $S_n \sim t^{(1+\zeta)/z} g(k_n/k_x)$, with $k_x\sim t^{-1/z}$. The scaling function $g(u)$ behaves as $g(u) \sim const.$ for $k_n \ll k_x$ and $g(u) \sim u^{-(1+\zeta)}$ for $k_n \gg k_x$, indicating that there exists a growing correlation length $\ell \sim k_x^{-1} \sim t^{1/z}$. When $\ell \sim L$, the saturation regime is reached and $S_n^\infty \sim k_n^{-(1+\zeta)}$ is independent of the time $t$.
The two-times generalization of the structure factor is defined as [@cuentas_EW] $$\label{eq:SqDttw}
S_n(\Delta t,t_w)=L\langle \left| c_n(t_w+\Delta t)-c_n(t_w) \right|^2 \rangle.$$ This definition allows for the characterization of the fluctuations of a given mode between $t_w$ and $t$. For the EW equation, the two-times structure factor can be written as a scaling function expressed in terms of the quotients $\Delta t/t_x$ and $t_w/t_x$, in analogy with the roughness scaling . However, the focus here will be set on how the structure factor behaves on the $t_w$-dependent saturation regime, which gives clear evidence of the growing correlation lengths. Then, for $\Delta t \gg t_x$, i.e. in the saturation regime, the structure factor is computed for different $t_w$ values as $$\label{eq:Sqtw}
S^\infty_n(t_w)=\lim_{\Delta t \gg t_x}L\langle \left| c_n(t_w+\Delta t)-c_n(t_w) \right|^2 \rangle.$$ It was recently reported for the case of the EW equation that $S^\infty_n(t_w) \sim k_w^{-(1+\zeta)} g_1(k_n/k_w)$, with $k_w \sim t_w^{-1/z}$ [@cuentas_EW]. The scaling function $g_1(u)$ behaves as $g_1(u) \sim (T_0+T) u^{-(1+\zeta)}$ for $k_n \ll k_w$ and $g_1(u) \sim 2T u^{-(1+\zeta)}$ for $k_n \gg k_w$. The particular wave vector $k_w$ precisely separates two regimes. While the regime with large wave vectors, $k_n \gg k_w$, is equilibrated at the working temperature, small wave vectors, $k_n \ll k_w$, still have memory of the initial temperature. Note that in contrast with the one-time structure factor , the two-times generalization contains information on the dynamic exponent in the saturation regime due to its $t_w$-dependence.
shows the structure factor $S^\infty_n(t_w)$ as a function of $k_n$ for different waiting times, obtained in the present work. Panels (a) and (b) correspond to $T_0=0$ and $T_0=5$, respectively. The structure factor clearly presents the $k_n^{-(1+\zeta)}$ decay, but changes between two asymptotes at a given $t_w$-dependent value. In order to emphasize this behavior, the insets show the same data plotted as $k_n^2\,S^\infty_n(t_w)$ against $k_n$. The crossover between the two asymptotes is the kind of behavior described by the scaling function $g_1(u)$. Indeed, the two asymptotic values are $T_0+T$ and $2T$ respectively, indicated with dotted lines in the insets. These values correspond to those previously obtained for the EW equation, again suggesting a strong connection between the non-stationary solution of the EW and KPZ equations in the saturation regime. This feature is stressed in , where the scaling of the structure factor with $k_n/k_w \sim k_n\;t_w^{1/z}$ is shown for the two initial conditions. The data are also compared with the EW equation solution [@cuentas_EW]. It can be seen that although the asymptotic values coincide, the intermediate dynamic regime does not perfectly match. This might be related to the fact that the dynamic exponent is also involved in the scaling function $g(u)$, which goes beyond both models having the same roughness exponent in (1+1) dimensions.
Scattering function {#sec:scattering-function}
-------------------
Another quantity which has proved to be very useful in the analysis of the non-equilibrium dynamics in systems of interacting particles is the incoherent scattering function [@Barrat; @Bonn; @Tanaka]. In the context of elastic lines it can be defined as [@cuentas_EW] $$\label{eq:scater}
C_q(\Delta t,t_w)=\frac{1}{L}
\sum_{i=1}^L \left\langle e^{iq\left[ \delta x_i(t_w + \Delta t)-\delta x_i(t_w) \right]} \right\rangle.$$ It has been shown that due to the gaussian character of the variables involved in the solution of the EW equation, the scattering function can be written in terms of the two-times roughness in a very simple way as $$\label{eq:scater-gauss}
C_q(\Delta t,t_w)=e^{-\frac{q^2}{2}\,w^2(\Delta t,t_w)},$$ which establishes a clear relation between scattering functions and diffusion-like correlations. In the growth regime, $\Delta t \ll t_x$, it has been already shown that the roughness behaves as $w^2 \sim t_w^\beta f_1(\Delta t/t_w)$, with $f_1(u) \sim c(T) u^\beta$ for $\Delta t \ll t_w$ and $f_1(u) \sim c_0(T_0,T) u^\beta$ for $\Delta t \gg t_w$, leading to a stretched exponential relaxation of the scattering function, i.e. $$\label{eq:scater-stretch}
C_q(\Delta t,t_w)=
\left\{
\begin{array}{lll}
e^{-\frac{1}{2}c(T)\,q^2\Delta t^\beta} & \mbox{for} & \Delta t \ll t_w,
\\
e^{-\frac{1}{2}c_0(T_0,T)\,q^2\Delta t^\beta} & \mbox{for} & \Delta t \gg t_w.
\end{array}
\right.$$ This equation involves two stretched exponentials sharing the same exponent $\beta$ but different prefactors. This relation trivially holds for gaussian variables, however its validity for the KPZ equation must be still tested.
\(a) presents the scattering function for the KPZ equation obtained numerically at different $q$ and $t_w$ values. The scattering function presents a strong $q$-dependent saturation regime, on which the $q^2$ factor of the exponential competes with the $L^\zeta$ factor coming from the saturation regime of the roughness. Therefore, at low $q$ values a saturation regime can be observed, while at large $q$ values the scattering function decays faster to a near-zero value. This is clearly observed for the selected values $q^2=0.025$ and $q^2=0.1$ in (a). The generic form of the $C_q$ curves is similar to the one obtained for the EW equation. In order to test the validity of , the scattering function for $q=0.05$ is plotted in (b) together with the values $e^{-\frac{q^2}{2}\,w^2}$ obtained using the results for the roughness presented in (a). The results in (b) prove that relation also holds for the KPZ equation. One could have expected this relation to hold in the saturation regime, since the steady state solution is the same for EW and KPZ equations. However, the fact that it also holds in [*all*]{} the dynamic $\Delta t$ and $t_w$-dependent growth regime is absolutely non-trivial, since the fluctuations of the KPZ are not expected to be gaussian distributed. The same results, i.e. the scattering function being directly obtained from the roughness through , were obtained for $T_0=5$ (not shown here).
Roughness distribution {#sec:roughness-distribution}
----------------------
It has been previously proposed that the roughness distribution function in the saturation regime scales with the average roughness as the only scaling parameter [@static-inter]. This proposition can be extended to all the dynamic regime in steady state, allowing to write the roughness distribution as [@Zoltan] $$\label{eq:Phi-AntalRacz} P\left( w^2;t \right)=\frac{1}{w^2(t)}\;\Phi \left[
\frac{w^2}{w^2(t)}; \frac{t}{L^z} \right].$$ Here, the fluctuating roughness $w^2$ should be distinguished from its average value $w^2(t)$, which explicitly contains the time dependence. The last argument in states that scaling works for curves with the same time scale $t/t_x$, thus accounting for finite size effects. This scaling relation can also be generalized to include the waiting time dependence by considering also the $t_w/t_x$ scale [@us; @cuentas_EW], thus leading to $$\label{eq:Phi-Dttw} P\left( w^2;\Delta t,t_w\right)=\frac{1}{w^2(\Delta
t,t_w)}\;\Phi\left[ \frac{w^2}{w^2(\Delta t,t_w)}; \frac{\Delta t}{L^z},
\frac{t_w}{L^z}\right].$$ This last scaling relation was analitically obtained for the EW equation [@cuentas_EW] and numerically obtained for a directed polymer in random media model [@Bustingorryetal].
Using the results obtained in the present work a test of the scaling relation for the KPZ equation can be performed. The result is shown in , obtained using the roughness distribution for three system sizes, $L=64$, $L=256$ and $L=1024$, and different times $\Delta t$ and $t_w$. In panel (a), the bare distribution functions are shown, while panel (b) shows the scaling function $\Phi(x)$, with $x=w^2/w^2(\Delta t,t_w)$. These results correspond to $T_0=0$. Since the difference between the selected system sizes is a factor four, the saturation time is scaled by a factor $4^z=8$. Then to keep the quotient $t_w/t_x$ fixed, the waiting time should also be scaled by the same factor $4^z=8$; the same holds for the value of $\Delta t$ if one wants to scale with $\Delta t/t_x$. It is shown that all the curves in panel (a) collapse into three sets of curves in panel (b), corresponding to different pairs of $(t_w/t_x,\Delta t/t_x)$ values, in agreement with the scaling relation . This scaling of the roughness distribution function also accounts for the saturation regime, leading to the known stationary solution. A direct comparison with the EW results in the dynamic regime is not possible here because the scaling factor is different; in the present case it should be $4^z=16$ with $z=2$ for EW. Instead, the stationary saturation solution of the EW case [@static-inter] is plotted in panel (b), with a continuous dashed line, showing again that the numerical solution of the KPZ equation asymptotically tends to the same distribution function in (1+1) dimensions[@Racz].
Conclusions {#sec:conclusions}
===========
The out-of-equilibrium dynamics of the KPZ equation has been studied in detail using numerical simulations. It corresponds to the relaxation of a non-linear interface from a given initial condition and it has been shown here that a complicated glassy dynamics emerges, as also does for the EW equation [@cuentas_EW]. The system size $L$ was incorporated in the analysis, thus allowing to reach finite size equilibration after a time $t_x$. Therefore, the out-of-equilibrium regime described here effectively corresponds to a pre-asymptotic regime before equilibrium is reached. One can imagine a very large system size whose equilibration time $t_x$ is much larger than the observation time, making the aging regime the relevant observable time regime. In a sense this is expected when quenched disorder is taken into account, where the dynamics becomes much slower and the equilibration time goes beyond the observation time. Thus, the main aging characteristics described here are of key importance for analyzing the non-equilibrium dynamics of problems which include other components like disorder, external forces or line-line interactions.
It has been shown here that the main characteristics of the out-of-equilibrium dynamics of the EW equation can be extended to the KPZ equation. One should only be careful of using the proper scaling exponents. For instance, from the scaling of the two-times roughness and structure factor the existence of the growing correlation length is revealed, which allows to rationalize the different time regimes. In terms of elapsed time $\Delta t$ and waiting time $t_w$, it is the relative value of the correlation lengths $\ell (\Delta t)$ and $\ell(t_w)$ which defines the scaling properties in the growth regime. When considering the saturation regime, the size of the system $\ell(t_x) \sim L$, associated to the saturation time $t_x$, should also be considered. Thus, it is the competition between these three length scales, $\ell (\Delta t)$, $\ell(t_w)$ and $\ell(t_x)$, and its relation to the corresponding times through the dynamic exponent $z$, which define the aging dynamics of the system.
Results concerning the scattering function have been also presented. This correlation function presents a well defined stretched exponential decay, which is indeed commonly observed in different glassy systems [@Leto]. Moreover, it was shown that the scattering function is directly related to the roughness through equation . This simple exponential relation was previously analytically obtained for the EW equation based on the gaussian character of the fluctuations [@cuentas_EW]. Although in the saturation regime the fluctuations in the KPZ equation are also gaussian distributed, in the growth regime this is not necessarily true. Thus, the fact that both scattering function and roughness are related in this simple way for the whole two-times dependent dynamics is a non-trivial result, posing new questions about the relation between scattering functions and displacements fluctuations out of equilibrium, a fact already pointed out in colloidal glass experiments [@Bonn].
Finally, the scaling of the roughness distribution has been studied. The scaling in equation for the distribution function, which depends on the relative time scales $\Delta t/L^z$ and $t_w/L^z$, has been tested here and a good collapse of the data was obtained. This indicates once more that the correct variables which include both dynamics and finite size effects are the relative scales between $\ell (\Delta t)$, $\ell(t_w)$ and $\ell(t_x)$.
Therefore, it has been highlighted here that the scaling relations analytically found for the EW equation are quite robust, allowing also for a good description of the out-of-equilibrium dynamics of the KPZ equation. While the present work focused on the numerical solution of the continuum equation, it could also be interesting to test these ideas in discrete models of interface dynamics. Finally, in order to further test these scaling relations, it would also be interesting to study higher dimensions for which the scaling exponents are different.
The author specially thanks to L.F. Cugliandolo for stimulating discussions and suggestions. The author also thanks to E. Katzav and M. Pleimling for interesting discussions. Financial support from the Swiss National Science Foundation under MaNEP and Division II is also acknowledged.
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[^1]: Note that for $\Delta t \ll t_x$ the roughness can be also written as $w^2 \sim \Delta t^\beta f_1'(\Delta t/t_w)$, with $f_1' \sim c(T)$ for $\Delta t \ll t_x$ and $f_1'(u) \sim c_0(T_0,T)$ for $\Delta t \gg t_w$. emphasizes the fact that this corresponds indeed to multiplicative aging.
|
---
abstract: 'A novel approach to calibrate the sensitivity of a differential thermometer, consisting of several thermocouples connected in series (thermopile), has been developed. The goal of this method is to increase the accuracy of small temperature difference measurements ($\Delta T \leq \SI{1}{\kelvin}$), without invoking higher sensor complexity. To this end, a method to determine the optimal temperature difference employed during the differential measurement of thermoelectric sensitivities has been developed. This calibration temperature difference is found at the minimum of combined measurement and linearization error for a given mean temperature. The developed procedure is demonstrated in an illustrative example calibration of a nine-junction thermopile. For mean temperatures between $\SI{-10}{\celsius}$ and $\SI{+15}{\celsius}$, the thermoelectric sensitivity was measured with an uncertainty of less than $\SI{\pm 2}{\percent}$. Subsequently, temperature differences as low as $\SI{0.01}{\kelvin}$ can be resolved, while the thermometer used for the example calibration was accurate only to $\SI{\pm0.3}{\kelvin}$. This and higher degrees of accuracy are required in certain research applications, for example to detect heat flux modulations in bifurcating fluidic systems.'
address: |
Institute for Nano- and Microfluidics,\
Center of Smart Interfaces, TU Darmstadt,\
Alarich-Weiss-Str. 10, 64287 Darmstadt, Germany
author:
- Tim Prangemeier
- Iman Nejati
- |
Andreas Müller,\
Philip Endres, Mario Fratzl
- Mathias Dietzel
bibliography:
- 'bib2ETF.bib'
title: Optimized thermoelectric sensitivity measurement for differential thermometry with thermopiles
---
thermopile calibration ,thermoelectric sensitivity measurement ,Seebeck coefficient measurement ,Seebeck effect ,differential thermometer ,calibration uncertainty analysis ,high-precision heat flux modulation measurement
This is the post-print authors’ version of the manuscript published in Experimental Thermal and Fluid Science. © Elsevier 2015. doi: 10.1016/j.expthermflusci.2015.01.018.\
This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.
Introduction {#sec:intro}
============
In 1826 A. C. Becquerel made the first recorded proposal to employ the Seebeck effect in thermometers and by 1902 thermocouples were commercially available [@Hunt1964]. Today, they are in widespread use due to their low cost, robustness, small size, simplicity, speed of response and large temperature range [@Childs2000]. This thermoelectric effect was discovered in the early 1820s by T. Seebeck [@vanHerwaarden1986; @Bell2008; @DiSalvo1999], whereby an electric potential is induced due to a temperature gradient in a thermopair [@Riffat2003; @Huang1990; @Martin2010]. It is often utilized in specialized devices, such as thermometry near absolute zero [@Armbruester1981; @Maeno1983] or in nanoscale devices [@Bakker2012].
While the absolute temperature is of interest in many applications, in others, such as monitoring nuclear reactors [@Hashemian2006], temperature difference measurements are required. For example, in the 1970s thermocouples were calibrated to measure temperature differences in aircraft engine oil as low as $\SI{2.5}{\kelvin}$ over an absolute temperature range of hundreds of Kelvin [@Eccles1973].
Thermocouples can be employed to directly determine a temperature difference in a single measurement (and without reference junction compensation) [@Huang1990; @Martin2010]. This circumvents the error propagation that would otherwise be encountered when differencing two separate temperature measurements. To increase sensitivity, multiple thermocouples can be connected in series forming a thermopile. For precise measurements of differential temperatures above $\SI{1}{\kelvin}$, @Huang1990 calibrated a thermopile with a high precision quartz thermometer $(T \pm \SI{0.04}{\kelvin})$ and a specialized method of signal conversion. An accuracy of $\pm \SI{0.07}{\kelvin}$ was achieved. When the calibrated thermopile is employed to measure differential temperatures, the mean temperature $T_{\text{\text{m}}}$ is used to account for the non-linearity of the calibration curve. This method is limited by the accuracy of the calibration thermometer [@Drnovsek1998].
Differential temperature measurements are of particular interest for heat transfer investigations, as heat flux cannot be measured directly [@Childs1999]. Nonetheless, differential thermometry can be used to relate the heat flux to the temperature gradient and the material properties (known heat resistance). Heat flux sensors based on this technique have been designed for a variety of applications ranging from industry to biological systems research, as well as radiometry for photo-voltaic and solar thermal energy studies [@Langley1999; @Ewing2010; @Baughn1986; @Park2012]. Heat flux uncertainty of $\SI{\pm 4}{\percent}$ and $\SI{\pm 7}{\percent}$ are reported by [@Baughn1986] and [@Park2012] respectively.
In many heat transfer applications, the heat resistance between two thermal reservoirs is not known a priori. In these cases, the heat flux can be determined by measuring the heating (or cooling) power required to maintain quasi-isothermal reservoir boundaries. For electric heat sources this is simply achieved by an electric-power measurement. For connectively cooled heat sink surfaces, on the other hand, this is typically accomplished by measuring the difference between the inlet and outlet temperature of the coolant in conjunction with its mass flux and material properties [@Koschmieder1974b]. A high degree of temperature uniformity of the cooled plate is required in many studies. The pattern symmetry of surface tension gradient driven Bénard-Marangoni convection, for example, is highly sensitive towards temperature non-uniformities. Therefore, accurate differential temperature measurements are required to detect the heat flux modulation caused by bifurcations points, while at the same time maintaining a thermal gradient across the plate, which is as small as possible in order to maintain quasi-isothermal boundary conditions. To this end, a thermoelectric circuit has been developed by [@Koschmieder1974b]; however, no uncertainty propagation analysis has been reported for the sensor calibration.
The thermoelectric sensitivity (TES) is employed in the above mentioned heat flux studies [@Langley1999; @Koschmieder1974b; @Ewing2010]. It is commonly measured with the differential method (described in [@Martin2010]) [@Ewing2010; @Wold1916; @Bidwell1922; @Weiss1956; @Burkov2001; @Zhou2005]. The uncertainty of the achieved TES measurement is dependent on the chosen temperature difference. To avoid non-linear effects and errors, it has in the past been recommended to choose a temperature difference on the order of a few percent of the mean temperature $T_{\text{\text{m}}}$ ($\Delta T/ T_{\text{\text{m}}} << 1$) [@Zhou2005; @Martin2010]. However, until now, there has been no report of any systematic study of the selection of the temperature difference employed during TES measurement. This shortcoming is addressed in this study.
While international standards for the calibration of thermocouples as thermometers exist, customized calibration techniques have been developed for specialized applications [@Huang1990; @Ballestrin2006b]. For example, calorimetric and radiometric calibrations of thermoelectric heat flux sensors and radiometers have been developed [@Ballestrin2004; @Ballestrin2006b]. For precise measurements of small differential temperatures, a method of calibration and signal conversion has been proposed for measurements with thermopiles [@Huang1990]. In this study we address the calibration of even smaller temperature differences, and propose a novel approach.
Subsequently, the above mentioned technique proposed by @Huang1990 is further extended to measure temperature differences below $\SI{1}{\kelvin}$. In addition to the mean temperature of the differential temperature measurement (taken into account by @Huang1990), the mean temperature during the calibration process is considered as well. Instead of the calibration of $U(T)$, the inverse of the TES is used as the calibration coefficient, and it is measured with the differential method described in @Martin2010. An analysis of the maximum uncertainty of this method is performed and a procedure to optimize the accuracy of the differential technique is developed. The novel method is employed over a range of temperatures ($\SI{-10}{\celsius}$ to $\SI{15}{\celsius}$) in an illustrative example and compared to the aforementioned technique. A higher differential temperature resolution is achieved, although a significantly less accurate calibration thermometer $(T \pm \SI{0.3}{\kelvin})$ was used. Finally, the steps of this method from calibration to small differential temperature measurement with high accuracy are summarized.
[p[15mm]{}p[110mm]{}]{} $\mathrm{\textbf{Nomenclature}}$ &\
\
$2h $ & applied calibration temperature difference ($\si{\kelvin}$)\
$2h_{\text{opt}}$ & optimal temperature difference ($ \si{\kelvin} $)\
$E_{\text{\text{m}}} $ & measurement uncertainty ($\si[per-mode = fraction]{ \volt \per \kelvin}$)\
$E_{\text{lin}} $ & linearization error ($\si[per-mode = fraction]{\volt \per \kelvin}$)\
$E_S $ & uncertainty of TES ($\si[per-mode = fraction]{\volt \per \kelvin}$)\
$E_{T,\text{cal}} $ & temperature uncertainty of calibration thermometer ($ \si{\kelvin} $)\
$S_{\text{\text{A}}},S_{\text{\text{B}}}$ & TES of materials $\text{\text{A}}$, $\text{\text{B}}$ ($\si[per-mode = fraction]{ \volt \per \kelvin}$)\
$S_{\text{\text{AB}}}$ & TES of material pairing $\text{\text{AB}}$ ($\si[per-mode = fraction]{\volt \per \kelvin}$)\
$S_{\text{st}} $ & TES from industry standard ($\si[per-mode = fraction]{\volt \per \kelvin}$)\
$S_{\text{st,K}} $ & standard K-type TES ($\si[per-mode = fraction]{\volt \per \kelvin}$)\
$S(T) $ & TES as a function of temperature ($\si[per-mode = fraction]{\volt \per \kelvin}$)\
$T$ & temperature ($\si{\celsius}$)\
$ T_{\text{\text{m}}} $ & mean temperature ($\si{\celsius}$)\
$ T_{\text{ref}}$ &reference temperature ($\si{\celsius}$)\
$\Delta T $ & temperature difference ($\si{\kelvin}$)\
$U(T,T_{\text{ref}}) $ &electric potential difference across thermopile ($\si{ \volt}$)\
$U(T_1,T_2) $ &electric potential difference across thermopile ($\si{ \volt}$)\
$ $ &\
\
\[tab:nomenclature\]
Fundamentals of thermocouples and thermopiles {#sec:theory}
=============================================
The TES or Seebeck coefficient $S_{\text{A}}(T)$ is a physical property of material A and is dependent on the local temperature $T$ [@Martin2010; @Huang1990]. As expressed by the Thomson relations (which essentially express microscopic reversibility), the Seebeck effect itself is a manifestation of the cross-correlation between thermal transport due to a gradient in electric potential on the one hand and charge separation induced by temperature differences on the other. Hence, when the material is exposed to an infinitesimal temperature difference $\mathrm{d}T$, an electric potential difference $\mathrm{d}U$ is induced: $$\label{eqn:dU}
\mathrm{d}U = S_{\text{A}}(T) \mathrm{d}T \mathrm{.}$$ The TES of a thermocouple is the difference between the sensitivities of materials A and B ($S_{\text{\text{AB}}} = S_A - S_B$). Thus, for a thermocouple ($N=1$) exposed to a finite temperature difference $T_2-T_1$, an electric potential difference $$\label{eqn:vt}
U(T_{1},T_{2}) = N \cdot \int_{T_1}^{T_2}{S_{\text{\text{AB}}}(T) \mathrm{d}T}\mathrm{,}$$ is induced [@Martin2010; @Bernhard2004]. In order to improve the signal-to-noise ratio, the potential difference can be augmented by aligning multiple ($N>1$) thermocouples in series. This assembly is typically referred to as a thermopile [@Huang1990] and is shown schematically in Fig. \[fig:thermocouple\].
In accordance with technical standards (@DIN60584, @EURAMET) thermopiles are commonly calibrated by comparison with a calibration thermometer. For later reference, we refer to this type of calibration as absolute calibration. One set of junctions is held at a constant reference temperature $T_{\text{ref}}$, while the other set of junctions is exposed to a range of $P$ different temperatures $T_i$ ($i=1,..,P$). The resulting calibration curve (with interpolated values between the $T_i$) is the potential difference $U(T,T_{\text{ref}})$ as a function of the temperature $T$ at the measurement junction. By definition $U(T_{\text{ref}},T_{\text{ref}})$ is zero at $T_{\text{ref}}$. Apart from this offset, the calibration curve is otherwise independent of $T_{\text{ref}}$ [@Huang1990; @Bernhard2004; @Bentley1998]. In general the calibration relationship is a non-linear function of the temperature $T$. For various standard type thermocouples (E, J, K, N and R) @Drebuschak2009 found $U(T,T_{\text{ref}})$ to be a linear function of $T$ at high temperatures and a quadratic function of $T$ at lower temperatures. Typically, the transition from the quadratic to the linear relation takes place between $\SI{250}{\kelvin}$ and $\SI{350}{\kelvin}$. In this study $U(T,T_{\text{ref}})$ is considered to increase monotonously with temperature, as is shown schematically in Fig. \[fig:ut\].
For the absolute calibration type, @Drnovsek1998 analyzed the various sources of uncertainty and found the error $E_{T,\text{cal}}$ associated with the accuracy of the thermometer used for calibration to be the limiting factor. Therefore, measurements that make use of this type of calibration have an uncertainty of at least $E_{\Delta T} \geq \sum E_{T,\text{cal}}$. The sum of calibration thermometry uncertainty stems from the measurement of $T_{\text{ref}}$ and $T_i$.
Theoretical derivation of optimized TES measurement method {#sec:sens}
==========================================================
In order to circumvent the uncertainty limitations imposed by the absolute calibration, the TES of the thermopile is employed as the calibration parameter in this study. For small differential temperatures, Eq. (\[eqn:vt\]) is linearized and the calibration relationship becomes: $$\label{eqn:linvt}
U(\Delta T,T_{\text{\text{m}}}) = {S(T_{\text{\text{m}}})} \cdot \Delta T \mathrm{.}$$ As is detailed below, this correlation is relatively robust towards uncertainty in the measurement of the mean temperature $T_{\text{\text{m}}}$. The TES of a K-type thermocouple is listed in Table \[tab:st-20\] (derived from @DIN60584). In the temperature range of interest in this study, the largest variation of TES due to a fluctuation of $T_{\text{\text{m}}}$ by $\SI{1}{ \kelvin}$ is found to be less than $\SI{0.2}{\percent}$ (between $\SI{-10}{ \celsius}$ and $\SI{-9}{ \celsius}$).
---------------------- ----------------
[temperature]{} [TES ]{}
\[1mm\] [ $T$ in ]{} [$S(T)$ in ]{}
$-10 $ $38.896$
\[0mm\] $-9$ $38.957$
\[1mm\] $-8$ $39.018$
\[1mm\] $-7$ $39.077$
\[1mm\] $-6$ $39.135$
\[1mm\]
---------------------- ----------------
: TES $S(T)$ for selected temperatures based on the industry standard @DIN60584.[]{data-label="tab:st-20"}
The TES $S(T_{\text{\text{m}}})$ of a thermopile can be experimentally determined by either the integral or the differential method [@Martin2010]. The integral (or large $\Delta T$) method involves differentiating an analytic expression of the calibration curve $U(T,T_{\text{ref}})$ [@Wood1988]. One drawback of this method is that small fluctuations of the reference temperature influence the accuracy of the absolute calibration curve $U(T,T_{\text{ref}})$. Furthermore, no objective method to evaluate the accuracy of the sensitivity exists [@Martin2010].
For the differential (small $\Delta T$) method, a temperature difference of $2h$ is applied around $T_{\text{\text{m}}}$ and the resulting potential difference is measured [@Pinisetty2011]. The sensitivity is then calculated as $$\label{eqn:stm}
S(T_{\text{\text{m}}}) = \frac{U(T_{\text{\text{m}}} + h)-U(T_{\text{\text{m}}}-h)}{2h} \mathrm{,}$$ where the numerator is the voltage $U(T_{\text{\text{m}}} + h,T_{\text{\text{m}}}-h)$ measured across the thermopile. Expression (\[eqn:stm\]) is effectively a second order central differencing scheme to calculate the gradient $U' (T_{\text{\text{m}}}) = S(T_{\text{\text{m}}})$ at $T_{\text{\text{m}}}$. This technique was already employed in 1916 by @Wold1916 and is still employed today [@Middleton1953; @Weiss1956; @Burkov2001; @Zhou2005; @Pinisetty2011]. Typical temperature differences $2h$ range from $\SI{1}{\kelvin}$ to $\SI{20}{\kelvin}$. While most studies offer no analysis of measurement uncertainty, @Burkov2001 achieved TES uncertainties on the order of $\pm \SI{4}{\percent}$ to $\pm \SI{10}{\percent}$. However, to the best of our knowledge, no study exists, which addresses the selection strategy of $2h$ itself.
Given the diametrical effect of the temperature difference $2h$ on the measurement uncertainty and linearization error, a procedure to select an optimal temperature difference $2h$ for the calibration is described in the following. To that end, the measurement uncertainty and linearization error are analyzed and an optimal value $2h$ is found, for which the combined error is minimized. At the same time the linearization error is quantified, allowing it to be monitored and limited relative to the measurement uncertainty.
Measurement uncertainty {#sec:me}
-----------------------
When experimentally determining the TES $S(T_{\text{\text{m}}})$ with Eq. (\[eqn:stm\]), the uncertainty of the temperature and voltage measurements is propagated. The maximum relative measurement uncertainty is given by: $$\label{eqn:me1}
\frac{E_{\text{m}}}{S(T_{\text{\text{m}}})} = \frac{E_{2h}}{2h} + \frac{E_U}{U(2h,T_{\text{\text{m}}})} \mathrm{,}$$ where $E_{2h}=2E_{T,\text{cal}}$ and $E_{U}$ are the measurement uncertainty of the applied temperature difference $2h$ and of the potential difference $U$, respectively. Herein, the voltage $U(2h,T_{\text{\text{m}}})$ is approximated by $U(2h,T_{\text{\text{m}}}) = 2h \cdot N\cdot S_{\text{st,K}}(T_{\text{\text{m}}})$, with values for $S _{\text{st,K}}(T_{\text{\text{m}}})$ taken from an industry standard e.g. @DIN60584. This leads to the measurement uncertainty of the sensitivity as a function of the applied (calibration) temperature difference $2h$: $$\label{eqn:me2}
E_{\text{m}}(h) = \frac{E_U+ E_{2h}\cdot N \cdot S_{\text{st,K}}(T_{\text{\text{m}}})}{2h} \mathrm{.}$$ Hence, the influence of the measurement uncertainty can be minimized by applying large temperature differences $2h$ for the calibration. This is depicted schematically in Fig. \[fig:me\].
Linearization error {#sec:le}
-------------------
Determination of $S(T_{\text{\text{m}}})$ according to Eq. (\[eqn:stm\]) approximates the slope of $U(T)$ at $T=T_{\text{\text{m}}}$ by means of linearization with a central difference scheme. Given the generally non-linear character of $U(T)$, neglecting the higher-order terms of the differencing scheme, introduces a linearization error. Using two Taylor series developed around $T_{\text{\text{m}}}$ and evaluated at $+h$ and $-h$ [@Schafer2006], one finds $$\label{eqn:le1}
U'(T_{\text{\text{m}}}) = \frac{U(T_{\text{\text{m}}} +h) - U(T_{\text{\text{m}}} - h)}{2h} - \frac{U'''(T_{\text{\text{m}}})h^2}{3!}-( ...)\mathrm{.}$$ Consequently, the linearization error reads
$$\label{eqn:le2}
E_{\text{lin}}(h)=\frac{U'''(T_{\text{\text{m}}})h^2}{3!}+ (...)\mathrm{ .}$$
Therefore, in contrast to the measurement uncertainty, the linearization error is minimal for small values of $2h$. The trade-off between the measurement and linearization error leads to an optimization problem. In the following, an optimal calibration temperature difference $2h_{\text{opt}}$, with a minimum of combined uncertainty, is derived.
Optimization of total thermoelectric sensitivity uncertainty {#sec:em}
------------------------------------------------------------
The sum of the measurement uncertainty (Eq. (\[eqn:me2\])) and linearization error (Eq. (\[eqn:le2\])) is an expression of the total uncertainty $E_{S}(h) = E_{\text{m}}(h)+E_{\text{lin}}(h)$ of the TES $S$. The minimum of $E_S(h)$ determines $h_{\text{opt}}$, i.e. ${\mathrm{d}E_S(h_{\text{opt}})}/{\mathrm{d}h} \stackrel{!}{=} 0$.
For an approximation of $U(T,T_{\text{ref}})$ with a polynomial of 3rd order, $h_{\text{opt}}$ can be found analytically: $$\label{eqn:hopt}
h_{\text{opt}} = \sqrt[3]{3 \frac{E_U+ E_{2h}\cdot N \cdot S_{\text{st}}(T_{\text{\text{m}}})}{2U'''(T_{\text{\text{m}}})}}\mathrm{.}$$ For higher-order polynomials $h_{\text{opt}}$ is computed numerically.
Sensitivity calibration curve {#sec:sc}
-----------------------------
The method described above yields the TES $S(T_{\text{\text{m}}})$ of a thermopile at a given mean temperature $T_{\text{\text{m}}}$ with minimized total uncertainty. Generally, thermopiles are calibrated for measurements over a range of mean temperatures. Therefore, the aforementioned procedure is repeated at intervals over the temperature range of interest, leading to a sensitivity calibration curve $S(T)$ as a function of temperature.
Illustrative calibration of a thermopile {#sec:illex}
========================================
The application of the above derived TES measurement method is demonstrated in the following illustrative example. The thermopile considered here is used for the detection of heat flux modulation in bifurcating fluidic systems. In this particular experimental setup, geometric boundary conditions limit the number of thermopile junctions and require the thermocouple wiring to be relatively long (over $\SI{0.5}{\meter}$). However, this method is applicable to thermopiles in general, and its effectiveness is expected to increase with the number of junctions (as later described in Section \[sec:disc\]).
Experimental setup and procedure {#sec:exp}
--------------------------------
The thermopile employed in this study was fabricated with $N = 9$ K-type thermocouples, each of $\SI{700}{mm}$ length, connected in series. Two calibration baths (Brookfield TC550 and Polyscience 1197P), with temperature stabilities of $\pm \SI{0.01}{\kelvin}$, were employed. A Pt100 resistance thermometer, with measurement uncertainty $E_{T,\text{cal}} = \pm \SI{0.3}{\kelvin}$, was utilized to measure the temperatures of the baths. The same thermometer was used for every measurement in order to reduce the effect of a systematic thermometer offset. A Fluke 8846A multimeter was used to measure the potential difference across the thermopile with an uncertainty of $E_U = \pm \left( \SI{3.5}{\micro \volt} + \SI{0.0025}{\percent} \hspace{1mm}U \right)$. The apparatus is shown schematically in Fig. \[fig:apparatus\].
[]{} \[fig:apparatus\]
The calibration curve $U(T,T_{\text{ref}})$ was recorded at intervals of $\SI{5}{\kelvin}$ over a temperature range from $\SI{-30}{\celsius}$ to $\SI{40}{\celsius}$. The reference temperature was $T_{\text{ref}}= \SI{10}{\celsius}$. Based on this and with the procedure described in Section \[sec:sens\], the optimal TES calibration temperature difference $2 h_{\text{opt}}$ was calculated for each sensitivity measurement. The TES was then measured in $\SI{5}{\kelvin}$ intervals between $\SI{-10}{\celsius}$ and $\SI{15}{\celsius}$. In some cases limitations on the minimum and maximum temperatures of the calibration baths hindered the application of $2h_{\text{opt}}$. In these cases an alternate calibration temperature difference $2h$ was employed. Four repetitions of the TES measurements were performed for each temperature interval.
Inhomogeneous thermocouple material in combination with temperature gradients can lead to voltage offsets [@Martin2010], for which the calibration relation expressed in Eq. (\[eqn:linvt\]) does not account for. However, no such offset was found when a strong temperature gradient was induced by exposing both ends of the thermopile to the same low temperature (approximately $\SI{-30}{\celsius}$), while the middle section remained at room temperature. Therefore, such offsets are deemed to be negligible throughout the measurements.
Results {#sec:results}
-------
The recorded calibration curve $U(T,T_{\text{ref}})$ is shown in Fig. \[fig:exput\]. A 5th order polynomial was fitted to the data with a least squares regression. Based on this polynomial, the measurement and linearization uncertainty was analyzed for every TES measurement (see Section \[sec:sens\]). For $T_{\text{\text{m}}} = \SI{0}{\celsius}$, an example of the errors as a function of the calibration temperature difference $2h$ is presented in Fig. \[fig:pler\]. The optimal step size is between $2h_{\text{opt}}=\SI{30}{\kelvin}$ and $2h_{\text{opt}} = \SI{50}{\kelvin}$, for the range of mean temperatures $T_{\text{\text{m}}}$ of interest in this study.
The experimental TES data points and the calibration curve $S(T)$ (fitted to the data by a least squares regression with a polynomial of 5th order) are shown in Fig. \[fig:expst\]. The relative, maximum uncertainty of $S(T)$ is the sum of the relative measurement uncertainty (Eq. (\[eqn:me1\])) and the relative linearization error $E_{\text{lin}}/S$ (Section \[sec:le\]). The largest relative, maximum uncertainty is found to be $E_S/S = \SI{2}{\percent}$.
In conjunction with the sensitivity calibration curve and Eq. (\[eqn:linvt\]), the calibrated thermopile can be employed to measure small differential temperatures $$\label{eqn:application}
\Delta T = \frac{U}{ {S(T_{\text{\text{m}}})}} \mathrm{,}$$ with $ U$ and $T_{\text{\text{m}}}$ being the measured potential difference and mean temperature respectively. As already discussed in Section \[sec:sens\], the TES is relatively robust toward uncertainties in the measurement of the mean temperature. For instance in the worst case scenario, uncertainty in $T_{\text{\text{m}}}$ of $\pm \SI{1}{\kelvin}$ would lead to a possible variation of $S$ by less than $\SI{0.2}{\percent}$. This is negligibly small in comparison to the overall TES uncertainty $E_S/S$ of $\SI{2}{\percent}$. Therefore, measuring $T_{\text{\text{m}}}$ with a level of uncertainty on the order of $\pm \SI{1}{\kelvin}$ suffices in this study.
The resulting maximum uncertainty of a temperature difference measurement is the sum of the relative sensitivity uncertainty and that due to the multimeter, namely $$\begin{aligned}
\label{eqn:aer}
E_{\Delta T} &= \left( \frac{E_S}{S}+ \frac{E_U}{U} \right) \Delta T \nonumber \\
&=\frac{E_S}{S} \Delta T+ \frac{E_U}{S} \mathrm{.}\end{aligned}$$ With the equipment used in this illustrative example, small differential temperatures ($\Delta T \leq \SI{1}{\kelvin}$) can be measured with an uncertainty of $E_{\Delta T} = \pm ( \SI{0.01}{\kelvin}+\SI{2}{\percent} \Delta T)$. The multimeter uncertainty ($E_{U}$) is the limiting factor with respect to differential temperature resolution. It contributes an error on the order of no less than $\SI{0.01}{\kelvin}$ to every measurement. In Table \[tab:terror\] the uncertainty is shown for various differential temperatures. For example, a temperature difference of $\SI{0.1}{\kelvin}$ can be measured with an uncertainty of $E_{\Delta T} = \pm \SI{0.01}{\kelvin}$.
------------------------------------------- --------------------------------------
[temperature difference]{} [maximum uncertainty]{}
\[1mm\] [ $\Delta T$ in $\si{\kelvin}$]{} [$E_{\Delta T}$ in $\si{\kelvin}$]{}
$0.01 $ $\pm 0.01$
\[0mm\] $0.05$ $\pm 0.01$
\[1mm\] $0.1$ $\pm 0.01$
\[1mm\] $1.0$ $\pm 0.03$
\[1mm\]
------------------------------------------- --------------------------------------
: Measurement uncertainty of calibrated thermopile for various differential temperatures.[]{data-label="tab:terror"}
Discussion {#sec:disc}
==========
The uncertainty of small differential temperature measurements with the sensitivity calibrated thermopile can be as low as $\pm \SI{0.01}{\kelvin}$. In comparison, using the calibration Pt100 thermometer ($T \pm \SI{0.3}{\kelvin}$) to measure a differential temperature results in uncertainty of no less than $\pm \SI{0.6}{\kelvin}$. The ratio of the calibration thermometer uncertainty for measuring temperature differences, $2E_{T,\text{cal}}$, and the calibrated thermopile uncertainty for differential temperatures, $E_{\Delta T}$, characterizes the increase in accuracy of the calibration process. The calibration procedure in this study yields a ratio of $2E_{T,\text{cal}}/E_{\Delta T} = 60$.
The sensitivity calibrated thermopile also outperforms other thermopile calibration and signal processing techniques for small temperature differences. In the method proposed by @Huang1990, the thermopile is calibrated by comparison to a calibration thermometer with measurement uncertainty of $\SI{0.04}{\kelvin}$. The smallest uncertainty of differential temperature measurements is reported to be $\SI{0.07}{\kelvin}$. Hence, a ratio of $2E_{T,\text{cal}}/E_{\Delta T} = 1.1$ is found for this calibration procedure. In this case, the ratio is based on the most probable error, in contrast to the larger, maximum uncertainty employed in this study. It is therefore somewhat larger than it would be using the maximum uncertainty. An overview of this comparison between the two methods is given in Table \[tab:hucomp\].
@Huang1990 proposed approach
------------------------------------------------ ------------------------------------------------------ -------------------------------------------------------
calibration coefficient $U(T)$ $S(T)$
\[0mm\] $\Delta T$ signal conversion [midpoint extrapolation ]{} linear
$\Delta T = f\left(U(T)_{| T_m \pm \Delta T}\right)$ $\Delta T = U / S(T_m)$
\[1mm\] instrumentation ${E_{T,cal}}$; ${E_U}$ $\pm \SI{0.04}{\kelvin}$; $\SI{\pm 5}{\micro \volt}$ $\pm \SI{0.3}{\kelvin}$; $\pm \SI{3.5}{\micro \volt}$
\[1mm\] max. $\Delta T$ resolution $\SI{0.01}{\kelvin}$
\[1mm\] ratio $2E_{T,cal}/E_{\Delta T}$ $1.1$ $60$
\[1mm\]
: Comparison of the method proposed by @Huang1990 and the novel approach proposed in this study. Methodological aspects are compared above the line, while the details of the illustrative examples are given below.[]{data-label="tab:hucomp"}
As is discussed in the previous section, the accuracy of small differential temperature measurements is limited in this study by the uncertainty of the TES $E_S$ and by the uncertainty associated with the employed multimeter $E_U$ (see Eq. (\[eqn:aer\])). The TES is determined with a relative uncertainty of $E_S/S = \pm 2 \si{\percent}$. Therefore, the corresponding absolute uncertainty of temperature differences scales with the measurand (the measured temperature difference). In contrast to this, the uncertainty due to the multimeter is effectively constant (effectively independent of $\Delta T$) and therefore limits the measurement of small temperature differences. With the multimeter employed in this study the limit is $\pm\SI{0.01}{\kelvin}$. This absolute uncertainty limit is proportional to the multimeter uncertainty. It can be attenuated by employing a more accurate multimeter in the $\Delta T$ measurement process, without the need for renewed calibration.
In general, the same can be achieved by employing thermopiles with a larger number of junctions $N$ (although this was not possible in this study), augmenting $U$. Both of these possibilities lead to reduced relative voltage uncertainty (second term ${E_U}/{U}$ in Eq. (\[eqn:aer\]) ). This term is inversely proportional to $N$. For thermopiles with a large number of junctions, this term can become small in comparison to the TES uncertainty (${E_S}/{S}$). The TES uncertainty stems from the calibration procedure and is generally independent of $N$ (given ${E_U}/U<<{E_{2h}}/{2h}$, which is generally the case). Hence, for large thermopile sensitivities, be it by increased multimeter sensitivity or an increase of thermopile sensitivity, the TES uncertainty limits the differential temperature measurement accuracy. Therefore, the method proposed here is particularly effective for devices with high sensitivities.
On the other hand, for the measurement of large temperature differences, the TES uncertainty is the dominant source of uncertainty. For the measurement of temperatures differences larger than the calibration temperature difference $2h$, the calibrated thermopile has lower accuracy than the calibration thermometer. This is no additional limitation to the procedure as it was explicitly developed for small differential temperatures and is limited to these by the linearity assumption associated with Eq. (\[eqn:application\]).
The novel contribution of the calibration procedure proposed here is the optimized TES measurement. For given apparatus this procedure leads to an optimal temperature difference $2h_{\text{opt}}$ for the differential determination of the TES with minimized uncertainty. This allows the use of larger calibration temperature differences than have been recommended in the past ($\Delta T/ T_{\text{\text{m}}} << 1$) [@Zhou2005; @Martin2010], thus reducing the measurement uncertainty. Furthermore, this method also gives an estimate of the magnitude of the linearization error. For instance, herein, the linearization error was less than one quarter the size of the measurement uncertainty. The application of this technique is not limited to the calibration of thermocouples or thermopiles. It can be applied to optimize the differential method of TES or Seebeck coefficient measurement in general.
To characterize the performance of the optimization procedure on the measurement of TES, we compare the achieved uncertainty using the optimized procedure ($\pm \SI{2}{\%}$) to an equivalent measurement with a commonly employed calibration temperature difference of $2h = \SI{10}{\kelvin}$ [@Wold1916; @Bidwell1922; @Weiss1956; @Burkov2001; @Zhou2005]. For this temperature difference and with the equipment available in this study, a TES uncertainty of $E^{2h=\SI{10}{\kelvin}}_S = \pm \SI{7}{\%}$ was found. Therefore, in this example, the optimized technique reduces the uncertainty by .
The reduction of the TES uncertainty also enhances the accuracy of subsequent differential temperature measurements with the calibrated thermopile. For example, a thermopile calibrated using $2h=\SI{10}{\kelvin}$ could measure a temperature difference of $\Delta T=\SI{0.1}{\kelvin}$ with an uncertainty of $\pm \SI{0.02}{\kelvin}$. In comparison, the uncertainty is halved for the thermopile calibrated in this study, which achieves $\Delta T=\SI{0.1}{\kelvin} \pm \SI{0.01}{\kelvin}$. Although this effect may not be crucial for the measurement of relatively large differential temperatures, the use of the optimization procedure is essential for the measurement of heat flux entering a quasi-isothermal heat sink.
Summary of proposed calibration method {#sec:summary}
--------------------------------------
Regardless of the illustrative example given in section \[sec:illex\] and addressed above, the general sensitivity calibration method to measure the TES can be summarized with the following steps:
1. Acquire the standard (absolute) calibration curve $U(T,T_{\text{ref}})$.
2. Calculate the optimal calibration temperature difference $2h_{\text{opt}}$ based on the minimization of the total TES error $E_S(h) = E_{\text{m}} + E_{\text{lin}}$.
3. Record the sensitivity calibration curve $S(T)$ using Eq. (\[eqn:stm\]) and $2h_{\text{opt}}$. Subsequently, analyze the uncertainty $E_S(h)$.
To measure differential temperatures with the calibrated thermopile the following steps need to be performed:
1. Measure the mean temperature $T_{\text{\text{m}}}$ to within $\pm \SI{1}{\kelvin}$ and select corresponding sensitivity $S(T_{\text{\text{m}}})$.
2. Measure the potential difference $U$ over the thermopile.
3. Calculate the differential temperature with Eq. (\[eqn:application\]) ($\Delta T = U/S(T_{\text{\text{m}}})$) and the uncertainty $E_{\Delta T}$ (Eq. (\[eqn:aer\])).
Steps $(1)$ to $(3)$ correspond to tiers $(1)$ to $(3)$ in Fig. \[fig:bsb\], while steps $(4)$ to $(6)$ are shown in tier $(4)$. Tiers (2) and (3) are repeated at intervals over the temperature range of interest.
Conclusions {#sec:conc}
===========
A novel, improved calibration method for small differential temperature sensors based on the Seebeck effect has been developed. To this end, a standard $U(T,T_{\text{ref}})$ calibration curve is recorded to determine an optimal calibration temperature difference $2h_{\text{opt}}$, which minimizes the combined measurement and linearization uncertainties. The overall thermoelectric sensitivity is then measured with the optimized differential (small $\Delta T$) method. Subsequently it is used as the inverse calibration coefficient $1/S$ of a linear calibration relationship $\Delta T = U / S$.
To demonstrate the proposed method, a thermopile has been fabricated and calibrated specifically for the measurement of small differential temperatures ($\Delta T \leq \SI{1}{\kelvin}$) with an uncertainty of $\pm ( \SI{0.01}{\kelvin}+\SI{2}{\percent} \Delta T)$. In the course of the calibration, the thermoelectric sensitivity has been measured with a maximum relative uncertainty of no more than $E_S/S=\pm \SI{2}{\percent}$. It has been found that in comparison to a measurement with a temperature difference commonly reported in the literature ($2h=\SI{10}{\kelvin}$), the achieved thermoelectric sensitivity uncertainty is $\SI{70}{\%}$ lower. For small differential temperatures the uncertainty of the sensitivity calibrated thermopile is $55$ times lower than that which would be achieved using an alternate signal conversion method and $60$ times lower than employing the calibration thermometer to difference two measured temperatures.
The optimized calibration method is not only applicable to thermopiles, but also to differential measurement of thermoelectric sensitivity or Seebeck coefficient in general. The highly accurate measurement of differential temperatures is especially of interest in the quantification of heat flux modulations through quasi-isothermal heat sinks.
Acknowledgements {#sec:ack .unnumbered}
================
The authors gratefully acknowledge the support received from Steffen Hardt. Funding was provided by the German research foundation (DFG grant no. DI 1689/1-1), which is kindly acknowledged.
Authors’ contributions: TP prepared and edited manuscript, conducted measurements, processed data and developed optimization procedure. IN edited manuscript and coordinated the study. AM developed optimization procedure and took meaasurements. MF and PE fabricated thermopile and performed data analysis. MD developed optimization procedure, edited mansucript and guided the study.
|
---
abstract: |
The goal of automatic is to translate spoken language to a continuous stream of sign language video at a level comparable to a human translator. If this was achievable, then it would revolutionise Deaf hearing communications. Previous work on predominantly isolated has shown the need for architectures that are better suited to the continuous domain of full sign sequences.
In this paper, we propose Progressive Transformers, a novel architecture that can translate from discrete spoken language sentences to continuous 3D skeleton pose outputs representing sign language. We present two model configurations, an end-to-end network that produces sign direct from text and a stacked network that utilises a gloss intermediary.
Our transformer network architecture introduces a counter that enables continuous sequence generation at training and inference. We also provide several data augmentation processes to overcome the problem of drift and improve the performance of models. We propose a back translation evaluation mechanism for , presenting benchmark quantitative results on the challenging dataset and setting baselines for future research.
author:
- Ben Saunders
- Necati Cihan Camgoz
- Richard Bowden
bibliography:
- 'bibliography.bib'
title: 'Progressive Transformers for End-to-End Sign Language Production'
---
Introduction
============
Sign language is the language of communication for the Deaf community, a rich visual language with complex grammatical structures. As it is their native language, most Deaf people prefer using sign as their main medium of communication, as opposed to the written form of spoken language. , converting spoken language to continuous sign sequences, is therefore essential in involving the Deaf in the predominantly spoken language of the wider world.
![Overview of the Progressive Transformer architecture, showing Text to Gloss to Pose (T2G2P) and Text to Pose (T2P) model configurations. (PT: Progressive Transformer, ST: Symbolic Transformer)[]{data-label="fig:architecture_overview"}](figures/17_Sym_Reg_Overview.pdf){width="0.9\linewidth"}
In this paper, we propose a Progressive Transformer model for that is trained on human translation data consisting of continuous sign pose sequences. An overview of our approach is shown in Figure \[fig:architecture\_overview\]. We evaluate two different configurations, first translating from spoken language to sign pose via gloss intermediary[^1] (T2G2P), as this has been shown to increase translation performance [@camgoz2018neural]. In the second configuration we go direct, translating end-to-end from spoken language to sign (T2P).
Our Progressive Transformer is the first model to translate from text to continuous sign pose in an end-to-end manner. Our novelties include an alternative formulation of transformer decoding for continuous variable sequences such as motion capture, where there is no easily defined vocabulary. Furthermore, we introduce a counter encoding into the network which allows prediction of sequence length and the ability to drive the timing at inference. We also propose several data augmentation methods that assist in reducing drift in model production.
In both configurations, to evaluate performance, we propose a back translation evaluation method for , using a back-end to translate back to spoken language (right hand block of Figure \[fig:architecture\_overview\]). We evaluate on the challenging dataset, presenting several benchmark results to underpin future research. We also share qualitative results to give further insight of the models performance to the reader, producing accurate sign pose sequences of an unseen text sentence.
The rest of this paper is organised as follows: In Section \[sec:related\_work\], we go over the previous research on and and the state-of-the-art in the field of machine translation. In Section \[sec:methodology\], we discuss our Progressive Transformer approach for . Section \[sec:quant\_experiments\] outlines the evaluation protocol and presents quantitative results, whilst Section \[sec:qual\_experiments\] showcases qualitative examples. Finally, we conclude the paper in Section \[sec:conc\] by discussing our findings and possible future work.
Related Work {#sec:related_work}
============
### Sign Language Recognition & Translation:
Sign language has been a focus of computer vision researchers for over 30 years [@bauer2000video; @starner1997real; @tamura1988recognition], primarily on isolated [@ozdemir2016isolated] and, relatively recently, the more demanding task of [@koller2015continuous]. However, in both cases the majority of work has relied on manual feature representations (based on the hands) [@cooper2012sign] and statistical temporal modelling [@vogler1999parallel]. The availability of larger datasets, such as [@forster2014extensions], have enabled the application of deep learning approaches such as [@koller2016deephand; @koller2016deepsign] and .
Distinct to , the task of was recently introduced by Camgoz et al. [@camgoz2018neural], aiming to directly translate sign videos to spoken language sentences. is more challenging than due to the differences in grammar and ordering between sign and spoken language. Encoder-decoder models are the current state-of-the-art in , trained end-to-end from spatial sign embeddings via an intermediate gloss representation [@camgoz2018neural].
### Sign Language Production:
Previous approaches to have extensively used animated avatars [@glauert2006vanessa; @karpouzis2007educational; @mcdonald2016automated] that can generate realistic sign production, but rely on phrase lookup and pre-generated sequences. has also been applied to [@kayahan2019hybrid; @kouremenos2018statistical], relying on static rules-based processing that can be difficult to encode.
Recently, deep learning approaches have been applied to the task of [@duarte2019cross; @xiao2020skeleton]. Stoll et al. present an initial model using a combination of and a [@stoll2018sign]. The authors break the problem into three separate processes that are trained independently, producing a concatenation of isolated 2D skeleton poses mapped from sign glosses via a look-up table. Contrary to Stoll et al., our paper focuses on automatic sign production and learning the mapping between text and skeleton pose sequences directly, instead of providing this a priori.
The closest work to this paper is that of Zelinka et al., who build a neural-network-based translator between text and synthesised skeletal pose [@zelinka2020neural]. Zelinka et al. produce a single sign for each word with a set size of 7 frames, generating sequences with a fixed length and ordering. In contrast, our model allows a dynamic length of output sign sequence, learning the correct length and ordering of each word from the data, whilst using a progress counter to determine the end of sequence generation. Unlike [@zelinka2020neural], who work on a proprietary dataset, we produce results on the publicly available , providing a benchmark for future research.
### Neural Machine Translation:
aims to learn a mapping between language sequences, generating a target sequence from a source sequence of another language. were first proposed to solve the sequence-to-sequence problem, with Kalchbrenner et al. [@kalchbrenner2013recurrent] introducing a single that iteratively applied a hidden state computation. Further models were later developed [@cho2014properties; @sutskever2014sequence] that introduced encoder-decoder architectures mapping both sequences to an intermediate embedding space. Bahdanau et al. [@bahdanau2014neural] overcame the bottleneck problem by adding an attention mechanism that facilitated a over the source sentence for the context most useful to the target word prediction.
Transformer networks [@vaswani2017attention], a recent breakthrough, are based solely on attention mechanisms, generating a representation of the entire source sequence with global dependencies. is used to model different weighted combinations of an input sequence, improving the representation power of the model. Transformers have achieved impressive results in many classic tasks such as language modelling [@dai2019transformer; @zhang2019ernie] and sentence representation [@devlin2018bert] alongside other domains including image captioning [@li2019entangled; @zhou2018end] and action recognition [@girdhar2019video].
Applying methods to continuous output tasks is a relatively underresearched problem. Encoder-decoder models and have been used to map text to a human action sequence [@ahn2018text2action; @plappert2018learning] whilst adversarial discriminators have enabled the production of realistic pose [@ginosar2019learning; @lee2019dancing]. In order to determine sequence length of continuous outputs, previous works have used a fixed output size [@zelinka2020neural], which limits the models flexibility, a binary end-of-sequence (EOS) flag [@graves2013generating] or a continuous representation of an EOS token [@mukherjee2019predicting]. Related to this work, transformer networks have been applied to many continuous output tasks such as image generation [@parmar2018image], music production [@huang2018music] and speech recognition [@chiu2018state; @povey2018time].
Progressive Transformers {#sec:methodology}
========================
In this section we introduce Progressive Transformers, which learn to translate spoken language sentences to continuous sign pose sequences. Our objective is to learn the conditional probability $p(Y|X)$ of producing a sequence of signs $Y = (y_{1},...,y_{U})$ with $U$ time steps, given a spoken language sentence $X = (x_{1},...,x_{T})$ with $T$ words. Gloss can also be used as intermediary supervision for the network, formulated as $Z = (z_{1},...,z_{N})$ with $N$ glosses, where the objective is then to learn the conditional probabilities $p(Z|X)$ and $p(Y|Z)$.
Producing a target sign sequence from a reference text sequence poses several challenges. Firstly, the sequences have drastically varying length, with the number of frames being much larger than the number of words ($U >> T$). The sequences also have a non-monotonic relationship due to the different vocabulary and grammar used in sign and spoken languages. Finally, the target signs inhabit a continuous vector space requiring a differing representation to the discrete space of text or gloss.
To address the production of continuous sign sequences, we propose a progressive transformer-based architecture that allows translation from a symbolic to a continuous sequence domain. We first formalise a Symbolic Transformer architecture, converting an input to a symbolic target feature space, as detailed in Figure \[fig:architecture\_details\]a. This is used in our model to convert from spoken language to gloss representation as an intermediary step before pose production, as seen in Figure \[fig:architecture\_overview\].
We then describe the Progressive Transformer architecture, translating from a symbolic input to a continuous output representation, as shown in Figure \[fig:architecture\_details\]b. We use this model for the production of realistic and understandable sign language sequences, either via gloss supervision in the model or direct from spoken language in our end-to-end model. To allow sequence length prediction of a continuous output, we introduce a counter mechanism that allows the model to track the progress of sequence generation. In the remainder of this section we describe each component of the architecture in detail.
Symbolic Transformer
--------------------
We build on the classic transformer [@vaswani2017attention], a model designed to learn the mapping between symbolic source and target languages. As per the standard pipeline [@mikolov2013distributed], we first embed the source, $x_{t}$, and target, $z_{n}$, tokens via a linear embedding layer, to represent the one-hot-vector in a higher-dimensional space where tokens with similar meanings are closer. Symbolic embedding, with weight, $W$, and bias, $b$, can be formulated as: $$\label{eq:word_embedding}
w_{t} = W^{x} \cdot x_{t} + b^{x},\;\;\;\; g_{n} = W^{z} \cdot z_{n} + b^{z}$$ where $w_{t}$ and $g_{n}$ are the vector representations of the source and target tokens.
Transformer networks do not have a notion of word order, as all source tokens are fed to the network simultaneously without positional information. To compensate for this and provide temporal ordering, we apply a temporal embedding layer after each input embedding. For the symbolic transformer, we apply positional encoding, as: $$\label{eq:word_PE}
\hat{w}_{t} = w_{t} + \textrm{PositionalEncoding}(t)$$ $$\label{eq:gloss_PE}
\hat{g}_{n} = g_{n} + \textrm{PositionalEncoding}(n)$$ where PositionalEncoding is a predefined sinusoidal function conditioned on the relative sequence position $t$ or $n$.
Our symbolic transformer model consists of an encoder-decoder architecture. The encoder first learns the contextual representation of the source sequence through self-attention mechanisms, understanding each input token in relation to the full sequence. The decoder then determines the mapping between the source and target sequences, aligning the representation sub-spaces and generating target predictions in an auto-regressive manner.
The symbolic encoder ($E_{S}$) consists of a stack of $L$ identical layers, each containing 2 sub-layers. Given the temporally encoded source embeddings, $\hat{w}_{t}$, a mechanism first generates a weighted contextual representation, performing multiple projections of scaled dot-product attention. This aims to learn the relationship between each token of the sequence and how relevant each time step is in the context of the full sequence. Formally, scaled dot-product attention outputs a vector combination of values, $V$, weighted by the relevant queries, $Q$, keys, $K$, and dimensionality, $d_{k}$: $$\label{eq:attention}
\textrm{Attention}(Q,K,V) = \text{softmax}(\frac{Q K^{T}}{\sqrt{d_{k}}})V$$
stacks parallel attention mechanisms in $h$ different mappings of the same queries, keys and values, each with varied learnt parameters. This allows different representations of the input to be generated, learning complementary information in different sub-spaces. The outputs of each head are then concatenated together and projected forward via a final linear layer, as: $$\begin{aligned}
\label{eq:multi_head_attention}
\textrm{MHA}(Q,K, & V) = [head_{1}, ... ,head_{h}] W^{O}, \nonumber \\
& \textrm{where} \medspace head_{i} = \textrm{Attention}(QW_{i}^{Q},KW_{i}^{K},VW_{i}^{V})\end{aligned}$$ and $W^{O}$,$W_{i}^{Q}$,$W_{i}^{K}$ and $W_{i}^{V}$ are weights related to each input variable.
The outputs of are then fed into the second sub-layer of a non-linear feed-forward projection. A residual connection [@he2016deep] and subsequent layer norm [@ba2016layer] is employed around each of the sub-layers, to aid training. The final symbolic encoder output can be formulated as: $$\label{eq:symbolic_encoder}
h_{t} = E_{S}(\hat{w}_{t} | \hat{w}_{1:T})$$ where $h_{t}$ is the contextual representation of the source sequence.
The symbolic decoder ($D_{S}$) is an auto-regressive architecture that produces a single token at each time-step. The positionally embedded target sequences, $\hat{g}_{n}$, are extracted and passed through an initial MHA self-attention layer similar to the encoder, with an extra masking operation. Alongside the fact that the targets are offset from the inputs by one position, the masking of future frames prevents the model from attending to subsequent time steps in the sequence.
The decoder contains a further sub-layer, which maps representations from the encoder and decoder and learns the alignment between the source and target sequences. The final sub-layer is a feed forward layer, with all sub-layers followed by a residual connection and layer normalisation as in the encoder. After all encoder layers are processed, a final non-linear feed forward layer is applied, with a softmax operation to generate the most likely output token at each time step. The output of the symbolic decoder can be formulated as: $$\label{eq:symbolic_decoder}
z_{n+1} = \operatorname*{argmax}_{i} D_{S}(\hat{g}_{n} | \hat{g}_{1:n-1} , h_{1:T} )$$ where $z_{n+1}$ is the output at time $n+1$, from a target vocabulary of size $i$.
Progressive Transformer
-----------------------
We now adapt our symbolic transformer architecture to cope with continuous outputs, so that it can be used to convert source sequences to a continuous target domain. In this work, Progressive Transformers (Figure \[fig:architecture\_details\]b) translate from the symbolic domains of gloss or text to continuous skeleton pose sequences that represent the motion of a signer producing a sentence of sign language. The model must produce skeleton pose outputs that can both express a realistic sign pose sequence and an accurate translation of the given input sequence.
We represent each sign frame, $y_{u}$, as a continuous vector of 3D joint positions of the signer. These continuous joint values are first passed through a linear embedding layer, allowing sign poses of similar content to be closely represented in the dense space. The continuous embedding layer can be formulated as: $$\label{eq:joint_embedding}
j_{u} = W^{y} \cdot y_{u} + b^{y}$$ where $j_{u}$ is the embedded 3D joint coordinates of each frame, $y_{u}$.
With continuous sign pose as a target, we apply a counter encoding layer as temporal embedding, determining the progress of sequence generation and allowing timing to be driven at inference. The counter, $c$, holds a continuous value between 0 and 1 and represents the frame position relative to the total sequence length. At training time, the joint embeddings, $j_{u}$, are concatenated with the respective counter value, $c_{u}$. The counter encoding is formulated as: $$\label{eq:counter_appending}
\hat{j}_{u} = [j_{u},\textrm{CounterEncoding}(u)]$$ where CounterEncoding is a function producing the counter value for frame $u$, and $\hat{j}_{u}$ is the concatenated counter joint embeddings. At each time-step, counter values are predicted alongside the skeleton pose, as shown in Figure \[fig:counter\_concatenation\], with sequence generation concluded once the counter reaches 1. This provides a way to determine the end of sequence without the use of a tokenised vocabulary.
The counter provides the model with information relating to the length and speed of each sign pose sequence, determining the sign duration. At inference, we drive the sequence generation by replacing the predicted counter value, $\hat{c}$, with the ground truth timing information, $c^{*}$, to produce a stable output sequence.
The Progressive Transformer model also consists of an encoder-decoder architecture. Due to the input coming from a symbolic source, the encoder has a similar setup to the symbolic transformer, learning a contextual representation of the input sequence. As the representation will ultimately be used for the end goal of , they must contain sufficient context to fully and accurately reproduce sign. Taking as input the temporally embedded source embeddings, $\hat{w}_{t}$, the encoder can be formulated as: $$\label{eq:progressive_encoder}
r_{t} = E_{S} (\hat{w}_{t} | \hat{w}_{1:T})$$ where $r_{t}$ is the encoded contextual representation.
The progressive decoder ($D_{P}$) is an auto-regressive model that produces a skeleton pose state at each time-step, alongside the counter value described above. Distinct from symbolic transformers, the progressive decoder produces continuous sequences that hold a sparse representation in a large continuous sub-space. The counter-concatenated joint embeddings, $\hat{j}_{u}$, as extracted as target input, representing the sign information of each frame.
A self-attention sub-layer is first applied, with target masking to avoid attending to future positions. A further mechanism is then used to map the symbolic representations from the encoder to the continuous domain of the decoder, learning the important alignment between spoken and sign languages.
A final feed forward sub-layer follows, with each sub-layer followed by a residual connection and layer normalisation as before. No softmax layer is used as the skeleton joint coordinates can be regressed directly and do not require stochastic prediction. The progressive decoder output can be formulated as: $$\label{eq:progressive_decoder}
[\hat{y}_{u+1},\hat{c}_{u+1}] = D_{P}(\hat{j}_{u} | \hat{j}_{1:u-1} , r_{1:T} )$$ where $\hat{y}_{u+1}$ corresponds to the 3D joint positions representing the produced sign pose of frame $u+1$ and $\hat{c}_{u+1}$ is the respective counter value. The decoder learns to generate one frame at a time until the predicted counter value reaches 1, determining the end of sequence. Once the full sign pose sequence is produced, the Progressive Transformer model is trained on the loss between the predicted sequence, $\hat{y}_{1:U}$, and the ground truth, $y_{1:U}^{*}$: $$\label{eq:loss_mse}
L_{MSE} = \frac{1}{u} \sum_{i=1}^{u} ( y_{1:U}^{*} - \hat{y}_{1:U} ) ^{2}$$
The progressive transformer outputs, $\hat{y}_{1:U}$, represent the 3D skeleton joint positions of each frame of a produced sign sequence. Animating a video from this sequence is then a trivial task, plotting the joints and connecting the relevant bones, with timing information provided from the counter. These 3D joints could subsequently be used to animate an avatar [@kipp2011sign; @mcdonald2016automated] or condition a [@isola2017image; @zhu2017unpaired].
Quantitative Experiments {#sec:quant_experiments}
========================
In this section we share our experimental setup and report experimental results. We first provide dataset and evaluation details, outlining back translation. We then evaluate both symbolic and progressive transformer models, demonstrating results of data augmentation and model configuration.
Sign Language Production Dataset {#sec:dataset}
--------------------------------
Forster et al. released [@forster2012rwth] and the extended [@forster2014extensions] as large video-based corpora containing parallel sequences of and spoken text extracted from German weather forecast recordings. These datasets are ideal for computational sign language research due to the provision of gloss level annotations, becoming the primary benchmark for both and .
In this work, we use the publicly available dataset introduced by Camgoz et al. [@camgoz2018neural], a continuous extension of the original . This corpus includes parallel sign videos and German translation sequences with redefined segmentation boundaries generated using the forced alignment approach of [@koller2016deepsign]. 8257 videos of 9 different signers are provided, with a vocabulary of 2887 German words and 1066 different sign glosses from a combined 835,356 frames.
We train our progressive transformer to generate sequences of 3D skeletons. 2D joint positions are first extracted from each video using OpenPose [@cao2018openpose]. We then utilise the skeletal model estimation improvements presented in [@zelinka2020neural] to lift the 2D joint positions to 3D. An iterative inverse kinematics approach is applied to minimise 3D pose whilst maintaining consistent bone length and correcting misplaced joints. Finally, we apply skeleton normalisation similar to [@stoll2018sign] and represent 3D joints as $x$, $y$ and $z$ coordinates. An example is shown in Figure \[fig:czech\_skel\].
Evaluation Details {#sec:evaluation_details}
------------------
In this work, we present back-translation as a means of evaluation. Back translation, the dual mapping between source and text sequences, has been predominantly used to generate new data and improve the performance of monolingual corpus translation [@gracca2019generalizing; @sennrich2015improving]. Recent work on used an discriminator to determine whether generated skeletons were identifiable [@xiao2020skeleton], but did not measure the translation performance. We utilise the state-of-the-art [@camgoz2018neural] as our back translation model, generating the spoken language prediction of a produced sign pose sequence. BLEU and ROUGE scores can then be computed on the predicted text sequences, providing BLEU n-grams from 1 to 4 for completeness.
In the following experiments, our symbolic and progressive transformer models are built with 2 layers, 8 heads and embedding size of 256. All parts of our network are trained with Xavier initialisation [@glorot2010understanding] and Adam optimization [@kingma2014adam] with default parameters and a learning rate of $10^{-3}$. Our code is based on Kreutzer et al.’s NMT toolkit, JoeyNMT [@JoeyNMT], and implemented using PyTorch [@paszke2017automatic].
Symbolic Transformer: Text to Gloss
-----------------------------------
Our first experiment measures the performance of the symbolic transformer architecture for sign language understanding. We train our symbolic transformer to predict the gloss representation of a source spoken language text. Table \[tab:text\_to\_gloss\_results\] shows our model achieves state-of-the-art results, significantly outperforming that of Stoll et al. [@stoll2018sign] who use an encoder-decoder network with 4 layers of 1000 . This supports our use of the proposed transformer architecture for sign language understanding.
\[tab:text\_to\_gloss\_results\]
Progressive Transformer: Gloss to Pose
--------------------------------------
In our next set of experiments, we evaluate our progressive transformer and its capability to produce a continuous sign pose sequence from a given symbolic input. As a baseline, we train a progressive transformer model to translate from gloss to sign pose, with results shown in Table \[tab:data\_augmentation\_results\] (Base).
We believe our base progressive model suffers from prediction drift, with erroneous predictions building over time. As transformer models are trained to predict the next time-step of all ground truth inputs, they are often not robust to noise in target inputs. At inference time, with predictions based off previous outputs, errors are propagated throughout the full sequence generation, quickly leading to poor quality production. The impact of drift is heightened due to the continuous distribution of the target skeleton poses. As neighbouring frames differ little in content, a model learns to just copy the previous ground truth input and receive a small loss penalty. We thus experiment with various data augmentation approaches in order to overcome drift and improve performance.
\[tab:data\_augmentation\_results\]
### Future Prediction
Our first data augmentation method is conditional future prediction, requiring the model to predict more than just the next frame in the sequence. Experimentally, we find the best performance comes from a prediction of all of the next 10 frames from the current time step. As can be seen in Table \[tab:data\_augmentation\_results\], prediction of future time steps increases performance from the base architecture. We believe this is because the model now cannot rely on just copying the previous frame, as there are more considerable changes to the skeleton positions in 10 frames time. The underlying structure and movement of signing has to be learnt, encoding how each gloss is represented and reproduced in the training data.
### Just Counter
Inspired by the memorisation capabilities of transformer models, we next experiment with a pure memorisation approach. Only the counter values are provided as target input to the model, as opposed to the usual full 3D joint positions. We show a further performance increase with this approach, considerably increasing the BLEU-4 score as shown in Table \[tab:data\_augmentation\_results\].
We believe the just counter model setup helps to allay the effect of drift, as the model now must learn to decode the target sign pose solely from the counter position, without relying on the ground truth joint embeddings it previously had access to. Setup is then identical at both training and inference, with the model only having to generalise to new data rather than new prediction inputs.
### Gaussian Noise
Our final augmentation experiment examines the effect of applying noise to the skeleton pose sequences during training, increasing the variety of training data in order to build a more robust model. For each joint, statistics on the positional distribution of the previous epoch are collected, with randomly sampled noise applied to the inputs of the next epoch. Applied noise is multiplied by a noise factor, $r_{n}$, with empirical validation suggesting $r_{n} = 5$ gives the best performance. An increase of Gaussian noise causes the model to become more robust to prediction inputs, as it must learn to correct the augmented inputs back to the target outputs. However, as $r_{n}$ increases even further, model performance starts to degrade.
Table \[tab:data\_augmentation\_results\] (FP & GN) shows that the best BLEU-4 performance comes from a combination of future prediction and Gaussian noise augmentation. The model must learn to cope with both multi-frame prediction and a noisy input, building a firm robustness to drift. We continue with this setup for further experiments.
\[tab:configuration\_results\]
Text2Pose v Text2Gloss2Pose
---------------------------
Our final experiment evaluates the two network configurations outlined in Figure \[fig:architecture\_overview\], sign pose production either direct from text or via a gloss intermediary. Text to Pose (T2P) consists of a single progressive transformer model with spoken language input, learning to jointly translate from the domain of spoken language to sign and subsequently produce meaningful sign representations. Text to Gloss to Pose (T2G2P) uses an initial symbolic transformer to convert to gloss, an important intermediary step to provide correct sign ordering [@camgoz2018neural], which is then input into a further progressive transformer to produce sign pose sequences.
As can be seen from Table \[tab:configuration\_results\], the T2P model outperforms that of T2G2P, with a BLEU-4 of 11.82 and 10.51 for dev and test respectively. This is surprising, as a large body of previous work has suggested that using gloss as intermediary helps networks learn [@camgoz2018neural]. However, we believe this is because there is more information available within spoken language compared to a gloss representation, with more tokens per sequence to predict from. Gloss representation can result in an bottleneck where important information is lost, whereas regressing directly from text provides more useful context.
The success of the T2P network shows that our progressive transformer model is powerful enough to complete two sub-tasks; firstly mapping spoken language sequences to a sign representation, then producing an accurate sign pose recreation. This is important for future scaling and application of the model architecture, as many sign language domains do not have gloss availability.
![Examples of produced sign pose sequences. The top row shows the spoken language input from the unseen validation set alongside English translation. The middle row presents our produced sign pose sequence from this text input, with the bottom row displaying the ground truth video for comparison[]{data-label="fig:qualitative_output"}](figures/Multi_Visual_Outputs.pdf){width="1.0\linewidth"}
Furthermore, our final BLEU-4 scores outperform the state-of-the-art in the similar Sign to Text task outlined in Camgoz et al. [@camgoz2018neural] (9.94 BLEU-4). Note: this is an unfair direct comparison, but it does provide an indication of model performance and the quality of the produced sign pose sequences.
Qualitative Experiments {#sec:qual_experiments}
=======================
In this section we report qualitative results for our progressive transformer model. We share snapshot examples of produced sign pose sequences in Figure \[fig:qualitative\_output\], with more examples provided in supplementary material. The unseen spoken language sequenceis shown as input alongside the sign pose sequence produced by our Progressive Transformer model, with ground truth video for comparison.
As can be seen from the provided examples, our model produces visually pleasing and realistic looking sign with a close correspondence to the ground truth video. Body motion is smooth and accurate, whilst hand shapes are meaningful if a little under-expressed. We find that the most difficult production occurs with proper nouns and specific entities, due to the lack of grammatical context and examples in the training data.
These examples show that regressing continuous sequences can be successfully achieved using an attention-based mechanism. The predicted joint locations for neighbouring frames are closely positioned, showing that the model has learnt the subtle movement of the signer. Smooth transitions between signs are produced, highlighting a difference from the discrete generation of spoken language.
Conclusions {#sec:conc}
===========
The production of continuous sign language sequences from spoken language is an important task to improve communication between the Deaf and hearing. Previous work has focused on producing concatenated isolated signs instead of continuous sign sequences. In this paper, we proposed Progressive Transformers, a novel architecture that translates from discrete spoken language sequences to continuous skeleton pose outputs representing sign. We presented two model configurations, an end-to-end network that produces sign direct from text and a stacked network that utilises a gloss intermediary. Additionally, we introduced a counter that allowed continuous sequence generation at training and inference.
We evaluated our approach on the challenging dataset, setting baselines for future research with a back translation evaluation mechanism. We found that regressing continuous sequences can be successfully achieved using a self-attention-based model, helping to learn the correct skeleton pose representation of a signer. Our experiments showed the importance of several data augmentation techniques to improve the performance of models. Furthermore, we have shown that a direct text to pose translation configuration can outperform a gloss intermediary model, meaning models are not limited to only training on data where expensive gloss annotation is available.
As future work, we would like to expand our network to multi-channel sign production, focusing on non-manual aspects of sign language such as body pose, facial expressions and mouthings. It would be interesting to condition a to produce sign videos, learning a prior for each sign represented in the data.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work received funding from the SNSF Sinergia project ‘SMILE’ (CRSII2 160811), the European Union’s Horizon2020 research and innovation programme under grant agreement no. 762021 ‘Content4All’ and the EPSRC project ‘ExTOL’ (EP/R03298X/1). This work reflects only the author’s view and the Commission is not responsible for any use that may be made of the information it contains.
Appendix
========
In this appendix we show further qualitative results for our progressive transformer model. We share snapshot examples of produced sign pose sequences in Figure \[fig:qualitative\_output\_1\] and \[fig:qualitative\_output\_2\]. The unseen spoken language sequence is shown as input alongside the sign pose sequence produced by our Progressive Transformer model, with ground truth skeleton pose and video provided for comparison. We also provide videos of produced sign pose sequences in the supplementary materials file, highlighting the quality of production against both ground truth video and skeleton pose.
Qualitative Results
-------------------
![Examples of produced sign pose sequences showing spoken language input, produced sign pose and ground truth pose and original video.[]{data-label="fig:qualitative_output_1"}](figures/Combined_1.jpg){width="0.9\linewidth"}
![Examples of produced sign pose sequences showing spoken language input, produced sign pose and ground truth pose and original video.[]{data-label="fig:qualitative_output_2"}](figures/Combined_2.jpg){width="0.9\linewidth"}
[^1]: Glosses are a written representation of sign, defined as minimal lexical items.
|
---
abstract: 'We succeeded in observing two large spicules simultaneously with the Atacama Large Millimeter/submillimeter Array (ALMA), the Interface Region Imaging Spectrograph (IRIS), and the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory. One is a spicule seen in the IRIS Mg II slit-jaw images and AIA 304Å images (MgII/304Å spicule). The other one is a spicule seen in the 100GHz images obtained with ALMA (100GHz spicule). Although the 100GHz spicule overlapped with the MgII/304Å spicule in the early phase, it did not show any corresponding structures in the IRIS Mg II and AIA 304Å images after the early phase. It suggests that the spicules are individual events and do not have a physical relationship. To obtain the physical parameters of the 100GHz spicule, we estimate the optical depths as a function of temperature and density using two different methods. One is using the observed brightness temperature by assuming a filling factor, and the other is using an emission model for the optical depth. As a result of comparing them, the kinetic temperature of the plasma and the number density of ionized hydrogens in the 100GHz spicule are $\sim$6800 K and $\rm 2.2\times10^{10} \ cm^{-3}$. The estimated values can explain the absorbing structure in the 193Å image, which appear as a counterpart of the 100GHz spicule. These results suggest that the 100GHz spicule presented in this paper is classified to a macrospicule without a hot sheath in former terminology.'
author:
- Masumi Shimojo
- Tomoko Kawate
- 'Takenori J. Okamoto'
- Takaaki Yokoyama
- Noriyuki Narukage
- Taro Sakao
- Kazumasa Iwai
- 'Gregory D. Fleishman'
- Kazunari Shibata
bibliography:
- 'Alma\_Spicule2019.bib'
title: Estimating the temperature and density of a spicule from 100 GHz data obtained with ALMA
---
Introduction {#sec:intro}
============
A spicule is one of the building blocks of the solar atmosphere and a key-phenomenon for understanding the heating of corona and chromosphere. It is believed that spicules provide the energy and mass for forming the hot atmospheric layers and solar wind. Observations of spicules on the solar limb have been carried out since the 1870s [@1870sepd.book.....S] and visible and ultraviolet chromospheric lines (e.g., H$\rm\alpha$, Ca II K) are used for most observations. Due to the recent chromospheric observations with a high-spatial resolution from solar observing satellites, our knowledge of a spicule is rapidly renewing . On the other hand, it is hard to derive physical parameters of the spicules from theses chromospheric lines because of deviating from the local thermodynamic equilibrium (LTE) for these lines. Some authors determined temperature and density of spicules and their distribution, but the results of them depend on the complicated forward modeling of the radiation from non-LTE medium.
Theoretical studies of spicules have also been widely performed, such as slow shock model [e.g. @1961ApJ...134..347O; @1982SoPh...75...99S; @1982SoPh...78..333S; @1982ApJ...257..345H; @1988ApJ...327..950S; @1990ApJ...349..647S; @2006ApJ...647L..73H; @2015ApJ...812L..30I], Alfven wave model . Realistic radiative MHD simulation models have also been developed recently [e.g., @2017Sci...356.1269M; @2017ApJ...848...38I], though more detailed physical analyses are necessary to understand the basic mechanism of splicule formation even for simulation results (See also [@2000SoPh..196...79S] and [@2012SSRv..169..181T] for the review of spicule theories.) To confirm whether the physical processes in the simulations occur in the solar atmosphere, comparisons between the simulations and the observations are essential. For this purpose, it is required to derive physical quantities of spicules from the observations, but it is not so easy as mentioned above.
Since the millimeter waves emitted from the chromospheric plasma satisfies the LTE condition, it is relatively easier than other chromospheric lines to derive the physical parameters. On the other hand, achieving a high-spatial resolution with millimeter waves require an interferometer with long baselines, and no one had investigated spicules with these wavelength ranges until recently. The Atacama Large Millimeter/submillimeter Array [ALMA; @2009IEEEP..97.1463W] is the largest interferometer in the world for observing astronomical objects with millimeter and submillimeter waves, and started scientific solar observations in 2016. The spatial resolution of ALMA for observing the Sun is $\sim$ 2 with 100 GHz and is useful for investigating spicules. The ALMA data already show spicules and plasmoid eruptions .
We obtained the observing time in ALMA Cycle 4 and succeeded in observing spicules with ALMA as well as the Interface Region Imaging Spectrograph [IRIS; @2014SoPh..289.2733D], and the Atmospheric Imaging Assembly [AIA; @2012SoPh..275...17L] onboard the Solar Dynamics Observatory [SDO; @2012SoPh..275....3P] simultaneously. In this paper, we present results of the coordinated observation, estimate the temperature and density of the spicules from the ALMA data, and discuss what phenomena in the past observations correspond to spicules observed at 100 GHz.
Observations {#sec:obs}
============
The ALMA observatory started offering solar observations with Band 3 and Band 6 at Cycle 4, which is the 5th open-use observing period from October 2016 to September 2017. We succeeded in obtaining the observing time in Cycle 4, and our observation was carried out between 14:22 – 16:13UT on 26 April 2017. We used the Band3 receiver and the Time Division Mode (TDM) of the correlator. Hence, the frequency of the 1st local oscillator is 100 GHz [@2016CyclePG].
We synthesized one solar image from the visibility data obtained in each integration set whose duration is 2 seconds in the correlator, i.e., the time cadence of the images is 2 seconds. We used the full spectral range obtained with the TDM to enhance the signal-to-noise ratio in an image. Therefore, a synthesized image shows a distribution of intensity at 100 GHz with a bandwidth of about 8 GHz. To obtain synthesized images with high dynamic range, we carried out the self-calibration only for the phase after the standard calibration described in [@2017SoPh..292...87S]. The self-calibration has 5 steps with the different accumulating period to obtain the solution from 10 minutes for the first step to 2 seconds for the last step. The CLEAN deconvolution method is used for the image synthesis with the Clark algorithm and Briggs weighting scheme on CASA 5.4.0 [@2007ASPC..376..127M]. During the five-steps self-calibration, we used the CLEAN model created in the previous self-calibration as the initial guess of the synthesis. The field of view of the synthesized images is about 1. The sizes of the synthesized beam are 2.67 and 1.88 along the major and minor axes, respectively. Although the singe-dish observations were carried out with the interferometric observations for obtaining the full-sun images [@2017SoPh..292...88W], we did not combine the synthesized images and full-sun images because it is not established how to deal with the solar limb in the full-sun image for the combining. Hence, in this study, we only deal with relative brightness temperatures. They are roughly the same as the differences from the averaged brightness temperature of the field of view. To remove jitter motion caused by the remaining phase error, we spatially shift the images based on the offsets estimated by the auto-correlation of the images.
The coordinated observation with IRIS was executed during the observing period. IRIS obtained slit-jaw images of Mg II (2796Å) bands with a cadence and a spatial resolution of 40 s and 0.40, respectively. Although the spectra of the Mg II lines are obtained with IRIS during the period, we do not use the spectral data because the slit positions do not cover the phenomena described in the next section. We also use 304Å (He II) and 193Å (Fe XII) band images taken with AIA. They provide context filtergrams with a cadence of 12 s. The spatial resolution is 1.5.
For the co-alignment of the images, we compared between the AIA 304Å images and the IRIS Mg II images, and aligned the Mg II images to AIA 304 Å images. We aligned the ALMA images with AIA ones only by using the coordinate information in the data header because the pointing accuracy of ALMA (within 0.6 rms) is sufficient for the current analysis [@ALMACycle4TH].
The spicules observed with ALMA, IRIS, and AIA/SDO {#sec:spic}
==================================================
The panels in Figure \[fig:all\] show spicules in the 100GHz, 304Å, 193Å, and Mg II images. The fields of view of them are the same. In the Mg II image, the solar limb is located around X=956, and elongated structures of most spicules cannot be recognized beyond X=963. In the 304Å image, we cannot see the counterpart of the photosphere, and the limb at 304 Å is located around the apexes of the spicules seen in the Mg II images. The limb in the 100GHz image is located around X=960and is very similar to that in the 193Å image, as reported by [@2018ApJ...863...96Y]. We found two large spicules whose apexes are significantly higher than the others seen in the Mg II images. The region that these large spicules appear is indicated by the red boxes in Figure \[fig:all\]. We concentrate on the phenomena seen in this region and describe them in Figure \[fig:zoom\] and Movie 1.
A large spicule appears at around 14:37:50UT above the limb of the Mg II and 304Å images (Figure \[fig:zoom\] and Movie 1). Because the shapes of the spicules in these images resemble each other, we call the large spicule hereafter “Mg II/304Å spicule”. In the 304Å images, the Mg II/304Å spicule reached the maximum height at around 14:39UT and the height from the limb seen in the Mg II images are $\sim$20($\sim$15,000 km), and the maximum width is about 4 – 5 (2,900 – 3,700 km). The rising velocity of the Mg II/304Å spicule is about 70 km $\rm s^{-1}$ (the green line in Figure \[fig:ts\] ). We cannot find the counterpart of the spicule in 100GHz and 193Å images.
About one minute later from the start of the Mg II/304Å spicule (14:38:40UT), another large spicule appeared above the limb in the 100 GHz images (Movie 1). We call the spicule “100GHz spicule” in this paper. As pointed out in [@2017SoPh..292...87S], the brightness just above the solar limb in the synthesized images is decreasing gradually with the distance from the limb. The brightness profile does not show the actual brightness profile of the limb that should show steeply decreasing at the limb. It is caused by the lack of baselines. In other words, the number of ALMA’s antennas is not enough for synthesizing images of the solar limb. To remove the influence of the artificial structure, we estimated the brightness temperature of the 100GHz spicule from subtracting the brightness at the pre-event, from the brightness of the events, pixel by pixel. As a result, the peak enhancement of the brightness temperature caused by the spicule is about 240 K. This signal level is much larger than the noise, i.e. $\sim$2.6 K (Shimojo et al. 2017b). An absorbing structure appeared in the 193Å images, and its shape is similar to the spicule in the 100GHz image. The rising velocity of the spicule seen in the 100GHz and 193Å images is about 40 km $\rm s^{-1}$ (the yellow line in Figure \[fig:ts\]). The 100GHz spicule reached the maximum height of $\sim$15 ($\sim$11,000 km) from the limb seen in the Mg II images at 14:41UT. The obtained width in 100 GHz image is 2 ($\sim$1,500 km), but the actual width can be narrower than the value because the width is comparable to the size of the synthesized beam.
The 100GHz spicule is overlapped by the Mg II/304Å spicule at 14:38:40UT, as shown in the upper panels of Figure \[fig:zoom\]. Nevertheless, we cannot find any remarkable enhancements in Mg II and 304Å images that correspond to the 100GHz spicule. After 14:43UT (the lower panels in Figure \[fig:zoom\]), there is no overlapped region between the 100GHz spicule and Mg II/304Å spicule. Moreover, the apparent velocity of the 100 GHz spicule ($\sim$40 km s$^{-1}$) is significantly slower than the Mg II/304A spicule ($\sim$70 km s$^{-1}$), as shown in Figure \[fig:ts\]. The facts would suggest that the spicules are individual events and do not have a physical relationship.
Considering the typical formation temperatures of the Mg II band [5000 – 16,000 K: @2014SoPh..289.2733D] and 304Å band [$\sim$ 50,000 K: @2012SoPh..275...17L], the temperature of the 100GHz spicule should be lower than about 10,000 K because we cannot find its counterpart in Mg II and 304Å images. Can the plasma with such temperature and density be explained by the observed structure at 100 GHz? To answer the question, we estimate the optical depths of the spicule at 100 GHz from observations and an emission model, and examine the temperature and density of the 100GHz spicule. Before the estimations, we must consider a filling factor because the spatial resolution of the 100GHz images is not enough for resolving the spicule. The spatial resolution of AIA is not enough either, so that we assume that the width of the 100GHz spicule is 0.5. Since the apparent width of the 100GHz spicule is 2, the filling factor of the spicule in the 100GHz image is roughly 0.25. Thus, we use 960 K (= 240 K/0.25) as the brightness temperature of the 100GHz spicule. From the observed enhancement of the brightness temperature, the optical depth of the spicule as a function of electron temperature ($\tau = - log_{e}(1-T_{b}/T)$ $\tau$: optical depth, , $T$: electron temperature, $T_{b}$: brightness temperature) is shown by the black solid line in Figure \[fig:od\], based on the following assumptions: 1) the background emission is negligibly small because the phenomenon locates in the off-limb region, 2) the temperature range of the 100GHz spicule is lower than 10,000K, 3) the emission from the spicule is the thermal free-free emission (electron-proton free-free) from the medium satisfied the LTE condition. On the other hand, by assuming the temperature and density of the spicule and the line-of-sight diameter of the spicule is 0.5 (same as the width), we derived the optical depth of the thermal free-free emission from the emission model . The model functions indicate the colored lines in Figure \[fig:od\]. The black line estimates from the observations and the colored line of the model cross when we assume the kinetic temperature of the plasma and the number density of ionized hydrogens to be $\sim$6800 K and $\rm2.2\times10^{10} \ cm^{-3}$ respectively and the values indicate the physical parameters of the 100GHz spicule.
[@1998ApJ...504L.127Z] presented that the absorption at the limb in 195Å images show the chromosphere, where the number density of neutral hydrogens is $\rm 10^{10} cm^{-3}$. Therefore, for explaining the absorbing counterpart of the 100GHz spicule in the 193Å images, the number density of neutral hydrogens should be similar or larger than the number density of the electrons in the temperature range that the thermal free-free emission is dominant. According to the lower-right panel of Figure 1 in , the temperature of the 100GHz spicule should be 4000 – 7000 K. The temperature range is consistent with the temperature derived above, and thus, justifies our estimation. Moreover, the total number density of hydrogens, which is the sum of the number densities of ionized and neutral hydrogens, is assumed $\rm 10^{11} cm^{-3}$ in the lower-right panel of Figure 1 in Rutten (2017). Hence, the mass density of the 100GHz spicule might be about $\rm \sim10^{-13} g \ cm^{-3}$ that is consistent with the previous results of the infrared observations [@2000SoPh..196...79S].
Discussion {#sec:disc}
==========
The spicules described in this paper would be categorized into “macrospicule” formerly because their sizes and velocities are larger than the typical values of spicules. Therefore, it is appropriate to compare our results with the properties of macrospicules, rather than those of spicules. Macrospicules are most often visible in EUV transition-region lines, such as in He II 304 Å, and consist of a cool core and hot sheath . Since the Mg II/304Å spicule has a cool component revealed in the Mg II image and a hotter component shown in the He II 304Å images, it has the typical properties of macrospicules described in the preceding papers.
On the other hand, for the 100 GHz spicule, we do not find its counterpart in Mg II/304Å images. [@1991ApJ...376L..25H] presented that the brightness temperature of macrospicules at 4.8, 8.5, and 15 GHz can be explained by an empirical model that macrospicules consist of a cool core at $\sim$8000K surrounding by a hot sheath at $\rm 1 $ – $\rm 2 \times 10^{5}$ K. The temperature of the cool core is similar to the 100GHz spicule, and there is no significant signal of hotter lines around the 100GHz spicule. Hence, the 100GHz spicule described in this paper is a macrospicule without a hot sheath that is already reported by some authors .
The height of the limb in 100GHz images is similar to the average heights of the typical spicules seen in Mg II images, and the limb seen in 304Å images. It suggests that the 100GHz limb is covered by multiple spicules in Mg II and He II images. Hence, we cannot identify a hot sheath around the 100GHz spicules, but we cannot argue that a spicule without a hot sheath is common in 100GHz images. To reveal the properties of typical spicules seen in mm-wave images, we need the higher resolution enough for resolving spicules, which will be realized with the higher observing frequency and/or long baselines of ALMA.
This paper makes use of the following ALMA data: ADS/JAO.ALMA \#2016.1.00070.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), MOST, ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre. SDO is part of NASA’s Living With a Star Program. The authors are supported by JSPS KAKENHI Grants: M.S. is by JP17K0539, T.K is by JP15H05814 and JP17K14314, T.J.O is by JP16K17663 (PI: T.J.O.) and JP25220703 (PI: S. Tsuneta), T.Y. is by JP15H03640. K.I. is by JP18H04442. G.F. was supported in part by NSF grant AST-1820613 to the New Jersey Institute of Technology. The study was started from the ALMA workshop “ALMA-Sol-CDAW19” held in January 2019 that is supported by the ALMA project, NAOJ.
|
---
author:
- 'M. Zorotovic, M.R. Schreiber, S.G. Parsons'
date: 'Received: 19 June 2014/ Accepted: 21 July 2014'
title: |
The evolution of the self-lensing binary KOI-3278:\
evidence of extra energy sources during CE evolution
---
Introduction
============
Post-common-envelope binaries (PCEBs) consisting of a white dwarf (WD) and a main-sequence (MS) companion are close-binary stars with orbital periods typically shorter than a day. Their discovery [@kraft58-1] immediately raised the question of their origin, because the progenitor of the WD must have been much bigger than the separation of the two stars in the currently observed system. Based on the pioneering works of @paczynski76-1 and @webbink84-1 the puzzle now seems to be solved. If the initially more massive star fills its Roche lobe as a giant and if the mass ratio $q=M_{\mathrm{donor}}/M_{\mathrm{gainer}}$ exceeds a critical value $q_{crit}$[^1], dynamically unstable mass transfer is generated. This leads to the formation of a common envelope (CE) engulfing both the core of the primary (the future WD) and the secondary star. The CE is expelled at the expense of orbital energy and angular momentum leaving behind a short period PCEB consisting of the compact core of the primary and the secondary star.
Despite significant recent progress [@ricker+taam12-1], numerical calculations still fail to simultaneously cover the large range of time and spatial scales involved in CE evolution and to make detailed predictions for the parameters of the emerging PCEB. Therefore, a simple energy equation relating the binding energy of the envelope to the change in orbital energy parametrized with the so-called CE efficiency ($\alpha_{\mathrm{CE}}$) is normally used to predict the outcome of CE evolution. Such an approach requires observational constraints on the efficiency parameter.
Recent surveys of PCEBs have established large samples of close binaries containing a WD and an M-dwarf companion [e.g., @nebot-gomez-moranetal11-1; @parsonsetal13-1]. These samples have been proved useful to understand several aspects of close-compact-binary evolution [e.g., @zorotovicetal10-1]; however, they only contain low-mass secondary stars. The predicted significant population of PCEBs containing a WD plus a massive (${\raisebox{-0.4ex}{${\stackrel{>}{\scriptstyle \sim}}$}}1{\mbox{$M_{\odot}$}}$) secondary star [see, e.g., @zorotovicetal14-1] has not yet been identified. This is because such a massive MS star completely outshines the WD at all wavelengths longer than UV. Finding and analyzing the evolutionary history of these PCEBs is crucial not only because the CE efficiency may depend on the mass of the secondary, as speculated by e.g., @politano+weiler07-1, but also because these systems may hold the key to understanding one of the oldest problems in astrophysics: the progenitor problem for supernovae TypeIa [SNIa, see, e.g., @wang+han12-1 for a recent review]. PCEBs with a massive MS secondary star are the progenitors of the two most popular channels proposed towards SNIa. In the single-degenerate channel [@whelan+Iben73-1] these PCEBs start thermal-timescale mass transfer which allows the WD mass to grow until it eventually explodes as a SNIa. In the double-degenerate channel [@webbink84-1] a WD with a close and massive companion (either a PCEB or a close binary emerging from stable mass transfer) evolves into a CE phase which leaves behind a double-degenerate system that may finally merge and, in the case of two C/O WDs with a total mass exceeding the Chandrasekhar limit, produce a SNIa explosion.
Recently, the second WD with a close and massive companion star has been identified (after IKPeg). This system, KOI-3278, identified using data from the Kepler spacecraft [@kruse+agol14-1], is remarkable not only for its long orbital period (88.18 days), but also because it is eclipsing. This combination of long period and high inclination results in a spectacular five-hour pulse once every orbit, caused by the 0.634C/O WD acting as a gravitational lens as it passes in front of its 1.042MS companion.
Given the small separation and the masses of the two stars, KOI-3278 must have evolved through a CE. Assuming that stable mass transfer occurred in KOI-3278 would require a large critical mass ratio ($q\sim1.5$) [*[and]{}*]{} strong wind mass loss of the WD progenitor on the AGB. While this configuration cannot be completely excluded, the resulting stable mass transfer could not have reduced the binary orbital period to the measured 88.18 days. Thus, KOI-3278 is the second PCEB containing a massive companion star.
Here we reconstruct the evolutionary history of KOI-3278 to derive constraints on the CE efficiency and predict its future to evaluate whether it might be the first progenitor of a double WD that will be formed through two CE phases.
Constraints on CE evolution from KOI-3278
=========================================
In its simplest form, the energy equation describing CE evolution can be expressed as $$\label{eq:alpha}
E_\mathrm{bind} = \alpha_{\mathrm{CE}}\Delta E_\mathrm{orb}.$$ The most basic assumption is to approximate the binding energy only by the gravitational energy of the envelope, $$\label{eq:Egr}
E_\mathrm{bind} = E_\mathrm{gr}=-\frac{G M_\mathrm{1} M_\mathrm{1,e}}{\lambda R_\mathrm{1}},$$ where $M_\mathrm{1}$, $M_\mathrm{1,e}$, and $R_\mathrm{1}$ are the total mass, envelope mass, and radius of the primary star, and $\lambda$ is a binding energy parameter that depends on the structure of the primary star. Simulations of PCEBs [e.g., @dekool+ritter93-1; @willems+kolb04-1; @politano+weiler06-1] have been performed assuming different values of $\alpha_{\mathrm{CE}}$ and assuming $\lambda = 0.5$ or $1.0$. However, keeping $\lambda$ constant is not a very realistic assumption for all types of possible primaries as was pointed out by e.g., @dewi+tauris00-1. Very loosely bound envelopes in more evolved stars can reach much higher values, especially if the recombination energy $U_\mathrm{rec}$ available within the envelope supports the ejection process. Therefore, a more realistic form for the binding energy equation is $$\label{eq:Eball}
E_\mathrm{bind}=\int_{M_\mathrm{1,c}}^{M_\mathrm{1}}-\frac{G m}{r(m)}dm + \alpha_{\mathrm{rec}}\int_{M_\mathrm{1,c}}^{M_\mathrm{1}}U_\mathrm{rec}(m),$$ where $\alpha_\mathrm{rec}$ is the efficiency of using recombination energy, i.e., the fraction of recombination energy that contributes to the ejection process. The effects of the extra energy source can be included in the $\lambda$ parameter by equating Eqs.\[eq:Egr\] and\[eq:Eball\].
Previous observational constraints on CE efficiencies
-----------------------------------------------------
While the above straightforward energy equation accurately describes the basic idea of CE evolution, it requires observational constraints to estimate the efficiencies. Several attempts to provide such constraints have been made using PCEBs consisting of a WD and a late-type (M dwarf) companion. In @zorotovicetal10-1 we have shown that the evolutionary history of the identified PCEBs can be reconstructed assuming that both efficiencies are in the range of $0.2-0.3$. The case for such relatively small efficiencies has recently been strengthened by @toonen+nelemans13-1 and @camachoetal2014-1 who performed binary population models of PCEBs taking into account selection effects affecting the observed samples. While the relative contributions of recombination and orbital energy remain unclear [e.g., @rebassa-mansergasetal12-1], the small values of the CE efficiencies are also in agreement with first tentative results obtained from numerical simulations of the CE phase [@ricker+taam12-1].
However, we have no information whether these values hold for larger secondary masses. While the decrease of $\alpha_{\mathrm{CE}}$ with increasing secondary masses proposed by @demarcoetal11-1 seems unlikely [@zorotovicetal11-2], the efficiencies may perhaps increase with secondary mass. The parameters of the only previously known PCEB with a G-type secondary, i.e., IKPeg (${\mbox{$P_\mathrm{orb}$}}=21.722\,d$, ${\mbox{$M_\mathrm{WD}$}}=1.19{\mbox{$M_{\odot}$}}$, ${\mbox{$M_\mathrm{sec}$}}=1.7{\mbox{$M_{\odot}$}}$), indicate that we can probably not simply apply the constraints for PCEBs with M-dwarfs to larger secondary masses. As shown by e.g., @davisetal10-1 and @zorotovicetal10-1, IKPeg is the only PCEB that requires additional energy sources to be at work during CE evolution. Despite the potential importance of IKPeg for our understanding of CE evolution, we cannot develop evolutionary theories based on just one system. Every new PCEB with a massive secondary therefore needs to be carefully analyzed.
A note of caution for BSE users
-------------------------------
In their discovery paper, @kruse+agol14-1 used the binary star evolution (BSE) code from @hurleyetal02-1 and found a possible evolutionary path for KOI-3278 assuming $\alpha_{\mathrm{CE}} = 0.3$ and $\lambda=0.2$. However, if we run BSE with the initial parameters obtained by @kruse+agol14-1 with these values, the binary system does not survive the CE phase. This discrepancy is easily explained by taking a closer look at the evolution of BSE. In its original version, the code requested a fixed value for $\lambda$ as an input parameter. However, the code was frequently updated over the years and a function to compute the value of $\lambda$ was included[^2]. This change is not described in the README file and is not commented in the main code (*bse.f*). However, digging into the code it becomes clear that in the current version the input parameter called “*lambda*” represents the fraction of the recombination (ionization) energy that is included to compute the real value of $\lambda$, i.e., $\alpha_\mathrm{rec}$. If the user still wants to use a fixed value for $\lambda$, the input value must be negative (e.g., if one wants to use $\lambda=0.2$, the input parameter should be $-0.2$).
The result obtained by @kruse+agol14-1 thus only shows that the evolutionary history of KOI-3278 can be understood if recombination energy significantly contributes to expelling the envelope. Given the long orbital period of KOI-3278, however, the crucial question is if it represents the second system after IKPeg that [*[requires]{}*]{} additional energy sources to contribute during CE evolution.
Reconstructing KOI-3278
-----------------------
Given the importance of understanding the evolution of PCEBs with massive secondaries, we here properly reconstruct the evolution of KOI-3278. We use the BSE code from @hurleyetal02-1 to identify possible progenitors of KOI-3278 and investigate whether additional energy sources are required to understand its evolutionary history. Our reconstruction algorithm is described in detail in @zorotovicetal11-1.
We first try to reconstruct the CE phase allowing the stellar parameters to vary within the 1-$\sigma$ uncertainties of the measured stellar parameters (as given by @kruse+agol14-1) but without considering additional energy sources. Interestingly, as in the case of IKPeg, we do not find possible progenitors for KOI-3278 without violating energy conservation, i.e., for any possible progenitor of the WD, the CE could not have been expelled by the use of orbital energy alone. This represents an important result as KOI-3278 is only the second PCEB with a massive secondary star and, in contrast to all PCEBs with low-mass secondary stars, both these systems require $\alpha_{\mathrm{rec}}>0$. Figure\[f-spdist\] illustrates our finding. It shows the maximum orbital period that a system can have if the only energy source used to expel the envelope is the orbital energy ($\alpha_{\mathrm{rec}}=0$ and $\alpha_{\mathrm{CE}}=1$) as a function of secondary mass, orbital period, and WD mass. The two systems with massive companions, KOI-3278 and IKPeg, are the only two that require extra energy sources. If a fraction of the recombination energy is assumed to contribute to expelling the envelope, the evolutionary history of both systems can be reconstructed. Assuming $\alpha_{\mathrm{CE}}=\alpha_{\mathrm{rec}}=0.25$ [@zorotovicetal10-1] we derive initial masses of $M_{1,i} = 2.450{\mbox{$M_{\odot}$}}$ and $M_{2,i} = 1.034{\mbox{$M_{\odot}$}}$ and an initial orbital period of $P_{orb,i} \sim 1300\,d$ for the progenitor of KOI-3278, which is similar to the values obtained by @kruse+agol14-1 for $\alpha_{\mathrm{CE}}=0.3$ and $\alpha_{\mathrm{rec}}=0.2$.
Predicting the future of KOI-3278
=================================
While reconstructing the evolution of KOI-3278 provides new information about the efficiencies of CE evolution, the future of close binaries consisting of WDs and massive companions is equally important as these systems may either enter a second CE phase, which may lead to a double WD (DWD), or start thermal-timescale mass transfer. These configurations represent the two classical channels towards SNIa, i.e. the double- and single-degenerate channel. Which of the two channels is taken by a given system depends on the timescale of nuclear evolution of the secondary and the timescale of orbital angular momentum loss until the secondary fills its Roche lobe.
If the PCEB has a relatively long orbital period, the secondary is likely to evolve off the MS and fill its Roche lobe as a giant. This configuration leads to dynamically unstable mass transfer if $q>q_{\mathrm{crit}}\sim1-1.5$. It depends then on the CE efficiencies and the orbital period at the onset of CE evolution if the system survives the CE phase or if the two stars merge. In the first case, a DWD is formed. The two WDs lose angular momentum because of gravitational radiation and if this DWD has a total mass exceeding the Chandrasekhar limit it may produce a SNIa.
If, on the other hand, the PCEB has a short orbital period, angular momentum loss due to magnetic braking and gravitational radiation can cause the secondary to fill its Roche lobe while it is still on the MS. This will lead to thermal-timescale mass transfer (for $q\,{\raisebox{-0.4ex}{${\stackrel{>}{\scriptstyle \sim}}$}}\,1$) and the systems appear as a super-soft X-ray source, i.e., the mass-transfer rate is high enough to generate stable hydrogen burning on the surface of the WD, allowing the WD mass to grow. It may explode as SNIa if it reaches the Chandrasekhar limit.
We predict the future of KOI-3278 using BSE and the current system parameters as derived in @kruse+agol14-1. Given the age of the system and the mass of the secondary, the latter will evolve off the MS and fill its Roche lobe during the first giant branch, in $\sim9\,Gyr$. At that moment the orbital period will still be $\sim80\,d$, the secondary will have a total mass of $1.009{\mbox{$M_{\odot}$}}$ with a core-mass of $0.332{\mbox{$M_{\odot}$}}$. Given the mass ratio of the system, mass transfer will be dynamically unstable and lead to CE evolution. KOI-3278 will survive this second CE phase and form a DWD, a C/O WD plus a He WD, for almost any value of the efficiencies (even if $\alpha_{\mathrm{rec}} = 0$, $\alpha_{\mathrm{CE}}$ needs to be only larger than $0.043$[^3] to avoid a merger). This makes KOI-3278 the first known progenitor of a DWD formed by two CEs.
In Fig.\[f-dwd\] we relate the future DWD parameters of KOI-3278 to the currently known sample of DWDs. The two surveys that have identified most of the currently known DWDs are the SPY survey [@napiwotzkietal03-1] and the ELM survey [@kilicetal10-1]. We compiled the DWDs from both surveys using the tables provided in @nelemansetal05-1 and @brownetal12-1 [@brownetal13-1]. With an orbital period of $0.251^{+2.154}_{-0.234}\,d$ and a mass of $0.332{\mbox{$M_{\odot}$}}$ for the WD that will form during the second CE phase, KOI-3278 will become a very typical DWD that will evolve towards shorter orbital periods driven by orbital angular momentum loss due to gravitational radiation. Given the mass ratio, and depending on the strength of spin-orbit coupling, the binary may either become an AMCVn system or, more probably, merge [see @marshetal04-1 their figure1].
Conclusion
==========
Understanding the evolution of the two known PCEBs containing a G-type secondary star requires additional sources of energy, such as recombination energy, to contribute during CE evolution. This may indicate that a larger fraction of the total available energy is used to expel the envelope. In other words, at least one of the efficiencies may increase with secondary mass. If this can be confirmed, the population of PCEBs with F- and G-type secondaries will be dominated by long orbital period systems (${\mbox{$P_\mathrm{orb}$}}\sim\,2-100\,d$) and most secondaries will evolve into giants before the second phase of mass transfer and may, such as KOI-3278, survive a second CE. This, finally, may imply that the double-degenerate channel towards SNIa is more likely to occur than the single-degenerate channel.
We thank Fondecyt for their support under the grants 3130559 (MZ), 1141269 (MRS), and 3140585 (SGP).
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[^1]: Early calculations led to $q_{crit}\sim1$ [@webbink88-1] while more recent works do not exclude values as high as $q_{crit}=1.5$ [@passyetal12-1; @woods+ivanova11-1].
[^2]: the function called *celamf* can be found in the file called *zfuncs.f* and it was recently published in @claeysetal14-1.
[^3]: This limit should be slightly larger if we take into account that the WD that emerges from the CE phase is probably bloated compared to a cool WD (as used in BSE).
|
---
abstract: 'We give the centralizers of irreducible representations from a finitely generated group $\Gamma$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $\chi^i(\Gamma,PSL(p,\mathbb{C}))$. When $\Gamma$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of algebraic singularities of the irreducible part of the character variety.'
author:
- 'Clément Guérin[^1]'
title: 'Bad irreducible subgroups and singular locus for character varieties in $PSL(p,\mathbb{C})$'
---
**Keywords : **Representation variety $\cdot$ Character variety $\cdot$ Irreducible representations $\cdot$ Centralizer of irreducible representations $\cdot$ Orbifolds $\cdot$ Fuchsian groups representations
**Mathematics Subject Classification (2000)** 20C15 $\cdot$ 15A21
Introduction {#intro}
============
Let $G$ be a complex reductive group, e.g. $GL(n,\mathbb{C})$, $SL(n,\mathbb{C})$ or $PSL(n,\mathbb{C})$. A proper subgroup $P$ of $G$ is *parabolic* if $G/P$ is a complete variety. A subgroup $H$ of $G$ is *irreducible* if $H$ is not contained in a parabolic subgroup of $G$. A subgroup $H$ of $G$ is *completely reducible*, if for any parabolic subgroup $P$ of $G$ containing $H$, there is a Levi subgroup $L$ of $P$ such that $H$ is contained in $L$.
Throughout the paper, if $G$ is a group, $Z(G)$ denotes its center and if $H$ is a subgroup of $G$, $Z_G(H)$ denotes the centralizer of $H$ in $G$. If $H$ is an irreducible subgroup of $G$, we say that $H$ is *good* if $Z_G(H)=Z(G)$ and *bad* else. Sikora proved (see [@Sik], Corollary 17) that a completely reducible subgroup $H$ of $G$ is irreducible if and only if its centralizer $Z_G(H)$ in $G$ is a finite extension of $Z(G)$.
Let $\Gamma$ be a finitely generated group, a representation from $\Gamma$ to $G$ is *irreducible* (resp. *completely reducible*, *good*, *bad*) if $\rho(\Gamma)$ is. While Schur’s Lemma implies that all irreducible representations in $SL(n,\mathbb{C})$ or $GL(n,\mathbb{C})$ are good, there are many examples in the literature of bad representations in other complex reductive algebraic groups (see Proposition 3.32 in [@F-L]). Roughly speaking, our goal in this paper is to study bad representations from a finitely generated group to $PSL(p,\mathbb{C})$ when $p$ is a prime number. We make a brief recall of some notions on representation/character varieties (see [@Sik] and also [@L-M] for a complete exposition).
The set of representations from $\Gamma$ to $G$ is denoted $\operatorname{Hom}(\Gamma,G)$. It is an affine algebraic set. Indeed, if $\Gamma$ is generated by $r$ elements, $\operatorname{Hom}(\Gamma,G)$ can be cut out from $G^r$ by polynomial equations. As a result, there is a *universal representation algebra* $R(\Gamma,G)$ (see Paragraph 5 in [@Sik]) such that $\operatorname{Hom}(\Gamma,G)$ corresponds to the set of maximal ideals of $R(\Gamma,G)$. Remark that $R(\Gamma,G)$ may not be reduced (i.e. it could non-trivial contain nilpotent elements). As a result, it is useful to define the *schematic representation variety* $\operatorname{\bf{Hom}}(\Gamma,G)$ as the spectrum of the ring $R(\Gamma,G)$. By definition, the set of $\mathbb{C}$-points of $\operatorname{\bf{Hom}}(\Gamma,G)$ is $\operatorname{Hom}(\Gamma,G)$. From Section \[badsub\] to Section \[surfgrpcase\] included, we will only be interested in the $\mathbb{C}$-points of the representation variety. However, the tangent spaces computed in Section \[orbalgsing\], are, a priori, tangent to the schematic representation variety $\operatorname{\bf{Hom}}(\Gamma,G)$ and it is necessary to introduce this scheme for this reason. There are two natural topologies on $\operatorname{Hom}(\Gamma,G)$. One is the Zariski topology, the other is the transcendental topology (induced by the topology of $G$ as a complex group). In this paper, we will mostly consider the second topology.
We denote $\operatorname{Hom}^i(\Gamma,G)$ the set of irreducible representations from $\Gamma$ to $G$. Since being non-irreducible is a closed condition, the set $\operatorname{Hom}^i(\Gamma,G)$ is open in $\operatorname{Hom}(\Gamma,G)$ (see Proposition 27 in [@Sik]).
The adjoint group $G/Z(G)$ acts on $\operatorname{Hom}(\Gamma,G)$ by conjugation. If $\rho:\Gamma\rightarrow G$ is a representation, its conjugacy class will be denoted $[\rho]$. Roughly speaking, the character variety is the quotient of $\operatorname{Hom}(\Gamma,G)$ by this conjugation action. However, in order to obtain an interesting geometric structure on the quotient we need to consider the GIT quotient $\operatorname{Hom}(\Gamma,G)//(G/Z(G))$. It is only defined when $G$ is reductive. This quotient will be called the *character variety* of $\Gamma$ into $G$ and will be denoted $\chi(\Gamma,G)$. By definition, there is an algebraic map $\Psi: \operatorname{Hom}(\Gamma,G)\rightarrow \chi(\Gamma,G)$ which induces by duality an isomorphism between $\mathbb{C}[\chi(\Gamma,G)]$ and $\mathbb{C}[\operatorname{Hom}(\Gamma,G)]^G=(R(\Gamma,G)/\sqrt{(0)})^G$. Likewise, one may define the *schematic character variety* $\mathfrak{X}(\Gamma,G)$ as the spectrum of $\mathbb{C}[\operatorname{\bf{Hom}}(\Gamma,G)]^G=R(\Gamma,G)^G$.
It turns out that $\chi(\Gamma,G)$ can be identified to the set of closed orbits in $\operatorname{Hom}(\Gamma,G)$ for the conjugation action. The projection $\Psi$ sends a representation $\rho$ to the unique closed orbit contained in the topological closure of $[\rho]$. Since $[\rho]$ is closed if and only if $\rho$ is completely reducible, $\Psi$ restricted to the subset of completely reducible representations is the usual projection on a quotient space.
In particular, if we denote $\chi^i(\Gamma,G)$ the subset of conjugacy classes of irreducible representations in $\chi(\Gamma,G)$, then $\chi^i(\Gamma,G)$ can be identified to the usual quotient $\operatorname{Hom}^i(\Gamma,G)/(G/Z(G))$. Remark that the action of $G/Z(G)$ on $\operatorname{Hom}^i(\Gamma,G)$ is proper by Proposition 1.1 in [@J-M]. In our paper, the *singular locus of the character variety* of $\Gamma$ into $G$ is the set $\chi^i_{Sing}(\Gamma,G)$ of conjugacy classes of bad representations.
A closed point $x$ in a (possibly non-reduced) algebraic scheme $X$ is if it belongs to a unique irreducible component and the dimension of its Zariski-tangent space coincides with the dimension of the unique irreducible component containing it. A point $x$ in $X$ is an *algebraic singularity* in $X$ if it is not a simple point. In particular, $\chi^i(\Gamma,G)$ or $\mathfrak{X}^i(\Gamma,G)$ may contain algebraic singularities.
Remark that being a simple point of the schematic representation (resp. character) variety is stronger than being a simple point of the representation (resp. character) variety, see Paragraph 9 in [@Sik]. Following its terminology, we say that a representation is *scheme smooth* (resp. *smooth*) if it is a simple point of $\operatorname{\bf{Hom}}(\Gamma,G)$ (resp. $\operatorname{Hom}(\Gamma,G)$). We use the same terminology for conjugacy class of representations.
The question whether the singular locus of the character variety coincides with the set of algebraic singularities of the irreducible part of the (schematic) character variety or not is discussed in Section \[orbalgsing\]. Proposition \[singvarcar\] states that for $G=PSL(p,\mathbb{C})$ and $\Gamma$ a free group of rank $\geq 2$ or a closed surface group of genus $\geq 2$, these sets are the same. This statement cannot be generalized to any finitely generated group $\Gamma$ and complex reductive group $G$ because of Examples \[algnotorb\] and \[orbnotalg\].
When $\Gamma$ is a free group over $l\geq2$ generators, then $\operatorname{Hom}(\Gamma,G)=G^l$ and $\operatorname{Hom}^i(\Gamma,G)$ being open in it, is a manifold. When $\Gamma$ is a closed surface group of genus $g\geq 2$ and $G$ is reductive, it can be shown that $\operatorname{Hom}^i(\Gamma,G)$ is a manifold (all its points are scheme smooth, c.f. Proposition 37 in [@Sik], see also Paragraph 1.2 in [@Gol]). Since the action of $G/Z(G)$ on $\operatorname{Hom}^i(\Gamma,G)$ is proper and its stabilizers are finite, the quotient space $\chi^i(\Gamma,G)$ is, in these cases, an orbifold. The singular locus we defined above is the orbifold locus of $\chi^i(\Gamma,G)$, i.e. the set of points with a non-trivial local isotropy. This geometric interpretation as the set of orbifold singularities of an orbifold is our first reason for studying this singular locus. However, it appeared that this locus was well defined for any finitely generated group and that most results we obtained were true for any finitely generated group, even if $\chi^i(\Gamma,G)$ is not an orbifold.
From Section \[badsub\] to the end of the paper, we only focus on the case $G=PSL(p,\mathbb{C})$ with $p$ a prime number. For some proofs, it is necessary to deal with the case $p=2$ separately from the case $p$ odd. When this is the case, we will systematically assume $p$ odd, leaving $p=2$ to the reader. Anyway, for $p=2$, most of these results have already been obtained by Heusener and Porti in [@H-P].
In the sequel we will need a few notations. Fix a prime number $p$. Denote $\pi$ the natural projection from $SL(p,\mathbb{C})$ to $PSL(p,\mathbb{C})$. To simplify some proofs, it is easier to see matrices $M\in SL(p,\mathbb{C})$ as automorphisms of $\mathbb{C}^{\mathbb{Z}/p}$ with the canonical basis indexed by $\mathbb{Z}/p$. In particular their lines and rows will be indexed by $0,\dots, p-1$.
For any matrix $M\in SL(p,\mathbb{C})$, we will denote $\overline{M}$ its projection in $PSL(p,\mathbb{C})$. $D$ will denote the subgroup of diagonal matrices in $SL(p,\mathbb{C})$ and $\xi$ will be a primitive $p^{\text{th}}$ root of the unity. When $p$ is odd, the diagonal matrix $D(\xi)$ is defined by $(D(\xi))_{i,i}:=\xi^{i}$ for $0\leq i\leq p-1$. To any bijection $\sigma$ of $\mathbb{Z}/p$, we associate the corresponding permutation matrix $M_{\sigma}\in GL(p,\mathbb{C})$. Finally, $c$ will denote the cyclic permutation $(0,1,\dots,p-1)$ of $\mathbb{Z}/p$. When $p=2$, we define $D(\xi):=\begin{pmatrix}\sqrt{-1}&0\\0&-\sqrt{-1}\end{pmatrix}$ and $M_c:= \begin{pmatrix}0&\sqrt{-1}\\ \sqrt{-1}&0\end{pmatrix}$. We prove, in Section \[badsub\] :
[theorem]{}[classifthm]{}\[classifcentr\] Let $p$ be a prime number and $\overline{H}$ be a bad subgroup of $PSL(p,\mathbb{C})$. Then, there are two cases :
- $Z_{PSL(p,\mathbb{C})}(H)$ is isomorphic to $\mathbb{Z}/p\times\mathbb{Z}/p$. In which case $\overline{H}$ is conjugate to the group $\langle \overline{D(\xi)}\rangle\times \langle\overline{M_c}\rangle$ and $Z_{PSL(p,\mathbb{C})}(\overline{H})=\overline{H}$.
- $Z_{PSL(p,\mathbb{C})}(H)$ is isomorphic to $\mathbb{Z}/p$. In which case $\overline{H}$ is conjugate to $\overline{K}\rtimes \langle\overline{M_c}\rangle$ where $\overline{K}$ is a non-trivial subgroup of $\overline{D}$, different from $\langle \overline{D(\xi)}\rangle$ and invariant by the action of $\langle\overline{M_c}\rangle$, in particular $Z_{PSL(p,\mathbb{C})}(\overline{H})$ is conjugate to $\langle \overline{D(\xi)}\rangle$.
For $p=2$, this is implied by Remark 3.11 and Remark 4.3 in [@H-P]. As a result, any bad representation from $\Gamma$ to $PSL(p,\mathbb{C})$ is conjugate to a representation into $\overline{D}\rtimes \langle\overline{M_c}\rangle$. Therefore, the natural inclusion of $\operatorname{Hom}^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$ into $\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))$ induces a surjective map $\varphi$ from $\chi^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$ onto $\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))$. This will give a parametrization of the singular locus.
For any $\overline{\rho}$ in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$, define $\mathcal{H}_{\overline{\rho}}^i:=\{\rho\in \operatorname{Hom}^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)\mid q\circ \rho=\overline{\rho}\}$ and $\overline{\mathcal{H}_{\overline{\rho}}}^i$ be its projection on the character variety. Then $\chi^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$ is the union of all $\overline{\mathcal{H}_{\overline{\rho}}}^i$. In section \[diagbyfin\], we give a cohomological description for $\overline{\mathcal{H}_{\overline{\rho}}}^i$. In Section \[secinj\], we prove :
[theorem]{}[homeobarHrho]{}\[homeobarHrho\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $\overline{\rho}$ be a non-trivial morphism in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$. Then $\varphi_{|\overline{\mathcal{H}_{\overline{\rho}}}^i}$ is a homeomorphism onto its image.
As a result, $\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))$ is a union of a finite number of $\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)$ whose geometry is given by cohomology groups. At the end of Section \[secinj\], we briefly justify that we only need to chose, for any normal subgroup $K$ of index $p$ in $\Gamma$, one $\overline{\rho}_K$ in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$ whose kernel is $K$ to get $\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))$. To sum up : $$\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))=\bigcup_{\substack{K\triangleleft \Gamma\\ \text{$[\Gamma:K]=p$} } }\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$$
The subset $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ will be called the *pseudo-component* associated to $K$. When $\Gamma$ is a free group these are the irreducible components of the singular locus (this is not true for all $\Gamma$). In Section \[secint\], we study the possible intersections between different pseudo-components. Recall that the $p$-rank of $\Gamma$ is by definition $\dim_{\mathbb{Z}/p}(\Gamma^{Ab}/p\Gamma^{Ab})$.
[theorem]{}[combi]{}\[combi\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number. Let $r$ be the $p$-rank of $\Gamma$. Then :
1. $\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))$ is the union of $\frac{p^r-1}{p-1}$ pseudo-components.
2. The intersection of two different pseudo-components is finite of cardinal $p-1$. All its elements are conjugacy classes of abelian irreducible representations.
3. Conversely, any conjugacy class of abelian irreducible representations belongs to exactly $p+1$ pseudo-components.
4. There are exactly $\frac{(p^r-1)(p^{r-1}-1)}{p^2-1}$ conjugacy classes of abelian irreducible representations from $\Gamma$ to $PSL(p,\mathbb{C})$.
One can compare this theorem to the well-known example when $p=2$ and $\Gamma=\mathbb{F}_2=\langle a,b\rangle$ is a free group over $2$ generators (see Example 4.4 in [@H-P] or Corollary 10 in [@Sik15]).
\[F2PSL2\] First, the natural map $\pi^*$ from $SL(2,\mathbb{C})^2=\operatorname{Hom}(\mathbb{F}_2,SL(2,\mathbb{C}))$ to $PSL(2,\mathbb{C})^2=\operatorname{Hom}(\mathbb{F}_2,PSL(2,\mathbb{C}))$ is onto. The automorphism group of this orbifold cover is the Klein group $(\mathbb{Z}/2)^2$ : $(i,j)\cdot (A,B):=((-1)^iA,(-1)^jB)$.
On the other hand, an old result of Vogt states that $\chi(\mathbb{F}_2,SL(2,\mathbb{C}))=\mathbb{C}^3$ by sending $[\rho]$ to $(\operatorname{tr}(\rho(a)),\operatorname{tr}(\rho(b)),\operatorname{tr}(\rho(ab)))$.
Since $\pi^*$ induces a map from $\chi(\mathbb{F}_2,SL(2,\mathbb{C}))=\mathbb{C}^3$ to $\chi(\mathbb{F}_2,PSL(2,\mathbb{C}))$, we can identify $\chi(\mathbb{F}_2,PSL(2,\mathbb{C}))$ to $\chi(\mathbb{F}_2,SL(2,\mathbb{C}))/K$ where the action of $K$ is given by $(i,j)\cdot(x,y,z):=((-1)^ix,(-1)^jy,(-1)^{i+j}z) $.
Computing the invariant ring $\mathbb{C}[x,y,z]^{(\mathbb{Z}/2)^2}$, one identifies $\chi(\mathbb{F}_2,PSL(2,\mathbb{C}))$ to $\{(X,Y,Z,T)\in\mathbb{C}^4\mid T^2=XYZ\}$ and explicitly computes $\pi^*(x,y,z)$ to be $(x^2,y^2,z^2,xyz)$. Removing three non-irreducible points, the set of branch points for $\pi^*$ is, by definition, the singular locus of the orbifold $\chi^i_{Sing}(\mathbb{F}_2,PSL(2,\mathbb{C}))$.
The set of branch points is easily computed to be the union of $X=Z=T=0$, $Y=Z=T=0$ and $X=Y=T=0$. Each of them (minus one non-irreducible point) corresponds to one of the three pseudo-components in the theorem above. Any two of them intersect in one single point whose coordinates are $X=Y=Z=T=0$. It corresponds to the abelian irreducible representation $a\mapsto \overline{D(\xi)}$, $b\mapsto \overline{M_c}$ and thus fully agrees with the aforementioned theorem.
When $\Gamma=\mathbb{F}_l$ is a free group of rank $l\geq 2$, we prove in Section \[freegrpcase\] (see Corollary \[orblocfree\]) that $\chi^i_{Sing}(\mathbb{F}_l,PSL(p,\mathbb{C}))$ is connected of dimension $(p-1)(l-1)$.
In [@F-L-R], Florentino, Lawton and Ramras compute higher homotopy groups of the irreducible part of free groups character varieties into $G_n:=SL(n,\mathbb{C})$ or $GL(n,\mathbb{C})$ : $\pi_k(\chi^i(\mathbb{F}_l,G_n))$ (see Theorem 5.4 in loc. cit.).
In Remark 5.7, they conjecture that for any complex reductive group $G$ and $l$ big enough, $\pi_2(\chi^{good}(\mathbb{F}_l,G))=\pi_1(G/Z(G))$ provided that we can bound above the dimension of bad representations. The dimension count we obtained directly leads to the validity of the conjecture when $G=PSL(p,\mathbb{C})$ for $(l-1)(p-1)\geq 2$.
In Section \[surfgrpcase\], we study the case when $\Gamma=\pi_1(\Sigma_g)$ is a closed surface group of genus $g\geq 2$. Corollary \[conncomporblocsurfgrp\] states that $\chi^i_{Sing}(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$ has $p$ connected components (given by the Euler invariant). Jun Li’s Theorem in [@Li] implies that $\pi_0(\chi(\pi_1(\Sigma_g),G))$ is equal to $\pi_1(G)$. Therefore, when $G=PSL(p,\mathbb{C})$, each connected component of $\chi(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$ contains one unique connected component of $\chi^i_{Sing}(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$.
Bad subgroups in $PSL(p,\mathbb{C})$ {#badsub}
====================================
In this section, we classify, up to conjugation, bad subgroups in $PSL(p,\mathbb{C})$. More precisely, we prove :
The main idea to prove this theorem is to pull-back the problem in $SL(p,\mathbb{C})$ using $\pi$. For $A\in SL(p,\mathbb{C})$ and $\lambda\in \mathbb{C}$, we denote $E_{\lambda}(A)=\{v\in \mathbb{C}^{\mathbb{Z}/p}\mid Av=\lambda v\}$.
\[eigspace\] Let $n\geq 1$ and $A,B$ be two matrices in $GL(n,\mathbb{C})$. Assume $[A,B]=\lambda I_n$ where $\lambda \in \mathbb{C}^*$.Then, for $\mu \in \mathbb{C}$, we have $BE_{\mu}(A)=E_{\lambda\mu}(A)$.
The assumption implies that $BA=\lambda^{-1}AB$. For any $v\in \mathbb{C}^{\mathbb{Z}/n}$, we have the following equivalences : $v\in E_{\mu}(A)\Leftrightarrow Av=\mu v\Leftrightarrow BAv=\mu Bv\Leftrightarrow ABv=\lambda \mu Bv$ i.e. $Bv\in E_{\lambda\mu}(A)$. Therefore, $BE_{\mu}(A)=E_{\lambda\mu}(A)$.
This directly leads to :
\[central\] Let $p$ be a prime number and $\overline{H}$ be a bad subgroup of $PSL(p,\mathbb{C})$. Then, up to conjugation, $\overline{D(\xi)}$ belongs to $Z_{PSL(p,\mathbb{C})}(\overline{H})$.
Assume $p$ odd. Let $H$ be $\pi^{-1}(\overline{H})$ and $U$ be $\pi^{-1}(Z_{PSL(p,\mathbb{C})}(\overline{H}))$. Since $Z_{PSL(p,\mathbb{C})}(\overline{H})$ is non-trivial, it follows that $U$ contains an element $u$ which is not central. Furthermore, since $H$ is an irreducible subgroup of $SL(p,\mathbb{C})$, its centralizer is $Z(SL(p,\mathbb{C}))$ by Schur’s lemma.
As a result, there exists $h_0\in H$ such that $[h_0,u]\neq I_p$. However, since $\overline{h_0}$ and $\overline{u}$ commute, it follows that $[h_0,u]$ belongs to $Z(SL(p,\mathbb{C}))$. Therefore, there exists $0<k<p$ such that $[h_0,u]=\xi^kI_p$.
Applying Lemma \[eigspace\], $h_0$ acts on the spectrum of $u$ by multiplying by $\xi^k$. Let $\mu$ be an eigenvalue of $u$, since $\xi^k$ is a non-trivial primitive $p^{\text{th}}$ root of the unity, $u$ has $p$ different eigenvalues $\mu,\xi^k\mu,\dots,\xi^{k(p-1)}\mu$. Finally, since $u$ is a matrix of dimension $p$, its eigenspaces have dimension $1$ and $u$ is conjugate to the diagonal matrix with $\mu,\xi^k\mu,\dots,\xi^{k(p-1)}\mu$ on the diagonal. Since $\det(u)=1$ and $p$ is odd, we see that $\mu$ is a $p^{\text{th}}$ root of the unity. Therefore, $u$ is conjugate to $D(\xi)$.
Because of this proposition, we compute the centralizer of $\overline{D(\xi)}$ in $PSL(p,\mathbb{C})$.
\[centrdxi\] Let $p$ be a prime number. Then $Z_{PSL(p,\mathbb{C})}(\overline{D(\xi)})=\overline{D}\rtimes \langle \overline{M_c}\rangle$ where $\langle \overline{M_c}\rangle$ acts on $\overline{D}$ by conjugation.
Assume $p$ odd. Let $U:=\pi^{-1}(Z_{PSL(p,\mathbb{C})}(\overline{D(\xi)}))$. For $0\leq k\leq p-1$, we have $[D(\xi),M_c^k]=\xi^k I_p$ by straightforward computations. It follows that $U$ contains both the group $D$ and the matrix $M_c$. Conversely, if $u\in U$ then $[D(\xi),u]=\xi^k I_p$ for some $k$ and therefore : $$\begin{aligned}
[D(\xi),uM_c^{-k}]&=D(\xi)uM_c^{-k}D(\xi)^{-1}M_c^ku^{-1}\\
&=D(\xi)uD(\xi)^{-1}\xi^{-k}u^{-1}\text{ since $[D(\xi),M_c^{-k}]=\xi^{-k} I_p$}\\
&=\xi^{-k}[D(\xi),u]=I_p\text{.}\end{aligned}$$
Hence, $u$ is the product of an element in the centralizer of $D(\xi)$ in $SL(p,\mathbb{C})$ (which is $D$, since $D(\xi)$ is diagonal with pairwise distinct eigenvalues) with some power of $M_c$. As a result, $U=D\rtimes \langle M_c\rangle$ and projecting this equality in $PSL(p,\mathbb{C})$, we have $Z_{PSL(p,\mathbb{C})}(\overline{D(\xi)})=\overline{D}\rtimes \langle \overline{M_c}\rangle$.
When $p=2$, the proof is slightly different since there is a non-trivial intersection between $D$ and $\langle M_c\rangle$, however $\overline{D}$ and $\langle\overline{M_c}\rangle$ still have a trivial intersection.
\[centralconj\] Let $p$ be a prime number and $\overline{H}$ be a bad subgroup of $PSL(p,\mathbb{C})$. Then, there is a non-trivial subgroup $\overline{K}$ of $\overline{D}$ such that $\overline{H}$ is conjugate to $\overline{K}\rtimes\langle \overline{M_c}\rangle$.
We denote $q$ the natural projection $\overline{D}\rtimes \langle \overline{M_c}\rangle\to \langle \overline{M_c}\rangle$. Combining Proposition \[central\] and Lemma \[centrdxi\], $\overline{H}$ is conjugate to a subgroup of $\overline{D}\rtimes \langle \overline{M_c}\rangle$. We identify $\overline{H}$ to this subgroup of $\overline{D}\rtimes \langle \overline{M_c}\rangle$.
If $q(\overline{H})$ were trivial then $\overline{H}$ would be contained in $\overline{D}$ which is not irreducible, therefore $q(\overline{H})$ is not trivial. Let $x$ be in $\pi^{-1}(\overline{H})$ such that $q(\overline{x})=\overline{M_c}$. Then :
$$x=\begin{pmatrix}\lambda_0&&\\&\ddots&\\&&\lambda_{p-1}\end{pmatrix}M_c\text{.}$$
If $s$ denotes a diagonal matrix where $s_{i,i}=\lambda_0^{-1}\dots\lambda_{i}^{-1}$ for $0\leq i\leq p-1$ then $sxs^{-1}=M_c$. If $t:=(\det(s))^{1/p}s$ then $txt^{-1}=M_c$ and $t$ is diagonal of determinant $1$. Therefore, $\overline{H}$ is conjugate in $\overline{D}\rtimes \langle \overline{M_c}\rangle$ to a subgroup $\overline{H'}$ of $\overline{D}\rtimes \langle \overline{M_c}\rangle$ which contains $\overline{M_c}$. Since $\overline{M_c}$ belongs to $\overline{H'}$, we have that $\overline{H'}=\operatorname{Ker}(q_{|\overline{H'}})\rtimes \langle \overline{M_c}\rangle$. Since $\overline{H'}$ needs to be irreducible, it is clear that $\operatorname{Ker}(q_{|\overline{H'}})$ cannot be trivial.
Similar to Lemma \[centrdxi\] (although it is written in a different way since it will be used with Lemma \[freeMc\] later), we have :
\[centrMc\] Let $p$ be a prime number and $g\in SL(p,\mathbb{C})$ such that $[g,M_c]\in Z(SL(p,\mathbb{C}))$. Then, $g=P(M_c)D(\xi)^k$ where $P\in\mathbb{C}[X]$ and $k\geq 0$.
Assume $p$ odd. Since $M_c$ has pairwise distinct eigenvalues (like $D(\xi)$, its minimal polynomial is $X^p-1$), any matrix commuting with $M_c$ can be written as a polynomial in $M_c$. Like in Lemma \[centrdxi\], if $[g,M_c]=\xi^k I_p$ then $[gD(\xi)^{-k},M_c]=I_p$ and therefore, what is written above implies that there is a polynomial $P$ such that $gD(\xi)^{-k}=P(M_c)$ and we are done.
\[freeMc\]
Let $p$ be a prime number. Let $(d_0,\dots,d_{p-1})$ and $(d_0',\dots,d_{p-1}')$ be two $p$-tuples of complex diagonal matrices. Then :
$$\sum_{j=0}^{p-1}d_jM_c^j= \sum_{j=0}^{p-1}d_j'M_c^j\Rightarrow \forall \text{ }0\leq j\leq p-1\text{, } d_j=d_j'\text{.}$$
Assume $p$ odd. For $0\leq j\leq p-1$, and for $0\leq i,k\leq p-1$, we have :
$$(d_jM_c^j)_{i,k}=\left\lbrace\begin{array}{ll} (d_j)_{i,i}&\text{ if $i-k=j$ mod $p$}\\0&\text{ else.}\end{array}\right.$$
As a result :
$$\left(\sum_{j=0}^{p-1}d_jM_c^j\right)_{i,k}=(d_{k-i\text{ mod } p})_{i,i}\text{.}$$
Applying this expression to both $(d_0,\dots,d_{p-1})$ and $(d_0',\dots,d_{p-1}')$, we easily see that the equality of the sum of matrices in the assumption implies the equalities $d_j=d_j'$ for $j=0,\dots,p-1$.
\[centrKMc\]
Let $p$ be a prime number, $\overline{K}$ be a non-trivial subgroup of $\overline{D}$ which is stable by the conjugation action of $\overline{M_c}$. Then :
$$Z_{PSL(p,\mathbb{C})}(\overline{K}\rtimes\langle\overline{M_c}\rangle)=\left\lbrace\begin{array}{ll} \langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle &\text{ if } \overline{K}=\langle \overline{D(\xi)}\rangle \\
\langle \overline{D(\xi)}\rangle &\text{ else.}\\
\end{array}\right.$$
Since $\overline{D(\xi)}$ commutes with $\overline{D}\rtimes\langle\overline{M_c}\rangle$, $\langle \overline{D(\xi)}\rangle$ is, in any case, included in $Z_{PSL(p,\mathbb{C})}(\overline{K}\rtimes\langle\overline{M_c}\rangle)$.
Let $\overline{x}$ be in $Z_{PSL(p,\mathbb{C})}(\overline{K}\rtimes\langle\overline{M_c}\rangle)$. By Lemma \[centrMc\], there exist complex numbers $a_0$,…, $a_{p-1}$ and $0\leq k\leq p-1$ such that $x=(a_0+a_1M_c+\cdots+a_{p-1}M_c^{p-1})D(\xi)^k$. Let $\overline{d}$ be in $\overline{K}$. Then, there is $0\leq t\leq p-1$ such that $dxd^{-1}=\xi^tx $. We also have :
$$dxd^{-1}=\left(\sum_{j=0}^{p-1}a_jdM_c^jd^{-1} \right)D(\xi)^k=\left(\sum_{j=0}^{p-1}\underbrace{a_jdM_c^jd^{-1}M_c^{-j}}_{\text{ diagonal}}M_c^j \right)D(\xi)^k\text{.}$$
Therefore, we apply Lemma \[freeMc\] to get $a_jdM_c^jd^{-1}M_c^{-j}=a_j\xi^t$ for $j=0,\dots,p-1$. Assume $a_j$ is non-zero for some $j>0$, then $[d,M_c^j]$ belongs to $Z(SL(p,\mathbb{C}))$ and therefore, $\overline{d}$ commutes with $\overline{M_c}^j$. Since $j\neq 0$ and $\overline{M_c}$ is of order $p$, $\overline{d}$ commutes with $\overline{M_c}$. Since $d$ is diagonal, Lemma \[centrMc\] implies that $\overline{d}$ belongs to $\langle \overline{D(\xi)}\rangle$.
As a result, if $\overline{K}$ is not equal to $\langle \overline{D(\xi)}\rangle$ then $a_j=0$ for all $j>0$ and $\overline{x}$ belongs to $\langle \overline{D(\xi)}\rangle$. It follows that the centralizer of $\overline{K}\rtimes\langle\overline{M_c}\rangle$ in $PSL(p,\mathbb{C})$ is $\langle \overline{D(\xi)}\rangle$.
If $\overline{K}=\langle \overline{D(\xi)}\rangle$ then taking $d=D(\xi)$ above, we have $a_j[D(\xi),M_c^j]=a_j\xi^t$. Therefore $a_j(\xi^j-\xi^t)=0$. As a result, $a_j=0$ if $j\neq t$ and therefore $\overline{x}$ belongs to $ \langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle $. We proved that the centralizer of $ \langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle $ is contained in $ \langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle $, since this group is abelian, its centralizer in $PSL(p,\mathbb{C})$ is itself.
\[irred\] Let $\overline{K}$ be a non-trivial $\langle\overline{M_c}\rangle$-invariant subgroup of $\overline{D}$. Then $\overline{K} \rtimes\langle\overline{M_c}\rangle$ is completely reducible. Since they also have finite centralizers (using this proposition), Corollary 17 in [@Sik] implies that they are irreducible.
It is a direct consequence of Proposition \[centralconj\] which states that any bad subgroup $\overline{H}$ is conjugate to $\overline{K}\rtimes \langle\overline{M_c}\rangle$ where $\overline{K}$ is a non-trivial subgroup of $\overline{D}$ and of Proposition \[centrKMc\] which gives their centralizer.
Our next order of business is to describe $\chi^i_{Sing}(\Gamma, PSL(p,\mathbb{C}))$ when $\Gamma$ is a finitely generated group. Denote $\operatorname{Hom}^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$ the set of irreducible representations from $\Gamma$ to $PSL(p,\mathbb{C})$ whose image is contained in $\overline{D}\rtimes \langle \overline{M_c}\rangle$. By definition, this set is included in $\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C})) $. We denote $\iota$ the inclusion. It induces a map $\varphi$ on the character variety and we have the following commutative diagram : $$\xymatrix{\operatorname{Hom}^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle) \ar[rr]^{\iota}\ar[d]^{\textnormal{mod }\overline{D}\rtimes \langle \overline{M_c}\rangle}&&\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C})) \ar[d]^{\textnormal{mod }PSL(p,\mathbb{C})}\\ \chi^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)\ar[rr]^{\varphi} &&\chi^i(\Gamma,PSL(p,\mathbb{C}))}$$
From Theorem \[classifcentr\], we immediately deduce the following corollary :
\[singular\]
Let $p$ be a prime number and $\Gamma$ be a finitely generated group. Then $\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))=\varphi(\chi^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle))$.
Therefore, to describe the singular locus, it suffices to describe $\chi^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$ (Section \[diagbyfin\]) and the behavior of $\varphi$ (Sections \[secinj\] and \[secint\]).
The character variety into a virtually abelian semidirect product {#diagbyfin}
=================================================================
This section will be devoted to the description of $\chi^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$. This requires group cohomology and its computation in terms of cochains. Definitions and results needed for this paper are given in Appendix \[grocoh\].
We recall that $\langle \overline{M_c}\rangle$ naturally acts on $\overline{D}$ by conjugation. Let $q$ be the natural projection of the semidirect product $\overline{D}\rtimes \langle \overline{M_c}\rangle$ onto $\langle \overline{M_c}\rangle$.
Given a group morphism $\overline{\rho}$ in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$, we remark that it makes of $\overline{D}$ a multiplicative $\Gamma$-module since $\langle \overline{M_c}\rangle$ acts by conjugation on $\overline{D}$. When it is necessary to specify the action, we will denote $\overline{D}_{\overline{\rho}}$ this $\Gamma$-module.
For any $\overline{\rho}$ in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$, define $\mathcal{H}_{\overline{\rho}}:=\{\rho\in \operatorname{Hom}(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)\mid q\circ \rho=\overline{\rho}\}$. The representation variety is a disjoint union of such sets :
$$\operatorname{Hom}(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)=\bigcup_{\overline{\rho}\in \operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)}\mathcal{H}_{\overline{\rho}}\text{.}$$
Furthermore :
\[Hrho\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $\overline{\rho}$ be in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$. Then the following map is a well-defined homeomorphism
$$f:\left|\begin{array}{clc}Z^1(\Gamma,\overline{D}_{\overline{\rho}})&\longrightarrow& \mathcal{H}_{\overline{\rho}}\\u&\longmapsto& \left(\gamma\mapsto u(\gamma)\overline{\rho}(\gamma)\right)\end{array}\right.\text{.}$$
Any group morphism $\rho$ from $\Gamma$ to $\overline{D}\rtimes \langle \overline{M_c}\rangle $ can be uniquely written as the product of a map $u_{\rho}:\Gamma\rightarrow \overline{D}$ and $q\circ \rho$. In particular, for any $\rho\in \mathcal{H}_{\overline{\rho}}$ : $\rho(\cdot)=u_{\rho}(\cdot)\overline{\rho}(\cdot)$.
It boils down to understand under which conditions on $u:\Gamma \rightarrow \overline{D}$ the map $\rho$ from $\Gamma$ to $\overline{D}\rtimes \langle \overline{M_c}\rangle$ defined as the product of $u$ and $\overline{\rho}$ is a group morphism. For two elements $\gamma,\gamma'$ in $\Gamma$, we compute : $$\begin{aligned}
&\rho(\gamma\gamma')=u(\gamma\gamma')\overline{\rho}(\gamma\gamma')\\
\text{ and }&\rho(\gamma)\rho(\gamma')=u(\gamma)\overline{\rho}(\gamma)u(\gamma')\overline{\rho}(\gamma')=u(\gamma)\gamma\cdot u(\gamma')\overline{\rho}(\gamma)\overline{\rho}(\gamma')\text{.}\end{aligned}$$
As a result, $\rho$ is a group morphism if and only if $u(\gamma\gamma')=u(\gamma) \gamma\cdot u(\gamma') $ for all $\gamma,\gamma'$ in $\Gamma$, i.e. if and only if $u$ is a $1$-cocycle from $\Gamma$ to $\overline{D}_{\overline{\rho}}$. It follows that :
$$\begin{array}{clc}Z^1(\Gamma,\overline{D}_{\overline{\rho}})&\longrightarrow& \mathcal{H}_{\overline{\rho}}\\u&\longmapsto& \left(\gamma\mapsto u(\gamma)\overline{\rho}(\gamma)\right)\end{array}$$
is a well-defined bijection. This map is continuous and its inverse is the push-forward of the projection on the first factor for $\overline{D}\rtimes \langle \overline{M_c}\rangle$ which is, also continuous.
There is a similar decomposition for the character variety. For $g\in \overline{D}\rtimes \langle \overline{M_c}\rangle$ and $\rho$ a representation from $\Gamma$ to $ \overline{D}\rtimes \langle \overline{M_c}\rangle$, $q\left(g\rho(\cdot) g^{-1}\right)=q(\rho(\cdot))$. As a result, $\overline{\mathcal{H}_{\overline{\rho}}}:=\{[\rho]\in \chi(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)\mid q\circ \rho=\overline{\rho}\}$ is well-defined and :
$$\chi(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)=\bigcup_{\overline{\rho}\in \operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)}\overline{\mathcal{H}_{\overline{\rho}}}\text{.}$$
Since the actions of $\Gamma$ and $\langle \overline{M_c}\rangle$ on $\overline{D}$ commute, the conjugation action of $\langle \overline{M_c}\rangle$ on $\overline{D}$ induces an action on $H^1(\Gamma,\overline{D}_{\overline{\rho}})$ (by acting on the coefficients).
\[barHrho\]Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $\overline{\rho}$ be in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$. Then the following map is a well-defined homeomorphism
$$\overline{f}:\left|\begin{array}{clc}H^1(\Gamma,\overline{D}_{\overline{\rho}})/\langle \overline{M_c}\rangle&\longrightarrow& \overline{\mathcal{H}_{\overline{\rho}}}\\ \left[u\right]\text{ mod }\langle \overline{M_c}\rangle&\longmapsto& \left[\gamma\mapsto u(\gamma)\overline{\rho}(\gamma)\right]\end{array}\right.\text{.}$$
Let $u_1$ and $u_2$ be in $Z^1(\Gamma,\overline{D}_{\overline{\rho}})$ and $\rho_i:=u_i\overline{\rho}$ be the corresponding element in $\mathcal{H}_{\overline{\rho}}$. Then $\rho_1$ is conjugate to $\rho_2$ in $\overline{D}\rtimes \langle \overline{M_c}\rangle$ if and only if there is $g=\overline{d}\text{ }\overline{M_c}^k$ in $\overline{D}\rtimes \langle \overline{M_c}\rangle$ such that $g\rho_1(\gamma)g^{-1}=\rho_2(\gamma)$ for all $\gamma\in \Gamma$. This is equivalent to $u_2(\gamma)=\overline{d}\gamma\cdot \overline{d}^{-1}\overline{M_c}^ku_1(\gamma)\overline{M_c}^{-k}$ for all $\gamma\in \Gamma$. Therefore, $\rho_1$ is conjugate to $\rho_2$ if and only if there exists $k$ such that $u_2\left(\overline{M_c}^k\cdot u_1\right)^{-1}$ belongs to $B^1(\Gamma,\overline{D}_{\overline{\rho}})$.
With this equivalence, we deduce that the bijection $f$ in proposition \[Hrho\] induces the wanted bijection :
$$\overline{f}:\left|\begin{array}{clc}H^1(\Gamma,\overline{D}_{\overline{\rho}})/\langle \overline{M_c}\rangle&\longrightarrow& \overline{\mathcal{H}_{\overline{\rho}}}\\ \left[u\right]\text{ mod }\langle \overline{M_c}\rangle&\longmapsto& \left[\gamma\mapsto u(\gamma)\overline{\rho}(\gamma)\right]\end{array}\right.\text{.}$$
Remark that $\overline{f}$ makes the following diagram commute :
$$\xymatrix{Z^1(\Gamma,\overline{D}_{\overline{\rho}})\ar[rr]^f\ar[d]&&\mathcal{H}_{\overline{\rho}}\ar[d]\\
H^1(\Gamma,\overline{D}_{\overline{\rho}})/\langle \overline{M_c}\rangle\ar[rr]^{\overline{f}}&& \overline{\mathcal{H}_{\overline{\rho}}}
}$$
Since $f$ is continuous and the projection from $Z^1(\Gamma,\overline{D}_{\overline{\rho}})$ to $H^1(\Gamma,\overline{D}_{\overline{\rho}})/\langle \overline{M_c}\rangle$ is open, it follows that $\overline{f}$ is continuous. Likewise, since $f^{-1}$ is continuous and the projection of $\mathcal{H}_{\overline{\rho}}$ onto $\overline{\mathcal{H}_{\overline{\rho}}}$ is open, it follows that $\overline{f}^{-1}$ is continuous.
Let $\overline{\rho}$ be a morphism from $\Gamma$ to $\langle \overline{M_c}\rangle$. We will need later a more explicit computation of $H^1(\Gamma,\overline{D}_{\overline{\rho}})$, when we fix particular examples of $\Gamma$.
\[Transgr\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $\overline{\rho}$ be a non-trivial morphism from $\Gamma$ to $\langle \overline{M_c}\rangle$. Let $\gamma_0\in \Gamma$ verifying $\overline{\rho}(\gamma_0)=\overline{M_c}$. Then the restriction map from $\Gamma$ to $K$ induces an homeomorphism between $$H^1(\Gamma,\overline{D}_{\overline{\rho}})\text{ and } \left\lbrace f\in \operatorname{Hom}(\operatorname{Ker}(\overline{\rho}),\overline{D})\left|\begin{array}{l}f(\gamma_0^p)=\overline{I_p}\\f(\gamma_0\gamma\gamma_0^{-1})=\overline{M_c}\cdot f(\gamma)\text{, } \forall \gamma\in K\end{array}\right. \right\rbrace\text{.}$$
To simplify the proof, we denote $K:=\operatorname{Ker}(\overline{\rho})$. We have an exact sequence of groups $\xymatrix{1\ar[r]& K\ar[r]& \Gamma\ar[r]^{\overline{\rho}}&\langle \overline{M_c}\rangle\ar[r]&1}$. Since $\Gamma$ acts on $\overline{D}$ via $\overline{\rho}$, we may write the first terms of the Inflation-Restriction sequence (see Proposition \[InfRes\]). $$\xymatrix{H^1(\langle \overline{M_c}\rangle,\overline{D}_{\overline{\rho}}^K)\ar[r]^{Inf}&H^1(\Gamma,\overline{D}_{\overline{\rho}})\ar[r]^{Res}&H^1(K,\overline{D}_{\overline{\rho}})^{\langle \overline{M_c}\rangle}\ar[r]^{T}&H^2(\langle \overline{M_c}\rangle,\overline{D}_{\overline{\rho}}^K) }\text{.}$$
The subgroup $K$ acts trivially on $\overline{D}$ so that $H^1(K,\overline{D}_{\overline{\rho}})=\operatorname{Hom}(K,\overline{D})$ by Lemma \[H1triv\]. The Inflation-Restriction sequence becomes : $$\xymatrix{H^1(\langle \overline{M_c}\rangle,\overline{D})\ar[r]^{Inf}&H^1(\Gamma,\overline{D}_{\overline{\rho}})\ar[r]^{Res}&\operatorname{Hom}(K,\overline{D})^{\langle \overline{M_c}\rangle}\ar[r]^{T}&H^2(\langle \overline{M_c}\rangle,\overline{D}) }\text{.}
\label{InflaRestri}$$
Applying Lemma \[H1cycl\], with $G$ being $\langle \overline{M_c}\rangle$, $g$ being $\overline{M_c}$ and $M:=\overline{D}$ : $$H^1(\langle \overline{M_c}\rangle,\overline{D})=\frac{\operatorname{Ker}(\operatorname{Norme}_{\overline{D}})}{\operatorname{Im}(\operatorname{Trace}_{\overline{D}})}
\label{H1cyclic}$$
Let $b$ be a diagonal matrix with coefficients $b_0,\dots,b_{p-1}$ on its diagonal : $$b(M_c\cdot b)^{-1}= \begin{pmatrix}b_{0}b_{p-1}^{-1}&&\\&\ddots&\\&&b_{p-1}b_{p-2}^{-1}\end{pmatrix} \text{.}$$
Let $c\in D$ then define $b_{p-1-i}:=c_{0,0}\cdots c_{i ,i}$. Since $c_{0,0}\cdots c_{p-1,p-1}=\det(c)=1$, an induction on $i$ shows that $b_ib_{i-1}^{-1}=c_{i,i}$. Multiplying $b$ by $\det(b)^{1/p}$ (this does not change the equation), we may even assume that $\det(b)=1$ and then $\operatorname{Trace}_{\overline{D}}(\overline{b})=\overline{b}(\overline{M_c}\cdot\overline{b})^{-1}=\overline{c}$.
Therefore, $\operatorname{Im}(\operatorname{Trace}_{\overline{D}})=\overline{D}$. Equation \[H1cyclic\] implies that $H^1(\langle \overline{M_c}\rangle,\overline{D})$ is trivial. The exact sequence \[InflaRestri\] implies that the restriction morphism $\operatorname{Res}$ is an isomorphism between $H^1(\Gamma,\overline{D}_{\overline{\rho}})$ and the kernel of the Transgression map $T$. Furthermore, the restriction map is clearly continuous and open on its image.
From Appendix \[grocoh\], we know that the action of $\langle \overline{M_c}\rangle$ over $\operatorname{Hom}(K,\overline{D})$ used to define $\operatorname{Hom}(K,\overline{D})^{\langle \overline{M_c}\rangle}$ in the Inflation-Restriction sequence is defined by : $$\left(\overline{M_c}\cdot f\right)(\gamma):=\overline{M_c}\cdot f(\gamma_0^{-1}\gamma\gamma_0) \text{, for $\gamma\in K$ and $f\in \operatorname{Hom}(K,\overline{D})$.}$$
Finally, Paragraph 10.2 in [@D-H-W] contains an explicit formula to compute the Transgression map when the kernel acts trivially (their formula is written additively, here it is written multiplicatively). If $f\in \operatorname{Hom}(K,\overline{D})^{\langle \overline{M_c}\rangle}$ then $T(f)$ is the cohomology class of the following $2$-cocycle on $\langle \overline{M_c}\rangle$ :
$$t(f)(\overline{M_c^k},\overline{M_c^l})=\left\lbrace
\begin{array}{cl}
\overline{I_p} & \mbox{if $0\leq k,l\leq p-1$ and $k+l<p$}\\
f(\gamma_0^p)^{-1}&\mbox{if $0\leq k,l\leq p-1$ and $k+l\geq p$}
\end{array}
\right.
\label{transgexpli}$$
Assume that $f$ is in the kernel of the Transgression map then $t(f)$ is a $2$-coboundary in $B^2(\langle \overline{M_c}\rangle,\overline{D}_{\overline{\rho}})$ ; i.e. there is a map $g:\langle \overline{M_c}\rangle\rightarrow \overline{D}$ such that for all $k,l$ in $\{0,\dots,p-1\}$ : $t(f)(\overline{M_c^k},\overline{M_c^l})=g(\overline{M_c^k})\overline{M_c^k}\cdot g(\overline{M_c^l}) g(\overline{M_c^{l+k}})^{-1}$. In particular, for $k=1$ and $l<p-1$ : $g(\overline{M^{l+1}})=g(\overline{M_c})\overline{M_c}\cdot g(\overline{M_c^l})$. An induction shows that $g(\overline{M_c^{l}})=g(\overline{M_c})\dots\overline{M_c^{l-1}}\cdot g(\overline{M_c})$. For $k=1$ and $l=p-1$ : $$f(\gamma_0^p)^{-1}=g(\overline{M_c})\overline{M_c}\cdot g(\overline{M_c^{p-1}})=\underbrace{g(\overline{M_c})\dots\overline{M_c^{p-1}}\cdot g(\overline{M_c})}_{\operatorname{Norme}_{\overline{D}}(g(\overline{M_c}))}=\overline{I_p} \text{.}$$ Therefore, if $f$ is in $\operatorname{Ker}(T)$ then $f(\gamma_0^p)$ is trivial. Conversely, using Equation \[transgexpli\], if $f(\gamma_0^p)$ is trivial then $f$ is in $\operatorname{Ker}(T)$. Combining this with the explicit definition of the action of $\langle \overline{M_c}\rangle$ on $\operatorname{Hom}(K,\overline{D})$ :
$$\operatorname{Ker}(T)= \left\lbrace f\in \operatorname{Hom}(K,\overline{D})\left|\begin{array}{l}f(\gamma_0^p)=\overline{I_p}\\f(\gamma_0\gamma\gamma_0^{-1})=\overline{M_c}\cdot f(\gamma)\text{, } \forall \gamma\in K\end{array}\right. \right\rbrace$$
Since we have already proved that the restriction morphism is an homeomorphism between $H^1(\Gamma,\overline{D}_{\overline{\rho}})$ and $\operatorname{Ker}(T)$, we are done.
\[Hirred\] If $\overline{\rho}:\Gamma\to\langle \overline{M_c}\rangle$ is trivial, then any representation in $\mathcal{H}_{\overline{\rho}}$ has its image included in $\overline{D}$ and therefore cannot be irreducible. If $\overline{\rho}:\Gamma\to\langle \overline{M_c}\rangle$ is non-trivial then a representation $\rho$ in $\mathcal{H}_{\overline{\rho}}$ is irreducible if and only if it is not conjugate to $\langle\overline{M_c}\rangle$ (Remark \[irred\]), if and only if $\operatorname{Ker}(q_{|\rho(\Gamma)})$ is not trivial. It follows that a representation $\rho$ belonging to $\mathcal{H}_{\overline{\rho}}$ is irreducible if and only if its corresponding $1$-cocycle (via the correspondence of proposition \[Hrho\]) is not a $1$-coboundary.
\[decomposi\] We denote $\mathcal{H}_{\overline{\rho}}^i$ the set of irreducible representations in $\mathcal{H}_{\overline{\rho}}$ and $\overline{\mathcal{H}_{\overline{\rho}}}^i$ their conjugacy classes up to conjugation by $\overline{D}\rtimes \langle \overline{M_c}\rangle$. We have :
$$\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))=\bigcup_{\substack{\overline{\rho}\in \operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)\\\overline{\rho}\textnormal{ non-trivial}}}\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^{i})
\label{decompbadlocus}$$
In the next section we will prove that $\varphi$ restricted to $\overline{\mathcal{H}_{\overline{\rho}}}^{i}$ is an homeomorphism onto its image. Therefore, Propositions \[barHrho\] and \[Transgr\] will eventually give a topological understanding of the singular locus.
A domain of injectivity for $\varphi$ {#secinj}
=====================================
We first need to prove :
\[injectivity\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $\overline{\rho}$ be a non-trivial morphism in $\operatorname{Hom}(\Gamma,\langle \overline{M_c}\rangle)$ and $\rho,\rho'$ be two irreducible representations in $\mathcal{H}_{\overline{\rho}}$. If there exists $g\in PSL(p,\mathbb{C})$ such that $g\cdot \rho=\rho'$ then $g\in \overline{D}\rtimes\langle\overline{M_c}\rangle$.
Let $\gamma_0\in \Gamma$ such that $\overline{\rho}(\gamma_0)=M_c$. Since $\rho,\rho'$ are in $\mathcal{H}_{\overline{\rho}}$ there exists $d_0,d_0',\dots,d_{p-1},d_{p-1}'$ such that :
$$\rho(\gamma_0)=\overline{\begin{pmatrix}d_0&&\\&\ddots&\\&&d_{p-1}\end{pmatrix}}\overline{M_c}\textnormal{ and } \rho'(\gamma_0)=\overline{\begin{pmatrix}d_0'&&\\&\ddots&\\&&d_{p-1}'\end{pmatrix}}\overline{M_c}\text{.}$$
Like in proposition \[centralconj\], there are $g_1$ and $g_2\in \overline{D}$ verifying $g_1\rho(\gamma_0)g_1^{-1}=\overline{M_c}$ and $g_2\rho'(\gamma_0)g_2^{-1}=\overline{M_c}$. Since $g\cdot \rho=\rho'$, $\overline{M_c}=(g_1^{-1}g_2g)\overline{M_c}(g_1^{-1}g_2g)^{-1}$.
Let us denote $\overline{h}:=g_1^{-1}g_2g$. We have just seen that $\overline{h}$ centralizes $\overline{M_c}$. Lemma \[centrMc\] implies that there are $a_0,\dots, a_{p-1}\in\mathbb{C}$ and an integer $s\geq 0$ such that : $$h=\sum_{k=0}^{p-1}a_kM_c^kD(\xi)^s\text{.}$$ Since $g_1\cdot\rho$ is irreducible, there is $\gamma\in \Gamma$ such that $g_1\cdot\rho(\gamma)$ is diagonal and not trivial. Remark that $\gamma$ belongs to $\operatorname{Ker}(\overline{\rho})$ and therefore $g_2\cdot\rho'(\gamma)$ is diagonal as well. Let $h_1,h_2\in D$ verifying $\overline{h_1}=g_1\cdot\rho(\gamma)$ and $\overline{h_2}=g_2\cdot\rho(\gamma)$. By definition of $h$, there is $l\geq 0$ verifying $hh_1h^{-1}= \xi^lh_2$. Hence : $$\sum_{k=0}^{p-1}a_kM_c^kD(\xi)^sh_1=
\xi^l\sum_{k=0}^{p-1}a_kh_2M_c^kD(\xi)^s\text{.}$$
Applying Lemma \[freeMc\], we have $a_kM_c^kD(\xi)^sh_1=a_kh_2M_c^kD(\xi)^s$, for $0\leq k\leq p-1$. Since $D(\xi)^s$ commutes with $h_1$ : $a_k(M_c^kh_1-\xi^lh_2M_c^k)=0$ holds for $0\leq k\leq p-1$. Assume there are $0\leq k_1<k_2\leq p-1$ verifying $a_{k_1}\neq 0$ and $a_{k_2}\neq 0$. Then $h_1=\xi^lM_c^{-k_1}h_2M_c^{k_1}$ and $\xi^lh_2=M_c^{k_2}h_1M_c^{-k_2}$ and therefore : $$h_1=M_c^{k_2-k_1}h_1M_c^{k_1-k_2}\text{.}$$
Since $M_c^{k_2-k_1}=M_{c^{k_2-k_1}}$ and $c^{k_2-k_1}$ is a cyclic permutation of order $p$, the equation above implies that $h_1=g_1\cdot\rho(\gamma)$ is trivial, which is a contradiction. As a result, there is a unique $k$ such that $a_k\neq 0$ and $\overline{h}$ belongs to $\overline{D}\rtimes\langle\overline{M_c} \rangle$. Since $g=g_2^{-1}g_1\overline{h}$, $g$ also belongs to $\overline{D}\rtimes\langle\overline{M_c} \rangle$.
To finish the proof of Theorem \[homeobarHrho\], we need to deal with the topology :
Proposition \[injectivity\] proves that $\varphi_{|\overline{\mathcal{H}_{\overline{\rho}}}^i}$ is injective. We recall the diagram
$$\xymatrix{\operatorname{Hom}^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle) \ar[rr]^{\iota}\ar[d]^{\textnormal{mod }\overline{D}\rtimes \langle \overline{M_c}\rangle=\psi_1}&&\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C})) \ar[d]^{\textnormal{mod }PSL(p,\mathbb{C})=\psi_2}\\ \chi^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)\ar[rr]^{\varphi} &&\chi^i(\Gamma,PSL(p,\mathbb{C}))}$$
We will denote $\psi_1$ the projection mod $\overline{D}\rtimes \langle \overline{M_c}\rangle$ and $\psi_2$ the projection mod $PSL(p,\mathbb{C})$. Since they are projections by a topological group action, $\psi_1$ and $\psi_2$ are both continuous and open. Since $\iota$ is induced by the inclusion of $\overline{D}\rtimes \langle \overline{M_c}\rangle$ into $PSL(p,\mathbb{C})$ which is a homeomorphism onto its image, $\iota$ is also a homeomorphism onto its image and $\operatorname{Im}(\iota)$ is a closed subset of $PSL(p,\mathbb{C})$. Since $\psi_2$ is continuous and $\psi_1$ is open, $\varphi$ is continuous and, in particular, $\varphi_{|\overline{\mathcal{H}_{\overline{\rho}}}^i}$ is continuous. Our next order of business is to show that $\varphi_{|\overline{\mathcal{H}_{\overline{\rho}}}^i}$ is open in its image. We restrict the diagram to $\mathcal{H}_{\overline{\rho}}^i$ :
$$\xymatrix{
\mathcal{H}_{\overline{\rho}}^i\ar[rrrr]^{\iota_{|\mathcal{H}_{\overline{\rho}}^i}} \ar[d]_{\psi_{1|\mathcal{H}_{\overline{\rho}}^i}} &&&&\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C})) \ar[d]_{\psi_2}\\ \overline{\mathcal{H}_{\overline{\rho}}}^i\ar[rrrr]^{\varphi_{|\overline{\mathcal{H}_{\overline{\rho}}}^i}} &&&&\chi^i(\Gamma,PSL(p,\mathbb{C}))
}$$
Since each $\mathcal{H}_{\overline{\rho}}^i$ is closed and $\operatorname{Hom}^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$ is a finite disjoint union of such $\mathcal{H}_{\overline{\rho}}^i$’s, each $\mathcal{H}_{\overline{\rho}}^i$ is open in $\operatorname{Hom}^i(\Gamma,\overline{D}\rtimes \langle \overline{M_c}\rangle)$. It follows that $\psi_1$ restricted to $\mathcal{H}_{\overline{\rho}}^i$ is still continuous and open and $\iota$ restricted to $\mathcal{H}_{\overline{\rho}}^i$ is still an homeomorphism onto its image. The group $PSL(p,\mathbb{C})$ acts properly on $\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))$ (Proposition 1.1 in [@J-M]), i.e. the following function is proper :
$$\zeta :
\left|
\begin{array}{rcl}
PSL(p,\mathbb{C})\times \operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))&\longrightarrow& \operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))^2\\
(g,\rho)&\longmapsto& (g\cdot \rho,\rho) \\
\end{array}
\right.\text{.}$$
Since both $PSL(p,\mathbb{C})$ and $\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))$ are locally compact and Hausdorff, the function $\zeta$ is closed. For any closed set $F$ in $\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))$, the set $\zeta(PSL(p,\mathbb{C})\times F)$ is closed. Therefore, its projection on the first coordinate $PSL(p,\mathbb{C})\cdot F$ is closed in $\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))$.
Let $U$ be an open subset of $\overline{\mathcal{H}_{\overline{\rho}}}^i$. We want to prove that $\varphi(U)$ is open in $\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)$. Denote $U_0:=(\psi_{1|\mathcal{H}_{\overline{\rho}}^i})^{-1}(U)$ which is open by continuity of $\psi_1$. Then $\iota(U_0)$ is open in $\iota(\mathcal{H}_{\overline{\rho}}^i)$, hence there exists an open set $V$ in $\operatorname{Hom}^i(\Gamma,PSL(p,\mathbb{C}))$ such that $V\cap \iota(\mathcal{H}_{\overline{\rho}}^i)=\iota(U_0) $.
Let $F:=V^c\cap \iota(\mathcal{H}_{\overline{\rho}}^i)$. Since $V^c$ is closed and $\iota(\mathcal{H}_{\overline{\rho}}^i)$ is closed, $F$ is closed, and using properness as explained above, its saturation $PSL(p,\mathbb{C})\cdot F$ is closed. Let $V_0$ be $V-PSL(p,\mathbb{C})\cdot F$ then $V_0$ is open.
By definition, $V_0\cap\iota(\mathcal{H}_{\overline{\rho}}^i)$ is contained in $V\cap \iota(\mathcal{H}_{\overline{\rho}}^i)$. Conversely, let $\rho$ be in $\iota(U_0)= V\cap \iota(\mathcal{H}_{\overline{\rho}}^i)$, we want to show that $\rho$ is not conjugate to an element of $F$ (this will imply that $\rho$ belongs to $V_0\cap\iota(\mathcal{H}_{\overline{\rho}}^i)$) and we do it by contradiction. Assume that $\rho\in PSL(p,\mathbb{C})\cdot F$, i.e. there is $g\in PSL(p,\mathbb{C})$ such that $g\cdot \rho\in F$. Since $\rho$ and $g\cdot \rho$ are both irreducible and valued in $ \overline{D}\rtimes\langle\overline{M_c}\rangle$ by definition, Proposition \[injectivity\] implies that $g$ belongs to $\overline{D}\rtimes \langle \overline{M_c}\rangle$. Since $\rho$ belongs to $\iota(U_0)$ and $g\cdot\rho$ does not, $U_0$ is not stable by the action of $ \overline{D}\rtimes\langle\overline{M_c}\rangle$, which is in contradiction with the very definition of $U_0$. Therefore $V_0\cap\iota(\mathcal{H}_{\overline{\rho}}^i)=V\cap \iota(\mathcal{H}_{\overline{\rho}}^i)$. Since $\psi_2\circ\iota=\varphi\circ\psi_1$, we have $\varphi(U)=\psi_2(\iota(U_0))=\psi_2(V\cap \iota(\mathcal{H}_{\overline{\rho}}^i))$. Therefore $\varphi(U)=\psi_2(V_0\cap \iota(\mathcal{H}_{\overline{\rho}}^i))$.
It remains to show that $\psi_2(V_0\cap \iota(\mathcal{H}_{\overline{\rho}}^i))=\psi_2(V_0)\cap \psi_2( \iota(\mathcal{H}_{\overline{\rho}}^i))$. The left-hand side is clearly contained in the right-hand side. Since $V_0$ is stable (by definition) by the action of $PSL(p,\mathbb{C})$, we have the other inclusion and therefore, we have the equality.
Finally $\varphi(U)$ is the intersection of $\psi_2(V_0)$ which is open (since $\psi_2$ is open) and $\psi_2( \iota(\mathcal{H}_{\overline{\rho}}^i))=\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)$. Therefore $\varphi(U)$ is open in $\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)$. The function $\varphi_{\overline{\mathcal{H}_{\overline{\rho}}}^i}$ is continuous and open in its image, since it is also injective, it is an homeomorphism onto its image.
As a result, the singular locus of a character variety in $PSL(p,\mathbb{C})$ is a finite union of topological spaces, namely the $\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)$’s, whose topology is given by Propositions \[barHrho\] and \[Transgr\]. Before studying the intersections between these spaces, we remark that some of them may be equal to each other.
\[identifvarphiHrho\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $\overline{\rho},\overline{\rho}'$ be two non-trivial elements in $\operatorname{Hom}(\Gamma,\langle\overline{M_c}\rangle)$. Then, $\operatorname{Ker}(\overline{\rho})=\operatorname{Ker}(\overline{\rho}')$ if and only if there is $\phi\in\operatorname{Aut}(\langle\overline{M_c}\rangle)$ verifying $\overline{\rho}'=\phi\circ\overline{\rho}$. When it is true, $\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)=\varphi(\overline{\mathcal{H}_{\overline{\rho}'}}^i)$.
The equivalence in this lemma is straightforward.
Let $\phi\in\operatorname{Aut}(\langle\overline{M_c}\rangle)$, then $\phi$ is uniquely determined by $l\in (\mathbb{Z}/p)^*$ where $\phi(\overline{M_c})=\overline{M_c}^l$. We defined $c$ to be the permutation of $\mathbb{Z}/p$ sending $i$ to $i+1$. Let us now define the permutation $\sigma_l$ of $\mathbb{Z}/p$ sending $i$ to $l\times i$. We easily see that $\sigma_l\circ c(i)=l\times i+l=c^l\circ \sigma_l(i)$. Therefore $\sigma_lc\sigma_l^{-1}=c^l $ and $M_{\sigma_l}M_{c}M_{\sigma_l}^{-1}=M_c^l $. Let $M:=M_{\sigma_l}$ if $\det(M_{\sigma_l})=1$ and $-M_{\sigma_l}$ if $\det(M_{\sigma_l})=-1$. It follows that the conjugation by $\overline{M}$ sends $\iota(\mathcal{H}_{\overline{\rho}}^i)$ to $\iota(\mathcal{H}_{\phi\circ \overline{\rho}}^i)$. As a result $\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)=\varphi(\overline{\mathcal{H}_{\phi\circ \overline{\rho}}}^i)$.
\[newdecomposi\] Fix a finitely generated group $\Gamma$ and a prime number $p$. Then, for any normal subgroup $K$ of index $p$ in $\Gamma$, fix one group morphism $\overline{\rho}_K$ in $\operatorname{Hom}(\Gamma,\langle\overline{M_c}\rangle)$ verifying $K=\operatorname{Ker}(\overline{\rho})$. Lemma \[identifvarphiHrho\] and the decomposition of the singular locus in Equation \[decompbadlocus\] implies the following decomposition :
$$\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))=\bigcup_{\substack{K\triangleleft \Gamma\\ \text{$[\Gamma:K]=p$} } }\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\text{.}
\label{decompbadlocus2}$$
We shall call $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ the *pseudo-component* associated to $K$. When $\Gamma$ is a free group, each of these pseudo-components is an irreducible component of $\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))$ (Remark \[irredcompfree\]). However, in general it has no reason to be an irreducible component of the singular locus (the surface group is a counter example, see Remark \[irredcompsurf\]).
The intersection pattern in the singular locus {#secint}
==============================================
In this section, we give a description of the intersection pattern between the different pseudo-components defined above. The next lemma is straightforward and will be used several times throughout this section.
\[notprop\] Let $\Gamma$ be a group, $p$ be a prime number and $K,K'$ be two different normal subgroups of index $p$ in $\Gamma$. Then $\Gamma/K\cap K'$ is isomorphic to $(\mathbb{Z}/p)^2$.
\[abelirred\] Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $K,K'$ be two different normal subgroups of index $p$ in $\Gamma$. Let $\rho:\Gamma\to PSL(p,\mathbb{C})$ be a representation whose conjugacy class belongs to $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$. Then $\operatorname{Ker}(\rho)=K\cap K'$ and $\rho(\Gamma)$ is conjugate to $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$.
By definition, $\rho$ is both conjugate to a representation in $\iota(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ and to a representation in $\iota(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$. Let $g,h\in PSL(p,\mathbb{C})$ verifying that $g\rho g^{-1}$ (resp. $h\rho h^{-1}$) belongs to $\iota(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ (resp. $\iota(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$). This implies that $g\rho(K) g^{-1}\leq \overline{D}$ and $h\rho(K') h^{-1}\leq \overline{D}$. In particular, $\rho(K)$ and $\rho(K')$ are both abelian.
Since $K$ and $K'$ are different and both of index $p$ in $\Gamma$, there is $\gamma_1\in K'\cap K^c$ verifying $g\rho(\gamma_1)g^{-1}=\overline{d_1}\text{ }\overline{M_c}$ with $\overline{d_1}\in\overline{D}$. Let $\gamma\in K\cap K'$, since $g\rho(K')g^{-1}$ is abelian, $g\rho(\gamma_1)g^{-1} $ and $g\rho(\gamma)g^{-1}$ commute. Since $g\rho(\gamma)g^{-1}\in\overline{D}$, it also commutes with $\overline{d_1}$ and, therefore, it commutes with $\overline{M_c}$. By Lemma \[centrMc\], an element in $\overline{D}$ commuting with $\overline{M_c}$ necessarily belongs to $\langle \overline{D(\xi)}\rangle$. Thus, $g\rho(K\cap K')g^{-1}$ is included in $\langle \overline{D(\xi)}\rangle$. Likewise, $h\rho(K\cap K')h^{-1}$ is included in $\langle \overline{D(\xi)}\rangle$.
Assume $K\cap K'$ is not included in $\operatorname{Ker}(\rho)$. Then, the groups $g\rho(K\cap K')g^{-1}$ and $h\rho(K\cap K')h^{-1}$ are both equal to $\langle\overline{D(\xi)}\rangle$. Thus, $hg^{-1}$ normalizes $\langle\overline{D(\xi)}\rangle$.
Let $0<k<p$ be an integer such that $hg^{-1}\overline{D(\xi)}gh^{-1}=\overline{D(\xi)}^k$ and $M_{\sigma}$ be a permutation matrix such that $M_{\sigma}D(\xi)^kM_{\sigma}^{-1}=D(\xi)$. Then $\overline{M_{\sigma}}hg^{-1}$ commutes with $\overline{D(\xi)}$, whence (Lemma \[centrdxi\]) belongs to $\overline{D}\rtimes\langle\overline{M_c}\rangle$. Therefore, $hg^{-1}$ normalizes $\overline{D}$ and, in particular $K=K'$ which is a contradiction. Therefore $\operatorname{Ker}(\rho)=K\cap K'$.
Lemma \[notprop\] states that $\Gamma/K\cap K'$ is isomorphic to $(\mathbb{Z}/p)^2$. Therefore, $\rho(\Gamma)$ is abelian irreducible. Theorem \[classifcentr\] implies that any abelian irreducible subgroup of $PSL(p,\mathbb{C})$ is conjugate to $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$, whence the result.
\[paraminter\] Assume $\Gamma$, $p$, $K$ and $K'$ are given like in the preceding proposition. This proposition implies that, up to conjugation, any representation $\rho$ whose conjugacy class belongs to $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ factorizes through the projection to $\Gamma/K\cap K'$ in an isomorphism between $\Gamma/K\cap K'$ and $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$. Therefore, to count the number of elements in $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$, it suffices to count the number of isomorphisms between $\Gamma/K\cap K'$ and $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ up to conjugation in $PSL(p,\mathbb{C})$.
Let $p$ be a prime number. The group $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ is a $\mathbb{Z}/p$-vector space of dimension $2$ and has the following $\mathbb{Z}/p$-basis $( \overline{D(\xi)},\overline{M_c})$. Once the basis is fixed, it is natural to identify its automorphism group to $GL(2,\mathbb{Z}/p)$.
\[countZp2\] Let $p$ be a prime number. Then, the subgroup of $\operatorname{Aut}(\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle)$ induced by the normalizer of $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ in $PSL(p,\mathbb{C})$ is $SL(2,\mathbb{Z}/p)$.
Assume $p$ odd. We denote $\operatorname{Aut}_0(\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle)$ the subgroup of automorphisms of $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ which can be realized as the conjugation by an element of $PSL(p,\mathbb{C})$.
Let $V_1:=\lambda V_0$ where $V_0:=(\xi^{ij})_{0\leq i,j\leq p-1}$ is the Vandermonde matrix and $\lambda$ be $\det(V_0)^{-\frac{1}{p}}=(-1)^{\frac{p-1}{2p}}/p$. Direct computations lead to $V_0D(\xi)V_0^{-1}=M_c^{-1}$ and $V_0M_cV_0^{-1}=D(\xi)$. Therefore $\overline{V_1}\text{ }\overline{D(\xi)}\text{ }\overline{V_1}^{-1}=\overline{M_c}^{-1}$ and $ \overline{V_1}\text{ }\overline{M_c}\text{ }\overline{V_1}^{-1}=\overline{D_{\xi}}$. Thus $\overline{V_1}$ normalizes $\langle \overline{D(\xi)}\rangle\times \langle \overline{M_c}\rangle$ and induces by conjugation the automorphism $\begin{pmatrix}0&1\\ -1&0\end{pmatrix}$ on $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$.
Let $S$ be a diagonal matrix where $S_{i,i}=\xi^{\frac{i(i+1)}{2}}$ for $i=0,\dots,p-1$. Then $\det(S)=\xi^{\frac{(p-1)p(p+1)}{6}}=1$ since $p$ is odd. Since $S$ is diagonal, it commutes with $D(\xi)$ therefore $\overline{S}\text{ }\overline{D(\xi)}\text{ }\overline{S}^{-1}=\overline{D(\xi)}$. Direct computations show that $SM_cS^{-1}=D(\xi)M_c$. Therefore $\overline{S}\text{ }\overline{M_c}\text{ }\overline{S}^{-1}=\overline{M_c}\text{ }\overline{D(\xi)}$. Whence $\overline{S}$ normalizes $\langle \overline{D(\xi)}\rangle\times \langle \overline{M_c}\rangle$ and induces by conjugation the automorphism $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ on $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$.
Therefore, the subgroup generated by $\begin{pmatrix}0&1\\ -1&0\end{pmatrix}$ and $\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ is contained in $\operatorname{Aut}_0(\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle)$ . Since these matrices generate $SL(2,\mathbb{Z}/p)$, this group is contained in $\operatorname{Aut}_0(\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle)$.
Conversely, let $\psi$ be in $\operatorname{Aut}_0(\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle)$. We denote $d$ its determinant. We denote $\psi_d:=\begin{pmatrix}1&0\\ 0&d\end{pmatrix}$. Since $\det(\psi_d^{-1}\psi)=1$, it follows that the automorphism $\psi_d$ also belongs to $\operatorname{Aut}_0(\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle)$.
Let $g$ be in $PSL(p,\mathbb{C})$ verifying for all $x\in \langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ that $gxg^{-1}=\psi_d(x)$. Then $g$ commutes with $\overline{D(\xi)}$. Furthermore $g\overline{M_c}g^{-1}=\overline{M_c}^d$. Since Lemma \[centrdxi\] implies that $g$ belongs to $\overline{D}\rtimes \langle\overline{M_c}\rangle$, we may project this equality onto $ \langle\overline{M_c}\rangle$ to get that $\overline{M_c}=\overline{M_c}^d$ and $d=1$. As a result $\det(\psi)=1$ and we are done.
Therefore, we can count the number of elements in the intersection of two pseudo-components :
\[countinter\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $K,K'$ be two different normal subgroups of index $p$ in $\Gamma$. Then $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ is finite of cardinal $p-1$.
Following Remark \[paraminter\], there are as many elements in $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ as there are isomorphisms between $\Gamma/K\cap K'$ and $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ up to conjugation.
We fix $\psi_0$ such isomorphism. Since the natural action of $\operatorname{Aut}(\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle)$ on the set of isomorphisms between $\Gamma/K\cap K'$ and $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ is transitive, it suffices to understand under which condition $\psi\circ\phi_0$ is conjugate to $\phi_0$. Lemma \[countZp2\] implies that $\psi\circ\phi_0$ is conjugate to $\phi_0$ if and only if $\psi\in SL(2,\mathbb{Z}/p)$. Therefore, there are $|GL(2,\mathbb{Z}/p)|/|SL(2,\mathbb{Z}/p)|=p-1$ elements in $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$.
\[abirredK\]
Let $\Gamma$ be a finitely generated group, $p$ be a prime number and $\rho$ be an abelian irreducible representation from $\Gamma$ to $PSL(p,\mathbb{C})$. Then the conjugacy class of $\rho$ belongs to exactly $p+1$ pseudo-components.
Let $K$ be a normal subgroup of index $p$ in $\Gamma$. We first show that the conjugacy class of $\rho$ belongs to $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ if and only if $\operatorname{Ker}(\rho)$ is contained in $K$.
If some conjugate $g\rho g^{-1}$ of $\rho$ belongs to $\iota(\mathcal{H}_{\overline{\rho}_K}^i)$, then $\operatorname{Ker}(\rho)$ will be contained in $\operatorname{Ker}\left(q\circ\left(g\rho g^{-1}\right)\right)=K$.
Assume that $\operatorname{Ker}(\rho)$ is contained in $K$. Since $\rho$ is abelian irreducible, Theorem \[classifcentr\] implies that $\rho$ is conjugate to a representation $g\rho g^{-1}$ from $\Gamma$ onto $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$.
Since $K$ strictly contains $\operatorname{Ker}(\rho)$, it follows that $g\rho (K)g^{-1}$ is a subgroup of index $p$ in $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$. The group $SL(2,\mathbb{Z}/p)$ acts transitively by automorphism on the subgroups of index $p$ in $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$.
Therefore, Lemma \[countZp2\] implies that there is $h\in PSL(p,\mathbb{C})$ normalizing the group $\langle \overline{D(\xi)}\rangle \times\langle\overline{M_c}\rangle$ and verifying that $hg\rho(K)g^{-1}h^{-1}=\langle \overline{D(\xi)}\rangle $.
Thus the representation $hg\rho(hg)^{-1}$ is still a representation in $\overline{D}\rtimes\langle\overline{M_c}\rangle$ and $\operatorname{Ker}\left(q\circ \left(hg\rho(hg)^{-1}\right)\right)=K$. Therefore, its conjugacy class belongs to $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$.
Using this equivalence, there are as many pseudo-components containing the conjugacy class of $\rho$ as there are normal subgroups of index $p$ containing $\operatorname{Ker}(\rho)$. This is the number of subgroups of index $p$ in $\Gamma/\operatorname{Ker}(\rho)$ which is isomorphic to $(\mathbb{Z}/p)^2$. Thus, there are $p+1$ of them.
We sum up this combinatorial information in one single theorem. Before that, we recall that if $\Gamma$ is a group then $\Gamma^{Ab}/p\Gamma^{Ab}$ is a $\mathbb{Z}/p$-vector space. Its dimension is called the *$p$-rank* of $\Gamma$.
1. In Remark \[newdecomposi\], we justified that $\chi^i_{Sing}(\Gamma,PSL(p,\mathbb{C}))$ was the union of pseudo-components associated to $K$ for $K$ normal subgroups of index $p$ in $\Gamma$. Since the set of such normal subgroups is in bijection with the set of subgroups of index $p$ in $\Gamma^{Ab}/p\Gamma^{Ab}$, we have $\frac{p^r-1}{p-1}$ of them.
2. It is the statement of Proposition \[abelirred\] and Corollary \[countinter\].
3. It is the statement of Proposition \[abirredK\].
4. On one hand, for each unordered pair $\{K,K'\}$ of different normal subgroups of index $p$ in $\Gamma$, there are $p-1$ conjugacy classes of abelian irreducible representations from $\Gamma$ to $PSL(p,\mathbb{C})$ in $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ and there are $\begin{pmatrix}\frac{p^r-1}{p-1}\\2\end{pmatrix}$ such unordered pairs.
On the other hand, for each conjugacy class abelian irreducible representation from $\Gamma$ to $PSL(p,\mathbb{C})$, there are $\begin{pmatrix}p+1\\2\end{pmatrix}$ unordered pairs $\{K,K'\}$ of different normal subgroups of index $p$ in $\Gamma$ for which the conjugacy class of this representation is contained in $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$.
Let $N_{\Gamma,p}$ be the number of conjugacy classes of abelian irreducible representations. Then :
$$N_{\Gamma,p}\begin{pmatrix}p+1\\2\end{pmatrix}=\begin{pmatrix}\frac{p^r-1}{p-1}\\2\end{pmatrix}(p-1)\text{, } N_{\Gamma,p}\frac{(p+1)p}{2}=\frac{(p^r-1)(p^r-p)}{2(p-1)^2}(p-1)$$
Therefore, the number of conjugacy classes of abelian irreducible representations from $\Gamma$ to $PSL(p,\mathbb{C})$ is $\frac{(p^r-1)(p^r-p)}{(p-1)p(p+1)}=\frac{(p^r-1)(p^{r-1}-1)}{p^2-1}$.
In this theorem, the combinatorial information about the singular locus of the character variety only depends on the $p$-rank of $\Gamma$. To understand its topology, Section \[diagbyfin\] tells us that we need to compute cohomology groups. This is what we will do for free groups in Section \[freegrpcase\] and for surface groups in Section \[surfgrpcase\].
The free group case {#freegrpcase}
===================
Define $\Gamma:=\mathbb{F}_l$ to be the free group of rank $l\geq 2$. We recall that the abelianization of $\mathbb{F}_l$ is $\mathbb{Z}^l$. Furthermore the canonical map from $\operatorname{Aut}(\mathbb{F}_l)$ to $\operatorname{Aut}((\mathbb{F}_l)^{Ab})=GL(l,\mathbb{Z})$ is known to be surjective (c.f. [@L-S]). Since the set of normal subgroups $K$ of index $p$ is in bijective correspondence with the set of subgroups of index $p$ in $(\mathbb{F}_l)^{Ab}=\mathbb{Z}^l$, it follows that $\operatorname{Aut}(\mathbb{F}_l)$ acts transitively on the set of normal subgroups of index $p$ in $\Gamma$. Using this, we have :
\[onesinglphiKfree\]
Let $l\geq 2$ and $K,K'$ be two normal subgroups of index $p$ in $\mathbb{F}_l$. Then $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ and $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ are homeomorphic to each other.
Let $\phi$ be an element in $\operatorname{Aut}(\mathbb{F}_l)$ such that $\phi(K)=K'$. Then the precomposition by $\phi^{-1}$ provides a homeomorphism from $\iota(\mathcal{H}_{\overline{\rho}_K})$ to $\iota(\mathcal{H}_{\overline{\rho}_{K'}})$ whose continuous inverse is the precomposition by $\phi$. This homeomorphism induces, on the character variety, an homeomorphism between $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ and $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$.
\[freegrponeK\] This proposition implies that we only need to focus on one single pseudo-component. Here we are going to explain, how to construct a particular normal subgroup of index $p$ in $\mathbb{F}_l$ using topological coverings. What we are mostly interested in is the expression of the generators of $K$ as words in the generators of $\mathbb{F}_l$. We begin with a disk $D$ and remove from this disk a smaller disk with the same center. Then we remove along a radius another $l-1$ holes. After a rotation of the disk by an angle of $\frac{2\pi}{p}$, we remove again along a radius another $l-1$ holes. Doing this $p$ times we construct a surface $Y$ with $(l-1)p+1$ holes. Since, by construction, this surface is invariant by a rotation $r$ of order $p$ we may quotient out $Y$ by $\langle r\rangle$ to construct a new surface $X$ with exactly $l$ holes. See Figure \[pict1\].
Define $\psi:Y\rightarrow X$ the projection identifying points in $Y$ modulo $\langle r\rangle$. This is a Galois cover of $X$ of order $p$. Therefore, $K:=\pi_1(Y)$ is a normal subgroup of index $p$ in $\pi_1(X)$. Since $X$ is a disk with $l$ holes, the group $\pi_1(X)$ is freely generated by loops around each hole, we will denote them $x_1,\dots, x_{l}$.
On the other hand, a system of free generators for $K$ is given by loops around the $(l-1)p+1$ holes which can be identified to $x_1^p$ and $x_1^{i}x_jx_1^{-i}$ with $0\leq i\leq p-1$ and $2\leq j\leq l$ (see Figure \[pict1\]).
(0,7) ellipse (6 and 2.7); (0,7) ellipse (1.2 and 0.5); (2,7) ellipse (0.3 and 0.175); (5.5,7) ellipse (0.3 and 0.175); (1.5,7.75) ellipse (0.3 and 0.175); (3.5,8.75) ellipse (0.3 and 0.175); (0,7) ellipse(6 and 2.7) (0,7) ellipse (1.2 and 0.5) (2,7) ellipse(0.3 and 0.175) (5.5,7) ellipse(0.3 and 0.175) (1.5,7.75) ellipse(0.3 and 0.175) (3.5,8.75) ellipse (0.3 and 0.175);;
at (0,7.5)[$\blacktriangleleft$]{}; at (2,7.175)[$\blacktriangleleft$]{}; at (5.5,7.175)[$\blacktriangleleft$]{}; at (1.5,7.925)[$\blacktriangleleft$]{}; at (3.5,8.925)[$\blacktriangleleft$]{}; (2.4,7)–(5.1,7); (1.7,7.85)–(3.3,8.65);
at (3.75,6.6) \[xscale=1,yscale=1,rotate=0\] [$\underbrace{\quad\quad\quad\quad\quad\quad\quad\quad\quad}$]{}; at (3.75,6.2)[$l-1$ holes]{};
at (2.1,8.55) \[xscale=1,yscale=1,rotate=-152\] [$\underbrace{\quad\quad\quad\quad\quad\quad\quad}$]{}; at (2,8.85)\[rotate=28\][$l-1$ holes]{};
at (-1.2,7.45) [$x_1^p$]{}; at (2.45,7.2) [$x_2$]{}; at (5.1,7.2) [$x_l$]{}; at (2.6,7.75) [$x_1x_2x_1^{-1}$]{}; at (3.7,8.4) [$x_1x_lx_1^{-1}$]{};
(6.15,7).. controls (6,7.8) and (5.8,8) .. (5.28,8.5) .. controls (4.5,9) .. (4.2,9.1); at (4.2,9.8) [ rotation of angle $\frac{2\pi}{p}$]{};
(1,7.9)..controls (0,8.2) and (-2.5,7.8) .. (-2.5,7)..controls (-2.5,6.2) and (0,5.8) .. (1.5,6.2); at (-3,7)[$p$ lines of holes]{};
at (-4.2,9.8)\[xscale=2,yscale=2,rotate=0\][Surface $Y$]{};
(0,7) –(0,6.5); (0,6.5) – (0,4.3); (0,4.3)–(0,-1);
at (1.2,3)[projection $\psi$]{};
at (-4.2,1.8)\[xscale=2,yscale=2,rotate=0\][Surface $X$]{};
(0,-1) ellipse (6 and 2.7); (0,-1) ellipse (1.2 and 0.5); (2,-1) ellipse (0.3 and 0.175); (5.5,-1) ellipse (0.3 and 0.175); (0,-1) ellipse(6 and 2.7) (0,-1) ellipse (1.2 and 0.5) (2,-1) ellipse(0.3 and 0.175) (5.5,-1) ellipse(0.3 and 0.175);;
at (0,-1.5)[$\blacktriangleright$]{}; at (2,-0.825)[$\blacktriangleleft$]{}; at (5.5,-0.825)[$\blacktriangleleft$]{}; (2.4,-1)–(5.1,-1);
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Since $\langle\overline{M_c}\rangle$ acts by conjugation on $\overline{D}$, it also acts on $\overline{D}^k$ by simultaneously acting on each coordinates. The next proposition relies on the results of Section \[diagbyfin\], especially the explicit computations of cohomology groups.
\[Hrhofreegroup\]
Let $l\geq 2$, $p$ be a prime number. Then any pseudo-component in $\chi^i_{Sing}(\mathbb{F}_l,PSL(p,\mathbb{C}))$ is homeomorphic to $\left(\overline{D}^{l-1}-\{(\overline{I_p},\dots,\overline{I_p})\}\right)/ \langle \overline{M_c}\rangle$.
Using Proposition \[onesinglphiKfree\], we only construct the homeomorphism for a unique pseudo-component. Let $x_1,\dots,x_l$ be a system of free generators for $\mathbb{F}_l$. Then Remark \[freegrponeK\] justifies that the subgroup $K_0$ generated by $X:=x_1^p$ and $Y_{i,j}:=x_1^{i}x_jx_1^{-i}$ with $0\leq i\leq p-1$ and $2\leq j\leq l$ is freely generated by these generators and is a normal subgroup of index $p$ in $\mathbb{F}_l$.
According to Theorem \[homeobarHrho\], $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i)$ is homeomorphic to $\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i$. By Proposition \[barHrho\], we need to compute $H^1(\mathbb{F}_l,\overline{D}_{\overline{\rho}_{K_0}})$. Following the notation of Proposition \[Transgr\], we choose $\gamma_0:=x_1$. This proposition states that there is an homeomorphism between
$$H^1(\mathbb{F}_l,\overline{D}_{\overline{\rho}_{K_0}})\text{ and } \left\lbrace f\in \operatorname{Hom}(K_0,\overline{D})\left|\begin{array}{l}f(x_1^p)=\overline{I_p}\\f(x_1\gamma x_1^{-1})=\overline{M_c}\cdot f(\gamma)\text{, } \forall \gamma\in K_0\end{array}\right. \right\rbrace\text{.}$$
Identifying $f$ to the tuples $(f(X),(f(Y_{i,j})))$ of images of the free generators of $K_0$, $\operatorname{Hom}(K_0,\overline{D})$ is equal to $\overline{D}^{1+p(l-1)}$. Furthermore, $f(x_1^p)=\overline{I_p}$ means that $f(X)=\overline{I_p}$. Whereas $f(x_1\gamma x_1^{-1})=\overline{M_c}\cdot f(\gamma)$, for all $\gamma\in K_0$ is equivalent to $f(Y_{i+1,j})=\overline{M_c}\cdot f(Y_{i,j})$ for $j=2,\dots,l$ and $i=0,\dots,p-2$.
Therefore, the map sending $f$ to $(f(Y_{0,2}),\dots,f(Y_{0,l}))$ defines a natural identification between the space above and $\overline{D}^{l-1}$. As a result, there is an homeomorphism between $H^1(\Gamma,\overline{D}_{\overline{\rho}_{K_0}})$ and $\overline{D}^{l-1}$. This leads to an homeomorphism between $H^1(\Gamma,\overline{D}_{\overline{\rho}_{K_0}})/\langle \overline{M_c}\rangle$ and $\overline{D}^{l-1}/\langle \overline{M_c}\rangle$. By Proposition \[barHrho\], $\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}$ is homeomorphic to $\overline{D}^{l-1}/\langle \overline{M_c}\rangle$. Finally, in Remark \[Hirred\], we have seen that $\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}$ has only one point which is a conjugacy class of non-irreducible representations. It is associated to the zero element in $H^1(\Gamma,\overline{D}_{\overline{\rho}_{K_0}})/\langle \overline{M_c}\rangle$. As a result, $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i)$ is homeomorphic to $\left(\overline{D}^{l-1}-\{(\overline{I_p},\dots,\overline{I_p})\}\right)/ \langle \overline{M_c}\rangle$.
\[orblocfree\]
Let $l\geq 2$ and $p$ be a prime number. Then the singular locus of $\chi^i(\mathbb{F}_l,PSL(p,\mathbb{C}))$ is connected of dimension $(p-1)(l-1)$.
Equation \[decompbadlocus2\] implies that the singular locus $\chi^i_{Sing}(\mathbb{F}_l,PSL(p,\mathbb{C}))$ is a union of pseudo-components. By Theorem \[combi\], any two pseudo-components intersect. Since they are all connected by Proposition \[Hrhofreegroup\], the union itself is connected.
Proposition \[Hrhofreegroup\] also implies that the dimension of a pseudo-component is $(p-1)(l-1)$. Since the intersection between any two pseudo-components is a finite number of point and we have a finite number of these sets, the topological dimension of $\chi^i_{Sing}(\mathbb{F}_l,PSL(p,\mathbb{C}))$ is also $(p-1)(l-1)$.
\[irredcompfree\]
Since $\left(\overline{D}^{l-1}-\{(\overline{I_p},\dots,\overline{I_p})\}\right)/ \langle \overline{M_c}\rangle$ is irreducible for the Zariski topology, the correspondence given in Proposition \[Hrhofreegroup\] implies that the pseudo-components are the irreducible components of $\chi^i_{Sing}(\mathbb{F}_l,PSL(p,\mathbb{C}))$.
The closed surface group case {#surfgrpcase}
=============================
Let $\Sigma_g$ be a closed surface of genus $g\geq 2$. We are studying the singular locus of the character variety of the fundamental group of $\pi_1(\Sigma_g)$ in $PSL(p,\mathbb{C})$. Basically, our study will follow the lines of the free group case. From the Dehn-Nielsen-Baer theorem (see [@F-M] theorem 8.1), we know that the orientation preserving mapping class group of a surface is a subgroup of index $2$ in $\operatorname{Out}(\pi_1(\Sigma_g))$. We denote $\operatorname{Out}^+(\pi_1(\Sigma_g))$ this subgroup of index $2$ and $\operatorname{Aut}^+(\pi_1(\Sigma_g))$ the corresponding subgroup of automorphisms.
The set of normal subgroups $K$ of index $p$ in $\pi_1(\Sigma_g)$ is in bijective correspondence with the set of subgroups of index $p$ in $\pi_1(\Sigma_g)^{Ab}=\mathbb{Z}^{2g}$. Furthermore, it is known that $Sp(2g,\mathbb{Z})$ acts transitively on these subgroups. Since the canonical map from $\operatorname{Out}^+(\pi_1(\Sigma_g))$ on $Sp(2g,\mathbb{Z})$ is known to be surjective (c.f. [@F-M], theorem 6.4), it follows that $\operatorname{Aut}^+(\pi_1(\Sigma_g))$ acts transitively on the set of normal subgroups of index $p$ in $\pi_1(\Sigma_g)$. Using this, we have :
\[onesinglphiKsurfgrp\]
Let $g\geq 2$ and $K,K'$ be two normal subgroups of index $p$ in $\pi_1(\Sigma_g)$. Then, there exists an automorphism $\phi$ in $\operatorname{Aut}^+(\pi_1(\Sigma_g))$ such that the precomposition by $\phi$ induces an homeomorphism between $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ and $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$.
The same proof as for Proposition \[onesinglphiKfree\].
\[surfgrponeK\] This proposition implies that we only need to focus on one single pseudo-component. Let $a_1,b_1$,…,$a_g,b_g$ be the standard generators of $\pi_1(\Sigma_g)$ : $$\pi_1(\Sigma_g)=\langle a_1,b_1,\dots,a_g,b_g\mid \prod_{i=1}^g[a_i,b_i]=1\rangle\text{.}$$ Then define $K$ to be the group generated $a_1^p$, $b_1$, $a_1^{i}a_ja_1^{-i}$, $a_1^{i}b_ja_1^{-i}$ for $0\leq i\leq p-1$ and $2\leq j\leq g$. Replacing the disks in Figure \[pict1\] by tori, one may identify $K$ to the fundamental group of a $p$-fold Galois cover of $\Sigma_g$. As a result, $K$ is a normal subgroup of index $p$ in $\pi_1(\Sigma_g)$. Furthermore $K$ is the fundamental group of a closed topological surface of genus $1+(g-1)p$ and $\pi_1(K)^{Ab}$ is isomorphic to $\mathbb{Z}^{2+2(g-1)p}$ freely generated by the images of $a_1^p$, $b_1$, $a_1^{i}a_ja_1^{-i}$, $a_1^{i}b_ja_1^{-i}$ for $0\leq i\leq p-1$ and $2\leq j\leq g$.
\[Hrhosurfgroup\]
Let $g\geq 2$, $\Sigma_g$ be a closed surface of genus $g$ and $p$ be a prime number. Then any pseudo-component in $\chi^i_{Sing}(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$ is homeomorphic to $\left(\langle \overline{D(\xi)}\rangle\times \overline{D}^{2(g-1)}-\{(\overline{I_p},\dots,\overline{I_p})\}\right)/ \langle \overline{M_c}\rangle$.
Using Proposition \[onesinglphiKsurfgrp\], we only construct the homeomorphism for a unique pseudo-component. Let $a_1,b_1$,…,$a_g,b_g$ be the standard generators of $\pi_1(\Sigma_g)$. Then, Remark \[surfgrponeK\] justifies that the subgroup $K_0$ generated by $A:=a_1^p$, $B:=b_1$, $A_{i,j}:=a_1^{i}a_ja_1^{-i}$ and $B_{i,j}:=a_1^{i}b_ja_1^{-i}$ with $0\leq i\leq p-1$ and $2\leq j\leq g$ is a normal subgroup of index $p$ in $\pi_1(\Sigma_g)$. Furthermore, its abelianization is freely generated by the images of $A$, $B$, $A_{i,j}$ and $B_{i,j}$.
According to Theorem \[homeobarHrho\], $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i)$ is homeomorphic to $\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i$. By Proposition \[barHrho\], we need to compute $H^1(\pi_1(\Sigma_g),\overline{D}_{\overline{\rho}_{K_0}})$. Following the notation of Proposition \[Transgr\], we choose $\gamma_0:=a_1$. This proposition states that there is an homeomorphism between $H^1(\pi_1(\Sigma_g),\overline{D}_{\overline{\rho}_{K_0}})\text{ and }$ $$\left\lbrace f\in \operatorname{Hom}(K_0,\overline{D})\left|\begin{array}{l}f(a_1^p)=\overline{I_p}\\f(a_1\gamma a_1^{-1})=\overline{M_c}\cdot f(\gamma)\text{, } \forall \gamma\in K_0\end{array}\right. \right\rbrace\text{.}$$
Identifying $f$ to the tuples of images $(f(X),f(Y),(f(X_{i,j})),f(Y_{i,j}))$ of the free generators of $K_0^{Ab}$, $\operatorname{Hom}(K_0,\overline{D})$ is equal to $\overline{D}^{1+p(g-1)}$. $f(a_1^p)=\overline{I_p}$ means that $f(A)=\overline{I_p}$. Remark that $$a_1^{-1}Ba_1=a_1^{-1}b_1a_1=b_1(a_1b_1)^{-1}[b_1,a_1](a_1b_1)=b_1(a_1b_1)^{-1}\prod_{j=2}^g[a_i,b_i]a_1b_1\text{.}$$ Therefore $f(a_1Ba_1^{-1})=f(B)$. Whence $f(a_1\gamma a_1^{-1})=\overline{M_c}\cdot f(\gamma)$, for all $\gamma\in K_0$ is equivalent to $f(B)=f(a_1Ba_1^{-1})=\overline{M_c}\cdot f(B)$, $f(A_{i+1,j})=\overline{M_c}\cdot f(A_{i,j})$ and $f(B_{i+1,j})=\overline{M_c}\cdot f(B_{i,j})$ for $j=2,\dots,l$ and $i=0,\dots,p-2$.
Thus, the map sending $f$ to $(f(B),f(A_{0,2}),f(B_{0,2}),\dots,f(A_{0,g}),f(B_{0,g}))$ defines a natural identification between the space above and $\langle \overline{D(\xi)}\rangle\times\overline{D}^{2(g-1)}$. As a result, there is an homeomorphism between $H^1(\Gamma,\overline{D}_{\overline{\rho}_{K_0}})$ and $\langle \overline{D(\xi)}\rangle\times\overline{D}^{2(g-1)}$.
The end of the proof is similar to the free group case (Proposition \[Hrhofreegroup\]).
Unlike the free group case, neither the pseudo-components nor the singular locus are connected. To have a more refined statement, we introduce an invariant. Let $g\geq 2$ and $\Sigma_g$ be a closed surface of genus $g$ whose fundamental group is generated by $a_1,b_1,\dots,a_g,b_g$ verifying one single relation $[a_1,b_1]\cdots[a_g,b_g]=1$.
If $\rho$ is a representation from $\pi_1(\Sigma_g)$ to $PSL(p,\mathbb{C})$, we arbitrarily choose for each $\gamma\in \pi_1(\Sigma_g)$, $\hat{\rho}(\gamma)\in SL(p,\mathbb{C})$ such that $\overline{\hat{\rho}(\gamma)}=\rho(\gamma)$. Since $\rho$ is a representation in $PSL(p,\mathbb{C})$ we have that : $$e(\rho):=\prod_{i=1}^g[\hat{\rho}(a_i),\hat{\rho}(b_i)]\in Z(SL(p,\mathbb{C}))\text{.}$$ It is a straightforward verification that $e(\rho)$ only depends on $\rho$ (and not on the chosen lifts). It is called the *Euler invariant*. It is invariant by conjugation. We shall see, in the next lemma, that it is also invariant by the action of $\operatorname{Aut}^+(\pi_1(\Sigma_g))$.
\[mcginv\]
Let $g\geq 2$, $\Sigma_g$ be a closed surface of genus $g\geq 2$, $\phi\in \operatorname{Aut}^+(\pi_1(\Sigma_g))$ and $\rho\in \operatorname{Hom}(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$. Then $e(\rho\circ \phi)=e(\rho)$.
Let $\phi$ be in $\operatorname{Aut}^+(\pi_1(\Sigma_g))$ and $\rho\in \operatorname{Hom}(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$. Consider the central exact sequence defining $PSL(p,\mathbb{C})$ :
$$\xymatrix{1\ar[r]&Z(SL(p,\mathbb{C}))\ar[r]&SL(p,\mathbb{C})\ar[r]&PSL(p,\mathbb{C})\ar[r]&1}$$
Let $[z]\in H^2(PSL(p,\mathbb{C}),Z(SL(p,\mathbb{C})))$ representing this exact sequence.
Then, $[\rho^*z]$ in $H^2(\pi_1(\Sigma_g),Z(SL(p,\mathbb{C})))$ is invariant by $Aut^+(\pi_1(\Sigma_g))$ since it leaves invariant the volume form generating $H^2(\Gamma,\mathbb{Z})$. Using Poincaré duality, $H^2(\pi_1(\Sigma_g),Z(SL(p,\mathbb{C})))$ and $H^0(\pi_1(\Sigma_g),Z(SL(p,\mathbb{C}))^*)=Z(SL(p,\mathbb{C}))^*$ are in natural dual pairing. It gives an isomorphism between $H^2(\pi_1(\Sigma_g),Z(SL(p,\mathbb{C})))$ and $Z(SL(p,\mathbb{C}))$. One can check that through this isomorphism, $[\rho^*z]$ will be sent to $e(\rho)$. Whence the result.
It leads to the following topological result.
\[topsurfgrpHrhoK\]
Let $g\geq 2$, $\Sigma_g$ be a closed surface of genus $g$, $p$ be a prime number and $K$ be a normal subgroup of index $p$ in $\pi_1(\Sigma_g)$. Then $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K}}}^i)$ has exactly $p$ connected components which are the fibers over the Euler invariant. Furthermore
- $e^{-1}(I_p)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K}}}^i)$ is homeomorphic to $(\overline{D}^{2(g-1)}-\{(1,\dots, 1)\})/\langle \overline{M_c}\rangle$.
- for $1\leq k\leq p-1$, $e^{-1}(\xi^kI_p)\cap \varphi(\overline{\mathcal{H}_{\overline{\rho}_{K}}}^i)$ is homeomorphic to $\overline{D}^{2(g-1)}/\langle \overline{M_c}\rangle$.
Proposition \[onesinglphiKsurfgrp\] which states that $\operatorname{Aut}^+(\pi_1(\Sigma_g))$ acts transitively on components and Lemma \[mcginv\] which states that the Euler invariant is invariant by action of $\operatorname{Aut}^+(\pi_1(\Sigma_g))$ imply that we only need to do it for one normal subgroup of index $p$ in $\pi_1(\Sigma_g)$. Define $a_1,b_1,\dots, a_g,b_g$ to be its standard generators and, like before define $K_0$ to be the subgroup generated by $a_1^p,b_1$, $a_1^{i}a_ja_1^{-i}$, $a_1^{i}b_ja_1^{-i}$ for $1\leq i\leq p$ and $2\leq j\leq g$.
Define a group morphism $\rho_k:\pi_1(\Sigma_g)\rightarrow \overline{D}\rtimes\langle \overline{M_c}\rangle$ by $\rho_k(a_1):=\overline{M_c} $, $\rho_k(b_1):=\overline{D(\xi)}^k$ and $\rho_k(a_i)=\rho_k(b_i)= \overline{D(\xi)}$ for $2\leq i\leq g$. Since $\rho_k(a_1)$ and $\rho_k(a_2)$ generate an irreducible subgroup of $PSL(p,\mathbb{C})$, $\rho_k$ is irreducible. Finally, $\operatorname{Ker}(q\circ\rho_k)$ contains $K_0$ and is therefore equal to it because they have the same index in $\pi_1(\Sigma_g)$. Therefore, $[\rho_k]$ belongs to $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i)$.
Taking the following lifts $\hat{\rho_k}(a_1):=M_c$, $\hat{\rho_k}(b_1):=D(\xi)^k$, $\hat{\rho_k}(a_i):=D(\xi)$ and $\hat{\rho_k}(b_i):=D(\xi)$ for $i\geq 2$, the Euler invariant of $\rho_k$ is $e(\rho_k)=\xi^{-k}I_p$. As a result, the map $e:\varphi(\overline{\mathcal{H}_{\overline{\rho}}}^i)\rightarrow Z(SL(p,\mathbb{C}))$ is onto. Whence $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i)$ has at least $p$ connected components (the $p$ fibers over $e$). Proposition \[Hrhosurfgroup\] implies that it has exactly $p$ connected components. Therefore, the $p$ connected components of $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i)$ are its fibers over $e$.
Remark that, through the homeomorphism of Proposition \[Hrhosurfgroup\], which sends $\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K_0}}}^i)$ to $\left(\langle \overline{D(\xi)}\rangle\times \overline{D}^{2(g-1)}-\{(\overline{I_p},\dots,\overline{I_p})\}\right)/ \langle \overline{M_c}\rangle$, the conjugacy class of $\rho_k$ is sent to $(\overline{D(\xi)}^k,\overline{D(\xi)},\dots,\overline{D(\xi)})$.
As a result, the connected component $e^{-1}(I_p)$ of the conjugacy class of $\rho_0$ is homeomorphic to $\left( \overline{D}^{2(g-1)}-\{(\overline{I_p},\dots,\overline{I_p})\}\right)/ \langle \overline{M_c}\rangle$. For $1\leq k\leq p-1$, the connected component $e^{-1}(\xi^{-k}I_p)$ of the conjugacy class of $\rho_k$ is homeomorphic to $\overline{D}^{2(g-1)}/ \langle \overline{M_c}\rangle$.
\[numbconcomp\] Applying this corollary, $\chi_{Sing}^i(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$ has at least $p$ connected components given by their Euler invariants. Our next order of business is to show that each fiber above the Euler invariant is connected.
Basically, the idea would be to apply Theorem \[combi\] (like in the free group case). However, it might happen that $\varphi(\overline{H_{\overline{\rho}_K}}^i)\cap\varphi(\overline{H_{\overline{\rho}_{K'}}}^i)\cap e^{-1}(\xi^k I_p)$ is empty. The necessary and sufficient conditions for this set to be not empty are given in Proposition \[interempty\]. To do so, it is necessary to consider $\pi_1(\Sigma_g)^{Ab}/p\pi_1(\Sigma_g)^{Ab}$ as a symplectic space.
By *symplectic (vector) space*, we mean a vector space endowed with a symplectic bilinear form. Let $(E_0,\omega)$ be a symplectic space with a linear basis $(x_1,y_1,\dots,x_g,y_g)$. It is a symplectic basis if $\omega(x_i,x_j)=0=\omega(y_i,y_j)$ for all $i,j$ and $\omega(x_i,y_j)=1$ if $i=j$ and $0$ else. We will use the fact that any vector space endowed with a symplectic bilinear form admits a symplectic basis and any linear endomorphism sending a symplectic basis to a symplectic basis is a symplectic transformation. We denote $Sp(E_0,\omega)$ the group of symplectic transformations for $(E_0,\omega)$.
Furthermore, if $E$ is a subspace of codimension $2$ in $E_0$ then either $\omega_{|E}$ is non-degenerate or it has a kernel of dimension $2$ (because $\omega$ is non-degenerate). In the first case, we say that $E$ is *non-degenerate*, in the second case we say that $E$ is *degenerate*.
\[twoorbits\]
Let $g\geq 1$ and $(E_0,\omega)$ be a symplectic $K$-linear space of dimension $2g$. Then, there are, two orbits in $\left\lbrace (E,E')\left|\begin{array}{c}E,E'\text{ are hyperplanes in $E_0$}\\ E\neq E'\end{array}\right.\right\rbrace$ for the action of $Sp(E_0,\omega)$ depending whether $E\cap E'$ is degenerate or non-degenerate.
Let $(x_1,y_1,\dots,x_g,y_g)$ be a symplectic basis for $E_0$. Let $E,E'$ be two hyperplanes in $E_0$ verifying $E\neq E'$.
If $E\cap E'$ is non-degenerate we shall find a symplectic transformation sending $E$ to $\operatorname{Span}(x_1,x_2,y_2,\dots,x_g,y_g)$ and $E'$ to $\operatorname{Span}(y_1,x_2,y_2,\dots,x_g,y_g)$.
If $E\cap E'$ is degenerate we shall find a symplectic transformation sending $E$ to $\operatorname{Span}(y_1,x_2,y_2,x_3,y_3\dots,x_g,y_g)$ and $E'$ to $\operatorname{Span}(x_1,y_1,y_2,x_3,y_3\dots,x_g,y_g)$.
Assume $F:=E\cap E'$ is non-degenerate. Let $(x_2',y_2',\dots, x_g',y_g')$ be a symplectic basis of $F$. Let $u$ be in $E$ such that $E=\operatorname{Span}(u)\oplus F$. Since $F$ is non-degenerate, there exists $f_0\in F$ such that $\omega(u,\cdot)_{|F}=\omega(f_0,\cdot)_{|F}$. Therefore $v:=u-f_0$ is orthogonal to $F$ and $E=\operatorname{Span}(v)\oplus^{\perp} F$. Likewise, there is $v'\in E'$ verifying $E'=\operatorname{Span}(v')\oplus^{\perp} F$. Since $E_0=\operatorname{Span}(v,v')\oplus^{\perp} F$ and $E_0$ and $F$ are both non-degenerate, $\operatorname{Span}(v,v')$ is non degenerate, whence $\omega(v,v')\neq 0$.
Let $v'':=\omega(v,v')^{-1} v'$ then $(v,v'',x_2',y_2',\dots,x_g',y_g')$ is a symplectic basis of $E_0$. Therefore, the linear map $\psi$ sending this symplectic basis to the initial one $(x_1,y_1,\dots,x_g,y_g)$ is the wanted symplectic transformation.
Assume $F:=E\cap E'$ is degenerate. Let $K:=F^{\perp}$. Then $F=K\oplus^{\perp} F_0$ where $F_0$ is non-degenerate. Let $(x_3',y_3',\dots, x_g',y_g')$ be a symplectic basis of $F_0$.
Let $u$ be in $E$ such that $E=\operatorname{Span}(u)\oplus F$. Since $F_0$ is non-degenerate, there exists $f_0\in F_0$ such that $\omega(u,\cdot)_{|F_0}=\omega(f_0,\cdot)_{|F_0}$. Therefore $v:=u-f_0$ is orthogonal to $F_0$ and $E=(\operatorname{Span}(v)\oplus K)\oplus^{\perp} F_0$. Likewise, there is $v'\in E'$ verifying $E'=(\operatorname{Span}(v')\oplus K)\oplus^{\perp} F_0$. We remark that $E_0=(\operatorname{Span}(v,v')\oplus K)\oplus^{\perp} F_0$ and therefore $\operatorname{Span}(v,v')\oplus K$ is non-degenerate.
Since $\operatorname{Span}(v,v')\cap F$ is trivial and $F=K^{\perp}$, there is $y_1'\in K$ verifying $\omega(v,y_1')=1$ and $\omega(v',y_1')=0$. If $v'':=v'-\omega(v,v')y_1'$, then $v''$ is both orthogonal to $v'$ and $y_1'$. Furthermore $E''=(\operatorname{Span}(v'')\oplus K)\oplus^{\perp} F_0$. Finally, there is $y_2'\in K$ verifying $\omega(v'',y_2')=1$ and $\omega(v,y_2')=0$. Therefore $(v,y_1',v'',y_2',x_3',y_3',\dots, x_g',y_g')$ is a symplectic basis and the linear map sending this symplectic basis to $(x_1,y_1,\dots,x_g,y_g)$ is the wanted symplectic transformation.
\[irredcompsurf\] Let $g\geq 2$. Then $\pi_1(\Sigma_g)^{Ab}/p\pi_1(\Sigma_g)^{Ab}$ is isomorphic to $(\mathbb{Z}/p)^{2g}$. We denote $\Phi$ the natural projection of $\pi_1(\Sigma_g)$ onto $E_{p,g}:=(\mathbb{Z}/p)^{2g}$. Remark that there is a natural symplectic form $\omega$ on $E_{p,g}$ which is invariant by the action of $\operatorname{Aut}^+(\pi_1(\Sigma_g))$. If $(a_1,b_1,\dots,a_g,b_g)$ is a standard system of generator for $\pi_1(\Sigma_g)$ then $(\Phi(a_1),\Phi(b_1),\dots,\Phi(a_g),\Phi(b_g))$ is naturally a symplectic basis of $E_{p,g}$ for $\omega$.
\[interempty\]
Let $g\geq 2$, $p$ be a prime number, $\Sigma_g$ be a closed surface of genus $g$, $K,K'$ be two different normal subgroups of index $p$ in $\pi_1(\Sigma_g)$ and $k$ be an integer between $0$ and $p-1$. Then, the cardinal of $e^{-1}(\xi^kI_p)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ is :
$$\left\lbrace\begin{array}{cl}1&\text{ if $k\neq 0$ and $\Phi(K\cap K')$\text{ is non-degenerate}}\\ 0&\text{ if $k=0$ and $\Phi(K\cap K')$\text{ is non-degenerate}}\\ 0&\text{ if $k\neq 0$ and $\Phi(K\cap K')$\text{ is degenerate}}\\ p-1&\text{ if $k=0$ and $\Phi(K\cap K')$\text{ is degenerate}}\end{array}\right.$$
Lemma \[twoorbits\] justifies that we only need to show this property for one example when $\Phi(K\cap K')$ is non-degenerate and one example when $\Phi(K\cap K')$ is degenerate. We denote $a_1,b_1,\dots,a_g,b_g$ a system of standard generators for $\pi_1(\Sigma_g)$.
The non-degenerate case. We define $K:=\Phi^{-1}(E)$ and $K':=\Phi^{-1}(E')$ where
$$\begin{array}{l}E:=\operatorname{Span}(\Phi(b_1),\Phi(a_2),\Phi(b_2),\dots,\Phi(a_g),\Phi(b_g))\\ E':=\operatorname{Span}(\Phi(a_1),\Phi(a_2),\Phi(b_2),\dots,\Phi(a_g),\Phi(b_g))\end{array}$$
Then $K$ and $K'$ are both normal subgroup of index $p$ in $\pi_1(\Sigma_g)$. Furthermore $\Phi(K\cap K')$ is the subspace of $E_{p,g}$ generated by $(\Phi(a_2),\Phi(b_2),\dots,\Phi(a_g),\Phi(b_g))$. Since this is a symplectic family, it follows that $\Phi(K\cap K')$ is non-degenerate.
For $1\leq k\leq p-1$, define a group morphism $\rho_k$ from $\pi_1(\Sigma_g)$ to $PSL(p,\mathbb{C})$ with $\rho_k(a_1):=\overline{M_c}$, $\rho_k(b_1):=\overline{D(\xi)}^k$ and $\rho(K\cap K')$ is trivial. The image of $\rho_k$ is irreducible and its conjugacy class belongs to $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$.
Finally, $e(\rho_k)=[M_c,D(\xi)^k]=\xi^{-k}I_p$. As a result, $\rho_1$,…, $\rho_{p-1}$ leads to $p-1$ different points in $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$. Theorem \[combi\] implies that $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ contains $p-1$ elements. Whence the result.
The degenerate case. We define $K:=\Phi^{-1}(E)$ and $K':=\Phi^{-1}(E')$ where :
$$\begin{array}{l}E:=\operatorname{Span}(\Phi(b_1),\Phi(a_2),\Phi(b_2),\dots,\Phi(a_g),\Phi(b_g))\\ E':=\operatorname{Span}(\Phi(a_1),\Phi(b_1),\Phi(b_2),\Phi(a_3),\Phi(b_3),\dots,\Phi(a_g),\Phi(b_g))\text{.}\end{array}$$
Then $K$ and $K'$ are both normal subgroup of index $p$ in $\pi_1(\Sigma_g)$. Furthermore $\Phi(K\cap K')$ is the subspace of $E_{p,g}$ generated by $\Phi(b_1)$, $\Phi(b_2)$, $\Phi(a_3)$, $\Phi(b_3)$,…, $\Phi(a_g)$, $\Phi(b_g)$. Since $\Phi(b_1)$ is orthogonal to $\Phi(K\cap K')$, $\Phi(K\cap K')$ is degenerate.
For $1\leq k\leq p-1$, define a group morphism $\rho_k$ from $\pi_1(\Sigma_g)$ to $PSL(p,\mathbb{C})$ with $\rho_k(a_1):=\overline{M_c}$, $\rho_k(a_2):=\overline{D(\xi)}^k$ and $\rho(K\cap K')$ is trivial. The image of $\rho_k$ is irreducible and its conjugacy class belongs to $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$.
Finally, $e(\rho_k)=I_p$. Remark that if $\rho_k$ is conjugate to $\rho_{k'}$ then the bases of $\langle\overline{D(\xi)}\rangle \times \langle\overline{M_c}\rangle$ : $(\overline{M_c},\overline{D(\xi)}^k)$ and $(\overline{M_c},\overline{D(\xi)}^{k'})$ are obtained by the action of the normalizer of $\langle\overline{D(\xi)}\rangle \times \langle\overline{M_c}\rangle$. Lemma \[countZp2\] implies $k=k'$. Therefore $\rho_1$,…, $\rho_{p-1}$ leads to $p-1$ different points in $\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_{K'}}}^i)$ which all have a trivial Euler invariant. Theorem \[combi\] implies that their conjugacy classes are the only points in the intersection, whence the result.
The final result relies on the following geometric lemma.
\[sympgeom\]
Let $E$ and $E'$ be two different hyperplanes of $V:=(\mathbb{Z}/p)^{2g}$ with $g\geq 2$. Let $\omega$ be a symplectic form on $V$. We have two cases :
- If $E\cap E'$ is non-degenerate, there is an hyperplane $E_0$ such that $E\cap E_0$ and $E'\cap E_0$ are both degenerate.
- If $E\cap E'$ is degenerate, there is an hyperplane $E_0$ such that $E\cap E_0$ and $E'\cap E_0$ are both non-degenerate.
Let $(x_1,y_1,\dots,x_g,y_g)$ be a symplectic basis of $V$. Lemma \[twoorbits\] justifies that it suffices to prove the lemma for one example in each case.
Let $F$ be the subspace generated by $x_2,y_2$,…,$x_g,y_g$, $E$ be $\operatorname{Span}(x_1)\oplus F$ and $E'$ be $\operatorname{Span}(y_1)\oplus F$. Then $E\cap E'=F$ is non-degenerate. Let $E_0$ be the subspace generated by $x_1,y_1$, $x_2$, $x_3,y_3$,…,$x_g,y_g$. Then $x_2$ is a degenerate vector of $E_0\cap E$ and $E_0\cap E'$. Whence the result in this case.
Let $F_0$ be the subspace generated by $y_1,y_2,x_3,y_3$,…,$x_g,y_g$, $E$ be $\operatorname{Span}(x_1)\oplus F$ and $E'$ be $\operatorname{Span}(x_2)\oplus F$. Then $E\cap E'=F$ is degenerate. Let $E_0$ be the subspace generated by $x_1,x_2,y_1+y_2,x_3,y_3,\dots,x_g,y_g$. Then $E_0\cap E$ and $E_0 \cap E'$ are both non-degenerate. Whence the result in this case.
Applying Corollary \[topsurfgrpHrhoK\], $e^{-1}(\xi^kI_p)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ is connected of dimension $2(g-1)(p-1)$. Using Proposition \[interempty\] and Lemma \[sympgeom\], this leads to :
\[conncomporblocsurfgrp\]
Let $g\geq 2$ and $\Sigma_g$ be a closed Riemann surface of genus $g$. The orbifold locus of $\chi^i(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$ has $p$ connected components given by the fibers for the Euler invariant and its dimension is $2(g-1)(p-1)$.
\[surfgrpirredcomp\]
Considering the Zariski topology, we see that for each $0\leq k\leq p-1$ and each normal subgroup $K$ of index $p$ in $\pi_1(\Sigma_g)$, $e^{-1}(\xi^kI_p)\cap\varphi(\overline{\mathcal{H}_{\overline{\rho}_K}}^i)$ is an irreducible component of $\chi_{Sing}^i(\pi_1(\Sigma_g),PSL(p,\mathbb{C}))$.
Orbifold singularities and algebraic singularities {#orbalgsing}
==================================================
Like we said in the introduction, the singular locus $\chi^i_{Sing}(\Gamma,G)$ should be understood as orbifold singularities since, when $\Gamma$ is a free group or a surface group, it is the set of orbifold singularities of a well-defined orbifold. A priori, it does not necessarily coincide with the locus of algebraic singularities of $\chi^i(\Gamma,G)$. In Proposition \[singvarcar\], we prove that these two notions coincide when $\Gamma$ is a free group of rank $\geq 2$ or a closed surface group of genus $g\geq 2$ and $G=PSL(p,\mathbb{C})$ with $p$ prime.
To do this, we recall some facts about tangent spaces to representation varieties and character varieties. If $\Gamma$ is a finitely generated group , $G$ is a complex algebraic group, $\mathfrak{g}$ is its Lie algebra and $\rho:\Gamma\to G$ is a representation, then $\mathfrak{g}$ is a $\Gamma$-module with the action given by $Ad\circ\rho$. This $\Gamma$-module will be denoted $\mathfrak{g}_{Ad\circ\rho}$.
In [@Sik], Proposition 34 (see also [@L-M], Chapter 2) Sikora proves that the Zariski tangent space to $\operatorname{\bf{Hom}}(\Gamma,G)$ at $\rho$ (denoted $T_{\rho}\operatorname{\bf{Hom}}(\Gamma,G)$) can be identified to this space of $1$-cocycles $Z^1(\Gamma,\mathfrak{g}_{Ad\circ\rho})$.
\[similar\] The proof of Proposition \[Hrho\] is similar to Sikora’s proof. The reason is that the Zariski tangent space $T_{\rho}\operatorname{\bf{Hom}}(\Gamma,G)$ can be identified to the fiber of $\operatorname{Hom}(\Gamma,G(\mathbb{C}[\epsilon]))$ above $\rho$ where $G(\mathbb{C}[\epsilon])$ is naturally isomorphic to $\mathfrak{g}\rtimes_{Ad}G$.
One can show that the Zariski tangent space to the orbit $[\rho]$ at $\rho$ is exactly $B^1(\Gamma,\mathfrak{g}_{Ad\circ\rho})$. For instance, Weil (in [@Wei]) uses this to prove that $H^1(\Gamma,\mathfrak{g}_{Ad\circ\rho})=0$ implies the local rigidity of $\rho$ (i.e. a neighborhood of $\rho$ in the representation variety is contained in the conjugacy class of $\rho$). In particular, any finite group representation is locally rigid.
However, in general, the Zariski tangent space at $[\rho]$ to the schematic character variety $\mathfrak{X}(\Gamma,G)$ is different from $H^1(\Gamma,\mathfrak{g}_{Ad\circ\rho})$ (e.g. in [@Ben]). When $\rho$ is scheme smooth, the link between this tangent space and $H^1(\Gamma,\mathfrak{g}_{Ad\circ\rho})$ has been studied in [@Sik], Paragraph 13 by Sikora (see also Proposition 5.2 in [@H-P]) using Luna’s Étale Slice Theorem. Theorem 53 in [@Sik] states that if $\rho$ is a scheme smooth completely reducible representation then :
$$\dim\left(T_0 \left(H^1(\Gamma,\mathfrak{g}_{Ad\circ\rho})//Z_G(\rho)\right)\right)=\dim\left( T_{[\rho]}X(\Gamma,G)\right)=\dim\left( T_{[\rho]}\mathfrak{X}(\Gamma,G)\right)
\label{dimesptgt}$$
To prove Proposition \[singvarcar\], we need two lemmas :
\[dimirred\]
Let $\Gamma$ be either a free group or a closed surface group and $G$ be a complex reductive algebraic group. Then, for any irreducible representation $\rho$ from $\Gamma$ to $G$, $\dim(\chi^i(\Gamma,G))=\dim(H^1(\Gamma,\mathfrak{g}_{Ad\circ\rho}))$.
If $\Gamma$ is a free group with $l\geq 2$ generators then $Z^1(\Gamma,\mathfrak{g}_{Ad\circ\rho}))=\mathfrak{g}^l$ and $B^1(\Gamma,\mathfrak{g}_{Ad\circ\rho}))=\mathfrak{g}/\mathfrak{g}^{Ad\circ\rho(\Gamma)}$. Since $\rho(\Gamma)$ is irreducible $\mathfrak{g}^{Ad\circ\rho(\Gamma)}=\operatorname{Lie}(Z(G))$. $$\dim(H^1(\Gamma,\mathfrak{g}_{Ad\circ\rho}))=\dim(\mathfrak{g}^l)-\dim(\mathfrak{g})+\dim(Z(G))=(l-1)\dim(G)+\dim(Z(G))$$ $$\dim(\chi^i(\Gamma,G))=\dim(\operatorname{Hom}(\Gamma,G))-\dim(G/Z(G))=(l-1)\dim(G)+\dim(Z(G))\text{.}$$
Whence the result in the free group case. For the surface group case, it is also classical but more difficult, see [@Gol].
The next lemma is well-known but its proof is written for the convenience of the reader. Our interest in cyclic quotients (vector spaces quotiented by a cyclic group action) comes from Theorem \[classifcentr\] which implies that centralizers of non-abelian bad representations are cyclic.
\[cyclicaction\] Let $G$ be a cyclic group of prime order $p$ generated by $g$ and assume that we have a linear action of $G$ on $\mathbb{C}^N$ where $N\geq 0$. Then $0$ mod $G$ is an algebraic singularity of the cyclic quotient $\mathbb{C}^N//G$ if and only if $\operatorname{codim}(\operatorname{Fix}(g))>1$.
Up to some change of basis, there are integers $0\leq a_1\leq \cdots\leq a_N\leq p-1$ such that the action of $G$ on $\mathbb{C}^N$ is given by $g\cdot (v_1,\dots,v_N)=(\xi^{a_1}v_1,\dots,\xi^{a_N}v_N)$ where $\xi$ is a primitive $p$-th root of the unity. To study $\mathbb{C}^N//G$, we only need to compute its coordinate ring $B$. Let $A:=\mathbb{C}[x_1,\dots,x_N]$ be the coordinate ring of $\mathbb{C}^N$. Then $B$ is the subring of $A$ of invariants by the action of $G$ : $B=A^G$.
If $\operatorname{codim}(\operatorname{Fix}(g))=0$, then $A=B$ and $\mathbb{C}^N//G$ is everywhere smooth. If $\operatorname{codim}(\operatorname{Fix}(g))=1$, then $B=\mathbb{C}[x_1,\dots,x_{N-1},x_N^{p}]$. In this case $B$ is isomorphic to $A$ and, again, $\mathbb{C}^N//G$ is everywhere smooth.
If $\operatorname{codim}(\operatorname{Fix}(g))>1$, then let $k<N-1$ be such that $a_k=0$ and $a_{k+1}>0$. Define $\mathfrak{m}_0:=(x_1,\dots,x_N)$ in $A$ and $\mathfrak{n}_0:=B\cap \mathfrak{m}_0$ the maximal ideal in $B$ associated to $0$ mod $G$. Then $\mathfrak{n}_0$ contains $x_1,\dots,x_k,x_{k+1}^p,\dots,x_N^p$. Let $u$ (resp. $v$) be the inverse of $a_{N-1}$ (resp $a_N$) modulo $p$, $x_{N-1}^{u}x_N^{p-v}$ is invariant by $G$. Therefore it belongs to $\mathfrak{n}_0$ and we have $N+1$ elements in $\mathfrak{n}_0$ which are linearly independent in the Zariski tangent space $\mathfrak{n}_0/\mathfrak{n}_0^2$ to $\mathbb{C}^N//G$ at $0$ mod $G$. Whence $0$ mod $G$ is an algebraic singularity.
The proof of the following proposition is a generalization of the proof of Proposition 5.8 in [@H-P].
\[singvarcar\] Let $\Gamma$ be either a free group of rank $\geq 2$ or a surface group of genus $g\geq 2$, $p$ be a prime number and $\rho:\Gamma\to PSL(p,\mathbb{C})$ be an irreducible representation. Then $[\rho]$ is an algebraic singularity in the schematic character variety if and only if $\rho$ is bad.
Let $\rho$ be a good representation, since $\Gamma$ is a free group or a closed surface group, $\rho$ is scheme smooth. Corollary 50 in [@Sik] implies that $[\rho]$ is scheme smooth as well since $\rho$ has trivial centralizer.
Conversely, assume $\rho$ is a bad representation. Up to conjugation (Theorem \[classifcentr\]) we may assume that $\rho$ is a representation into $\overline{D}\rtimes\langle\overline{M_c}\rangle$. Let $K$ be the kernel of $q\circ \rho$. For the moment, we assume that $\rho$ is not abelian and therefore its centralizer is $\langle\overline{D(\xi)}\rangle$. In order to prove that $[\rho]$ is an algebraic singularity we prove that $\dim\left( T_{[\rho]}\mathfrak{X}(\Gamma,PSL_p)\right)> \dim\mathfrak{X}(\Gamma,PSL_p)=\dim\chi(\Gamma,PSL(p,\mathbb{C}))$.
We denote $\mathfrak{sl}_p(\mathbb{C})$ the Lie algebra of $PSL(p,\mathbb{C})$. Following Equation \[dimesptgt\], we need to compute $H^1(\Gamma,\mathfrak{sl}_p(\mathbb{C})_{Ad\circ\rho})$. For $0\leq i,j\leq p-1$, we denote $E_{i,j}$ the matrix with a $1$ at the $(i,j)$-th entry and $0$ everywhere else. Let $\mathfrak{d}_0$ be the Lie algebra of trace free diagonal matrices in $\mathfrak{sl}_p(\mathbb{C})$ and for $k=1,\dots,p-1$, define $$\mathfrak{d}_k:=\bigoplus_{i=0}^{p-1}\mathbb{C}E_{i,i+k}\text{.}$$ Since for $0\leq k\leq p-1$, $\mathfrak{d}_k$ is stable by the action of $\overline{D}$ and $\overline{M_c}$, the decomposition of $\mathfrak{sl}_p(\mathbb{C})=\mathfrak{d}_0\oplus\dots\oplus \mathfrak{d}_{p-1}$ as $\mathbb{C}$-vector space is also a decomposition as $\Gamma$-module. $$H^1(\Gamma,\mathfrak{sl}_p(\mathbb{C})_{Ad\circ\rho})= H^1(\Gamma,\mathfrak{d}_{0,Ad\circ\rho})\oplus \bigoplus_{k=1}^{p-1}H^1(\Gamma,\mathfrak{d}_{k,Ad\circ\rho})\text{.}
\label{decompH1}$$ For $1\leq k\leq p-1$, $0\leq i\leq p-1$, $Ad(\overline{D(\xi)})\cdot E_{i,i+k}=\xi^{k}E_{i,i+k}$. Therefore, the action of $\overline{D(\xi)}$ on $\mathfrak{d}_k$ is the multiplication by $\xi^k$ and so is the induced action of $\overline{D(\xi)}$ on $H^1(\Gamma,\mathfrak{d}_{k,Ad\circ\rho})$.
We recognize a cyclic quotient. Using Lemma \[cyclicaction\], we only need to prove that $\dim(H^1(\Gamma,\mathfrak{sl}_p(\mathbb{C})_{Ad\circ\rho}))-\dim (H^1(\Gamma,\mathfrak{d}_{0,Ad\circ\rho}))>1$ to end up with an algebraic singularity at the origin.
If $\Gamma=\mathbb{F}_r$, then $\dim \left(Z^1(\mathbb{F}_r,\mathfrak{d}_{k,Ad\circ\rho})\right)=r\dim(\mathfrak{d}_k)=rp$. Since $\rho$ is irreducible, $\mathfrak{sl}_p(\mathbb{C})^{\mathbb{F}_r}=\{0\}$. Therefore $\mathfrak{d}_k^{\mathbb{F}_r}=\{0\}$ and $\dim \left(B_1(\mathbb{F}_r,\mathfrak{d}_{k,Ad\circ\rho})\right)=p-0=p$. Therefore $\dim \left(H^1(\mathbb{F}_r,\mathfrak{d}_{k,Ad\circ\rho})\right)=(r-1)p\geq 2$.
At the end of Paragraph $6$ in [@Wei], Weil gives an explicit formula for the dimension of the first cohomology groups for Fuchsian groups without parabolic elements acting on $\mathbb{C}$-vector spaces. Applying this to the surface group case, $\dim\left(H^1(\pi_1(\Sigma_g),\mathfrak{d}_{k,Ad\circ\rho})\right)=(2g-2)p\geq 2$ for $1\leq k\leq p-1$.
In any case, Lemma \[cyclicaction\] implies that $$\dim \left(T_0 \left(H^1(\Gamma,\mathfrak{sl}_{p}(\mathbb{C})_{Ad\circ\rho})//\langle \overline{D(\xi)}\rangle\right)\right)> \dim \left(H^1(\Gamma,\mathfrak{sl}_p(\mathbb{C})_{Ad\circ\rho})\right)\text{.}$$ Using Lemma \[dimirred\] and Sikora’s expression for the dimension of the tangent space at $[\rho]$ (Equation \[dimesptgt\]) which applies because $\rho$ is scheme smooth, $[\rho]$ is an algebraic singularity of the schematic character variety.
In the free group or closed surface group case, any abelian irreducible representation is a limit of non-abelian bad representations. Since each of them is an algebraic singularity of the character variety and the set of algebraic singularities is closed, any conjugacy class of bad representations is an algebraic singularity.
\[reduced\]
Let $X$ be a (possibly non-reduced) scheme and $x$ be a closed point of $X$. A point $x$ is *reduced* if its local ring does not contain non-trivial nilpotent elements. A representation (resp. conjugacy class of representations) is scheme smooth if and only if it is both smooth and reduced. Using Corollary 55 in [@Sik], one sees that we can forget the “schematic” in the preceding proposition.
In general, there is no link between the set of algebraic singularities for the irreducible part of the character variety and the singular locus of the character variety. The next example is based on the idea of Sikora, c.f. [@Sik], Example 42.
\[algnotorb\] Let $\rho_1$ be the trivial representation of $\mathbb{Z}^2$ into $SL(2,\mathbb{C})$ and $\rho_2$ be the unique irreducible representation of the symmetric group $S_3$ into $SL(2,\mathbb{C})$. Let $\Gamma$ be the free product of $\mathbb{Z}^2$ and $S_3$ and $\rho:=\rho_1*\rho_2$. Then $\rho$ is an irreducible representation of $\Gamma$ into $SL(2,\mathbb{C})$.
We denote $\operatorname{Hom}(\Gamma,SL(2,\mathbb{C}))_0$ the set of representations in $\operatorname{Hom}(\Gamma,SL(2,\mathbb{C}))$ such that their restriction to $S_3$ is conjugate to $\rho_2$. Since $\rho_2$ is good and locally rigid, $\operatorname{Hom}(\mathbb{Z}^2,SL(2,\mathbb{C}))\times \{\rho_2\}$ is an étale slice for the $PSL(2,\mathbb{C})$-action by conjugation on $\operatorname{Hom}(\Gamma,SL(2,\mathbb{C}))_0$. Therefore, the tangent space to the character variety at $[\rho]$ is simply the tangent space at $\rho_1$ to $\operatorname{Hom}(\mathbb{Z}^2,SL(2,\mathbb{C}))$.
Example 42 in [@Sik] justifies that $\rho_1$ is an algebraic singularity of the representation variety $\operatorname{Hom}(\mathbb{Z}^2,SL(2,\mathbb{C}))$. Therefore, $[\rho]$ is an algebraic singularity of $\chi^i(\Gamma,SL(2,\mathbb{C}))$. However, $\rho$ is a good representation.
\[orbnotalg\] Let $\Gamma=\mathbb{Z}/3\times \mathbb{Z}/3$ and $G=PSL(3,\mathbb{C})$. Define $\rho$ to be an isomorphism between $\Gamma$ and the unique (up to conjugation) abelian irreducible group of $G$. Being locally rigid, $\rho$ is scheme smooth. Equation \[dimesptgt\] implies that $\dim\left( T_{[\rho]}\mathfrak{X}(\Gamma,G)\right)=0$ and therefore $[\rho]$ is necessarily scheme smooth as well. However, $\rho$ is an abelian irreducible representation and, in particular, is bad.
In the proof of Proposition \[singvarcar\], we used the hypothesis that $\Gamma$ is a free group or a closed surface group for two different things. The first was to insure that irreducible representations are scheme smooth (this is what fails in Example \[algnotorb\]). The second was to insure that cohomology groups of $\Gamma$ into $\mathfrak{d}_k$ are big enough to use Lemma \[cyclicaction\] (this is what trivially fails in Example \[orbnotalg\] because $\Gamma$ is finite). The modular group in $PSL(2,\mathbb{C})$ leads to a non-locally rigid example where the second condition fails, see Proposition \[PSL(2,Z)\].
\[PSL(2,Z)\]
$\chi^i(PSL(2,\mathbb{Z}),PSL(p,\mathbb{C}))$ is a manifold for any prime $p>3$.
The singular locus of $\chi^i(PSL(2,\mathbb{Z}),PSL(3,\mathbb{C}))$ is a singleton. Its unique point is not scheme smooth.
The singular locus of $\chi^i(PSL(2,\mathbb{Z}),PSL(2,\mathbb{C}))$ is a singleton. Its unique point is scheme smooth.
First, $PSL(2,\mathbb{Z})$ is isomorphic to $\mathbb{Z}/2*\mathbb{Z}/3=\langle a, b\mid a^2=1=b^3\rangle$. Therefore $$\operatorname{\bf{Hom}}(PSL(2,\mathbb{Z}),PSL_p)=\operatorname{\bf{Hom}}(\mathbb{Z}/2,PSL_p)\times \operatorname{\bf{Hom}}(\mathbb{Z}/3,PSL_p)\text{.}$$ Since $\operatorname{\bf{Hom}}(\mathbb{Z}/2,PSL_p)$ and $ \operatorname{\bf{Hom}}(\mathbb{Z}/3,PSL_p)$ are both smooth (because any closed point is locally rigid whence scheme smooth), $\operatorname{\bf{Hom}}(PSL(2,\mathbb{Z}),PSL_p)$ is also smooth and therefore the open subvariety $\operatorname{Hom}^i(PSL(2,\mathbb{Z}),PSL(p,\mathbb{C}))$ is also smooth. As a result, $\chi^i(PSL(2,\mathbb{Z}),PSL(p,\mathbb{C}))$ is an orbifold and conjugacy classes of good representations are necessarily scheme smooth.
When $p>3$, there is no normal subgroup of index $p$ in $PSL(2,\mathbb{Z})$. Equation \[decompbadlocus2\] directly implies that the singular locus is empty, whence the orbifold $\chi^i(PSL(2,\mathbb{Z}),PSL(p,\mathbb{C}))$ is actually a manifold.
$p=3$. There is only one normal subgroup of index $3$ in $PSL(2,\mathbb{Z})$ (it is normally generated by $a$). Any bad morphism from $PSL(2,\mathbb{Z})$ in $PSL(3,\mathbb{C})$ will then be conjugate to $\beta_3$ : $a\mapsto \overline{\begin{pmatrix}-1&&\\&-1&\\&&1\end{pmatrix}}$, $b\mapsto \overline{M_c}$ because $a$ needs to be of order $2$. By definition, there is a single point in $\chi^i_{Sing}(PSL(2,\mathbb{Z}),PSL(3,\mathbb{C}))$.
$p=2$. Likewise, there is only one conjugacy class of bad representations from $PSL(2,\mathbb{Z})$ in $PSL(2,\mathbb{C})$ which is given by $\beta_2$ : $a\mapsto \overline{M_c}$, $b\mapsto \overline{\begin{pmatrix}e^{\frac{2\sqrt{-1}\pi}{3}}&\\&e^{-\frac{2\sqrt{-1}\pi}{3}}\end{pmatrix}}$. There is a single point in $\chi^i_{Sing}(PSL(2,\mathbb{Z}),PSL(2,\mathbb{C}))$.
To study, the singularity $[\beta_p]$ in the character variety one does the exact same thing as in Proposition \[singvarcar\]. Since $\dim (H^1(PSL(2,\mathbb{Z}),\mathfrak{d}_{k,Ad\circ\beta_p}))=1$ for $p=2,3$ and $1\leq k\leq p-1$, we may apply the criterion of Lemma \[cyclicaction\] and Sikora’s formula (Equation \[dimesptgt\]). This leads to the wanted result.
There are more general questions related to the algebraic singularities of the (schematic) character variety.
- Completely reducible representations which are not irreducible are usually algebraic singularities of free groups character varieties (see for instance Theorem 3.21 in [@F-L]). But there are, in some cases, non irreducible representations whose conjugacy class is “accidentally” smooth on the (schematic) character variety. It is highlighted in Remark 3.22 in loc. cit., one could think of $\chi(\mathbb{Z},SL(n,\mathbb{C}))=\mathbb{C}^{n-1}$ or $\chi(\mathbb{F}_2,SL(2,\mathbb{C}))=\mathbb{C}^3$ whose all points, even the non-irreducible ones, are smooth.
- The method in the proof of Proposition \[singvarcar\] virtually generalizes to any Fuchsian groups (by *Fuchsian group*, we mean discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ such that $\mathbb{H}^2/\Gamma$ has finite volume, see [@Wei]). Indeed, for such groups, irreducible representations are scheme smooth in $PSL(p,\mathbb{C})$, which allows us to use Sikora’s formula (Equation \[dimesptgt\]). It would certainly be interesting to look more closely to non-Fuchsian examples (e.g. Example \[algnotorb\]).
- Does Proposition \[singvarcar\] remain true if $PSL(p,\mathbb{C})$ is replaced by any complex reductive group $G$ (in the free group case, it is related to Conjecture 3.34 in [@F-L]) ?
Group cohomology {#grocoh}
================
The aim of this section is to give a brief overview of the results needed in this paper for the cohomology of groups. For a complete overview of the theorems and results about group cohomology, the reader is advised to read [@Bro]. For this section, the modules will be noted additively (note that in Section \[diagbyfin\], the modules are multiplicative and additive in Section \[orbalgsing\]).
Let $G$ be a group and $M$ be an additive $G$-module. We begin with a few definitions for cocycles of degree $1$ or $2$ (compare with Chapter III, Paragraph 1, Example 3 in loc. cit.).
The set of fixed points for the $G$-action of $M$ will be denoted $M^G$.
A *$1$-cocycle* from $G$ to $M$ is a map $z:G\to M$ such that for $g,h\in G$, we have $z(gh)=z(g)+g\cdot z(h)$.
A *$1$-coboundary* from $G$ to $M$ is a map $b:G\to M$ such that there is $m\in M$ verifying for $g\in G$, $b(g)=m-g\cdot m$.
A *$2$-cocycle* from $G$ to $M$ is a map $z:G^2\to M$ such that for $g,h,k\in G$, we have $z(g,h)=g\cdot z(h,k)- z(gh,k)+z(g,hk)$.
A *$2$-coboundary* from $G$ to $M$ is a map $b:G^2\to M$ such that there is a map $m:G\to M$ verifying for $g,h\in G$, $b(g,h)=m(g)+g\cdot m(h)-m(gh)$.
For $i=1,2$. The set of $i$-cocycles (resp. $i$-coboundaries) is denoted $Z^i(G,M)$ (resp. $B^i(G,M)$). It is straightforward to check that $B^i(G,M)$ is contained in $Z^i(G,M)$. Remark that they are both abelian subgroups of $C^i(G,M)$ (the set of maps from $G^i$ to $M$, also known as the set of $i$-cochains in this context). The *$i$-th cohomology group* is the quotient $H^i(G,M):=Z^i(G,M)/B^i(G,M)$. If $z$ is a $i$-cocycle, $[z]$ denotes its image in $H^i(G,M)$.
Remark that $H^i(G,M)$ is actually defined for all $i\in \mathbb{N}$ but we will only use the first and the second cohomology groups.
These low-degree cohomology groups are useful in algebra (read Chapter IV in loc. cit. for applications to group extensions), especially using cochains. In Section \[diagbyfin\] and Section \[orbalgsing\], this is the link between the first cohomology group and semidirect product which is used (see Remark \[similar\]). The first example of computation, which directly follows from the definition, is an easy exercise which is left to the reader.
\[H1triv\]
Let $G$ be a group and $M$ be a trivial $G$-module (that is $G$ acts trivially on $M$) then $H^1(G,M)=\operatorname{Hom}(G,M)$.
In a list of four papers, Fox introduced free differential calculus (the second paper of this list is [@Fox]). It can be used to compute low-degree cohomology of groups (see [@Gol] for instance) especially when the module is a vector space. In general, computing cohomology groups proves to be difficult using its definition with cochains and one need to use a more abstract approach.
Let $G$ be a cyclic group with generator $g$ and order $n$. We define two maps related to the $G$-module $M$ : $$\operatorname{Norme}_M:\left|\begin{array}{lcc}M&\longrightarrow& M^G\\x&\longmapsto & \sum_{i=0}^{n-1}g^i\cdot x\end{array}\right. \text{ and } \operatorname{Trace}_M:\left|\begin{array}{lcc}M&\longrightarrow& M\\x&\longmapsto &x-g\cdot x\end{array}\right.$$ Then, we have in Chapter III, Paragraph 1, Example 2 in [@Bro] :
\[H1cycl\]
Let $G$ be a cyclic group with generator $g$ and order $n$ and $M$ be a $G$-module, then $H^1(G,M)$ is isomorphic to $\operatorname{Ker}(\operatorname{Norme}_M)/\operatorname{Im}(\operatorname{Trace}_M)$.
Although we will not use more than what is contained in this lemma, it should be acknowledged that the cohomology of cyclic groups is $2$-periodic and the lemma mentioned above is a simple consequence of this fact (see Chapter VI, Paragraph 9 in loc. cit. for more details).
The following lemma is particularly interesting in Section \[orbalgsing\] where the modules considered are $\mathbb{C}$-vector spaces (see Chapter III, Corollary 10.2 in loc. cit.).
\[cohfin\]
Let $G$ be a finite group and $M$ be a $G$-module. If the multiplication by $|G|$ is invertible in $M$ (e.g. if $M$ is a vector space over a field of characteristic $0$) then $H^1(G,M)=H^2(G,M)=0$.
The last result we will need is a way to relate the cohomology of a group with the cohomology of its normal subgroups. Before stating the proposition, we need to make a few comments. Start with $G$ a group and a normal subgroup $N$ of $G$. Let $Q$ be the quotient group $G/N$ with its natural projection $p:G\rightarrow Q$. If $M$ is a $G$-module then $M$ is also a $N$-module (by simply restricting the action) and $M^N$ is a $Q$-module (since $M^N$ is both a $G$-module and a trivial $N$-module).
For $q\in Q$, we arbitrarily choose $x_q\in G$ such that $x_q$ mod $N= q$. If $z$ belongs to $Z^1(N,M)$, then one can see that the map $z_q: n\mapsto x_q\cdot z(x_q^{-1}nx_q)$ is also an element of $Z^1(N,M)$. Furthermore, if $z$ is in $B^1(N,M)$, then $z_q$ is also in $B^1(N,M)$. Therefore, we have a map from $H^1(N,M)$ to itself sending $[z]$ to $[z_q]$. One can check that this defines an action of $Q$ on $H^1(N,M)$ which does not depend on the lifts $x_q$ chosen.
If $z$ is a $1$-cocycle from $G$ to $M$, then its restriction $z'$ to $N$ is also a $1$-cocycle. Furthermore, $[z']$ can be shown to be a fixed points for the $Q$-action on $H^1(N,M)$. We denote $\operatorname{Res}[z]:=[z']$. This defines a map $\operatorname{Res}: H^1(G,M)\to H^1(N,M)^Q$ called the *restriction map*.
If $w$ is a $1$-cocycle from $Q$ to $M^N$, then its push-forward by $p$, $z:=w\circ p: G\to M$ belongs to $Z^1(G,M)$ and we denote $\operatorname{Inf}[w]:=[z]$. This defines a map $\operatorname{Inf}: H^1(Q,M^N)\to H^1(G,M)$ called the *inflation map*.
The result we are interested in is the decomposition of the cohomology of $G$ in $M$ using the cohomology of $Q$ and $N$. The decomposition is called the Hochschild-Serre Spectral Sequence see Chapter VII, Theorem 6.3 in [@Bro] (remark that the spectral sequence is given for the homology but also works for the cohomology). For us, the most useful part of this spectral sequence are its first terms which give the Inflation-Restriction sequence, see Chapter VII, Corollary 6.4 in loc. cit. :
\[InfRes\] Let $G$ be a group and $M$ be a $G$-module. Assume we have the following exact sequence $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$. Then there is a map $T: H^1(N,M)^Q\to H^2(Q,M^N)$ called the *Transgression map* such that the following exact sequence is exact :
$$\xymatrix{1\ar[r]&H^1(Q,M^N)\ar[r]^{\operatorname{Inf}}&H^1(G,M)\ar[r]^{\operatorname{Res}}&H^1(N,M)^Q\ar[r]^{T}&H^2(Q,M^N)}$$
\[DHWexplicit\] It is possible to explicitly write down the Transgression map $T$ defined in the proposition, see Paragraph 10.2 in [@D-H-W].
[Buc99]{}
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[^1]: University of Luxembourg, Campus Kirchberg Mathematics Research Unit, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg. *e-mail : [email protected]*
|
---
abstract: 'This paper presents a dual stage EKF (Extended Kalman Filter)-based algorithm for the real-time and robust stereo VIO (visual inertial odometry). The first stage of this EKF-based algorithm performs the fusion of accelerometer and gyroscope while the second performs the fusion of stereo camera and IMU. Due to the sufficient complementary characteristics between accelerometer and gyroscope as well as stereo camera and IMU, the dual stage EKF-based algorithm can achieve a high precision of odometry estimations. At the same time, because of the low dimension of state vector in this algorithm, its computational efficiency is comparable to previous filter-based approaches. We call our approach DS-VIO (dual stage EKF-based stereo visual inertial odometry) and evaluate our DS-VIO algorithm by comparing it with the state-of-art approaches including OKVIS, ROVIO, VINS-MONO and S-MSCKF on the EuRoC dataset. Results show that our algorithm can achieve comparable or even better performances in terms of the RMS error.'
author:
- 'Xiaogang Xiong$^{1}$, Wenqing Chen$^{1}$, Zhichao Liu$^{2}$ and Qiang Shen$^{3}$[^1] [^2] [^3]'
bibliography:
- 'VIO\_Chen.bib'
title: '**DS-VIO: Robust and Efficient Stereo Visual Inertial Odometry based on Dual Stage EKF** '
---
INTRODUCTION
============
In GPS-denied environments, for instance, indoors and in urban canyons, it is essential for mobile robot platforms such as micro aerial vehicles (MAVs) to know their own pose for operations. In recent years, this pose estimation problem is popularly solved by the combination of visual information from cameras and measurements from an Inertial Measurement Unit (IMU), which is usually referred to as Visual Inertial Odometry (VIO)[@Sun_2018_Robust; @zheng2018pi]. Compared to lidar based approaches, VIO requires only a lightweight and small-size sensor package, making it the preferred technique for mobile robot platforms.
Realtime VIO solutions are critical for robots to response quickly to corresponding environments. At the same time, computation efficiency of VIO solutions is also important for robots with limited payloads and computation power such as MAVs. It is well-known that filter-based VIO solutions are generally more computationally efficient than that of optimization based methods [@Sun_2018_Robust] while stereo cameras are more robust to hostile environments compared with monocular camera. In order to solve the above problem of pose estimation in a realtime and robust manner, we propose a new VIO solution based on the extended Kalman filer (EKF) and information from stereo cameras in this paper.
To achieve a precise and robust pose estimation, the algorithm we present here is a dual stage EKF-based VIO solution, which is referred as to DS-VIO. It is consisted of two EKF filters and each stage has one EKF filter. We demonstrate that our algorithm perform comparable or even higher precision than other state-of-art VIO algorithms. The primary contribution of our work is a new filter-based VIO framework that contains a dual stage EKF filter. The first stage perform the sensor fusion of the measurements of accelerometer and gyroscope. The second stage performs the fusion of the measurements of IMU and stereo camera. The dual stage of EKF allows the VIO solution achieve higher precision and robustness. To show the effects, we provide detailed comparisons between our DS-VIOS with other state-of-art open-source VIO approaches including OKVIS, ROVIO, VINS-MONO and S-MSCKF on the EuRoC dataset through experiments. The results demonstrate that our DS-VIO is able to achieve similar or even better performance than these state-of-art VIO approaches in term of precision and robustness.
The rest of the paper is orgnized as follows: Section \[Sec:Related\_Work\] introduces the related work. Section \[Sec:First\_Stage\] and Section \[Sec:Second\_Stage\] provide the mathematical details of the first and second stage of EKF filter. Section \[Sec:Dual\] describes the enforcement mechanism of dual stage EKF filter. The experiments comparing our DS-VIO with state of the art open-source VIO approaches is conducted in Section \[Sec:Experiments\]. Finally, Section \[Sec:Conclusion\] draws some conclusions of this paper.
RELATED WORK {#Sec:Related_Work}
============
There are a large number of work which fuses measurements from cameras and IMUs to perform pose estimations, including VIO and VI-SLAM (Simultaneous Localization and Mapping). The related work is discussed from three different aspects: ([1]{}) loosely or tightly coupled solutions. ([2]{}) filter-based or optimization-based solutions. ([3]{}) direct or indirect solutions.
Loosely coupled solutions process the measurements from IMUs and cameras separately. Some methods process the images for computing relative-motion estimates between consecutive poses firstly [@Ma2012Robust; @Tardif2010A; @Weiss2011Real] and then fuse the result with IMU measurements. In contrast, in [@Matthies2012Fully; @Oskiper2007Visual], IMU measurements are used to compute rotation estimates and fuse the result with an visual estimation algorithm. Loosely coupled solutions lead to a reduction in computational cost and information loss [@Li2014Visual]. However, the tightly coupled solutions can achieve higher accuracy [@Konolige2008FrameSLAM; @Konolige2010Large; @mourikis2007multi; @qin2018vins; @sun2018robust]. In this work, we are interested in tightly coupled solutions and the proposed DS-EKF is also a tightly coupled solution.
Existing tightly coupled solutions can be divided into filter-based or optimization-based solutions. The latter generally attains higher accuracy because re-linearization at each iteration can better deal with their nonlinear measurement models, like [@Leutenegger2014Keyframe; @Lupton2012Visual; @qin2018vins; @Usenko2016Direct]. However, the procedure for multiple iterations lead to heavy calculation burden. As for filter-based solutions, to date, the majority of the proposed algorithms are EKF-based methods like [@Jones2011Visual; @Kelly2008Combined; @Kelly2009Visual; @Kleinert2010Inertial; @mourikis2007multi; @Pinies2007Inertial; @sun2018robust], also, Uncented Kalman Filter in [@Kelly2011Visual], and Particle Filter in [@yap2011particle]. In order to improve the accuracy of filter-based solutions, some researchers focus on the consistency of estimator. The works in [@Huang2009A; @Huang2010Observability] present the First Estimate Jacobian EKF (FEJ-EKF). Observability Constraint EKF (OC-EKF) is presented in [@Hesch2017Consistency; @Roumeliotis2016Observability; @sun2018robust]. The key idea of FEJ-EKF is to choose the same or first-ever available estimations for all the state variables as the linearization points. However, OC-EKF guarantees observability of the linearized system by ensuring the rank of nullspace of the nonlinearized system not changed after linearized. The OC-EKF is applied in our DS-VIO algorithm.
The vast majority of pose estimation methods have relied on the usage of point features, which are detected and tracked in the images like [@mourikis2007multi; @qin2018vins]. The methods in [@Tarrio2016Realtime; @Yu2017Edge] employ edge information while the method in [@zheng2018pi] employs both point and line features. All the above methods are classified as indirect solutions. The methods in [@Forster2015IMU; @Zheng2017Photometric] present a direct method which employ the image-intensity measurements directly. The direct methods exploit more information from the images than that of indirect methods.
FIRST STAGE EKF FILTER DESCRIPTION {#Sec:First_Stage}
==================================
The mathematical description in this paper follows the formulations in [@mourikis2007multi]. The gyroscope state is defined as: $$\begin{aligned}
\textbf X_g &=& [ {_G^I}{\overline {\textbf q} }{^T} \quad \textbf b{_g^T} ]^T \nonumber\end{aligned}$$ where ${_G^I}{\overline {\textbf q} }$ is the unit quaternion describing the rotation from inertial frame (${G}$) to the body frame (${I}$), $ \textbf b{_g}\in {\mathbb{R}}^{3\times 1}$ describes the bias of gyroscope measurement. Here, the body frame (${I}$) is assumed to be fixed to IMU frame.
Process Model {#Sec:Model}
-------------
The state equation in the time-continuous form is given as follows:
\[Gyro\_cont\_model\] $$\begin{aligned}
{_G^I}{ {\dot { \overline {\textbf q } }}} &=& \frac{1}{2} \bm{\Omega} ( { \bm{\omega}} (t)) {_G^I}{\overline {\textbf q}}(t) \\
{\dot {\textbf b} } _g(t) &= &{\textbf n}_{wg}(t)\end{aligned}$$
where $ \bm{\omega} = [\omega_x \ \omega_y \ \omega_z ]^T$ is the rotational velocity in the IMU frame, $$\begin{aligned}
\bm{\Omega} ( { \bm{\omega}}) =
\left[\begin{array}{cccc}
-(\bm{\omega} \times) & \bm{\omega} \\
- \bm{\omega} ^T & 0
\end{array}\right]
,
(\bm{\omega} \times) =
\left[\begin{array}{cccc}
0 & - \omega_z & \omega_y \\
\omega_z & 0 & -\omega_x \\
\omega_y & \omega_x & 0
\end{array}\right] \nonumber\end{aligned}$$ and the gyroscope bias ${\textbf b} _g$ is modeled as a random walk process with random walk rate $ {\textbf n}_{wg} $. The gyroscope measurements $ \omega_m$ is: $$\begin{aligned}
\bm{ \omega}_m = \bm{\omega} +\textbf b_g +\textbf n_g
\end{aligned}$$ By applying the expectation operator in , we can get the estimates of the evolving gyroscope state:
\[State\_estimation\] $$\begin{aligned}
{_G^I}{\dot{ \hat{\overline {\textbf q } } } } &=& \frac{1}{2} \bm{\Omega} ( \hat { \bm{\omega} } ) {_G^I} {\hat{ \bar { \textbf q} }} \label{State_estimation_a} \\
{\dot{\hat {\textbf b } }} _g &=& \textbf 0_{3\times1} \label{State_estimation_b}\end{aligned}$$
where $\hat { \bm{ \omega }}={ \bm{\omega}}_m - \hat {\textbf b}_g $.
To propagate the uncertainty of the state, the discrete time state transition matrix should be computed, as for , $$\begin{aligned}
\nonumber {\Phi}_{k_q}& =&\mathrm{exp} \biggl( \int_{t_k}^{t_{k+1}} \frac{1}{2} \bm{\Omega} ( { \hat {\bm{\omega}}(\tau)} ) d\tau \biggr)_{4\times4} \approx (\textbf I + \frac{t_k}{2} \bm{\Omega} )\end{aligned}$$ where $t_k{\stackrel{\Delta}{=}}t(k)$ and $h:=t_{k+1}-t_{k}$ is the time interval between two IMU measurements. As for , in a similar manner, one has $$\nonumber {\Phi}_{k_b} = \textbf 0_{3\times3}.$$ Therefore, we can get $$\begin{aligned}
\nonumber \textbf X_g(k+1) &=&
\left[\begin{array}{cccc}
{_G^I}{\overline {\textbf {q}} (k+1)} \\
\textbf b{_g}(k+1)
\end{array}\right]
=
\left[\begin{array}{cccc}
{\Phi}_{k_q} {_G^I}{\overline {\textbf {q}} (k)} \\
{\Phi}_{k_b} \textbf b{_g}(k)
\end{array}\right]+
\textbf w(k).\end{aligned}$$ Therefore, one can obtain the following expression: $$\label{Gyro_disc_state}
\textbf X_g(k+1) \approx \textbf H_p \textbf X_g +\textbf w(k)$$ where $\textbf H_p$ is the Jacobian of process model with respect to the gyroscope state, which is shown in Appendix, $\textbf w(k)$ is a $7 \times 1$ vector describe the process noise, of which the covariance matrix is as follows: $$\textbf Q_{1,k} =
\left[\begin{array}{cccc}
\sigma{_{q}^2}\textbf I_4 & \textbf 0_{3\times 3} \\
\textbf 0_{4\times 4} & \sigma{_{b}^2}\textbf I_3
\end{array}\right].$$ Finally, the propagated covariance of the gyroscope state is described as: $$\textbf P_{G|G_{k+1|k}} = \textbf H_p \textbf P_{G|G_{k|k}} \textbf H_p ^T + \textbf Q_{1,k}$$
Measurement Model
-----------------
The first correction stage uses data from the accelerometers to correct the gyroscope state, the measurement is, $$\begin{aligned}
\textbf z(k) &=& \textbf a(k) = [a_x(k) \ a_y(k) \ a_z(k) ]^T\end{aligned}$$ According to [@Sabatelli2013A], $\hat {\textbf z}(k)$ can be presented by $$\hat{\textbf z}(k) = R_n^b
\left[\begin{array}{cccc}
0 \\
0 \\
-g
\end{array}\right] =-g
\left[\begin{array}{cccc}
2q_1q_3 - 2q_0q_2 \\
2q_0q_1 + 2q_2q_3 \\
q_0^2 - q_1^2 -q_2^2 +q_3^2
\end{array}\right]$$ where $g$ is the constant g-force acceleration, and $$\begin{gathered}
R_n^b=
\left[ \begin{matrix}
q_0^2 + q_1^2 -q_2^2 - q_3^2 & 2(q_1q_2+q_0q_3) \\
2(q_1q_2-q_0q_3) &q_0^2 - q_1^2 + q_2^2 - q_3^2 \\
2(q_1q_3 +q_0q_2) & 2(q_2q_3-q_0q_1)
\end{matrix} \right.
\\
\left.
\begin{matrix}
& & 2(q_1q_3 - q_0q_2) \\
& & 2(q_2q_3 + q_0q_1) \\
& &q_0^2 - q_1^2 - q_2^2 + q_3^2
\end{matrix}
\right]. \nonumber\end{gathered}$$ The residual of the measurement can be approximated as $$\label{residual}
\textbf r(k) = \textbf z(k) - \hat{\textbf z}(k) = \textbf H_g\textbf X_g +\textbf v(k)$$ where $\textbf H_g$ is the Jacobian of measurement with respect to the gyroscope state, which is shown in Appendix, $\textbf v(k)\in {\mathbb{R}}^{3 \times 1}$ describes the measurement noise, of which the covariance matrix is $\sigma{_{a}^2}\textbf I_3$,
SECOND STAGE EKF FILTER DESCRIPTION {#Sec:Second_Stage}
===================================
The evolving IMU state is defined as follows: $$\begin{aligned}
\nonumber \textbf X_I = [ {_G^I}{\overline {\textbf q} }{^T} \quad \textbf b{_g^T} \quad {^G}{\textbf v}{_I^T} \quad \textbf b{_a^T} \quad {^G}{\textbf p}{_I^T} ]^T\end{aligned}$$ where $ {^G}{\textbf v}{_I} $ and $ {^G}{\textbf p}{_I}\in \mathbb{R}^{3\times 1}$ are vectors describing the IMU position and velocity in frame ${G}$, and $ \textbf b{_a}\in \mathbb{R}^{3\times 1} $ describes the bias of accelerometer measurement. The bias is modeled as random walk processes and the random walk rate is $ {\textbf n}_{wa} $. The IMU error-state is defined as: $$\begin{aligned}
\widetilde{ \textbf X}_I=[ \bm{ \delta} \bm {\theta}_I^T \quad \widetilde{ \textbf b}_g^T \quad {^G}\widetilde {\textbf v}_I^T \quad \widetilde {\textbf b}_a^T \quad {^G} \widetilde{ \textbf p}_I^T ]^T \nonumber \end{aligned}$$ where the standard additive error defined as $\tilde {x}{\stackrel{\Delta}{=}}x-\hat{x}$ with $ x\in\{\textbf b{_g^T}, {^G}{\textbf v}{_I^T}, \textbf b{_a^T},{^G}{\textbf p}{_I^T} \}$ and $ \hat x\in\{\hat{ \textbf b}{_g^T}, {^G}{\hat{ \textbf v}}{_I^T}, \hat {\textbf b}{_a^T},{^G}{\hat {\textbf p}}{_I^T} \}$ is used for position, velocity and bias, respectively, and $ \bm{ \delta} \bm {\theta}_I^T $ represents the quaternion error [@mourikis2007multi; @Bloesch2015Robust]. Ultimately, we add the $N$ camera poses in the IMU state, we can get the EKF state vector, at time-step, it can be defined as: $$\textbf X_E= [ \textbf X_{I_k}^T \quad {_G^{C_1}}{\overline {\textbf q}}^T \quad ^G{\textbf p}{_{C_1}^T} \quad \cdots \quad {_G^{C_N}}{\overline {\textbf q}}^T \quad ^G{\textbf p}{_{C_N}^T}]^T$$ where the ${_G^{C_i}}{\overline {\textbf q}}^T, {^G{\textbf p}}{_{C_i}^T}$ ,$i=1,2\cdots N$ is the camera attitude and position, respectively. The EKF error state vevtor is defined as: $$\begin{aligned}
\tilde{\textbf X}_E&=& [ \tilde{\textbf X} _{I_k}^T \quad { \bm{\delta \theta} } {_{C_1}^T} \quad {^G{\widetilde{ \textbf p}}{_{C_1}^T}} \quad \cdots \quad { \bm{\delta \theta} } {_{C_N}^T} \quad {^G{\widetilde{ \textbf p}}{_{C_N}^T} }]^T \nonumber\end{aligned}$$
Process Model {#process-model}
-------------
The continuous-time system model is described as :
\[IMU\_cont\_model\] $$\begin{aligned}
{_G^I}{ {\dot { \overline {\textbf q } }}} &=& \frac{1}{2} \Omega ( { \omega} (t)) {_G^I}{\overline {\textbf q}}(t) \\
{\dot {\textbf { b}} } _g(t)& =& {\textbf n}_{wg}(t) \\
{^G}{\dot {\textbf v} }_I(t)& =& {^G}{\textbf a}(t) \\
{\dot{\textbf b} }_a(t) &= &{\textbf n}_{wa}(t) \\
{^G}\dot{\textbf{ p}} _I(t) &=& {^G}{\textbf v}_I(t)\end{aligned}$$
where $^G{\textbf a}$ is the body acceleration in the global frame ($G$). The accelerometer measurement $ \textbf a_m$ is: $$\begin{aligned}
\nonumber
\textbf a_m = C({_G^I}\overline {\textbf q})(^G{\textbf a}-{^G{\textbf g}})+\textbf b_a+\textbf n_a\end{aligned}$$ where $C(\cdot)$ denotes a rotational matrix, $\textbf n_g$ and $\textbf n_a$ are measurement noise, $^G{\textbf g}$ is gravitational acceleration expressed int the local frame. From to , we can get the equations for propagating the estimates of the evolving IMU state by applying expectation operator:
\[IMU\_cont\_model\] $$\begin{aligned}
{_G^I}{\dot{ \hat{\overline {\textbf q } } } }& =& \frac{1}{2} \Omega ( \hat { \omega} ) {_G^I} {\hat{ \bar { \textbf q} }}\\
{\dot{\hat {\textbf b } }} _g & =& \textbf 0_{3\times1} \\
{^G}{\dot {\hat{ \textbf v}}}_I &=& C({_G^I}\hat{\bar{\textbf q } })^T {\hat {\textbf a} } +^G{\textbf g} \\
{\dot{ \hat{ \textbf b}}}_a &= &\textbf 0_{3\times1} \\
{^G}{\dot {\hat{\textbf p}}}_I &=& {^G}{\hat {\textbf v}_I}\end{aligned}$$
where $\hat { \textbf a} =
\textbf a_m - \hat {\textbf b}_a$. Therefore, the linearized continuous-time model for the IMU error-state is: $$\label{linearized}
{ \dot { \widetilde {\textbf X}}_{I} } = {\textbf F} { \widetilde {\textbf X}_{I} } +{\textbf G}{\textbf n}_{I}$$ where ${\textbf n}_I = [ {\textbf n}{_g^T} \quad {\textbf n}{_{wg}^T} \quad {\textbf n}{_a^T} \quad {\textbf n}{_{wa}^T} ]^T$, $\textbf F$ and $\textbf G$ are shown in appendix. The state equation model can be computed as follows: $$\widetilde{ \textbf X}_{E}(k+1) = {\Phi}_k \widetilde {\textbf X}_{E}(k) + \textbf Q_{2,k}$$ where the computation of transform matrix $ {\Phi}_k $ and the discrete time noise covariance $ \textbf Q_k$ can be found in [@sun2018robust]. So the propagated covariance of the IMU state is described as: $$\begin{aligned}
\nonumber
\textbf P_{I|I_{k+1|k}} = \Phi _{k}\textbf P_{I|I_{k|k}} \Phi {_k^T} +\textbf Q_{2,k}\end{aligned}$$ where the covariance matrix $\textbf P_{I|I_{k|k}}$ and $\textbf P_{I|I_{k+1|k}}$ are described as: $$\begin{aligned}
\nonumber
\textbf P_{E|E_{k|k}} &=&
\left[\begin{array}{cccc}
\textbf P_{I|I_{k|k}} & \textbf P_{I|C_{k|k}} \\
\textbf P_{C|I_{k|k}} &\textbf P_{C|C_{k|k}}
\end{array}\right]\\
\textbf P_{E|E_{k+1|k}}& =&
\left[\begin{array}{cccc}
\textbf P_{I|I_{k+1|k}} & \Phi _{k}\textbf P_{I|C_{k|k}} \\
\textbf P_{C|I_{k|k}}\Phi {_k^T} &\textbf P_{C|C_{k|k}}
\end{array}\right]. \nonumber\end{aligned}$$ When recording a new camera, we should add the camera pose into the state by using the following expressions: $$\begin{aligned}
_G^C{\hat {\overline{\textbf q}}} &=& _I^C{\bar {\textbf q}} \bigotimes {_G^I}{\overline {\hat{\textbf q}} } \nonumber \\
^G{\hat{\textbf p}}_C & =& ^G{\hat{\textbf p}}_I + C({_G^I}{\hat {\bar {\textbf q}}})^T{^I}{\textbf p}_C \nonumber\end{aligned}$$ The augmented covariance is given by: $$\label{augmented_cov}
\textbf P_{k|k} \leftarrow
\left[\begin{array}{cccc}
\textbf I_{6N+15} \\ \textbf J
\end{array}\right]
\textbf P_{k|k}
\left[\begin{array}{cccc}
\textbf I_{6N+15} \\ \textbf J
\end{array}\right] ^T$$ where $\textbf J$ is shown in Appendix.
Measurement Model {#Subsec:Model}
-----------------
The second correction stage employs data from the images to correct the IMU state, of which the measurement model we adopted here follows [@mourikis2007multi; @sun2018robust]. It is based the fact that static features in the wold can be observed from multiple camera poses showing constraints among all these camera poses. Consider the case of a single feature $f_j$ observed by the stereo cameras at time step $i$. Assume this feature are observed by both left and right cameras, and the left and right cameras poses are represented as $\left(_G^{C_{i,1}}\textbf q, ^G \textbf p_{C_{i,1}}\right)$ and $\left(_G^{C_{i,2}}\textbf q, ^G \textbf p_{C_{i,2}}\right)$. For the feature $f_j$ at time step $i$, the stereo measurement is, $$\begin{aligned}
\nonumber
z_i^j &=&
\left[\begin{array}{cccc}
u_{i,1}^j \\ v_{i,1}^j \\u_{i,2}^j \\ v_{i,2}^j
\end{array}\right] =
\left[\begin{array}{cccc}
^{C_{i,1}}Z_j & \textbf 0_{2\times 2} \\ \textbf 0_{2\times 2} &^{C_{i,2}}Z_j
\end{array}\right]
\left[\begin{array}{cccc}
^{C_{i,1}}X_j \\ ^{C_{i,1}}Y_j \\ ^{C_{i,2}}X_j \\ ^{C_{i,2}}Y_j
\end{array}\right]\end{aligned}$$ where $ \left[\begin{array}{cccc} ^{C_{i,k}}X_j & ^{C_{i,k}}Y_j & ^{C_{i,k}}Y_j \end{array}\right]^T $, $k = 1,2$ represent the positions of feature $f_j$ in the left and right camera frame are given by, $$^{C_i} {\textbf p}{_f{_j}} = \left[\begin{array}{cccc} ^{C_{i,k}}X_j \\ ^{C_{i,k}}Y_j \\ ^{C_{i,k}}Y_j \end{array}\right] = C(_G^{C_{i,k}}\textbf q ) ( ^G {\textbf p} _{f_{j}} - ^G{\textbf p} _{C_{i,k}} )$$ where $^G {\textbf p} _{f_{j}}$ is the position of feature $f_j$ in the golbal frame, it can be computed the least square method shown in [@mourikis2007multi].
Once $^G {\textbf p} _{f_{j}}$ is obtained, the measurement residual can be computed by: $$\nonumber
{\textbf r}{_i^j} ={\textbf z}{_i^j} - \hat {\textbf z}{_i^j}.$$ By linearizing about the estimates for the camera poses and feature position, the residual can be approximated as: $${\textbf r}{_i^j} \approx \textbf H_{X_{i}}^j \widetilde{ {\textbf X} } +\textbf H_{f_{i}}^j {^G {\widetilde{ \textbf p}} _{ f_{j}}} + \textbf n_i^j$$ where $\textbf n_i^j$ is the noise of the measurement, $\textbf H_{X_{i}}^j$ and $\textbf H_{f_{i}}^j$ are the Jacobian matrixes of the measurement ${\textbf z}{_i^j}$ with respect to the state and the feature position, respectively, of which the value of these two Jacobian matrixes can be found in [@sun2018robust], ${^G {\widetilde{ \textbf p}} _{ f_{j}}}$ is the error in the position estimate of $f_j$. By stacking all the observations of feature $f_j$, one can obtain: $$\nonumber
{\textbf r}{^j} \approx \textbf H_{X}^j \widetilde{ {\textbf X}} +\textbf H_{f}^j {^G {\widetilde {\textbf p}} _{ f_{j}}} + \textbf n^j.$$ Because the position of feature $^G {\textbf p} _{f_{j}}$ is computed by using the state estimate $\textbf X$, the error ${^G {\widetilde {\textbf p}} _{ f_{j}}}$ is correlated with the error $\widetilde {\textbf X}$. The form of residual cannot be directly used for update in the EKF. By projecting ${\textbf r}{^j}$ to the nullspace of $\textbf H_{f}^j$, one has $$\nonumber
{\textbf r_o}{^j} = \textbf A^T{\textbf r}{^j} \approx \textbf A^T \textbf H_{X}^j \widetilde{ {\textbf X}} + \textbf A^T \textbf n^j = \textbf H_{X,o}^j \widetilde {\textbf X} + \textbf n_o^j.$$ where $\textbf A$ denotes the unitary matrix whose columns form the basis of the left nullspace of $\textbf H_{f}^j$ [@mourikis2007multi].
DUAL STAGE EKF MECHANISM {#Sec:Dual}
========================
The structure of the proposed dual stage of EKF filter is shown in Fig.\[fig1\_structure\_block\]. The first stage of EKF is designed to combine the measurements from accelerometer and gyroscope. By implementing the first stage EKF filter, data from accelerometer is used as a corrective measure by taking into account the gravitational force to curb the error of the orientation estimate. The role of the second stage EKF is fusing measurements from IMU and stereo cameras. The images from stereo cameras can present a natural complement to IMU to to curb the errors and drift.
As shown in Fig.\[fig1\_structure\_block\], the first stage of EKF is executed immediately as the IMU measurements are acquired. For each IMU measurement received, one has:
- **Propagation:** The rotational velocity ${\bm{\omega}_m}$ is used to propagate the gyroscope state $ \textbf X_g$, and covariance matrix $\textbf P_{G|G}$.
- **update:** The body acceleration $\textbf a_m$ is used to perform an EKF update.
The second stage of EKF contains the first stage, and the first stage is regarded as the propagate part in the second stage EKF:
- **Propagation:** Whenever a new IMU measurement is received, it propagates the IMU state $\textbf X_I$ and covariance matrix $\textbf P_{E|E}$.
- **Image registration:** Whenever new stereo images are acquired, it (1) augments the IMU state and the corresponding covariance matrix; (2) operates image frontend processes, including feature extraction and feature matching.
- **update:** EKF update is performed when (1) a feature has been tracked in a number of images (the number is 3 in our algorithm) is no longer detected; (2) camera poses will be discarded when the largest number of camera poses in the IMU state has been reached.
We present a new strategy to discard old camera poses in the above **update**. The oldest non-keyframe is discarded according to two criteria whenever new stereo images are received. The two criteria from [@qin2018vins] is employed here for keyframe selection. The first one is the average parallax apart from the previous keyframe and the second one is tracking quality.
EXPERIMENTS {#Sec:Experiments}
===========
In this section, we perform two experiments to evaluate the proposed DS-VIO algorithm. In the first experiment, we compare the proposed algorithm with other state-of-the-art algorithms on public dataset by analyzing the Root Mean Square Error (RMSE) metrics. In the second experiment, we compare the proposed DS-VIO with S-MSCKF in the indoor environment. S-MSCKF is selected because it is also one of stereo and EKF filter based approaches.
Dataset Comparison
------------------
The proposed DS-VIO algorithm (stereo-filter) is compared with the state of the art approaches including OKVIS [@Leutenegger2014Keyframe] (stereo optimization), ROVIO [@Bloesch2015Robust] (monocular filter), VINS-MONO [@qin2018vins] (monocular optimization) and S-MSCKF [@sun2018robust] (stereo filter) on the EuRoC dataset [@Burri2016The]. These methods are different combinations of monocular, stereo, filter-based, and optimization-based methods. Among these methods, S-MSCKF is a tightly-coupled filtering-based stereo VIO algorithm which is closely related to our work. During the experiment, only the images from the left camera are used for monocular camera based algorithms like ROVIO and VINS-MONO. In order to perform a convictive experiment, only the performance of VIO is conducted and the functionality of loop closure is disabled for VINS-MONO. For each algorithm, its performance is evaluated for repeating experiments ten times and the mean value is treated as the result.
The EuRoC MAV Dataset is a collection of visual-inertial datasets collected on-board a Micro Aerial Vehicle (MAV). It contains synchronized 20Hz stereo images and 200Hz IMU messages, accurate motion and structure ground-truth. Some parts of the dataset exhibit very dynamic motions and different image brightness, which renders stereo matching and feature tracking more challenging.
![Root Mean Square Error (RMSE) results []{data-label="fig2_comparison"}](euroc.png){width="3.2in"}
Figure \[fig2\_comparison\] shows the RMSE results of our proposed DS-VIO algorithm and other state of the art algorithms including OKVIS, ROVIO, VINS-MONO and S-MSCKF on the EuRoC datasets. Because both our proposed method and S-MSCKF employ the KLT optical flow algorithm for feature tracking and stereo matching, they do not work properly on “V2\_03" dataset. As pointed out in [@sun2018robust], the rapid change of brightness in “V2\_03" dataset causes failures in the stereo feature matching. On the rest datasets, our method achieves comparable performance with other methods. For the most similar method S-MSCKF, we can see that our method has better performance than that of S-MSCKF on all the datasets.
Indoor Experiment
-----------------
In the indoor experiment, for the sake of fairness, we only compare S-MSCKF with our DS-VIO because both algorithms are stereo-filter based approaches. We choose rectangular corridor in our laboratory building as the experiment area. We encounter low light and texture-less condition in the corridor environment, as shown in Fig.\[fig3\_indoor\_Environment\]. The sensor suite we use is ZED-mini, which contains stereo cameras (30hz) and an IMU (800hz). With ZED-mini in our hands, we walk along the rectangular corridor for a circle around 1m/s.
Fig.\[fig4\_comparison\_results\] shows the comparison results between our DS-VIO and S-MSCKF. Fig.4(a) shows the trajectories in the $xy$ plane and Fig.4(b) shows the trajectories in three dimensions. The blue line represents trajectory of DS-VIO while the dotted line represents the trajectory of S-MSCKF. One can that the blue trajectory is a nonstandard rectangle but the dotted trajectory is far away from a rectangle. This comparison shows that the proposed DS-VIO achieves a smaller cumulative error than that S-MSCKF. One can attribute the archived senior performance to the first stage of EKF filter of the proposed DS-VIO, which employs the complementary characteristics between accelerometer and gyroscope.
CONCLUSIONS {#Sec:Conclusion}
===========
In this paper, we present a robust and efficient filter-based stereo VIO. It employs a dual stage of EKF to perform the state estimation. This dual stage EKF filter employ the complementary characters of IMU and stereo cameras as well as accelerometer and gyroscope. The accuracy and robustness of the proposed VIO is demonstrated by experiments of the EuRoC MAV Dataset and indoor environment by comparing with the state of the art VIO algorithms. The further work should explore how to achieve better accuracy and efficiency by selecting feature points which have some certain characters.
APPENDIX {#appendix .unnumbered}
========
The $\textbf H_p$ in is, $$\nonumber
\textbf H_p =
\left[\begin{array}{ccccccc}
\textbf H_{p1} & \textbf H_{p2}\\
\textbf 0_{3\times4} & \textbf I_{3}
\end{array}\right]$$ where $$\begin{aligned}
\nonumber \textbf H_{p1} & =&
\left[\begin{array}{ccccccc}
1 & -0.5\hat \omega_x t & -0.5\hat \omega_y t & -0.5\hat \omega_z t \\
0.5\hat \omega_x t & 1 & 0.5\hat \omega_z t & -0.5\hat \omega_y t \\
0.5\hat \omega_y t & - 0.5\hat \omega_z t & 1 & 0.5\hat \omega_x t \\
0.5\hat \omega_z t & 0.5\hat \omega_y t & - 0.5\hat \omega_x t &1
\end{array}\right]\end{aligned}$$ and $$\begin{aligned}
\nonumber \textbf H_{p2} & =&
\left[\begin{array}{ccccccc}
0.5q_1 t & 0.5q_2 t & 0.5q_3 t \\
-0.5q_0 t & 0.5q_3 t & -0.5q_2 t \\
-0.5q_3 t & -0.5q_2 t & 0.5q_1 t \\
0.5q_2 t & - 0.5q_1 t & -0.5q_0 t
\end{array}\right],\end{aligned}$$ $ \hat {\bm{\omega}} = [\hat\omega_x \ \hat\omega_y \ \hat \omega_z ]^T$ is the estimation of rotational velocity.
The $\textbf H_g$ in is, $$\nonumber
\textbf H_g =
\left[\begin{array}{ccccccc}
2gq_2 & -2gq_3 & 2gq_0 & -2gq_1 & 0&0&0\\
-2gq_1 & -2gq_0 & -2gq_3 & -2gq_2 & 0&0&0\\
-2gq_0 & 2gq_1 & 2gq_2 & 2gq_3 & 0&0&0\\
\end{array}\right]$$ where $g$ is the g-force acceleration. The $\textbf F$ and $\textbf G$ in are as follows: $$\begin{aligned}
\nonumber \textbf F && = \\
&&\left[\begin{array}{ccccc}
-(\hat { \omega} \times) & -{\textbf I}_3 & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3}\\
{\textbf 0}_{3 \times 3} & -{\textbf I}_3 & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} \\
-C({_G^I}\hat {\bar {\textbf q}})^T \cdot (\hat {\textbf a} \times) & \textbf 0_{3\times 3} & \textbf0_{3\times 3} &- C({_G^I}\bar {\textbf q})^T & \textbf 0_{3\times 3} \\
{\textbf 0}_{3 \times 3} & -{\textbf I}_3 & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} \\
{\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & -{\textbf I}_3 & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3}
\end{array}\right] \nonumber\end{aligned}$$ and $$\nonumber
\textbf G =
\left[\begin{array}{ccccc}
-{\textbf I}_3 & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3}\\
{\textbf 0}_{3 \times 3} & {\textbf I}_3 & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} \\
\textbf 0_{3\times 3} &\textbf 0_{3\times 3} &- C({_G^I}\bar{\textbf q})^T & \textbf 0_{3\times 3} \\
{\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf I}_3 \\
{\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3} & {\textbf 0}_{3 \times 3}
\end{array}\right].$$ The $\textbf J$ in is follows: $$\nonumber
\textbf J =
\left[\begin{array}{ccccc}
C({_I^C}{ {\bar {\textbf q}}}) & \textbf 0_{3\times 9} &\textbf 0_{3\times 3} &\textbf 0_{6N} \\
(C({_G^I}{ {\hat {\textbf q}}})^I{\textbf p}_I ) \times & \textbf 0_{3\times 9} & \textbf I_3 & \textbf 0_{6N}
\end{array}\right].$$
[^1]: $^{1}$ X. Xiong and W. Chen are with Faculty of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen, P.R. China [[email protected]]{}
[^2]: $^{2}$ UBTECH Robotics, Shenzhen, China
[^3]: $^{3}$ Arizona State University, School for Engineering of Matter, Transport and Energy, USA [[email protected]]{}
|
---
abstract: 'In the context of the (generalized) Delta Conjecture and its compositional form, D’Adderio, Iraci, and Wyngaerd recently stated a conjecture relating two symmetric function operators, $D_k$ and $\Theta_k$. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the $D_k$ operators.'
author:
- Marino Romero
bibliography:
- 'ThetaOperatorsBib.bib'
title: A proof of the Theta Operator Conjecture
---
[^1]
Introduction
============
In what follows, we will assume the reader is familiar with symmetric functions and plethystic substitution. For a standard symmetric function reference, there is Macdonald’s book [@Macdonald]. For some of the plethystic identities shown here, we will mostly reference [@explicit] and [@positivity]. We will also adopt the notation and conventions in [@fiveterm]. Garsia, Haiman, and Tesler defined in [@explicit] a family of plethystic operators $\{D_k\}_{k\in \mathbb{Z}}$ by setting $$D_k F[X] = F\left[X + \frac{M}{z} \right] \operatorname{Exp}[-zX] \Big|_{z^k}$$ where $$\operatorname{Exp}[X] = \sum_{n \geq 0} h_n[X] = \exp\left(\sum_{k \geq 1} \frac{p_k}{k} \right)$$ is the plethystic exponential and $M=(1-q)(1-t)$. In the definition of $D_k$, we would have $$\operatorname{Exp}[-zX] = \sum_{k \geq 0 } (-z)^{k} e_k[X].$$ For every partition $\mu$, set $$\Pi_\mu = \prod_{(i,j) \in \mu/(1)} (1-q^it^j)$$ and define the linear operator $\Pi$ by setting $$\Pi \operatorname{\widetilde{H}}_\mu = \Pi_\mu \operatorname{\widetilde{H}}_\mu.$$ To get a compositional refinement of the (generalized) Delta Conjecture [@delta], D’Adderio, Iraci, and Wyngaerd [@theta] define $$\Theta_k = \Pi~ \underline{e}_k^* ~\Pi^{-1}$$ where $\underline{f}^*$ is multiplication by $f^*=f[X/M]$.\
Conjecture 10.3 in [@theta] asserts that $$[\Theta_k, D_1] = \sum_{i=1}^k (-1)^{i} D_{i+1} \Theta_{k-i}.$$ Multiplying both sides by $(-1)^k$ and expanding $[\Theta_k, D_1]$ as $\Theta_k D_1 - D_1 \Theta_k$ gives $$(-1)^k\Theta_k D_1= D_1 (-1)^k\Theta_k + \sum_{i=1}^k D_{i+1} (-1)^{k-i} \Theta_{k-i}.$$ Let us set $\operatorname{\overline{\Theta}}_k = (-1)^k \Theta_k$, $$\begin{aligned}
\operatorname{\overline{\Theta}}(z) = \sum_{k \geq 0} z^k \operatorname{\overline{\Theta}}_k && \text{ and } && D_+(z) = \sum_{k \geq 1} z^k D_k.\end{aligned}$$ Then note that $$D_+(z) \operatorname{\overline{\Theta}}(z) |_{z^{k+1}} = D_1 \operatorname{\overline{\Theta}}_k + D_2 \operatorname{\overline{\Theta}}_{k-1} + \cdots + D_{k+1} \operatorname{\overline{\Theta}}_0.$$ The conjecture can then be rewritten as $$z \operatorname{\overline{\Theta}}(z) D_1 = D_+(z) \operatorname{\overline{\Theta}}(z).$$ This is what we will prove.
The $D_k$ operators
===================
Another way to write the $D_k$ operators is using the translation and multiplication operators $\operatorname{\mathcal{T}}_Y $ and $\operatorname{\mathcal{P}}_Z$, defined for any two expressions $Y$ and $Z$ by setting $$\begin{aligned}
\operatorname{\mathcal{T}}_Y F[X] = F[X+Y] && \text{ and } && \operatorname{\mathcal{P}}_Z F[X] = \operatorname{Exp}[ZX]F[X].\end{aligned}$$ Then following the definition of $D_k$, we have as in [@explicit] $$\sum_{-\infty < k < \infty } z^k D_k = \operatorname{\mathcal{P}}_{-z} \operatorname{\mathcal{T}}_{\frac{M}{z}}.$$
The proof
=========
For any symmetric function $F$, define the linear operator $\Delta_F$ by setting $$\begin{aligned}
\Delta_F \operatorname{\widetilde{H}}_\mu = F[B_\mu] \operatorname{\widetilde{H}}_\mu, && \text{ where} && B_\mu = \sum_{(i,j) \in \mu} q^i t^j. \end{aligned}$$ This is the Delta eigenoperator for the modified Macdonald basis, defined in [@positivity]. We will now follow the notation from Garsia and Mellit’s five-term relation [@fiveterm]. First note that if we set $$\Delta_u = \sum_{n \geq 0} (-u)^n\Delta_{e_n},$$ then we have $$\begin{aligned}
\Delta_u \operatorname{\widetilde{H}}_\mu = \operatorname{\widetilde{H}}_\mu \prod_{(i,j)\in \mu} (1-uq^it^j) && \text{ and } && \Delta_{u}^{-1} = \sum_{ n \geq 0 } u^n \Delta_{h_n}.\end{aligned}$$ Furthermore, $\Pi = \Delta_u/(1-u) |_{u=1},$ and we can write $$\operatorname{\overline{\Theta}}_k = \Delta_u (-1)^k \underline{e}_k^* \Delta_u^{-1} \Big|_{u=1}.$$ Even though $\Delta_1^{-1}$ on its own is not well defined, one can still write in this case that $ \operatorname{\overline{\Theta}}(z) =\Delta_1 \operatorname{\mathcal{P}}_{-\frac{z}{M}} \Delta_1^{-1}$. We will not need this since the $u$’s will vanish in our calculations. For this reason we will now, instead, consider the unspecialized operator $$\operatorname{\widetilde{\Theta}}_k = \Delta_v (-1)^k \underline{e}_k^* \Delta_v^{-1}$$ and let $$\operatorname{\widetilde{\Theta}}(z) = \sum_{k \geq 0}z^k \operatorname{\widetilde{\Theta}}_k = \Delta_v \operatorname{\mathcal{P}}_{-\frac{z}{M}} \Delta_{v}^{-1}$$ for some monomial $v$. We will show that $$z\operatorname{\widetilde{\Theta}}(z) D_1 = D_+(z) \operatorname{\widetilde{\Theta}}(z),$$ or rather $$z\operatorname{\widetilde{\Theta}}(z) D_1 \operatorname{\widetilde{\Theta}}(z)^{-1} = D_+(z).$$ In other words, we want to show that $$D_+(z) = z \Delta_v \operatorname{\mathcal{P}}_{-\frac{z}{M}} \Delta_v^{-1} D_1 \Delta_v \operatorname{\mathcal{P}}_{\frac{z}{M}} \Delta_v^{-1}.$$
Next, for $\mu \vdash n$, we set $\nabla \operatorname{\widetilde{H}}_\mu = \Delta_u \operatorname{\widetilde{H}}_\mu |_{u^n} = (-1)^n q^{n(\mu')} t^{n(\mu)} \operatorname{\widetilde{H}}_\mu.$ With this convention, we have $D_1 = \nabla \underline{e}_1 \nabla^{-1}$ [@explicit]. We can then rewrite the conjecture by substituting for $D_1$, giving $$D_+(z) = z \Delta_v \operatorname{\mathcal{P}}_{-\frac{z}{M}} \Delta_v^{-1} \nabla \underline{e}_1 \nabla^{-1} \Delta_v \operatorname{\mathcal{P}}_{\frac{z}{M}} \Delta_v^{-1}.$$
One of the main results from the five-term relation is the following identity:
For any two monomials $u$ and $v$, we have $$\nabla^{-1} \operatorname{\mathcal{T}}_{uv} \nabla = \Delta_v^{-1} \operatorname{\mathcal{T}}_u \Delta_{v} \operatorname{\mathcal{T}}_u^{-1}.$$
The dual version is given by translating $\operatorname{\mathcal{T}}_z$ to $\operatorname{\mathcal{P}}_{-z/M}$ and reversing the order: $$\nabla \operatorname{\mathcal{P}}_{-\frac{uv}{M}} \nabla^{-1} = \operatorname{\mathcal{P}}_{\frac{u}{M}} \Delta_v \operatorname{\mathcal{P}}_{-\frac{u}{M} } \Delta_{v}^{-1}.$$ We get $$\Delta_v \operatorname{\mathcal{P}}_{-\frac{z}{M} } \Delta_{v}^{-1} = \operatorname{\mathcal{P}}_{-\frac{z}{M}}\nabla \operatorname{\mathcal{P}}_{-\frac{zv}{M}} \nabla^{-1}$$ and the inverse formula $$\Delta_v \operatorname{\mathcal{P}}_{\frac{z}{M} } \Delta_{v}^{-1} = \nabla \operatorname{\mathcal{P}}_{\frac{zv}{M}} \nabla^{-1} \operatorname{\mathcal{P}}_{\frac{z}{M}}.$$
Substituting this in our conjectured formula, we get $$\begin{aligned}
\Delta_v \operatorname{\mathcal{P}}_{-\frac{z}{M}} \Delta_v^{-1} \nabla \underline{e}_1 \nabla^{-1} \Delta_v \operatorname{\mathcal{P}}_{\frac{z}{M}} \Delta_v^{-1} && =
& \operatorname{\mathcal{P}}_{-\frac{z}{M}}\nabla \operatorname{\mathcal{P}}_{-\frac{zv}{M}} \nabla^{-1} \nabla \underline{e}_1 \nabla^{-1} \nabla \operatorname{\mathcal{P}}_{\frac{zv}{M}} \nabla^{-1} \operatorname{\mathcal{P}}_{\frac{z}{M}} \\
&& = &
\operatorname{\mathcal{P}}_{-\frac{z}{M}}\nabla \operatorname{\mathcal{P}}_{-\frac{zv}{M}} \underline{e}_1\operatorname{\mathcal{P}}_{\frac{zv}{M}} \nabla^{-1} \operatorname{\mathcal{P}}_{\frac{z}{M}}
\\
&& = & \operatorname{\mathcal{P}}_{-\frac{z}{M}}\nabla \underline{e}_1 \nabla^{-1} \operatorname{\mathcal{P}}_{\frac{z}{M}}
\\ &&=& \operatorname{\mathcal{P}}_{-\frac{z}{M}}D_1 \operatorname{\mathcal{P}}_{\frac{z}{M}} .\end{aligned}$$ We can now write the final form of the conjecture. It is an extension of Proposition 1.6 in [@explicit].
$$D_+(z) = \sum_{k \geq 1} z^k D_k = z \operatorname{\mathcal{P}}_{-\frac{z}{M}} D_1 \operatorname{\mathcal{P}}_{\frac{z}{M}}$$ or $$D_+(z) = z\operatorname{\mathcal{P}}_{-\frac{z}{M}} \nabla \underline{e}_1 \nabla^{-1} \operatorname{\mathcal{P}}_{\frac{z}{M}}$$
We will show that for any $r$ $$\operatorname{\mathcal{P}}_{- \frac{z}{M}} D_r \operatorname{\mathcal{P}}_{\frac{z}{m}} = \sum_{k \geq 0} z^k D_{k+r}.$$ In particular for $r=1$, this proves the theorem. Following [@explicit], for any two expressions $Y$ and $Z$, we have $$\begin{aligned}
\operatorname{\mathcal{T}}_Y \operatorname{\mathcal{P}}_Z ~ F[X] & = \operatorname{\mathcal{T}}_Y ~ \operatorname{Exp}[XZ] F[X] \\
& = \operatorname{Exp}[(X+Y)Z] F[X+Y] \\
& = \operatorname{Exp}[YZ] \operatorname{Exp}[XZ] F[X+Y] \\
& = \operatorname{Exp}[YZ] \operatorname{\mathcal{P}}_Z \operatorname{\mathcal{T}}_Y ~ F[X],\end{aligned}$$ meaning $$\operatorname{\mathcal{T}}_Y \operatorname{\mathcal{P}}_Z = \operatorname{Exp}[YZ] \operatorname{\mathcal{P}}_Z \operatorname{\mathcal{T}}_Y.$$ Therefore, we have $$\begin{aligned}
\operatorname{\mathcal{P}}_{-\frac{z}{M}} \operatorname{\mathcal{P}}_{-u} \operatorname{\mathcal{T}}_{\frac{M}{u}} \operatorname{\mathcal{P}}_{\frac{z}{M}} & = \operatorname{\mathcal{P}}_{-\frac{z}{M}} \operatorname{\mathcal{P}}_{-u} \operatorname{Exp}\left[\frac{z}{u}\right] \operatorname{\mathcal{P}}_{\frac{z}{M}}\operatorname{\mathcal{T}}_{\frac{M}{u}} \\
& = \frac{1}{1-z/u} \operatorname{\mathcal{P}}_{-u} \operatorname{\mathcal{T}}_{\frac{M}{u}}.\end{aligned}$$ The coefficient of $z^{k} u^r$ (with $k \geq 0$) on the right hand side of the last equality is $$\operatorname{\mathcal{P}}_{-u}\operatorname{\mathcal{T}}_{\frac{M}{u}} \Big |_{u^{k+r}} = D_{k+r}.$$ Equating coefficients on both sides, we have $$\operatorname{\mathcal{P}}_{-\frac{z}{M}} D_r \operatorname{\mathcal{P}}_{\frac{z}{M}} \Big|_{z^{k}} = D_{k+r}.$$
[\
Marino Romero\
University of Pennsylvania\
Department of Mathematics\
[*E-mail*]{}: `[email protected]` ]{}
[^1]: This work was supported by NSF DMS1902731
|
---
abstract: 'The Painlevé-III equation with parameters $\Theta_0=n+m$ and $\Theta_\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\infty;m)=1$ whenever $n\in\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in [@BothnerMS18], we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\to+\infty$ with $m\in\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is filled with a locally uniform lattice of poles and zeros, and the density of the poles and zeros is small near the boundary of $E$ but blows up near the origin, which is the only fixed singularity of the Painlevé-III equation. In both the interior and exterior domains we provide accurate asymptotic formulæ for $u_n(x;m)$ that we compare with $u_n(x;m)$ itself for finite values of $n$ to illustrate their accuracy. We also consider the exceptional cases where $m$ is a half-integer, showing that the poles and zeros of $u_n(x;m)$ now accumulate along only one or the other of two “eyebrows”, i.e., exterior boundary arcs of $E$.'
address:
- 'Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, United States'
- 'Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, United States'
author:
- Thomas Bothner
- 'Peter D. Miller'
title: 'Rational Solutions of the Painlevé-III Equation: Large Parameter Asymptotics'
---
[^1]
Introduction
============
Generic solutions of the six Painlevé equations cannot be expressed in terms of elementary functions, hence the common terminology of *Painlevé transcendents* for the general solutions of these famous equations. However, it is also known that all of the Painlevé equations except for the Painlevé-I equation admit solutions expressible in terms of classical special functions (e.g., Airy solutions for Painlevé-II, or Bessel solutions for Painlevé-III) as well as rational solutions, both of which occur for certain isolated values of the auxiliary parameters (each Painlevé equation except Painlevé-I is actually a family of differential equations indexed by one or more complex parameters). Rational solutions of Painlevé equations have attracted interest partly because they are known to occur in several diverse applications such as the description of equilibrium configurations of fluid vortices [@Clarkson09] and of particular solutions of soliton equations [@Clarkson06], electrochemistry [@BassNRS10], parametrization of string theories [@Johnson06], spectral theory of quasi-exactly solvable potentials [@ShapiroT14], and the description of universal wave patterns [@BuckinghamM12]. In several of these applications it is interesting to consider the behavior of the rational Painlevé solutions when the parameters in the equation become large (possibly along with the independent variable); as the degree of the rational function is tied to the parameters via Bäcklund transformations, in this limit algebraic representations of rational solutions become unwieldy and hence less attractive than analytical ones as a means for extracting asymptotic behaviors. Recent progress on the analytical study of large-degree rational Painlevé solutions includes [@BertolaB15; @BuckinghamM14; @BuckinghamM15; @MillerS17] for Painlevé-II and [@Buckingham17; @MasoeroR18] for Painlevé-IV. Both of these equations have the property that there is no fixed singular point except the point at infinity. On the other hand, the Painlevé-III equation is the simplest of the Painlevé equations having a finite fixed singular point (at the origin). This paper is the second in a series beginning with [@BothnerMS18] concerning the large-degree asymptotic behavior of rational solutions to the Painlevé-III equation, which we take in the generic form $$\frac{{\mathrm{d}}^2u}{{\mathrm{d}}x^2}=\frac{1}{u}\left(\frac{{\mathrm{d}}u}{{\mathrm{d}}x}\right)^2-\frac{1}{x}\frac{{\mathrm{d}}u}{{\mathrm{d}}x} +
\frac{4\Theta_0 u^2 + 4(1-\Theta_\infty)}{x} + 4u^3-\frac{4}{u},\ \ \ \ x\in\mathbb{C}.
\label{eq:PIII}$$ It is convenient to represent the constant parameters $\Theta_0$ and $\Theta_\infty$ in the form $$\Theta_0=n+m\quad\text{and}\quad\Theta_\infty=m-n+1.
\label{eq:Thetas-m-n}$$ It is known that if $n\in\mathbb{Z}$, there exists a unique rational solution $u(x)=u_n(x;m)$ of that tends to $1$ as $x\to\infty$. The odd reflection $u(x)=-u_n(-x;m)$ provides a second distinct rational solution. Similarly, if $m\in\mathbb{Z}$, there are two rational solutions tending to $\pm{\mathrm{i}}$ as $x\to\infty$, namely $u(x)=\pm{\mathrm{i}}u_m(\pm{\mathrm{i}}x;n)$, while if neither $m$ nor $n$ is an integer, has no rational solutions at all. If only one of $m$ and $n$ is an integer, then there are exactly two rational solutions; however if both $m\in\mathbb{Z}$ and $n\in\mathbb{Z}$ there are exactly four distinct rational solutions: $u_n(x;m)$, $-u_n(-x;m)$, ${\mathrm{i}}u_m({\mathrm{i}}x;n)$, and $-{\mathrm{i}}u_m(-{\mathrm{i}}x;n)$.
Representations of $u_n(x;m)$
-----------------------------
### Algebraic representation {#sec:algebraic-representation}
It has been shown [@Clarkson03; @ClarksonLL16; @Umemura99] that $u_n(x;m)$ admits the representation $$u_n(x;m)=\frac{s_n(x;m-1)s_{n-1}(x;m)}{s_n(x;m)s_{n-1}(x;m-1)};\quad u_{-n}(x;m)=\frac{1}{u_n(x;m)},\quad n\in\mathbb{Z}_{\ge 0},
\label{eq:un-exact-fraction}$$ where $\{s_n(x;m)\}_{n=-1}^{\infty}$ are polynomials in $x$ with coefficients polynomial in $m$ that are defined by the recurrence formula $$s_{n+1}(x;m):=\frac{(4x+2m+1)s_n(x;m)^2-s_n(x;m)s_n'(x;m)-x\left(s_n(x;m)s_n''(x;m)-s_n'(x;m)^2\right)}{2s_{n-1}(x;m)}
\label{eq:sn-recurrence}$$ and the initial conditions $s_{-1}(x;m)=s_0(x;m)=1$. The polynomials $\{s_n(x;m)\}_{n=-1}^\infty$ are frequently called the *Umemura polynomials*, although in [@Umemura99] Umemura originally considered instead related functions that are polynomials in $1/x$. For $n$ not too large, the recurrence relation provides an effective computational strategy to obtain the poles and zeros of $u_n(x;m)$. The rational function $u_n(x;m)$ has the following symmetry: $$u_n(-x;m)=\frac{1}{u_n(x;-m)}.
\label{eq:u-n-exact-symmetry}$$ This follows from the fact that $u(x)\mapsto u(-x)^{-1}$ is a symmetry of – corresponding to the parameter mapping $(m,n)\mapsto (-m,n)$. Since this symmetry preserves rationality and asymptotics $u\to 1$ as $x\to\infty$, it descends from general solutions to the particular solution $u_n(x;m)$ as written in .
### Analytic representation {#sec:analytic-representation}
The goal of this paper is to study $u_n(x;m)$ when $n$ is a large positive integer and $m$ is a fixed complex number. The representation is useful to determine numerous properties of the rational Painlevé-III solutions, however when $n$ is large another representation becomes more preferable. To explain this alternate representation, we first define some $x$-dependent arcs in an auxiliary complex $\lambda$-plane as follows. Given $x\in\mathbb{C}$ with $x\neq 0$ and $|\mathrm{Arg}(x)|<\pi$, there is an intersection point ${{p}}$ and four oriented arcs ${{L^\infty_\squareurblack}}$, ${{L^0_\squareurblack}}$, ${{L^\infty_\squarellblack}}$, and ${{L^0_\squarellblack}}$ such that:
- The arc ${{L^\infty_\squareurblack}}$ originates from $\lambda=\infty$ in such a direction that ${\mathrm{i}}x\lambda$ is negative real and terminates at $\lambda={{p}}$, the arc ${{L^0_\squareurblack}}$ begins at $\lambda={{p}}$ and terminates at $\lambda=0$ in a direction such that $-{\mathrm{i}}x\lambda^{-1}$ is negative real, and the net increment of the argument of $\lambda$ along ${{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$ is $$\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)\pm 2\pi.
\label{eq:increment-argument-red}$$ The ambiguity of the sign in will be explained below (see Remark \[rmk:surgery\]).
- The arc ${{L^\infty_\squarellblack}}$ originates from $\lambda=\infty$ in such a direction that $-{\mathrm{i}}x\lambda$ is negative real and terminates at $\lambda={{p}}$, the arc ${{L^0_\squarellblack}}$ begins at $\lambda={{p}}$ and terminates at $\lambda=0$ in a direction such that ${\mathrm{i}}x\lambda^{-1}$ is negative real, and the net increment of the argument of $\lambda$ along ${{L^\infty_\squarellblack}}\cup{{L^0_\squarellblack}}$ is $$\Delta\arg(\squarellblack)=2\mathrm{Arg}(x).
\label{eq:increment-argument-blue}$$
- The arcs ${{L^\infty_\squareurblack}}$, ${{L^0_\squareurblack}}$, ${{L^\infty_\squarellblack}}$, and ${{L^0_\squarellblack}}$ do not otherwise intersect.
See Figure \[fig:y-5-Exp-i-pi-over-4\] below for an illustration in the case of $\mathrm{Arg}(x)=\tfrac{1}{4}\pi$. We define a single-valued branch of the argument function $\lambda\mapsto\arg(\lambda)$, henceforth denoted ${{\arg_\squarellblack(\lambda)}}$, by first selecting ${{L^\infty_\squarellblack}}\cup{{L^0_\squarellblack}}$ as the branch cut, and then defining ${{\arg_\squarellblack(\lambda)}}=0$ for sufficiently large positive $\lambda$ when $\mathrm{Im}(x)>0$ and ${{\arg_\squarellblack(\lambda)}}=\pi$ for sufficiently large negative $\lambda$ when $\mathrm{Im}(x)<0$. It is easy to see that this definition is consistent for $x>0$ but there is a jump across the negative real $x$-axis. We define an associated branch of the complex logarithm $\log(\lambda)$ by setting ${{\log_\squarellblack(\lambda)}}:=\ln|\lambda|+{\mathrm{i}}\,{{\arg_\squarellblack(\lambda)}}$. Then, given $q\in\mathbb{C}$, the corresponding branch of the power function $\lambda^q$ will be denoted by ${{\lambda^{q}_\squarellblack}}:={\mathrm{e}}^{q\,{{\log_\squarellblack(\lambda)}}}$. Finally, we denote by $L$ the union of the four oriented arcs ${{L^\infty_\squareurblack}}$, ${{L^0_\squareurblack}}$, ${{L^\infty_\squarellblack}}$, and ${{L^0_\squarellblack}}$, and define the function $$\varphi(\lambda):=\lambda-\lambda^{-1},\ \ \ \lambda\in\mathbb{C}\setminus\{0\}.
\label{eq:varphi}$$ The following Riemann-Hilbert problem was formulated in [@BothnerMS18 Sec. 1.2]. Here and below we follow the convention that subscripts $+$/$-$ refer to boundary values taken on a jump contour from the left/right, and $\sigma_3:=\mathrm{diag}[1,-1]$ denotes a standard Pauli spin matrix.
Given parameters $m\in\mathbb{C}$ and $n=0,1,2,3,\dots$, as well as $x\in\mathbb{C}\setminus\{0\}$ with $|\mathrm{Arg}(x)|<\pi$, seek a $2\times 2$ matrix function $\lambda\mapsto\mathbf{Y}(\lambda)=\mathbf{Y}_n(\lambda;x,m)$ with the following properties.
- **Analyticity:** $\lambda\mapsto\mathbf{Y}(\lambda)$ is analytic in the domain $\lambda\in\mathbb{C}\setminus L$. It takes continuous boundary values on $L\setminus\{0\}$ from each maximal domain of analyticity.
- **Jump conditions:** The boundary values $\mathbf{Y}_\pm(\lambda)$ are related on each arc of $L$ by the following formulæ: $$\mathbf{Y}_+(\lambda)
=\mathbf{Y}_-(\lambda)\begin{bmatrix}
1 & \displaystyle -\frac{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}}{\Gamma(\tfrac{1}{2}-m)}\lambda^n{\mathrm{e}}^{{\mathrm{i}}x\varphi(\lambda)}\\
0 & 1
\end{bmatrix},\quad \lambda\in {{L^0_\squareurblack}},
\label{eq:Yjump-1}$$ $$\mathbf{Y}_+(\lambda)
=\mathbf{Y}_-(\lambda)\begin{bmatrix}
1 & \displaystyle \frac{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}}{\Gamma(\tfrac{1}{2}-m)}\lambda^n{\mathrm{e}}^{{\mathrm{i}}x\varphi(\lambda)}\\
0 & 1
\end{bmatrix},\quad \lambda\in {{L^\infty_\squareurblack}},
\label{eq:Yjump-2}$$ $$\mathbf{Y}_+(\lambda)
=\mathbf{Y}_-(\lambda)\begin{bmatrix}
1 & 0\\
\displaystyle \frac{\sqrt{2\pi}({{\lambda^{(m+1)/2}_\squarellblack}})_+({{\lambda^{(m+1)/2}_\squarellblack}})_-}{\Gamma(\tfrac{1}{2}+m)}\lambda^{-n}{\mathrm{e}}^{-{\mathrm{i}}x\varphi(\lambda)} & 1
\end{bmatrix},\quad \lambda\in {{L^\infty_\squarellblack}},
\label{eq:Yjump-3}$$ $$\mathbf{Y}_+(\lambda)
=\mathbf{Y}_-(\lambda)\begin{bmatrix}
-{\mathrm{e}}^{2\pi{\mathrm{i}}m} & 0\\
\displaystyle \frac{\sqrt{2\pi}({{\lambda^{(m+1)/2}_\squarellblack}})_+({{\lambda^{(m+1)/2}_\squarellblack}})_-}{\Gamma(\tfrac{1}{2}+m)}\lambda^{-n}{\mathrm{e}}^{-{\mathrm{i}}x\varphi(\lambda)} & -{\mathrm{e}}^{-2\pi {\mathrm{i}}m}\end{bmatrix}, \quad \lambda\in {{L^0_\squarellblack}}.
\label{eq:Yjump-4}$$
- **Asymptotics:** $\mathbf{Y}(\lambda)\to\mathbb{I}$ as $\lambda\to\infty$. Also, the matrix function $\mathbf{Y}(\lambda){{\lambda^{-(\Theta_0+\Theta_\infty)\sigma_3/2}_\squarellblack}}=\mathbf{Y}(\lambda){{\lambda^{-(m+\tfrac{1}{2})\sigma_3}_\squarellblack}}$ has a well-defined limit as $\lambda\to 0$ (the same limit from each side of $L$).
\[rhp:renormalized\]
Given any choice of sign in , the sign may be reversed by a surgery performed on ${{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$ for any given value of $x\neq 0$, $|\mathrm{Arg}(x)|<\pi$ which leaves the conditions of Riemann-Hilbert Problem \[rhp:renormalized\] invariant. The surgery consists of bringing ${{L^\infty_\squareurblack}}$ together (with the same orientation) with ${{L^0_\squareurblack}}$ in some small arc. The jump for $\mathbf{Y}$ cancels on this small arc because the jump matrices in – are inverses of each other; thus, up to some relabeling, one has effectively changed the sign in . In [@BothnerMS18] the choice of sign in was tied to the sign of $\mathrm{Im}(x)$ due to the derivation of Riemann-Hilbert Problem \[rhp:renormalized\] from direct/inverse monodromy theory, however the above surgery argument shows that the sign is in fact arbitrary. The freedom to choose this sign will be important later when the solution of Riemann-Hilbert Problem \[rhp:renormalized\] is constructed for large $n$. \[rmk:surgery\]
It turns out that if Riemann-Hilbert Problem \[rhp:renormalized\] is solvable for some $x \in\mathbb{C}\setminus\{0\}$, then we may define corresponding matrices $\mathbf{Y}_1^\infty(x)$ and $\mathbf{Y}_0^0(x)$ by expanding $\mathbf{Y}(\lambda)=\mathbf{Y}_n(\lambda;x,m)$ for large and small $\lambda$, respectively: $$\mathbf{Y}(\lambda)=\mathbb{I}+\mathbf{Y}_1^\infty(x)\lambda^{-1}+\mathcal{O}(\lambda^{-2}),\quad\lambda\to\infty;\quad\mathbf{Y}_1^\infty(x)=[Y_{1,jk}^\infty(x)]_{j,k=1}^2$$ and $$\mathbf{Y}(\lambda){{\lambda^{-(m+\tfrac{1}{2})\sigma_3}_\squarellblack}}=\mathbf{Y}_0^0(x)+\mathcal{O}(\lambda),\quad\lambda\to 0;\quad\mathbf{Y}_0^0(x)=[Y_{0,jk}^0(x)]_{j,k=1}^2.$$ Then, according to [@BothnerMS18 Theorem 1], an alternate formula for the rational solution $u_n(x;m)$ of the Painlevé-III equation is $$u_n(x;m)=\frac{-{\mathrm{i}}Y^\infty_{1,12}(x)}{Y^0_{0,11}(x)Y^0_{0,12}(x)},
\label{eq:u-n-from-Y-formula}$$ where we have suppressed the parametric dependence on $n\in\mathbb{Z}$ and $m\in\mathbb{C}$ on the right-hand side.
Results and outline of paper {#sec:results}
----------------------------
A good way to introduce our results is to first explain a simple formal asymptotic calculation. Since we are interested in solutions $u=u_n(x;m)$ of with parameters written in the form when $n$ is large, and since numerical experiments such as those in [@BothnerMS18 Sec. 2] suggest that the largest poles and zeros of $u_n(x;m)$ lie at a distance $|x|$ from the origin proportional to $n$ with a local spacing that neither grows nor shrinks with $n$, it is natural to introduce a complex parameter $y\neq 0$ and a new independent variable $w\in\mathbb{C}$ by setting $x=ny+w$. It follows that if $u(x)$ solves –, then ${{p}}(w):=-{\mathrm{i}}u(x=ny+w)$ satisfies $$\frac{{\mathrm{d}}^2{{p}}}{{\mathrm{d}}w^2}=\frac{1}{{{p}}}\left(\frac{{\mathrm{d}}{{p}}}{{\mathrm{d}}w}\right)^2 + \frac{4{\mathrm{i}}}{y}({{p}}^2-1)-4{{p}}^3+\frac{4}{{{p}}}+\mathcal{O}(n^{-1})$$ in which the $\mathcal{O}(n^{-1})$ symbol absorbs several terms each of which is explicitly proportional to $n^{-1}\ll 1$. Dropping these formally small terms leads to an autonomous second-order equation which is amenable to classical analysis: $$\frac{{\mathrm{d}}^2\dot{{{p}}}}{{\mathrm{d}}w^2}=\frac{1}{\dot{{{p}}}}\left(\frac{{\mathrm{d}}\dot{{{p}}}}{{\mathrm{d}}w}\right)^2+\frac{4{\mathrm{i}}}{y}(\dot{{{p}}}^2-1)-4\dot{{{p}}}^3+\frac{4}{\dot{{{p}}}},
\label{eq:autonomous}$$ where $\dot{{{p}}}$ denotes a formal approximation to ${{p}}$. Solutions of the equation[^2] can be classified as follows:
- Equilibrium solutions $\dot{{{p}}}\equiv\text{constant}$. Generically with respect to $y$ there are four such equilibria: $\dot{{{p}}}\equiv \pm 1$ and $$\dot{{{p}}}\equiv {{p}}_0^\pm(y):=\frac{{\mathrm{i}}}{2y}\mp{\mathrm{i}}\sqrt{\frac{1}{4y^2}+1},$$ where to be precise we take the square roots to be equal to $1$ at $y=\infty$ and to be analytic in $y$ except on a line segment branch cut connecting the branch points $y=\pm\tfrac{1}{2}{\mathrm{i}}$ in the $y$ parameter plane. Note that of these four, the unique equilibrium that tends to $-{\mathrm{i}}$ as $y\to\infty$ (as would be consistent with $u=u_n(x;m)\to 1$ as $x\to\infty$) is $\dot{{{p}}}\equiv {{p}}_0^+(y)$.
- Non-equilibrium solutions. These can be obtained by integrating to find a first integral. Thus, provided $\dot{{{p}}}(w)$ is non-constant, we may write in the equivalent form $$\left(\frac{{\mathrm{d}}\dot{{{p}}}}{{\mathrm{d}}w}\right)^2 = \frac{16}{y^2}P(\dot{{{p}}};y,C),\quad P(\dot{{{p}}};y,C):=-\frac{y^2}{4}\dot{{{p}}}^4+\frac{{\mathrm{i}}y}{2}\dot{{{p}}}^3 + C\dot{{{p}}}^2 + \frac{{\mathrm{i}}y}{2}\dot{{{p}}}-\frac{y^2}{4},
\label{eq:dotV-ODE}$$ in which $C$ is a constant of integration. There are two types of non-equilibrium solutions:
- If $C$ is generic given $y$ such that $P$ has $4$ distinct roots, then all non-constant solutions of are (doubly-periodic) elliptic functions of $w$ with elliptic modulus depending on $C$ and $y$.
- If $C=C(y)$ is such that the quartic $P$ has fewer than $4$ distinct roots, then the higher-order roots are necessarily equilibrium solutions of and all non-constant solutions of are (singly-periodic) trigonometric functions of $w$.
Our rigorous analysis of $u_n(x;m)$ in the large-$n$ limit shows that all of the above types of solutions of the approximating equation play a role. In order to begin to explain our results, first observe that if $x$ is replaced with $ny+w$, then for large $n$, the dominant factors in the off-diagonal elements of the jump matrices in Riemann-Hilbert Problem \[rhp:renormalized\] are the exponentials ${\mathrm{e}}^{\pm nV(\lambda;y)}$, where $$V(\lambda;y):= -\log(\lambda)-{\mathrm{i}}y \varphi(\lambda).
\label{eq:V-define}$$ The fact that $V$ is multi-valued is not important because ${\mathrm{e}}^{\pm nV(\lambda;y)}$ is single-valued whenever $n\in\mathbb{Z}$. However, $\mathrm{Re}(V(\lambda;y))$ is certainly single-valued for $\lambda\in\mathbb{C}\setminus\{0\}$ and $y\in\mathbb{C}$. For simplicity, in the rest of the paper we write ${{p}}(y):={{p}}_0^+(y)$. Since ${{p}}(y)$ is analytic and non-vanishing in its domain of definition, the left-hand side of the equation $$\mathrm{Re}(V({{p}}(y);y))=0
\label{eq:critical}$$ defines a harmonic function in the complex $y$-plane omitting the vertical branch cut of ${{p}}(y)$ connecting the branch points $\pm\tfrac{1}{2}{\mathrm{i}}$. Therefore, determines a curve in the latter domain that turns out to be the union of four analytic arcs: two rays on the imaginary axis connecting the branch points $y=\pm\tfrac{1}{2}{\mathrm{i}}$ to $y=\pm{\mathrm{i}}\infty$ respectively, an arc in the right half-plane joining the two branch points, and its image under reflection through the imaginary axis. The union of the latter two arcs is the boundary of a compact and simply-connected eye-shaped set denoted $E$ containing the origin $y=0$. The eye $E$ is symmetric with respect to reflection through the origin as well as both the real and imaginary axes. See Figure \[fig:partialE\] below. Our first result is then the following.
Fix $m\in\mathbb{C}$ and let $K\subset\mathbb{C}\setminus E$ be bounded away from $E$, i.e., $\mathrm{dist}(y,E)>0$. Then $$u_n(ny;m)={\mathrm{i}}{{p}}(y) +\mathcal{O}(n^{-1}),\quad n\to+\infty,\quad y\in K,
\label{eq:u-outside}$$ where the error estimate is uniform for $y\in K$. \[theorem:outside\]
Thus, $u_n$ is approximated by the unique equilibrium solution of that tends to $-{\mathrm{i}}$ as $y\to\infty$, provided that $y$ lies outside the eye $E$. Since ${{p}}(y)$ is analytic and non-vanishing as a function of $y$ bounded away from $E$, the uniform convergence immediately implies the following.
Fix $m\in\mathbb{C}$ and let $K$ be as in the statement of Theorem \[theorem:outside\]. Then $u_n(\cdot;m)$ has no zeros or poles in the set $nK$ for $n$ sufficiently large. \[corollary:outside-no-poles-or-zeros\]
As an application of these results, let $y\in\mathbb{C}\setminus E$ and let $C_y$ denote a positively-oriented loop surrounding the point $y$. Then, from Cauchy’s integral formula it follows that, as $n\rightarrow+\infty$, $$\frac{{\mathrm{d}}^ju_n}{{\mathrm{d}}x^j}(ny;m)=\frac{1}{n^j}\frac{{\mathrm{d}}^ju_n}{{\mathrm{d}}y^j}(ny;m)=\frac{j!}{2\pi{\mathrm{i}}n^j}\oint_{C_y}\frac{u_n(ny';m)\,{\mathrm{d}}y'}{(y'-y)^{j+1}}={\mathrm{i}}n^{-j}\frac{{\mathrm{d}}^j{{p}}}{{\mathrm{d}}y^j}(y) + \mathcal{O}(n^{-j-1}), \quad j=1,2,3,\dots,
\label{eq:u-prime-outside}$$ where to evaluate the integral we used . It is easy to see that the error term enjoys similar uniformity properties as in Theorem \[theorem:outside\].
Next, we let $E_\mathrm{L}$ (resp., $E_\mathrm{R}$) denote the part of the interior of $E$ lying in the open left (resp., right) half-plane, compare again Figure \[fig:partialE\]. We now develop an asymptotic formula for $u_n(x;m)$ when $n^{-1}x\in E_\mathrm{R}$ and $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$. Since $E_\mathrm{L}$ and $E_\mathrm{R}$ are related by reflection through the origin, by the symmetry this formula will also be sufficient to describe $u_n(x;m)$ for large $n$ when $n^{-1}x\in E_\mathrm{L}$, because $-m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ whenever $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$. Given $m\in\mathbb{C}\setminus (\mathbb{Z}+\tfrac{1}{2})$, in – below we define complex-valued functions $\mathcal{Z}_n^\bullet(y,w;m)$, $\mathcal{Z}_n^\circ (y,w;m)$, $\mathcal{P}_n^\bullet(y,w;m)$, $\mathcal{P}_n^\circ(y,w;m)$, and $N(y)$, whose real and imaginary parts are smooth but non-analytic functions of the real and imaginary parts of $y\in E_\mathrm{R}$ and which are entire functions of $w\in\mathbb{C}$, with $N:E_\mathrm{R}\to\mathbb{C}$ non-vanishing. These functions depend crucially on a smooth but non-analytic function $C=C(y)$ defined on $E_\mathrm{R}$ by a procedure described in Sections \[sec:Boutroux-y-small\] and \[sec:Boutroux-degenerate\], and also on a related smooth function $B=B(y)$ with $\mathrm{Re}(B(y))<0$ defined by . In detail, compare ,$$\begin{split}
\mathcal{Z}_n^\bullet(y,w;m)&=\Theta(\mathscr{z}_n^\bullet(y,w;m)-{\mathrm{i}}\pi-\tfrac{1}{2}B(y),B(y)),\ \ \ \ \
\mathcal{Z}_n^\circ(y,w;m)=\Theta(\mathscr{z}_n^\circ(y,w;m)+{\mathrm{i}}\pi+\tfrac{1}{2}B(y),B(y)),\\
\mathcal{P}_n^\bullet(y,w;m)&=\Theta(\mathscr{p}_n^\bullet(y,w;m) +{\mathrm{i}}\pi +\tfrac{1}{2}B(y),B(y)),\ \ \ \ \
\mathcal{P}_n^\circ(y,w;m)=\Theta(\mathscr{p}_n^\circ(y,w;m)-{\mathrm{i}}\pi -\tfrac{1}{2}B(y),B(y)),
\end{split}$$ in which $\Theta(z,B)$ denotes the Riemann theta function defined by , and in which the complex-valued phases $\mathscr{z}_n^\bullet$, $\mathscr{z}_n^\circ$, $\mathscr{p}_n^\bullet$, and $\mathscr{p}_n^\circ$ are well-defined affine linear functions of $w\in\mathbb{C}$ and $n\in\mathbb{Z}_{\ge 0}$ with coefficients and constant terms that are smooth functions of $y\in\mathbb{R}$ depending parametrically on $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$. We then define $$\dot{u}_n(y,w;m):=N(y)\frac{\mathcal{Z}_n^\bullet(y,w;m)\mathcal{Z}_n^\circ(y,w;m)}{\mathcal{P}_n^\bullet(y,w;m)\mathcal{P}_n^\circ(y,w;m)},\quad y\in E_\mathrm{R},\quad w\in\mathbb{C},\quad m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2}),
\label{eq:udot-elliptic}$$ excluding isolated exceptional values of $(y,w)\in E_\mathrm{R}\times\mathbb{C}$ for which the denominator vanishes.
Fix $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$. For each $n\in\mathbb{Z}_{\ge 0}$ and each $y\in E_\mathrm{R}$, the function $\dot{{{p}}}(w):=-{\mathrm{i}}\dot{u}_n(y,w;m)$ is a non-equilibrium elliptic function solution of in the form with integration constant $C=C(y)$. If $\epsilon>0$ is an arbitrarily small fixed number and $K_y\subset E_\mathrm{R}$ and $K_w\subset\mathbb{C}$ are compact sets, then $$u_n(ny+w;m)=\dot{u}_n(y,w;m)+\mathcal{O}(n^{-1}),\quad n\to+\infty,
\label{eq:elliptic-asymptotics-ER}$$ holds uniformly on the set of $(y,w,n)$ defined by the conditions $y\in K_y$, $w\in K_w$ such that $$\begin{split}
\mathrm{dist}(\mathscr{z}_n^\bullet(y,w;m),2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z})&\ge\epsilon,\ \ \ \ \ \ \
\mathrm{dist}(\mathscr{z}_n^\circ(y,w;m),2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z})\ge\epsilon,\\
\mathrm{dist}(\mathscr{p}_n^\bullet(y,w;m),2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z})&\ge\epsilon,\ \ \ \ \ \ \
\mathrm{dist}(\mathscr{p}_n^\circ(y,w;m),2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z})\ge\epsilon.
\end{split}
\label{eq:cheese-carve-outs}$$ Under the same conditions and with the same sense of convergence, $$u_n(-(ny+w);-m)=\frac{1}{\dot{u}_n(y,w;m)}+\mathcal{O}(n^{-1}),\quad n\to+\infty,
\label{eq:elliptic-asymptotics-EL}$$ which provides asymptotics of $u_n(ny;m)$ when $y\in E_\mathrm{L}$. \[theorem:eye\]
The formula follows from with the use of the symmetry (and that $\dot{u}_n(y,w;m)$ is bounded and bounded away from zero on the indicated set, as it happens). Thus, provided that $n^{-1}x$ lies in either domain $E_\mathrm{L}$ or $E_\mathrm{R}$ and $m$ is *not* a half-integer, the rational Painlevé-III function $u_n(x;m)$ is locally approximated by a non-equilibrium elliptic function solution of the differential equation . Note that the fact that the leading term on the right-hand side of is an elliptic function follows from the first statement of Theorem \[theorem:eye\] and the fact that the integrated form admits the symmetry $(\dot{p},C,y,w)\mapsto (-\dot{p}^{-1},C,-y,-w)$.
The fact that in and we are approximating a function of a single complex variable $x=ny+w$ with a function of two independent complex variables $(y,w)$ deserves some explanation. Indeed, given $x$ there are many different choices of parameters $(y,w)$ for which $x=ny+w$, so the form of $\dot{u}_n(y,w;m)$ actually gives a family of approximations for the same quantity. The variable $w$ captures the local properties of the rational function $u_n(x;m)$; it is the scale on which $u_n(x;m)$ resembles a fixed elliptic function. On the other hand the variable $y$ captures the way that the elliptic modulus depends on the point of observation within the eye $E$ and unlike the meromorphic dependence on $w$, $\dot{u}_n(y,w;m)$ is a decidedly non-analytic function of $y$. If we approximate $u_n(x;m)$ by setting $w=0$ and letting $y$ vary, we obtain a globally accurate (on $K_y$) approximation that is unfortunately not analytic in $y$. However if we fix $y\in E_\mathrm{R}$ and let $w$ vary, we obtain a locally accurate ($w\in K_w$, so $x-ny=w=\mathcal{O}(1)$ as $n\to+\infty$) approximation that is an exact elliptic function depending only parametrically on $y$. \[remark:two-vars\]
If in any of the conditions we put $\epsilon=0$, then the corresponding phase agrees with a point of the lattice $2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}$ and the associated factor in the definition of $\dot{u}_n(y,w;m)$ vanishes. For $\epsilon>0$, each condition in defines a “swiss-cheese”-like region in the variables $(y,w)$ given $n\in\mathbb{Z}_{\ge 0}$ and $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ with holes centered at points corresponding to lattice points. In fact, if $y\in E_\mathrm{R}$ is also fixed, then the lattice $2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}$ is a uniform lattice and each of the conditions in omits from the complex $w$-plane the union of disks of radius $\epsilon$ centered at the lattice points. On the other hand, if instead it is $w\in\mathbb{C}$ that is fixed, then each of the conditions omits from the complex $y$-plane neighborhoods of diameter proportional to $\epsilon n^{-1}$ containing the points in a set that can be roughly characterized as a curvilinear grid of spacing proportional to $n^{-1}$.
Fix $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ and a compact set $K_y\subset E_\mathrm{R}$. If $\{y_n\}_{n=N}^\infty\subset K_y$ is a sequence such that $\mathcal{Z}_n^\bullet(y_n,0;m)=0$ for $n=N, N+1,\dots$ (or such that $\mathcal{Z}_n^\circ(y_n,0;m)=0$ for $n=N,N+1,\dots$), then for each sufficiently small $\epsilon>0$ there is exactly one simple zero, and possibly a group of an equal number of additional zeros and poles, of $u_n(ny;m)$ within $|y-y_n|<\epsilon n^{-1}$ for $n$ sufficiently large. Likewise, if $\{y_n\}_{n=N}^\infty\subset K_y$ is a sequence such that $\mathcal{P}_n^\bullet(y_n,0;m)=0$ for $n=N, N+1,\dots$ (or such that $\mathcal{P}_n^\circ(y_n,0;m)=0$ for $n=N,N+1,\dots$), then for each sufficiently small $\epsilon>0$ there is exactly one simple pole, and possibly a group of an equal number of additional zeros and poles, of $u_n(ny;m)$ within $|y-y_n|<\epsilon n^{-1}$ for $n$ sufficiently large. \[corollary:eye-zeros-and-poles:better\]
The proof of this result depends on Theorem \[theorem:eye\] and some additional technical properties of the zeros of the factors in the formula and will be given in Section \[sec:properties-of-udot-elliptic\]. The proof is based on an index argument, which computes the net number of zeros over poles within a small disk. For this reason, we cannot rule out the possible attraction of one or more pole-zero pairs of the rational function $u_n(x;m)$, in excess of a simple zero (or pole), toward a given zero (or singularity) of the approximating function. However, we do not observe any such “excess pairing” in practice. One approach to ruling out any excess pairing would be to compare against precise counts of the zeros and poles of $u_n(x;m)$ as documented in [@ClarksonLL16]. However, such a comparison would require accurate approximations in domains that completely cover the eye $E$ without overlaps. In this paper we avoid analyzing $u_n(x;m)$ near the origin, the corners $y=\pm\tfrac{1}{2}{\mathrm{i}}$, and the “eyebrows” (except in the special case $m\in\mathbb{Z}+\tfrac{1}{2}$; see below). These are projects for the future. Although for these reasons there remains some ambiguity about the distribution of poles and zeros of the rational function $u_n(x;m)$, our analysis gives very detailed information about the distribution of singularities and zeros of the approximation $\dot{u}_n(y,w;m)$. In particular, we have the following.
Let $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$. There is a continuous function $\rho:E_\mathrm{R}\to\mathbb{R}_+$, $\rho\in L^1_\mathrm{loc}(E_\mathrm{R})$, such that for any compact set $K\subset E_\mathrm{R}$, $$\lim_{n\to+\infty}\frac{1}{n^2}\#\{y\in K,\;\dot{u}_n(y,0;m)=0\}=\lim_{n\to+\infty}\frac{1}{n^2}\#\{y\in K,\;\dot{u}_n(y,0;m)=\infty\}=\int_K\rho(y)\,{\mathrm{d}}A(y),
\label{eq:density-integral}$$ where ${\mathrm{d}}A(y)$ denotes Lebesgue measure in the $y$-plane. The density $\rho$ is independent of $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ and satisfies $\rho(y)\to 0$ as $y\to\partial E_\mathrm{R}\setminus\{0\}$ and $\rho(r{\mathrm{e}}^{{\mathrm{i}}\theta})=h(\theta)r^{-1}+o(r^{-1})$ as $r\downarrow 0$ for some function $h:(-\pi/2,\pi/2)\to\mathbb{R}_+$. \[theorem:density\]
We would expect that the same statement holds with $\dot{u}_n(y,0;m)$ replaced by $u_n(ny;m)$, but this would require ruling out the excess pairing phenomenon mentioned above. The density function $\rho(y)$ is defined in below, and the proof of Theorem \[theorem:density\] is given in Section \[sec:properties-of-udot-elliptic\]. Although the proof of Theorem \[theorem:density\] does not allow us to consider sets $K$ that depend on $n$ in any serious way, the assumtion that holds when $K$ is the disk of radius $n^{-2}$ centered at the origin leads to the prediction that this disk contains $\mathcal{O}(1)$ zeros/singularities of $\dot{u}_n(y,0;m)$ consistent with the empirical observation that the smallest zeros and poles of $u_n(x;m)$ scale like $n^{-1}$ in the $x$-plane [@BothnerMS18].
While the asymptotic approximations of the rational Painlevé-III function $u_n(x;m)$ for $n^{-1}x\in E_\mathrm{L}\cup E_\mathrm{R}$ are much more complicated than the simple formula ${\mathrm{i}}{{p}}(n^{-1}x)$ valid for $n^{-1}x\in\mathbb{C}\setminus E$, they are easily implemented numerically, once the necessary ingredients developed as part of the proof of Theorem \[theorem:eye\] are incorporated. To quantitatively illustrate the accuracy of the approximations described in Theorems \[theorem:outside\] and \[theorem:eye\], we compare $u_n(x;m)$ with its approximations for $x$ restricted to a real interval that bisects $E$ in Figures \[fig:RealPlots-m0\]–\[fig:RealPlots-miOver5\].
![Comparison of $u_n(ny;m)$ (blue curve) with its approximations over the interval $-0.5<y<0.5$ for $m=0$ with $n=10$ (left) and $n=20$ (right). The points where this interval intersects $\partial E$ are shown with vertical gray lines. The approximation $\dot{u}_n(y,0;m)$ of Theorem \[theorem:eye\] is plotted in between the gray lines with black broken curves. The dotted curve is the analytic continuation into $E$ from the right of the outer approximation ${\mathrm{i}}{{p}}(y)$ described in Theorem \[theorem:outside\]. Likewise, the dash/dotted curve is the meromorphic continuation into $E$ from the left of the same outer approximation.[]{data-label="fig:RealPlots-m0"}](RealPlot-n10-m0-R.pdf "fig:"){width="49.00000%"}![Comparison of $u_n(ny;m)$ (blue curve) with its approximations over the interval $-0.5<y<0.5$ for $m=0$ with $n=10$ (left) and $n=20$ (right). The points where this interval intersects $\partial E$ are shown with vertical gray lines. The approximation $\dot{u}_n(y,0;m)$ of Theorem \[theorem:eye\] is plotted in between the gray lines with black broken curves. The dotted curve is the analytic continuation into $E$ from the right of the outer approximation ${\mathrm{i}}{{p}}(y)$ described in Theorem \[theorem:outside\]. Likewise, the dash/dotted curve is the meromorphic continuation into $E$ from the left of the same outer approximation.[]{data-label="fig:RealPlots-m0"}](RealPlot-n20-m0-R.pdf "fig:"){width="49.00000%"}
![As in Figure \[fig:RealPlots-m0\] but for $m=1$.[]{data-label="fig:RealPlots-m1"}](RealPlot-n10-m1-R.pdf "fig:"){width="49.00000%"}![As in Figure \[fig:RealPlots-m0\] but for $m=1$.[]{data-label="fig:RealPlots-m1"}](RealPlot-n20-m1-R.pdf "fig:"){width="49.00000%"}
![As in Figures \[fig:RealPlots-m0\]–\[fig:RealPlots-m1\] but for $m=\tfrac{1}{5}{\mathrm{i}}$. Here the top row compares the real parts and the bottom row compares the imaginary parts (the graph of $\mathrm{Im}(u_n(ny;m))$ is shown with a brown curve).[]{data-label="fig:RealPlots-miOver5"}](RealPlot-n10-miOver5-R.pdf "fig:"){width="49.00000%"}![As in Figures \[fig:RealPlots-m0\]–\[fig:RealPlots-m1\] but for $m=\tfrac{1}{5}{\mathrm{i}}$. Here the top row compares the real parts and the bottom row compares the imaginary parts (the graph of $\mathrm{Im}(u_n(ny;m))$ is shown with a brown curve).[]{data-label="fig:RealPlots-miOver5"}](RealPlot-n20-miOver5-R.pdf "fig:"){width="49.00000%"}\
![As in Figures \[fig:RealPlots-m0\]–\[fig:RealPlots-m1\] but for $m=\tfrac{1}{5}{\mathrm{i}}$. Here the top row compares the real parts and the bottom row compares the imaginary parts (the graph of $\mathrm{Im}(u_n(ny;m))$ is shown with a brown curve).[]{data-label="fig:RealPlots-miOver5"}](RealPlot-n10-miOver5-I.pdf "fig:"){width="49.00000%"}![As in Figures \[fig:RealPlots-m0\]–\[fig:RealPlots-m1\] but for $m=\tfrac{1}{5}{\mathrm{i}}$. Here the top row compares the real parts and the bottom row compares the imaginary parts (the graph of $\mathrm{Im}(u_n(ny;m))$ is shown with a brown curve).[]{data-label="fig:RealPlots-miOver5"}](RealPlot-n20-miOver5-I.pdf "fig:"){width="49.00000%"}
In these figures, we found it compelling to plot the approximate formula ${\mathrm{i}}{{p}}(y)$ of Theorem \[theorem:outside\] continued into the eye $E$ from the left and right, even though we have no basis for comparing the graphs of these (reciprocal) continuations with that of $u_n(ny;m)$ when $y\in E$. Indeed, in some situations these graphs appear to form quite accurate upper or lower envelopes of the wild modulated elliptic oscillations of $u_n(ny;m)$ that occur when $y\in E$ and that are captured with locally uniform accuracy by $\dot{u}_n(y,0;m)$. We have no explanation for these somewhat imprecise observations, but we find them interesting and note that similar phenomena occur for the rational solutions of the Painlevé-II equation (also without explanation) as was noted in [@BuckinghamM14].
Now, we go into the complex $y$-plane where we can illustrate both the shape of the eye $E$ and the phenomenon of attraction of poles and zeros of $u_n(ny;m)$ to the left ($E_\mathrm{L}$) and right ($E_\mathrm{R}$) halves. In these figures, the zeros and poles of the rational Painlevé function $u_n(ny;m)$ are plotted with the following convention (as in our earlier paper [@BothnerMS18]):
- Zeros of $u_n(x;m)$ that are also zeros of $s_n(x;m-1)$: blue filled dots.
- Zeros of $u_n(x;m)$ that are also zeros of $s_{n-1}(x;m)$: blue unfilled dots.
- Poles of $u_n(x;m)$ that are also zeros of $s_n(x;m)$: red filled dots.
- Poles of $u_n(x;m)$ that are also zeros of $s_{n-1}(x;m-1)$: red unfilled dots.
In addition to displaying the overall attraction of the poles and zeros to the eye domain $E$, the plots in Figures \[fig:m0-ZerosPlus\]–\[fig:m1Over4-PolesMinus\] are also intended to demonstrate the remarkable accuracy of the approximation of Theorem \[theorem:eye\] in capturing the locations of individual poles and zeros as described in Corollary \[corollary:eye-zeros-and-poles:better\]. As described in Section \[sec:properties-of-udot-elliptic\] below, each of the four factors in the fraction on the right-hand side of has zeros that may be characterized as the intersection points of integral level curves of two different functions (see and below) defined on $E_\mathrm{R}$ (and via the symmetry , $E_\mathrm{L}$). We plot the families of level curves for each of the four factors in separate figures in order to demonstrate another phenomenon that is evident but for which we have no good explanation: the zeros of the separate factors in the approximation $\dot{u}_n(y,0;m)$ as defined by appear to correspond precisely to the actual zeros of the four polynomial factors in the formula for the rational Painlevé-III function $u_n(ny;m)$. This coincidence is what motivates the superscript notation ($\bullet$ versus $\circ$) on the four factors in ; the zeros of the factors with superscript $\bullet$ (resp., $\circ$) apparently correspond in the limit $n\to+\infty$ to filled (resp., unfilled) dots.
![The black curves including the vertical segment form the boundary of the left ($E_\mathrm{L}$) and right ($E_\mathrm{R}$) halves of the eye $E$. The light blue curves are $\alpha_n^{0,+}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{0,+}(y,0,m)\in\mathbb{Z}$ (dotted) plotted in the $y$-plane; see for definitions of these functions. These plots are for $m=0$ and $n=5$ (left), $n=10$ (center), and $n=20$ (right). The blue/red dots are the actual zeros/poles of $u_n(ny;m)$ (filled for zeros of $s_n$ and unfilled for zeros of $s_{n-1}$). Note how the unfilled blue dots are attracted toward the intersections of the curves, which are the zeros of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{Z}_n^\circ(y,0;m)$.[]{data-label="fig:m0-ZerosPlus"}](n5m0-ZerosPlus.pdf "fig:"){width="30.00000%"}![The black curves including the vertical segment form the boundary of the left ($E_\mathrm{L}$) and right ($E_\mathrm{R}$) halves of the eye $E$. The light blue curves are $\alpha_n^{0,+}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{0,+}(y,0,m)\in\mathbb{Z}$ (dotted) plotted in the $y$-plane; see for definitions of these functions. These plots are for $m=0$ and $n=5$ (left), $n=10$ (center), and $n=20$ (right). The blue/red dots are the actual zeros/poles of $u_n(ny;m)$ (filled for zeros of $s_n$ and unfilled for zeros of $s_{n-1}$). Note how the unfilled blue dots are attracted toward the intersections of the curves, which are the zeros of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{Z}_n^\circ(y,0;m)$.[]{data-label="fig:m0-ZerosPlus"}](n10m0-ZerosPlus.pdf "fig:"){width="30.00000%"}![The black curves including the vertical segment form the boundary of the left ($E_\mathrm{L}$) and right ($E_\mathrm{R}$) halves of the eye $E$. The light blue curves are $\alpha_n^{0,+}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{0,+}(y,0,m)\in\mathbb{Z}$ (dotted) plotted in the $y$-plane; see for definitions of these functions. These plots are for $m=0$ and $n=5$ (left), $n=10$ (center), and $n=20$ (right). The blue/red dots are the actual zeros/poles of $u_n(ny;m)$ (filled for zeros of $s_n$ and unfilled for zeros of $s_{n-1}$). Note how the unfilled blue dots are attracted toward the intersections of the curves, which are the zeros of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{Z}_n^\circ(y,0;m)$.[]{data-label="fig:m0-ZerosPlus"}](n20m0-ZerosPlus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-ZerosPlus\] but here the light blue curves are $\alpha_n^{0,-}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{0,-}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the filled blue dots are attracted toward the intersections of the curves, which are now the zeros of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{Z}_n^\bullet(y,0;m)$.[]{data-label="fig:m0-ZerosMinus"}](n5m0-ZerosMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but here the light blue curves are $\alpha_n^{0,-}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{0,-}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the filled blue dots are attracted toward the intersections of the curves, which are now the zeros of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{Z}_n^\bullet(y,0;m)$.[]{data-label="fig:m0-ZerosMinus"}](n10m0-ZerosMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but here the light blue curves are $\alpha_n^{0,-}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{0,-}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the filled blue dots are attracted toward the intersections of the curves, which are now the zeros of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{Z}_n^\bullet(y,0;m)$.[]{data-label="fig:m0-ZerosMinus"}](n20m0-ZerosMinus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-ZerosPlus\] but here the light red curves are $\alpha_n^{\infty,+}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{\infty,+}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the filled red dots are attracted toward the intersections of the curves, which are the singularities of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{P}_n^\bullet(y,0;m)$.[]{data-label="fig:m0-PolesPlus"}](n5m0-PolesPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but here the light red curves are $\alpha_n^{\infty,+}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{\infty,+}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the filled red dots are attracted toward the intersections of the curves, which are the singularities of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{P}_n^\bullet(y,0;m)$.[]{data-label="fig:m0-PolesPlus"}](n10m0-PolesPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but here the light red curves are $\alpha_n^{\infty,+}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{\infty,+}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the filled red dots are attracted toward the intersections of the curves, which are the singularities of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{P}_n^\bullet(y,0;m)$.[]{data-label="fig:m0-PolesPlus"}](n20m0-PolesPlus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-ZerosPlus\] but here the light red curves are $\alpha_n^{\infty,-}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{\infty,-}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the unfilled red dots are attracted toward the intersections of the curves, which are now the singularities of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{P}_n^\circ(y,0;m)$.[]{data-label="fig:m0-PolesMinus"}](n5m0-PolesMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but here the light red curves are $\alpha_n^{\infty,-}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{\infty,-}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the unfilled red dots are attracted toward the intersections of the curves, which are now the singularities of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{P}_n^\circ(y,0;m)$.[]{data-label="fig:m0-PolesMinus"}](n10m0-PolesMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but here the light red curves are $\alpha_n^{\infty,-}(y,0,m)\in\mathbb{Z}$ (solid) and $\beta_n^{\infty,-}(y,0,m)\in\mathbb{Z}$ (dotted); see for definitions of these functions. Note how the unfilled red dots are attracted toward the intersections of the curves, which are now the singularities of $\dot{u}_n(y,0;m)$ arising from roots of $\mathcal{P}_n^\circ(y,0;m)$.[]{data-label="fig:m0-PolesMinus"}](n20m0-PolesMinus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-ZerosPlus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-ZerosPlus"}](n5m4iOver5-ZerosPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-ZerosPlus"}](n10m4iOver5-ZerosPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-ZerosPlus"}](n20m4iOver5-ZerosPlus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-ZerosMinus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-ZerosMinus"}](n5m4iOver5-ZerosMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosMinus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-ZerosMinus"}](n10m4iOver5-ZerosMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosMinus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-ZerosMinus"}](n20m4iOver5-ZerosMinus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-PolesPlus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-PolesPlus"}](n5m4iOver5-PolesPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesPlus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-PolesPlus"}](n10m4iOver5-PolesPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesPlus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-PolesPlus"}](n20m4iOver5-PolesPlus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-PolesMinus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-PolesMinus"}](n5m4iOver5-PolesMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesMinus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-PolesMinus"}](n10m4iOver5-PolesMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesMinus\] but for $m=\tfrac{4}{5}{\mathrm{i}}$.[]{data-label="fig:m4iOver5-PolesMinus"}](n20m4iOver5-PolesMinus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-ZerosPlus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-ZerosPlus"}](n5m1Over4-ZerosPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-ZerosPlus"}](n10m1Over4-ZerosPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosPlus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-ZerosPlus"}](n20m1Over4-ZerosPlus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-ZerosMinus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-ZerosMinus"}](n5m1Over4-ZerosMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosMinus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-ZerosMinus"}](n10m1Over4-ZerosMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-ZerosMinus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-ZerosMinus"}](n20m1Over4-ZerosMinus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-PolesPlus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-PolesPlus"}](n5m1Over4-PolesPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesPlus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-PolesPlus"}](n10m1Over4-PolesPlus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesPlus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-PolesPlus"}](n20m1Over4-PolesPlus.pdf "fig:"){width="30.00000%"}
![As in Figure \[fig:m0-PolesMinus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-PolesMinus"}](n5m1Over4-PolesMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesMinus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-PolesMinus"}](n10m1Over4-PolesMinus.pdf "fig:"){width="30.00000%"}![As in Figure \[fig:m0-PolesMinus\] but for $m=\tfrac{1}{4}$.[]{data-label="fig:m1Over4-PolesMinus"}](n20m1Over4-PolesMinus.pdf "fig:"){width="30.00000%"}
Another feature of the plots in Figures \[fig:m0-ZerosPlus\]–\[fig:m1Over4-PolesMinus\] is that only one pole or zero is evidently attracted to each crossing point of the curves, which suggests that the excess pairing phenomenon that cannot be ruled out by our index-based proof of Corollary \[corollary:eye-zeros-and-poles:better\] does in fact not occur. Finally, these plots illustrate the most important properties of the pole/zero density function $\rho(y)$ described in Theorem \[theorem:density\], namely the infinite density at the origin and the dilution of poles/zeros near the boundaries of $\partial E_\mathrm{L}$ and $\partial E_\mathrm{R}$ (which include the imaginary axis vertically bisecting $E$).
Clearly, when $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ there are many poles and zeros in the domains $E_\mathrm{L}$ and $E_\mathrm{R}$ when $n$ is large, and in this situation we say that the *eye is open*. On the other hand, the large-$n$ asymptotic behavior of $u_n(x;m)$ when $n^{-1}x$ is in a neighborhood of the eye $E$ is *completely different* than described above when $m\in\mathbb{Z}+\tfrac{1}{2}$. We refer to the closures (i.e., including endpoints) of the arcs of $\partial E_\mathrm{L}$ and $\partial E_\mathrm{R}$ in the open left and right half-planes respectively as the “eyebrows” of the eye $E$, denoting them by ${{\partial E^0_\squarellblack}}$ and ${{\partial E^\infty_\squareurblack}}$, respectively. Our first result is that, in a sense, the eye is closed when $m\in\mathbb{Z}+\tfrac{1}{2}$.
Suppose that $m=-(k+\tfrac{1}{2})$ for $k\in\mathbb{Z}_{\ge 0}$. Let $K\subset\mathbb{C}\setminus{{\partial E^\infty_\squareurblack}}$ be bounded away from ${{\partial E^\infty_\squareurblack}}$, i.e., $\mathrm{dist}(y,{{\partial E^\infty_\squareurblack}})>0$. Then $$\frac{1}{u_n(ny;m)}=\frac{1}{{\mathrm{i}}{{p^\infty_\squareurblack}}(y)} + \mathcal{O}(n^{-1}),\quad n\to+\infty,\quad y\in K,$$ where ${{p^\infty_\squareurblack}}(y)$ denotes the meromorphic continuation of ${{p}}(y)$ from a neighborhood of $y=\infty$ to the maximal domain $\mathbb{C}\setminus{{\partial E^\infty_\squareurblack}}$ as a non-vanishing function whose only singularity is a simple pole at the origin $y=0$, and the error estimate is uniform for $y\in K$. Likewise, if $m=k+\tfrac{1}{2}$ for $k\in\mathbb{Z}_{\ge 0}$ and $K\subset\mathbb{C}\setminus{{\partial E^0_\squarellblack}}$ is bounded away from ${{\partial E^0_\squarellblack}}$, then $$u_n(ny;m)={\mathrm{i}}{{p^0_\squarellblack}}(y)+\mathcal{O}(n^{-1}),\quad n\to+\infty,\quad y\in K,$$ where ${{p^0_\squarellblack}}(y)$ denotes the analytic continuation of ${{p}}(y)$ from a neighborhood of $y=\infty$ to the maximal domain $\mathbb{C}\setminus{{\partial E^0_\squarellblack}}$ as a function whose only zero is simple and lies at the origin, and the error estimate is uniform for $y\in K$. \[theorem:closed-eye-equilibrium\]
The functions ${{p^\infty_\squareurblack}}(y)$ and ${{p^0_\squarellblack}}(y)$ both agree with ${{p}}(y)$ for $y\in\mathbb{C}\setminus E$, and they are reciprocals of one another when $y\in E$. Theorem \[theorem:closed-eye-equilibrium\] is proved in Section \[sec:Half-Integer-Away-From-Edge\]. Note that this result is consistent with Theorem \[theorem:outside\], which does not require any condition on $m\in\mathbb{C}$. Moreover, it gives a far-reaching generalization of Theorem \[theorem:outside\] for the special case of $m\in\mathbb{Z}+\tfrac{1}{2}$. The uniform nature of the convergence implies that $u_n(ny;m)$ can have no poles or zeros in $K$ for sufficiently large $n$, unless the set $K$ contains the origin, in which case an index argument predicts a unique simple pole near the origin for $m=-(k+\tfrac{1}{2})$ and a unique simple zero near the origin for $m=k+\tfrac{1}{2}$. However, it is proven in [@ClarksonLL16] that there is a simple pole or zero *exactly* at the origin if $n$ is sufficiently large (given $k\in\mathbb{Z}_{\ge 0}$). Therefore, we have the following.
Suppose that $m=-(k+\tfrac{1}{2})$, $k\in\mathbb{Z}_{\ge 0}$. If $K\subset\mathbb{C}$ is bounded away from ${{\partial E^\infty_\squareurblack}}$, then $u_n(\cdot;m)$ has no zeros or poles in the set $nK$ for $n$ sufficiently large, except for a simple pole at the origin. On the other hand, if $m=k+\tfrac{1}{2}$, $k\in\mathbb{Z}_{\ge 0}$ and $K\subset\mathbb{C}$ is bounded away from ${{\partial E^0_\squarellblack}}$, then $u_n(\cdot;m)$ has no zeros or poles in the set $nK$ for $n$ sufficiently large, except for a simple zero at the origin. \[corollary:half-integer-m-outside-zeros-poles\]
This result can be combined with Theorem \[theorem:closed-eye-equilibrium\] to show immediately as in that the convergence of $u_n(ny;m)$ for $y\in K$ extends to all derivatives. Corollary \[corollary:half-integer-m-outside-zeros-poles\] also shows that if $m\in\mathbb{Z}+\tfrac{1}{2}$, all of the poles/zeros but one are attracted toward one or the other of the eyebrows as $n\to+\infty$, depending on the sign of $m$; this is what we mean when we say that the *eye is closed*. Counting arguments suggest it is reasonable that the poles and zeros should be organized near curves rather than in a two-dimensional area such as $E_\mathrm{L}\cup E_\mathrm{R}$ in this case. Indeed, in [@ClarksonLL16] it is also shown that the total number of zeros and poles of $u_n(x;m)$ scales as $n$ as $n\to+\infty$ when $m\in\mathbb{Z}+\tfrac{1}{2}$, while for $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ the number scales as $n^2$. Our methods allow for the following precise statement concerning the nature of convergence of the poles/zeros to one or the other of the eyebrows for $m\in\mathbb{Z}+\tfrac{1}{2}$. The following results refer to a “tubular neighborhood” $T$ of the eyebrow ${{\partial E^\infty_\squareurblack}}$ defined as follows: for sufficiently small positive constants $\delta_1$ and $\delta_2$, $$T=T_{\delta_1,\delta_2}:=\left\{y\in\mathbb{C}: |\arg(y)|\le\frac{\pi}{2}-\delta_1,\;|\mathrm{Re}(V({{p}}(y);y))|\le\delta_2\right\}.
\label{eq:T-define}$$ Since points on the eyebrow ${{\partial E^\infty_\squareurblack}}$ satisfy $\mathrm{Re}(V({{p}}(y));y)=0$, the set $T$ contains points on both sides of ${{\partial E^\infty_\squareurblack}}$, and the angular condition bounds the set $T$ away from the endpoints $y=\pm\tfrac{1}{2}{\mathrm{i}}$ of ${{\partial E^\infty_\squareurblack}}$. Note that $V({{p}}(y)^{-1};y)=-V({{p}}(y);y)\pmod{2\pi{\mathrm{i}}}$.
Let $m=-(\tfrac{1}{2}+k)$, $k\in\mathbb{Z}_{\ge 0}$, and let $T$ be as defined in . Then the following asymptotic formulæ hold in which the error terms are uniform on the indicated sub-domains of $T$ from which small discs of radius proportional to an arbitrarily small multiple of $n^{-1}$ centered at each zero or pole of the indicated approximation are excised:
- If $y\in T$ with $\mathrm{Re}(V({{p}}(y)^{-1};y))\le -\tfrac{1}{2}kn^{-1}\ln(n)$, then $u_n(ny;m)=\dot{u}_n+\mathcal{O}(n^{-1})$ where $\dot{u}_n$ is given explicitly by .
- For $\ell=1,\dots,k$,
- If $y\in T$ with $-\tfrac{1}{2}(k-2\ell+2)n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le -\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)$, then $u_n(ny;m)=\dot{u}_n+\mathcal{O}(n^{-1/2})$ where $\dot{u}_n$ is given explicitly by .
- If $y\in T$ with $-\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)$, then $u_n(ny;m)=\dot{u}_n+\mathcal{O}(n^{-1/2})$ where $\dot{u}_n$ is given explicitly by or .
- If $y\in T$ with $-\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell)n^{-1}\ln(n)$, then $u_n(ny;m)=\dot{u}_n+\mathcal{O}(n^{-1/2})$ where $\dot{u}_n$ is given explicitly by .
- If $y\in T$ with $\mathrm{Re}(V({{p}}(y)^{-1};y))\ge \tfrac{1}{2}kn^{-1}\ln(n)$, then $u_n(ny;m)=\dot{u}_n+\mathcal{O}(n^{-1})$ where $\dot{u}_n$ is given explicitly by .
These results imply corresponding asymptotic formulæ for $u_n(ny;m)$ if $m=\tfrac{1}{2}+k$, $k\in\mathbb{Z}_{\ge 0}$ by the exact symmetry ; in particular the eyebrow near which the asymptotics are nontrivial is then the left one, ${{\partial E^0_\squarellblack}}$. \[thm:edge-formulae\]
The inequalities on $y$ in the statement of the theorem describe a dissection of $T$ into finitely-many (depending on $k$) “layers” roughly parallel to the right eyebrow ${{\partial E^\infty_\squareurblack}}$ and overlapping at their common boundaries. The order of the layers as written in the theorem corresponds to $y$ crossing ${{\partial E^\infty_\squareurblack}}$ from inside $E$ to outside, and the “interior” layers described by the index $\ell$ are each of width proportional to $n^{-1}\ln(n)$. The approximation $\dot{u}_n$ assigned to each layer is a fractional linear (Möbius) function of $n^\beta{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}$ where the power $\beta$ and the coefficients of the linear expressions in the numerator/denominator depend on the layer. The latter coefficients are relatively slowly-varying functions of $y$ alone that are explicitly built from ${{p}}(y)$, and hence the dominant local behavior in any given layer is essentially trigonometric with respect to $y$. We wish to stress that, unlike the approximation formula whose ingredients involve implicitly-defined functions of $y\in E_\mathrm{R}$ and elements of algebraic geometry, the approximation $\dot{u}_n$ in each layer is an elementary function of $V(\lambda;y)$ and ${{p}}(y)$. In particular, it is easy to check that when $y$ is in the innermost or outermost layers but bounded away from ${{\partial E^\infty_\squareurblack}}$ (the “overlap domain”), Theorem \[thm:edge-formulae\] is consistent with Theorem \[theorem:closed-eye-equilibrium\].
The analogue of Corollary \[corollary:eye-zeros-and-poles:better\] in the present context is the following.
Let $m=-(k+\tfrac{1}{2})$, $k\in\mathbb{Z}_{\ge 0}$, and let $T$ be defined as in . If $\{y_n\}_{n=N}^\infty\subset T$ is a sequence for which $y_n$ is a zero of $\dot{u}_n$ for all $n\ge N$, then for each $\epsilon>0$ sufficiently small there is exactly one simple zero, and possibly a group of an equal number of additional zeros and poles, of $u_n(ny;m)$ within $|y-y_n|<\epsilon n^{-1}$ for $n$ sufficiently large. Likewise, if $\{y_n\}_{n=N}^\infty\subset T$ is a sequence for which $y_n$ is a pole of $\dot{u}_n$ for all $n\ge N$, then for each $\epsilon>0$ sufficiently small there is exactly one simple pole, and possibly a group of an equal number of additional zeros and poles, of $u_n(ny;m)$ within $|y-y_n|<\epsilon n^{-1}$ for $n$ sufficiently large. \[corollary:eyebrow-zeros-and-poles\]
As before, we suspect that with additional work one should be able to preclude the excess pairing phenomenon, so that the poles and zeros of $u_n(x;m)$ and its approximation $\dot{u}_n$ are in one-to-one correspondence. Now in each layer of $T$, the poles and zeros of $\dot{u}_n$ are easily seen to lie exactly along certain explicit curves roughly parallel to the eyebrow.
Suppose that $m=-(\tfrac{1}{2}+k)$, $k\in\mathbb{Z}_{\ge 0}$ and let $T$ be as in . The zeros and poles of the piecewise-meromorphic approximating function $\dot{u}_n$ on $T$ lie on a system of $4k+2$ non-intersecting curves roughly parallel to the eyebrow ${{\partial E^\infty_\squareurblack}}$. From left-to-right, these are:
- a curve of poles given by
- a curve of zeros given by
- For $\ell=1,\dots,k$,
- a curve of zeros given by
- a curve of poles given by
- a curve of poles given by
- a curve of zeros given by .
Analogous results hold for the approximation to $u_n(ny;m)$ for $m=\tfrac{1}{2}+k$, $k\in\mathbb{Z}_{\ge 0}$, obtained from $\dot{u}_n$ via the symmetry ($y\mapsto -y$, $m\mapsto -m$, $\dot{u}_n\mapsto \dot{u}_n^{-1}$). \[theorem:eyebrow-curves\]
Corollary \[corollary:eyebrow-zeros-and-poles\] and Theorem \[theorem:eyebrow-curves\] are proved in Section \[sec:eyebrow-zeros-and-poles\]. To illustrate the accuracy of these results, we compare the exact locations of zeros and poles of $u_n(ny;m)$ for $m=-(k+\tfrac{1}{2})$, $k\in\mathbb{Z}_{\ge 0}$, with the curves described in Theorem \[theorem:eyebrow-curves\] in Figures \[fig:EdgeCurves-k0\]–\[fig:EdgeCurves-k2\].
![The pole (red) and zero (blue) curves of $\dot{u}$ for $k=0$ and $n=5,10,20$ from left-to-right, shown together with the actual poles (red dots) and zeros (blue dots) of $u_n(ny;-(\tfrac{1}{2}+k))$ and the eyebrow ${{\partial E^\infty_\squareurblack}}$ (black curve).[]{data-label="fig:EdgeCurves-k0"}](EdgePlot-k0-n5.pdf "fig:"){width="0.3\linewidth"}![The pole (red) and zero (blue) curves of $\dot{u}$ for $k=0$ and $n=5,10,20$ from left-to-right, shown together with the actual poles (red dots) and zeros (blue dots) of $u_n(ny;-(\tfrac{1}{2}+k))$ and the eyebrow ${{\partial E^\infty_\squareurblack}}$ (black curve).[]{data-label="fig:EdgeCurves-k0"}](EdgePlot-k0-n10.pdf "fig:"){width="0.3\linewidth"}![The pole (red) and zero (blue) curves of $\dot{u}$ for $k=0$ and $n=5,10,20$ from left-to-right, shown together with the actual poles (red dots) and zeros (blue dots) of $u_n(ny;-(\tfrac{1}{2}+k))$ and the eyebrow ${{\partial E^\infty_\squareurblack}}$ (black curve).[]{data-label="fig:EdgeCurves-k0"}](EdgePlot-k0-n20.pdf "fig:"){width="0.3\linewidth"}
![As in Figure \[fig:EdgeCurves-k0\] but for $k=1$.[]{data-label="fig:EdgeCurves-k1"}](EdgePlot-k1-n5.pdf "fig:"){width="0.3\linewidth"}![As in Figure \[fig:EdgeCurves-k0\] but for $k=1$.[]{data-label="fig:EdgeCurves-k1"}](EdgePlot-k1-n10.pdf "fig:"){width="0.3\linewidth"}![As in Figure \[fig:EdgeCurves-k0\] but for $k=1$.[]{data-label="fig:EdgeCurves-k1"}](EdgePlot-k1-n20.pdf "fig:"){width="0.3\linewidth"}
![As in Figure \[fig:EdgeCurves-k0\] but for $k=2$.[]{data-label="fig:EdgeCurves-k2"}](EdgePlot-k2-n5.pdf "fig:"){width="0.3\linewidth"}![As in Figure \[fig:EdgeCurves-k0\] but for $k=2$.[]{data-label="fig:EdgeCurves-k2"}](EdgePlot-k2-n10.pdf "fig:"){width="0.3\linewidth"}![As in Figure \[fig:EdgeCurves-k0\] but for $k=2$.[]{data-label="fig:EdgeCurves-k2"}](EdgePlot-k2-n20.pdf "fig:"){width="0.3\linewidth"}
In addition to illustrating the accuracy of the approximation by $\dot{u}_n$, these figures demonstrate another phenomenon for which we do not yet have an explanation: for any given curve, the poles/zeros attracted are those contributed by exactly one of the four polynomial factors in . Furthermore, there appears again to be no excess pairing of poles and zeros.
Evidently, the large-$n$ asymptotic behavior of $u_n(x;m)$ is completely different for $m=\pm (k+\tfrac{1}{2})$, $k\in\mathbb{Z}_{\ge 0}$, and for $m=\pm (k+\tfrac{1}{2})+\epsilon$, however small $\epsilon\neq 0$ is. In other words, even crude aspects of the large-$n$ asymptotic behavior of $u_n(x;m)$ for $n^{-1}x$ in a neighborhood of the eye $E$ fail to be uniformly valid with respect to the second parameter $m$ near half-integer values of the latter. Thus, given $m\in\mathbb{C}$, the eye is either open or closed in the large-$n$ limit. On the other hand, the polynomials $s_n(x;m)$ in the formula are actually polynomials in both arguments $x$ and $m$ [@ClarksonLL16], and in this sense the limits of $n\to+\infty$ and $m\to \mathbb{Z}+\tfrac{1}{2}$ do not commute. Capturing the process of the closing of the eye requires connecting $m$ with $n$ in a suitable *double-scaling limit* so that $m$ tends to a given half-integer as $n\to+\infty$. In a subsequent paper, we will show that in the right double-scaling limit, *all three types of solutions of the autonomous model equation play a role in describing $u_n(ny;m)$ as $n\to+\infty$.*
Spectral Curve and $g$-function
===============================
When $n$ is large, the exponential factors ${\mathrm{e}}^{\pm nV(\lambda;y)}$ appearing in the jump conditions – need to be balanced in general by some compensating factors that can be used to control exponential growth. We therefore introduce a “$g$-function” $g(\lambda;y)$ that is taken to be bounded and analytic in $\mathbb{C}\setminus L$ with $g(\lambda;y)\to g_\infty(y)$ as $\lambda\to\infty$ for some $g_\infty(y)$ to be determined, and we set $$\mathbf{M}_n(\lambda;y,m):={\mathrm{e}}^{ng_\infty(y)\sigma_3}\mathbf{Y}_n(\lambda;ny,m){\mathrm{e}}^{-ng(\lambda;y)\sigma_3}.
\label{eq:Y-M-g-function}$$ Thus, representing – in the general form $\mathbf{Y}_{n+}(\lambda;x,m)=\mathbf{Y}_{n-}(\lambda;x,m)\mathbf{V}(\lambda;x,m)$, we obtain the corresponding jump conditions for $\mathbf{M}_n(\lambda;y,m)$ in the form $\mathbf{M}_{n+}(\lambda;y,m)=\mathbf{M}_{n-}(\lambda;y,m){\mathrm{e}}^{ng_-(\lambda;y)\sigma_3}\mathbf{V}(\lambda;ny,m){\mathrm{e}}^{-ng_+(\lambda;y)\sigma_3}$. Noting that $${\mathrm{e}}^{ng_-(\lambda;y)\sigma_3}\mathbf{V}(\lambda;ny,m){\mathrm{e}}^{-ng_+(\lambda;y)\sigma_3}=
\begin{bmatrix}
{\mathrm{e}}^{-n(g_+(\lambda;y)-g_-(\lambda;y))}V_{11}(\lambda;ny,m) & {\mathrm{e}}^{n(g_+(\lambda;y)+g_-(\lambda;y))}V_{12}(\lambda;ny,m)\\
{\mathrm{e}}^{-n(g_+(\lambda;y)+g_-(\lambda;y))}V_{21}(\lambda;ny,m) & {\mathrm{e}}^{n(g_+(\lambda;y)-g_-(\lambda;y))}V_{22}(\lambda;ny,m)\end{bmatrix}$$ we place the following conditions on $g$. We want $g$ to be chosen so that $L$ can be deformed and then split into several arcs along each of which one of the following alternatives holds (recall that $V$ is defined by ):
- $g_+(\lambda;y)-g_-(\lambda;y)={\mathrm{i}}K$ where $K\in\mathbb{R}$ is constant (implying that $g'(\lambda;y)$ has no jump discontinuity across the arc), and $\mathrm{Re}(2g_\pm(\lambda;y)-V(\lambda;y))<0$, or
- $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)={\mathrm{i}}K$ where $K\in\mathbb{R}$ is constant (implying that $g_+'(\lambda;y)+g_-'(\lambda;y)-V'(\lambda;y)=0$ holds along the arc), while $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$ on both sides of the arc, or
- $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)={\mathrm{i}}K$ where $K\in\mathbb{R}$ is constant (implying that $g_+'(\lambda;y)+g_-'(\lambda;y)-V'(\lambda;y)=0$ holds along the arc), while $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ on both sides of the arc, or
- $g_+(\lambda;y)-g_-(\lambda;y)={\mathrm{i}}K$ where $K\in\mathbb{R}$ is constant (implying that $g'(\lambda;y)$ has no jump discontinuity across the arc), and $\mathrm{Re}(2g_\pm(\lambda;y)-V(\lambda;y))>0$.
The real constant $K$ will generally be different in each maximal arc.
The spectral curve and its degenerations
----------------------------------------
If we assume that $g'(\lambda;y)$ has a finite number of arcs of discontinuity along $L\setminus\{0\}$, then obviously $(g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y))_+=(g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y))_-$ except along these arcs. Along the arcs of discontinuity where instead the condition $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)={\mathrm{i}}K$ holds, by differentiation along the arc we have $(g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y))_+=-(g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y))_-$. It follows that $(g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y))^2$ is an analytic function of $\lambda$ except at $\lambda=0$, which is the only singularity of $V'(\lambda;y)$. Now since $g'(\lambda;y)=\mathcal{O}(\lambda^{-2})$ as $\lambda\to\infty$ and $g'(\lambda;y)=\mathcal{O}(1)$ as $\lambda\to 0$, it follows that $$g'(\lambda;y)-\frac{1}{2}V'(\lambda;y)=\begin{cases} \displaystyle \frac{{\mathrm{i}}y}{2}\lambda^{-2} + \frac{1}{2}\lambda^{-1} + \mathcal{O}(1),&\quad\lambda\to 0\smallskip\\
\displaystyle \frac{{\mathrm{i}}y}{2} + \frac{1}{2}\lambda^{-1} + \mathcal{O}(\lambda^{-2}),&\quad\lambda\to\infty
\end{cases},
\label{eq:gprimeminushalfVprime-asymp}$$ and hence if $y\neq 0$, $$\left(g'(\lambda;y)-\frac{1}{2}V'(\lambda;y)\right)^2 = \begin{cases}
\displaystyle -\frac{y^2}{4}\lambda^{-4} + \frac{{\mathrm{i}}y}{2}\lambda^{-3}+\mathcal{O}(\lambda^{-2}),&\quad
\lambda\to 0\smallskip\\
\displaystyle -\frac{y^2}{4}+\frac{{\mathrm{i}}y}{2}\lambda^{-1}+\mathcal{O}(\lambda^{-2}),&\quad\lambda\to\infty
\end{cases}.
\label{eq:gprimeminushalfVprime-squared-asymp}$$
Therefore, if $y=0$, Liouville’s theorem shows that $$g'(\lambda;0)-\frac{1}{2}V'(\lambda;0)=\frac{1}{2}\lambda^{-1},$$ while if $y\neq 0$ we necessarily have that $$\left(g'(\lambda;y)-\frac{1}{2}V'(\lambda;y)\right)^2=\frac{1}{\lambda^4}P(\lambda;y,C),
\label{eq:tildeP4}$$ where $P(\cdot;y,C)$ is the quartic polynomial defined by and it only remains to determine $C$. Since the zero locus of $P(\lambda;y,C)$ is obviously symmetric with respect to the involution $\lambda\mapsto \lambda^{-1}$, the following configurations for $P(\lambda;y,C)$ include all possibilities, given that $y\neq 0$:
- All four roots coincide, in which case the four-fold root must lie at either $\lambda=1$ or $\lambda=-1$, i.e., $P(\lambda;y,C)=-\tfrac{1}{4}y^2(\lambda\mp 1)^4 = -\tfrac{1}{4}y^2\lambda^4\pm y^2\lambda^3 -\tfrac{3}{2}y^2\lambda^2 \pm y^2\lambda -\tfrac{1}{4}y^2$. Comparing with , we see that this situation occurs only if $y=\pm\tfrac{1}{2}{\mathrm{i}}$, and then only if also $C=-\tfrac{3}{2}y^2=\tfrac{3}{8}$. In this case, since $P(\lambda;y,C)$ is a perfect square, we have either $g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y)=\tfrac{1}{2}{\mathrm{i}}y(1\mp\lambda^{-1})^2=\tfrac{1}{2}{\mathrm{i}}y(1\mp 2\lambda^{-1}+\lambda^{-2})$ or $g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y)=-\tfrac{1}{2}{\mathrm{i}}y(1\mp\lambda^{-1})^2=-\tfrac{1}{2}{\mathrm{i}}y(1\mp 2\lambda^{-1}+\lambda^{-2})$. Since $g'(\lambda;y)=\mathcal{O}(\lambda^{-2})$ as $\lambda\to\infty$, only the former is consistent with , and then we see that in fact $g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y)=-\tfrac{1}{2}V'(\lambda;y)$, i.e., $g'(\lambda;y)=0$ in this case, which implies that $g(\lambda;y)=g_\infty(y)$. This case turns out to be relevant exactly for $y=\pm\tfrac{1}{2}{\mathrm{i}}$.
- There are two double roots that are exchanged[^3] by the involution, in which case there is a number ${{p}}\neq \pm 1$ such that $P(\lambda;y;C)=-\tfrac{1}{4}y^2(\lambda-{{p}})^2(\lambda-{{p}}^{-1})^2 =
-\tfrac{1}{4}y^2\lambda^4+\tfrac{1}{2}y^2({{p}}+{{p}}^{-1})\lambda^3 -\tfrac{1}{4}y^2({{p}}^2+4+{{p}}^{-2})\lambda^2 +\tfrac{1}{2}y^2({{p}}+{{p}}^{-1})\lambda -\tfrac{1}{4}y^2$. Comparing with shows that this is possible for all $y\neq 0$, provided that ${{p}}$ is determined as a function of $y$ up to reciprocation by ${{p}}+{{p}}^{-1}={\mathrm{i}}y^{-1}$ and then $C$ is given the value $C=-\tfrac{1}{4}y^2({{p}}^2+4+{{p}}^{-2})=-\tfrac{1}{4}(2y^2-1)$. In this case, $P(\lambda;y,C)$ is again a perfect square and hence either $g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y)=\tfrac{1}{2}{\mathrm{i}}y(1-{{p}}\lambda^{-1})(1-{{p}}^{-1}\lambda^{-1})$ or $g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y)=-\tfrac{1}{2}{\mathrm{i}}y(1-{{p}}\lambda^{-1})(1-{{p}}^{-1}\lambda^{-1})$. Only the former is consistent with given that $g'(\lambda;y)=\mathcal{O}(\lambda^{-2})$ as $\lambda\to\infty$ and again we deduce that $g'(\lambda;y)=0$ and hence also $g(\lambda;y)=g_\infty(y)$. This turns out to be the case corresponding to $y\in\mathbb{C}\setminus E$.
- There is one double root and two simple roots, with the double root being fixed by the involution and hence occurring at $\lambda=\pm 1$ and the simple roots being permuted by the involution and hence being given by $\lambda=\lambda_0$ and $\lambda=\lambda_0^{-1}$ for some $\lambda_0\neq\pm 1$. Thus $P(\lambda;y,C)=-\tfrac{1}{4}y^2(\lambda\mp 1)^2(\lambda-\lambda_0)(\lambda-\lambda_0^{-1})=-\tfrac{1}{4}y^2\lambda^4 +\tfrac{1}{4}y^2(\lambda_0\pm 2+\lambda_0^{-1})\lambda^3-\tfrac{1}{2}y^2(1\pm(\lambda_0+\lambda_0^{-1}))\lambda^2 +\tfrac{1}{4}y^2(\lambda_0\pm 2+\lambda_0^{-1})\lambda-\tfrac{1}{4}y^2$. Comparing with shows that this configuration is possible for all $y\neq 0$, provided that $\lambda_0$ is determined up to reciprocation by $\lambda_0+\lambda_0^{-1}=2{\mathrm{i}}y^{-1}\mp 2$ and that $C$ is assigned the value $C=-\tfrac{1}{2}y^2(1\pm (\lambda_0+\lambda_0^{-1}))=\tfrac{1}{2}y^2\mp {\mathrm{i}}y$. This case turns out to be relevant only when $y\in (E\cap{\mathrm{i}}\mathbb{R})\setminus\{\pm\tfrac{1}{2}{\mathrm{i}}\}$.
- There are four simple roots, none of which equal[^4] $1$ or $-1$, in which case for some $\lambda_0$ and $\lambda_1$ with $\lambda_0^2\neq 1$, $\lambda_1^2\neq 1$, $\lambda_1\neq\lambda_0$ and $\lambda_1\neq\lambda_0^{-1}$, we have $P(\lambda;y,C)=-\tfrac{1}{4}y^2(\lambda-\lambda_0)(\lambda-\lambda_0^{-1})(\lambda-\lambda_1)(\lambda-\lambda_1^{-1})=-\tfrac{1}{4}y^2\lambda^4+\tfrac{1}{4}y^2(\lambda_0+\lambda_0^{-1}+\lambda_1+\lambda_1^{-1})\lambda^3-\tfrac{1}{4}y^2((\lambda_0+\lambda_0^{-1})(\lambda_1+\lambda_1^{-1})+2)\lambda^2 + \tfrac{1}{4}y^2(\lambda_0+\lambda_0^{-1}+\lambda_1+\lambda_1^{-1})\lambda-\tfrac{1}{4}y^2$. Comparing with shows that this case is possible for all $y\neq 0$ with arbitrary $C$, and that then $\lambda_0$ and $\lambda_1$ are determined up to reciprocation and exchange by the identities $\lambda_0+\lambda_0^{-1}+\lambda_1+\lambda_1^{-1}=2{\mathrm{i}}y^{-1}$ and $(\lambda_0+\lambda_0^{-1})(\lambda_1+\lambda_1^{-1})=-2-4Cy^{-2}$. This turns out to be the case for $y\in E_\mathrm{L}\cup E_\mathrm{R}$.
Note that in either of the cases that $P(\lambda;y,C)$ is not a perfect square it is necessary to take care in placing the branch cuts of the square root to obtain $g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y)$ from $(g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y))^2=\lambda^{-4}P(\lambda;y,C)$ in order that the asymptotic relations hold rather than just .
Boutroux integral conditions {#sec:Boutroux}
----------------------------
In order to ensure that the constant $K$ associated with each distinguished arc of $L$ is real, it is necessary in the above case (iv) to impose further conditions. Given $y$ and $C$ such that this is the case, let $\Gamma=\{(\lambda,\mu):\,\mu^2=\lambda^{-4}P(\lambda;y,C)\}$ be the genus-$1$ Riemann surface or algebraic variety associated with the equation $\mu^2=\lambda^{-4}P(\lambda;y,C)$ in $\mathbb{C}^2$ with coordinates $(\lambda,\mu)$. Let $(\mathfrak{a},\mathfrak{b})$ be a canonical homology basis on $\Gamma$ and take concrete representatives that do not pass through the preimages on $\Gamma$ of each of the two points $\lambda=0$ or $\lambda=\infty$. Then we impose the *Boutroux conditions* $$\mathfrak{B}_\mathfrak{a}(u,v;y):=\mathrm{Re}\left(\oint_\mathfrak{a}\mu\,{\mathrm{d}}\lambda\right)=0\quad\text{and}\quad
\mathfrak{B}_\mathfrak{b}(u,v;y):=\mathrm{Re}\left(\oint_\mathfrak{b}\mu\,{\mathrm{d}}\lambda\right)=0,
\label{eq:Boutroux}$$ where $C=u+{\mathrm{i}}v$, i.e., $u:=\mathrm{Re}(C)$ and $v:=\mathrm{Im}(C)$. It follows from that the differential $\mu\,{\mathrm{d}}\lambda$ has real residues at the two points of $\Gamma$ over $\lambda=0$ and the two points over $\lambda=\infty$; therefore taken together the conditions do not depend on the choice of homology basis. We expect that the two real conditions should determine $u$ and $v$ as functions of $y\in\mathbb{C}$. Differentiation of the algebraic identity relating $\mu$ and $\lambda$ gives $$2\mu\frac{\partial\mu}{\partial u}=\frac{1}{\lambda^2}\quad\text{and}\quad 2\mu\frac{\partial\mu}{\partial v} = \frac{{\mathrm{i}}}{\lambda^2}$$ from which it follows (since the paths $\mathfrak{a}$ and $\mathfrak{b}$ may be locally taken to be independent of $y$ and $C$) that $$\frac{\partial\mathfrak{B}_\mathfrak{a,b}}{\partial u}(u,v;y) = \frac{1}{2}\mathrm{Re}\left(\oint_{\mathfrak{a},\mathfrak{b}}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right)\quad\text{and}\quad
\frac{\partial\mathfrak{B}_\mathfrak{a,b}}{\partial v}(u,v;y)=-\frac{1}{2}\mathrm{Im}\left(\oint_{\mathfrak{a},\mathfrak{b}}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right).$$ Therefore, the Jacobian determinant of the equations equals $$\begin{split}
\det\left(\frac{\partial(\mathfrak{B}_\mathfrak{a},\mathfrak{B}_\mathfrak{b})}{\partial (u,v)}\right) &= -\frac{1}{4}\mathrm{Re}\left(\oint_\mathfrak{a}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right)\mathrm{Im}\left(\oint_\mathfrak{b}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right) +\frac{1}{4}
\mathrm{Re}\left(\oint_\mathfrak{b}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right)\mathrm{Im}\left(\oint_\mathfrak{a}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right)\\
&=\frac{1}{4}\mathrm{Im}\left(\left[\oint_\mathfrak{a}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right]\left[\oint_\mathfrak{b}\frac{{\mathrm{d}}\lambda}{\mu\lambda^2}\right]^*\right).
\end{split}
\label{eq:Jacobian}$$ Noting that $\mu^{-1}\lambda^{-2}{\mathrm{d}}\lambda$ is a nonzero differential spanning the (one-dimensional) vector space of holomorphic differentials on $\Gamma$, it follows from [@Dubrovin81 Chapter II, Corollary 1] that the above Jacobian is strictly negative under the assumption that the four roots of $P(\lambda;y,C)$ are distinct. Thus, an application of the implicit function theorem allows us to extend any solution of the integral conditions for which $P(\lambda;y_0,u_0+{\mathrm{i}}v_0)$ has distinct roots to a neighborhood of $y_0$ on which $u$ and $v$ are smooth real-valued functions of $y$ satisfying $u(y_0)=u_0$ and $v(y_0)=v_0$. In fact, one can show that the Jacobian determinant *blows up* as the spectral curve degenerates, and it is in this way that the implicit function theorem ultimately fails.
Asymptotics of $u_n(ny;m)$ for $y\in \mathbb{C}\setminus E$ {#sec:outside}
===========================================================
In this section, we study Riemann-Hilbert Problem \[rhp:renormalized\] with $x=ny$ (i.e., we set $w=0$) and assume that $y$ lies in a neighborhood of $y=\infty$ to be determined.
Placement of arcs of $L$ and determination of $\partial E$ {#sec:placement-of-arcs}
----------------------------------------------------------
We first show that for $y$ sufficiently large in magnitude, we may take $C=-\tfrac{1}{4}(2y^2-1)$ and hence $P(\lambda;y,C)$ has two double roots; therefore the spectral curve is reducible leading to $g(\lambda;y)=g_\infty(y)$ for a suitable value of the latter constant. For $y$ large we take the double root ${{p}}={{p}}(y)$ satisfying ${{p}}+{{p}}^{-1}={\mathrm{i}}y^{-1}$ to be the branch for which ${{p}}(y)=-{\mathrm{i}}(1-\tfrac{1}{2}y^{-1}+\mathcal{O}(y^{-2}))$ as $y\to\infty$. Then, we choose $g_\infty(y):=\tfrac{1}{2}V({{p}}(y);y)$. Thus, $$\begin{split}
2g(\lambda;y)-V(\lambda;y)&=2g_\infty(y)-V(\lambda;y)=V({{p}}(y);y)-V(\lambda;y)\\
&=\log(\lambda)-\log({{p}}(y))+{\mathrm{i}}y(\lambda-\lambda^{-1})-{\mathrm{i}}y({{p}}(y)-{{p}}(y)^{-1})
={\mathrm{i}}y(\lambda+2{\mathrm{i}}-\lambda^{-1}) + \mathcal{O}(1),
\end{split}$$ where the $\mathcal{O}(1)$ error term applies in the limit $y\to\infty$ uniformly for $\lambda$ in compact subsets of $\mathbb{C}\setminus\{0\}$. Taking into account that $2g(\lambda;y)-V(\lambda;y)$ has a double zero at $\lambda={{p}}(y)$, one can show that if $|y|$ is sufficiently large, taking the common intersection point of all four contour arcs to be the point ${{p}}(y)$, it is possible to arrange the arcs so that $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ (resp., $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$) holds on ${{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$ (resp., on ${{L^\infty_\squarellblack}}\cup{{L^0_\squarellblack}}$), with the inequality being strict except at the intersection point $\lambda={{p}}(y)$, compare Figure \[fig:y-5-Exp-i-pi-over-4\].
![For $y=5{\mathrm{e}}^{{\mathrm{i}}\pi/4}$, the domain where $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ is shaded in red, and the domain where $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$ is shaded in blue. Left panel: the $\lambda$-plane. Right panel: the $\lambda^{-1}$-plane. The unit circle is shown with a dashed curve in each plot. Suitable contour arcs matching the scheme described in Section \[sec:analytic-representation\] including the argument increment conditions – (for one choice of the arbitrary sign in ) are also shown.[]{data-label="fig:y-5-Exp-i-pi-over-4"}](y-5-Exp-i-pi-over-4-preview.pdf)
The function ${{p}}(y)$ has an analytic continuation from the neighborhood of $y=\infty$ to the maximal domain $y\in\mathbb{C}\setminus I$, where $I$ denotes the imaginary segment connecting the two branch points $\pm\tfrac{1}{2}{\mathrm{i}}$. As $y$ is brought in from the point at infinity, it remains possible to place the arcs of the contour $L$ as described above at least until either $y$ meets the branch cut $I$ of ${{p}}(y)$ or the topology of the zero level set of $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))$ changes. The latter occurs precisely when the only other critical point $\lambda={{p}}(y)^{-1}$ moves onto the zero level set; since $\mathrm{Re}(V(\lambda^{-1};y))=-\mathrm{Re}(V(\lambda;y))$, whenever both ${{p}}(y)$ and ${{p}}(y)^{-1}$ lie on the same level of $\mathrm{Re}(V(\lambda;y))$ we necessarily have $\mathrm{Re}(V({{p}}(y);y))=0$. The set of $y\in\mathbb{C}\setminus I$ where the latter condition holds true is plotted in Figure \[fig:partialE\].
![The curves in the $y$-plane where $\mathrm{Re}(V({{p}}(y);y))=0$. The branch cut $I$ of ${{p}}(y)$ (gray line) joins the two junction points. The dots correspond to plots in subsequent figures. The domain $E$ is defined as the bounded region between the two black curves, bisected by the branch cut of ${{p}}(y)$. The sign of $\mathrm{Re}(V({{p}}(y);y))$ is indicated in each region. In particular, $E_\mathrm{R}$ (resp., $E_\mathrm{L}$) is the bounded region where $\mathrm{Re}(V({{p}}(y);y))>0$ (resp., $\mathrm{Re}(V({{p}}(y);y))<0$) holds.[]{data-label="fig:partialE"}](EyePlot-Annotated.pdf)
Because $y\in{\mathrm{i}}\mathbb{R}\setminus I$ implies that $|{{p}}(y)|=1$, it is easy to confirm that indeed $\mathrm{Re}(V({{p}}(y);y))=0$ for such $y$, see also Figure \[fig:partialE\]. The rest of the points comprise a closed curve $\partial E$ with two smooth arcs meeting at the branch points $\pm \tfrac{1}{2}{\mathrm{i}}$ and bounding the eye-shaped domain $E$ defined in Section \[sec:results\].
The following figures illustrate how the domains such as shown in Figure \[fig:y-5-Exp-i-pi-over-4\] change as the value of $y$ varies near the arcs of the curve shown in Figure \[fig:partialE\]. Figure \[fig:y-real\]
![The domain where $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ in red and where $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$ in blue for $y=0.381372$ (left column), $y=0.331372$ (middle column), and $y=0.281372$ (right column), corresponding to the green, amber, and red dots, respectively, on the real axis in Figure \[fig:partialE\]. The top row shows a neighborhood of the unit disk in the $\lambda$-plane, while the bottom row shows the exterior of the unit disk in the $\lambda^{-1}$-plane. In the plots in the right-hand column, the level curve has broken and it is no longer possible to place the contour arc ${{L^\infty_\squareurblack}}$ connecting ${{p}}(y)$ and $\infty$ completely within the red region. This phase transition, which apparently occurs only on the right edge of the domain $E$, is only relevant if the jump matrix on ${{L^\infty_\squareurblack}}$ is not the identity, i.e., if $m-\tfrac{1}{2}\notin\mathbb{Z}_{\ge 0}$. These plots show contours with $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)-2\pi=-2\pi$. The other choice $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)+2\pi=2\pi$ would also be compatible with the sign chart.[]{data-label="fig:y-real"}](y-real-plot-horizontal-annotated-preview.pdf)
concerns the three points on the real axis and Figure \[fig:y-diag\]
![As in Figure \[fig:y-real\] except for $y=0.414768{\mathrm{e}}^{-3\pi{\mathrm{i}}/4}$ (left column), $y=0.364768{\mathrm{e}}^{-3\pi{\mathrm{i}}/4}$ (middle column), and $y=0.314768{\mathrm{e}}^{-3\pi{\mathrm{i}}/4}$ (right column), corresponding to the green, amber, and red dots, respectively, on the diagonal in Figure \[fig:partialE\]. In the plots in the right-hand column, the level curve has broken and it is no longer possible to place the contour arc ${{L^0_\squarellblack}}$ connecting ${{p}}(y)$ and $0$ completely within the blue region. This phase transition, which apparently occurs only on the left edge of the domain $E$, is only relevant if the jump matrix on ${{L^0_\squarellblack}}$ is not the identity, i.e., if $-m-\tfrac{1}{2}\notin\mathbb{Z}_{\ge 0}$. These plots show contours with $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)+2\pi=\pi/2$. In this case, the other choice of $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)-2\pi$ could only be arranged by a surgery of ${{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$ that would result in contours incompatible with the sign chart.[]{data-label="fig:y-diag"}](y-diag-plot-horizontal-annotated-preview.pdf)
concerns the three points on the diagonal. Figure \[fig:y-imag\]
![As in Figure \[fig:y-real\] except for $y=-0.05+0.55{\mathrm{i}}$ (left column), $y=0.55{\mathrm{i}}$ (middle column), and $y=0.05+0.55{\mathrm{i}}$ (right column), corresponding to the three green dots near the positive imaginary axis in Figure \[fig:partialE\]. The topological change in sign chart has no effect on the placement of the contours. The configurations with $\mathrm{Re}(y)\le 0$ require the choice $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)-2\pi\approx -\pi$. On the other hand, the configuration with $\mathrm{Re}(y)>0$, although pictured here with $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)-2\pi$, admits surgery of the contours ${{L^0_\squareurblack}}$ and ${{L^\infty_\squareurblack}}$ within the domain $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ and hence the choice $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)+2\pi$ is also possible.[]{data-label="fig:y-imag"}](y-imag-plot-horizontal-annotated-preview.pdf)
shows that although there is a topological change in the level curve as $y$ crosses the imaginary axis in the exterior of $E$, this does not obstruct the placement of the contour arcs of $L$. On the other hand, the topological change that occurs when $y$ lies along the arc of $\partial E$ in the right half-plane (resp., left half-plane) only obstructs placement of the arc ${{L^\infty_\squareurblack}}$ (resp., ${{L^0_\squarellblack}}$) and therefore we write $\partial E$ as the union of two closed arcs: $\partial E={{\partial E^\infty_\squareurblack}}\cup{{\partial E^0_\squarellblack}}$. Note that the surgery allowing for a sign change $\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)-2\pi\leftrightarrow\Delta\arg(\squareurblack)=2\mathrm{Arg}(x)+2\pi$ (see Remark \[rmk:surgery\]) is compatible with the sign-chart/contour placement scheme provided that the domain $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ consists of a single component. If it consists of two components, then the contours ${{L^0_\squareurblack}}$ and ${{L^\infty_\squareurblack}}$ necessarily lie in distinct components and the surgery becomes impossible. The former holds in the exterior of $E$ for $\mathrm{Re}(y)>0$ and the latter for $\mathrm{Re}(y)\le 0$.
Parametrix construction {#sec:parametrix-for-outside}
-----------------------
Let $y$ be fixed outside of $E$, let $\delta>0$ be a fixed sufficiently small (given $y$) constant, and let $D$ denote the simply-connected neighborhood of $\lambda={{p}}(y)$ defined by the inequality $|2g(\lambda;y)-V(\lambda;y)|<\delta^2$. We will define a parametrix $\dot{\mathbf{M}}_n(\lambda;y,m)$ for $\mathbf{M}_n(\lambda;y,m)$ in by a piecewise formula: $$\dot{\mathbf{M}}_n(\lambda;y,m)=\begin{cases} \dot{\mathbf{M}}^{\mathrm{out}}(\lambda;y,m),\quad & \lambda\in\mathbb{C}\setminus(L\cup\overline{D})\\
\dot{\mathbf{M}}_n^{\mathrm{in}}(\lambda;y,m),\quad &\lambda\in D\setminus L.
\end{cases}
\label{eq:M-dot-piecewise}$$
Noting that the jump matrix for $\mathbf{M}_n(\lambda;y,m)$ converges uniformly on $L\setminus D$ (with exponential accuracy) to $\mathbb{I}$ except on ${{L^0_\squarellblack}}$, where the limit is instead $-{\mathrm{e}}^{2\pi{\mathrm{i}}m\sigma_3}$, and that $\mathbf{M}_n(\lambda;y,m){{\lambda^{-(m+1/2)\sigma_3}_\squarellblack}}$ should have a limit as $\lambda\to 0$, we define $\dot{\mathbf{M}}^{\mathrm{out}}(\lambda;y,m)$ as the following diagonal matrix: $$\dot{\mathbf{M}}^{\mathrm{out}}(\lambda;y,m):= \left[\frac{\lambda}{\lambda-{{p}}(y)}\right]^{(m+\tfrac{1}{2})\sigma_3},\ \ \lambda\in\mathbb{C}\setminus(L\cup\overline{D}),
\label{eq:M-dot-out}$$ where the branch cut is taken to be ${{L^0_\squarellblack}}$ and the branch is chosen such that the right-hand side tends to $\mathbb{I}$ as $\lambda\to\infty$. In order to define $\dot{\mathbf{M}}_n^{\mathrm{in}}(\lambda;y,m)$, we will find a certain canonical matrix function that satisfies exactly the jump conditions of $\mathbf{M}_n(\lambda;y,m)$ within the neighborhood $D$ and then we will multiply the result on the left by a matrix holomorphic in $D$ to arrange a good match with $\dot{\mathbf{M}}^{\mathrm{out}}(\lambda;y,m)$ on $\partial D$. For the first part, we introduce a conformal mapping $W:D\to\mathbb{C}$ by the following relation: $$W^2 = 2g(\lambda;y)-V(\lambda;y),\ \ \ \ \lambda\in D.
\label{eq:w-definition}$$ Because $2g(\lambda;y)-V(\lambda;y)$ is a locally analytic function vanishing precisely to second order[^5] at $\lambda={{p}}(y)$, the relation defines two different analytic functions of $\lambda$ both vanishing to first order at $\lambda={{p}}(y)$. We choose the analytic solution $W=W(\lambda;y)$ that is negative real in the direction tangent to ${{L^0_\squarellblack}}$. Then we deform the arcs of $L$ within $D$ so that in this neighborhood ${{L^0_\squarellblack}}$ and ${{L^\infty_\squarellblack}}$ correspond exactly to negative and positive real values of $W$, while ${{L^0_\squareurblack}}$ and ${{L^\infty_\squareurblack}}$ correspond exactly to negative and positive imaginary values of $W$. By definition of $D$, its image $W(D;y)$ under $W$ is the disk of radius $\delta$ centered at the origin, see Figure \[fig:Outside-Local\].
![The neighborhood $D$ and its image under the conformal mapping $W=W(\lambda)=W(\lambda;y)$.[]{data-label="fig:Outside-Local"}](Outside-Local.pdf)
Consider the matrix $\mathbf{N}_n(\lambda;y,m)$ defined in terms of $\mathbf{M}_n(\lambda;y,m)$ for $\lambda\in D$ by $\mathbf{N}_n(\lambda;y,m):=
\mathbf{M}_n(\lambda;y,m)d(\lambda;y,m)^{\sigma_3}$, where $$d(\lambda;y,m):=\begin{cases} {{\lambda^{-\tfrac{1}{2}(m+1)}_\squarellblack}}(-{\mathrm{e}}^{-{\mathrm{i}}\pi m}),&\mathrm{Im}(W(\lambda;y))>0,\; \mathrm{Re}(W(\lambda;y))\neq 0,\smallskip\\
{{\lambda^{-\tfrac{1}{2}(m+1)}_\squarellblack}},&\mathrm{Im}(W(\lambda;y))<0,\;\mathrm{Re}(W(\lambda;y))\neq 0.
\end{cases}
\label{eq:scalar-d-define}$$ Recalling the precise definition of ${{\lambda^{\pm\tfrac{1}{2}(m+1)}_\squarellblack}}$, with its cut along ${{L^\infty_\squarellblack}}\cup{{L^0_\squarellblack}}$, it follows that $d(\lambda;y,m)$ can be continued to the whole domain $D$ as an analytic nonvanishing function. The jump conditions satisfied by $\mathbf{N}_n(\lambda;y,m)$ within $D$ are then the following: $$\mathbf{N}_{n+}(\lambda;y,m)=\mathbf{N}_{n-}(\lambda;y,m)\begin{bmatrix}1 & \displaystyle -\frac{\sqrt{2\pi}}{\Gamma(\tfrac{1}{2}-m)}{\mathrm{e}}^{nW(\lambda;y)^2}\\
0 & 1\end{bmatrix},\quad \lambda\in {{L^0_\squareurblack}}\cap D,
\label{eq:N-jump-first}$$ $$\mathbf{N}_{n+}(\lambda;y,m)=\mathbf{N}_{n-}(\lambda;y,m)\begin{bmatrix}1 & \displaystyle \frac{\sqrt{2\pi}{\mathrm{e}}^{2\pi{\mathrm{i}}m}}{\Gamma(\tfrac{1}{2}-m)}{\mathrm{e}}^{nW(\lambda;y)^2}\\
0 & 1\end{bmatrix},\quad \lambda\in {{L^\infty_\squareurblack}}\cap D,$$ $$\mathbf{N}_{n+}(\lambda;y,m)=\mathbf{N}_{n-}(\lambda;y,m)\begin{bmatrix}1 & 0\\
\displaystyle -\frac{\sqrt{2\pi}{\mathrm{e}}^{-{\mathrm{i}}\pi m}}{\Gamma(\tfrac{1}{2}+m)}{\mathrm{e}}^{-nW(\lambda;y)^2} & 1\end{bmatrix},\quad\lambda\in{{L^\infty_\squarellblack}}\cap D,$$ and $$\mathbf{N}_{n+}(\lambda;y,m)=\mathbf{N}_{n-}(\lambda;y,m)\begin{bmatrix}-{\mathrm{e}}^{2\pi{\mathrm{i}}m} & 0\\
\displaystyle -\frac{\sqrt{2\pi}{\mathrm{e}}^{-{\mathrm{i}}\pi m}}{\Gamma(\tfrac{1}{2}+m)}{\mathrm{e}}^{-nW(\lambda;y)^2} & -{\mathrm{e}}^{-2\pi{\mathrm{i}}m}\end{bmatrix},\quad\lambda\in{{L^0_\squarellblack}}\cap D.
\label{eq:N-jump-last}$$ Although we will only use its values for $\lambda\in\mathbb{C}\setminus D$, the outer parametrix $\dot{\mathbf{M}}^{\mathrm{out}}(\lambda;y,m)$ has a convenient representation also for $\lambda\in D$ in terms of the conformal coordinate $W=W(\lambda)$: $$\dot{\mathbf{M}}^{\mathrm{out}}(\lambda;y,m)=f(\lambda;y,m)^{\sigma_3}W(\lambda;y)^{-(m+\tfrac{1}{2})\sigma_3},\quad \lambda\in D,
\label{eq:scalar-f-define}$$ where the power function of $W$ refers to the principal branch cut for $W<0$, and where $f(\lambda;y,m)$ is holomorphic and nonvanishing in $D$. Now letting $\zeta:=n^{1/2}W(\lambda;y)$, we define precisely a matrix $\mathbf{P}(\zeta;m)$ as the solution of the following model Riemann-Hilbert problem.
Given any $m\in\mathbb{C}$, seek a $2\times 2$ matrix function $\zeta\mapsto\mathbf{P}(\zeta;m)$ with the following properties:
- **Analyticity:** $\zeta\mapsto\mathbf{P}(\zeta;m)$ is analytic for $\mathrm{Re}(\zeta^2)\neq 0$, taking continuous boundary values on the four rays of $\mathrm{Re}(\zeta^2)=0$ oriented as in the right-hand panel of Figure \[fig:Outside-Local\].
- **Jump conditions:** The boundary values $\mathbf{P}_\pm(\zeta;m)$ taken on the four rays of $\mathrm{Re}(\zeta^2)=0$ satisfy the following jump conditions (cf., –): $$\mathbf{P}_+(\zeta;m)=\mathbf{P}_-(\zeta;m)\begin{bmatrix}1 & \displaystyle -\frac{\sqrt{2\pi}}{\Gamma(\tfrac{1}{2}-m)}{\mathrm{e}}^{\zeta^2}\\0 & 1\end{bmatrix},\quad \arg(\zeta)=-\frac{\pi}{2},
\label{eq:N-zeta-jump-first}$$ $$\mathbf{P}_+(\zeta;m)=\mathbf{P}_-(\zeta;m)\begin{bmatrix}1 & \displaystyle \frac{\sqrt{2\pi}{\mathrm{e}}^{2\pi{\mathrm{i}}m}}{\Gamma(\tfrac{1}{2}-m)}{\mathrm{e}}^{\zeta^2}\\0 & 1\end{bmatrix},\quad \arg(\zeta)=\frac{\pi}{2},$$ $$\mathbf{P}_+(\zeta;m)=\mathbf{P}_-(\zeta;m)\begin{bmatrix}1 & 0\\\displaystyle -\frac{\sqrt{2\pi}{\mathrm{e}}^{-{\mathrm{i}}\pi m}}{\Gamma(\tfrac{1}{2}+m)}{\mathrm{e}}^{-\zeta^2} & 1\end{bmatrix},\quad \arg(\zeta)=0,$$ and $$\mathbf{P}_+(\zeta;m)=\mathbf{P}_-(\zeta;m)\begin{bmatrix}-{\mathrm{e}}^{2\pi{\mathrm{i}}m} & 0\\
\displaystyle -\frac{\sqrt{2\pi}{\mathrm{e}}^{-{\mathrm{i}}\pi m}}{\Gamma(\tfrac{1}{2}+m)}{\mathrm{e}}^{-\zeta^2} & -{\mathrm{e}}^{-2\pi{\mathrm{i}}m}\end{bmatrix},\quad \arg(-\zeta)=0.
\label{eq:N-zeta-jump-last}$$
- **Asymptotics:** $\mathbf{P}(\zeta;m)$ is required to satisfy the normalization condition $$\lim_{\zeta\to\infty}\mathbf{P}(\zeta;m)\zeta^{(m+\tfrac{1}{2})\sigma_3}=\mathbb{I}.
\label{eq:N-zeta-normalize}$$ Here, $\zeta^p$ refers to the principal branch.
\[rhp:ParabolicCylinder\]
This problem will be solved in all details in Appendix \[app:PC\]. From it, we define the inner parametrix $\dot{\mathbf{M}}_n^{\mathrm{in}}(\lambda;y,m)$ as follows: $$\dot{\mathbf{M}}_n^{\mathrm{in}}(\lambda;y,m):=d(\lambda;y,m)^{\sigma_3}n^{\tfrac{1}{2}(m+\tfrac{1}{2})\sigma_3}f(\lambda;y,m)^{\sigma_3}\mathbf{P}(n^{1/2}W(\lambda;m);m)d(\lambda;y,m)^{-\sigma_3},\quad \lambda\in D\setminus L.$$ As shown in Appendix \[app:PC\], $\mathbf{P}(\zeta;m)\zeta^{(m+\tfrac{1}{2})\sigma_3}$ has a complete asymptotic expansion in descending powers of $\zeta$ as $\zeta\to\infty$ (see ). Taking into account the explicit leading terms from the expansion and using the fact that $W(\lambda;y)$ is bounded away from zero for $\lambda\in\partial D$, we get $$\dot{\mathbf{M}}_n^{\mathrm{in}}(\lambda;y,m)\dot{\mathbf{M}}^\mathrm{out}(\lambda;y,m)^{-1}=
n^{m\sigma_3/2}\begin{bmatrix}1+\mathcal{O}(n^{-1}) & a(\lambda;y,m)+\mathcal{O}(n^{-1})\\
\mathcal{O}(n^{-1}) & 1+\mathcal{O}(n^{-1})\end{bmatrix}n^{-m\sigma_3/2},\quad n\to+\infty,\quad\lambda\in\partial D,
\label{eq:Mismatch-0}$$ holding uniformly for the indicated values of $\lambda\in \partial D$, where $$a(\lambda;y,m):=-{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}2^md(\lambda;y,m)^2f(\lambda;y,m)^2W(\lambda;y)^{-1},\quad\lambda\in\partial D.$$
Error analysis and proof of Theorem \[theorem:outside\] {#sec:error-outside}
-------------------------------------------------------
To compare the unknown $\mathbf{M}_n(\lambda;y,m)$ with its parametrix $\dot{\mathbf{M}}_n(\lambda;y,m)$, note the constant conjugating factors in and consider the matrix function $\mathbf{F}_n(\lambda;y,m)$ defined by $\mathbf{F}_n(\lambda;y,m):=n^{-m\sigma_3/2}\mathbf{M}_n(\lambda;y,m)\dot{\mathbf{M}}_n(\lambda;y,m)^{-1}n^{m\sigma_3/2}$, which is well-defined for $\lambda\in \mathbb{C}\setminus (L\cup\partial D)$. This matrix satisfies a jump condition of the form $\mathbf{F}_{n+}(\lambda;y,m)=\mathbf{F}_{n-}(\lambda;y,m)(\mathbb{I}+\text{exponentially small})$ as $n\to+\infty$ uniformly for $\lambda\in L$ (in fact, $\mathbf{F}_{n+}(\lambda;y,m)=\mathbf{F}_{n-}(\lambda;y,m)$ exactly if $\lambda\in L\cap D$). To see this for $\lambda\in L\setminus\overline{D}$, note that $$\mathbf{F}_{n+}(\lambda;y,m)=\mathbf{F}_{n-}(\lambda;y,m)n^{-m\sigma_3/2}\dot{\mathbf{M}}^\mathrm{out}_-(\lambda;y,m)
\mathbf{V}_n(\lambda;y,m)\dot{\mathbf{V}}(\lambda;y,m)^{-1}\dot{\mathbf{M}}^\mathrm{out}_-(\lambda;y,m)^{-1}n^{m\sigma_3/2},\quad\lambda\in L\setminus\overline{D},$$ where $\mathbf{M}_{n+}(\lambda;y,m)=\mathbf{M}_{n-}(\lambda;y,m)\mathbf{V}_n(\lambda;y,m)$ and $\dot{\mathbf{V}}(\lambda;y,m)$ is the corresponding jump matrix for $\dot{\mathbf{M}}^\mathrm{out}(\lambda;y,m)$, which is just the diagonal part of $\mathbf{V}_n(\lambda;y,m)$ and which hence reduces to the identity matrix except on ${{L^0_\squarellblack}}$. The desired result then follows because $\mathbf{V}_n(\lambda;y,m)\dot{\mathbf{V}}(\lambda;y,m)^{-1}-\mathbb{I}$ is exponentially small in the limit $n\to +\infty$, while $\dot{\mathbf{M}}^\mathrm{out}_-(\lambda;y,m)$ and its inverse are independent of $n$ and bounded because $\lambda$ is excluded from $D$. Finally, for $\lambda\in\partial D$, taken with clockwise orientation, we have $$\mathbf{F}_{n+}(\lambda;y,m)=\mathbf{F}_{n-}(\lambda;y,m)n^{-m\sigma_3/2}
\dot{\mathbf{M}}_n^{\mathrm{in}}(\lambda;y,m)
\dot{\mathbf{M}}^{\mathrm{out}}(\lambda;y,m)^{-1}n^{m\sigma_3/2},\quad\lambda\in\partial D.
\label{eq:Fjump-circle}$$ The jump contour for $\mathbf{F}_n(\lambda;y,m)$ (and also for the related matrix $\mathbf{E}_n(\lambda;y,m)$ defined below) in a typical case of $y\in\mathbb{C}\setminus E$ is shown in Figure \[fig:OutsideError\].
![The jump contour for $\mathbf{F}_n(\lambda;y,m)$ and $\mathbf{E}_n(\lambda;y,m)$ for $y=0.364768{\mathrm{e}}^{-3\pi{\mathrm{i}}/4}\in\mathbb{C}\setminus E$ corresponding to the left-most column of Figure \[fig:y-diag\]. The jump contour is the union of the cyan and red arcs (which comprise $L\setminus \overline{D}$), and the green circle $\partial D$, which is taken to have clockwise orientation for the purposes of defining the boundary values.[]{data-label="fig:OutsideError"}](OutsideError-preview.pdf)
Taking into account the leading term on the upper off-diagonal in , we define a parametrix for $\mathbf{F}_n(\lambda;y,m)$ as a triangular matrix independent of $n$: $$\dot{\mathbf{F}}(\lambda;y,m):=\mathbb{I}+\frac{1}{2\pi{\mathrm{i}}}\oint_{\partial D}
\begin{bmatrix}
0 & a(\xi;y,m)\\
0 & 0\end{bmatrix}\frac{{\mathrm{d}}\xi}{\xi-\lambda} \quad\text{(clockwise orientation for $\partial D$)}.
\label{eq:dot-F-def}$$ Since $d(\cdot;y,m)$ and $f(\cdot;y,m)$ are analytic and nonvanishing within $D$, and since $W(\cdot;y)$ is univalent on $D$ with $W({{p}}(y);y)=0$, the above Cauchy integral can be evaluated by residues. In particular, if $\lambda\in\mathbb{C}\setminus\overline{D}$, then $$\dot{\mathbf{F}}(\lambda;y,m)=\begin{bmatrix}
1 & {\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}2^md({{p}}(y);y,m)^{2}f({{p}}(y);y,m)^{2}W'({{p}}(y);y)^{-1}({{p}}(y)-\lambda)^{-1} \\0 & 1\end{bmatrix},
\quad\lambda\in\mathbb{C}\setminus\overline{D}.
\label{eq:F-dot-outside}$$ Note that $\dot{\mathbf{F}}(\lambda;y,m)$ is analytic for $\lambda\in\mathbb{C}\setminus\partial D$, $\dot{\mathbf{F}}(\lambda;y,m)\to\mathbb{I}$ as $\lambda\to\infty$, and across $\partial D$ satisfies (by the Plemelj formula from ) the jump condition $$\dot{\mathbf{F}}_+(\lambda;y,m)=\dot{\mathbf{F}}_-(\lambda;y,m)\begin{bmatrix}1 & a(\lambda;y,m)\\0&1\end{bmatrix},\quad\text{$\lambda\in\partial D$ with clockwise orientation.}
\label{eq:Fdot-circle-jump}$$
At last, we consider the matrix $\mathbf{E}_n(\lambda;y,m):=\mathbf{F}_n(\lambda;y,m)\dot{\mathbf{F}}(\lambda;y,m)^{-1}$. The matrix $\mathbf{E}_n(\lambda;y,m)$ is analytic for $\lambda\in\mathbb{C}\setminus (L\cup\partial D)$, tends to the identity as $\lambda\to\infty$, and takes continuous boundary values from each component of its domain of analyticity, including at the origin. The jump conditions satisfied by $\mathbf{E}_n(\lambda;y,m)$ are as follows. Firstly, since $\mathbf{F}_n(\lambda;y,m)$ extends continuously to the arcs of $L\cap D$ while $\dot{\mathbf{F}}(\lambda;y,m)^{-1}$ is analytic for $\lambda\in\mathbb{C}\setminus\partial D$, it follows by Morera’s theorem that $\mathbf{E}_n(\lambda;y,m)$ is in fact analytic on the arcs of $L\cap D$. For $\lambda\in L\setminus\overline{D}$, $\dot{\mathbf{F}}(\lambda;y,m)$ is analytic with analytic inverse, both of which are bounded; since $\mathbf{F}_{n+}(\lambda;y,m)=\mathbf{F}_{n-}(\lambda;y,m)(\mathbb{I}+\text{exponentially small})$, it follows that as $n\to+\infty$, $\mathbf{E}_{n-}(\lambda;y,m)^{-1}\mathbf{E}_{n+}(\lambda;y,m)-\mathbb{I}$ is small beyond all orders uniformly for bounded $m\in\mathbb{C}$. Finally, for $\lambda\in\partial D$, $$\begin{split}
\mathbf{E}_{n+}(\lambda;y,m)&=\mathbf{F}_{n+}(\lambda;y,m)\dot{\mathbf{F}}_+(\lambda;y,m)^{-1}\\
&=\mathbf{F}_{n-}(\lambda;y,m)\begin{bmatrix}1+\mathcal{O}(n^{-1}) & a(\lambda;y,m)+\mathcal{O}(n^{-1})\\
\mathcal{O}(n^{-1}) & 1+\mathcal{O}(n^{-1})\end{bmatrix}\begin{bmatrix}1 & -a(\lambda;y,m)\\0 & 1\end{bmatrix}\dot{\mathbf{F}}_-(\lambda;y,m)^{-1}\\
&=\mathbf{E}_{n-}(\lambda;y,m)\dot{\mathbf{F}}_-(\lambda;y,m)\begin{bmatrix}1+\mathcal{O}(n^{-1}) & a(\lambda;y,m)+\mathcal{O}(n^{-1})\\
\mathcal{O}(n^{-1}) & 1+\mathcal{O}(n^{-1})\end{bmatrix}\begin{bmatrix}1 & -a(\lambda;y,m)\\0 & 1\end{bmatrix}\dot{\mathbf{F}}_-(\lambda;y,m)^{-1}\\
&=\mathbf{E}_{n-}(\lambda;y,m)(\mathbb{I}+\mathcal{O}(n^{-1})),\quad n\to+\infty,\quad\lambda\in\partial D,
\end{split}
\label{eq:E-jump}$$ where the $\mathcal{O}(n^{-1})$ terms are uniform on $\partial D$. Here, we used , , and on the second line. The jump contour for $\mathbf{E}_n(\lambda;y,m)$ is therefore exactly the same as that for $\mathbf{F}_n(\lambda;y,m)$; see Figure \[fig:OutsideError\]. From these considerations, we see that uniformly for $(y,m)$ in compact subsets of $(\mathbb{C}\setminus \overline{E})\times\mathbb{C}$, $\mathbf{E}_n(\lambda;y,m)$ satisfies the conditions of a small-norm Riemann-Hilbert problem for $|n|$ sufficiently large, and the unique solution satisfies $\mathbf{E}_n(\lambda;y,m)=\mathbb{I}+\mathcal{O}(n^{-1})$ uniformly for $\lambda\in\mathbb{C}\setminus (L\cup\partial D)$. Moreover, $\mathbf{E}_n(\lambda;y,m)$ is well-defined at $\lambda=0$ with $\mathbf{E}_n(0;y,m)=\mathbb{I}+\mathcal{O}(n^{-1})$ as $n\to +\infty$, and $\mathbf{E}_n(\lambda;y,m)=\mathbb{I}+\mathbf{E}_{n,1}(y,m)\lambda^{-1}+\mathcal{O}(\lambda^{-2})$ as $\lambda\to\infty$ with $\mathbf{E}_{n,1}(y,m)=\mathcal{O}(n^{-1})$ as $n\to +\infty$. Now, we have the exact identity $$\begin{split}
\mathbf{M}_n(\lambda;y,m)&=n^{m\sigma_3/2}\mathbf{F}_n(\lambda;y,m)n^{-m\sigma_3/2}\dot{\mathbf{M}}_n(\lambda;y,m)\\ &=n^{m\sigma_3/2}\mathbf{E}_n(\lambda;y,m)\dot{\mathbf{F}}(\lambda;y,m)n^{-m\sigma_3/2}\dot{\mathbf{M}}_n(\lambda;y,m),
\end{split}$$ and therefore from and $g(\lambda;y)=g_\infty(y)=\tfrac{1}{2}V({{p}}(y);y)$, we get $$\mathbf{Y}_n(\lambda;ny,m)={\mathrm{e}}^{-nV({{p}}(y);y)\sigma_3/2}n^{m\sigma_3/2}\mathbf{E}_n(\lambda;y,m)\dot{\mathbf{F}}(\lambda;y,m)n^{-m\sigma_3/2}\dot{\mathbf{M}}_n(\lambda;y,m){\mathrm{e}}^{nV({{p}}(y);y)\sigma_3/2}.
\label{eq:Y-representation-I}$$ Now recall the formula for the rational solution $u=u_n(x;m)$ of the Painlevé-III equation . Since to calculate the quantities in this formula we only need $\mathbf{Y}_n(\lambda;ny,m)$ for $\lambda$ in neighborhoods of the origin and infinity, we can safely replace $\dot{\mathbf{M}}^{(n)}(\lambda;y,m)$ in by the diagonal outer parametrix $\dot{\mathbf{M}}^\mathrm{out}(\lambda;y,m)$ which commutes with $n^{-m\sigma_3/2}$. Therefore, if $\mathbf{Q}_{n}(\lambda;y,m):=\mathbf{E}_n(\lambda;y,m)\dot{\mathbf{F}}(\lambda;y,m)\dot{\mathbf{M}}^\mathrm{out}(\lambda;y,m)$, then can be rewritten as $$u_n(ny;m)=\frac{-{\mathrm{i}}\displaystyle\lim_{\lambda\to\infty}\lambda Q_{n,12}(\lambda;y,m)}{\displaystyle\left[\lim_{\lambda\to 0}Q_{n,11}(\lambda;y,m){{\lambda^{-(m+1/2)}_\squarellblack}}\right]
\left[\lim_{\lambda\to 0}Q_{n,12}(\lambda;y,m){{\lambda^{m+1/2}_\squarellblack}}\right]}.
\label{eq:u-from-Y-rewrite}$$ Now all three factors of $\mathbf{Q}_n(\lambda;y,m)$ tend to $\mathbb{I}$ as $\lambda\to\infty$ but $\dot{\mathbf{M}}^\mathrm{out}(\lambda;y,m)$ is also diagonal, $$\begin{split}
\lim_{\lambda\to\infty}\lambda Q_{n,12}(\lambda;y,m)&=\lim_{\lambda\to\infty}\lambda \dot{F}_{12}(\lambda;y,m)+E_{n,1,12}(y,m)\\
&=-{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}2^md({{p}}(y);y,m)^2f({{p}}(y);y,m)^2W'({{p}}(y);y)^{-1}+\mathcal{O}(n^{-1})
\end{split}$$ where in the second line we used and $\mathbf{E}_{n,1}(y,m)=\mathcal{O}(n^{-1})$. Also, since $\dot{\mathbf{M}}^\mathrm{out}(\lambda;y,m){{\lambda^{-(m+1/2)\sigma_3}_\squarellblack}}$ tends to a limit of the form $h(y,m)^{\sigma_3}$ as $\lambda\to 0$, where $h(y,m)\neq 0$, $$\begin{gathered}
\left[\lim_{\lambda\to 0}Q_{n,11}(\lambda;y,m){{\lambda^{-(m+1/2)}_\squarellblack}}\right]
\left[\lim_{\lambda\to 0}Q_{n,12}(\lambda;y,m){{\lambda^{m+1/2}_\squarellblack}}\right]\\
\begin{aligned}
&=\left[
E_{n,11}(0;y,m)\dot{F}_{11}(0;y,m)+E_{n,12}(0;y,m)\dot{F}_{21}(0;y,m)\right]
\left[E_{n,11}(0;y,m)\dot{F}_{12}(0;y,m)+E_{n,12}(0;y,m)\dot{F}_{22}(0;y,m)\right]\\
&=\left[E_{n,11}(0;y,m)\right]\left[E_{n,11}(0;y,m){\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}2^md({{p}}(y);y,m)^2f({{p}}(y);y,m)^2W'({{p}}(y);y)^{-1}{{p}}(y)^{-1}+ E_{n,12}(0;y,m)\right]\\
&={\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}2^md({{p}}(y);y,m)^2f({{p}}(y);y,m)^2w'({{p}}(y);y)^{-1}{{p}}(y)^{-1}+\mathcal{O}(n^{-1}),
\end{aligned}\end{gathered}$$ where in the third line we used and in the fourth line we used $\mathbf{E}_n(0;y,m)=\mathbb{I}+\mathcal{O}(n^{-1})$. Using these results in then gives the asymptotic formula and completes the proof of Theorem \[theorem:outside\].
Asymptotics of $u_n(ny+w;m)$ for $y\in E$ and $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ {#sec:interior}
================================================================================================
To study $u_n(x;m)$ for values of $x$ corresponding to the interior of $E$, we wish to capture two different effects: (i) the rapid oscillation visible in plots showing a locally regular pattern of poles and zeros on a microscopic length scale $\Delta x\sim 1$ and (ii) the gradual modulation of this pattern over macroscopic length scales $\Delta x\sim n$. To separate these scales, we write $x=ny+w$ as described in Section \[sec:results\]. As mentioned in Remark \[remark:two-vars\], considering $u_n(ny+w;m)$ as a function of $w$ for fixed $y\in E$ captures the microscopic behavior of $u_n$, while setting $w=0$ and considering $u_n(ny;m)$ as a function of $y$ captures instead the macroscopic behavior of $u_n$. A similar approach to the rational solutions of the Painlevé-II equation was taken in [@BuckinghamM14]. In this section we will develop an approximation of $u_n(ny+w;m)$ that depends not on the combination $ny+m$ but rather separately on $y$ and $m$ in such a way as to explicitly separate these scales. In particular, it will turn out that the approximation is meromorphic in $w$ for each fixed $y$ but generally is not analytic at all in $y$.
Spectral curves satisfying the Boutroux integral conditions for $y\in E$ {#sec:Boutroux-in-E}
------------------------------------------------------------------------
We tie the spectral curve to the value $y$ of the macroscopic coordinate and compensate for nonzero values of the microscopic coordinate $w$ later in the construction of a parametrix.
### Solving the Boutroux integral conditions for $y$ small {#sec:Boutroux-y-small}
To construct a $g$-function for $y$ small, we assume that the spectral curve corresponds to a polynomial $P(\lambda;y,C)$ with four distinct roots. We write $y$ in polar form as $y=r{\mathrm{e}}^{{\mathrm{i}}\theta}$ and we write $C$ in the form $C=y\tilde{C}$. For $r>0$ we may divide the equations through by $\sqrt{r}$ and consider instead the *renormalized Boutroux integral conditions* $$\tilde{\mathfrak{B}}_\mathfrak{a}(\tilde{C};r,\theta):=\mathrm{Re}\left(\oint_\mathfrak{a}\tilde{\mu}\,{\mathrm{d}}\lambda\right)=0\quad\text{and}\quad
\tilde{\mathfrak{B}}_\mathfrak{b}(\tilde{C};r,\theta):=\mathrm{Re}\left(\oint_\mathfrak{b}\tilde{\mu}\,{\mathrm{d}}\lambda\right)=0
\label{eq:renormalized-Boutroux}$$ where $$\tilde{\mu}^2 = {\mathrm{e}}^{{\mathrm{i}}\theta}\left[\frac{1}{2}{\mathrm{i}}\lambda^{-1}+\tilde{C}\lambda^{-2}+\frac{1}{2}{\mathrm{i}}\lambda^{-3}\right]-\frac{1}{4}r{\mathrm{e}}^{2{\mathrm{i}}\theta}(1+\lambda^{-4}).
\label{eq:renormalized-rho}$$ Note that if $\tilde{u}:=\mathrm{Re}(\tilde{C})$ and $\tilde{v}:=\mathrm{Im}(\tilde{C})$, then just as in one has that $$\det\left(\frac{\partial(\tilde{\mathfrak{B}}_\mathfrak{a},\tilde{\mathfrak{B}}_\mathfrak{b})}{\partial (\tilde{u},\tilde{v})}\right) =\frac{1}{4}\mathrm{Im}\left(\left[\oint_\mathfrak{a}\frac{{\mathrm{d}}\lambda}{\tilde{\mu}\lambda^2}\right]\left[\oint_\mathfrak{b}\frac{{\mathrm{d}}\lambda}{\tilde{\mu}\lambda^2}\right]^*\right)
\label{eq:renormalized-Jacobian}$$ which is nonzero as long as $\tilde{\mu}$ (see the algebraic relation ) has distinct branch points on the Riemann sphere of the $\lambda$-plane, see [@Dubrovin81 Chapter II, Corollary 1]. We now first set $r=0$ and attempt to determine $\tilde{C}$ as a function of $\theta$. It is convenient to then reduce the cycle integrals in to contour integrals connecting pairs of branch points in the finite $\lambda$-plane, and since when $r=0$ the differential $\tilde{\mu}\,{\mathrm{d}}\lambda$ has a double pole with zero residue (in an appropriate local coordinate) at the branch point $\lambda=0$ we can integrate by parts to transfer “half” of the double pole to each of the finite nonzero roots of $\tilde{\mu}^2$ which (again in appropriate local coordinates) are simple zeros of $\tilde{\mu}\,{\mathrm{d}}\lambda$. In this way we obtain conditions equivalent to for $r=0$ involving a differential that is holomorphic at all three branch points in the finite $\lambda$-plane. These conditions are the following: $$\tilde{\mathfrak{B}}^0_\mathfrak{a}(\tilde{C};\theta):=\mathrm{Re}\left(\oint_\mathfrak{a}\tilde{\mu}_0\,{\mathrm{d}}\lambda\right)=0\quad\text{and}\quad
\tilde{\mathfrak{B}}^0_\mathfrak{b}(\tilde{C};\theta):=\mathrm{Re}\left(\oint_\mathfrak{b}\tilde{\mu}_0\,
{\mathrm{d}}\lambda\right)=0,
\label{eq:Boutroux-r=0}$$ where $$\tilde{\mu}_0^2:={\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}\frac{(\lambda-{\mathrm{i}}\tilde{C})^2}{\lambda(\lambda^2-2{\mathrm{i}}\tilde{C}\lambda+1)}.$$ The desired simplification is then that the cycle integrals in over $\mathfrak{a}$ and $\mathfrak{b}$ may be replaced (up to a harmless factor of $2$) by path integrals from $\lambda=0$ to the two roots of the quadratic $\lambda^2-2{\mathrm{i}}\tilde{C}\lambda+1$ respectively.
If ${\mathrm{e}}^{{\mathrm{i}}\theta}=1$, we may solve in this simplified form by assuming $\tilde{C}$ to be real and positive. Indeed, then the roots of $\lambda^2-2{\mathrm{i}}\tilde{C}\lambda+1$ are the values $\lambda={\mathrm{i}}(\tilde{C}\pm\sqrt{\tilde{C}^2+1})$ which lie on the positive and negative imaginary axes. It is easy to see that when $\theta=0$, $\tilde{\mu}_0^2>0$ holds for purely imaginary $\lambda$ between $\lambda={\mathrm{i}}(\tilde{C}-\sqrt{\tilde{C}^2+1})$ and $\lambda=0$. Therefore it is immediate that $$\mathrm{Re}\left(\int_0^{{\mathrm{i}}(\tilde{C}-\sqrt{\tilde{C}^2+1})}\tilde{\mu}_0\,{\mathrm{d}}\lambda\right)=0,\quad \theta=0,\quad \tilde{C}>0.$$ The remaining Boutroux integral condition then reduces under the hypotheses $\theta=0$ and $\tilde{C}>0$ to a purely real-valued integral condition on $\tilde{C}$: $$J(\tilde{C}):=\int_0^{\tilde{C}+\sqrt{\tilde{C}^2+1}}\frac{t-\tilde{C}}{\sqrt{t}\sqrt{1+2\tilde{C}t-t^2}}\,{\mathrm{d}}t=0$$ Obviously $\lim_{\tilde{C}\downarrow 0}J(\tilde{C})$ exists and the limit is positive. Also, by rescaling $t=\tilde{C}s$, $$J(\tilde{C})=-\sqrt{\tilde{C}}\int_0^1\frac{1-s}{\sqrt{s}\sqrt{\tilde{C}^{-2}+2s-s^2}}\,{\mathrm{d}}s + \sqrt{\tilde{C}}\int_1^2\frac{s-1}{s\sqrt{2-s}}\,{\mathrm{d}}s + o(\sqrt{\tilde{C}}),\quad \tilde{C}\to\infty,$$ and clearly the first term is the dominant one so $J(\tilde{C})<0$ for large positive $\tilde{C}$. Also, by direct calculation, $$J'(\tilde{C})=-\frac{1}{2}\int_0^{\tilde{C}+\sqrt{\tilde{C}^2+1}}\frac{{\mathrm{d}}t}{\sqrt{t}\sqrt{1+2\tilde{C}t-t^2}}<0,\quad \tilde{C}>0,$$ so there exists a unique simple root $\tilde{C}_0>0$ of $J(\tilde{C})$. Numerical computation shows that $\tilde{C}_0\approx 0.860437$.
If ${\mathrm{e}}^{{\mathrm{i}}\theta}=-1$, we can invoke the symmetry $\lambda\mapsto -\lambda$ and $\tilde{C}\mapsto -\tilde{C}$ of $\tilde{\mu}_0^2$ to deduce that the equations hold for $\tilde{C}=-\tilde{C}_0\approx -0.860437$.
When $r=0$, the elliptic curve given by has distinct branch points on the Riemann sphere unless $\tilde{C}=\pm{\mathrm{i}}$, and hence the Jacobian of the equations is nonzero for ${\mathrm{e}}^{{\mathrm{i}}\theta}=\pm 1$. The solution of the $r=0$ system can therefore be continued to other values of ${\mathrm{e}}^{{\mathrm{i}}\theta}$ until the condition $\tilde{C}\neq\pm{\mathrm{i}}$ is violated. It is easy to check that $\tilde{C}=\pm{\mathrm{i}}$ is consistent with only for ${\mathrm{e}}^{{\mathrm{i}}\theta}=\mp{\mathrm{i}}$. Therefore the solutions of the $r=0$ system obtained for ${\mathrm{e}}^{{\mathrm{i}}\theta}=\pm 1$ can be uniquely continued by the implicit function theorem to fill out an infinitesimal circle surrounding the origin $y=0$ with the possible exception of its intersection with the imaginary axis. Fixing any phase factor ${\mathrm{e}}^{{\mathrm{i}}\theta}\neq \pm{\mathrm{i}}$, we can then continue the solution of the full (rescaled) system to small $r>0$ (in fact, also for $r<0$, although the solution is not relevant), and the radial continuation can only be obstructed if branch points collide.
### Degenerate spectral curves satisfying the Boutroux integral conditions {#sec:Boutroux-degenerate}
The only possible values of $y\in\mathbb{C}$ for which all four roots of $P(\lambda;y,C)$ coincide are $y=\pm\frac{1}{2}{\mathrm{i}}$, which lie on the boundary of $E$. For all $y\in\mathbb{C}$ it is possible to have either a pair of distinct double roots or a double root and two simple roots, provided $C$ is appropriately chosen as a function of $y$. We will now show that these degenerate configurations are inconsistent with the Boutroux integral conditions , which have to be interpreted in a limiting sense, provided that $y$ lies in the interior of $E$ but does not also lie on the imaginary axis.
Consider first a nearly degenerate configuration of roots in which two simple roots of $P$ are very close to a point $\lambda={{p}}$ and two reciprocal simple roots are very close to $\lambda={{p}}^{-1}$. Then we may choose the cycle $\mathfrak{a}$ to encircle the pair of roots near, say, $\lambda={{p}}$. As the spectral curve degenerates with the cycle $\mathfrak{a}$ fixed, we may observe that $\mu$ becomes in the limit an analytic function of $\lambda$ in the interior of $\mathfrak{a}$ and therefore $\oint_\mathfrak{a}\mu\,{\mathrm{d}}\lambda\to 0$ and hence $\mathrm{Re}(\oint_\mathfrak{a}\mu\,{\mathrm{d}}\lambda)\to 0$, so one of the Boutroux integral conditions is automatically satisfied in the limit. The cycle $\mathfrak{b}$ should then be chosen to connect the small branch cut near $\lambda={{p}}$ with the small reciprocal branch cut near $\lambda={{p}}^{-1}$. In the limit that the spectral curve degenerates and $\mu^2$ becomes a perfect square, the second Boutroux integral condition becomes $$\mathrm{Re}\left(\oint_\mathfrak{b}\mu\,{\mathrm{d}}\lambda\right)\to 2\mathrm{Re}\left(\int_{{{p}}}^{{{p}}^{-1}}\frac{{\mathrm{i}}y}{2}\frac{(\lambda-{{p}})(\lambda-{{p}}^{-1})}{\lambda^2}\,{\mathrm{d}}\lambda\right)=\mathrm{Re}(V({{p}};y)-V({{p}}^{-1};y))$$ where ${{p}}+{{p}}^{-1}={\mathrm{i}}y^{-1}$. *The condition on $y\in\mathbb{C}$ that this quantity vanishes is precisely that either $y\in\partial E$ or $y$ lies on the imaginary axis outside of $E$. Therefore the Boutroux conditions cannot be satisfied by such a degenerate spectral curve if $y$ is in the interior of $E$.*
Next consider a nearly degenerate configuration in which a pair of simple roots of $P$ lie very close to $\lambda=\pm 1$ and another pair of reciprocal simple roots tend to distinct reciprocal limits satisfying whose sum is $2{\mathrm{i}}y^{-1}\mp 2$. Again taking $\mathfrak{a}$ to surround the coalescing pair of roots shows that $\mathrm{Re}(\oint_\mathfrak{a}\mu\,{\mathrm{d}}\lambda)\to 0$ in the limit. Then, in the same limit, up to signs, $$\mathrm{Re}\left(\oint_\mathfrak{b}\mu\,{\mathrm{d}}\lambda\right)\to 2\mathrm{Re}\left(\int_{\lambda^\pm}^{\pm 1}\frac{{\mathrm{i}}y}{2}\frac{\lambda\mp 1}{\lambda^2}r^\pm(\lambda;y)\,{\mathrm{d}}\lambda\right)=:F^\pm(y),$$ where $\lambda^\pm+(\lambda^\pm)^{-1}=2{\mathrm{i}}y^{-1}\mp 2$ and $r^\pm(\lambda;y)^2=(\lambda-\lambda^\pm)(\lambda-(\lambda^\pm)^{-1})$ with $r^\pm$ having a branch cut connecting the two roots of $r^\pm(\lambda;y)^2$ and, say, $r^\pm=\lambda+\mathcal{O}(1)$ as $\lambda\to\infty$. It is easy to show that $F^+(y)=0$ for $y$ on the segment between $y=0$ and $y=\tfrac{1}{2}{\mathrm{i}}$, and that $F^-(y)=0$ for $y$ on the segment between $y=0$ and $y=-\tfrac{1}{2}{\mathrm{i}}$. However, neither function $F^\pm(y)$ vanishes identically, so the equations $F^\pm(y)=0$ define a system of curves in the complex $y$-plane. The only branches of these curves in the interior of $E$ lie on the imaginary axis as illustrated in Figure \[fig:FplusFminus\].
![The locus $F^+(y)=0$ (orange) and $F^-(y)=0$ (green). The curve $\partial E$ is shown in black and the sub-domains $E_\mathrm{L}$ and $E_\mathrm{R}$ of $E\setminus ((\partial E)\cup{\mathrm{i}}\mathbb{R})$ are indicated.[]{data-label="fig:FplusFminus"}](FplusFminus.pdf)
Therefore, continuation along radial paths of the Boutroux conditions from the infinitesimal semicircles about the origin in the right and left half-planes defines a unique spectral curve for each $y\in E_\mathrm{L}\cup E_\mathrm{R}$, recalling that $E_\mathrm{L}$ ($E_\mathrm{R}$) is the part of the interior of $E$ in the open left (right) half-plane.
Stokes graph and construction of the $g$-function
-------------------------------------------------
For the rest of Section \[sec:interior\] we will be concerned with the approximation of $u_n(ny+w;m)$ for large $n$ when $m\in\mathbb{C}\setminus (\mathbb{Z}+\tfrac{1}{2})$ and $w$ is bounded, while $y\in E_\mathrm{L}\cup E_\mathrm{R}$. Actually, due to the exact symmetry , it is sufficient to assume that $y\in E_\mathrm{R}$, as $E_\mathrm{L}$ is the reflection through the origin of $E_\mathrm{R}$. Thus we assume for the rest of Section \[sec:interior\] that $y\in E_\mathrm{R}$ and at the end invoke to extend the results to $y\in E_\mathrm{L}$.
Given $y\in E_\mathrm{R}$, let $C=C(y)$ be determined by the procedure described in Section \[sec:Boutroux-in-E\] so that the Boutroux conditions are satisfied. The *Stokes graph* of $y$ is the system of arcs (*edges*) in the complex $\lambda$-plane emanating from the four distinct roots of $P(\lambda;y,C(y))$ (*vertices*, when taken along with $\lambda=0,\infty$) along which the condition $(g'(\lambda;y)-\tfrac{1}{2}V'(\lambda;y))^2\,{\mathrm{d}}\lambda^2=\lambda^{-4}P(\lambda;y,C(y))\,{\mathrm{d}}\lambda^2<0$ holds. The Boutroux conditions imply that the Stokes graph is connected. In particular, each pair of roots of $P(\lambda;y,C(y))$ that coalesce at $\partial E$ is directly connected by an edge of the Stokes graph. Denoting the union of these two edges by $\Sigma^\mathrm{out}(y)$, let $R(\lambda;y)$ be the function analytic for $\lambda\in\mathbb{C}\setminus\Sigma^\mathrm{out}(y)$ that satisfies $R(\lambda;y)^2=P(\lambda;y,C(y))$ and $R(\lambda;y)=\tfrac{1}{2}{\mathrm{i}}y\lambda^2+\mathcal{O}(\lambda)$ as $\lambda\to\infty$. According to and , $g'(\lambda;y)$ may then be defined by $$g'(\lambda;y)=\frac{1}{2}V'(\lambda;y)+\frac{R(\lambda;y)}{\lambda^2},\quad\lambda\in\mathbb{C}\setminus\Sigma^\mathrm{out}(y),\quad y\in E_\mathrm{R}.
\label{eq:gprime-E}$$ Note that the apparent singularity at $\lambda=0$ is removable, and $g'(\lambda;y)$ is integrable at $\lambda=\infty$. Figures \[fig:StokesGraphReal\], \[fig:StokesGraphUpperArc\], and \[fig:StokesGraphLowerArc\] below illustrate how the Stokes graph varies with $y\in E_\mathrm{R}$.
![The Stokes graph and sign charts for $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))$ (red/blue for negative/positive) for $y=0.16$ (left column), $y=0.24$ (center column), and $y=0.32$ (right column). Observe as expected that for small $y$ there are roots close to $\lambda=0$ and $\lambda=\infty$, while as $y\to\partial E$ there are two coalescing pairs of roots, each pair connected by an edge of the graph.[]{data-label="fig:StokesGraphReal"}](y-real-plot-Eye-preview.pdf)
![As in Figure \[fig:StokesGraphReal\], but for $y=0.3$ (left column), $y=0.3{\mathrm{e}}^{{\mathrm{i}}\pi/4}$ (center column), and $y=0.3{\mathrm{e}}^{99\pi{\mathrm{i}}/200}$ (right column). Observe that as $y$ approaches the positive imaginary axis (where $F^+(y)=0$) from within $E_\mathrm{R}$, a pair of roots coalesce at $\lambda=1$.[]{data-label="fig:StokesGraphUpperArc"}](y-upper-arc-Eye-preview.pdf)
![As in Figure \[fig:StokesGraphReal\], but for $y=0.3$ (left column), $y=0.3{\mathrm{e}}^{-{\mathrm{i}}\pi/4}$ (center column), and $y=0.3{\mathrm{e}}^{-99\pi{\mathrm{i}}/200}$ (right column). Observe that as $y$ approaches the negative imaginary axis (where $F^-(y)=0$) from within $E_\mathrm{R}$, a pair of roots coalesce at $\lambda=-1$.[]{data-label="fig:StokesGraphLowerArc"}](y-lower-arc-Eye-preview.pdf)
A comparison of the top and bottom rows of these figures illustrates the fact that the Stokes graph of $y\in E_\mathrm{R}$ is invariant while $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))$ changes sign under the involution $\lambda\mapsto \lambda^{-1}$.
Given the Stokes graph, we may lay over the arcs $\Sigma^\mathrm{out}(y)$ and in the complement of the Stokes graph a contour $L$ consisting of arcs ${{L^\infty_\squareurblack}}$, ${{L^0_\squareurblack}}$, ${{L^\infty_\squarellblack}}$, and ${{L^0_\squarellblack}}$ that satisfy the increment-of-argument conditions –. There are two topologically distinct cases differentiated by the sign $\mathrm{Im}(y)$, as illustrated in Figure \[fig:StokesGraph\] for $y\in E_\mathrm{R}$ with $\mathrm{Im}(y)>0$
![The Stokes graph (black curves) for $y=0.2+0.25{\mathrm{i}}\in E_\mathrm{R}$. Suitable contour arcs matching the argument increment conditions – are also shown. As in preceding figures, the sign of $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))$ is indicated with red (negative) and blue (positive) shading.[]{data-label="fig:StokesGraph"}](y-0p2PLUS0p25i-preview.pdf)
and in Figure \[fig:StokesGraphLowerRight\] for $y\in E_\mathrm{R}$ with $\mathrm{Im}(y)<0$.
![As in Figure \[fig:StokesGraph\] but for $y=0.2-0.25{\mathrm{i}}\in E_\mathrm{R}$.[]{data-label="fig:StokesGraphLowerRight"}](y-0p2MINUS0p25i-preview.pdf)
If $y\in E_\mathrm{R}$ with $\mathrm{Im}(y)=0$, we may use either configuration and obtain consistent results because as a rational function $u_n(x;m)$ is single-valued. In the rest of this section, we will for simplicity suppose frequently that $y\in E_\mathrm{R}\setminus\mathbb{R}$ simply for the convenience of being able to speak of contour $L$ as a well-defined notion. The vertices of the Stokes graph on the Riemann sphere are the four roots of $P(\lambda;y,C(y))$, each of which has degree $3$, and the points $\{0,\infty\}$, each of which has degree $2$.
The solution $\mathbf{Y}$ of Riemann-Hilbert Problem \[rhp:renormalized\] depends parametrically on $x=ny+w$, and when we consider $y\in E_\mathrm{R}$ we are introducing a $g$-function $g$ that depends on $y$ but not on $w$. Therefore, in this setting the analogue of the definition is instead $$\mathbf{M}_n(\lambda;y,w,m):={\mathrm{e}}^{ng_\infty(y)\sigma_3}\mathbf{Y}_n(\lambda;ny+w,m){\mathrm{e}}^{-ng(\lambda;y)\sigma_3},
\label{eq:Y-M-Eye}$$ i.e., the matrix $\mathbf{M}$ related to $\mathbf{Y}$ via will depend on both $y$ and $w$ as independent parameters.
### The $g$-function and its properties
When $y\in E_\mathrm{R}\setminus\mathbb{R}$, the self-intersection point of $L$ is identified with the root of $P(\lambda;y,C(y))$ adjacent to $0$ in the Stokes graph. Therefore, for $y\in E_\mathrm{R}\setminus\mathbb{R}$, the arcs ${{L^0_\squarellblack}}$ and ${{L^0_\squareurblack}}$ each connect two distinct vertices of the Stokes graph, while ${{L^\infty_\squarellblack}}$ joins three consecutive vertices and ${{L^\infty_\squareurblack}}$ joins four consecutive vertices. We break these latter arcs at the intermediate vertices; thus ${{L^\infty_\squarellblack}}={{L^{\infty,1}_\squarellblack}}\cup{{L^{\infty,2}_\squarellblack}}$ and ${{L^\infty_\squareurblack}}={{L^{\infty,1}_\squareurblack}}\cup{{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,3}_\squareurblack}}$ with the components ordered by orientation away from $\infty$ and where ${{L^{\infty,2}_\squarellblack}}$ and ${{L^{\infty,2}_\squareurblack}}$ are the two disjoint components of $\Sigma^\mathrm{out}(y)$. The different sub-arcs are illustrated in Figures \[fig:StokesGraph\] and \[fig:StokesGraphLowerRight\].
With these definitions, $g(\lambda;y)$ is determined up to an integration constant by and the condition that $g(\lambda;y)$ is analytic for $\lambda\in\mathbb{C}\setminus(\Sigma^\mathrm{out}(y)\cup{{L^{\infty,3}_\squareurblack}})$. Then, assuming that the branch cut of $\log(\lambda)$ in is disjoint from the contour $L$, $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)$ is constant along the two arcs of $\Sigma^\mathrm{out}(y)$, and we choose the integration constant (given the arbitrary choice of overall branch for $\log(\lambda)$ in ) so that $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)=0$ holds as an identity for $\lambda\in{{L^{\infty,2}_\squareurblack}}\subset\Sigma^\mathrm{out}(y)$. In particular, $g(\infty;y)$ is well-defined mod $2\pi{\mathrm{i}}\mathbb{Z}$. The Stokes graph of $y$ then coincides with the zero level set of the function $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))$. In Figures \[fig:StokesGraph\] and \[fig:StokesGraphLowerRight\], the region where $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ is shaded red while the region where $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$ is shaded blue. The advantage of placing the arcs of $L$ in relation to the Stokes graph of $y$ as shown in Figures \[fig:StokesGraph\] and \[fig:StokesGraphLowerRight\] is that the following conditions hold:
- For $\lambda\in{{L^0_\squareurblack}}$, $g_+(\lambda;y)=g_-(\lambda;y)$ and $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$.
- For $\lambda\in{{L^0_\squarellblack}}$, $g_+(\lambda;y)=g_-(\lambda;y)$ and $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$.
- For $\lambda\in{{L^{\infty,1}_\squareurblack}}$, $g_+(\lambda;y)=g_-(\lambda;y)$ and $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$.
- For $\lambda\in{{L^{\infty,2}_\squareurblack}}$, $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)=0$ (by choice of integration constant) and $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$ on both left and right sides of ${{L^{\infty,2}_\squareurblack}}$.
- For $\lambda\in{{L^{\infty,3}_\squareurblack}}$, we can use to deduce that $$g_+(\lambda;y)-g_-(\lambda;y)=-\oint\frac{R(\ell;y)}{\ell^2}\,{\mathrm{d}}\ell,\quad\lambda\in {{L^{\infty,3}_\squareurblack}},$$ where the integration is over a counterclockwise-oriented loop surrounding ${{L^{\infty,2}_\squareurblack}}$. As this loop can be interpreted as one of the homology cycles $(\mathfrak{a},\mathfrak{b})$ on the Riemann surface of the equation $\mu^2=\lambda^{-4}P(\lambda;y,C(y))$, by the Boutroux conditions we therefore have $g_+(\lambda;y)-g_-(\lambda;y)={\mathrm{i}}K_1$ where $K_1\in\mathbb{R}$ is a real constant (independent of $\lambda\in{{L^{\infty,3}_\squareurblack}}$, but depending on $y\in E_\mathrm{R}$). Also for $\lambda\in{{L^{\infty,3}_\squareurblack}}$ we have $\mathrm{Re}(g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y))<0$.
- For $\lambda\in{{L^{\infty,1}_\squarellblack}}$, $g_+(\lambda;y)=g_-(\lambda;y)$ and $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))>0$.
- For $\lambda\in{{L^{\infty,2}_\squarellblack}}$, since $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)=0$ holds on ${{L^{\infty,2}_\squareurblack}}$, integration of along ${{L^{\infty,3}_\squareurblack}}$ gives $$g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)=2\int_{{L^{\infty,3}_\squareurblack}}\frac{R(\ell;y)}{\ell^2}\,{\mathrm{d}}\ell.
\label{eq:LooksDifferentFor-m-NearNegativeHalfInteger}$$ The right-hand side can also be identified with a cycle integral on the Riemann surface, so by the Boutroux conditions we deduce that $g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y)={\mathrm{i}}K_2$ holds on ${{L^{\infty,2}_\squarellblack}}$, where $K_2=K_2(y)\in\mathbb{R}$ is a real constant. Also $\mathrm{Re}(2g(\lambda;y)-V(\lambda;y))<0$ holds on either side of the arc ${{L^{\infty,2}_\squarellblack}}$.
Szegő function
--------------
The Szegő function is a kind of lower-order correction to the $g$-function. Its dual purpose is to remove the weak $\lambda$-dependence from the jump matrices on $\Sigma^\mathrm{out}(y)$ for $\mathbf{M}_n(\lambda;y,w,m)$ defined in while simultaneously repairing the singularity at the origin captured by the condition that $\mathbf{M}_n(\lambda;y,w,m){{\lambda^{-(m+1/2)\sigma_3}_\squarellblack}}$ must be well-defined at $\lambda=0$. We write the scalar Szegő function $S(\lambda;y,m)$ in the form of an exponential: $S(\lambda;y,m)={\mathrm{e}}^{L(\lambda;y,m)}$ where $L(\lambda;y,m)$ is bounded except near the origin and is analytic for $\lambda\in\mathbb{C}\setminus (\Sigma^\mathrm{out}(y)\cup{{L^0_\squarellblack}})$. The Szegő function is then used to define a new unknown $\mathbf{N}_n(\lambda;y,w,m)$, by the formula $$\mathbf{N}_n(\lambda;y,w,m):=S(\infty;y,m)^{-\sigma_3}\mathbf{M}_n(\lambda;y,w,m)S(\lambda;y,m)^{\sigma_3}={\mathrm{e}}^{-L(\infty;y,m)\sigma_3}\mathbf{M}_n(\lambda;y,w,m){\mathrm{e}}^{L(\lambda;y,m)\sigma_3}.$$ To define the Szegő function, we insist that the boundary values taken by $L(\lambda;y,m)$ on the arcs of its jump contour $\Sigma^\mathrm{out}(y)\cup {{L^0_\squarellblack}}$ are related as follows:
- For $\lambda\in {{L^0_\squarellblack}}$, $L_+(\lambda;y,m)-L_-(\lambda;y,m)=-2\pi{\mathrm{i}}(m+\tfrac{1}{2})$.
- For $\lambda\in{{L^{\infty,2}_\squareurblack}}$, $L_+(\lambda;y,m)+L_-(\lambda;y,m)=\tfrac{1}{2}\ln(2\pi)-\log(\Gamma(\tfrac{1}{2}-m))-(m+1){{\log_\squarellblack(\lambda)}}-\tfrac{1}{2}{\mathrm{i}}\pi$.
- For $\lambda\in {{L^{\infty,2}_\squarellblack}}$, $L_+(\lambda;y,m)+L_-(\lambda;y,m)=-(m+1){{\langle\log_\squarellblack(\lambda)\rangle}}+\tfrac{1}{2}{\mathrm{i}}\pi +\gamma(y,m)$.
Here $\log(\Gamma(\tfrac{1}{2}-m))$ is an arbitrary value of the (generally complex) logarithm, we recall that ${{\log_\squarellblack(\lambda)}}:=\ln|\lambda|+{\mathrm{i}}{{\arg_\squarellblack(\lambda)}}$, and ${{\langle\log_\squarellblack(\lambda)\rangle}}$ refers to the average of the two boundary values of ${{\log_\squarellblack(\lambda)}}$ taken on ${{L^{\infty,2}_\squarellblack}}$. Also, $\gamma(y,m)$ is a constant to be determined so that $L(\lambda;y,m)$ tends to a well-defined limit $L(\infty;y,m)$ as $\lambda\to\infty$. Writing $L(\lambda;y,m)=R(\lambda;y)k(\lambda;y,m)$ and solving for $k$ using the Plemelj formula we obtain $$\begin{gathered}
k(\lambda;y,m)=-(m+\tfrac{1}{2})\int_{{L^0_\squarellblack}}\frac{{\mathrm{d}}\ell}{R(\ell;y)(\ell-\lambda)}\\
{}+\frac{1}{2\pi{\mathrm{i}}}\int_{{L^{\infty,2}_\squareurblack}}\frac{\tfrac{1}{2}\ln(2\pi)-\log(\Gamma(\tfrac{1}{2}-m))-(m+1){{\log_\squarellblack(\ell)}}-\tfrac{1}{2}{\mathrm{i}}\pi}{R_+(\ell;y)(\ell-\lambda)}\,{\mathrm{d}}\ell\\{}+\frac{1}{2\pi{\mathrm{i}}}
\int_{{L^{\infty,2}_\squarellblack}}\frac{-(m+1){{\langle\log_\squarellblack(\ell)\rangle}}+\tfrac{1}{2}{\mathrm{i}}\pi+\gamma(y,m)}{R_+(\ell;y)(\ell-\lambda)}\,{\mathrm{d}}\ell.\end{gathered}$$ Since $R(\lambda;y)=\mathcal{O}(\lambda^2)$ as $\lambda\to\infty$, we need $k(\lambda;y,m)=\mathcal{O}(\lambda^{-2})$ in the same limit, which gives the condition determining $\gamma(y,m)$: $$\begin{gathered}
0=-(m+\tfrac{1}{2})\int_{{L^0_\squarellblack}}\frac{{\mathrm{d}}\ell}{R(\ell;y)}\\
{}+\frac{1}{2\pi{\mathrm{i}}}\int_{{L^{\infty,2}_\squareurblack}}\frac{\tfrac{1}{2}\ln(2\pi)-\log(\Gamma(\tfrac{1}{2}-m))-(m+1){{\log_\squarellblack(\ell)}}-\tfrac{1}{2}{\mathrm{i}}\pi}{R_+(\ell;y)}\,{\mathrm{d}}\ell\\
{}+\frac{1}{2\pi{\mathrm{i}}}\int_{{L^{\infty,2}_\squarellblack}}\frac{-(m+1){{\langle\log_\squarellblack(\ell)\rangle}}+\tfrac{1}{2}{\mathrm{i}}\pi+\gamma(y,m)}{R_+(\ell;y)}\,{\mathrm{d}}\ell.\end{gathered}$$ Note that the coefficient of $\gamma(y,m)$ is necessarily nonzero as a complete elliptic integral of the first kind. We note also the identity $$\int_{{L^{\infty,2}_\squareurblack}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)} = -\int_{{L^{\infty,2}_\squarellblack}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)},
\label{eq:loop-swap}$$ from which it follows that $$\gamma(y,m)=\tfrac{1}{2}\ln(2\pi)-\log(\Gamma(\tfrac{1}{2}-m))-{\mathrm{i}}\pi +\tilde{\gamma}(y,m)
\label{eq:gamma-gammatilde}$$ where $$\begin{gathered}
\tilde{\gamma}(y,m):=\left(\int_{{{L^{\infty,2}_\squarellblack}}}\frac{{\mathrm{d}}\lambda}{R_+(\lambda;y)}\right)^{-1}\left[2\pi{\mathrm{i}}(m+\tfrac{1}{2})\int_{{{L^0_\squarellblack}}}\frac{{\mathrm{d}}\lambda}{R(\lambda;y)} \right.\\
\left. {}+ (m+1)\left(\int_{{{L^{\infty,2}_\squareurblack}}}\frac{{{\log_\squarellblack(\lambda)}}\,{\mathrm{d}}\lambda}{R_+(\lambda;y)} +
\int_{{{L^{\infty,2}_\squarellblack}}}\frac{{{\langle\log_\squarellblack(\lambda)\rangle}}\,{\mathrm{d}}\lambda}{R_+(\lambda;y)}\right)\right].\end{gathered}$$ Since $k(\lambda;y,m)$ exhibits negative one-half power singularities at each of the four roots of $P(\lambda;y,C(y))$, $L(\lambda;y,m)$ is bounded near these points. Near the origin, we have $L(\lambda;y,m)=-(m+\tfrac{1}{2}){{\log_\squarellblack(\lambda)}} + \mathcal{O}(1)$, and therefore $\mathbf{N}_n(\lambda;y,w,m)$ is bounded near $\lambda=0$.
The jump conditions satisfied by $\mathbf{N}_n(\lambda;y,w,m)$ on the arcs of $L$ when $y\in E_\mathrm{R}\setminus\mathbb{R}$ are then as follows: $$\mathbf{N}_{n+}(\lambda;y,w,m)=\mathbf{N}_{n-}(\lambda;y,w,m)\begin{bmatrix}
1 & \displaystyle\frac{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}{\mathrm{e}}^{-2L(\lambda;y,m)}{\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)}}{\Gamma(\tfrac{1}{2}-m)}
{\mathrm{e}}^{n(2g(\lambda;y)-V(\lambda;y))}\\
0 & 1\end{bmatrix},\quad \lambda\in{{L^{\infty,1}_\squareurblack}}.
\label{eq:NLInftyRedOne}$$ $$\begin{gathered}
\mathbf{N}_{n+}(\lambda;y,w,m)=\\
\mathbf{N}_{n-}(\lambda;y,w,m)\begin{bmatrix}
{\mathrm{e}}^{-{\mathrm{i}}n K_1(y)} & \displaystyle\frac{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}{\mathrm{e}}^{-2L(\lambda;y,m)}{\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)}}{\Gamma(\tfrac{1}{2}-m)}
{\mathrm{e}}^{n(g_+(\lambda;y)+g_-(\lambda;y)-V(\lambda;y))}\\
0 & {\mathrm{e}}^{{\mathrm{i}}n K_1(y)}\end{bmatrix},\quad \lambda\in{{L^{\infty,3}_\squareurblack}}.\end{gathered}$$ $$\begin{gathered}
\mathbf{N}_{n+}(\lambda;y,w,m)=\\
\mathbf{N}_{n-}(\lambda;y,w,m)\begin{bmatrix}
1 & 0\\
\displaystyle\frac{\sqrt{2\pi}({{\lambda^{(m+1)/2}_\squarellblack}})_+({{\lambda^{(m+1)/2}_\squarellblack}})_-{\mathrm{e}}^{2L(\lambda;y,m)}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)}}{\Gamma(\tfrac{1}{2}+m)}{\mathrm{e}}^{-n(2g(\lambda;y)-V(\lambda;y))} & 1\end{bmatrix},\quad \lambda\in{{L^{\infty,1}_\squarellblack}}.\end{gathered}$$ $$\mathbf{N}_{n+}(\lambda;y,w,m)=\mathbf{N}_{n-}(\lambda;y,w,m)\begin{bmatrix}
1 & \displaystyle -\frac{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}{\mathrm{e}}^{-2L(\lambda;y,m)}{\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)}}{\Gamma(\tfrac{1}{2}-m)}
{\mathrm{e}}^{n(2g(\lambda;y)-V(\lambda;y))}\\
0 & 1\end{bmatrix},\quad\lambda\in {{L^0_\squareurblack}}.$$ $$\begin{gathered}
\mathbf{N}_{n+}(\lambda;y,w,m)=\\
{}\mathbf{N}_{n-}(\lambda;y,w,m)\begin{bmatrix}
1 & 0\\
\displaystyle\frac{\sqrt{2\pi}({{\lambda^{(m+1)/2}_\squarellblack}})_+({{\lambda^{(m+1)/2}_\squarellblack}})_-{\mathrm{e}}^{L_+(\lambda;y,m)+L_-(\lambda;y,m)}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)}}{\Gamma(\tfrac{1}{2}+m)}{\mathrm{e}}^{-n(2g(\lambda;y)-V(\lambda;y))} & 1\end{bmatrix},\\
\lambda\in{{L^0_\squarellblack}}.
\label{eq:NLZeroBlue}\end{gathered}$$ $$\begin{gathered}
\mathbf{N}_{n+}(\lambda;y,w,m)\begin{bmatrix}1 & 0\\\displaystyle
-\frac{\Gamma(\tfrac{1}{2}-m){\mathrm{e}}^{2L_+(\lambda;y,m)}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}}{\mathrm{e}}^{-n(2g_+(\lambda;y)-V(\lambda;y))} & 1\end{bmatrix}=\\
\mathbf{N}_{n-}(\lambda;y,w,m)\begin{bmatrix}1 & 0\\\displaystyle
\frac{\Gamma(\tfrac{1}{2}-m){\mathrm{e}}^{2L_-(\lambda;y,m)}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}}{\mathrm{e}}^{-n(2g_-(\lambda;y)-V(\lambda;y))} & 1\end{bmatrix}{\mathrm{i}}\sigma_1{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)\sigma_3}
,\quad\lambda\in{{L^{\infty,2}_\squareurblack}}.
\label{eq:NLInftyRedTwo}\end{gathered}$$ $$\begin{gathered}
\mathbf{N}_{n+}(\lambda;y,w,m)\begin{bmatrix}1 & \displaystyle-{\mathrm{e}}^{{\mathrm{i}}\pi(m+1)}\frac{\Gamma(\tfrac{1}{2}+m){\mathrm{e}}^{-2L_+(\lambda;y,m)}{\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}({{\lambda^{m+1}_\squarellblack}})_+}{\mathrm{e}}^{n(2g_+(\lambda;y)-V(\lambda;y))}\\0 & 1\end{bmatrix}=\\
\mathbf{N}_{n-}(\lambda;y,w,m)
\begin{bmatrix}
1 & \displaystyle{\mathrm{e}}^{-{\mathrm{i}}\pi (m+1)}\frac{\Gamma(\tfrac{1}{2}+m){\mathrm{e}}^{-2L_-(\lambda;y,m)}{\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}({{\lambda^{m+1}_\squarellblack}})_-}{\mathrm{e}}^{n(2g_-(\lambda;y)-V(\lambda;y))}\\
0 & 1
\end{bmatrix}{\mathrm{i}}\sigma_1{\mathrm{e}}^{\delta(y,m)\sigma_3}{\mathrm{e}}^{-{\mathrm{i}}n K_2(y)\sigma_3}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)\sigma_3},\\\lambda\in{{L^{\infty,2}_\squarellblack}},
\label{eq:NLInftyBlueTwo}\end{gathered}$$ where $${\mathrm{e}}^{\delta(y,m)}:=-2\cos(\pi m){\mathrm{e}}^{\tilde{\gamma}(y,m)},
\label{eq:e-to-the-delta}$$ in which the product $\Gamma(\tfrac{1}{2}-m)\Gamma(\tfrac{1}{2}+m)$ has been eliminated using [@DLMF Eq. 5.5.3].
Referring to , it is the fact that ${\mathrm{e}}^{\delta(y,m)}=0$ and hence $\delta(y,m)$ is undefined when $m\in\mathbb{Z}+\tfrac{1}{2}$ that excludes the latter values from consideration in this section and hence in the statements of Theorem \[theorem:eye\], Corollary \[corollary:eye-zeros-and-poles:better\], and Theorem \[theorem:density\].
Steepest descent. Outer model problem and its solution
------------------------------------------------------
### Steepest descent and the derivation of the outer model Riemann-Hilbert problem
For the steepest descent step, we take advantage of the factorization of the jump matrix evidenced in the formulæ and . Let ${{\Lambda^\pm_\squareurblack}}$ denote lens-shaped domains immediately to the left ($+$) and right ($-$) of ${{L^{\infty,2}_\squareurblack}}$. Define $$\mathbf{O}_n(\lambda;y,w,m):=\mathbf{N}_n(\lambda;y,w,m)\begin{bmatrix}1 & 0\\
\displaystyle\mp\frac{\Gamma(\tfrac{1}{2}-m){\mathrm{e}}^{2L(\lambda;y,m)}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}}{\mathrm{e}}^{-n(2g(\lambda;y)-V(\lambda;y))} & 1\end{bmatrix},\quad \lambda\in{{\Lambda^\pm_\squareurblack}}.$$ Similarly, let ${{\Lambda^\pm_\squarellblack}}$ denote lens-shaped domains immediately to the left ($+$) and right ($-$) of ${{L^{\infty,2}_\squarellblack}}$, and define $$\mathbf{O}_n(\lambda;y,w,m):=\mathbf{N}_n(\lambda;y,w,m)\begin{bmatrix}1 &
\displaystyle
\mp{\mathrm{e}}^{\pm{\mathrm{i}}\pi(m+1)}\frac{\Gamma(\tfrac{1}{2}+m){\mathrm{e}}^{-2L(\lambda;y,m)}{\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}{{\lambda^{m+1}_\squarellblack}}}{\mathrm{e}}^{n(2g(\lambda;y)-V(\lambda;y))}\\0 & 1\end{bmatrix},\quad\lambda\in{{\Lambda^\pm_\squarellblack}}.$$ For all other values of $\lambda$ for which $\mathbf{N}_n(\lambda;y,w,m)$ is well-defined, we simply set $\mathbf{O}_n(\lambda;y,w,m):=\mathbf{N}_n(\lambda;y,w,m)$. If we denote by $\partial{{\Lambda^\pm_\squareurblack}}$ (resp., $\partial{{\Lambda^\pm_\squarellblack}}$) the arc of the boundary of ${{\Lambda^\pm_\squareurblack}}$ (resp., ${{\Lambda^\pm_\squarellblack}}$) distinct from ${{L^{\infty,2}_\squareurblack}}$ (resp., ${{L^{\infty,2}_\squarellblack}}$), but with the same initial and terminal endpoints, then the boundary values taken by $\mathbf{O}_n(\lambda;y,w,m)$ on these arcs satisfy the jump conditions $$\mathbf{O}_{n+}(\lambda;y,w,m)=\mathbf{O}_{n-}(\lambda;y,w,m)\begin{bmatrix}
1 & 0\\
\displaystyle\frac{\Gamma(\tfrac{1}{2}-m){\mathrm{e}}^{2L(\lambda;y,m)}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}{{\lambda^{-(m+1)}_\squarellblack}}}{\mathrm{e}}^{-n(2g(\lambda;y)-V(\lambda;y))} & 1\end{bmatrix},\quad\lambda\in\partial{{\Lambda^\pm_\squareurblack}},$$ and $$\mathbf{O}_{n+}(\lambda;y,w,m)=\mathbf{O}_{n-}(\lambda;y,w,m)\begin{bmatrix}
1 & \displaystyle{\mathrm{e}}^{\pm{\mathrm{i}}\pi (m+1)}\frac{\Gamma(\tfrac{1}{2}+m){\mathrm{e}}^{-2L(\lambda;y,m)}{\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)}}{\sqrt{2\pi}{{\lambda^{m+1}_\squarellblack}}}{\mathrm{e}}^{n(2g(\lambda;y)-V(\lambda;y))}\\
0 & 1\end{bmatrix},\quad\lambda\in\partial{{\Lambda^\pm_\squarellblack}}.$$ The effect of the transformation from $\mathbf{N}_n(\lambda;y,w,m)$ to $\mathbf{O}_n(\lambda;y,w,m)$ is that the jump matrices for $\mathbf{O}_n(\lambda;y,w,m)$ on ${{L^{\infty,2}_\squareurblack}}$ and ${{L^{\infty,2}_\squarellblack}}$ are now simply off-diagonal matrices: $$\mathbf{O}_{n+}(\lambda;y,w,m)=\mathbf{O}_{n-}(\lambda;y,w,m){\mathrm{i}}\sigma_1{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)\sigma_3},\quad\lambda\in{{L^{\infty,2}_\squareurblack}},$$ and $$\mathbf{O}_{n+}(\lambda;y,w,m)=\mathbf{O}_{n-}(\lambda;y,w,m){\mathrm{i}}\sigma_1{\mathrm{e}}^{\delta(y,m)\sigma_3}{\mathrm{e}}^{-{\mathrm{i}}nK_2(y)\sigma_3}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)\sigma_3},\quad\lambda\in{{L^{\infty,2}_\squarellblack}}.$$ On all remaining arcs of $L$, the boundary values of $\mathbf{O}_n(\lambda;y,w,m)$ agree with those of $\mathbf{N}_n(\lambda;y,w,m)$, which are related by the jump conditions –. Finally, we note that $\lambda\mapsto\mathbf{O}_n(\lambda;y,w,m)$ is analytic for $\lambda\in \mathbb{C}\setminus\Sigma_\mathbf{O}$, where $\Sigma_\mathbf{O}:=L\cup\partial{{\Lambda^\pm_\squareurblack}}\cup\partial{{\Lambda^\pm_\squarellblack}}$, taking continuous boundary values from each component of its domain of analyticity, and satisfies $\mathbf{O}_n(\lambda;y,w,m)\to\mathbb{I}$ as $\lambda\to\infty$.
The placement of the arcs of $L$ relative to the Stokes graph of $y$ now ensures that all jump matrices converge exponentially fast to the identity as $n\to+\infty$ with the exception of those on the arcs ${{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,3}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}}$. The convergence holds uniformly on compact subsets of each open contour arc, as well as uniformly in neighborhoods of $\lambda=0$ and $\lambda=\infty$. Building in suitable assumptions about the behavior near the four roots of $P(\lambda;y,C(y))$, we postulate the following model Riemann-Hilbert problem as an asymptotic description of $\mathbf{O}_n(\lambda;y,w,m)$ away from these four points.
Given $n\in\mathbb{Z}$, $y\in E_\mathrm{R}\setminus\mathbb{R}$, $w\in\mathbb{C}$, and $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$, seek a $2\times 2$ matrix function $\lambda\mapsto\dot{\mathbf{O}}^\mathrm{out}(\lambda)=\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ with the following properties:
- **Analyticity:** $\lambda\mapsto\dot{\mathbf{O}}^{\mathrm{out}}(\lambda)$ is analytic in the domain $\lambda\in\mathbb{C}\setminus({{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,3}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}})$. It takes continuous boundary values on the three indicated arcs of $L$ except at the four endpoints $\lambda_j(y)$ at which we require that all four matrix elements are $\mathcal{O}((\lambda-\lambda_j(y))^{-1/4})$.
- **Jump conditions:** The boundary values $\dot{\mathbf{O}}^{\mathrm{out}}_\pm(\lambda)$ are related on each arc of the jump contour by the following formulæ: $$\dot{\mathbf{O}}^{\mathrm{out}}_+(\lambda)=\dot{\mathbf{O}}^{\mathrm{out}}_-(\lambda){\mathrm{i}}\sigma_1{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)\sigma_3},\quad\lambda\in{{L^{\infty,2}_\squareurblack}},$$ $$\dot{\mathbf{O}}^{\mathrm{out}}_+(\lambda)=\dot{\mathbf{O}}^{\mathrm{out}}_-(\lambda){\mathrm{e}}^{-{\mathrm{i}}n K_1(y)\sigma_3},\quad\lambda\in{{L^{\infty,3}_\squareurblack}},$$ and $$\dot{\mathbf{O}}^{\mathrm{out}}_+(\lambda)=\dot{\mathbf{O}}^{\mathrm{out}}_-(\lambda){\mathrm{i}}\sigma_1{\mathrm{e}}^{\delta(y,m)\sigma_3}{\mathrm{e}}^{-{\mathrm{i}}n K_2(y)\sigma_3}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)\sigma_3},\quad\lambda\in{{L^{\infty,2}_\squarellblack}}.$$
- **Asymptotics:** $\dot{\mathbf{O}}^{\mathrm{out}}(\lambda)\to\mathbb{I}$ as $\lambda\to\infty$.
\[rhp:outer-elliptic\]
The jump diagram for Riemann-Hilbert Problem \[rhp:outer-elliptic\] is illustrated in Figure \[fig:OuterEllipticR\]. The solution of this problem (see Section \[sec:elliptic-outer-solution\] below) is called the *outer parametrix*.
![The jump contour and jump conditions for Riemann-Hilbert Problem \[rhp:outer-elliptic\]. Note that we denote by $\lambda_0$ the vertex of the Stokes graph adjacent to $\infty$.[]{data-label="fig:OuterEllipticR"}](OuterEllipticR.pdf)
### Solution of the outer model Riemann-Hilbert problem {#sec:elliptic-outer-solution}
To solve Riemann-Hilbert Problem \[rhp:outer-elliptic\], first let $H(\lambda;y)$ be defined by the formula $$\begin{gathered}
H(\lambda;y):=\frac{R(\lambda;y)K_1(y)}{2\pi}\int_{{{L^{\infty,3}_\squareurblack}}}\frac{{\mathrm{d}}\ell}{R(\ell;y)(\ell-\lambda)}+\frac{R(\lambda;y)({\mathrm{i}}K_2(y)+\eta(y))}{2\pi{\mathrm{i}}}\int_{{{L^{\infty,2}_\squarellblack}}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)(\ell-\lambda)},\\\lambda\in\mathbb{C}\setminus({{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,3}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}}).
\label{eq:H-exponent}\end{gathered}$$ Here, $\eta(y)$ is uniquely determined so that $H(\infty;y)$ is well-defined: $${\mathrm{i}}K_1(y)\int_{{L^{\infty,3}_\squareurblack}}\frac{{\mathrm{d}}\ell}{R(\ell;y)} + ({\mathrm{i}}K_2(y)+\eta(y))\int_{{L^{\infty,2}_\squarellblack}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)}=0.
\label{eq:eta-defining-equation}$$ Unlike the real-valued quantities $K_1(y)$ and $K_2(y)$, $\eta(y)$ is complex-valued, and it is well-defined because its coefficient is a complete elliptic integral, necessarily nonzero. The boundary values taken by $H(\lambda;y)$ on its jump contour are related by the conditions $$H_+(\lambda;y)+H_-(\lambda;y)=0,\quad\lambda\in{{L^{\infty,2}_\squareurblack}},
\label{eq:Hjump-1}$$ $$H_+(\lambda;y)-H_-(\lambda;y)={\mathrm{i}}K_1(y),\quad\lambda\in {{L^{\infty,3}_\squareurblack}},$$ and $$H_+(\lambda;y)+H_-(\lambda;y)={\mathrm{i}}K_2(y)+\eta(y),\quad\lambda\in{{L^{\infty,2}_\squarellblack}}.
\label{eq:Hjump-3}$$ We also define a related function $h(\lambda;y)$ by $$h(\lambda;y):=\frac{1}{2}\varphi(\lambda)+\frac{R(\lambda;y)}{{\mathrm{i}}y\lambda}+\frac{\nu(y)R(\lambda;y)}{2\pi{\mathrm{i}}}\int_{{{L^{\infty,2}_\squarellblack}}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)(\ell-\lambda)},\quad
\lambda\in\mathbb{C}\setminus({{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}})
\label{eq:h-exponent}$$ (note that $h(\lambda;y)$ is analytic at $\lambda=0$ because $R(0;y)=\tfrac{1}{2}{\mathrm{i}}y$), in which $\nu(y)$ is a constant determined uniquely by setting to zero the coefficient of the dominant term proportional to $\lambda$ in the Laurent series of $h$ at $\lambda=\infty$, making $h(\infty;y)$ well-defined: $$1-\frac{\nu(y)y}{4\pi}\int_{{L^{\infty,2}_\squarellblack}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)}=0.
\label{eq:nu-defining-equation}$$ The analogues of the conditions – for $h$ are $$h_+(\lambda;y)+h_-(\lambda;y)=\varphi(\lambda),\quad\lambda\in{{L^{\infty,2}_\squareurblack}},$$ $$h_+(\lambda;y)-h_-(\lambda;y)=0,\quad\lambda\in{{L^{\infty,3}_\squareurblack}},$$ and $$h_+(\lambda;y)+h_-(\lambda;y)=\varphi(\lambda)+\nu(y),\quad\lambda\in{{L^{\infty,2}_\squarellblack}}.$$
It follows that the matrix $\mathbf{P}_n(\lambda;y,w,m)$ related to the solution $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ of Riemann-Hilbert Problem \[rhp:outer-elliptic\] by $$\mathbf{P}_n(\lambda;y,w,m):={\mathrm{e}}^{-(nH(\infty;y)+{\mathrm{i}}w h(\infty,y))\sigma_3}\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m){\mathrm{e}}^{(nH(\lambda;y)+{\mathrm{i}}w h(\lambda;y))\sigma_3}
\label{eq:POout}$$ has the same properties of analyticity, boundedness, and identity normalization at $\lambda=\infty$ as does $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$, but the jump conditions for $\mathbf{P}_n(\lambda;y,w,m)$ take the form $$\mathbf{P}_{n+}(\lambda;y,w,m)=\mathbf{P}_{n-}(\lambda;y,w,m){\mathrm{i}}\sigma_1,\quad\lambda\in{{L^{\infty,2}_\squareurblack}},
\label{eq:PLInftyRedTwo}$$ $$\mathbf{P}_{n+}(\lambda;y,w,m)=\mathbf{P}_{n-}(\lambda;y,w,m),\quad\lambda\in{{L^{\infty,3}_\squareurblack}},
\label{eq:PLInftyRedThree}$$ and $$\mathbf{P}_{n+}(\lambda;y,w,m)=\mathbf{P}_{n-}(\lambda;y,w,m){\mathrm{i}}\sigma_1{\mathrm{e}}^{(\delta(y,m)+{\mathrm{i}}w\nu(y))\sigma_3}{\mathrm{e}}^{n\eta(y)\sigma_3},\quad\lambda\in{{L^{\infty,2}_\squarellblack}}.
\label{eq:PLInftyBlueTwo}$$ The jump condition together with the continuity of the boundary values taken by $\mathbf{P}_n(\lambda;y,m)$ on ${{L^{\infty,3}_\squareurblack}}$ from both sides indicates that the domain of analyticity of $\mathbf{P}_n(\lambda;y,w,m)$ is precisely the “two-cut” contour $\Sigma^\mathrm{out}(y)={{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}}$.
Let $\mathfrak{a}$ denote a counterclockwise-oriented loop surrounding the cut ${{L^{\infty,2}_\squareurblack}}$, and define the *Abel mapping* $A(\lambda;y)$ by $$A(\lambda;y):=2\pi{\mathrm{i}}\left[\oint_\mathfrak{a}\frac{{\mathrm{d}}\ell}{R(\ell;y)}\right]^{-1}\int_{\lambda_0(y)}^\lambda\frac{{\mathrm{d}}\ell}{R(\ell;y)},\quad\lambda\in\mathbb{C}\setminus({{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,3}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}}),
\label{eq:Abel-def}$$ where $\lambda_0(y)$ is the vertex adjacent to $\infty$ on the Stokes graph of $y$ (hence the initial endpoint of ${{L^{\infty,2}_\squareurblack}}$). Note that $A(\lambda;y)$ is well-defined because $1/R(\lambda;y)$ is integrable at $\lambda=\infty$. The integral over the corresponding $\mathfrak{b}$-cycle (in the canonical homology basis determined from $\mathfrak{a}$) of the $\mathfrak{a}$-normalized holomorphic differential that is the integrand of $A(\lambda;y)$ is then given by $$B(y):=-4\pi{\mathrm{i}}\left[\oint_\mathfrak{a}\frac{{\mathrm{d}}\ell}{R(\ell;y)}\right]^{-1}\int_{{L^{\infty,3}_\squareurblack}}\frac{{\mathrm{d}}\ell}{R(\ell;y)}.
\label{eq:B-of-y}$$ Since $$\oint_\mathfrak{a}\frac{{\mathrm{d}}\ell}{R(\ell;y)}=-2\int_{{L^{\infty,2}_\squareurblack}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)} = 2\int_{{L^{\infty,2}_\squarellblack}}\frac{{\mathrm{d}}\ell}{R_+(\ell;y)},$$ with the second equality following from , we can use to write $\eta(y)$ in the form $$\eta(y)=-{\mathrm{i}}K_2(y)+{\mathrm{i}}K_1(y)\frac{B(y)}{2\pi{\mathrm{i}}}.
\label{eq:eta-again}$$ It is a general fact [@Dubrovin81] that $\mathrm{Re}(B(y))<0$, which implies that therefore $\mathrm{Re}(\eta(y))\neq 0$ unless $K_1(y)=0$. More concretely, by comparing with the Stokes graphs illustrated in Figure \[fig:StokesGraphReal\], it is easy to see that for $y>0$ in the domain $E$, $B(y)$ is real and strictly negative. The Abel mapping satisfies the following jump conditions: $$A_+(\lambda;y)+A_-(\lambda;y)=0,\quad\lambda\in{{L^{\infty,2}_\squareurblack}},
\label{eq:ALInftyRedTwo}$$ $$A_+(\lambda;y)-A_-(\lambda;y)=-2\pi{\mathrm{i}},\quad\lambda\in{{L^{\infty,3}_\squareurblack}},$$ and $$A_+(\lambda;y)+A_-(\lambda;y)=-B(y),\quad\lambda\in{{L^{\infty,2}_\squarellblack}}.
\label{eq:ALInftyBlueTwo}$$ We now recall the *Riemann theta function* $\Theta(z,B)$ defined by the series $$\Theta(z,B):=\sum_{k\in\mathbb{Z}}{\mathrm{e}}^{kz+\tfrac{1}{2}Bk^2},\quad z\in\mathbb{C},\quad \mathrm{Re}(B)<0.
\label{eq:Riemann-Theta}$$ In the notation of [@DLMF §20], $\Theta(z,B)=\theta_3(w|\tau)$ where $z=2{\mathrm{i}}w$ and $B=2\pi{\mathrm{i}}\tau$ (i.e., in the currently relevant genus-$1$ setting the Riemann theta function basically coincides with one of the Jacobi theta functions). For each $B$ in the left half-plane, $\Theta(z,B)$ is an entire function of $z$ with the automorphic properties $$\Theta(-z,B)=\Theta(z,B),\quad
\Theta(z+2\pi{\mathrm{i}},B)=\Theta(z,B),\quad\text{and}\quad\Theta(z\pm B,B)={\mathrm{e}}^{-\tfrac{1}{2}B}{\mathrm{e}}^{\mp z}\Theta(z,B).
\label{eq:automorphic}$$ The function $\Theta(z,B)$ has simple zeros only, at each of the lattice points $z=(j+\tfrac{1}{2})2\pi{\mathrm{i}}+ (k+\tfrac{1}{2})B$, $(j,k)\in\mathbb{Z}^2$. Given a point $\kappa\in\mathbb{C}\setminus({{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,3}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}})$ and a complex number $s$, consider the meromorphic functions defined by $$q^\pm(\lambda;\kappa,s,y):=\frac{\Theta(A(\lambda;y)\pm A(\kappa;y)\pm{\mathrm{i}}\pi\pm\tfrac{1}{2}B(y)-s,B(y))}{\Theta(A(\lambda;y)\pm A(\kappa;y)\pm{\mathrm{i}}\pi\pm\tfrac{1}{2}B(y),B(y))},\quad\lambda\in\mathbb{C}\setminus({{L^{\infty,2}_\squareurblack}}\cup{{L^{\infty,3}_\squareurblack}}\cup{{L^{\infty,2}_\squarellblack}}).
\label{eq:gpm-def}$$ In fact, $q^+(\lambda;\kappa,s,y)$ is analytic for $\lambda$ in its domain of definition, but $q^-(\lambda;\kappa,s,y)$ has a simple pole at $\lambda=\kappa$ as its only singularity (unless $s$ is an integer linear combination of $2\pi{\mathrm{i}}$ and $B(y)$ in which case the singularity is cancelled and $q^-(\lambda;\kappa,s,y)$ is analytic as well). Consider the matrix function $$\mathbf{Q}(\lambda;\kappa,s,y):=\begin{bmatrix}q^+(\lambda;\kappa,s,y) & -{\mathrm{i}}q^-(\lambda;\kappa,-s,y)\\
{\mathrm{i}}q^-(\lambda;\kappa,s,y) & q^+(\lambda;\kappa,-s,y)\end{bmatrix}.
\label{eq:Q-def}$$ Then from the jump conditions – and the automorphic properties , it is easy to check that $\mathbf{Q}(\lambda;\kappa,s,y)$ satisfies the jump conditions: $$\mathbf{Q}_+(\lambda;\kappa,s,y)=\mathbf{Q}_-(\lambda;\kappa,s,y)\begin{bmatrix}0 & -{\mathrm{i}}\\{\mathrm{i}}& 0\end{bmatrix},\quad\lambda\in{{L^{\infty,2}_\squareurblack}},
\label{eq:QLInftyRedTwo}$$ $$\mathbf{Q}_+(\lambda;\kappa,s,y)=\mathbf{Q}_-(\lambda;\kappa,s,y),\quad\lambda\in{{L^{\infty,3}_\squareurblack}},$$ and $$\mathbf{Q}_+(\lambda;\kappa,s,y)=\mathbf{Q}_-(\lambda;\kappa,s,y)\begin{bmatrix}0 & -{\mathrm{i}}{\mathrm{e}}^{s}\\{\mathrm{i}}{\mathrm{e}}^{-s} & 0\end{bmatrix},\quad \lambda\in{{L^{\infty,2}_\squarellblack}}.
\label{eq:QLInftyBlueTwo}$$
To construct $\mathbf{P}_n(\lambda;y,w,m)$ from $\mathbf{Q}$, we need to remove the pole from the off-diagonal elements of $\mathbf{Q}$ while slightly modifying the jump conditions on $\Sigma^\mathrm{out}(y)$. To this end, we observe that we have the freedom to introduce mild singularities into $\mathbf{P}_n(\lambda;y,w,m)$ at the four roots of $P(\cdot;y,C(y))$, here denoted $\lambda_0(y)$ (adjacent to $\infty$ in the Stokes graph of $y$), $\lambda_1(y)$ (adjacent to $\lambda_0(y)$ in the Stokes graph), $\lambda_0(y)^{-1}$, and $\lambda_1(y)^{-1}$. Let $\phi(\lambda;y)$ denote the unique function analytic for $\lambda\in\Sigma^\mathrm{out}(y)$ with $\phi(\lambda;y)\to 1$ as $\lambda\to\infty$ that satisfies $$\phi(\lambda;y)^4=\frac{(\lambda-\lambda_0(y))(\lambda-\lambda_1(y)^{-1})}{(\lambda-\lambda_1(y))(\lambda-\lambda_0(y)^{-1})}.
\label{eq:phi-to-the-fourth}$$ Then set $$f^\mathrm{D}(\lambda;y):=\frac{1}{2}(\phi(\lambda;y)+\phi(\lambda;y)^{-1}),\quad
f^\mathrm{OD}(\lambda;y):=\frac{1}{2{\mathrm{i}}}(\phi(\lambda;y)-\phi(\lambda;y)^{-1}),\quad\lambda\in\mathbb{C}\setminus\Sigma^\mathrm{out}(y).
\label{eq:qDqOD}$$ It is easy to see that on both arcs of $\Sigma^\mathrm{out}(y)$, the jump condition $\phi_+(\lambda;y)=-{\mathrm{i}}\phi_-(\lambda;y)$ holds. This implies the corresponding jump conditions $$f^\mathrm{D}_+(\lambda;y)=f^\mathrm{OD}_-(\lambda;y),\quad
f^\mathrm{OD}_+(\lambda;y)=-f^\mathrm{D}_-(\lambda;y),\quad\lambda\in\Sigma^\mathrm{out}(y).
\label{eq:qDOD}$$ The functions $\lambda\mapsto f^\mathrm{D}(\lambda;y)$ and $\lambda\mapsto f^\mathrm{OD}(\lambda;y)$ are analytic in their domain of definition, and they are bounded except near the four roots of $P(\lambda;y,C(y))$, where they exhibit negative one-fourth root singularities. Also, $$f^\mathrm{D}(\lambda;y)=1+\mathcal{O}(\lambda^{-2})\quad\text{and}\quad
f^\mathrm{OD}(\lambda;y)=\frac{1}{4}{\mathrm{i}}(\lambda_0(y)+\lambda_1(y)^{-1}-\lambda_1(y)-\lambda_0(y)^{-1})\lambda^{-1}+\mathcal{O}(\lambda^{-2}),\quad\lambda\to\infty.$$ Observe that $$\begin{split}
f^\mathrm{D}(\lambda;y)f^\mathrm{OD}(\lambda;y)&=\frac{1}{4{\mathrm{i}}\phi(\lambda;y)^2}(\phi(\lambda;y)^4-1)\\
&=\frac{(\lambda-\lambda_0(y))(\lambda-\lambda_1(y)^{-1})-(\lambda-\lambda_1(y))(\lambda-\lambda_0(y)^{-1})}{4{\mathrm{i}}\phi(\lambda;y)^2(\lambda-\lambda_1(y))(\lambda-\lambda_0(y)^{-1})}\\
&=\frac{(\lambda_1(y)+\lambda_0(y)^{-1}-\lambda_0(y)-\lambda_1(y)^{-1})(\lambda-\kappa(y))}{4{\mathrm{i}}\phi(\lambda;y)^2(\lambda-\lambda_1(y))(\lambda-\lambda_0(y)^{-1}),}
\end{split}$$ where $$\kappa(y):=\frac{\lambda_1(y)\lambda_0(y)^{-1}-\lambda_0(y)\lambda_1(y)^{-1}}{\lambda_1(y)+\lambda_0(y)^{-1}-\lambda_0(y)-\lambda_1(y)^{-1}}=\frac{\lambda_0(y)+\lambda_1(y)}{1+\lambda_0(y)\lambda_1(y)}.
\label{eq:mu-def}$$ Therefore the product $f^\mathrm{D}(\lambda;y)f^\mathrm{OD}(\lambda;y)$ has precisely one simple zero in its domain of definition, namely $\lambda=\kappa(y)$, and this value is either a zero of $f^\mathrm{D}(\lambda;y)$ or $f^\mathrm{OD}(\lambda;y)$ but not both. In the case that $y>0$, the roots of $P(\lambda;y,C(y))$ lie on the imaginary axis with $1<|\lambda_1(y)|<|\lambda_0(y)|$. It is easy to check that $\phi(\lambda;y)$ is positive on the imaginary axis excluding the jump contour $\Sigma^\mathrm{out}(y)$, which also implies that $f^\mathrm{D}(\lambda;y)>0$ for such $\lambda$. The inequality $1<|\lambda_1(y)|<|\lambda_0(y)|$ implies that $\kappa(y)$ is negative imaginary, and that $|\kappa(y)|>|\lambda_1(y)^{-1}|$. Thus $\kappa(y)$ lies below both intervals of the jump contour $\Sigma^\mathrm{out}(y)$ on the imaginary axis, and hence $f^\mathrm{D}(\kappa(y);y)>0$. It therefore follows that for $y>0$, $\kappa(y)$ is a zero of $f^\mathrm{OD}(\lambda;y)$. This will remain so as $y$ varies in $E_\mathrm{R}$ so long as $\kappa(y)$ does not pass through either arc of $\Sigma^\mathrm{out}(y)$. We proceed under the assumption that $\lambda=\kappa(y)$ is a simple zero of $f^\mathrm{OD}(\lambda;y)$, and indicate below how the procedure should be modified if $\kappa(y)$ should ever intersect $\Sigma^\mathrm{out}(y)$, a possibility which is difficult to rule out analytically, although we have never observed it numerically.
We may obtain $\mathbf{P}_n(\lambda;y,w,m)$ from $\mathbf{Q}(\lambda;\mu(y),s,y)$ by multiplying the diagonal elements by $f^\mathrm{D}(\lambda;y)$ and the off-diagonal elements by $\pm f^\mathrm{OD}(\lambda;y)$, and by normalizing the result via left-multiplication by a constant matrix: $$\begin{gathered}
\mathbf{P}_n(\lambda;y,w,m):=\begin{bmatrix} Q_{11}(\infty;\kappa(y),s,y)^{-1} & 0\\
0 & Q_{22}(\infty;\kappa(y),s,y)^{-1}\end{bmatrix}\\
{}\cdot
\begin{bmatrix} f^\mathrm{D}(\lambda;y)Q_{11}(\lambda;\kappa(y),s,y) & f^\mathrm{OD}(\lambda;y)Q_{12}(\lambda;\kappa(y),s,y)\\
-f^\mathrm{OD}(\lambda;y)Q_{21}(\lambda;\kappa(y),s,y) & f^\mathrm{D}(\lambda;y)Q_{22}(\lambda;\kappa(y),s,y)
\end{bmatrix}.
\label{eq:P-formula}\end{gathered}$$ Combining – with shows that $\mathbf{P}_n(\lambda;y,w,m)$ satisfies the prescribed jump conditions – provided that the free parameter $s$ is given the value $$s=s_n(y,w,m):=-\delta(y,m)-{\mathrm{i}}w\nu(y)-n\eta(y).
\label{eq:sn}$$ Since the zero $\lambda=\kappa(y)$ of $f^\mathrm{OD}(\lambda;y)$ cancels the simple pole of $Q_{12}(\lambda;\kappa(y),s,y)$ and $Q_{21}(\lambda;\kappa(y),s,y)$, the singularity is removable and hence $\mathbf{P}_n(\lambda;y,w,m)$ is indeed analytic for $\lambda\in\mathbb{C}\setminus\Sigma^\mathrm{out}(y)$ with negative one-fourth root singularities at the roots of $P(\lambda;y,C(y))$. Finally, the constant matrix pre-factor ensures the asymptotic normalization condition that $\mathbf{P}_n(\infty;y,w,m)=\mathbb{I}$. Now that $\mathbf{P}_n(\lambda;y,w,m)$ has been determined, we recover $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ using , , , and .
Finally we indicate what changes if $\kappa(y)$ passes through an arc of $\Sigma^\mathrm{out}(y)$ as $y$ varies in $E_\mathrm{R}$. It is easy to see that each time $\kappa(y)$ crosses an arc of $\Sigma^\mathrm{out}(y)$ transversely, the simple zero at $\lambda=\kappa(y)$ is exchanged between the functions $f^\mathrm{D}(\lambda;y)$ and $f^\mathrm{OD}(\lambda;y)$. To account for this correctly, one should define the value of $A(\kappa(y);y)$ appearing in by analytic continuation of the Abel mapping $A(\lambda;y)$ through the cuts, which has the effect of transferring the simple pole at $\lambda=\kappa(y)$ between the function $q^+(\lambda;\kappa(y),s,y)$ and $q^-(\lambda;\kappa(y),s,y)$ and hence between the off-diagonal and diagonal elements of $\mathbf{Q}(\lambda;\kappa(y),s,y)$. With this interpretation of $A(\kappa(y);y)$ the formula remains analytic in its domain of definition and yields the solution of Riemann-Hilbert Problem \[rhp:outer-elliptic\] through .
### Properties of the solution of the outer model Riemann-Hilbert problem
The constant pre-factor in also introduces singularities in the parameter space. In other words, $\mathbf{P}_n(\lambda;y,w,m)$ and hence also $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ will exist if and only if $Q_{11}(\infty;\kappa(y),s_n(y,w,m),y)Q_{22}(\infty;\kappa(y),s_n(y,w,m),y)\neq 0$. This is equivalent to the condition $\Theta(A(\infty;y)+A(\kappa(y);y)+{\mathrm{i}}\pi +\tfrac{1}{2}B(y)-s_n(y,w,m),B(y))\Theta(A(\infty;y)+A(\kappa(y);y)+{\mathrm{i}}\pi+\tfrac{1}{2}B(y)+s_n(y,w,m),B(y))\neq 0$. In other words, we see that Riemann-Hilbert Problem \[rhp:outer-elliptic\] has a unique solution provided that the parameters do not satisfy either (distinguished by a sign $\pm$) of the following conditions $$\begin{gathered}
\text{No solution of Riemann-Hilbert Problem~\ref{rhp:outer-elliptic}:}\\
A(\infty;y)+A(\kappa(y);y)\pm (\delta(y,m)+{\mathrm{i}}w\nu(y)+n\eta(y))\in 2\pi{\mathrm{i}}\mathbb{Z} + B(y)\mathbb{Z}.
\label{eq:outer-pole-divisor}\end{gathered}$$
For each $y\in E_\mathrm{R}$, the condition is independent of the choice of sign ($\pm$). \[lemma:one-divisor\]
Fix $y\in E_\mathrm{R}$. It is sufficient to show that $2A(\infty;y)+2A(\kappa(y);y)\in 2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}$. Let $\Gamma$ denote the hyperelliptic Riemann surface associated with the equation $\tau^2=P(\lambda;y,C(y))$, which we model as two copies (“sheets”) of the complex $\lambda$-plane identified along the two cuts making up $\Sigma^\mathrm{out}(y)$. Selecting one sheet on which $A(\lambda;y)$ is defined as in , we may extend the definition to the universal covering of $\Gamma$ by analytic continuation through the cuts or through ${{L^{\infty,3}_\squareurblack}}$. Then by taking the quotient of the continuation by the lattice of integer periods $\Lambda:=2\pi{\mathrm{i}}\mathbb{Z}+ B(y)\mathbb{Z}$, we arrive at a well-defined function on $\Gamma$ taking values in the corresponding Jacobian variety $\mathrm{Jac}(\Gamma):=\mathbb{C}/\Lambda$, the Abel map of $\Gamma$ denoted $\mathcal{A}(Q)$, $Q\in\Gamma$. Labeling the points $\lambda$ on the original sheet of definition of $A(\lambda;y)$ as $Q^+(\lambda)$, and their corresponding hyperelliptic involutes on the second sheet as $Q^-(\lambda)$, we observe that for any $\lambda$ not one of the four branch points of $\Gamma$, the equalities $A(\lambda;y)=\mathcal{A}(Q^+(\lambda))=-\mathcal{A}(Q^-(\lambda)))$ hold on $\mathrm{Jac}(\Gamma)$ because the base point of the integral in was chosen as a branch point. We may also take $\lambda=\infty$ in the above relations, and hence we have $2A(\infty;y)+2A(\kappa(y);y)=\mathcal{A}(Q^+(\infty))+\mathcal{A}(Q^+(\kappa(y)))-\mathcal{A}(Q^-(\infty))-\mathcal{A}(Q^-(\kappa(y)))$, which is usually written as $\mathcal{A}(\mathscr{D})$ for $\mathscr{D}:=Q^+(\infty)+Q^+(\kappa(y))-Q^-(\infty)-Q^-(\kappa(y))$ when the action of $\mathcal{A}$ is extended to divisors $\mathscr{D}$ as formal sums of points with integer coefficients. Therefore, $2A(\infty;y)+2A(\kappa(y);y)\in 2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}$ is equivalent to the condition that $\mathcal{A}(\mathscr{D})=0$ in $\mathrm{Jac}(\Gamma)$ for the indicated divisor $\mathscr{D}$. According to the Abel-Jacobi theorem, to establish this condition it suffices to construct a nonzero meromorphic function on $\Gamma$ with simple poles at $Q^-(\infty)$ and $Q^-(\kappa(y))$ and simple zeros at the hyperelliptic involutes $Q^+(\infty)$ and $Q^+(\kappa(y))$, with no other zeros or poles. The existence of such a function must take advantage of the formula , because the Riemann-Roch theorem asserts that the dimension of the linear space of meromorphic functions on the genus $g=1$ Riemann surface $\Gamma$ with divisor of the form $\mathscr{D}=Q^+(\infty)+Q^+(\zeta)-Q^-(\infty)-Q^-(\zeta)$ is $\mathrm{deg}(\mathscr{D})-g+1=0-1+1=0$ unless the divisor is *special*, implying that $\zeta$ is non-generic.
In order to construct the required function, let $\tau(Q^\pm(\lambda)):=\pm R(\lambda;y)$ define $\tau$ properly as a function on $\Gamma$, and consider the function $f:\Gamma\to\mathbb{C}$ given by $$f(Q):=\frac{\lambda(Q)^2+a\lambda(Q)+b + c\tau(Q)}{\lambda(Q)-\zeta}
\label{eq:f-of-P}$$ for constants $a$, $b$, $c$, and $\zeta$. The only possible singularities of this function are the two points on $\Gamma$ over $\lambda=\infty$ and the two points over $\lambda=\zeta$. Recall the roots of $P(\lambda;y,C(y))$: $\lambda_j=\lambda_j(y)$, $j=0,1$, and their reciprocals. As $Q\to Q^\pm(\infty)$, we have $\tau(Q)=\pm\tfrac{1}{2}{\mathrm{i}}y(\lambda(Q)^2 -\tfrac{1}{2}(\lambda_0+\lambda_1 +\lambda_0^{-1}+\lambda_1^{-1})\lambda(Q) + O(1))$, so to ensure that $f(Q^+(\infty))=0$ we must choose $$c:=\frac{2{\mathrm{i}}}{y} \quad\text{and}\quad a:=-\frac{1}{2}\left(\lambda_0+\lambda_1+\lambda_0^{-1}+\lambda_1^{-1}\right).
\label{eq:a-c}$$ With the above choice of $c$ it is also clear that $f(Q)=2\lambda(Q)+O(1)$ as $Q\to Q^-(\infty)$, so $f$ has a simple pole at $Q=Q^-(\infty)$. Given these choices and the divisor parameter $\zeta\in\mathbb{C}$, upon taking a generic value of $b$, $f(Q)$ will have simple poles at both $Q=Q^+(\zeta)$ and $Q=Q^-(\zeta)$. We may obviously choose $b$ uniquely such that $f(Q)$ is holomorphic at $Q=Q^+(\zeta)$: $$b:=-\zeta^2-a\zeta-c\tau(Q^+(\zeta))=-\zeta^2+\frac{1}{2}\left(\lambda_0+\lambda_1+\lambda_0^{-1}+\lambda_1^{-1}\right)\zeta-\frac{2{\mathrm{i}}}{y} R(\zeta;y).
\label{eq:b}$$ With $a,b,c$ determined for arbitrary fixed $\zeta$, there is no additional parameter available in the form to ensure that $f(Q^+(\zeta))=0$, a fact that is consistent with the Riemann-Roch argument given above.
So we take the point of view that $\zeta$ should be viewed as the additional parameter needed to guarantee that $f(Q^+(\zeta))=0$. Indeed, for this to be the case, the derivative with respect to $\lambda$ of the numerator in should vanish at $Q=Q^+(\zeta)$; we therefore require: $$2\zeta-\frac{1}{2}\left(\lambda_0+\lambda_1+\lambda_0^{-1}+\lambda_1^{-1}\right)=-\frac{2{\mathrm{i}}}{y}\frac{{\mathrm{d}}\tau}{{\mathrm{d}}\lambda}(Q^+(\zeta)) = -\frac{2{\mathrm{i}}}{y}\frac{{\mathrm{d}}R}{{\mathrm{d}}\lambda}(\zeta;y).
\label{eq:zeta-condition}$$ By implicit differentiation, $$\begin{gathered}
\frac{{\mathrm{d}}R}{{\mathrm{d}}\lambda}(\lambda;y)=-\frac{y^2}{8R(\lambda;y)}\big((\lambda-\lambda_0)(\lambda-\lambda_1)(\lambda-\lambda_0^{-1}) + (\lambda-\lambda_0)(\lambda-\lambda_1)(\lambda-\lambda_1^{-1})\\
{}+(\lambda-\lambda_0)(\lambda-\lambda_0^{-1})(\lambda-\lambda_1^{-1}) + (\lambda-\lambda_1)(\lambda-\lambda_0^{-1})(\lambda-\lambda_1^{-1})\big).
\label{eq:rho-prime}\end{gathered}$$ Substituting into and squaring both sides yields a cubic equation for $\zeta$ with solutions: $$\zeta=\kappa(y),\quad\zeta=\kappa(y)^{-1},\quad\text{and}\quad\zeta=0,$$ where $\kappa(y)$ is given by . These are precisely the three values of $\zeta\in\mathbb{C}$ for which the divisor $\mathscr{D}=Q^+(\infty)+Q^+(\zeta)-Q^-(\infty)-Q^-(\zeta)$ is special in the setting of the Riemann-Roch theorem. Selecting the desired solution $\zeta=\kappa(y)$, it remains only to confirm that holds *without squaring both sides*. But, since $-2a$ is the sum of roots of $P(\lambda;y,C(y))$, from we can also write $a=-{\mathrm{i}}y^{-1}$, and then since when $y>0$ we know that the branch cuts of $R(\lambda;y)$ lie on opposite halves of the imaginary axis and $\kappa(y)$ lies on the imaginary axis below both cuts, it follows that both sides of are negative imaginary for $\zeta=\kappa(y)$ and $y>0$. The persistence of for $\zeta=\kappa(y)$ as $y$ varies within $E_\mathrm{R}$ follows by analytic continuation, with the re-definition of $A(\kappa(y);y)$ as described in the last paragraph of Section \[sec:elliptic-outer-solution\], should $\kappa(y)$ pass through $\Sigma^\mathrm{out}(y)$ as both move in the complex $\lambda$-plane.
Numerical calculations allow us to find the exact lattice point corresponding to the sum of Abel maps appearing in the proof: $2A(\infty;y)+2A(\kappa(y);y)=-B(y)$ holds as an identity on $y\in E_\mathrm{R}$. In a similar way, one can also prove the identity $2A(0;y)-2A(\kappa(y);y)=2\pi{\mathrm{i}}$.
The parameter values excluded by the (equivalent) conditions are said to form the *Malgrange divisor* for Riemann-Hilbert Problem \[rhp:outer-elliptic\]. We have the following result.
Riemann-Hilbert Problem \[rhp:outer-elliptic\] has a unique solution with unit determinant provided that $n=0,1,2,3,\dots$, $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$, $y\in E_\mathrm{R}$, and $w\in\mathbb{C}$ do not lie in the Malgrange divisor . Moreover, for fixed $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ and $\epsilon>0$ arbitrarily small, $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ is uniformly bounded on the set of $(\lambda,n,y,w)$ satisfying $|w|\le K$ for some $K>0$ and the conditions $$\mathrm{dist}(\lambda,\{\lambda_0(y),\lambda_1(y),\lambda_0(y)^{-1},\lambda_1(y)^{-1}\})\ge\epsilon,
\label{eq:lambda-cheese}$$ $$\quad\mathrm{dist}(A(\infty;y)+A(\kappa(y);y)\pm (\delta(y,m)+{\mathrm{i}}w\nu(y)+n\eta(y)),2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z})\ge\epsilon,\quad\text{and}\quad \mathrm{dist}(y,\partial(E_\mathrm{R}))\ge\epsilon.
\label{eq:Swiss-Cheese}$$ \[lem:Outer-Bounded\]
Note that for fixed $n$ and $w=0$, the two conditions in bound $y$ within $E_\mathrm{R}$ by a distance $\epsilon$ from the boundary and also bound $y$ away from the points of the Malgrange divisor by a distance proportional to $\frac{\epsilon}{n}$, that is, an arbitrarily small fixed fraction of the spacing between the points of the divisor.
The uniqueness and unimodularity of the solution given existence are standard results. It remains to show the boundedness under the conditions $|w|\le K$, , and , which is not obvious because the solution formula for $\dot{\mathbf{O}}^{(n),\mathrm{out}}(\lambda;y,w,m)$ contains exponential factors and theta-function factors that grow exponentially with $n$, which is allowed to grow without bound. However, the conditions of Riemann-Hilbert Problem \[rhp:outer-elliptic\] only involve $n$ in the form of exponential factors ${\mathrm{e}}^{\pm{\mathrm{i}}n K_j(y)}$, $j=1,2$, which have unit modulus for all $n$ because $K_j(y)\in\mathbb{R}$ by the Boutroux conditions . The parameter space for Riemann-Hilbert Problem \[rhp:outer-elliptic\] is therefore a subset of a compact set even though $n$ can become unbounded. This fact leads to the claimed uniform boundedness. See [@BuckinghamM14 Proposition 8] for a similar argument with full details.
### Defining the approximation $\dot{u}_n(y,w;m)$
Reversing the substitutions $\mathbf{Y}_n\mapsto\mathbf{M}_n\mapsto\mathbf{N}_n\mapsto\mathbf{O}_n$ and using shows that the rational solution $u=u_n(x;m)$ of the Painlevé-III equation can be expressed for $y\in E_\mathrm{R}$ and $x=ny+w$ in terms of $\mathbf{O}_n(\lambda;y,w,m)$ in the form $$u_n(ny+w;m)=\frac{-{\mathrm{i}}Y^\infty_{1,12}(ny+w,m)}{Y^0_{0,11}(ny+w,m)Y^0_{0,12}(ny+w,m)}=\frac{-{\mathrm{i}}O^\infty_{n,1,12}(y,w,m)}{O^0_{n,0,11}(y,w,m)O^0_{n,0,12}(y,w,m)}.
\label{eq:u_n-O}$$ where $\mathbf{O}^0_{n,0}(y,w,m)=\mathbf{O}_n(0;y,w,m)$ and $\mathbf{O}^\infty_{n,1}(y,w,m)=\lim_{\lambda\to\infty}\lambda (\mathbf{O}_n(\lambda;y,w,m)-\mathbb{I})$. Note that here we do not have to exclude real values of $y$, because $u_n(ny+w;m)$ is rational in $y$ meaning that must be consistent for positive $y$ in $E_\mathrm{R}$. In Section \[sec:Error-elliptic\] we will show that under the conditions $|w|\le K$, and , the outer parametrix $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ is an accurate approximation of $\mathbf{O}_n(\lambda;y,w,m)$, from which $u_n(ny+w;m)$ can be extracted according to . This motivates the introduction of an explicit approximation for $u_n(ny+w;m)$ obtained by replacing $\mathbf{O}_n(\lambda;y,w,m)$ by its outer parametrix in : $$\dot{u}_n(y,w;m):=\frac{-{\mathrm{i}}\dot{O}^\infty_{n,1,12}(y,w,m)}{\dot{O}^0_{n,0,11}(y,w,m)\dot{O}^0_{n,0,12}(y,w,m)},
\label{eq:elliptic-dot-u-def}$$ where $\dot{\mathbf{O}}^0_{n,0}(y,w,m)=\dot{\mathbf{O}}_n^{\mathrm{out}}(0;y,w,m)$ and $\dot{\mathbf{O}}^\infty_{n,1}(y,w,m)=\lim_{\lambda\to\infty}\lambda (\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)-\mathbb{I})$. Using the formulæ developed in Section \[sec:elliptic-outer-solution\] for the outer parametrix then yields the formula for $\dot{u}_n(y,w;m)$, in which $$\begin{split}
\mathcal{Z}_n^\circ(y,w;m)&:=\Theta(A(\infty;y)+A(\kappa(y);y)+{\mathrm{i}}\pi+\tfrac{1}{2}B(y)-s_n(y,w,m),B(y))\\
\mathcal{Z}_n^\bullet(y,w;m)&:=\Theta(A(\infty;y)-A(\kappa(y);y)-{\mathrm{i}}\pi-\tfrac{1}{2}B(y)+s_n(y,w,m),B(y))\\
\mathcal{P}_n^\bullet(y,w;m)&:=\Theta(A(0;y)+A(\kappa(y);y)+{\mathrm{i}}\pi+\tfrac{1}{2}B(y)-s_n(y,w,m),B(y))\\
\mathcal{P}_n^\circ(y,w;m)&:=\Theta(A(0;y)-A(\kappa(y);y)-{\mathrm{i}}\pi-\tfrac{1}{2}B(y)+s_n(y,w,m),B(y))
\end{split}
\label{eq:four-factors}$$ and, using the fact that $\phi(0;y)^2=\lambda_0(y)\lambda_1(y)^{-1}$, $$N(y):=\frac{{\mathrm{i}}}{\kappa(y)}
\cdot
\frac{\Theta(A(0;y)+A(\kappa(y);y)+{\mathrm{i}}\pi+\tfrac{1}{2}B(y),B(y))\Theta(A(0;y)-A(\kappa(y);y)-{\mathrm{i}}\pi-\tfrac{1}{2}B(y),B(y))}{\Theta(A(\infty;y)+A(\kappa(y);y)+{\mathrm{i}}\pi+\tfrac{1}{2}B(y),B(y))\Theta(A(\infty;y)-A(\kappa(y);y)-{\mathrm{i}}\pi-\tfrac{1}{2}B(y),B(y))}.
\label{eq:N-of-y}$$ We recall that for $y\in E_\mathrm{R}$, $s_n(y,w,m)=-\delta(y,m)-{\mathrm{i}}w\nu(y)-n\eta(y)$. Observe that $N(y)$ is well-defined and nonzero for all $y\in E_\mathrm{R}$ and is independent of $n$ and $m$.
### The differential equation satisfied by $\dot{{{p}}}(w)=-{\mathrm{i}}\dot{u}_n(y,w;m)$ {#sec:DE-w}
Although $u_n(x;m)$ is a rational function of $x=ny+w$, the approximation $\dot{u}_n(y,w;m)$ is not a meromorphic function of $y$ because $C=C(y)$ is determined from the Boutroux equations , from which a direct computation shows that $\overline{\partial}C\neq 0$ in general, i.e., the real and imaginary parts of $C$ do not satisfy the Cauchy-Riemann equations with respect to the real and imaginary parts of $y$. On the other hand, since $s_n(y,w,m)$ is linear in $w$, it is obvious from with – that $\dot{u}_n(y,w;m)$ is a meromorphic function of $w$ for each fixed $y\in E_\mathrm{R}$. In order to establish the first statement of Theorem \[theorem:eye\], we will prove in this section that the related function $\dot{p}(w):=-{\mathrm{i}}\dot{u}_n(y,w;m)$ is in fact an elliptic function of $w$ satisfying the differential equation in which the constant $C=C(y)$ is determined from the Boutroux equations .
Rather than try to deal directly with the explicit formula , we argue indirectly from the conditions of Riemann-Hilbert Problem \[rhp:outer-elliptic\]. We first observe that the outer parametrix $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ satisfies a simple algebraic equation. Indeed, it is straightforward to check that the matrix $$\mathbf{G}(\lambda):=R(\lambda;y)\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)\sigma_3
\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)^{-1}
\label{eq:G-matrix-def}$$ is an entire function; its continuous boundary values match along the three arcs of the jump contour of $R$ and $\dot{\mathbf{O}}_n^{\mathrm{out}}$, and it is clearly bounded near the four roots of $R^2$, hence analyticity in the whole complex $\lambda$-plane follows by Morera’s theorem. Moreover, since $$R(\lambda;y)=\frac{1}{2}{\mathrm{i}}y\lambda^2+\frac{1}{2}\lambda +{\mathrm{i}}\frac{1-4C(y)}{4y}+ \mathcal{O}(\lambda^{-1}),\quad\lambda\to\infty,$$ Liouville’s theorem shows that $\mathbf{G}(\lambda)$ is a quadratic matrix polynomial in $\lambda$. Using the expansion $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)=\mathbb{I}+\lambda^{-1}\dot{\mathbf{O}}^\infty_{n,1}(y,w,m) + \lambda^{-2}\dot{\mathbf{O}}^\infty_{n,2}(y,w,m)+\mathcal{O}(\lambda^{-3})$ as $\lambda\to\infty$ shows that $$\mathbf{G}(\lambda)=\frac{1}{2}{\mathrm{i}}y\sigma_3\lambda^2 + \frac{1}{2}\left(\sigma_3+{\mathrm{i}}y\left[\dot{\mathbf{O}}^\infty_{n,1}(y,w,m),\sigma_3\right]\right)\lambda +
\mathbf{G}^\infty +
\mathcal{O}(\lambda^{-1}),
\quad\lambda\to\infty,
\label{eq:G-infty}$$ where $$\mathbf{G}^\infty:={\mathrm{i}}\frac{1-4C(y)}{4y}\sigma_3 +\frac{1}{2}\left[\dot{\mathbf{O}}^\infty_{n,1}+{\mathrm{i}}y\dot{\mathbf{O}}^\infty_{n,2},\sigma_3\right]-\frac{1}{2}{\mathrm{i}}y\left[\dot{\mathbf{O}}^\infty_{n,1},\sigma_3\right]\dot{\mathbf{O}}^\infty_{n,1}.
\label{eq:G0-def}$$ Also, using $$R(\lambda;y)=\frac{1}{2}{\mathrm{i}}y +\frac{1}{2}\lambda+\mathcal{O}(\lambda^2),\quad\lambda\to 0$$ and the expansion $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)=\dot{\mathbf{O}}^0_{n,0}(y,w,m)+\dot{\mathbf{O}}^0_{n,1}(y,w,m)\lambda+\mathcal{O}(\lambda^2)$ as $\lambda\to 0$ gives $$\mathbf{G}(\lambda)=\frac{1}{2}{\mathrm{i}}y\dot{\mathbf{O}}^0_{n,0}(y,w,m)\sigma_3\dot{\mathbf{O}}^0_{n,0}(y,w,m)^{-1} + \mathbf{G}_1^0\lambda+\mathcal{O}(\lambda^2),\quad\lambda\to 0,
\label{eq:G-zero}$$ where $$\mathbf{G}_1^0:=\frac{1}{2}\dot{\mathbf{O}}_{n,0}^0\sigma_3(\dot{\mathbf{O}}_{n,0}^0)^{-1} +\frac{1}{2}{\mathrm{i}}y\dot{\mathbf{O}}_{n,0}^0
\left[(\dot{\mathbf{O}}_{n,0}^0)^{-1}\dot{\mathbf{O}}_{n,1}^0,\sigma_3\right](\dot{\mathbf{O}}_{n,0}^0)^{-1}.
\label{eq:G1-def}$$ Therefore $\mathbf{G}(\lambda)$ is the quadratic matrix polynomial $$\mathbf{G}(\lambda)=\frac{1}{2}{\mathrm{i}}y\sigma_3\lambda^2 + \frac{1}{2}\left(\sigma_3+{\mathrm{i}}y\mathbf{A}(w)\right)\lambda+\frac{1}{2}{\mathrm{i}}y\mathbf{B}(w),
\label{eq:G-of-lambda}$$ where, suppressing explicit dependence on the parameters $y\in E$ and $m\in\mathbb{C}$, $$\mathbf{A}(w):=\left[\dot{\mathbf{O}}^\infty_{n,1}(y,w,m),\sigma_3\right]\quad \text{and}\quad
\mathbf{B}(w):=\dot{\mathbf{O}}^0_{n,0}(y,w,m)\sigma_3\dot{\mathbf{O}}^0_{n,0}(y,w,m)^{-1}.$$ These matrices have the forms $$\mathbf{A}(w)=\begin{bmatrix}0 & A_{12}(w)\\A_{21}(w) & 0\end{bmatrix}\quad\text{and}\quad
\mathbf{B}(w)=\begin{bmatrix}\beta(w) & B_{12}(w)\\B_{21}(w) & -\beta(w)\end{bmatrix}$$ where $$\det(\mathbf{B}(w))=-1\quad\implies\quad \beta(w)^2=1-B_{12}(w)B_{21}(w).
\label{eq:beta-identity}$$ Comparing the constant terms between the expansions and yields the identity $$\mathbf{G}^\infty=\frac{1}{2}{\mathrm{i}}y\mathbf{B}(w),
\label{eq:G0-eqn}$$ where $\mathbf{G}^\infty$ is given by , and comparing the terms proportional to $\lambda$ in the same expansions yields $$\mathbf{G}^0_{1}=\frac{1}{2}\left(\sigma_3+{\mathrm{i}}y \mathbf{A}(w)\right),
\label{eq:G1-eqn}$$ where $\mathbf{G}^0_{1}$ is given by . Since $\sigma_3^2=\mathbb{I}$, it is also clear from that the square of the matrix polynomial $\mathbf{G}(\lambda)$ is a multiple of the identity, i.e., a specific scalar polynomial: $$\mathbf{G}(\lambda)^2 = R(\lambda;y)^2\mathbb{I}=P(\lambda;y,C(y))\mathbb{I},
\label{eq:G-squared-P}$$ where $P$ is the quartic in . On the other hand, calculating the square directly from gives $$\begin{gathered}
\mathbf{G}(\lambda)^2 = \frac{1}{4}\left(-y^2\lambda^4 + 2{\mathrm{i}}y\lambda^3 + \left(1-y^2A_{12}(w)A_{21}(w)-2y^2\beta(w)\right)\lambda^2\right. \\{}\left.+ {\mathrm{i}}y\left(2\beta(w)+{\mathrm{i}}y(A_{12}(w)B_{21}(w)+A_{21}(w)B_{12}(w))\right)\lambda-y^2\right)\mathbb{I}.
\label{eq:G-squared-direct}\end{gathered}$$ Comparing the coefficient of $\lambda$ between and $P(\lambda;y,C(y))\mathbb{I}$ using yields the identity $$\beta(w)=1-\frac{1}{2}{\mathrm{i}}y(A_{12}(w)B_{21}(w)+A_{21}(w)B_{12}(w)).
\label{eq:lambda-1-identity}$$ Using to eliminate $\beta(w)$ from and comparing again with gives the identity $$\begin{gathered}
P(\lambda;y,C(y))=-\frac{1}{4}y^2\lambda^4 + \frac{1}{2}{\mathrm{i}}y\lambda^3 \\
+\frac{1}{4}\left(1-y^2A_{12}(w)A_{21}(w)-2y^2+{\mathrm{i}}y^3(A_{12}(w)B_{21}(w)+A_{21}(w)B_{12}(w))\right)\lambda^2
+\frac{1}{2}{\mathrm{i}}y\lambda -\frac{1}{4}y^2.
\label{eq:Poly-rewrite}\end{gathered}$$ We note that the coefficient of $\lambda^2$ here is actually independent of $w$, since according to it is given by $C=C(y)$, but the above expression is more useful in the context of the present discussion.
On the other hand, one may observe that the matrix $\mathbf{F}(\lambda;w):=\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m){\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)\sigma_3/2}$ satisfies jump conditions that are independent of $w\in\mathbb{C}$, and therefore $\mathbf{F}_w\mathbf{F}^{-1}$ is a function of $\lambda$ analytic except possibly at $\lambda=0$ where $\mathbf{F}$ has essential singularities. By expansion for large and small $\lambda$ and Liouville’s theorem, it follows that $\mathbf{F}_w\mathbf{F}^{-1}$ is a Laurent polynomial: $$\frac{\partial\mathbf{F}}{\partial w}(\lambda;w)\mathbf{F}(\lambda;w)^{-1}=
\frac{1}{2}{\mathrm{i}}\sigma_3\lambda +\frac{1}{2}{\mathrm{i}}\left[\dot{\mathbf{O}}^\infty_{n,1}(y,w,m),\sigma_3\right]-\frac{1}{2}{\mathrm{i}}\dot{\mathbf{O}}^0_{n,0}(y,w,m)\sigma_3\dot{\mathbf{O}}^0_{n,0}(y,w,m)^{-1}\lambda^{-1}.$$ Therefore, the outer parametrix $\mathbf{O}_n^{\mathrm{out}}(\lambda;y,w,m)$ itself satisfies the differential equation $$\begin{gathered}
\frac{\partial\dot{\mathbf{O}}_n^{\mathrm{out}}}{\partial w}(\lambda;y,w,m)=\frac{1}{2}{\mathrm{i}}\left[\sigma_3,\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)\right]\lambda + \frac{1}{2}{\mathrm{i}}\left[\dot{\mathbf{O}}^\infty_{n,1}(y,w,m),\sigma_3\right]\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m) \\{}+\frac{1}{2}{\mathrm{i}}\left(\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)\sigma_3-
\dot{\mathbf{O}}^0_{n,0}(y,w,m)\sigma_3\dot{\mathbf{O}}^0_{n,0}(y,w,m)^{-1}\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)\right)\lambda^{-1}.
\label{eq:outer-parametrix-ODE}\end{gathered}$$ Substituting the large-$\lambda$ expansion of $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ yields an infinite hierarchy of differential equations on the expansion coefficient matrices, the first member of which is $$\frac{{\mathrm{d}}\dot{\mathbf{O}}^\infty_{n,1}}{{\mathrm{d}}w}=\frac{1}{2}{\mathrm{i}}\left[\sigma_3,\dot{\mathbf{O}}^\infty_{n,2}\right] +
\frac{1}{2}{\mathrm{i}}\left[\dot{\mathbf{O}}^\infty_{n,1},\sigma_3\right]\dot{\mathbf{O}}^\infty_{n,1} +\frac{1}{2}{\mathrm{i}}\sigma_3 -\frac{1}{2}{\mathrm{i}}\dot{\mathbf{O}}^0_{n,0}\sigma_3(\dot{\mathbf{O}}^0_{n,0})^{-1}.
\label{eq:DE-infty-1}$$ Using the off-diagonal part of the identity we can eliminate the commutator $[\sigma_3,\dot{\mathbf{O}}^\infty_{n,2}]$, and therefore implies that $$\frac{{\mathrm{d}}\dot{\mathbf{O}}^\infty_{n,1}}{{\mathrm{d}}w}=\frac{1}{2y}\mathbf{A}(w)+\frac{1}{2}{\mathrm{i}}\left(\mathbf{A}(w)\dot{\mathbf{O}}_{n,1}^\infty\right)^\mathrm{D}+\frac{1}{2}{\mathrm{i}}\sigma_3+\frac{1}{2}{\mathrm{i}}(\mathbf{B}(w))^\mathrm{D}-{\mathrm{i}}\mathbf{B}(w),$$ where $(\cdot)^\mathrm{D}$ denotes the diagonal part of a matrix. Taking the commutator of this equation with $\sigma_3$ then yields $$\frac{{\mathrm{d}}\mathbf{A}}{{\mathrm{d}}w} = \frac{1}{2y}[\mathbf{A},\sigma_3] -{\mathrm{i}}[\mathbf{B},\sigma_3].
\label{eq:ODE-1}$$ Similarly, substituting into the small-$\lambda$ expansion of $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ and taking just the leading (constant) term gives the differential equation $$\frac{{\mathrm{d}}\dot{\mathbf{O}}^0_{n,0}}{{\mathrm{d}}w}=\frac{1}{2}{\mathrm{i}}[\dot{\mathbf{O}}^\infty_{n,1},\sigma_3]\dot{\mathbf{O}}^0_{n,0} +\frac{1}{2}{\mathrm{i}}\dot{\mathbf{O}}_{n,0}^{0}[(\dot{\mathbf{O}}_{n,0}^{0})^{-1}\dot{\mathbf{O}}^0_{n,1},\sigma_3].$$ Multiplying the identity on the right by $\dot{\mathbf{O}}_{n,0}^{0}$ allows $\dot{\mathbf{O}}_{n,1}^{0}$ to be eliminated from the right-hand side of the above differential equation, leading to $$\frac{{\mathrm{d}}\dot{\mathbf{O}}_{n,0}^0}{{\mathrm{d}}w}={\mathrm{i}}\mathbf{A}(w)\dot{\mathbf{O}}_{n,0}^0 +\frac{1}{2y}[\sigma_3,\dot{\mathbf{O}}_{n,0}^0].$$ This identity allows us to compute the derivative of $\mathbf{B}(w)$. Using also $\mathbf{B}(w)^2=\mathbb{I}$ yields the differential equation $$\frac{{\mathrm{d}}\mathbf{B}}{{\mathrm{d}}w}={\mathrm{i}}[\mathbf{A},\mathbf{B}]-\frac{1}{2y}[\mathbf{B},\sigma_3].
\label{eq:ODE-2}$$ The differential equations and obviously form a closed system on the matrices $\mathbf{A}(w)$ and $\mathbf{B}(w)$. From , we can express $\dot{p}(w):=-{\mathrm{i}}\dot{u}_n(y,w;m)$ in terms of the elements of $\mathbf{A}(w)$ and $\mathbf{B}(w)$ simply as $$\dot{p}(w)=-{\mathrm{i}}\dot{u}_n(y,w;m)=-\frac{A_{12}(w)}{B_{12}(w)}.
\label{eq:dot-p-AB}$$ Now we use and to differentiate $\dot{p}(w)$: $$\frac{{\mathrm{d}}\dot{p}}{{\mathrm{d}}w}=-2{\mathrm{i}}\beta(w)\frac{A_{12}(w)^2}{B_{12}(w)^2}+\frac{2}{y}\frac{A_{12}(w)}{B_{12}(w)}-2{\mathrm{i}}.$$ Therefore, using to eliminate $\beta(w)^2$, we find that $$\begin{gathered}
\frac{y^2}{16}\left(\frac{{\mathrm{d}}\dot{p}}{{\mathrm{d}}w}\right)^2=-\frac{1}{4}y^2 -\frac{1}{2}{\mathrm{i}}y\frac{A_{12}(w)}{B_{12}(w)} +
\left(\frac{1}{4}-\frac{1}{2}y^2\beta(w)\right)\frac{A_{12}(w)^2}{B_{12}(w)^2} \\
{}-\frac{1}{2}{\mathrm{i}}y\beta(w)\frac{A_{12}(w)^3}{B_{12}(w)^3}+\frac{1}{4}y^2\left(B_{12}(w)B_{21}(w)-1\right)\frac{A_{12}(w)^4}{B_{12}(w)^4}.\end{gathered}$$ Substituting $\lambda=\dot{p}$ with into gives $$\begin{gathered}
P(\dot{p};y,C(y))=-\frac{1}{4}y^2-\frac{1}{2}{\mathrm{i}}y\frac{A_{12}(w)}{B_{12}(w)}\\
{}+\frac{1}{4}\left(1-y^2A_{12}(w)A_{21}(w)-2y^2+{\mathrm{i}}y^3(A_{12}(w)B_{21}(w)+A_{21}(w)B_{12}(w))\right)\frac{A_{12}(w)^2}{B_{12}(w)^2}\\
{}-\frac{1}{2}{\mathrm{i}}y\frac{A_{12}(w)^3}{B_{12}(w)^3} -\frac{1}{4}y^2\frac{A_{12}(w)^4}{B_{12}(w)^4}.\end{gathered}$$ Subtracting these two identities yields $$\begin{gathered}
\frac{y^2}{16}\left(\frac{{\mathrm{d}}\dot{p}}{{\mathrm{d}}w}\right)^2-P(\dot{p};y,C(y))=\\
\left(-\frac{1}{2}y^2(\beta(w)-1)+\frac{1}{4}y^2A_{12}(w)A_{21}(w)-\frac{1}{4}{\mathrm{i}}y^3(A_{12}(w)B_{21}(w)+A_{21}(w)B_{12}(w))\right)
\frac{A_{12}(w)^2}{B_{12}(w)^2}\\
-\frac{1}{2}{\mathrm{i}}y(\beta(w)-1)\frac{A_{12}(w)^3}{B_{12}(w)^3}+\frac{1}{4}y^2B_{12}(w)B_{21}(w)\frac{A_{12}(w)^4}{B_{12}(w)^4}.\end{gathered}$$ Finally, eliminating $\beta(w)$ using yields the differential equation . Together with the fact that the four roots of $P(\lambda;y,C(y))$ are distinct by choice of $C(y)$ satisfying the Boutroux conditions on $E_\mathrm{R}$, this proves the first statement of Theorem \[theorem:eye\].
Airy-type parametrices {#sec:Airy}
----------------------
Local parametrices for the matrix $\mathbf{O}_n(\lambda;y,w,m)$ are needed in neighborhoods of each of the four roots of $P(\lambda;y,C(y))$, $\lambda=\lambda_0,\lambda_1,\lambda_1^{-1},\lambda_0^{-1}$, where we recall that by definition $\lambda_0$ is adjacent to $\infty$ and $\lambda_1$ is adjacent to $\lambda_0$ on the Stokes graph of $y\in E_\mathrm{R}\setminus\mathbb{R}$. Centering a disk of sufficiently small radius independent of $n$ at each of these points, a conformal map $W=W(\lambda)$ can be defined in each disk as indicated in Table \[tab:ConfMapR\].
------------------ -- ------------------------------- ------------------------------------------- ------------------------------------ -------------------------------
$W>0$ $\vphantom{\Big[}\arg(W)=\tfrac{2}{3}\pi$ $\arg(W)=-\tfrac{2}{3}\pi$ $W<0$
$\lambda_0$ $L^{\infty,1}_\squareurblack$ $\partial\Lambda^-_\squareurblack$ $\partial\Lambda^+_\squareurblack$ $L^{\infty,2}_\squareurblack$
$\lambda_1$ $L^{\infty,3}_\squareurblack$ $\partial\Lambda^+_\squareurblack$ $\partial\Lambda^-_\squareurblack$ $L^{\infty,2}_\squareurblack$
$\lambda_1^{-1}$ $L^{\infty,1}_\squarellblack$ $\partial\Lambda^-_\squarellblack$ $\partial\Lambda^+_\squarellblack$ $L^{\infty,2}_\squarellblack$
------------------ -- ------------------------------- ------------------------------------------- ------------------------------------ -------------------------------
: Conformal map data for $y\in E_\mathrm{R}\setminus\mathbb{R}$.[]{data-label="tab:ConfMapR"}
As indicated in this table, we assume that certain contours near $\lambda_0^{-1}$ are fused together within the corresponding disk, and that all contours are locally arranged to lie along straight rays in the $W$-plane emanating from the origin. Locally, the jump contours divide the $W$-plane into four sectors: $$S_\mathrm{I}:\; 0<\arg(W)<\frac{2}{3}\pi; \;\;
S_\mathrm{II}:\; \frac{2}{3}\pi<\arg(W)<\pi;\;\;
S_\mathrm{III}:\; -\pi<\arg(W)<-\frac{2}{3}\pi;\;\;
S_\mathrm{IV}:\; -\frac{2}{3}\pi<\arg(W)<0.
\label{eq:Airy-sectors}$$ In each case, the jump conditions satisfied by $\mathbf{O}_n(\lambda;y,w,m)$ can then be cast into a universal form by means of a substitution $$\mathbf{P}(\lambda):=\mathbf{O}_n(\lambda;y,w,m){\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)\sigma_3/2}{\mathrm{e}}^{-L(\lambda;y,m)\sigma_3}{{\lambda^{-(m+1)\sigma_3/2}_\squarellblack}}\mathbf{T}(\lambda),$$ where $\mathbf{T}(\lambda)$ is a piecewise-constant matrix defined in the four sectors of each disk as indicated in Table \[tab:T-in-ER\]. Note that the Boutroux conditions $K_j\in\mathbb{R}$, $j=1,2$, imply that $\mathbf{T}(\lambda)$ is uniformly bounded on compact sets with respect to $m\in\mathbb{C}$ and for arbitrary $n\in\mathbb{Z}_{\ge 0}$.
-- ----------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------
In Sector $S_\mathrm{I}$ In Sector $S_\mathrm{II}$ In Sector $S_\mathrm{III}$ In Sector $S_\mathrm{IV}$
$\vphantom{\Big[}c^{\sigma_3}$ $c^{\sigma_3}$ $c^{\sigma_3}$ $c^{\sigma_3}$
$\vphantom{\Big[}(c{\mathrm{e}}^{{\mathrm{i}}n K_1/2})^{\sigma_3}$ $(c{\mathrm{e}}^{{\mathrm{i}}n K_1/2})^{\sigma_3}$ $(c{\mathrm{e}}^{-{\mathrm{i}}n K_1/2})^{\sigma_3}$ $(c{\mathrm{e}}^{-{\mathrm{i}}n K_1/2})^{\sigma_3}$
$\vphantom{\Big[}(-c{\mathrm{e}}^{{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(-c{\mathrm{e}}^{{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{-{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{-{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$
$\vphantom{\Big[}(c{\mathrm{e}}^{-{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{-{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{{\mathrm{i}}\pi m/2}{\mathrm{e}}^{{\mathrm{i}}n K_1})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$
$\vphantom{\Big[}(c{\mathrm{e}}^{-{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{-{\mathrm{i}}\pi m/2}{\mathrm{e}}^{-{\mathrm{i}}n K_1})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$ $(c{\mathrm{e}}^{{\mathrm{i}}\pi m/2})^{\sigma_3}{\mathrm{i}}\sigma_1$
-- ----------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------
: The transformation $\mathbf{T}(\lambda)$ defined in the four sectors of the $W$-plane in each of the four disks for $y\in E_\mathrm{R}\setminus\mathbb{R}$.[]{data-label="tab:T-in-ER"}
The jump conditions satisfied by $\mathbf{P}(\lambda)$ in each case are most conveniently written in terms of the rescaled variable $\zeta = n^{2/3}W(\lambda)$: $$\begin{split}
\mathbf{P}_+(\lambda)&=\mathbf{P}_-(\lambda)\begin{bmatrix}1 & {\mathrm{e}}^{-\zeta^{3/2}}\\0 & 1\end{bmatrix},\quad \arg(\zeta)=0,\\
\mathbf{P}_+(\lambda)&=\mathbf{P}_-(\lambda)\begin{bmatrix}1 & 0\\{\mathrm{e}}^{\zeta^{3/2}} & 1\end{bmatrix},\quad\arg(\zeta)=\pm\frac{2}{3}\pi,\\
\mathbf{P}_+(\lambda)&=\mathbf{P}_-(\lambda)\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix},\quad
\arg(-\zeta)=0,
\end{split}
\label{eq:Airy-jumps}$$ where in each case the boundary values of $\mathbf{P}$ are defined with respect to orientation in the direction of increasing real part of $\zeta$, and where all powers of $\zeta$ are principal branches. We may make a similar transformation of the outer parametrix, noting that in each disk the matrix $$\dot{\mathbf{P}}^\mathrm{out}(\lambda):=\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,m){\mathrm{e}}^{{\mathrm{i}}w\varphi(\lambda)\sigma_3/2}{\mathrm{e}}^{-L(\lambda;y,m)\sigma_3}{{\lambda^{-(m+1)\sigma_3/2}_\squarellblack}}\mathbf{T}(\lambda),
\label{eq:O-out-local-transform}$$ is analytic except for $\arg(-\zeta)=0$ where it satisfies exactly the same jump condition as does $\mathbf{P}(\lambda)$. This fact, along with the fact that the matrix elements of $\dot{\mathbf{P}}^\mathrm{out}(\lambda)$ blow up at $W(\lambda)=0$ as negative one-fourth powers, implies that $\dot{\mathbf{P}}^\mathrm{out}(\lambda)$ can be written in the form $$\dot{\mathbf{P}}^\mathrm{out}(\lambda)=
\mathbf{H}_n(\lambda;y,w,m)W(\lambda)^{\sigma_3/4}\mathbf{V}=\mathbf{H}_n(\lambda;y,w,m)n^{-\sigma_3/6}\zeta^{\sigma_3/4}\mathbf{V},\quad \mathbf{V}:=\frac{1}{\sqrt{2}}\begin{bmatrix}1 & -{\mathrm{i}}\\-{\mathrm{i}}& 1\end{bmatrix},
\label{eq:local-H-define}$$ where $\mathbf{H}_n(\lambda;y,w,m)$ is a function of $\lambda$ that is analytic in the disk in question and uniformly bounded with respect to $m$ in compact subsets of $\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$ and $n\in\mathbb{Z}_{\ge 0}$, provided $|w|\le K$ for some $K>0$ and $y$ satisfy conditions such as enumerated in Lemma \[lem:Outer-Bounded\]. Noting that the boundary of each disk corresponds to $\zeta$ proportional to $n^{2/3}$, we wish to model the matrix function $\mathbf{P}(\lambda)$ by something that satisfies the jump conditions exactly and that matches with the terms $\zeta^{\sigma_3/4}\mathbf{V}$ coming from the outer parametrix when $\zeta$ is large. We are thus led to the the following model Riemann-Hilbert problem.
\[rhp:Airy\] Find a $2\times 2$ matrix function $\zeta\mapsto\mathbf{A}(\zeta)$ with the following properties:
- **Analyticity:** $\zeta\mapsto\mathbf{A}(\zeta)$ is analytic in the sectors $S_\mathrm{I}$, $S_\mathrm{II}$, $S_\mathrm{III}$, and $S_\mathrm{IV}$ of the complex $\zeta$-plane (see ), and takes continuous boundary values from each sector.
- **Jump conditions:** The boundary values $\mathbf{A}_\pm(\zeta)$ are related on each ray of the jump contour by the following formulæ (cf., ), $$\begin{split}
\mathbf{A}_+(\zeta)&=\mathbf{A}_-(\zeta)\begin{bmatrix}1 & {\mathrm{e}}^{-\zeta^{3/2}}\\0 & 1\end{bmatrix},\quad\arg(\zeta)=0,\\
\mathbf{A}_+(\zeta)&=\mathbf{A}_-(\zeta)\begin{bmatrix}1 & 0\\{\mathrm{e}}^{\zeta^{3/2}} & 1\end{bmatrix},\quad\arg(\zeta)=\pm\frac{2}{3}\pi,\\
\mathbf{A}_+(\zeta)&=\mathbf{A}_-(\zeta)\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix},\quad\arg(-\zeta)=0.
\end{split}
\label{eq:Airy-jumps-A}$$
- **Asymptotics:** $\mathbf{A}(\zeta)\mathbf{V}^{-1}\zeta^{-\sigma_3/4}\to\mathbb{I}$ as $\zeta\to\infty$.
This problem will be solved in all details in Appendix \[app:Airy\], where it will be shown that $\mathbf{A}(\zeta)\mathbf{V}^{-1}\zeta^{-\sigma_3/4}$ has a complete asymptotic expansion in descending integer powers of $\zeta$ as $\zeta\to\infty$, with the dominant terms being given by $$\mathbf{A}(\zeta)\mathbf{V}^{-1}\zeta^{-\sigma_3/4}=\mathbb{I}+\begin{bmatrix}\mathcal{O}(\zeta^{-3}) & \mathcal{O}(\zeta^{-1})\\ \mathcal{O}(\zeta^{-2}) & \mathcal{O}(\zeta^{-3})\end{bmatrix},\quad\zeta\to\infty.
\label{eq:Airy-norm-better}$$ In each disk we then build a local approximation of $\mathbf{O}_n(\lambda;y,w,m)$ by multiplying on the left by the holomorphic prefactor $\mathbf{H}_n(\lambda;y,w,m)n^{-\sigma_3/6}$ and on the right by the piecewise-analytic substitution relating $\mathbf{O}_n(\lambda;y,w,m)$ and $\mathbf{P}(\lambda)$: $$\dot{\mathbf{O}}_n^{\mathrm{in}}(\lambda;y,w,m):=\mathbf{H}_n(\lambda;y,w,m)n^{-\sigma_3/6}\mathbf{A}(n^{2/3}W(\lambda))\mathbf{T}(\lambda)^{-1}{{\lambda^{(m+1)\sigma_3/2}_\squarellblack}}{\mathrm{e}}^{L(\lambda;y,m)\sigma_3}{\mathrm{e}}^{-{\mathrm{i}}w\varphi(\lambda)\sigma_3/2},$$ where $W(\lambda)$ is the conformal map associated with the disk via Table \[tab:ConfMapR\], $\mathbf{T}(\lambda)$ is the unimodular transformation matrix given in Table \[tab:T-in-ER\], and $\mathbf{H}_n(\lambda;y,w,m)$ is associated with the outer parametrix and the disk in question via –.
Error analysis and proof of Theorem \[theorem:eye\] {#sec:Error-elliptic}
---------------------------------------------------
Let $\Sigma_\mathbf{O}$ denote the jump contour for the matrix function $\mathbf{O}_n(\lambda;y,w,m)$, which consists of the contour $L$ augmented with the lens boundaries $\partial\Lambda^\pm_\squareurblack$ and $\partial\Lambda^\pm_\squarellblack$. The *global parametrix* denoted $\dot{\mathbf{O}}_n(\lambda;y,w,m)$ is defined as $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ when $\lambda$ lies outside of all four disks, but instead as $\dot{\mathbf{O}}_n^{\mathrm{in}}(\lambda;y,w,m)$ within each disk (the precise definition is different in each disk as explained in Section \[sec:Airy\]). We wish to compare the global parametrix with the (unknown) matrix function $\mathbf{O}_n(\lambda;y,w,m)$, so we introduce the error matrix $\mathbf{E}_n(\lambda;y,w,m)$ defined by $\mathbf{E}_n(\lambda;y,w,m):=\mathbf{O}_n(\lambda;y,w,m)\dot{\mathbf{O}}_n(\lambda;y,w,m)^{-1}$. The maximal domain of analyticity of $\mathbf{E}_n(\lambda;y,w,m)$ is determined from those of the two factors; therefore $\mathbf{E}_n(\lambda;y,w,m)$ is analytic in $\lambda$ except along a jump contour consisting of (i) the part of $\Sigma_\mathbf{O}$ lying outside of all four disks and (ii) the boundaries of all four disks. That $\mathbf{E}_n(\lambda;y,w,m)$ can be taken to be an analytic function in the interior of each disk follows from the fact that the inner parametrices $\dot{\mathbf{O}}_n^{\mathrm{in}}(\lambda;y,w,m)$ satisfy exactly the same jump conditions locally as does $\mathbf{O}_n(\lambda;y,w,m)$ and an argument based on Morera’s theorem. The jump contour for $\mathbf{E}_n(\lambda;y,w,m)$ corresponding to the Stokes graph shown in Figure \[fig:StokesGraph\] is shown in Figure \[fig:InsideError\].
![The jump contour for $\mathbf{E}_n(\lambda;y,w,m)$ for $y=0.2+0.25{\mathrm{i}}\in E_\mathrm{R}$ (cf., Figure \[fig:StokesGraph\]). The jump contour consists of the red and cyan arcs (arcs of $L\setminus({{L^{\infty,2}_\squarellblack}}\cup{{L^{\infty,2}_\squareurblack}})$ and of the lens boundaries $\partial\Lambda_\squarellblack^\pm$ and $\partial\Lambda_\squareurblack^\pm$ restricted to the exterior of the four disks) and the green circles (the boundaries of the four disks, each of which is taken to have clockwise orientation for the purposes of defining the boundary values of $\mathbf{E}_n(\lambda;y,w,m)$). Note that the arcs of $\partial\Lambda_\squarellblack^\pm$ in the left-hand panel are oriented toward the upper left, while the arcs of $\partial\Lambda_\squareurblack^\pm$ in the right-hand panel are oriented toward the lower right.[]{data-label="fig:InsideError"}](InsideError-preview.pdf)
Let us assume that, given $m\in\mathbb{C}\setminus (\mathbb{Z}+\tfrac{1}{2})$ and any fixed constants $K>0$ and $\epsilon>0$, $w\in\mathbb{C}$ and $y\in E_\mathrm{R}$ are restricted according to $|w|\le K$ and the inequalities . Then by Lemma \[lem:Outer-Bounded\], $\dot{\mathbf{O}}_n(\lambda;y,w,m)$ is uniformly bounded whenever $\lambda$ lies outside all four disks (which both gives $\dot{\mathbf{O}}_n(\lambda;y,w,m)=\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ and guarantees the condition ). Since $\mathbf{O}_{n+}(\lambda;y,w,m)=\mathbf{O}_{n-}(\lambda;y,w,m)(\mathbb{I}+\text{exponentially small})$ holds on all arcs of $\Sigma_\mathbf{O}$ lying outside the disks and on which $\dot{\mathbf{O}}_n^{\mathrm{out}}(\lambda;y,w,m)$ is analytic, and since $\mathbf{O}_n(\lambda;y,w,m)$ and $\mathbf{O}_n^{\mathrm{out}}(\lambda;y,w,m)$ satisfy exactly the same jump conditions across all remaining arcs of $\Sigma_\mathbf{O}$ outside of all disks, it follows from Lemma \[lem:Outer-Bounded\] that $\mathbf{E}_{n+}(\lambda;y,w,m)=\mathbf{E}_{n-}(\lambda;y,w,m)(\mathbb{I}+\text{exponentially small})$ holds on all jump arcs for $\mathbf{E}_n(\lambda;y,w,m)$ with the exception of the four disk boundaries. Let the boundary of each disk be oriented in the clockwise direction. Then a computation shows that on each disk boundary, the matrix $\mathbf{E}_n(\lambda;y,w,m)$ satisfies the jump condition $\mathbf{E}_{n+}(\lambda;y,w,m)=\mathbf{E}_{n-}(\lambda;y,w,m)\mathbf{H}_n(\lambda;y,w,m)n^{-\sigma_3/6}\mathbf{A}(\zeta)\mathbf{V}^{-1}\zeta^{-\sigma_3/4}n^{\sigma_3/6}\mathbf{H}_n(\lambda;y,w,m)^{-1}$ where $\zeta=n^{2/3}W(\lambda)$, $W(\lambda)$ is the relevant conformal mapping from Table \[tab:ConfMapR\], $\mathbf{A}(\zeta)$ is the solution of Riemann-Hilbert Problem \[rhp:Airy\], and $\mathbf{H}_n(\lambda;y,w,m)$ is a bounded function with unit determinant. Applying the condition and using the fact that $W(\lambda)$ is bounded away from zero on the disk boundary then yields the estimate $\mathbf{E}_{n+}(\lambda;y,w,m)=\mathbf{E}_{n-}(\lambda;y,w,m)(\mathbb{I}+\mathcal{O}(n^{-1}))$ as $n\to+\infty$. Since $\mathbf{E}_n(\lambda;y,w,m)\to\mathbb{I}$ as $\lambda\to\infty$, it then follows that this matrix satisfies the conditions of a small-norm Riemann-Hilbert problem. This implies that (under the conditions of Lemma \[lem:Outer-Bounded\]) $\mathbf{E}_n(\lambda;y,w,m)$ exists for sufficiently large $n\in\mathbb{Z}_{>0}$ and satisfies: $$\mathbf{E}_n(0;y,w,m)=\mathbb{I}+\mathcal{O}(n^{-1})\quad\text{and}\quad
\lim_{\lambda\to\infty}\lambda(\mathbf{E}_n(\lambda;y,w,m)-\mathbb{I})=\mathcal{O}(n^{-1}),\quad n\to +\infty.$$ From this result, it follows that $u_n(ny+w;m)=\dot{u}_n(y,w;m)+\mathcal{O}(n^{-1})$ under the conditions which serve to bound the four factors in the fraction in away from zero. Combining this result valid for $y\in E_\mathrm{R}$ with the symmetry then concludes the proof of Theorem \[theorem:eye\].
Detailed properties of the approximation $\dot{u}_n(y,w;m)$. Proofs of Corollary \[corollary:eye-zeros-and-poles:better\] and Theorem \[theorem:density\] {#sec:properties-of-udot-elliptic}
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Given $n$ and $m$, the zeros of $\dot{u}_n(y,w;m)$ are the roots of the theta function factors in the numerator, namely the pairs $(y,w)$ for which $$\text{Zeros of $\dot{u}_n(y,w;m)$:\quad}A(\infty;y)\pm A(\kappa(y);y)\mp s_n(y,w,m)\in 2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}.
\label{eq:zeros-of-udot}$$ Note that the zeros of $\dot{u}_n(y,w;m)$ corresponding to taking the top sign in are also points of the Malgrange divisor , i.e., points at which the solution of Riemann-Hilbert Problem \[rhp:outer-elliptic\] fails to exist. On the other hand, the singularities of $\dot{u}_n(y,w;m)$ are the pairs $(y,w)$ that produce zeros of the denominator in , $$\text{Singularities of $\dot{u}_n(y,w;m)$:\quad}A(0;y)\pm A(\kappa(y);y)\mp s_n(y,w,m)\in 2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}.
\label{eq:poles-of-udot}$$ We hesitate to call these singularities “poles” for reasons to be explained in Section \[sec:DE-w\] below.
Given $n\in\mathbb{Z}_{\ge 0}$, $m\in\mathbb{C}\setminus (\mathbb{Z}+\tfrac{1}{2})$, $y\in E_\mathrm{R}$, and $w\in\mathbb{C}$, at most one of the four conditions in – holds, or equivalently at most one of the four factors $\mathcal{Z}_n^\bullet(y,w;m)$, $\mathcal{Z}_n^\circ(y,w;m)$, $\mathcal{P}_n^\bullet(y,w;m)$, or $\mathcal{P}_n^\circ(y,w;m)$ appearing in the formula vanishes. \[lem:disjoint-lattices\]
It suffices to show that none of $2A(\infty;y)$, $2A(0;y)$, $A(\infty;y)-A(0;y)$, nor $A(\infty;y)+A(0;y)$ lies in the lattice $\Lambda:=2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}$. As in the proof of Lemma \[lemma:one-divisor\], we introduce the Riemann surface $\Gamma=\Gamma(y)$ and its Abel mapping $\mathcal{A}:\Gamma\to\mathrm{Jac}(\Gamma)=\mathbb{C}/\Lambda$, extended to divisors in the usual way. Given that the base point is a branch point, it is equivalent to show that none of $\mathcal{A}(Q^+(\infty)-Q^-(\infty))$, $\mathcal{A}(Q^+(0)-Q^-(0))$, $\mathcal{A}(Q^+(\infty)-Q^+(0))$, nor $\mathcal{A}(Q^+(\infty)-Q^-(0))$ is mapped to $0\in\mathrm{Jac}(\Gamma)$. But by the Abel-Jacobi theorem, we just need to rule out the existence of a nonzero meromorphic function on $\Gamma$ having any of the divisors $\mathscr{D}=Q^+(\infty)-Q^-(\infty)$, $\mathscr{D}=Q^+(0)-Q^-(0)$, $\mathscr{D}=Q^+(\infty)-Q^+(0)$, or $\mathscr{D}=Q^+(\infty)-Q^-(0)$. However, in each case there is only one simple pole whose residue necessarily vanishes, so the desired meromorphic function would in fact be holomorphic with a zero somewhere on $\Gamma$, hence identically zero.
Each of the zero/singularity conditions – defines a regular lattice of points in the $w$-plane, and the minimum distance between points of the four lattices is exactly $$\begin{gathered}
\delta(y):=\frac{1}{|\nu(y)|}\min\big\{\mathrm{dist}(2A(\infty,y),\Lambda(y)),\mathrm{dist}(2A(0,y),\Lambda(y)),\\
\mathrm{dist}(A(\infty,y)-A(0,y),\Lambda(y)),\mathrm{dist}(A(\infty,y)+A(0,y),\Lambda(y))\big\}>0,\quad y\in E_\mathrm{R},\end{gathered}$$ where $\Lambda(y):=2\pi{\mathrm{i}}\mathbb{Z}+B(y)\mathbb{Z}$ where $\nu(y)$ is the function on $E_\mathrm{R}$ defined by . Note that $|\nu(y)|$ is continuous on $E_\mathrm{R}$ and hence the inequality above follows from Lemma \[lem:disjoint-lattices\]. Moreover, the minimum factor in $\delta(y)$ is continuous on $E_\mathrm{R}$, so it follows from compactness of $K_y\subset E_\mathrm{R}$ in the statement of Corollary \[corollary:eye-zeros-and-poles:better\] that $$\delta:=\inf_{n\ge N}\delta(y_n)\ge\inf_{y\in K_y}\delta(y) = \min_{y\in K_y}\delta(y)>0.$$ By definition of the sequence $\{y_n\}_{n=N}^\infty$, taking $y=y_n$ makes one of the four factors in the fraction on the right-hand side of vanish at $w=0$ for all $n=N,N+1,\dots$, while the roots of the other three factors are bounded away from $w=0$ in the $w$-plane by the distance $\delta>0$. Now choose $\epsilon=\tfrac{1}{3}\delta$ as the parameter in Theorem \[theorem:eye\]; the circle $|w|=\epsilon$ then lies on the boundary of the closed domain characterized by the inequalities . Letting $n\to+\infty$, Theorem \[theorem:eye\] implies that the winding numbers (indices) about the circle $|w|=\epsilon$ of the rational function $f(w):=u_n(ny_n+w;m)$ and the meromorphic function $g(w):=\dot{u}_n(y_n,w;m)$ necessarily agree for sufficiently large $n$. But since $\epsilon<\delta$ the index of $g(w)=\dot{u}_n(y_n,w;m)$ is $1$ ($-1$) if the sequence $\{y_n\}_{n=N}^\infty$ corresponds to roots of a factor of the numerator (denominator) in , and this common value of the index is precisely the net number of zeros less poles of the rational function $f(w)=u_n(ny_n+w;m)$ within the circle $|w|=\epsilon$. This establishes the third statement of Theorem \[theorem:eye\] and completes the proof.
Since $\mathrm{Re}(B(y))<0$ holds for $y\in E_\mathrm{R}$ and therefore $B(y)$ and $2\pi{\mathrm{i}}$ are necessarily linearly independent over the real numbers, we can resolve the left-hand sides of – into real multiples of the lattice periods: $$\begin{split}
A(\infty;y)\pm A(\kappa(y);y)\mp s_n(y,w,m)&= 2\pi{\mathrm{i}}\alpha_n^{0,\pm}(y,w,m) + B(y)\beta_n^{0,\pm}(y,w,m),\quad\text{where}\\
\alpha_n^{0,\pm}(y,w,m)&:=\frac{\mathrm{Im}(B(y)^*(A(\infty;y)\pm A(\kappa(y);y)\mp s_n(y,w,m)))}{2\pi\mathrm{Re}(B(y))}\\
\beta_n^{0,\pm}(y,w,m)&:=\frac{\mathrm{Re}(A(\infty;y)\pm A(\kappa(y);y)\mp s_n(y,w,m))}{\mathrm{Re}(B(y))},
\end{split}
\label{eq:Zero-Quantum}$$ and $$\begin{split}
A(0;y)\pm A(\kappa(y);y)\mp s_n(y,w,m)&= 2\pi{\mathrm{i}}\alpha_n^{\infty,\pm}(y,w,m) + B(y)\beta_n^{\infty,\pm}(y,w,m),\quad\text{where}\\
\alpha_n^{\infty,\pm}(y,w,m)&:=\frac{\mathrm{Im}(B(y)^*(A(0;y)\pm A(\kappa(y);y)\mp s_n(y,w,m)))}{2\pi\mathrm{Re}(B(y))}\\
\beta_n^{\infty,\pm}(y,w,m)&:=\frac{\mathrm{Re}(A(0;y)\pm A(\kappa(y);y)\mp s_n(y,w,m))}{\mathrm{Re}(B(y))}.
\end{split}
\label{eq:Pole-Quantum}$$ Thus, the zeros of $\dot{u}_n(y,w;m)$ satisfy the *quantization conditions* $$\text{Zeros of $\dot{u}_n(y,w;m)$:\quad}
\alpha_n^{0,\pm}(y,w,m)\in\mathbb{Z}\quad\text{and}\quad\beta_n^{0,\pm}(y,w,m)\in\mathbb{Z},
\label{eq:zeros-quantization}$$ and similarly $$\text{Singularities of $\dot{u}_n(y,w;m)$:\quad}
\alpha_n^{\infty,\pm}(y,w,m)\in\mathbb{Z}\quad\text{and}\quad\beta_n^{\infty,\pm}(y,w,m)\in\mathbb{Z}.
\label{eq:poles-quantization}$$ One way to parametrize points within the domain $E_\mathrm{R}$ is by choosing to set $w=0$ and thus $x=ny$ where $y$ ranges over $E_\mathrm{R}$. Using this parametrization, we can give the following.
Given $n$ and $m$, for each choice of sign $\pm$, the conditions (resp., ) for $w=0$ define a network of two families of curves whose common intersections locate the zeros (resp., singularities) of $\dot{u}_n(y,0;m)$ on $E_\mathrm{L}\cup E_\mathrm{R}$. Given $m\in\mathbb{C}\setminus (\mathbb{Z}+\tfrac{1}{2})$ it is particularly interesting to consider how the curves depend on $n\in\mathbb{Z}$ large and positive. For this, we observe that the only dependence on $n$ enters through $s_n(y,0,m)$; substituting from and gives, for $y\in E_\mathrm{R}$, $$\alpha_n^{0,\pm}(y,0,m)=\alpha_0^{0,\pm}(y,0,m)\mp \frac{nK_2(y)}{2\pi},\quad\alpha_0^{0,\pm}(y,0,m)=
\frac{\mathrm{Im}(B(y)^*(A(\infty;y)\pm A(\kappa(y);y)\pm\delta(y,m)))}{2\pi\mathrm{Re}(B(y))},
\label{eq:alpha-zero-n}$$ $$\beta_n^{0,\pm}(y,0,m)=\beta_0^{0,\pm}(y,0,m)\mp\frac{nK_1(y)}{2\pi},\quad\beta_0^{0,\pm}(y,0,m)=
\frac{\mathrm{Re}(A(\infty;y)\pm A(\kappa(y);y)\pm\delta(y,m))}{\mathrm{Re}(B(y))},$$ $$\alpha_n^{\infty,\pm}(y,0,m)=\alpha_0^{\infty,\pm}(y,0,m)\mp \frac{nK_2(y)}{2\pi},\quad\alpha_0^{\infty,\pm}(y,0,m)=
\frac{\mathrm{Im}(B(y)^*(A(0;y)\pm A(\kappa(y);y)\pm\delta(y,m)))}{2\pi\mathrm{Re}(B(y))},$$ $$\beta_n^{\infty,\pm}(y,0,m)=\beta_0^{\infty,\pm}(y,0,m)\mp\frac{nK_1(y)}{2\pi},\quad\beta_0^{\infty,\pm}(y,0,m)=
\frac{\mathrm{Re}(A(0;y)\pm A(\kappa(y);y)\pm\delta(y,m))}{\mathrm{Re}(B(y))}.
\label{eq:beta-infinity-n}$$ The simplified formulæ – show that when $n$ is large, the quantization conditions – determine a locally (with respect to $y$) uniform tiling of the $y$-plane by parallelograms each of which has area (measured in the $y$-coordinate) $A_\lozenge(y)(1+o(1))$ as $n\to+\infty$, where $$A_\lozenge(y)=\frac{4\pi^2}{n^2|\nabla K_1(y)\times\nabla K_2(y)|},\quad y\in E_\mathrm{R},$$ see Figure \[fig:parallelogram\]. By working in the $w$-plane rather than the $y$-plane, one can see that the area $A_\lozenge(y)$ is also proportional by a factor of $n^2$ to the Jacobian determinant .
![The local tiling of the $y$-plane by parallelograms of area $A_\lozenge(y)$.[]{data-label="fig:parallelogram"}](Parallelograms.pdf)
For each choice of sign $\pm$, one associates via (resp., ) exactly one zero (resp., pole) of $\dot{u}_n(y,0;m)$ with each parallelogram. Hence the densities (per unit $y$-area) of zeros and poles are exactly the same and are given by $n^2\rho(y)(1+o(1))$ as $n\to+\infty$, where $$\rho(y):=\frac{2}{n^2A_\lozenge(y)}=\frac{1}{2\pi^2}|\nabla K_1(y)\times\nabla K_2(y)|,\quad y\in E_\mathrm{R}.
\label{eq:zero-pole-density}$$ Note that since $K_1(y)$ and $K_2(y)$ are functions independent of $m\in\mathbb{C}\setminus(\mathbb{Z}+\tfrac{1}{2})$, the same is true of $\rho(y)$. The density $\rho(y)$ is a smooth nonnegative function on $E_\mathrm{R}$, but it vanishes on $\partial E_\mathrm{R}\setminus\{0\}$ and blows up as $y\to 0$. To prove the former, we may use the fact that $\rho(y)$ is inversely-proportional to the Jacobian determinant , which blows up as $y\to\partial E_\mathrm{R}\setminus\{0\}$ as mentioned in Section \[sec:Boutroux\]. To prove the blowup of $\rho(y)$ at the origin, we first express the gradients in polar coordinates $y=r{\mathrm{e}}^{{\mathrm{i}}\theta}$, and thus $$|\nabla K_1(y)\times\nabla K_2(y)| = \frac{1}{r}\left|\frac{\partial K_1}{\partial r}\frac{\partial K_2}{\partial\theta} -\frac{\partial K_1}{\partial \theta}\frac{\partial K_2}{\partial r}\right|,\quad y=r{\mathrm{e}}^{{\mathrm{i}}\theta}.
\label{eq:grad-cross-product-polar}$$ Next, we compute the partial derivatives near $r=0$. For this purpose, we recall the scalings introduced in Section \[sec:Boutroux-y-small\] to construct the solution of the Boutroux equations for small $r=|y|$. Thus, $C=y\tilde{C}(r,\theta)$, where for $|\theta|<\pi/2$, $$\tilde{C}^0(\theta):=\lim_{r\downarrow 0}\tilde{C}(r,\theta),\quad
\tilde{C}^0_{r}(\theta):=\lim_{r\downarrow 0}\tilde{C}_r(r,\theta),\quad\text{and}\quad
\tilde{C}^0_{\theta}(\theta):=\lim_{r\downarrow 0}\tilde{C}_\theta(r,\theta)=\tilde{C}^{0\prime}(\theta)$$ all exist (the subscripts $r$ and $\theta$ denote partial derivatives). For each such $\theta$, in the limit $r\downarrow 0$, $\lambda_0\to\infty$ while $\lambda_1$ converges to a nonzero limit. Since the integrands in the definitions of $K_j(y)$, $j=1,2$, are singular at $\lambda=0,\infty$, we first use the Cauchy theorem to rewrite $K_j(y)$ as contour integrals over contours that we may take to be independent of $r$ as $r\downarrow 0$. Since $\lambda_0\to\infty$ and $\lambda_0^{-1}\to 0$ as $r\downarrow 0$, it is necessary to account for some residues at $\lambda=0,\infty$, but from and it follows that these contributions are independent of $y$, so they will not play any role upon taking the required derivatives. Thus, $$K_1(y)= -\pi - {\mathrm{i}}\int_{C_1}\frac{R(\lambda;y)}{\lambda^2}\,{\mathrm{d}}\lambda,$$ where the original contour of integration (a counterclockwise-oriented path just enclosing the arc ${{L^{\infty,2}_\squareurblack}}$ with endpoints $\lambda_0\to\infty$ and $\lambda_1$) has been replaced with $C_1$, a counterclockwise-oriented closed path enclosing the arc ${{L^{\infty,2}_\squarellblack}}$ with endpoints $\lambda_0^{-1}\to 0$ and $\lambda_1^{-1}$ as well as the limit point $\lambda=0$. Likewise, $$K_2(y)= \pi -{\mathrm{i}}\int_{C_2}\frac{R(\lambda;y)}{\lambda^2}\,{\mathrm{d}}\lambda,$$ where $C_2$ is a contour consisting of two arcs joining $\lambda_1$ with $\lambda_1^{-1}$ such that $\lambda_0^{-1}$ and $\lambda=0$ are contained in the region between the two arcs, while $\lambda_0$ is excluded. With the help of a small additional contour deformation near $\lambda_1$ and $\lambda_1^{-1}$ in the case of $K_2$, both new contours may be taken to be locally independent of $y$ and hence derivatives may be computed by differentiation under the integral sign. Thus $$\frac{\partial K_j}{\partial r,\theta}(r{\mathrm{e}}^{{\mathrm{i}}\theta})=-\frac{{\mathrm{i}}}{2}\int_{C_j}\frac{\partial P}{\partial r,\theta}(\lambda;r{\mathrm{e}}^{{\mathrm{i}}\theta},r{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}(r,\theta))\frac{{\mathrm{d}}\lambda}{\lambda^2R(\lambda;r{\mathrm{e}}^{{\mathrm{i}}\theta})},\quad j=1,2.
$$ With the scaling of Section \[sec:Boutroux-y-small\], $P(\lambda;r{\mathrm{e}}^{{\mathrm{i}}\theta},r{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}(r,\theta))=-\tfrac{1}{4}r^2{\mathrm{e}}^{2{\mathrm{i}}\theta}\lambda^4 +\tfrac{1}{2}{\mathrm{i}}r{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda^3 + r{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}(r,\theta)\lambda^2 +\tfrac{1}{2}{\mathrm{i}}r{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda-\tfrac{1}{4}r^2{\mathrm{e}}^{2{\mathrm{i}}\theta}$, and therefore $$\frac{\partial K_j}{\partial r}(r{\mathrm{e}}^{{\mathrm{i}}\theta})=-\frac{{\mathrm{i}}}{4}\int_{C_j}\frac{-r{\mathrm{e}}^{2{\mathrm{i}}\theta}\lambda^2 +{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda + 2{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}(r,\theta) + 2r{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}_r(r,\theta) +{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda^{-1}-r{\mathrm{e}}^{2{\mathrm{i}}\theta}\lambda^{-2}}{R(\lambda;r{\mathrm{e}}^{{\mathrm{i}}\theta})}\,{\mathrm{d}}\lambda,\quad j=1,2,
\label{eq:Kj-r}$$ and $$\frac{\partial K_j}{\partial \theta}(r{\mathrm{e}}^{{\mathrm{i}}\theta})=-\frac{{\mathrm{i}}}{4}\int_{C_j}\frac{-{\mathrm{i}}r^2{\mathrm{e}}^{2{\mathrm{i}}\theta}\lambda^2 -r{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda + 2{\mathrm{i}}r{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}(r,\theta)+2r{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}_\theta(r,\theta) - r{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda^{-1}-{\mathrm{i}}r^2{\mathrm{e}}^{2{\mathrm{i}}\theta}\lambda^{-2}}{R(\lambda;r{\mathrm{e}}^{{\mathrm{i}}\theta})}\,{\mathrm{d}}\lambda,\quad j=1,2.
\label{eq:Kj-theta}$$ Now, given $\theta$, the following limit exists uniformly on $C_j$ (again after suitable small deformation near $\lambda_1$ and $\lambda_1^{-1}$ in the case of $C_2$): $$\lim_{r\downarrow 0} r^{-1/2}R(\lambda;r{\mathrm{e}}^{{\mathrm{i}}\theta})=\tilde{R}(\lambda;\theta),\quad\tilde{R}(\lambda;\theta)^2 = \frac{1}{2}{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda^3+{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}^0(\theta)\lambda^2 + \frac{1}{2}{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}\lambda,$$ where $\tilde{R}(\lambda;\theta)$ is analytic except on the limiting Stokes graph arcs $\Sigma^\mathrm{out}(y)$ and is well-defined by choosing the branch globally based on the above limit at any generic point $\lambda$. Similar uniform limits for the numerators in the integrands of – then show that $$\lim_{r\downarrow 0} r^{1/2}\frac{\partial K_j}{\partial r}(r{\mathrm{e}}^{{\mathrm{i}}\theta})=\frac{{\mathrm{e}}^{{\mathrm{i}}\theta}}{4}\int_{C_j}\frac{\lambda+\lambda^{-1}}{\tilde{R}(\lambda;\theta)}\,{\mathrm{d}}\lambda-\frac{{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}\tilde{C}^0(\theta)}{2}\int_{C_j}
\frac{{\mathrm{d}}\lambda}{\tilde{R}(\lambda;\theta)},\quad j=1,2$$ and $$\lim_{r\downarrow 0}r^{-1/2}\frac{\partial K_j}{\partial\theta}(r{\mathrm{e}}^{{\mathrm{i}}\theta})=\frac{{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}}{4}\int_{C_j}
\frac{\lambda+\lambda^{-1}}{\tilde{R}(\lambda;\theta)}\,{\mathrm{d}}\lambda -\frac{{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\theta}}{2}({\mathrm{i}}\tilde{C}^0(\theta)+\tilde{C}^0_\theta(\theta))\int_{C_j}\frac{{\mathrm{d}}\lambda}{\tilde{R}(\lambda;\theta)},\quad j=1,2.$$ Therefore, the following limit exists: $$\frac{1}{2\pi^2}\lim_{r\downarrow 0}\left[\frac{\partial K_1}{\partial r}\frac{\partial K_2}{\partial\theta}-\frac{\partial K_1}{\partial\theta}\frac{\partial K_2}{\partial r}\right]=-\frac{{\mathrm{i}}{\mathrm{e}}^{2{\mathrm{i}}\theta}\tilde{C}^0_\theta(\theta)}{16\pi^2}
\det\begin{bmatrix}\displaystyle\int_{C_1}\frac{\lambda + \lambda^{-1}}{\tilde{R}(\lambda;\theta)}\,{\mathrm{d}}\lambda & \displaystyle \int_{C_1}\frac{{\mathrm{d}}\lambda}{\tilde{R}(\lambda;\theta)}\\
\displaystyle \int_{C_2}\frac{\lambda + \lambda^{-1}}{\tilde{R}(\lambda;\theta)}\,{\mathrm{d}}\lambda & \displaystyle \int_{C_2}\frac{{\mathrm{d}}\lambda}{\tilde{R}(\lambda;\theta)}
\end{bmatrix}.$$ The absolute value of this limit is the quantity $h(\theta)$ referred to in the statement of Theorem \[theorem:density\].
The Special Case of $m\in\mathbb{Z}+\tfrac{1}{2}$ {#sec:m-half}
=================================================
As in Section \[sec:outside\], in this section we study Riemann-Hilbert Problem \[rhp:renormalized\] under the substitution $x=ny$, i.e., we set $w=0$.
Asymptotic behavior of $u_n(ny;m)$ for $y$ away from the distinguished eyebrow. Proof of Theorem \[theorem:closed-eye-equilibrium\] {#sec:Half-Integer-Away-From-Edge}
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If $m\in\mathbb{Z}+\tfrac{1}{2}$, then the jump matrix for $\mathbf{Y}(\lambda)$ simplifies dramatically. Indeed, if $m=\pm(\tfrac{1}{2}+k)$, $k=0,1,2,3,\dots$, then $\Gamma(\tfrac{1}{2}\mp m)^{-1}=0$. Therefore, if $m$ is a positive half-integer, in place of – we have simply $\mathbf{Y}_+(\lambda)=\mathbf{Y}_-(\lambda)$ for $\lambda\in {{L^0_\squareurblack}}\cup{{L^\infty_\squareurblack}}$, while if $m$ is a negative half-integer, in place of – we have $\mathbf{Y}_+(\lambda)=\mathbf{Y}_-(\lambda)$ for $\lambda\in{{L^0_\squarellblack}}\cup{{L^\infty_\squarellblack}}$. Now, we observe that the arc ${{\partial E^\infty_\squareurblack}}\subset \partial E$ in the right half-plane is the locus of values of $y$ for which the inequality $\mathrm{Re}(V(\lambda;y))>0$ necessarily breaks down at some point of the contour ${{L^\infty_\squareurblack}}$, and that this inequality is *only needed to control the generically nonzero off-diagonal element of the corresponding jump matrix for $\mathbf{Y}$*. Since this off-diagonal element vanishes for $m=\tfrac{1}{2}+k$, $k=0,1,2,3,\dots$, we see that in this case *the open arc ${{\partial E^\infty_\squareurblack}}\setminus\{\pm\tfrac{1}{2}{\mathrm{i}}\}$ is no obstruction to the continuation of the asymptotic expansions and into the domain $E$.* Likewise, for $m=-(\tfrac{1}{2}+k)$, $k=0,1,2,3,\dots$, the open arc ${{\partial E^0_\squarellblack}}\setminus\{\pm\tfrac{1}{2}{\mathrm{i}}\}$ is no obstruction to the continuation and into $E$.
The function ${{p}}(y)$ may be continued through its branch cut $I$ connecting $\pm\tfrac{1}{2}{\mathrm{i}}$ from the right. This continuation can be written in terms of the principal branch of the square root by the formula $${{p}}(y)=\frac{{\mathrm{i}}}{2y}-\frac{{\mathrm{i}}}{y}\left(y-\tfrac{1}{2}{\mathrm{i}}\right)^{1/2}\left(y+\tfrac{1}{2}{\mathrm{i}}\right)^{1/2},\quad \text{$\mathrm{Re}(y)>0$ or $|\mathrm{Im}(y)|>\tfrac{1}{2}$}.
\label{eq:lambda-0-continue}$$ Here the branch cuts of the two square-root factors emanate to the left from the corresponding roots $\pm\tfrac{1}{2}{\mathrm{i}}$, so the right-hand side is analytic in the interior domain $E$ with the possible exception of a simple pole at $y=0$. This particular continuation into $E$ through the open arc ${{\partial E^\infty_\squareurblack}}\setminus\{\pm\tfrac{1}{2}{\mathrm{i}}\}$ is precisely the function ${{p^0_\squarellblack}}(y)$, a function that has the arc ${{\partial E^0_\squarellblack}}\subset \partial E$ as its branch cut. Since $(\pm\tfrac{1}{2}{\mathrm{i}})^{1/2}=2^{-1/2}{\mathrm{e}}^{\pm{\mathrm{i}}\pi/4}$, it is easy to check that $\mathop{\mathrm{Res}}_{y=0}{{p^0_\squarellblack}}(y)=0$, so ${{p^0_\squarellblack}}(y)$ is analytic throughout the interior of $E$. We conclude that if $m=\tfrac{1}{2}+k$, $k=0,1,2,3,\dots$, the asymptotic formula in which ${{p}}(y)$ is simply replaced by ${{p^0_\squarellblack}}(y)$, is valid both for $y\in\mathbb{C}\setminus E$ as well as throughout the maximal domain of analyticity for ${{p^0_\squarellblack}}(y)$, namely $y\in\mathbb{C}\setminus{{\partial E^0_\squarellblack}}$. Likewise, the continuation of ${{p}}(y)$ through its branch cut from the left can be written as $$p(y)=\frac{{\mathrm{i}}}{2y}+\frac{{\mathrm{i}}}{y}\left(-\left(y-\tfrac{1}{2}{\mathrm{i}}\right)\right)^{1/2}\left(-\left(y+\tfrac{1}{2}{\mathrm{i}}\right)\right)^{1/2},\quad\text{$\mathrm{Re}(y)<0$ or $|\mathrm{Im}(y)|>\tfrac{1}{2}$,}$$ which is precisely the branch ${{p^\infty_\squareurblack}}(y)$ defined as a meromorphic function on the maximal domain $y\in\mathbb{C}\setminus{{\partial E^\infty_\squareurblack}}$, the only singularity of which is a simple pole at the origin $y=0$. For $y$ in the interior of the eye $E$, both ${{p^0_\squarellblack}}(y)$ and ${{p^\infty_\squareurblack}}(y)$ are well-defined, and we have the identity ${{p^\infty_\squareurblack}}(y)={{p^0_\squarellblack}}(y)^{-1}$ (and of course for $\lambda\in\mathbb{C}\setminus E$ the identity ${{p^\infty_\squareurblack}}(y)={{p^0_\squarellblack}}(y)={{p}}(y)$ holds). Due to the pole at the origin, if $m=-(\tfrac{1}{2}+k)$, $k=0,1,2,3,\dots$, the formula should be replaced with $u_n(ny;m)^{-1}=({\mathrm{i}}{{p^\infty_\squareurblack}}(y))^{-1}+\mathcal{O}(n^{-1})$ as $n\to+\infty$ which is valid uniformly for $y$ in compact subsets of $\mathbb{C}\setminus{{\partial E^\infty_\squareurblack}}$. This completes the proof of Theorem \[theorem:closed-eye-equilibrium\].
Asymptotic behavior of $u_n(ny;m)$ for $y$ near the distinguished eyebrow. Proof of Theorem \[thm:edge-formulae\]
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While to describe the asymptotic behavior of $u_n(ny;m)$ for $m=-(\tfrac{1}{2}+k)$ with $k\in\mathbb{Z}_{\ge 0}$ and $y$ bounded away from the eyebrow ${{\partial E^\infty_\squareurblack}}$ it was useful to introduce the analytic continuation ${{p^\infty_\squareurblack}}(y)$ of ${{p}}(y)$ from a neighborhood of $y=\infty$ to the maximal domain $\mathbb{C}\setminus{{\partial E^\infty_\squareurblack}}$, for $y$ near ${{\partial E^\infty_\squareurblack}}$ it is better to denote the two critical points of $V(\lambda;y)$ as ${{p}}(y)$ and ${{p}}(y)^{-1}$, both of which are analytic functions on all proper sub-arcs of the eyebrow ${{\partial E^\infty_\squareurblack}}$. We consider the matrix $\mathbf{M}_n(\lambda;y,m)$ with the simplest choice of $g$-function, namely $g(\lambda)\equiv 0$, which will treat the two critical points more symmetrically, as turns out to be appropriate for $y$ near the eyebrow ${{\partial E^\infty_\squareurblack}}$. It is then convenient to reformulate the Riemann-Hilbert conditions on $\mathbf{M}_n(\lambda;y,m)$ in the special case that $m=-(\tfrac{1}{2}+k)$ for $k\in\mathbb{Z}_{\ge 0}$. Since the jump on ${{L^\infty_\squarellblack}}\cup{{L^0_\squarellblack}}$ reduces to the identity in this case (see –), the jump contour for $\mathbf{M}_n^{(k)}(\lambda;y):=\mathbf{M}_n(\lambda;y,-(\tfrac{1}{2}+k))=\mathbf{Y}_n(\lambda;ny,-(\tfrac{1}{2}+k))$ is simply $L={{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$, and along the latter contour the factor ${{\lambda^{-(m+1)}_\squarellblack}}={{\lambda^{k-1/2}_\squarellblack}}$ appearing in the jump conditions – changes sign at the junction point between ${{L^\infty_\squareurblack}}$ and ${{L^0_\squareurblack}}$. Therefore, if we define a branch $\lambda_\infty^{k-1/2}$ analytic along $L$ and such that $\lambda_\infty^{k-1/2}={{\lambda^{k-1/2}_\squarellblack}}$ holds when $\lambda\in {{L^\infty_\squareurblack}}$, the Riemann-Hilbert problem for $\mathbf{M}_n^{(k)}(\lambda;y)$ can be written as follows.
Given parameters $n,k\in\mathbb{Z}_{\ge 0}$ as well as $y$ in a tubular neighborhood $T$ of ${{\partial E^\infty_\squareurblack}}$ as defined in , seek a $2\times 2$ matrix function $\lambda\mapsto\mathbf{M}_n^{(k)}(\lambda;y)$ with the following properties:
- **Analyticity:** $\lambda\mapsto\mathbf{M}_n^{(k)}(\lambda;y)$ is analytic in the domain $\lambda\in\mathbb{C}\setminus L$, $L:={{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$. It takes continuous boundary values on $L\setminus\{0\}$ from each maximal domain of analyticity.
- **Jump conditions:** The boundary values $\mathbf{M}^{(k)}_{n\pm}(\lambda;y)$ are related by $$\mathbf{M}^{(k)}_{n+}(\lambda;y)=\mathbf{M}^{(k)}_{n-}(\lambda;y)\begin{bmatrix}1 & \displaystyle\frac{\sqrt{2\pi}}{k!}\lambda_\infty^{k-1/2}{\mathrm{e}}^{-nV(\lambda;y)}\\0 & 1\end{bmatrix},\quad\lambda\in L.
\label{eq:M-edge-jump}$$
- **Asymptotics:** $\mathbf{M}_n^{(k)}(\lambda;y)\to\mathbb{I}$ as $\lambda\to\infty$ and $\mathbf{M}_n^{(k)}(\lambda;y)\lambda^{k\sigma_3}$ has a well-defined limit as $\lambda\to 0$.
\[rhp:edge\]
### Motivation: the special case of $k=0$
When $k=0$, Riemann-Hilbert Problem \[rhp:edge\] reduces from a multiplicative matrix problem to an additive scalar problem for the $12$-entry, and the explicit solution is obtained from the Plemelj formula: $$\mathbf{M}_n^{(0)}(\lambda;y)=\begin{bmatrix}1 & \displaystyle\frac{1}{{\mathrm{i}}\sqrt{2\pi}}\int_L\frac{\mu_\infty^{-1/2}{\mathrm{e}}^{-nV(\mu;y)}}{\mu-\lambda}\,{\mathrm{d}}\mu\\
0 & 1\end{bmatrix}.$$ Since $\mathbf{Y}_n(\lambda;ny,-\tfrac{1}{2})=\mathbf{M}_n^{(0)}(\lambda;y)$, applying gives the exact result $$u_n(ny;-\tfrac{1}{2})={\mathrm{i}}\frac{\displaystyle\int_L\lambda_\infty^{-1/2}{\mathrm{e}}^{-nV(\lambda;y)}\,{\mathrm{d}}\lambda}{\displaystyle\int_L\lambda_\infty^{-3/2}{\mathrm{e}}^{-nV(\lambda;y)}\,{\mathrm{d}}\lambda}.
\label{eq:u-edge-k-zero}$$ The large-$n$ asymptotic behavior of the rational solution $u_n(ny;-\tfrac{1}{2})$ is therefore reduced to the classical saddle-point expansion of two related contour integrals. When $y$ is close to the eyebrow ${{\partial E^\infty_\squareurblack}}$, $\mathrm{Re}(V({{p}}(y);y))\approx 0$, so the landscape of $\mathrm{Re}(-V(\lambda;y))$ in the $\lambda$-plane is similar to that shown in the central panels of Figure \[fig:y-real\], except in small neighborhoods of the two critical points $\lambda={{p}}(y),{{p}}(y)^{-1}$. In particular, for $\lambda$ bounded away from these two points, the contour $L={{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$ lies entirely in the red-shaded domain and hence $\mathrm{Re}(-V(\lambda;y))<0$ holds. This makes the corresponding contributions to the integrands in the numerator and denominator of exponentially small by comparison with the contributions from neighborhoods of the two saddle points. In a sense, this classical saddle point analysis can be embedded in a more general scheme that applies to Riemann-Hilbert Problem \[rhp:edge\] also for $k=1,2,3,\dots$.
In our previous paper on the subject of rational solutions of the Painlevé-III equation [@BothnerMS18] we observed that when $m\in\mathbb{Z}+\tfrac{1}{2}$ it is possible to reduce Riemann-Hilbert Problem \[rhp:renormalized\] to a linear algebraic Hankel system of dimension independent of $n$ in which the coefficients are contour integrals amenable to the classical method of steepest descent when $n$ is large such as those just considered above. We originally thought that these Hankel systems would provide the most efficient approach to the detailed analysis of $u_n(ny;m)$ for half-integral $m$, but it turns out that an approach based on more modern techniques of steepest descent for Riemann-Hilbert problems is more effective. We develop this approach in the following paragraphs.
### Modified outer parametrix
The same argument that focuses the contour integrals in on the critical points serves more generally to make the jump matrix in an exponentially small perturbation of the identity matrix except in neighborhoods of the critical points, which in turn suggests approximating $\mathbf{M}_n^{(k)}(\lambda;y)$ with a single-valued analytic function built to satisfy the required asymptotic conditions as $\lambda\to\infty$ and $\lambda\to 0$. Thus, given $k\in\mathbb{Z}_{\ge 0}$ and nonnegative integers $\alpha_1$ and $\alpha_2$ such that $$\alpha_1+\alpha_2 = -\left(m+\frac{1}{2}\right)=k,
\label{eq:alpha-partition-general-red}$$ we define an outer parametrix for $\mathbf{M}_n^{(k)}(\lambda;y)$ by the formula $$\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)=\lambda^{-k\sigma_3}(\lambda-{{p}}(y)^{-1})^{\alpha_1\sigma_3}(\lambda-{{p}}(y))^{\alpha_2\sigma_3}.
\label{eq:edge-outer}$$ This function is analytic for $\lambda\in\mathbb{C}\setminus\{0,{{p}}(y),{{p}}(y)^{-1}\}$, and satisfies the required asymptotic conditions in the sense that $\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)\to\mathbb{I}$ as $\lambda\to\infty$ and that $\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)\lambda^{k\sigma_3}$ is analytic at $\lambda=0$. The singularities in the outer parametrix at the critical points $\lambda={{p}}(y),{{p}}(y)^{-1}$ are needed to balance the local behavior of $\mathbf{M}_n^{(k)}(\lambda;y)$ which we turn to approximating next.
### Inner parametrices based on Hermite polynomials
As the tubular neighborhood $T$ containing $y$ excludes the branch points $\lambda=\pm\tfrac{1}{2}{\mathrm{i}}$, the two critical points remain distinct and hence simple, and both are analytic and nonvanishing functions of $y\in T$. To set up a uniform treatment of the two critical points, we may also refer to the critical points as $\lambda_1(y):={{p}}(y)^{-1}$ and $\lambda_2(y):={{p}}(y)$, which indicates the order in which neighborhoods of these points are visited as $\lambda$ traverses $L={{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$. Since $\lambda_j(y)$ are analytic and nonvanishing functions on $T$, to define $\lambda_\infty^{k-1/2}=\lambda^k\lambda_\infty^{-1/2}$ for $\lambda=\lambda_j(y)$ it suffices by analytic continuation to determine the value when $y=0.331372$ corresponding to the real midpoint of the eyebrow ${{\partial E^\infty_\squareurblack}}$. Thus, from the central panels in Figure \[fig:y-real\] we get that when $\lambda=\lambda_j(0.331372)$, $\lambda_\infty^{-1/2}$ lies in the right half-plane for both $j=1,2$. Let $D_j$ be simply-connected neighborhoods of $\lambda_j(y)$, $j=1,2$, respectively, and assume that these neighborhoods are sufficiently small but independent of $n$. Exploiting the fact that both critical points of $V$ are simple, we conformally map $D_j$ to a neighborhood of the origin via a conformal mapping $\lambda\mapsto W_j(\lambda;y)$, where $$V(\lambda;y)-V(\lambda_j(y);y)=W_j(\lambda;y)^2,\quad\lambda\in D_j,\quad j=1,2.$$ In this equation we make sure to choose branches of $\log(\lambda)$ in $V$ so that the left-hand side is a well-defined analytic function of $\lambda$ that vanishes to second order at the critical point $\lambda=\lambda_j(y)$. For small enough $D_j$, this relation defines $W_j(\lambda;y)$ as a conformal mapping up to a sign, which we select such that (possibly after some local adjustment of $L$ near the critical points) the image of the oriented arc $L\cap D_j$ is a real interval containing $W_j=0$ traversed in the direction of increasing $W_j$. We will need the precise value of $W_j'(\lambda_j(y);y)$, and by implicit differentiation one finds that $\tfrac{1}{2}V''(\lambda_j(y);y)=W_j'(\lambda_j(y);y)^2$. Now for $j=1,2$, $\tfrac{1}{2}V''(\lambda_j(y);y)$ is an analytic and non-vanishing function of $y$ on the tubular neighborhood $T$ in question, so to determine the $\lambda$-derivative $W_j'(\lambda_j(y);y)$ as an analytic function, it suffices to determine its value at any one point, say $y=0.331372$ where the eyebrow ${{\partial E^\infty_\squareurblack}}$ intersects the positive real $y$-axis. Here, from the central panels in Figure \[fig:y-real\] one can use the geometric interpretation of $W'_j(\lambda_j(y);y)^{-1}$ as the phase factor of the directed tangent to $L$ to deduce that $W_1'(\lambda_1(y);y)$ is positive imaginary while $W_2'(\lambda_2(y);y)$ is negative real when $y=0.331372$.
Given the conformal maps $W_j:D_j\to \mathbb{C}$, $j=1,2$, we define corresponding analytic and non-vanishing functions of $\lambda\in D_j$ denoted $f_j^{(\alpha_1,\alpha_2)}(\lambda;y)$ such that $$\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)=f_j^{(\alpha_1,\alpha_2)}(\lambda;y)^{\sigma_3}W_j(\lambda;y)^{\alpha_j\sigma_3},\quad\lambda\in D_j, \quad j=1,2.
\label{eq:F-j-define-red}$$ Likewise, the function $\lambda\mapsto \sqrt{2\pi}\lambda_\infty^{k-1/2}/k!$ admits analytic continuation from $L\cap D_j$ to all of $D_j$, and this function is non-vanishing on $D_j$ (taken sufficiently small but independent of $n$). Therefore, it has an analytic and non-vanishing square root, which we denote by $d_j(\lambda)$, $j=1,2$. Then, using the jump condition for the modified matrix $\mathbf{N}_n^{(k,j)}(\lambda;y):=\mathbf{M}_n^{(k)}(\lambda;y){\mathrm{e}}^{-nV(\lambda_j(y);y)\sigma_3/2}d_j(\lambda)^{\sigma_3}$ satisfies the local jump conditions $$\mathbf{N}^{(k,j)}_{n+}(\lambda;y)=\mathbf{N}^{(k,j)}_{n-}(\lambda;y)\begin{bmatrix}1 & {\mathrm{e}}^{-nW_j(\lambda;y)^2}\\0 & 1\end{bmatrix},\quad\lambda\in L\cap D_j,\quad j=1,2.$$ To define appropriate solutions of these jump conditions within the neighborhoods $D_j$ yielding inner parametrices matching well onto the outer parametrix when $\lambda\in\partial D_j$, we need to take into account the final factor on the right-hand side of . Thus writing $\zeta=n^{1/2}W_j(\lambda;y)$, we arrive at the following model Riemann-Hilbert problem.
Given $\alpha\in\mathbb{Z}_{\ge 0}$, seek a $2\times 2$ matrix function $\zeta\mapsto\mathbf{H}^{(\alpha)}(\zeta)$ with the following properties:
- **Analyticity:** $\zeta\mapsto\mathbf{H}^{(\alpha)}(\zeta)$ is analytic for $\mathrm{Im}(\zeta)\neq 0$, taking continuous boundary values on the real axis oriented left-to-right.
- **Jump conditions:** The boundary values $\mathbf{H}^{(\alpha)}_\pm(\zeta)$ taken on the real axis satisfy the following jump condition: $$\mathbf{H}^{(\alpha)}_+(\zeta)=\mathbf{H}^{(\alpha)}_-(\zeta)\begin{bmatrix}1 & {\mathrm{e}}^{-\zeta^2}\\0 & 1\end{bmatrix},\quad \zeta\in\mathbb{R}.$$
- **Asymptotics:** $\mathbf{H}^{(\alpha)}(\zeta)$ is required to satisfy the normalization condition $$\lim_{\zeta\to\infty}\mathbf{H}^{(\alpha)}(\zeta)\zeta^{-\alpha\sigma_3}=\mathbb{I}.
\label{eq:FIK-Hermite-normalize-red}$$
\[rhp:FIK-Hermite-red\]
This problem is well-known [@FokasIK91] to be solvable explicitly in terms of Hermite polynomials $\{H_j(\zeta)\}_{j=0}^\infty$ defined by the positivity of the leading coefficient, $H_j(\zeta)=h_j\zeta^j+\cdots$ for $h_j>0$, and the orthogonality conditions $$\int_\mathbb{R}H_j(\zeta)H_{j'}(\zeta){\mathrm{e}}^{-\zeta^2}\,{\mathrm{d}}\zeta=\delta_{jj'}.$$ Indeed the solution for $\alpha=0$ is explicitly $$\mathbf{H}^{(0)}(\zeta):=\begin{bmatrix}1&\displaystyle\frac{1}{2\pi{\mathrm{i}}}\int_\mathbb{R}\frac{{\mathrm{e}}^{-s^2}\,{\mathrm{d}}s}{s-\zeta}\\0 & 1\end{bmatrix},$$ and for positive degree, $$\mathbf{H}^{(\alpha)}(\zeta):=\begin{bmatrix} \displaystyle\frac{1}{h_\alpha}H_\alpha(\zeta) & \displaystyle
\frac{1}{2\pi{\mathrm{i}}h_\alpha}\int_\mathbb{R}\frac{H_\alpha(s){\mathrm{e}}^{-s^2}\,{\mathrm{d}}s}{s-\zeta}\\
\displaystyle -2\pi{\mathrm{i}}h_{\alpha-1}H_{\alpha-1}(\zeta) & \displaystyle -h_{\alpha-1}\int_\mathbb{R}\frac{H_{\alpha-1}(s){\mathrm{e}}^{-s^2}\,{\mathrm{d}}s}{s-\zeta}\end{bmatrix},\quad \alpha\ge 1.$$ From these formulæ we see that the normalization condition takes the more concrete form $$\mathbf{H}^{(\alpha)}(\zeta)\zeta^{-\alpha\sigma_3}=
\begin{cases}
\begin{bmatrix}1 & (2\pi{\mathrm{i}}h_0^2)^{-1}\zeta^{-1}+\mathcal{O}(\zeta^{-3})\\0 & 1\end{bmatrix},&\quad\alpha=0,\smallskip\\
\begin{bmatrix}1+O(\zeta^{-2}) & (2\pi{\mathrm{i}}h_\alpha^2)^{-1}\zeta^{-1}+\mathcal{O}(\zeta^{-3})\\
-2\pi{\mathrm{i}}h_{\alpha-1}^2\zeta^{-1}+\mathcal{O}(\zeta^{-3}) & 1+\mathcal{O}(\zeta^{-2})\end{bmatrix},&\quad\alpha\ge 1,
\end{cases}$$ in the limit $\zeta\to\infty$, where the error terms on the diagonal (resp., off-diagonal) are full asymptotic series in descending even (resp., odd) powers of $\zeta$ (terminating after finitely-many terms in the first column), and where the leading coefficients are explicitly given by [@DLMF Chapter 18] $$h_\alpha:=\frac{2^{\alpha/2}}{\pi^{1/4}\sqrt{\alpha!}},\quad\alpha=0,1,2,\dots,
\label{eq:Hermite-constants}$$ and by convention we define $h_{-1}:=0$. Now we define the inner parametrices by $$\begin{gathered}
\dot{\mathbf{M}}_n^{\mathrm{in},(\alpha_1,\alpha_2,j)}(\lambda;y):={\mathrm{e}}^{-nV(\lambda_j(y);y)\sigma_3/2}n^{-\alpha_j\sigma_3/2}d_j(\lambda)^{\sigma_3}f_j^{(\alpha_1,\alpha_2)}(\lambda;y)^{\sigma_3}\mathbf{H}^{(\alpha_j)}(n^{1/2}W_j(\lambda;y))d_j(\lambda)^{-\sigma_3}{\mathrm{e}}^{nV(\lambda_j(y);y)\sigma_3/2},\\
\lambda\in D_j,\quad j=1,2.\end{gathered}$$ Since the factors to the left of $\mathbf{H}^{(\alpha_j)}(\cdot)$ are analytic within $D_j$, it is easy to see that $\dot{\mathbf{M}}_n^{\mathrm{in},(\alpha_1,\alpha_2,j)}(\lambda;y)$ is analytic within $D_j$ except along $L\cap D_j$, where it exactly satisfies the jump condition . We also see easily that $$\begin{gathered}
\dot{\mathbf{M}}_n^{\mathrm{in},(\alpha_1,\alpha_2,j)}(\lambda;y)\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)^{-1}=
{\mathrm{e}}^{-nV(\lambda_j(y);y)\sigma_3/2}n^{-\alpha_j\sigma_3/2}d_j(\lambda)^{\sigma_3}f_j^{(\alpha_1,\alpha_2)}(\lambda;y)^{\sigma_3}\\
{}\cdot\mathbf{H}^{(\alpha_j)}(\zeta)\zeta^{-\alpha_j\sigma_3}\cdot
f_j^{(\alpha_1,\alpha_2)}(\lambda;y)^{-\sigma_3}d_j(\lambda)^{-\sigma_3}n^{\alpha_j\sigma_3/2}{\mathrm{e}}^{nV(\lambda_j(y);y)\sigma_3/2},
\label{eq:MinMoutEdge}\end{gathered}$$ where $\zeta:=n^{1/2}W_j(\lambda;y)$. Now, set $$A_j^{(\alpha_1,\alpha_2)}(\lambda;y):=\frac{d_j(\lambda)^2f_j^{(\alpha_1,\alpha_2)}(\lambda;y)^2}{2\pi{\mathrm{i}}h_{\alpha_j}^2W_j(\lambda;y)}
\quad\text{and}\quad
B_j^{(\alpha_1,\alpha_2)}(\lambda;y):= -\frac{2\pi{\mathrm{i}}h_{\alpha_j-1}^2}{d_j(\lambda)^2f_j^{(\alpha_1,\alpha_2)}(\lambda;y)^2W_j(\lambda;y)},\quad j=1,2.$$ These are meromorphic functions of $\lambda\in D_j$ with simple poles at $\lambda_j(y)$, and they are independent of $n$. Since $W_j(\lambda;y)$ is bounded away from zero when $\lambda\in\partial D_j$, restriction to the boundaries $\partial D_j$, $j=1,2$, gives $$\begin{gathered}
\dot{\mathbf{M}}_n^{\mathrm{in},(\alpha_1,\alpha_2,j)}(\lambda;y)\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)^{-1}={\mathrm{e}}^{-nV(\lambda_j(y);y)\sigma_3/2}n^{-\alpha_j\sigma_3/2}\\
\cdot\begin{bmatrix}1 + \mathcal{O}(n^{-1}) & A_j^{(\alpha_1,\alpha_2)}(\lambda;y)n^{-1/2}+\mathcal{O}(n^{-3/2})\\
B_j^{(\alpha_1,\alpha_2)}(\lambda;y)n^{-1/2}+\mathcal{O}(n^{-3/2}) & 1+\mathcal{O}(n^{-1})\end{bmatrix}\\
\cdot n^{\alpha_j\sigma_3/2}{\mathrm{e}}^{nV(\lambda_j(y);y)\sigma_3/2},\quad\lambda\in\partial D_j,\quad j=1,2
\label{eq:red-eyebrow-disk-boundaries}\end{gathered}$$ if $\alpha_j\ge 0$, while in the special case $\alpha_j=0$ we may also write $$\begin{gathered}
\dot{\mathbf{M}}_n^{\mathrm{in},(\alpha_1,\alpha_2,j)}(\lambda;y)\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)^{-1} = {\mathrm{e}}^{-nV(\lambda_j(y);y)\sigma_3/2}
\begin{bmatrix}1 & A_j^{(\alpha_1,\alpha_2)}(\lambda;y)n^{-1/2}+\mathcal{O}(n^{-3/2})\\0 & 1\end{bmatrix}{\mathrm{e}}^{nV(\lambda_j(y);y)\sigma_3/2},\\
\lambda\in \partial D_j,\quad\alpha_j=0.
\label{eq:red-eyebrow-disk-boundaries-alpha-0}\end{gathered}$$
### Initial global parametrix construction and comparison matrices {#sec:edge-comparison-matrices}
Given non-negative integers $\alpha_1,\alpha_2$ satisfying , the global parametrix for $\mathbf{M}_n^{(k)}(\lambda;y)$ is then defined by: $$\dot{\mathbf{M}}_n^{(\alpha_1,\alpha_2)}(\lambda;y):=\begin{cases}
\dot{\mathbf{M}}_n^{\mathrm{in},(\alpha_1,\alpha_2,1)}(\lambda;y),&\quad\lambda\in D_1\\
\dot{\mathbf{M}}_n^{\mathrm{in},(\alpha_1,\alpha_2,2)}(\lambda;y),&\quad\lambda\in D_2\\
\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y),&\quad\lambda\in\mathbb{C}\setminus\overline{D_1\cup D_2}.
\end{cases}$$
For later purposes, we will need to record the residues of $A_j^{(\alpha_1,\alpha_2)}(\lambda;y)$ and $B_j^{(\alpha_1,\alpha_2)}(\lambda;y)$ at $\lambda=\lambda_j$, where $W_j(\lambda;y)$ vanishes to first order. Thus, $$\mathop{\mathrm{Res}}_{\lambda=\lambda_j}A_j^{(\alpha_1,\alpha_2)}(\lambda;y)=\frac{d_j(\lambda_j)^2f_j^{(\alpha_1,\alpha_2)}(\lambda_j;y)^2}{2\pi{\mathrm{i}}h_{\alpha_j}^2W_j'(\lambda_j;y)}\quad\text{and}\quad
\mathop{\mathrm{Res}}_{\lambda=\lambda_j}B_j^{(\alpha_1,\alpha_2)}(\lambda;y)=
-\frac{2\pi{\mathrm{i}}h_{\alpha_j-1}^2}{d_j(\lambda_j)^2f_j^{(\alpha_1,\alpha_2)}(\lambda_j;y)^2W_j'(\lambda_j;y)},$$ and combining with and l’Hôpital’s rule gives $$f_j^{(\alpha_1,\alpha_2)}(\lambda_j;y)=\lambda_j^{-k}(\lambda_j-\lambda_{3-j})^{\alpha_{3-j}}W_j'(\lambda_j;y)^{-\alpha_j},\quad j=1,2.$$ Recalling the definition of $d_j(\lambda)$ as a square root of $\sqrt{2\pi}\lambda_\infty^{k-1/2}/k!$, we have $$\begin{split}
d_j(\lambda_j)^2f_j^{(\alpha_1,\alpha_2)}(\lambda_j;y)^2 &= \frac{\sqrt{2\pi}}{k!}\lambda_j^{-k}\lambda_{j,\infty}^{-1/2}(\lambda_2-\lambda_1)^{2\alpha_{3-j}}W_j'(\lambda_j;y)^{-2\alpha_j}\\ &=
\frac{2^{\alpha_j}\sqrt{2\pi}}{k!}\lambda_j^{-k}\lambda_{j,\infty}^{-1/2}(\lambda_2-\lambda_1)^{2\alpha_{3-j}}V''(\lambda_j;y)^{-\alpha_j}
\end{split}$$ where $\lambda_{j,\infty}^{-1/2}$ refers to the branch of the square root that lies in the right half-plane when $y=0.331372$ (the real point of ${{\partial E^\infty_\squareurblack}}$), continued analytically to all $y\in T$, and we used the identity $W_j'(\lambda_j;y)^2=\tfrac{1}{2}V''(\lambda_j;y)$ and the fact that $2\alpha_{3-j}$ is even. Combining this with finally gives $$\begin{split}
\mathop{\mathrm{Res}}_{\lambda=\lambda_j}A^{(\alpha_1,\alpha_2)}_j(\lambda;y)&=
\frac{\alpha_j!\lambda_j^{-k}\lambda_{j,\infty}^{-1/2}(\lambda_2-\lambda_1)^{2\alpha_{3-j}}V''(\lambda_j;y)^{-\alpha_j}}{{\mathrm{i}}k! V''(\lambda_j;y)^{1/2}}\\
\mathop{\mathrm{Res}}_{\lambda=\lambda_j}B^{(\alpha_1,\alpha_2)}_j(\lambda;y)&=
\frac{k!\lambda_j^k\lambda_{j,\infty}^{1/2}(\lambda_2-\lambda_1)^{-2\alpha_{3-j}}V''(\lambda_j;y)^{\alpha_j}}{{\mathrm{i}}(\alpha_j-1)!V''(\lambda_j;y)^{1/2}},
\end{split}
\label{eq:edge-A-B-residues}$$ where the analytic functions $V''(\lambda_1(y);y)^{1/2}$ and $V''(\lambda_2(y);y)^{1/2}$ are respectively positive imaginary and negative real when $y=0.331372$.
To compare $\mathbf{M}_n^{(k)}(\lambda;y)$ with its parametrix, we define two types of comparison matrices by $$\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y):={\mathrm{e}}^{nV(\lambda_j(y);y)\sigma_3/2}n^{(\alpha_j+1/2)\sigma_3/2}\mathbf{M}_n^{(k)}(\lambda;y)\dot{\mathbf{M}}_n^{(\alpha_1,\alpha_2)}(\lambda;y)^{-1}n^{-(\alpha_j+1/2)\sigma_3/2}{\mathrm{e}}^{-nV(\lambda_j(y);y)\sigma_3/2},\quad j=1,2.$$ Both types (i.e., for $j=1,2$) of comparison matrix have the property that they are analytic functions of $\lambda$ in both domains $D_1$ and $D_2$ (because the continuous boundary values of $\mathbf{M}_n^{(k)}(\lambda;y)$ and $\dot{\mathbf{M}}_n^{(\alpha_1,\alpha_2)}(\lambda;y)$ satisfy the same jump conditions there) and in the exterior domain except on the original jump contour $L={{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$. Moreover, it is easy to check that $\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)\to\mathbb{I}$ as $\lambda\to\infty$. The comparison matrices therefore satisfy the conditions of a Riemann-Hilbert problem specified by jump conditions across the part of $L$ exterior to the domains $D_1$ and $D_2$ and across the boundaries $\partial D_1$ and $\partial D_2$ of these domains. Jump contours for the comparison matrices are illustrated in Figures \[fig:EdgeError-within\] and \[fig:EdgeError-without\] (cf., Figure \[fig:y-real\]) for two different values of $y$ on either side of the eyebrow ${{\partial E^\infty_\squareurblack}}$.
![The jump contour for the comparison matrices $\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)$, and also for the final error matrix $\mathbf{E}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)$, for $y=0.33$, a point just to the left of the eyebrow ${{\partial E^\infty_\squareurblack}}$. The jump contour consists of the arcs of $L={{L^\infty_\squareurblack}}\cup{{L^0_\squareurblack}}$ outside the disks $D_1$ and $D_2$ (red), as well as the boundaries of the latter disks (green) that are oriented in the clockwise direction for the purposes of defining the boundary values taken thereon. The background is a contour plot of $\mathrm{Re}(V(\lambda;y))$, with pink shading for $\mathrm{Re}(V(\lambda;y))>0$ and blue shading for $\mathrm{Re}(V(\lambda;y))<0$.[]{data-label="fig:EdgeError-within"}](EdgeError-y-0p33-preview.pdf)
![As in Figure \[fig:EdgeError-within\], but for $y=0.333$, a point just to the right of the eyebrow ${{\partial E^\infty_\squareurblack}}$.[]{data-label="fig:EdgeError-without"}](EdgeError-y-0p333-preview.pdf)
Observe that the landscape of $\mathrm{Re}(V(\lambda;y))$ shown in these plots resembles, at least for $\lambda$ not too close to $\lambda_1$ or $\lambda_2$, that illustrated in the central panels of Figure \[fig:y-real\]. Therefore, when $y$ is close to the eyebrow ${{\partial E^\infty_\squareurblack}}$, if the tubular neighborhood $T$ is taken to be sufficiently thin (by choosing $\delta_2$ sufficiently small in ) given the domains $D_1$ and $D_2$, the jump condition satisfied by $\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)$ on the arcs of $L$ exterior to the latter domains has the form (because all red contours lie strictly within the pink-shaded region) $$\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n+}(\lambda;y)= \mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n-}(\lambda;y)(\mathbb{I}+\text{exponentially small}),\quad n\to+\infty,\quad \lambda\in L\setminus(D_1\cup D_2),$$ with the convergence being in the $L^p$ sense for every $p$ and holding uniformly for $y\in T$. Therefore, the essential jump conditions for $\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)$ are those across the domain boundaries $\partial D_1$ and $\partial D_2$. Taking these to be oriented in the clockwise sense, using – gives $$\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n+}(\lambda;y)=\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n-}(\lambda;y)
\begin{bmatrix}1+\mathcal{O}(n^{-1}) & A_j^{(\alpha_1,\alpha_2)}(\lambda;y)+\mathcal{O}(n^{-1})\\ \mathcal{O}(n^{-1}) & 1+\mathcal{O}(n^{-1})
\end{bmatrix},\quad\lambda\in\partial D_j$$ (in the special case that $\alpha_j=0$ the $\mathcal{O}(n^{-1})$ error terms in all but the $12$ entry of the jump matrix vanish identically), while $$\begin{gathered}
\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n+}(\lambda;y)=\\
\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n-}(\lambda;y)
\begin{bmatrix}1+\mathcal{O}(n^{-1}) & a_n^{(\alpha_1,\alpha_2,j)}(y)[A_{3-j}^{(\alpha_1,\alpha_2)}(\lambda;y)+\mathcal{O}(n^{-1})]\\
b_n^{(\alpha_1,\alpha_2,j)}(y)[B_{3-j}^{(\alpha_1,\alpha_2)}(\lambda;y)+\mathcal{O}(n^{-1})] & 1+\mathcal{O}(n^{-1})\end{bmatrix},\\
\lambda\in\partial D_{3-j}
\label{eq:Edge-F-jump-other-boundary}\end{gathered}$$ where $$a_n^{(\alpha_1,\alpha_2,j)}(y):={\mathrm{e}}^{n(V(\lambda_j(y);y)-V(\lambda_{3-j}(y);y))}n^{\alpha_j-\alpha_{3-j}}
\quad\text{and}\quad
b_n^{(\alpha_1,\alpha_2,j)}(y):={\mathrm{e}}^{n(V(\lambda_{3-j}(y);y)-V(\lambda_j(y);y))}n^{\alpha_{3-j}-\alpha_j-1}$$ (in the special case that $\alpha_{3-j}=0$ the $O(n^{-1})$ error terms in all but the $12$ entry of the jump matrix vanish identically, and in addition $B_{3-j}^{(\alpha_1,\alpha_2)}(\lambda;y)\equiv 0$ because by convention $h_{-1}=0$). Recalling that $\mathrm{Re}(V(\lambda_2(y);y))+\mathrm{Re}(V(\lambda_1(y);y))=0$ and $\alpha_1+\alpha_2=k$, upon suitable conditions on $y\in T$ the jump condition reduces to one of the following forms:
- Case I: If $\alpha_{3-j}=0$ (so also $\alpha_j=k$) and the inequality $$\mathrm{Re}(V(\lambda_j(y);y))\le-\frac{1}{2}\alpha_j\frac{\ln(n)}{n}=-\frac{1}{2}k\frac{\ln(n)}{n}
\label{eq:edge-inequality-3}$$ holds, then becomes $$\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n+}(\lambda;y)=\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n-}(\lambda;y)
\begin{bmatrix}1 & a_n^{(\alpha_1,\alpha_2,j)}(y)[A_{3-j}^{(\alpha_1,\alpha_2)}(\lambda;y)+\mathcal{O}(n^{-1})]\\0 & 1\end{bmatrix},\quad\lambda\in\partial D_{3-j}
\label{eq:edge-jump-reduce-3}$$ in which $a_n^{(\alpha_1,\alpha_2,j)}(y)=\mathcal{O}(1)$ as $n\to+\infty$.
- Case II$_\mathrm{a}$: If $\alpha_{3-j}>0$ (so also $\alpha_j<k$) and the inequalities $$\frac{1}{2}(k-2\alpha_j-\tfrac{1}{2})\frac{\ln(n)}{n}\le\mathrm{Re}(V(\lambda_j(y);y))\le\frac{1}{2}(k-2\alpha_j)\frac{\ln(n)}{n}
\label{eq:edge-inequality-1}$$ hold, then becomes $$\mathbf{F}_{n+}^{(\alpha_1,\alpha_2,j)}(\lambda;y)=\mathbf{F}_{n-}^{(\alpha_1,\alpha_2,j)}(\lambda;y)
\begin{bmatrix}1+\mathcal{O}(n^{-1}) & a_n^{(\alpha_1,\alpha_2,j)}(y)[A_{3-j}^{(\alpha_1,\alpha_2)}(\lambda;y)+\mathcal{O}(n^{-1})]\\ \mathcal{O}(n^{-1/2}) & 1+\mathcal{O}(n^{-1})\end{bmatrix},\quad\lambda\in\partial D_{3-j}
\label{eq:edge-jump-reduce-1}$$ in which $a_n^{(\alpha_1,\alpha_2,j)}(y)=\mathcal{O}(1)$ as $n\to+\infty$.
- Case II$_\mathrm{b}$: If $\alpha_{3-j}>0$ (so also $\alpha_j<k$) and the inequalities $$\frac{1}{2}(k-2\alpha_j-1)\frac{\ln(n)}{n}\le\mathrm{Re}(V(\lambda_j(y);y))\le \frac{1}{2}(k-2\alpha_j-\tfrac{1}{2})\frac{\ln(n)}{n}
\label{eq:edge-inequality-2}$$ hold, then becomes $$\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n+}(\lambda;y)=\mathbf{F}^{(\alpha_1,\alpha_2,j)}_{n-}(\lambda;y)
\begin{bmatrix}1+\mathcal{O}(n^{-1}) & \mathcal{O}(n^{-1/2})\\ b_n^{(\alpha_1,\alpha_2,j)}(y)[B_{3-j}^{(\alpha_1,\alpha_2)}(\lambda;y)+\mathcal{O}(n^{-1})] & 1+\mathcal{O}(n^{-1})\end{bmatrix},\quad\lambda\in\partial D_{3-j}
\label{eq:edge-jump-reduce-2}$$ in which $b_n^{(\alpha_1,\alpha_2,j)}(y)=\mathcal{O}(1)$ as $n\to+\infty$.
By varying the index $j=1,2$ as well as the choice of non-negative integers $\alpha_1+\alpha_2=k$, the above inequalities , , and on $y\in T$ actually cover the whole tubular neighborhood $T$. Indeed, given $k\ge 0$, begin by taking $\alpha_1=k$ and $\alpha_2=0$, and consider the comparison matrix $\mathbf{F}_n^{(k,0,1)}(\lambda;y)$. Assuming that $\mathrm{Re}(V(\lambda_1(y);y))\le -\tfrac{1}{2}kn^{-1}\ln(n)$, the inequality of Case I guarantees that governs the jump condition on $\partial D_2$. Then, for $\ell=1,\dots,k$,
- Take $\alpha_1=k-\ell+1>0$ and $\alpha_2=\ell-1<k$, and consider the comparison matrix $\mathbf{F}_n^{(k-\ell+1,\ell-1,2)}(\lambda;y)$. Assuming that $-\tfrac{1}{2}(k-2\ell+2)n^{-1}\ln(n)\le\mathrm{Re}(V(\lambda_1(y);y))\le -\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)$, the inequalities of Case II$_\mathrm{a}$ imply that governs the jump condition on $\partial D_1$. Assuming that $-\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)\le\mathrm{Re}(V(\lambda_1(y);y))\le -\tfrac{1}{2}(k-2\ell+1)$, the inequalities of Case II$_\mathrm{b}$ imply that governs the jump condition on $\partial D_1$.
- Now take $\alpha_1=k-\ell<k$ and $\alpha_2=\ell>0$, and consider the comparison matrix $\mathbf{F}_n^{(k-\ell,\ell,1)}(\lambda;y)$. Assuming that $-\tfrac{1}{2}(k-2\ell+1)n^{-1}\ln(n)\le\mathrm{Re}(V(\lambda_1(y);y))\le -\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)$, the inequalities of Case II$_\mathrm{b}$ imply that governs the jump condition on $\partial D_2$. Assuming that $-\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)\le\mathrm{Re}(V(\lambda_1(y);y))\le-\tfrac{1}{2}(k-2\ell)n^{-1}\ln(n)$, the inequalities of Case II$_\mathrm{a}$ imply that governs the jump condition on $\partial D_2$.
Finally, take $\alpha_1=0$ and $\alpha_2=k$, and consider the comparison matrix $\mathbf{F}_n^{(0,k,2)}(\lambda;y)$. Assuming that $\mathrm{Re}(V(\lambda_1(y);y))\ge \tfrac{1}{2}kn^{-1}\ln(n)$, the inequality of Case I then guarantees that governs the jump condition on $\partial D_1$.
### Modeling of comparison matrices
To determine the asymptotic behavior as $n\to+\infty$ of the various types of comparison matrices, it now becomes necessary to model the leading terms of the jump matrices, which generally do not decay to the identity on the domain boundaries $\partial D_1$ and $\partial D_2$, but that are guaranteed to be bounded by association of $y\in T$ with the appropriate indices $\alpha_1$, $\alpha_2$, and $j$ as described above. In Cases I and II$_\mathrm{a}$, the dominant terms in the jump matrices on $\partial D_1$ and $\partial D_2$ are both upper triangular matrices, while in Case II$_\mathrm{b}$ one jump matrix is upper triangular and the other is lower triangular. This situation requires two different types of parametrices, which we formulate as Riemann-Hilbert problems here.
Let distinct points $\lambda_1\neq\lambda_2$, $\lambda_j\neq 0$, $j=1,2$, be given with corresponding simply-connected neighborhoods $D_1$ and $D_2$ with $D_1\cap D_2=\emptyset$. For $j=1,2$, let $\phi_j$ be meromorphic on $D_j$ and continuous up to the boundary $\partial D_j$ with a simple pole at $\lambda_j$ as the only singularity in $D_j$. Seek a $2\times 2$ matrix function $\lambda\mapsto\dot{\mathbf{F}}(\lambda)$ with the following properties:
- **Analyticity:** $\lambda\mapsto\dot{\mathbf{F}}(\lambda)$ is analytic for $\lambda\in\mathbb{C}\setminus (\partial D_1\cup\partial D_2)$ and takes continuous boundary values from each side on $\partial D_1$ and $\partial D_2$.
- **Jump conditions:** The boundary values are related by the following jump conditions. Assuming clockwise orientation of $\partial D_j$, $j=1,2$, $$\dot{\mathbf{F}}_+(\lambda)=\dot{\mathbf{F}}_-(\lambda)\begin{bmatrix}1 & \phi_j(\lambda)\\
0 & 1\end{bmatrix},\quad\lambda\in \partial D_j,\quad j=1,2.$$
- **Asymptotics:** $\dot{\mathbf{F}}(\lambda)\to\mathbb{I}$ as $\lambda\to\infty$.
\[rhp:upper-upper\]
This problem always has a unique solution, which may be sought in the form $$\dot{\mathbf{F}}(\lambda)=\begin{bmatrix}1 & \dot{f}(\lambda)\\0 & 1\end{bmatrix}.$$ The conditions of Riemann-Hilbert Problem \[rhp:upper-upper\] descend to the conditions that the scalar function $\dot{f}(\lambda)$ be analytic for $\lambda\in\mathbb{C}\setminus(\partial D_1\cup\partial D_2)$ with $\dot{f}(\lambda)\to 0$ as $\lambda\to\infty$, and the jump conditions now take the additive form: $\dot{f}_+(\lambda)=\dot{f}_-(\lambda)+\phi_j(\lambda)$ holds for $\lambda\in\partial D_j$, $j=1,2$. It follows that $\dot{f}(\lambda)$ is given by the Plemelj formula $$\dot{f}(\lambda)=\frac{1}{2\pi{\mathrm{i}}}\oint_{\partial D_1}\frac{\phi_1(\mu)\,{\mathrm{d}}\mu}{\mu-\lambda} +
\frac{1}{2\pi{\mathrm{i}}}\oint_{\partial D_2}\frac{\phi_2(\mu)\,{\mathrm{d}}\mu}{\mu-\lambda},\quad\lambda\in\mathbb{C}\setminus(\partial D_1\cup\partial D_2).$$ The integrals may be evaluated explicitly, by residues. In the special case that $\lambda$ is exterior to both domains $D_j$, $j=1,2$, we therefore find $$\dot{\mathbf{F}}(\lambda)=\begin{bmatrix}1 & \displaystyle\frac{\Phi_1}{\lambda-\lambda_1} +\frac{\Phi_2}{\lambda-\lambda_2}\\0 & 1\end{bmatrix},\quad\lambda\in\mathbb{C}\setminus\overline{D_1\cup D_2},$$ where $\Phi_j$ denotes the residue of $\phi_j$ at $\lambda_j$, $j=1,2$. In particular, we see that $$\dot{\mathbf{F}}(\lambda)=\mathbb{I} + \lambda^{-1}\dot{\mathbf{F}}^\infty_1 + \mathcal{O}(\lambda^{-2}),\quad\lambda\to\infty\quad\text{and}\quad
\dot{\mathbf{F}}(\lambda)=\dot{\mathbf{F}}^0_0 + \mathcal{O}(\lambda),\quad\lambda\to 0,
\label{eq:dotF-expand}$$ in which we have $$\dot{F}^\infty_{1,12}=\Phi_1+\Phi_2,\quad\dot{F}^0_{0,11}=1,\quad\dot{F}^0_{0,12}=-\frac{\Phi_1}{\lambda_1}-\frac{\Phi_2}{\lambda_2}.
\label{eq:dot-F-elements-upper-upper}$$
Let distinct nonzero points $\lambda_\mathrm{U}\neq\lambda_\mathrm{L}$ be given with corresponding simply-connected neighborhoods $D_\mathrm{U}$ and $D_\mathrm{L}$ with $D_\mathrm{U}\cap D_\mathrm{L}=\emptyset$. Let $\phi_\mathrm{U}$ and $\phi_\mathrm{L}$ be meromorphic on $D_\mathrm{U}$ and $D_\mathrm{L}$ respectively and continuous up to the corresponding boundary curves, with simple poles only at $\lambda_\mathrm{U}$ and $\lambda_\mathrm{L}$ respectively. Seek a $2\times 2$ matrix function $\lambda\mapsto\dot{\mathbf{F}}(\lambda)$ with the following properties:
- **Analyticity:** $\lambda\mapsto\dot{\mathbf{F}}(\lambda)$ is analytic for $\lambda\in\mathbb{C}\setminus(\partial D_\mathrm{U}\cup\partial D_\mathrm{L})$ and takes continuous boundary values from each side on $\partial D_\mathrm{U}$ and $\partial D_\mathrm{L}$.
- **Jump conditions:** The boundary values are related by the following jump conditions. Assuming clockwise orientation of $\partial D_\mathrm{U}$, $$\dot{\mathbf{F}}_+(\lambda)=\dot{\mathbf{F}}_-(\lambda)\begin{bmatrix}1 & \phi_\mathrm{U}(\lambda)\\0 & 1\end{bmatrix},\quad\lambda\in\partial D_\mathrm{U},
\label{eq:dot-F-jump-U}$$ and assuming clockwise orientation of $\partial D_\mathrm{L}$, $$\dot{\mathbf{F}}_+(\lambda)=\dot{\mathbf{F}}_-(\lambda)\begin{bmatrix}1 & 0\\\phi_\mathrm{L}(\lambda) & 1\end{bmatrix},\quad\lambda\in\partial D_\mathrm{L}.$$
- **Asymptotics:** $\dot{\mathbf{F}}(\lambda)\to\mathbb{I}$ as $\lambda\to\infty$.
\[rhp:F-dot-eyebrow\]
This problem is a generalization of the one that characterizes the soliton solutions of AKNS systems [@BealsC84]. Unlike Riemann-Hilbert Problem \[rhp:upper-upper\] this problem is only conditionally solvable. Letting $\Phi_\mathrm{U}$ denote the residue of $\phi_\mathrm{U}$ at $\lambda_\mathrm{U}$, and $\Phi_\mathrm{L}$ that of $\phi_\mathrm{L}$ at $\lambda_\mathrm{L}$, this problem has a unique solution if and only if $$\Delta:=\Phi_\mathrm{U}\Phi_\mathrm{L}+(\lambda_\mathrm{U}-\lambda_\mathrm{L})^2\neq 0.
\label{eq:edge-denominator}$$ The solution is a rational function in the domain exterior to $D_\mathrm{U}\cup D_\mathrm{L}$: $$\dot{\mathbf{F}}(\lambda)=\mathbb{I}+\frac{1}{\lambda-\lambda_\mathrm{U}}\frac{(\lambda_\mathrm{U}-\lambda_\mathrm{L})\Phi_\mathrm{U}}{\Delta}\begin{bmatrix}
0 & \lambda_\mathrm{U}-\lambda_\mathrm{L}\\0 & \Phi_\mathrm{L}\end{bmatrix}
+\frac{1}{\lambda-\lambda_\mathrm{L}}\frac{(\lambda_\mathrm{L}-\lambda_\mathrm{U})\Phi_\mathrm{L}}{\Delta}
\begin{bmatrix}\Phi_\mathrm{U} & 0\\\lambda_\mathrm{L}-\lambda_\mathrm{U} & 0\end{bmatrix},\quad
\lambda\in\mathbb{C}\setminus\overline{D_\mathrm{U}\cup D_\mathrm{L}}.
\label{eq:dotF-edge-outside}$$ This formula determines $\dot{\mathbf{F}}(\lambda)$ in the domains $D_\mathrm{U}$ and $D_\mathrm{L}$ by the jump conditions; Laurent expansion of the interior boundary value $\dot{\mathbf{F}}_-(\lambda)$ shows in each case that its only singularity is removable. Moreover, shows that expansions of the form again hold whenever the solution exists, in which $$\dot{F}^\infty_{1,12}= \frac{(\lambda_\mathrm{U}-\lambda_\mathrm{L})^2\Phi_\mathrm{U}}{\Delta},\quad
\dot{F}^0_{0,11}=1-\frac{(\lambda_\mathrm{L}-\lambda_\mathrm{U})\Phi_\mathrm{L}\Phi_\mathrm{U}}{\lambda_\mathrm{L}\Delta},\quad
\text{and}\quad
\dot{F}^0_{0,12}=-\frac{(\lambda_\mathrm{U}-\lambda_\mathrm{L})^2\Phi_\mathrm{U}}{\lambda_\mathrm{U}\Delta}.
\label{eq:dot-F-elements-upper-lower}$$
### Final error analysis and asymptotic formulæ for $u_n(ny;-(\tfrac{1}{2}+k))$ {#sec:Edge-error-analysis}
Suppose that $y$ is such that, after proper association of the data of Riemann-Hilbert Problem \[rhp:upper-upper\] or \[rhp:F-dot-eyebrow\] with the leading terms of the jump matrices for the comparison matrix $\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)$, $\dot{\mathbf{F}}(\lambda)$ exists and is bounded as $n\to+\infty$ (no condition in the case of Riemann-Hilbert Problem \[rhp:upper-upper\]). Then it is easy to check that the error matrix $\mathbf{E}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y):=\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)\dot{\mathbf{F}}(\lambda)^{-1}$ satisfies the conditions of a small-norm Riemann-Hilbert problem formulated relative to a jump contour such as shown in Figures \[fig:EdgeError-within\] and \[fig:EdgeError-without\], with the result that $$\mathbf{E}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)=\mathbb{I}+\lambda^{-1}\mathbf{E}^\infty_{n,1}(y) + \mathcal{O}(\lambda^{-2}),\quad\lambda\to\infty\quad\text{and}\quad \mathbf{E}_n(\lambda;y)=\mathbf{E}^0_{n,0}(y) + \mathcal{O}(\lambda),\quad\lambda\to 0,$$ where $\mathbf{E}^\infty_{n,1}(y)=\mathcal{O}(n^{-1/2})$ and $\mathbf{E}^0_{n,0}(y)=\mathbb{I}+\mathcal{O}(n^{-1/2})$ as $n\to+\infty$ uniformly for $y\in T$ (if Case I holds, we may replace $\mathcal{O}(n^{-1/2})$ in both estimates with $\mathcal{O}(n^{-1})$). Note that for $\lambda$ in the exterior of the domain $D_1\cup D_2$, we have the exact identity $$\begin{split}
\mathbf{Y}_n(\lambda;ny,-(\tfrac{1}{2}+k))&=\mathbf{M}_n^{(k)}(\lambda;y)\\
&=\delta_n(y)^{\sigma_3}\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)\delta_n(y)^{-\sigma_3}\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)\\&=\delta_n(y)^{\sigma_3}\mathbf{F}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)\delta_n(y)^{-\sigma_3}\\
&=\delta_n(y)^{\sigma_3}\mathbf{E}_n^{(\alpha_1,\alpha_2,j)}(\lambda;y)\dot{\mathbf{F}}(\lambda)\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)\delta_n(y)^{-\sigma_3},\quad\lambda\in\mathbb{C}\setminus\overline{D_1\cup D_2},
\end{split}$$ where $\delta_n(y)\neq 0$ is independent of $\lambda$ and takes a different form in different parts of the tubular neighborhood $T$ containing $y$, the hypothesis on $\lambda$ ensures that $\dot{\mathbf{M}}_n^{(\alpha_1,\alpha_2)}(\lambda;y)=\dot{\mathbf{M}}^{\mathrm{out},(\alpha_1,\alpha_2)}(\lambda;y)$, and we used the fact that the outer parametrix is diagonal. Consequently, from , we arrive at the following approximate formula for $u_n(ny;-(\tfrac{1}{2}+k))$ valid for large $n$: $$u_n(ny;-(\tfrac{1}{2}+k))=\frac{-{\mathrm{i}}\dot{F}^\infty_{1,12}+\mathcal{O}(n^{-1/2})}{\dot{F}^0_{0,11}\dot{F}^0_{0,12}+\mathcal{O}(n^{-1/2})},\quad n\to+\infty,$$ which holds uniformly for $y\in T$ for which $\dot{\mathbf{F}}(\lambda)$ exists and is bounded (in Case II$_\mathrm{b}$ the denominator $\Delta$ should be bounded away from zero). In Case I, the error terms can be replaced with $\mathcal{O}(n^{-1})$. Therefore, if $y\in T$ is such that Case I or II$_\mathrm{a}$ holds, from we have $$u_n(ny;-(\tfrac{1}{2}+k))={\mathrm{i}}\frac{\lambda_1\lambda_2(\Phi_1+\Phi_2)+\mathcal{O}(n^{-1})}{\lambda_2\Phi_1+\lambda_1\Phi_2 + \mathcal{O}(n^{-1})},\quad n\to+\infty,\quad \text{if Case I holds,}
\label{eq:u-upper-upper-CaseI}$$ $$u_n(ny;-(\tfrac{1}{2}+k))={\mathrm{i}}\frac{\lambda_1\lambda_2(\Phi_1+\Phi_2)+\mathcal{O}(n^{-1/2})}{\lambda_2\Phi_1+\lambda_1\Phi_2 + \mathcal{O}(n^{-1/2})},\quad n\to+\infty,\quad \text{if Case II$_\mathrm{a}$ holds,}
\label{eq:u-upper-upper}$$ while if instead Case II$_\mathrm{b}$ holds and $\Delta$ is bounded away from zero, from and we have, for any given $\delta>0$ independent of $n$, $$u_n(ny;-(\tfrac{1}{2}+k))={\mathrm{i}}\frac{\lambda_\mathrm{U}\lambda_\mathrm{L}((\lambda_\mathrm{L}-\lambda_\mathrm{U})^2+\Phi_\mathrm{U}\Phi_\mathrm{L})+\mathcal{O}(n^{-1/2})}{\lambda_\mathrm{L}(\lambda_\mathrm{L}-\lambda_\mathrm{U})^2+\lambda_\mathrm{U}\Phi_\mathrm{U}\Phi_\mathrm{L}+ \mathcal{O}(n^{-1/2})},\quad n\to+\infty,\quad\text{if Case II$_\mathrm{b}$ holds and $|\Delta|\geq \delta>0$.}
\label{eq:u-upper-lower}$$ Due to the inequalities that characterize Cases I, II$_\mathrm{a}$, and II$_\mathrm{b}$, the leading terms in the numerator and denominator are bounded as $n\to+\infty$ in each case. If in addition the leading terms in the denominator are bounded away from zero, one can extract a leading approximation $\dot{u}_n$ of $u_n(ny;-(\tfrac{1}{2}+k))$ with a small absolute error: $$u_n(ny;-(\tfrac{1}{2}+k))=\dot{u}_n+\begin{cases}\mathcal{O}(n^{-1}),\quad& \text{(Case I)}\\
\mathcal{O}(n^{-1/2}),\quad&\text{(Case II$_\mathrm{a}$)},\end{cases}\quad
\dot{u}_n:={\mathrm{i}}\frac{\Phi_1+\Phi_2}{\lambda_2\Phi_1+\lambda_1\Phi_2},
\label{eq:dot-u-CaseI-IIa}$$ if $|\lambda_2\Phi_1+\lambda_1\Phi_2|\ge\delta>0$ and for Case II$_\mathrm{b}$, $$u_n(ny;-(\tfrac{1}{2}+k))=\dot{u}_n+\mathcal{O}(n^{-1/2}),\quad \dot{u}_n:=
{\mathrm{i}}\frac{(\lambda_\mathrm{L}-\lambda_{\mathrm{U}})^2+\Phi_\mathrm{U}\Phi_\mathrm{L}}{\lambda_\mathrm{L}(\lambda_\mathrm{L}-\lambda_\mathrm{U})^2+\lambda_\mathrm{U}\Phi_\mathrm{U}\Phi_\mathrm{L}},
\label{eq:dot-u-CaseIIb}$$ if $|\Delta|\ge\delta>0$ and also $|\lambda_\mathrm{L}(\lambda_\mathrm{L}-\lambda_\mathrm{U})^2+\lambda_\mathrm{U}\Phi_\mathrm{U}\Phi_\mathrm{L}|\ge\delta>0$. Note that in this case the zeros of $\Delta$ (where Riemann-Hilbert Problem \[rhp:F-dot-eyebrow\] fails to be solvable) correspond to zeros of the approximation $\dot{u}_n$. In deriving the above formulæ for $\dot{u}_n$ we used the fact that $\lambda_1\lambda_2=\lambda_\mathrm{U}\lambda_\mathrm{L}=1$.
### Concrete formulæ for $\dot{u}_n$ in domains covering the tubular neighborhood $T$ {#sec:edge-formulae}
Now recall that $\lambda_1(y)={{p}}(y)^{-1}$ and $\lambda_2(y)={{p}}(y)$. We construct $\dot{u}_n$ for each $y\in T$ according to the scheme described at the end of Section \[sec:edge-comparison-matrices\].
Suppose first that $\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}kn^{-1}\ln(n)$. Then we are to consider the comparison matrix $\mathbf{F}_n^{(k,0,1)}(\lambda;y)$ which corresponds to Case I and hence the formula with $$\Phi_1:=\mathop{\mathrm{Res}}_{\lambda={{p}}(y)^{-1}}A_1^{(k,0)}(\lambda;y) = \frac{{{p}}(y)^k{{p}}(y)_\infty^{1/2}V''({{p}}(y)^{-1};y)^{-k}}{{\mathrm{i}}V''({{p}}(y)^{-1};y)^{1/2}}$$ and $$\Phi_2:=a_n^{(k,0,1)}(y)\mathop{\mathrm{Res}}_{\lambda={{p}}(y)}A_2^{(k,0)}(\lambda;y)=
{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}n^k\frac{{{p}}(y)^{-k}{{p}}(y)_\infty^{-1/2}({{p}}(y)^{-1}-{{p}}(y))^{2k}}{{\mathrm{i}}k! V''({{p}}(y);y)^{1/2}}$$ where we used the identity $V({{p}}(y);y)=-V({{p}}(y)^{-1};y)\pmod{2\pi{\mathrm{i}}}$. Since only the ratio of the residues $\Phi_1$ and $\Phi_2$ enters into the formula , it is possible to remove all ambiguity of branches of square roots as follows. It is straightforward to check that whenever $\lambda$ is such that $V'(\lambda;y)=0$ we have $V''(\lambda;y)=\lambda^{-2}(\lambda^2-1)(\lambda^2+1)^{-1}$ and therefore the identity $V''({{p}}(y)^{-1};y)=-{{p}}(y)^4V''({{p}}(y);y)$ holds for $y\in T$. Taking square roots and carefully determining the sign to choose for consistency when $y=0.331372\in{{\partial E^\infty_\squareurblack}}$, this identity implies that $V''({{p}}(y)^{-1};y)^{1/2}={\mathrm{i}}{{p}}(y)^2V''({{p}}(y);y)^{1/2}$. Furthermore, ${{p}}(y)_\infty^{1/2}/{{p}}(y)_\infty^{-1/2}={{p}}(y)$, so all details of the “$\infty$” branch of the power functions disappears from the ratio of residues. Using these facts, we arrive at a simple formula for $\dot{u}_n$ valid for $\mathrm{Re}(V({{p}}(y)^{-1};y))\le -\tfrac{1}{2}kn^{-1}\ln(n)$: $$\dot{u}_n={\mathrm{i}}{{p}}(y)\frac{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}n^{-k}k!{{p}}(y)^{-1}({{p}}(y)^{-1}+{{p}}(y))^k({{p}}(y)^{-1}-{{p}}(y))^{-3k}}{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}n^{-k}k!{{p}}(y)({{p}}(y)^{-1}+{{p}}(y))^k({{p}}(y)^{-1}-{{p}}(y))^{-3k}},\quad
\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\frac{1}{2}k\frac{\ln(n)}{n}.
\label{eq:dot-u-UU-left}$$ We observe that as $y$ moves away from ${{\partial E^\infty_\squareurblack}}$ into the interior of $E$, the exponentials tend to zero and $\dot{u}_n\approx{\mathrm{i}}{{p}}(y)^{-1}$ consistent with Theorem \[theorem:closed-eye-equilibrium\].
Now let $\ell$ be an integer varying from $\ell=1$ to $\ell=k$; we must now analyze four corresponding sub-cases depending on $y\in T$. First we are to consider the comparison matrix $\mathbf{F}_n^{(k-\ell+1,\ell-1,2)}(\lambda;y)$. Assuming the inequalities $-\tfrac{1}{2}(k-2\ell+2)n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)$, we are in Case II$_\mathrm{a}$ with $$\begin{split}
\Phi_1&=a_n^{(k-\ell+1,\ell-1,2)}(y)\mathop{\mathrm{Res}}_{\lambda={{p}}(y)^{-1}}A_1^{(k-\ell+1,\ell-1)}(\lambda;y) \\ &= {\mathrm{e}}^{-2nV({{p}}(y)^{-1};y)}n^{2\ell-k-2}
\frac{(k-\ell+1)!{{p}}(y)^k{{p}}(y)_\infty^{1/2}({{p}}(y)^{-1}-{{p}}(y))^{2\ell-2}V''({{p}}(y)^{-1};y)^{\ell-k-1}}{{\mathrm{i}}k!V''({{p}}(y)^{-1};y)^{1/2}}
\end{split}$$ and $$\Phi_2=\mathop{\mathrm{Res}}_{\lambda={{p}}(y)}A_2^{(k-\ell+1,\ell-1)}(\lambda;y) =
\frac{(\ell-1)!{{p}}(y)^{-k}{{p}}(y)_\infty^{-1/2}({{p}}(y)^{-1}-{{p}}(y))^{2k-2\ell+2}
V''({{p}}(y);y)^{-\ell+1}}{{\mathrm{i}}k!V''({{p}}(y);y)^{1/2}}.$$ Applying similar arguments to express the ratio of residues appearing in in terms of integer powers of ${{p}}(y)$ gives $$\begin{gathered}
\dot{u}_n={\mathrm{i}}{{p}}(y)\frac{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^{\ell-1}n^{2\ell-k-2}\frac{(k-\ell+1)!}{(\ell-1)!}
{{p}}(y)^{-1}({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell+2}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k-6}}{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^{\ell-1}n^{2\ell-k-2}\frac{(k-\ell+1)!}{(\ell-1)!}
{{p}}(y)({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell+2}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k-6}},\\
-\frac{1}{2}(k-2\ell+2)\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le
-\frac{1}{2}(k-2\ell+\tfrac{3}{2})\frac{\ln(n)}{n}.
\label{eq:dot-u-UU-first}\end{gathered}$$ Comparing and in the case $\ell=1$ shows that the same formula for $\dot{u}_n$ holds over the whole range of values $\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-\tfrac{1}{2})n^{-1}\ln(n)$ although different comparison matrices are involved in the derivation. Continuing with studying the same comparison matrix but now assuming that $-\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell+1)n^{-1}\ln(n)$, we are in Case II$_\mathrm{b}$ with $\lambda_\mathrm{U}=\lambda_2={{p}}(y)$ and $\lambda_\mathrm{L}=\lambda_1={{p}}(y)^{-1}$, with corresponding residues $$\Phi_\mathrm{U}=\mathop{\mathrm{Res}}_{\lambda={{p}}(y)}A_2^{(k-\ell+1,\ell-1)}(\lambda;y) =
\frac{(\ell-1)!{{p}}(y)^{-k}{{p}}(y)_\infty^{-1/2}({{p}}(y)^{-1}-{{p}}(y))^{2k-2\ell+2}
V''({{p}}(y);y)^{-\ell+1}}{{\mathrm{i}}k!V''({{p}}(y);y)^{1/2}},$$ and $$\begin{split}
\Phi_\mathrm{L}&=b_n^{(k-\ell+1,\ell-1,2)}\mathop{\mathrm{Res}}_{\lambda={{p}}(y)^{-1}}B_1^{(k-\ell+1,\ell-1)}(\lambda;y)\\
&={\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}n^{k-2\ell+1}\frac{k!{{p}}(y)^{-k}{{p}}(y)_\infty^{-1/2}({{p}}(y)^{-1}-{{p}}(y))^{2-2\ell}V''({{p}}(y)^{-1};y)^{k-\ell+1}}{{\mathrm{i}}(k-\ell)!V''({{p}}(y)^{-1};y)^{1/2}}.
\end{split}$$ After some simplification of the product $\Phi_\mathrm{U}\Phi_\mathrm{L}$ of the residues along the lines indicated above, the applicable formula for $\dot{u}_n$ becomes $$\begin{gathered}
\dot{u}_n={\mathrm{i}}{{p}}(y)^{-1}\frac{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^\ell n^{2\ell-k-1}\frac{(k-\ell)!}{(\ell-1)!}
{{p}}(y)({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell+1}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k-3}}
{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^\ell n^{2\ell-k-1}\frac{(k-\ell)!}{(\ell-1)!}
{{p}}(y)^{-1}({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell+1}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k-3}},\\
-\frac{1}{2}(k-2\ell+\tfrac{3}{2})\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le -\frac{1}{2}(k-2\ell+1)\frac{\ln(n)}{n}.
\label{eq:dot-u-UL-1}\end{gathered}$$ Switching now to the comparison matrix $\mathbf{F}_n^{(k-\ell,\ell,1)}(\lambda;y)$, we assume that the inequalities $-\tfrac{1}{2}(k-2\ell-1)n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le -\tfrac{1}{2}(k-2\ell-\tfrac{1}{2})n^{-1}\ln(n)$ hold, which again imply Case II$_\mathrm{b}$ but now with $\lambda_\mathrm{U}=\lambda_1={{p}}(y)^{-1}$ and $\lambda_\mathrm{L}=\lambda_2={{p}}(y)$, and corresponding residues $$\Phi_\mathrm{U}=\mathop{\mathrm{Res}}_{\lambda={{p}}(y)^{-1}} A_1^{(k-\ell,\ell)}(\lambda;y)=
\frac{(k-\ell)!{{p}}(y)^k{{p}}(y)_\infty^{1/2}({{p}}(y)^{-1}-{{p}}(y))^{2\ell}V''({{p}}(y)^{-1};y)^{\ell-k}}{{\mathrm{i}}k!V''({{p}}(y)^{-1};y)^{1/2}}$$ and $$\begin{split}
\Phi_\mathrm{L}&=b_n^{(k-\ell,\ell,1)}\mathop{\mathrm{Res}}_{\lambda={{p}}(y)}B_2^{(k-\ell,\ell)}(\lambda;y)\\
&={\mathrm{e}}^{-2nV({{p}}(y)^{-1};y)}n^{2\ell-k-1}\frac{k!{{p}}(y)^k{{p}}(y)_\infty^{1/2}({{p}}(y)^{-1}-{{p}}(y))^{2\ell-2k}V''({{p}}(y);y)^{\ell}}{{\mathrm{i}}(\ell-1)!V''({{p}}(y);y)^{1/2}}.
\end{split}$$ Thus becomes $$\begin{gathered}
\dot{u}_n={\mathrm{i}}{{p}}(y)^{-1}\frac{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^\ell n^{2\ell-k-1}\frac{(k-\ell)!}{(\ell-1)!}
{{p}}(y)({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell+1}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k-3}}
{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^\ell n^{2\ell-k-1}\frac{(k-\ell)!}{(\ell-1)!}
{{p}}(y)^{-1}({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell+1}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k-3}},\\
-\frac{1}{2}(k-2\ell+1)\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le -\frac{1}{2}(k-2\ell+\tfrac{1}{2})\frac{\ln(n)}{n}.
\label{eq:dot-u-UL-2}\end{gathered}$$ Comparing and , we observe that the approximate formula $\dot{u}_n$ is the same over the whole range $-\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)$ over which Case II$_\mathrm{b}$ applies with different comparison matrices. Continuing with the same comparison matrix we now assume the inequalities $-\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell)n^{-1}\ln(n)$ and find that Case II$_a$ applies once again with residues given by $$\Phi_1=\mathop{\mathrm{Res}}_{\lambda={{p}}(y)^{-1}}A_1^{(k-\ell,\ell)}(\lambda;y)=
\frac{(k-\ell)!{{p}}(y)^k{{p}}(y)_\infty^{1/2}({{p}}(y)^{-1}-{{p}}(y))^{2\ell}V''({{p}}(y)^{-1};y)^{\ell-k}}{{\mathrm{i}}k!V''({{p}}(y)^{-1};y)^{1/2}}$$ and $$\begin{split}
\Phi_2&=a_n^{(k-\ell,\ell,1)}(y)\mathop{\mathrm{Res}}_{\lambda={{p}}(y)}A_2^{(k-\ell,\ell)}(\lambda;y)\\
&={\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}n^{k-2\ell}\frac{\ell!{{p}}(y)^{-k}{{p}}(y)_\infty^{-1/2}({{p}}(y)^{-1}-{{p}}(y))^{2k-2\ell}V''({{p}}(y);y)^{-\ell}}{{\mathrm{i}}k!V''({{p}}(y);y)^{1/2}}.
\end{split}$$ Hence from we get $$\begin{gathered}
\dot{u}_n={\mathrm{i}}{{p}}(y)\frac{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^\ell n^{2\ell-k}\frac{(k-\ell)!}{\ell!}{{p}}(y)^{-1}({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k}}
{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^\ell n^{2\ell-k}\frac{(k-\ell)!}{\ell!}{{p}}(y)({{p}}(y)^{-1}+{{p}}(y))^{k-2\ell}({{p}}(y)^{-1}-{{p}}(y))^{6\ell-3k}},\\
-\frac{1}{2}(k-2\ell+\tfrac{1}{2})\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le
-\frac{1}{2}(k-2\ell)\frac{\ln(n)}{n}.
\label{eq:dot-u-UU-last}\end{gathered}$$ We observe that and agree upon replacing $\ell$ with $\ell+1$ in the former.
Having completed the above four cases with $\ell=k$, it remains only to turn to the comparison matrix $\mathbf{F}_n^{(0,k,2)}(\lambda;y)$ and assume the inequality $\mathrm{Re}(V({{p}}(y)^{-1};y))\ge \tfrac{1}{2}kn^{-1}\ln(n)$. This corresponds to Case I with residues $$\Phi_1=a_n^{(0,k,2)}(y)\mathop{\mathrm{Res}}_{\lambda={{p}}(y)^{-1}}A_1^{(0,k)}(\lambda;y)
={\mathrm{e}}^{-2nV({{p}}(y)^{-1};y)}n^k\frac{{{p}}(y)^k{{p}}(y)_\infty^{1/2}({{p}}(y)^{-1}-{{p}}(y))^{2k}}{{\mathrm{i}}k!V''({{p}}(y)^{-1};y)^{1/2}}$$ and $$\Phi_2=\mathop{\mathrm{Res}}_{\lambda={{p}}(y)}A_2^{(0,k)}(\lambda;y)=
\frac{{{p}}(y)^{-k}{{p}}(y)_\infty^{-1/2}V''({{p}}(y);y)^{-k}}{{\mathrm{i}}V''({{p}}(y);y)^{1/2}}.$$ Using these in gives $$\dot{u}_n={\mathrm{i}}{{p}}(y)\frac{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^kn^k\frac{1}{k!}{{p}}(y)^{-1}
({{p}}(y)^{-1}+{{p}}(y))^{-k}({{p}}(y)^{-1}-{{p}}(y))^{3k}}{{\mathrm{e}}^{2nV({{p}}(y)^{-1};y)}-{\mathrm{i}}(-1)^kn^k\frac{1}{k!}{{p}}(y)
({{p}}(y)^{-1}+{{p}}(y))^{-k}({{p}}(y)^{-1}-{{p}}(y))^{3k}},\quad
\mathrm{Re}(V({{p}}(y)^{-1};y))\ge\frac{1}{2}k\frac{\ln(n)}{n}.
\label{eq:dot-u-UU-right}$$ If $k=0$, the formulæ and agree and define $\dot{u}_n$ by the same formula for all $y\in T$. However if $k>0$, then agrees with with $\ell=k$, showing that the latter formula defines the approximation $\dot{u}_n$ over the whole range $\mathrm{Re}(V({{p}}(y)^{-1};y))\ge\tfrac{1}{2}(k-\tfrac{1}{2})n^{-1}\ln(n)$. Also, as $y$ moves out of $T$ into the exterior of $E$, we have $\mathrm{Re}(V({{p}}(y)^{-1};y))>0$, so as $n\to+\infty$ we get $\dot{u}_n\approx {\mathrm{i}}{{p}}(y)$, which is again consistent with Theorem \[theorem:closed-eye-equilibrium\]. Combining these formulæ with the convergence results described in Section \[sec:Edge-error-analysis\] and the exact symmetry to extend the results to $m=\tfrac{1}{2}+k$, $k\in\mathbb{Z}_{\ge 0}$ completes the proof of Theorem \[thm:edge-formulae\].
### Detailed asymptotics for poles and zeros. Proofs of Corollary \[corollary:eyebrow-zeros-and-poles\] and Theorem \[theorem:eyebrow-curves\] {#sec:eyebrow-zeros-and-poles}
Each of the formulæ for $\dot{u}_n$ described in Section \[sec:edge-formulae\] is a different meromorphic function of $y$ whose accuracy as an approximation of $u_n(ny;-(\tfrac{1}{2}+k))$ holds in an absolute sense for $y$ in a certain curvilinear strip roughly parallel to the eyebrow ${{\partial E^\infty_\squareurblack}}$ and of width proportional to $n^{-1}\ln(n)$. The absolute accuracy of the approximation depends on the assumption that $y$ is bounded away from each pole and zero of $\dot{u}_n$ by a distance proportional to $n^{-1}$ by an arbitrarily small constant. It is easy to see that this distance is an arbitrarily small fraction of the spacing between nearest poles or zeros of $\dot{u}_n$. This allows one to compute the index (winding number) of $u_n(ny;-(\tfrac{1}{2}+k))$ about a small circle containing just one pole or zero of $\dot{u}_n$ and hence deduce that the index is $-1$ or $1$ respectively.
According to and the discussion following , the zeros and poles of $\dot{u}_n$ in the left-most sub-domain of $T$ given by the inequality $\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-\tfrac{1}{2})n^{-1}\ln(n)$ lie exactly on the respective curves $$\text{Zero curve of $\dot{u}_n$ for $\displaystyle\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\frac{1}{2}(k-\tfrac{1}{2})\frac{\ln(n)}{n}$:}\quad
{\mathrm{e}}^{2n\mathrm{Re}(V({{p}}(y)^{-1};y))}=\frac{k!|{{p}}(y)^{-1}+{{p}}(y)|^k}{n^{k}|{{p}}(y)^{-1}-{{p}}(y)|^{3k}}|{{p}}(y)|^{-1},
\label{eq:left-zero-curve}$$ $$\text{Pole curve of $\dot{u}_n$ for $\displaystyle\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\frac{1}{2}(k-\tfrac{1}{2})\frac{\ln(n)}{n}$:}\quad
{\mathrm{e}}^{2n\mathrm{Re}(V({{p}}(y)^{-1};y))}=\frac{k!|{{p}}(y)^{-1}+{{p}}(y)|^k}{n^{k}|{{p}}(y)^{-1}-{{p}}(y)|^{3k}}|{{p}}(y)|.
\label{eq:left-pole-curve}$$ Note that because $|{{p}}(y)|<1$ holds for all $y\in{{\partial E^\infty_\squareurblack}}$, the zero curve lies to the right of the pole curve. Then, for $\ell=1,\dots,k$, by and , the zeros and poles of $\dot{u}_n$ in the domain $-\tfrac{1}{2}(k-2\ell+\tfrac{3}{2})n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)$ lie exactly on the respective curves $$\begin{gathered}
\text{Zero curve of $\dot{u}_n$ for $\displaystyle-\frac{1}{2}(k-2\ell+\tfrac{3}{2})\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\frac{1}{2}(k-2\ell+\tfrac{1}{2})\frac{\ln(n)}{n}$:}\\
{\mathrm{e}}^{2n\mathrm{Re}(V({{p}}(y)^{-1};y))}=\frac{(k-\ell)!|{{p}}(y)^{-1}+{{p}}(y)|^{k-2\ell+1}}{(\ell-1)!n^{k-2\ell+1}|{{p}}(y)^{-1}-{{p}}(y)|^{3k-6\ell+3}}|{{p}}(y)|,
\label{eq:mid-UL-zero-curve}\end{gathered}$$ $$\begin{gathered}
\text{Pole curve of $\dot{u}_n$ for $\displaystyle-\frac{1}{2}(k-2\ell+\tfrac{3}{2})\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\frac{1}{2}(k-2\ell+\tfrac{1}{2})\frac{\ln(n)}{n}$:}\\
{\mathrm{e}}^{2n\mathrm{Re}(V({{p}}(y)^{-1};y))}=\frac{(k-\ell)!|{{p}}(y)^{-1}+{{p}}(y)|^{k-2\ell+1}}{(\ell-1)!n^{k-2\ell+1}|{{p}}(y)^{-1}-{{p}}(y)|^{3k-6\ell+3}}|{{p}}(y)|^{-1},
\label{eq:mid-UL-pole-curve}\end{gathered}$$ (the zero curve lies to the left of the pole curve) and from and , the zeros and poles of $\dot{u}_n$ in the adjacent domain $-\tfrac{1}{2}(k-2\ell+\tfrac{1}{2})n^{-1}\ln(n)\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\tfrac{1}{2}(k-2\ell-\tfrac{1}{2})n^{-1}\ln(n)$ lie exactly on the respective curves $$\begin{gathered}
\text{Zero curve of $\dot{u}_n$ for $\displaystyle-\frac{1}{2}(k-2\ell+\tfrac{1}{2})\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\frac{1}{2}(k-2\ell-\tfrac{1}{2})\frac{\ln(n)}{n}$:}\\
{\mathrm{e}}^{2n\mathrm{Re}(V({{p}}(y)^{-1};y))}=\frac{(k-\ell)!|{{p}}(y)^{-1}+{{p}}(y)|^{k-2\ell}}{\ell!n^{k-2\ell}|{{p}}(y)^{-1}-{{p}}(y)|^{3k-6\ell}}|{{p}}(y)|^{-1},
\label{eq:mid-UU-zero-curve}\end{gathered}$$ $$\begin{gathered}
\text{Pole curve of $\dot{u}_n$ for $\displaystyle-\frac{1}{2}(k-2\ell+\tfrac{1}{2})\frac{\ln(n)}{n}\le\mathrm{Re}(V({{p}}(y)^{-1};y))\le-\frac{1}{2}(k-2\ell-\tfrac{1}{2})\frac{\ln(n)}{n}$:}\\
{\mathrm{e}}^{2n\mathrm{Re}(V({{p}}(y)^{-1};y))}=\frac{(k-\ell)!|{{p}}(y)^{-1}+{{p}}(y)|^{k-2\ell}}{\ell!n^{k-2\ell}|{{p}}(y)^{-1}-{{p}}(y)|^{3k-6\ell}}|{{p}}(y)|
\label{eq:mid-UU-pole-curve}\end{gathered}$$ (again the zero curve lies to the right of the pole curve). Finally, according to , the zeros and poles of $\dot{u}_n$ in the right-most sub-domain of $T$ given by the inequality $\mathrm{Re}(V({{p}}(y)^{-1};y))\ge \tfrac{1}{2}(k+\tfrac{1}{2})n^{-1}\ln(n)$ lie along the latter curves in the terminal case of $\ell=k$.
Solution of Riemann-Hilbert Problem \[rhp:ParabolicCylinder\] {#app:PC}
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Derivation of differential equation
-----------------------------------
Suppose that $\mathbf{P}(\zeta;m)$ satisfies the conditions of Riemann-Hilbert Problem \[rhp:ParabolicCylinder\]. It is easy to check that $\det(\mathbf{P}(\zeta;m))=1$ for all $\zeta$. Then the related matrix $\mathbf{R}(\zeta;m):=\mathbf{P}(\zeta;m){\mathrm{e}}^{\zeta^2\sigma_3/2}$ has the same analyticity domain and is equally regular up to the jump contour $\mathrm{Re}(\zeta^2)=0$. It satisfies jump conditions across the four rays of the jump contour that are direct analogues of –, except that the $\zeta$-dependent exponential factors ${\mathrm{e}}^{\pm \zeta^2}$ have been removed from the off-diagonal elements of the jump matrices. Thus, $\mathbf{R}(\zeta;m)$ satisfies jump conditions that are independent of $\zeta$ on each ray. Differentiating these jump conditions with respect to $\zeta$ then shows that $\mathbf{R}'(\zeta;m)$ satisfies exactly the same jump conditions as does $\mathbf{R}(\zeta;m)$ and hence (using the fact that $\det(\mathbf{R}(\zeta;m))=1$ for $\mathrm{Re}(\zeta^2)\neq 0$) $\mathbf{R}'(\zeta;m)\mathbf{R}(\zeta;m)^{-1}$ can be continued to the jump contour unambiguously, defining an entire function of $\zeta$. Supposing for the moment that the normalization condition holds in the stronger sense that $$\mathbf{P}(\zeta;m)\sim(\mathbb{I} + \zeta^{-1}\mathbf{P}^{\infty}_1(m) + \cdots)\zeta^{-(m+\tfrac{1}{2})\sigma_3},\quad\zeta\to\infty$$ with $\mathbf{P}^{\infty}_1(m)$ being the same for all four sectors and with the indicated asymptotic series being differentiable term-by-term, the entire function $\mathbf{R}'(\zeta;m)\mathbf{R}(\zeta;m)^{-1}$ satisfies $$\mathbf{R}'(\zeta;m)\mathbf{R}(\zeta;m)^{-1}=\sigma_3\zeta + [\mathbf{P}^{\infty}_1(m),\sigma_3] + \mathcal{O}(\zeta^{-1}),\quad\zeta\to\infty,$$ and hence by Liouville’s theorem, $\mathbf{R}'(\zeta;m)\mathbf{R}(\zeta;m)^{-1}=\sigma_3\zeta + [\mathbf{P}^{\infty}_1(m),\sigma_3]$ exactly. In other words, $\mathbf{R}(\zeta;m)$ satisfies the following differential equation: $$\frac{{\mathrm{d}}\mathbf{R}}{{\mathrm{d}}\zeta}=\begin{bmatrix}\zeta & \alpha\\\beta & -\zeta\end{bmatrix}\mathbf{R},\quad \alpha:=-2P^{\infty}_{1,12}(m),\quad\beta:=2P^{\infty}_{1,21}(m).
\label{eq:R-matrix-PC-ODE}$$ Rescaling by $t:=\zeta\sqrt{2}$ and eliminating the second row shows that any element $R_{1k}$, $k=1,2$, of the first row satisfies the differential equation of parabolic cylinder functions (see [@DLMF Eq. 12.2.2]) $$\frac{{\mathrm{d}}^2R_{1k}}{{\mathrm{d}}t^2} - \left(\frac{1}{4}t^2 + a\right)R_{1k}=0,\quad a:=\frac{1}{2}(1+\alpha\beta), \quad k=1,2.$$ According to [@DLMF §12.2(i)], we will take $R_{1k}$ as a linear combination of an appropriate “numerically satisfactory” pair of solutions in each of the four sectors: $$R_{1k}(\zeta;m)=\alpha A_k^\mathrm{I}U(a,\sqrt{2}\zeta) + \alpha B_k^\mathrm{I}U(-a,-{\mathrm{i}}\sqrt{2}\zeta),\quad 0\le\arg(\zeta)\le\frac{\pi}{2},
\label{eq:R1-I}$$ $$R_{1k}(\zeta;m)=\alpha A_k^\mathrm{II}U(-a,-{\mathrm{i}}\sqrt{2}\zeta) + \alpha B_k^\mathrm{II}U(a,-\sqrt{2}\zeta),\quad \frac{\pi}{2}\le\arg(\zeta)\le\pi,$$ $$R_{1k}(\zeta;m)=\alpha A_k^\mathrm{III}U(a,-\sqrt{2}\zeta) + \alpha B_k^\mathrm{III}U(-a,{\mathrm{i}}\sqrt{2}\zeta),\quad -\pi\le\arg(\zeta)\le -\frac{\pi}{2},$$ and $$R_{1k}(\zeta;m)=\alpha A_k^\mathrm{IV}U(-a,{\mathrm{i}}\sqrt{2}\zeta)+\alpha B_k^\mathrm{IV}U(a,\sqrt{2}\zeta),\quad -\frac{\pi}{2}\le\arg(\zeta)\le 0.
\label{eq:R1-IV}$$ From the first row of we then find the corresponding matrix elements $R_{2k}(\zeta;m):=\alpha^{-1}(R_{1k}'(\zeta;m)-\zeta R_{1k}(\zeta;m))$. Hence using [@DLMF Eqs. 12.8.2–12.8.3], from – we get $$R_{2k}(\zeta;m)=-\sqrt{2}A_k^\mathrm{I}U(a-1,\sqrt{2}\zeta)-{\mathrm{i}}\sqrt{2}(a-\tfrac{1}{2})B_k^\mathrm{I}U(1-a,-{\mathrm{i}}\sqrt{2}\zeta),\quad 0\le\arg(\zeta)\le\frac{\pi}{2},
\label{eq:R2-I}$$ $$R_{2k}(\zeta;m)=-{\mathrm{i}}\sqrt{2}(a-\tfrac{1}{2})A_k^\mathrm{II}U(1-a,-{\mathrm{i}}\sqrt{2}\zeta) +\sqrt{2}B_k^\mathrm{II}U(a-1,-\sqrt{2}\zeta),\quad \frac{\pi}{2}\le\arg(\zeta)\le\pi,$$ $$R_{2k}(\zeta;m)=\sqrt{2}A_k^\mathrm{III}U(a-1,-\sqrt{2}\zeta)+{\mathrm{i}}\sqrt{2}(a-\tfrac{1}{2})B_k^\mathrm{III}U(1-a,{\mathrm{i}}\sqrt{2}\zeta),\quad -\pi\le\arg(\zeta)\le-\frac{\pi}{2},$$ $$R_{2k}(\zeta;m)={\mathrm{i}}\sqrt{2}(a-\tfrac{1}{2})A_k^\mathrm{IV}U(1-a,{\mathrm{i}}\sqrt{2}\zeta) -\sqrt{2}B_k^\mathrm{IV}U(a-1,\sqrt{2}\zeta),\quad -\frac{\pi}{2}\le\arg(\zeta)\le 0.
\label{eq:R2-IV}$$ Note that in addition to the sixteen coefficients $A_k^S$ and $B_k^S$, $k=1,2$, $S=\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}$, it remains to determine also the parameters $\alpha$ and $\beta$.
Selection of solutions and parameter determination
--------------------------------------------------
Now we impose that the matrix $\mathbf{R}(\zeta;m)$ satisfy the leading-order normalization condition implied by . For this purpose, given the choice of basis made above in each sector $S$, it is sufficient to use the large-$z$ asymptotic expansion for $U(a,z)$ given by [@DLMF Eq. 12.9.1], which implies that $U(a,z)=e^{-z^2/4}z^{-a-1/2}(1+\mathcal{O}(z^{-2}))$ as $z\to\infty$ with $|\arg(z)|< 3\pi/4$. Thus, in order to avoid unwanted exponential growth it is necessary to choose: $$A_1^\mathrm{I}=B_2^\mathrm{I}=0,\quad A_2^\mathrm{II}=B_1^\mathrm{II}=0,\quad A_1^\mathrm{III}=B_2^\mathrm{III}=0,\quad A_2^\mathrm{IV}=B_1^\mathrm{IV}=0.$$ With these choices, all four elements of $\mathbf{R}(\zeta;m){\mathrm{e}}^{-\zeta^2\sigma_3/2}\zeta^{(m+\tfrac{1}{2})\sigma_3}$ are bounded by a power of $\zeta$ as $\zeta\to\infty$. Determining the parameter $a=\tfrac{1}{2}(1+\alpha\beta)$ in terms of $m$ explicitly by $$a=-m$$ is then necessary to ensure the existence of a finite limit as $\zeta\to\infty$. By examination of the diagonal elements in the four sectors, we then deduce that $\mathbf{R}(\zeta;m){\mathrm{e}}^{-\zeta^2\sigma_3/2}\zeta^{(m+\tfrac{1}{2})\sigma_3}=\mathbb{I}+\mathcal{O}(\zeta^{-2})$ as $\zeta\to\infty$ in all directions, provided that the remaining eight nonzero coefficients are determined as follows: $$\begin{gathered}
B_1^\mathrm{I}=A_1^\mathrm{II}=\alpha^{-1}2^{\tfrac{1}{4}-\tfrac{1}{2}a}{\mathrm{e}}^{{\mathrm{i}}\pi(\tfrac{1}{2}a-\tfrac{1}{4})},\quad
B_1^\mathrm{III}=A_1^\mathrm{IV}=\alpha^{-1}2^{\tfrac{1}{4}-\tfrac{1}{2}a}{\mathrm{e}}^{-{\mathrm{i}}\pi(\tfrac{1}{2}a-\tfrac{1}{4})},\\
A_2^\mathrm{I}=B_2^\mathrm{IV}=-2^{\tfrac{1}{2}a-\tfrac{3}{4}},\quad B_2^\mathrm{II}=2^{\tfrac{1}{2}a-\tfrac{3}{4}}{\mathrm{e}}^{{\mathrm{i}}\pi(\tfrac{1}{2}-a)},\quad A_2^\mathrm{III}=2^{\tfrac{1}{2}a-\tfrac{3}{4}}{\mathrm{e}}^{-{\mathrm{i}}\pi(\tfrac{1}{2}-a)}.
\end{gathered}$$ The only remaining ambiguity concerns the precise values of $\alpha$ and $\beta$ such that $\tfrac{1}{2}(1+\alpha\beta)=a=-m$. This ambiguity is resolved by resorting to the jump conditions in Riemann-Hilbert Problem \[rhp:ParabolicCylinder\]. Here, we make use of the connection formula $U(a,z)=\pm{\mathrm{i}}{\mathrm{e}}^{\pm{\mathrm{i}}\pi a}U(a,-z)+\sqrt{2\pi}{\mathrm{e}}^{\pm{\mathrm{i}}\pi(a-1/2)/2}U(-a,\pm{\mathrm{i}}z)/\Gamma(a+1/2)$, see [@DLMF Eq. 12.2.19]. With the help of this formula, it is straightforward to check that the jump conditions are satisfied by $\mathbf{P}(\zeta;m):=\mathbf{R}(\zeta;m){\mathrm{e}}^{-\zeta^2\sigma_3/2}$, provided that $\alpha$ is given by: $$\alpha={\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}2^{m+1}.$$ Then from $a=\tfrac{1}{2}(1+\alpha\beta)=-m$ we get $$\beta={\mathrm{i}}{\mathrm{e}}^{-{\mathrm{i}}\pi m}2^{-m}(m+\tfrac{1}{2}).$$
Refined asymptotics of $\mathbf{P}(\zeta)$
------------------------------------------
The matrix $\mathbf{P}(\zeta;m)=\mathbf{R}(\zeta;m){\mathrm{e}}^{-\zeta^2\sigma_3/2}$ clearly satisfies all of the conditions of Riemann-Hilbert Problem \[rhp:ParabolicCylinder\], and it is easy to show that there can be only one solution. Here we give the complete asymptotic expansion of the solution $\mathbf{P}(\zeta;m)$ in the large-$\zeta$ limit, giving information beyond the normalization condition . Indeed, using [@DLMF Eq. 12.9.1], we find that $$\mathbf{P}(\zeta;m)\zeta^{(m+\tfrac{1}{2})\sigma_3}\sim\begin{bmatrix}
\displaystyle\sum_{j=0}^\infty\frac{(\tfrac{1}{2}+m)_{2j}}{4^jj!\zeta^{2j}} &
\displaystyle - {\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}2^m\zeta^{-1}\sum_{j=0}^\infty \frac{(-1)^j(\tfrac{1}{2}-m)_{2j}}{4^jj!\zeta^{2j}}\\
\displaystyle {\mathrm{i}}{\mathrm{e}}^{-{\mathrm{i}}\pi m}2^{-m-1}(m+\tfrac{1}{2})\zeta^{-1}\sum_{j=0}^\infty \frac{(\tfrac{3}{2}+m)_{2j}}{4^jj!\zeta^{2j}} &
\displaystyle\sum_{j=0}^\infty\frac{(-1)^j(-\tfrac{1}{2}-m)_{2j}}{4^jj!\zeta^{2j}}
\end{bmatrix},\quad\zeta\to\infty
\label{eq:N-expansion}$$ uniformly in all directions of the complex plane, including along the sector boundaries. Notably, the expansion coefficients do not depend on the sector in which the solution is analyzed, a fact that also follows from the exponential decay of the off-diagonal elements of the jump matrices in Riemann-Hilbert Problem \[rhp:ParabolicCylinder\].
Solution of Riemann-Hilbert Problem \[rhp:Airy\] {#app:Airy}
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Derivation of differential equation
-----------------------------------
Reasoning as in Appendix \[app:PC\], one checks that if $\mathbf{A}(\zeta)$ satisfies Riemann-Hilbert Problem \[rhp:Airy\], then the related matrix $\mathbf{S}(\zeta):=\mathbf{A}(\zeta){\mathrm{e}}^{-\zeta^{3/2}\sigma_3/2}$ has unit determinant and satisfies jump conditions analogous to except that the exponential factors in the jump matrices for $\arg(\zeta)=0$ and $\arg(\zeta)=\pm \tfrac{2}{3}\pi$ have been cancelled. It follows by differentiation of the resulting constant jump matrices with respect to $\zeta$ that $\mathbf{S}'(\zeta)\mathbf{S}(\zeta)^{-1}$ is an entire function of $\zeta$. Assuming that $\mathbf{A}(\zeta)\mathbf{V}^{-1}\zeta^{-\sigma_3/4}\sim\mathbb{I} + \zeta^{-1}\mathbf{A}_1^\infty + \cdots$ as $\zeta\to\infty$ with the coefficient $\mathbf{A}_1^\infty$ being the same for each of the four sectors of analyticity and with the asymptotic series being differentiable term-by-term, it follows that $$\mathbf{S}'(\zeta)\mathbf{S}(\zeta)^{-1}=\frac{3{\mathrm{i}}}{4}\begin{bmatrix}A^\infty_{1,21} & A^\infty_{1,22}-A^\infty_{1,11}-\zeta\\1 & -A^\infty_{1,21}\end{bmatrix}+\mathcal{O}(\zeta^{-1}),\quad\zeta\to\infty.$$ By Liouville’s theorem, we derive the first-order system of differential equations satisfied by $\mathbf{S}(\zeta)$: $$\frac{{\mathrm{d}}\mathbf{S}}{{\mathrm{d}}\zeta}=\frac{3{\mathrm{i}}}{4}\begin{bmatrix}A^\infty_{1,21} & A^\infty_{1,22}-A^\infty_{1,11}-\zeta\\1 & -A^\infty_{1,21}\end{bmatrix}\mathbf{S}.
\label{eq:Airy-system}$$ Rescaling by $t=(\tfrac{3}{4})^{2/3}(\zeta-c)$, where $c:=(A^\infty_{1,21})^2+A^\infty_{1,22}-A^\infty_{1,11}$ and eliminating the first row of $\mathbf{S}$ it follows that the elements of the second row of $\mathbf{S}$ are solutions of Airy’s equation [@DLMF Eqn. 9.2.1] $$\frac{{\mathrm{d}}^2S_{2k}}{{\mathrm{d}}t^2}-\zeta S_{2k}=0,\quad k=1,2.
\label{eq:Airy}$$ We represent the second-row elements of $\mathbf{S}$ as linear combinations of “numerically satisfactory” solutions (see [@DLMF Table 9.2.1]) appropriate for each sector: $$S_{2k}(\zeta)=A_k^\mathrm{I}\mathrm{Ai}(t) + B_k^\mathrm{I}\mathrm{Ai}({\mathrm{e}}^{-2\pi{\mathrm{i}}/3}t),\quad
0\le\arg(\zeta)\le\frac{2}{3}\pi,$$ $$S_{2k}(\zeta)=A_k^\mathrm{II}\mathrm{Ai}({\mathrm{e}}^{-2\pi{\mathrm{i}}/3}t)+B_k^\mathrm{II}\mathrm{Ai}({\mathrm{e}}^{2\pi{\mathrm{i}}/3}t),
\quad \frac{2}{3}\pi\le\arg(\zeta)\le\pi,$$ $$S_{2k}(\zeta)=A_k^\mathrm{III}\mathrm{Ai}({\mathrm{e}}^{-2\pi{\mathrm{i}}/3}t)+B_k^\mathrm{III}\mathrm{Ai}({\mathrm{e}}^{2\pi{\mathrm{i}}/3}t),\quad -\pi\le\arg(\zeta)\le-\frac{2}{3}\pi,$$ $$S_{2k}(\zeta)=A_k^\mathrm{IV}\mathrm{Ai}(t)+B_k^\mathrm{IV}\mathrm{Ai}({\mathrm{e}}^{2\pi{\mathrm{i}}/3}t),\quad
-\frac{2}{3}\pi\le\arg(\zeta)\le 0.$$ Once the sixteen coefficients $A_k^S$ and $B_k^S$, $k=1,2$, $S=\mathrm{I},\mathrm{II},\mathrm{III},\mathrm{IV}$ have been determined, the first row elements of $\mathbf{S}(\zeta)$ will then be determined from the first-order system . The parameter $c$ involved in the relation connecting $\zeta$ with $t$ also needs to be determined.
Selection of solutions
----------------------
We now consider the normalization condition on $\mathbf{A}(\zeta)$ which in terms of $\mathbf{S}(\zeta)$ reads $\mathbf{S}(\zeta){\mathrm{e}}^{\zeta^{3/2}\sigma_3/2}\mathbf{V}^{-1}\zeta^{-\sigma_3/4}=\mathbb{I}+\mathcal{O}(\zeta^{-1})$ as $\zeta\to\infty$. Imposing this condition in each sector with the help of the well-known asymptotic formula [@DLMF Eqn. 9.7.5] for $\mathrm{Ai}(t)$ valid for large $t$, we find that in order to avoid unwanted exponential growth it is necessary to choose $$B_1^\mathrm{I}=A_2^\mathrm{I}=0,\quad A_1^\mathrm{II}=B_2^\mathrm{II}=0,\quad
B_1^\mathrm{III}=A_2^\mathrm{III}=0,\quad B_1^\mathrm{IV}=A_2^\mathrm{IV}=0.
\label{eq:Airy-zero-coeffs}$$ While these conditions remove the terms exhibiting the most rapid exponential growth as $\zeta\to\infty$, there are still subdominant exponentially growing terms that, since $\det(\mathbf{S}(\zeta))=1$, can only be removed if the parameter $c$ is made to vanish: $c=0$. Assuming and $c=0$, the quantity $\mathbf{S}(\zeta){\mathrm{e}}^{\zeta^{3/2}\sigma_3/2}\mathbf{V}^{-1}\zeta^{-\sigma_3/4}$ tends to a finite limit as $\zeta\to\infty$ but with a leading error term proportional to $\zeta^{-1/2}$. Removing this term and imposing the condition that the finite limit in question is $\mathbb{I}$ gives two additional conditions per sector that determine all eight remaining coefficients.
Thus, the second row of the matrix $\mathbf{S}(\zeta)$ is uniquely determined in each of the four sectors simply from the differential equation and by imposing the desired asymptotic behavior for large $\zeta$. The first row is then determined from the second using and $c=0$, and finally once $\mathbf{S}(\zeta)$ is known in all four sectors, $\mathbf{A}(\zeta)=\mathbf{S}(\zeta){\mathrm{e}}^{-\zeta^{3/2}\sigma_3/2}$. The resulting formulæ are as follows, in which $t=(\tfrac{3}{4})^{2/3}\zeta$ because $c=0$: $$\mathbf{A}(\zeta):=\sqrt{2\pi}\left(\frac{4}{3}\right)^{\sigma_3/6}\begin{bmatrix}-\mathrm{Ai}'(t) & {\mathrm{e}}^{2\pi{\mathrm{i}}/3}\mathrm{Ai}'(t{\mathrm{e}}^{-2\pi{\mathrm{i}}/3})\\
-{\mathrm{i}}\mathrm{Ai}(t) & {\mathrm{i}}{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{-2\pi{\mathrm{i}}/3})\end{bmatrix}{\mathrm{e}}^{2t^{3/2}\sigma_3/3},\quad 0<\arg(\zeta)<\frac{2}{3}\pi,$$ $$\mathbf{A}(\zeta):=\sqrt{2\pi}\left(\frac{4}{3}\right)^{\sigma_3/6}\begin{bmatrix}
{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}\mathrm{Ai}'(t{\mathrm{e}}^{2\pi{\mathrm{i}}/3}) & {\mathrm{e}}^{2\pi{\mathrm{i}}/3}\mathrm{Ai}'(t{\mathrm{e}}^{-2\pi{\mathrm{i}}/3})\\
{\mathrm{i}}{\mathrm{e}}^{2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{2\pi{\mathrm{i}}/3}) & {\mathrm{i}}{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{-2\pi{\mathrm{i}}/3})
\end{bmatrix}{\mathrm{e}}^{2t^{3/2}\sigma_3/3},\quad\frac{2}{3}\pi<\arg(\zeta)<\pi,$$ $$\mathbf{A}(\zeta):=\sqrt{2\pi}\left(\frac{4}{3}\right)^{\sigma_3/6}\begin{bmatrix}
{\mathrm{e}}^{2\pi{\mathrm{i}}/3}\mathrm{Ai}'(t{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}) & -{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}\mathrm{Ai}'(t{\mathrm{e}}^{2\pi{\mathrm{i}}/3})\\
{\mathrm{i}}{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}) & -{\mathrm{i}}{\mathrm{e}}^{2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{2\pi{\mathrm{i}}/3})
\end{bmatrix}{\mathrm{e}}^{2t^{3/2}\sigma_3/3},\quad -\pi<\arg(\zeta)<-\frac{2}{3}\pi,$$ $$\mathbf{A}(\zeta):=\sqrt{2\pi}\left(\frac{4}{3}\right)^{\sigma_3/6}\begin{bmatrix}
-\mathrm{Ai}'(t) & -{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}\mathrm{Ai}'(t{\mathrm{e}}^{2\pi{\mathrm{i}}/3})\\
-{\mathrm{i}}\mathrm{Ai}(t) & -{\mathrm{i}}{\mathrm{e}}^{2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{2\pi{\mathrm{i}}/3})\end{bmatrix}
{\mathrm{e}}^{2t^{3/2}\sigma_3/3},\quad -\frac{2}{3}\pi<\arg(\zeta)<0.$$ The jump conditions relating the boundary values of $\mathbf{A}(\zeta)$ in Riemann-Hilbert Problem \[rhp:Airy\] are now seen to simply be a consequence of the connection formula $\mathrm{Ai}(t)+{\mathrm{e}}^{-2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{-2\pi{\mathrm{i}}/3})+{\mathrm{e}}^{2\pi{\mathrm{i}}/3}\mathrm{Ai}(t{\mathrm{e}}^{2\pi{\mathrm{i}}/3})=0$ (see [@DLMF Eqn. 9.2.12]).
Refined asymptotics of $\mathbf{A}(\zeta)$
------------------------------------------
Using the known asymptotic expansions of $\mathrm{Ai}(t)$ and $\mathrm{Ai}'(t)$ (see [@DLMF Eqns. 9.7.5–9.7.6]), it is easy to show that $\mathbf{A}(\zeta)\mathbf{V}^{-1}\zeta^{-\sigma_3/4}$ has a complete asymptotic expansion in integer powers of $\zeta$ as $\zeta\to\infty$, and that the leading error term is characterized by the formula .
[99]{} L. Bass, J. J. Nimmo, C. Rogers, W. K. Schief, “Electrical structures of interfaces: a Painlevé II model,” *Proc. R. Soc. A* **466**, 2117–2136, 2010. R. Beals and R. R. Coifman, “Scattering and inverse scattering for first order systems,” *Comm. Pure Appl. Math.* **37**, 39–90, 1984. M. Bertola and T. Bothner, “Zeros of large degree Vorob’ev-Yablonski polynomials via a Hankel determinant identity,” *Int. Math. Research Notices* **2015**, 9330–9399, 2015. T. J. Bothner, P. D. Miller, and Y. Sheng, “Rational solutions of the Painlevé-III equation,” to appear in *Stud. Appl. Math*, 2018. DOI: 10.1111/sapm.12220. `arXiv:1801.04360`. R. J. Buckingham, “Large-degree asymptotics of rational Painlevé-IV functions associated to generalized Hermite polynomials,” to appear in *Int. Math. Research Notices*, 2018. `arXiv:1706.09005`. R. J. Buckingham and P. D. Miller, “The sine-Gordon equation in the semiclassical limit: critical behavior near a separatrix,” *J. Anal. Math.* **118**, 397–492, 2012. R. J. Buckingham and P. D. Miller, “Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour,” *Nonlinearity* **27**, 2489–2577, 2014. R. J. Buckingham and P. D. Miller, “Large-degree asymptotics of rational Painlevé-II functions: critical behaviour,” *Nonlinearity* **28**, 1539–1596, 2015. P. A. Clarkson, “The third Painlevé equation and associated special polynomials,” *J. Phys. A: Math. Gen.* **36**, 9507–9532, 2003. P. A. Clarkson, “Special polynomials associated with rational solutions of the Painlevé equations and applications to soliton equations,” *Comput. Meth. Funct. Theory* **6**, 329–401, 2006. P. A. Clarkson, “Vortices and polynomials,” *Stud. Appl. Math.* **123**, 37–62, 2009. P. A. Clarkson, C.-K. Law, C.-H. Lin, “An algebraic proof for the Umemura polynomials for the third Painlevé equation,” `arXiv:1609.00495`, 2016. B. A. Dubrovin, “Theta functions and non-linear equations,” *Russ. Math. Surveys* **36**, 11–92, 1981. A. S. Fokas, A. R. Its, and A. V. Kitaev, “Discrete Painlevé equations and their appearance in quantum gravity,” *Comm. Math. Phys.* **142**, 313–344, 1991. C. V. Johnson, “String theory without branes,” `arXiv:hep-th/0610223`, 2006. D. Masoero and P. Roffelsen, “Poles of Painlevé IV rationals and their distribution,” *SIGMA* **14**, 002 (49 pages), 2018. P. D. Miller and Y. Sheng, “Rational solutions of the Painlevé-II equation revisited,” *SIGMA* **13** 065 (29 pages), 2017. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., NIST Digital Library of Mathematical Functions, `http://dlmf.nist.gov/`, Release 1.0.14, 2016. B. Shapiro and M. Tater, “On spectral asymptotics of quasi-exactly solvable quartic and Yablonskii-Vorob’ev polynomials,” `arXiv:1412.3026`, 2014. H. Umemura, “100 years of the Painlevé equation,” *Sugaku* **51**, 395–420, 1999 (in Japanese).
[^1]: T. B. acknowledges support of the AMS and the Simons Foundation through a travel grant and P. D. M. is supported by the National Science Foundation under grants DMS-1513054 and DMS-1812625. The authors are grateful to Y. Sheng for useful conversations.
[^2]: More properly, it is a family of equations parametrized by $y\in\mathbb{C}\setminus\{0\}$.
[^3]: That it is impossible to have two double roots that are fixed individually by the involution can be seen as follows. It would be necessary to have one double root at $\lambda=1$ and another double root at $\lambda=-1$, and therefore $P(\lambda;y;C)=-\tfrac{1}{4}y^2(\lambda^2-1)^2=-\tfrac{1}{4}y^2\lambda^4+\tfrac{1}{2}y^2\lambda^2 -\tfrac{1}{4}y^2$. Comparing with shows that this situation cannot occur for $y\neq 0$.
[^4]: If there are four roots and one of them is $\lambda=\pm 1$, then the others are $\lambda=\mp 1$, $\lambda=\lambda_0$ and $\lambda=\lambda_0^{-1}$ with $\lambda_0^2\neq 1$. Thus $P(\lambda;y,C)=-\tfrac{1}{4}y^2(\lambda^2-1)(\lambda-\lambda_0)(\lambda-\lambda_0^{-1})=-\tfrac{1}{4}y^2\lambda^4+\tfrac{1}{4}y^2(\lambda_0+\lambda_0^{-1})\lambda^3-\tfrac{1}{4}y^2(\lambda_0+\lambda_0^{-1})\lambda +\tfrac{1}{4}y^2$. Comparing with shows that this case is not possible for $y\neq 0$.
[^5]: Indeed, $2g''({{p}}(y);y)-V''({{p}}(y);y)=-{{p}}(y)^{-2}-2{\mathrm{i}}y{{p}}(y)^{-3}=(1-{{p}}(y)^2){{p}}(y)^{-2}(1+{{p}}(y)^2))^{-1}$ can only vanish if ${{p}}(y)=\pm 1$ which corresponds to $y=\pm \tfrac{1}{2}{\mathrm{i}}$.
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---
abstract: 'In 1935, Erdős and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane which definitely contain an increasing subset of $m$ points or a decreasing subset of $k$ points (as ordered by their $x$-coordinates). We consider their result from an on-line game perspective: Let points be determined one by one by player A first determining the $x$-coordinate and then player B determining the $y$-coordinate. What is the minimum number of points such that player A can force an increasing subset of $m$ points or a decreasing subset of $k$ points? We introduce this as the *Erdős-Szekeres on-line number* and denote it by $\text{ESO}(m,k)$. We observe that $\text{ESO}(m,k) < (m-1)(k-1)+1$ for $m,k \ge 3$, provide a general lower bound for $\text{ESO}(m,k)$, and determine $\text{ESO}(m,3)$ up to an additive constant.'
author:
- 'Kirk Boyer[^1]'
- 'Lauren M. Nelsen'
- 'Luke L. Nelsen[^2]'
- Florian Pfender
- 'Elizabeth Reiland[^3]'
- 'Ryan Solava[^4]'
bibliography:
- 'ESO.bib'
title: 'Erdős-Szekeres On-Line'
---
Introduction
============
In [@ErdosSzekeres1935], Erdős and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane (ordered by their $x$-coordinates) that guarantees an increasing (in terms of $y$-coordinates) subset of $m$ points or a decreasing subset of $k$ points. We refer to this number as the *Erdős-Szekeres number* and denote it by $\operatorname{ES}(m,k)$. Their theorem has since seen several proofs, as well as related results with random or algorithmic themes (see [@Steele1995]).
“On-line” refers to a process in which an entire structure is not known, and instead decisions must be made with limited information. On-line graph coloring was most notably developed in [@GyarfasLehel1988] by Gyarfás and Lehel; for more developments which have sprung from this topic, we refer the reader to [@Kierstead1998; @Paulusma2016]. We consider the question of Erdős and Szekeres in an on-line setting with the following game: Let points be determined one by one with player A first determining the $x$-coordinate and then player B determining the $y$-coordinate. The question we want to answer is the following. What is the minimum number of points such that player A can force an increasing subset of $m$ points or a decreasing subset of $k$ points? We refer to this number as the *Erdős-Szekeres on-line number* and denote it by $\operatorname{ESO}(m,k)$.
In Section \[sec:prelim\] we introduce necessary definitions, a table of small results, and prove the following weak but general upper bound:
\[thm:ESO<ES\] $\operatorname{ESO}(m,k) \le (m-1)(k-1)$ for all $m,k \ge 3$.
If both players are playing uniformly at random, it is not difficult to see that the game for $m=k$ typically ends after about $\frac12m^2$ turns. Considering random play often gives good intuition for bounds in deterministic play, and this random intuition would suggest that the bound in Theorem \[thm:ESO<ES\] is off by a factor $2$.
We establish such a lower bound for the Erdős-Szekeres on-line number in Section \[sec:mkLB\]:
\[thm:mkLB\] $\operatorname{ESO}(m,k) \ge \lfloor\frac{k}{2}\rfloor (m-k+5) -3$ for $m \ge k \ge 4$.
Notice the unusual behavior of this bound depending on the parity of $k$. We achieve this lower bound by considering a related game which restricts the choices for player B. For this related game, we show that the leading term in Theorem \[thm:mkLB\] is correct for every fixed $k$, and the dependence on the parity of $k$ can be seen in the proof. In fact, we conjecture that this leading term is correct for the original game as well.
\[con\] For all $m\ge k$, we have $\operatorname{ESO}(m,k)=\lfloor\frac{k}{2}\rfloor m + O(k^2)+o(mk)$.
For the values $k=1$ and $k=2$, it is trivial to determine $\operatorname{ESO}(m,k)$. The first interesting value is $k=3$. We find evidence for our conjecture by providing strategies for player A (Section \[sec:m3UB\]) and player B (Section \[sec:m3LB\]), obtaining the following result.
\[thm:m3\] $\operatorname{ESO}(m,3) = m + (6m)^{\frac13} + O(1).$ Specifically, $$m + (6m)^{\frac13} - 2 < \operatorname{ESO}(m,3) < m + (6m)^{\frac13} + 3.$$
Finally, we mention some variations of this on-line game in Section \[sec:conclusion\].
Preliminary Definitions and Observations {#sec:prelim}
========================================
We begin with a formal definition of the *Erdős-Szekeres on-line number*.
*$x(p)$, $y(p)$, Up-runs and down-runs.*\
Given a point $p \in \mathbb{R}^2$, we denote the $x$-coordinate of $p$ by $x(p)$ and the $y$-coordinate by $y(p)$. When we denote a set of points by $p_1, \dots, p_n$, we assume that $x(p_i)<x(p_{i+1})$ for all $1\le i\le n-1$. Consider a set of points ${\mathcal{C}}= \{p_1,\ldots,p_n\}$. If $y(p_i)\le y(p_{i+1})$ for all $1\le i\le n-1$, then we say that ${\mathcal{C}}$ is an *$n$-up-run*, or simply an *up-run*. If $y(p_i)> y(p_{i+1})$ for all $1\le i\le n-1$, then we say that ${\mathcal{C}}$ is an *$n$-down-run*, or simply a *down-run*.
[max totalsize=[6cm]{}[6cm]{},center,padding=1em]{}
\(1) at (0,1.5) ; (2) at (.5,3.5) ; (3) at (1.5,2.5) ; (4) at (2,1) ; (5) at (2.5,0) ; (6) at (3,2.5) ;
at (\[shift=[(135:.6)]{}\]1) [$r$]{}; at (\[shift=[(45:.6)]{}\]2) [$s$]{}; at (\[shift=[(45:.6)]{}\]3) [$t$]{}; at (\[shift=[(45:.6)]{}\]4) [$u$]{}; at (\[shift=[(45:.6)]{}\]5) [$v$]{}; at (\[shift=[(45:.6)]{}\]6) [$w$]{};
\(1) edge node (3); (3) edge node (6);
\(2) edge node (3); (3) edge node (4); (4) edge node (5);
*The Erdős-Szekeres on-line game, $A_{m,k}$.*\
Let $m,k \ge 1$. In each step of the game $A_{m,k}$, player A chooses a value $\hat{x} \in (0,1)$ and then player B chooses a value $\hat{y} \in (0,1)$, forming a point $(\hat{x},\hat{y})$ in $(0,1) \times (0,1)$. The game ends when after some turn there is either an $m$-up-run or a $k$-down-run. Let player A have the objective of ending the game in the fewest number of turns, and player B in the greatest number of turns. We denote by $\operatorname{ESO}(m,k)$ the number of turns a game of $A_{m,k}$ will take when both players play optimally, which we call the *Erdős-Szekeres on-line number*.
(-3,-3) rectangle (3,3); (1) at (0.5,-3) ; (2) at (0.5,3) ; at (\[shift=[(-30:1)]{}\]2) [$x = \hat{x}$]{};
at (5,0) [$\longrightarrow$]{};
(7,-3) rectangle (13,3); (11) at (10.5,-3) ; (12) at (10.5,3) ; (13) at (7,-2) ; (14) at (13,-2) ; at (\[shift=[(30:1)]{}\]13) [$y = \hat{y}$]{};
\(5) at (-.5,2) ; (6) at (-2,-.5) ; (7) at (2,.5) ;
\(15) at (9.5,2) ; (16) at (8,-.5) ; (17) at (12,.5) ; (18) at (10.5,-2) ;
at (\[shift=[(35:1)]{}\]18) [$(\hat{x},\hat{y})$]{};
\(1) edge node (2); (11) edge node (12);
\(13) edge node (14);
Clearly, $\operatorname{ESO}(m,k) \ge \min\{m,k\}$. Note that in an optimal strategy, neither player needs to repeat a previously played $x$- or $y$-value; when providing a strategy for one player, we will assume that the opposing player does not repeat a value already chosen. By reflecting any instance of $A_{m,k}$ about the line $x = \frac{1}{2}$, we also see that $\operatorname{ESO}(m,k) = \operatorname{ESO}(k,m)$. Hence we assume that $m \ge k$ unless otherwise noted.
Observe that, after the first turn, player B always has a choice which will not increase the length of a longest up-run. To see this, without loss of generality assume that there is a point $p$ immediately to the right of the $x$-value player A has chosen. Then player B can choose $y(p)+\varepsilon$ for a sufficiently small, positive $\varepsilon$ such that $p$ and the new point are interchangeable in any up-run containing one of them. Similarly, player B always has a choice which will not increase the length of a longest down-run. Thus $\operatorname{ESO}(m,k)$ is strictly increasing in both $m$ and $k$ when $m,k \ge 2$.
We also have $\operatorname{ESO}(m,k) \le \operatorname{ES}(m,k)$ and thus by the theorem of Erdős and Szekeres [@ErdosSzekeres1935] we know that $\operatorname{ESO}(m,k) \le (m-1)(k-1)+1$. It is clear that $\operatorname{ESO}(m,k) = \operatorname{ES}(m,k)$ when $k \in \{1,2\}$, but on the other hand it is easy (and perhaps a fun exercise for the reader) to show that $\operatorname{ESO}(3,3) = 4 < \operatorname{ES}(3,3) = 5$. In fact, equality holds only for $k \le 2$. We prove this after some preliminary definitions:
*Quadrants of a point.*\
Let ${\mathcal{P}}$ be a point set in $(0,1) \times (0,1)$ and let $p' \in {\mathcal{P}}$. Then the *north-east quadrant of $p'$* is $\operatorname{NE}(p') = \{ p \in {\mathcal{P}}: y(p') < y(p) ~ \& ~ x(p') < x(p) \}$. The *north-west quadrant*, *south-west quadrant* and *south-east quadrant of $p'$* are defined similarly and are denoted by $\operatorname{NW}(p')$, $\operatorname{SW}(p')$ and $\operatorname{SE}(p')$, respectively.
(0,0) rectangle (7,7); (11) at (0,4) ; (12) at (7,4) ; (13) at (4,0) ; (14) at (4,7) ;
\(1) at (1,1) ; (2) at (2,5) ; (3) at (3,6) ; (4) at (4,4) ; (5) at (5,3) ; (6) at (6,2) ;
at (\[shift=[(40:.6)]{}\]1) [$p_1$]{}; at (\[shift=[(40:.6)]{}\]2) [$p_2$]{}; at (\[shift=[(40:.6)]{}\]3) [$p_3$]{}; at (\[shift=[(40:.6)]{}\]4) [$p_4$]{}; at (\[shift=[(40:.6)]{}\]5) [$p_5$]{}; at (\[shift=[(40:.6)]{}\]6) [$p_6$]{};
at (0,0)\[shift=[(35:.45)]{}\] [SW]{}; at (7,0)\[shift=[(155:.45)]{}\] [SE]{}; at (0,7)\[shift=[(-35:.45)]{}\] [NW]{}; at (7,7)\[shift=[(-155:.45)]{}\] [NE]{};
\(11) edge node (12); (13) edge node (14);
With these definitions, we are now ready to prove Theorem \[thm:ESO<ES\]. Our proof is an adaptation of Seidenberg’s proof that $\operatorname{ES}(n,n) \le (n-1)^2 +1$ (see [@Steele1995]).
Assume ${\mathcal{P}}$ is a set of $(m-1)(k-1)-1$ points that have already been played. If ${\mathcal{P}}$ contains an $m$-up-run or a $k$-down-run, then we are done; so suppose otherwise.
Label each point $p \in {\mathcal{P}}$ with $(i,j)$, where $i$ is the length of the longest up-run in ${\mathcal{P}}$ with $p$ as its left-most point and $j$ is the length of the longest down-run in ${\mathcal{P}}$ with $p$ as its left-most point. Since there are no $m$-up-runs and no $k$-down-runs, the set of labels of ${\mathcal{P}}$ is a subset of $[m-1] \times [k-1]$.
Observe that each of the labels are distinct as for any points $p,q$ with $x(p)<x(q)$, either the first coordinate of $q$’s label is larger than the first coordinate of $p$’s label or the second coordinate of $q$’s label is larger than the second coordinate of $p$’s label. Hence only one element from $[m-1] \times [k-1]$ is missing as a label.
**Case 1:** The missing label is not $(m-1,k-1)$. Then there is some point $q$ that is the left-most point of both an $(m-1)$-up-run and a $(k-1)$-down-run. If player A plays to the left of $q$, then player B will choose a $y$-value that results in either an $m$-up-run or a $k$-down-run.
**Case 2:** The missing label is $(m-1,k-1)$. Then there is some point $q$ that is labeled with $(1,1)$. Now let ${\mathcal{P}}^\uparrow = \{ p \in {\mathcal{P}}: \operatorname{NE}(p) = \varnothing \}$. Observe that ${\mathcal{P}}^\uparrow$ is the set of points whose labels have “1” in the first coordinate. Since $(m-1,k-1)$ is the only missing label, the set of labels of points in ${\mathcal{P}}^\uparrow$ is $\{ (1,j) \}_{j=1}^{k-1}$. Hence $|{\mathcal{P}}^\uparrow|=k-1$. Also observe that no two points from ${\mathcal{P}}^\uparrow$ form a 2-up-run and thus ${\mathcal{P}}^\uparrow$ is a $(k-1)$-down-run with $q$ as the right-most point. Making similar observations about the set ${\mathcal{P}}_\downarrow = \{ p \in {\mathcal{P}}: \operatorname{SE}(p) = \varnothing \}$ shows that ${\mathcal{P}}_\downarrow$ is an $(m-1)$-up-run with $q$ as the right-most point. If player A plays to the right of $q$, then player B will choose a $y$-value that results in either an $m$-up-run or a $k$-down-run.
Although Theorem \[thm:ESO<ES\] establishes that this on-line number of the Erdős-Szekeres problem is distinct from their classical result when $m,k \ge 3$, it does not provide a precise sense of what $\operatorname{ESO}(m,k)$ is in general. In addition to the degenerate cases, we list some small results obtained via dynamic programming[^5] in Table \[tab:smallresults\].
$(m,k)$, $m\ge k$ $(m,1)$ $(m,2)$ $(3,3)$ $(4,3)$ $(5,3)$ $(6,3)$ $(7,3)$ $(4,4)$ $(5,4)$
--------------------------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
$\operatorname{ESO}(m,k)$ 1 $m$ 4 6 7 9 10 8 11
$\operatorname{ES}(m,k)$ 1 $m$ 5 7 9 11 13 10 13
: Some Small Exact Results
\[tab:smallresults\]
A General Lower Bound for $\operatorname{ESO}(m,k)$ {#sec:mkLB}
===================================================
In this section, we study a closely related game $B_{m,k}$ (for $k \ge 2$). The game is precisely the same as $A_{m,k}$, except that player B is always restricted to choose from the set $\{1,2,\ldots,k-1\}$ (referred to as *tiers*). Note that a $k$-down-run is impossible in this game, and hence player A’s objective is to force an $m$-up-run while player B’s is to avoid one. Call the minimum number of moves in which player A can force a win in this game $B(m,k)$. Player B can use a strategy from this game to play the previous game $A_{m,k}$. Therefore, $\operatorname{ESO}(m,k)\ge B(m,k)$ for all $m$ and $k$. Hence Theorem \[thm:mkLB\] is corollary to Proposition \[prop:BgameLB2\].
For the game $B_{m,k}$, we will use a few more terms.
*${\mathcal{L}}_x$, ${\mathcal{R}}_x$, Separated*\
At any point during the game, we denote by ${\mathcal{L}}_x$ the set of points $p$ to the left of $x$, i.e. $x(p) < x$. Similarly, we denote by ${\mathcal{R}}_x$ the set of points to the right of $x$. Two $x$-values $x_1$ and $x_2$ are *separated* by $t$ points if $|{\mathcal{R}}_{x_1} \cap {\mathcal{L}}_{x_2}| + |{\mathcal{R}}_{x_2} \cap {\mathcal{L}}_{x_1}| = t$. To say that points $p_1$ and $p_2$ are *separated* is to say that $x(p_1)$ and $x(p_2)$ are separated.
\[prop:BgameUB\] $$B(m,k) \le \left\lfloor\frac{k}{2}\right\rfloor (m-1)+1.$$
We describe a strategy for player A. Player A always chooses an $x$-value $\hat{x}$ such that the $y$-values of points in ${\mathcal{L}}_{\hat{x}}$ are all at most $\frac{k}{2}$, and the $y$-values of points in ${\mathcal{R}}_{\hat{x}}$ are all greater than $\frac{k}{2}$. By following this strategy from the first turn, player A will always be able to choose such an $\hat{x}$. Let $\ell$ be the size of a longest up-run with $y$-values at most $\frac{k}{2}$, and let $r$ be the size of a longest up-run with $y$-values greater than $\frac{k}{2}$. Due to player A’s strategy, these two up-runs together form an $(\ell+r)$-up-run. Applying Erdős-Szekeres to each of the parts separately yields that there are at most $\ell\left\lfloor\frac{k}{2}\right\rfloor$ points with $y$-values at most $\frac{k}{2}$, and at most $r\left\lfloor\frac{k-1}{2}\right\rfloor$ points with $y$-values greater than $\frac{k}{2}$ at any time. Plugging in $\ell+r=m-1$ at the penultimate turn yields the claimed upper bound.
Since $B(m,k) \ge m$, Proposition \[prop:BgameUB\] implies that $B(m,2) = B(m,3) = m$ for all $m$. For the lower bound, we first prove the following, weaker version of Proposition \[prop:BgameLB2\].
\[prop:BgameLB1\] $$B(m,k) \ge \left\lfloor\frac{k}{2}\right\rfloor (m-k+1)+k-1.$$
We describe a strategy for player B. At every step, in response to player A choosing $\hat{x}$, player B plays any value in $\{1,\dots ,k-1\}$ such that any other point in the same tier is separated from $\hat{x}$ by at least $\left\lfloor \frac{k}{2}\right\rfloor -1$ points. Now let ${\mathcal{S}}= \{ p_1, \dots, p_m \}$ be an $m$-up-run. Since ${\mathcal{S}}$ can have at most $k-2$ points $p_i$ such that $y(p_{i+1}) \ne y(p_i)$, there are at least $(m-1)-(k-2)=m-k+1$ points $p_i$ such that $p_{i+1}$ is in the same tier. Since points in the same tier must be separated by at least $\left\lfloor \frac{k}{2}\right\rfloor -1$ points, this accounts for $m+\left(\lfloor\frac{k}{2}\rfloor -1\right) (m-k+1)$ total points played.
By avoiding the bottom tiers when player A chooses an $x$-value to the far left and avoiding the top tiers when player A chooses an $x$-value to the far right, player B can adjust the strategy given above to guarantee a few more points.
\[prop:BgameLB2\] Let $k \ge 4$. If $k$ is even, then $\displaystyle B(m,k) \ge \frac{k}{2} (m-k+5) -3$.\
If $k$ is odd, then $\displaystyle B(m,k) \ge \frac{k-1}{2} (m-k+6) -3$.
We describe a strategy for player B. First assume that $k$ is even. Suppose player A chooses $\hat{x}$. Now, let $c = \max\{1, \frac{k}{2} -|{\mathcal{L}}_{\hat{x}}| \}$ and let $d = \min\{ k-1, \frac{k}{2} + |{\mathcal{R}}_{\hat{x}}| \}$. Then player B plays any value in $\{c,\dots ,d\}$ such that any other point in the same tier is separated from $\hat{x}$ by at least $\frac{k}{2} -1$ points. Thus for any point $p$, we have both $|{\mathcal{L}}_{x(p)}| \ge \max\left\{ 0, \frac{k}{2}-y(p) \right\}$ and $|{\mathcal{R}}_{x(p)}| \ge \max\left\{ 0, y(p) - \frac{k}{2} \right\}$ at any time after $p$ has been played. Now let ${\mathcal{S}}= \{ p_1, \dots, p_m \}$ be an $m$-up-run. Observe that there are at least $(m-1)-\big( y(p_m) - y(p_1) \big)$ pairs of consecutive points in ${\mathcal{S}}$ which are in the same tier. Then counting points in ${\mathcal{S}}$, points separating consecutive points in ${\mathcal{S}}$ in the same tier, and points to the left and right of ${\mathcal{S}}$ yields at least $$m + \Big[ m-1 -\Big(y(p_m) - y(p_1)\Big) \Big] \left( \frac{k}{2} -1 \right) + \max\left\{ 0, \frac{k}{2}-y(p_1) \right\} + \max\left\{ 0, y(p_m) - \frac{k}{2} \right\}$$ total points. The above expression is minimized when $y(p_m)-y(p_1) = k-2$, which means that at least $m + \big[ m-1 -(k-2) \big] \left( \frac{k}{2} -1 \right) + k-2 = \frac{k}{2} (m-k+5) -3$ points have been played.
If $k$ is odd, then player B can use the strategy for $k-1$ (by never choosing the top tier) to obtain the claimed bound.
Establishing an Upper Bound for $\operatorname{ESO}(m,3)$ {#sec:m3UB}
=========================================================
We provide an upper bound for $\operatorname{ESO}(m,3)$ by describing a strategy for player A. This strategy (see Definition \[def:Astrategy\]) will assess the state of the game and decide which of several strategy “modes" to use at that time. We begin by defining some terms used throughout this section and the next, and then define the different strategy modes. We finish by analyzing the full strategy in Lemma \[lemma:m3UB\].
*Columns, Rows, Notches*\
Suppose a game of $A_{m,k}$ is underway with a set of points ${\mathcal{P}}= \{p_1, \dots, p_t\}$ having been played. Let $p_0 = 0$ and $p_{t+1}=1$. Then each of the intervals $\{\big(x(p_i), x(p_{i+1})\big)\}_{i=0}^t$ is a set of equivalent choices for player A’s next turn, called a *column*. When we say that player A plays in a column $(x(p_i),x(p_{i+1}))$, we mean that player A chooses any $x$-value in $(x(p_i),x(p_{i+1}))$; for formality, without loss of generality we may assume that player A chooses the $x$-value $\frac{x(p_i)+x(p_{i+1})}{2}$.
Similarly, a *row* is an interval between consecutive $y$-values of ${\mathcal{P}}$ from which all choices that player $B$ makes are equivalent given the state of the game.
A *notch* is a unit of measurement between columns or between rows with respect to a point and a subset of points. Let $p \in {\mathcal{P}}$, let ${\mathcal{A}}\subseteq {\mathcal{P}}$, and let $\tilde{{\mathcal{A}}} = {\mathcal{A}}\cup \{p\}$. Let $p_1, \dots, p_s$ be an ordering of $\tilde{{\mathcal{A}}}$ where $p_i = p$. Consider the columns induced by $\tilde{{\mathcal{A}}}$, say $I_j = (x(p_j),x(p_{j+1}))$ for $0 \le j \le t$. Then for each $j < i$, we say that the column $I_j$ is $i-j-1$ *notches left of $p$ with respect to ${\mathcal{A}}$*, or simply $i-j-1$ *notches left of $p$* if ${\mathcal{A}}$ is understood. For each $j \ge i$, we say that the column $I_j$ is $j-i$ *notches right of $p$*.
Similarly, we speak of rows being some number of notches *above* or *below* $p$ (with respect to ${\mathcal{A}}$). (See Figure \[fig:notches\].)
(0,0) rectangle (6,6); (1a) at (0,1) ; (1b) at (6,1) ; (2a) at (0,2) ; (2b) at (6,2) ; (3a) at (0,3.3) ; (3b) at (6,3.3) ; (4a) at (0,4) ; (4b) at (6,4) ; (5a) at (0,5.2) ; (5b) at (6,5.2) ; (qa) at (0,2.6) ; (qb) at (6,2.6) ;
\(1) at (1,1) ; (2) at (2,2) ; (3) at (3,3.3) ; (4) at (4,4) ; (5) at (5,5.2) ; (q) at (1.2,2.6) ; (qh) at (2.2,4.5) ;
at (\[shift=[(-40:.6)]{}\]1) [$v_1$]{}; at (\[shift=[(-40:.6)]{}\]2) [$v_2$]{}; at (\[shift=[(-40:.6)]{}\]3) [$v_3$]{}; at (\[shift=[(-40:.6)]{}\]4) [$v_4$]{}; at (\[shift=[(-40:.6)]{}\]5) [$v_5$]{}; at (\[shift=[(145:.6)]{}\]q) [$q$]{}; at (\[shift=[(140:.6)]{}\]qh) [$\hat{q}$]{};
at (\[shift=[(10:2.2)]{}\]5b) [3 notches above $q$]{}; at (\[shift=[(16:2.3)]{}\]4b) [2 notches above $q$]{}; at (\[shift=[(10:2.2)]{}\]3b) [1 notch above $q$]{}; at (\[shift=[(9:2.2)]{}\]qb) [0 notches above $q$]{}; at (\[shift=[(-10:2.2)]{}\]qb) [0 notches below $q$]{}; at (\[shift=[(-16:2.3)]{}\]2b) [1 notch below $q$]{}; at (\[shift=[(-16:2.3)]{}\]1b) [2 notches below $q$]{};
(1a) edge node (1b); (2a) edge node (2b); (3a) edge node (3b); (4a) edge node (4b); (5a) edge node (5b); (qa) edge node (qb);
*“$f$-Middling" Strategy Mode*\
In the game $A_{m,3}$, we have the initial conditions ${\mathcal{S}}= {\mathcal{N}}= {\mathcal{W}}= \varnothing$, $(a_x,b_x) = (a_y,b_y) = (0,1)$ and ${\texttt{t}}= 0$, where ${\mathcal{S}}$ is the point set in the *active segment* $(a_x,b_x) \times (a_y,b_y)$ forming an up-run, ${\mathcal{N}}$ is an up-run of *saved* points previously in ${\mathcal{S}}$, ${\mathcal{W}}$ is the set of “wasted" points not extending the up-run, and [`t`]{} is the number of times player B has chosen to not extend ${\mathcal{S}}$ into a longer up-run. The parameter $f$ is a sequence $\{f_i\}_{i=1}^\infty$ of nonnegative integers given ahead of time. As player A, do the following:
1. \[Mstep:play\] Play in a middlemost column of ${\mathcal{S}}$ with respect to $(a_x,b_x) \times (a_y,b_y)$, say the interval $( x(p_\ell), x(p_r) )$. (Possibly with $p_\ell = (0,0)$ or $p_r = (1,1)$ being artificial points.) Player B chooses a row, creating the new point $q$. Assume the game is not over and thus that $y(q) \in (a_y,b_y)$.
2. If $q$ extended ${\mathcal{S}}$ to be a longer up-run, then return to Step \[Mstep:play\]. Otherwise, $q$ is above $p_r$ or below $p_\ell$; increment [`t`]{}. If $|\operatorname{NE}(q) \cap {\mathcal{S}}| \le f_{\texttt{t}}$ or $|\operatorname{SW}(q) \cap {\mathcal{S}}| \le f_{\texttt{t}}$, then exit the strategy mode now. Add $q$ to ${\mathcal{W}}$. Go to Step \[Mstep:pointright\] if $q$ is above $p_r$ and go to Step \[Mstep:pointleft\] if $q$ is below $p_\ell$.
3. \[Mstep:pointright\] Let $\hat{q} = \operatorname*{argmax}_{p \in \operatorname{SE}(q) \cap {\mathcal{S}}} x(p)$ and redefine $a_x := x(\hat{q})$ and $a_y := y(\hat{q})$. Then update ${\mathcal{S}}$ and ${\mathcal{N}}$ by deleting $(\operatorname{SW}(\hat{q}) \cup \{\hat{q}\}) \cap {\mathcal{S}}$ from ${\mathcal{S}}$ and adding these points to ${\mathcal{N}}$. Return to Step \[Mstep:play\].
4. \[Mstep:pointleft\] Let $\hat{q} = \operatorname*{argmin}_{p \in \operatorname{NW}(q) \cap {\mathcal{S}}} x(p)$ and redefine $b_x := x(\hat{q})$ and $b_y := y(\hat{q})$. Then update ${\mathcal{S}}$ and ${\mathcal{N}}$ by deleting $(\operatorname{NE}(\hat{q}) \cup \{\hat{q}\}) \cap {\mathcal{S}}$ from ${\mathcal{S}}$ and adding these points to ${\mathcal{N}}$. Return to Step \[Mstep:play\].
We illustrate the $f$-Middling strategy mode in the Appendix.
*Barb, “Playing the Barb" Strategy Mode*\
An *$(s, t)$-barb*, or simply *barb*, is a subset ${\mathcal{B}}= \{w,z\} {\mathbin{\mathaccent\cdot\cup}}{\mathcal{U}}{\mathbin{\mathaccent\cdot\cup}}{\mathcal{V}}\subseteq {\mathcal{P}}$ such that $w,z$ form a down-run, ${\mathcal{U}}$ and ${\mathcal{V}}$ are (possibly empty) $s$- and $t$-up-runs, respectively, and also $x(w) < \max_{u \in {\mathcal{U}}} x(u) < \min_{v \in {\mathcal{V}}} x(v) < x(z)$ and $\max_{u \in {\mathcal{U}}} y(u) < y(z) < y(w) < \min_{v \in {\mathcal{V}}} y(v)$. We say that $w$ and $z$ are the *spikes* of ${\mathcal{B}}$, that ${\mathcal{U}}$ is the *bottom wire* of ${\mathcal{B}}$, and that ${\mathcal{V}}$ is the *top wire* of ${\mathcal{B}}$. If one of ${\mathcal{U}}$ or ${\mathcal{V}}$ is empty, we drop the appropriate inequality conditions and say that ${\mathcal{B}}$ is a *half-barb*. We say that player A *plays the barb* by doing the following:
1. \[PBstep:play\] Let $a := \max_{p \in {\mathcal{U}}\cup \{w\}} x(p)$ and $b := \min_{p \in {\mathcal{V}}\cup \{z\}} x(p)$. Play in the column $(a,b)$ with respect to ${\mathcal{B}}$. Player B chooses a row, creating the new point $q$.
2. Assuming the game is not over, $q$ is in the row zero notches above $w$ or zero notches below $z$ with respect to ${\mathcal{B}}$. If the former, add $q$ to ${\mathcal{V}}$; if the latter, add $q$ to ${\mathcal{U}}$. Return to Step \[PBstep:play\].
(0,0) rectangle (12,8);
(u1) at (1,1) ; (u2) at (2,1.4) ; (u3) at (3,2.2) ; (u4) at (4,2.9) ; (u5) at (5,3.4) ; (v1) at (7.5,5.5) ; (v2) at (8,6) ; (v3) at (9.5,6.7) ; (v4) at (11,7) ; (w) at (4.5,5) ; (z) at (8.5,4) ;
at (\[shift=[(120:.5)]{}\]u1) [$u_1$]{}; at (\[shift=[(120:.5)]{}\]u2) [$u_2$]{}; at (\[shift=[(120:.5)]{}\]u3) [$u_3$]{}; at (\[shift=[(120:.5)]{}\]u4) [$u_4$]{}; at (\[shift=[(120:.5)]{}\]u5) [$u_5$]{};
at (\[shift=[(-60:.5)]{}\]v1) [$v_1$]{}; at (\[shift=[(-60:.5)]{}\]v2) [$v_2$]{}; at (\[shift=[(-60:.5)]{}\]v3) [$v_3$]{}; at (\[shift=[(-60:.5)]{}\]v4) [$v_4$]{};
at (\[shift=[(90:.5)]{}\]w) [$w$]{}; at (\[shift=[(-90:.5)]{}\]z) [$z$]{};
(u1) edge node (u2); (u2) edge node (u3); (u3) edge node (u4); (u4) edge node (u5); (u5) edge node (z);
(u5) edge node (v1);
\(w) edge node (v1); (v1) edge node (v2); (v2) edge node (v3); (v3) edge node (v4);
\(w) edge node (z);
The following observation illustrates the usefulness of barbs for player A.
If a game of $A_{m,3}$ is underway with an $(s,t)$-barb ${\mathcal{B}}$, then the game will end after at most $m - s -t$ more turns if player A plays the barb ${\mathcal{B}}$.
*“$w$-Barb" Strategy Mode*\
Suppose a game of $A_{m,3}$ is underway with the point set ${\mathcal{P}}= {\mathcal{U}}{\mathbin{\mathaccent\cdot\cup}}\{r_1,q_1\} {\mathbin{\mathaccent\cdot\cup}}{\mathcal{V}}_1$ such that ${\mathcal{B}}_1 = \{r_1,q_1\} \cup {\mathcal{V}}_1$ is a half-barb with spikes $r_1,q_1$ (where $x(r_1) < x(q_1)$) and top wire ${\mathcal{V}}_1$, such that ${\mathcal{U}}= \operatorname{SW}(r_1)$, and such that ${\mathcal{U}}$ is an up-run. Let $\hat{r}_1 = \operatorname*{argmin}_{p \in \operatorname{NW}(q_1)\setminus\{r_1\}} y(p)$. The parameter $w$ is given as a positive integer. As player A, do the following:
1. \[Bstep:play\] Let $i$ be maximum such that ${\mathcal{B}}_i$ has been defined; play in the column $(x(r_i),x(\hat{r}_i))$. Player B chooses a row, creating the new point $q$. Assuming the game is not over, we have $y(q) < y(\hat{r}_i)$.
2. If $q$ is zero notches above $r_i$ with respect to ${\mathcal{B}}_i$, add $q$ to $\mathcal{V}_i$, then redefine $\hat{r}_i := q$ and go to Step \[Bstep:play\]. Otherwise, $q$ is below $r_i$, say by $d$ notches with respect to ${\mathcal{U}}$. If $d \ge w - i$ and $d > 0$, then go to Step \[Bstep:stepdown\]. Otherwise, $d < w - i$ or $d = 0$; go to Step \[Bstep:playbarb\].
3. \[Bstep:stepdown\] Consider $q_i$ a lost point. Rename $q$ to be $q_{i+1}$, define $r_{i+1} := \operatorname*{argmin}_{p \in \operatorname{NW}(q_{i+1})} y(p)$, define $\hat{r}_{i+1} := \operatorname*{argmin}_{p \in \operatorname{NW}(q_{i+1})\setminus\{r_{i+1}\}} y(p)$, define ${\mathcal{V}}_{i+1} : = {\mathcal{V}}_i {\mathbin{\mathaccent\cdot\cup}}\operatorname{NW}(q_{i+1}) \setminus \{ r_{i+1} \}$, and define the half-barb ${\mathcal{B}}_{i+1} := \{r_{i+1},q_{i+1}\} {\mathbin{\mathaccent\cdot\cup}}{\mathcal{V}}_{i+1}$. Return to Step \[Bstep:play\].
4. \[Bstep:playbarb\] Consider $\operatorname{NW}(q)$ as lost points. Let $\hat{{\mathcal{U}}} = \operatorname{SW}(q) \cup \{q\}$ and let ${\mathcal{B}}= \{r_i,q_i\} \cup \hat{{\mathcal{U}}} \cup {\mathcal{V}}_i$. Then ${\mathcal{B}}$ is a barb with spikes $r_i$ and $q_i$, top wire ${\mathcal{V}}_i$ and bottom wire $\hat{{\mathcal{U}}}$; play the barb ${\mathcal{B}}$ until the game finishes.
We illustrate a step of the $w$-Barb strategy mode in the Appendix.[^6]
\[lemma:m3barbbound\] If player A adopts the $w$-Barb strategy mode in a game of $A_{m,3}$ with $w \ge \left\lfloor\sqrt{\max\{2|{\mathcal{U}}|-\frac{15}{4},0\}}+\frac32\right\rfloor +1$, then the game ends after a total of at most $m+w$ turns.
Since the game would end just as soon if a 3-down-run is formed, we may assume that the game ends once an $m$-up-run is formed. Observe that if the game ends after reaching Step \[Bstep:playbarb\], then the $d+1$ points in $\operatorname{NW}(q)$ and the $i$ points $q_1, \dots, q_i$ are the only points not in the $m$-up-run containing ${\mathcal{V}}_i \cup \hat{{\mathcal{U}}}$. Thus a total of $m+d+i+1$ turns were taken. Since $d+1 \le w-i$, at most $m+w$ turns were taken.
If the game ends without ever reaching Step \[Bstep:playbarb\], then it is because ${\mathcal{V}}_i \cup {\mathcal{U}}$ is an $m$-up-run containing all points except $q_1, \dots, q_i$, making a total of $m+i$ turns taken. Now observe that for any $2 \le j < w$ we have $|\operatorname{NW}(q_j) \setminus \operatorname{NW}(q_{j-1})| \ge w-j$ and also that $\mathop{\cdot\hspace*{-7.5pt}\bigcup}_{j=2}^i \Big( \operatorname{NW}(q_j) \setminus \operatorname{NW}(q_{j-1}) \Big) = \left(\bigcup_{j=2}^i \operatorname{NW}(q_j)\right) \setminus \operatorname{NW}(q_{j-1}) \subseteq {\mathcal{U}}$ for all $i \ge 2$. Hence if $i = w+1$, then $|{\mathcal{U}}| \ge \sum_{j=2}^i \Big| \operatorname{NW}(q_j) \setminus \operatorname{NW}(q_{j-1}) \Big| \ge 1+1+\sum_{j=2}^{w-1} (w-j) = \frac{(w-\frac{3}{2})^2 +\frac{15}{4}}{2}$. Since $w > \sqrt{\max\{2|{\mathcal{U}}|-\frac{15}{4},0\}}+\frac32$, this implies $|{\mathcal{U}}| \ge\frac{(w-\frac{3}{2})^2 +\frac{15}{4}}{2} > |{\mathcal{U}}|$, a contradiction. Thus $i \le w$ always, and therefore at most $m+w$ turns are taken in this case as well.
\[def:Astrategy\]*$f$-Combined Strategy*\
Let $f$ be given. As player A in the game $A_{m,3}$, do the following:
1. \[Cstep:middling\] (Middling Mode) Begin by playing the $f$-Middling strategy mode. Play until the game ends or until the strategy mode terminates, making note of ${\mathcal{N}}$.
2. \[Cstep:transitiona\] (Transition) If the $f$-Middling mode strategy mode terminated, then in the active segment $(a_x,b_x) \times (a_y,b_y)$ we have an up-run ${\mathcal{S}}$ and a point $q$ in a middlemost column $(x(p_\ell),x(p_r))$ with respect to ${\mathcal{S}}$, where $|\operatorname{NE}(q) \cap {\mathcal{S}}| \le f_{\texttt{t}}$ or $|\operatorname{SW}(q) \cap {\mathcal{S}}| \le f_{\texttt{t}}$ and without loss of generality we may assume $y(q) < y(p_\ell)$.\
If $q$ is at least one notch below $p_\ell$ with respect to ${\mathcal{S}}$, then let ${\mathcal{U}}:= \operatorname{SW}(q) \cap {\mathcal{S}}$, $r_1 := \operatorname*{argmin}_{p \in \operatorname{NW}(q) \cap {\mathcal{S}}} y(p)$, $q_1 := q$, ${\mathcal{V}}_1 := \operatorname{NE}(r_1) \cap {\mathcal{S}}$ and go to Step \[Cstep:barb\].\
Else, $q$ is zero notches below $p_\ell$ with respect to ${\mathcal{S}}$; proceed to Step \[Cstep:transitionb\].
3. \[Cstep:transitionb\] (Transition) Play in the column $(x(p_\ell),x(q))$. Player B chooses a row, creating the new point $\hat{q}$. Assuming the game is not over, we have $a_y < y(\hat{q}) < b_y$ and may assume without loss of generality that either $\hat{q}$ is zero notches above $p_\ell$ with respect to ${\mathcal{S}}$ or at least one notch below $q$ with respect to ${\mathcal{S}}$.\
If the former, then let ${\mathcal{U}}:= \operatorname{SW}(p_\ell) \cap {\mathcal{S}}$, $r_1 := p_\ell$, $q_1 := q$, ${\mathcal{V}}_1 := \operatorname{NE}(p_\ell) \cap {\mathcal{S}}$. Also add $\hat{q}$ to ${\mathcal{V}}_1$. Go to Step \[Cstep:barb\].\
If the latter, then $\hat{q}$ is say $k$ notches below $q$ with respect to ${\mathcal{S}}$. Then let ${\mathcal{U}}:= \operatorname{SW}(\hat{q}) \cap {\mathcal{S}}$, $r_1 := \operatorname*{argmin}_{p \in \operatorname{NW}(\hat{q}) \cap {\mathcal{S}}} y(p)$, $q_1 := \hat{q}$, and ${\mathcal{V}}_1 := \operatorname{NE}(r_1) \cap {\mathcal{S}}$. Also add $q$ to ${\mathcal{W}}$. Go to Step \[Cstep:barb\].
4. \[Cstep:barb\] (Barb Mode) Given ${\mathcal{U}}$, $r_1$, $q_1$ and ${\mathcal{V}}_1$, play $A_{m-|{\mathcal{N}}|,3}$ by adopting the $w$-Barb strategy mode in $(a_x,b_x) \times (a_y,b_y)$ with $w = \left\lfloor\sqrt{\max\{2|{\mathcal{U}}|-\frac{15}{4},0\}}+\frac32\right\rfloor +1$ until the game ends.
\[lemma:m3UB\] Suppose player A uses the $f$-Combined Strategy in the game $A_{m,3}$ with $f=\{f_i\}_{i=1}^\infty$. Suppose also that there exists $T \ge 2$ meeting the following conditions:
1. \[cond:transition\] $\sum_{i=1}^T (f_i+2) +(f_T+1) \ge m$, and
2. \[cond:barb\] $i + \left\lfloor\sqrt{\max\{2|{\mathcal{U}}|-\frac{15}{4},0\}}+\frac32\right\rfloor \le T$ for all $i \in [T-1]$.
Then the game ends after a total of at most $m+T+1$ turns.
Suppose the game has ended, again assuming that it was not because a 3-down-run was formed. We proceed by case analysis.
Suppose the game ended while still in Step \[Cstep:middling\]. This means that each time ${\texttt{t}}$ was incremented in the $f$-Middling strategy mode (say to $j$), at least $f_j+2$ points were added to ${\mathcal{N}}$ from ${\mathcal{S}}$ ($\operatorname{SW}(\hat{q}) \cup \{\hat{q}\}$ or $\operatorname{NE}(\hat{q}) \cup \{\hat{q}\}$) and at least $f_j+1$ points were kept in ${\mathcal{S}}$. Since there was no $m$-up-run at this time, we have $\sum_{i=1}^j (f_i+2) +(f_j+1) \le |{\mathcal{N}}|+|{\mathcal{S}}| < m$. Since this is true for all $j \in [{\texttt{t}}]$ but not for $j=T$ (by Condition \[cond:transition\]), we have ${\texttt{t}}< T$. Since the points played in the game were either in ${\mathcal{N}}\cup {\mathcal{S}}$ which now form an $m$-up-run or the ${\texttt{t}}$ points in ${\mathcal{W}}$, we have $m+{\texttt{t}}$ points played. Thus at most $m+T-1$ turns were taken.
It is also possible that the game ended while in Steps \[Cstep:transitiona\] and \[Cstep:transitionb\], specifically when the last point $\hat{q}$ was played in the “former" case of Step \[Cstep:transitionb\]. Again, the total number of points played is $m+{\texttt{t}}$. Observe that if Step \[Cstep:transitiona\] is reached, then ${\texttt{t}}\le T$. This is because $\sum_{i=1}^{{\texttt{t}}-1} (f_i+2) +(f_{{\texttt{t}}-1}+1) < m$, which cannot be true for ${\texttt{t}}= T+1$ by Condition \[cond:transition\]. Thus at most $m+T$ turns were taken.
Finally, suppose the game ended while in Step \[Cstep:barb\]. First note that all points played outside of $(a_x,b_x) \times (a_y,b_y)$ were played before reaching Step \[Cstep:transitiona\] and totaled $|{\mathcal{N}}|+|{\mathcal{W}}|$, where $|{\mathcal{W}}|={\texttt{t}}-1$ because we exited the $f$-Middling strategy mode. All points played within $(a_x,b_x) \times (a_y,b_y)$ were either a single point added to ${\mathcal{W}}$ in Step \[Cstep:transitionb\] or part of the subgame $A_{m-|{\mathcal{N}}|,3}$ in Step \[Cstep:barb\]. By Lemma \[lemma:m3barbbound\], the subgame took at most $m-|{\mathcal{N}}|+ \left\lfloor\sqrt{\max\{2f_{\texttt{t}}-\frac{15}{4},0\}}+\frac32\right\rfloor +1$ turns since we had $|{\mathcal{U}}| \le f_{\texttt{t}}$ upon entering Step \[Cstep:barb\]. Altogether, then, $(|{\mathcal{N}}|+{\texttt{t}}-1)+(1)+(m-|{\mathcal{N}}|+\left\lfloor\sqrt{\max\{2f_{\texttt{t}}-\frac{15}{4},0\}}+\frac32\right\rfloor +1) = m+{\texttt{t}}+\left\lfloor\sqrt{\max\{2f_{\texttt{t}}-\frac{15}{4},0\}}+\frac32\right\rfloor +1$ points were played. As noted in a previous case, ${\texttt{t}}< T$ because we reached Step \[Cstep:transitiona\]. Thus by Condition \[cond:barb\] at most $m + T +1$ turns were taken.
\[cor:m3UB\] Let $m \ge 3$ be given, and let $T$ be the least integer satisfying the inequality $(T-1)^3 +17(T-1) \ge 6(m-3)$. Then $\operatorname{ESO}(m,3) \le m + T +1$.
Define $f = \{f_i\}_{i=1}^\infty$ by $f_i = \frac{(T-\frac12-i)^2+\frac74}{2}$ for $i \le T-1$ and by $f_i = 0$ for $i \ge T$. Observe that $f$ satisfies Condition \[cond:barb\] from Lemma \[lemma:m3UB\] by definition since the the first $T-1$ terms are the maximum integers satisfying the inequality condition. To see that $f$ satisfies Condition \[cond:transition\], it suffices to observe that $\sum_{i=1}^{T-1} (f_i+2) = \frac16 \left[ (T-1)^3 +17(T-1) \right]$. This implies $\sum_{i=1}^T (f_i+2) +(f_T +1) = \frac16 \left[ (T-1)^3 +17(T-1) \right] +3 \ge m$.
Applying $T = \lceil \sqrt[3]{6(m-3)} \rceil +1$ to Corollary \[cor:m3UB\] yields $$\operatorname{ESO}(m,3) \le m + \lceil \sqrt[3]{6(m-3)} \rceil +2 < m + \sqrt[3]{6(m-3)} +3 < m + \sqrt[3]{6m} +3.$$
Establishing a Tighter Lower Bound for $\operatorname{ESO}(m,3)$ {#sec:m3LB}
================================================================
To provide a lower bound for $\operatorname{ESO}(m,3)$, we now give a strategy for player B. First, we provide some technical definitions.
*$h(x)$, $h^{\uparrow n}(x)$, $h_{\downarrow n}(x)$ given ${\mathcal{C}}$, $c$, and $d$*\
Consider a real number $x \in (0,1)$, an integer $n$, an up-run ${\mathcal{C}}$ with distinct $y$-values, and real numbers $0 \le c < d \le 1$.
We define $h(x)$ to be a $y$-value such that $c<y<d$ and such that ${\mathcal{C}}\cup\{(x,y)\}$ is an up-run with distinct $y$-values. Let ${\mathcal{C}}= \{ p_1, \dots, p_t \}$ and let $p_0 = (0,0)$ and $p_{t+1} = (1,1)$. Let $j = \max\{ i : x(p_i) < x\}$. If $c \ge y(p_{j+1})$ or $d \le y(p_j)$, then $h(x)$ is undefined. Otherwise, let $h(x) = \max\big\{c, y(p_j)\big\} + \big(x-x(p_j)\big) \dfrac{\min\big\{d, y(p_{j+1})\big\} -y(p_j)}{x(p_{j+1}) - \max\big\{c, y(p_j)\big\} }$.
We define $h^{\uparrow n}(x)$ to be a $y$-value $n$ notches above $\big( x, h(x) \big)$ with respect to ${\mathcal{C}}$ such that $c \le y < d$. Let $(a,b)$ be the row $\max\{0,n\}$ notches above $\big( x, h(x) \big)$ with respect to ${\mathcal{C}}$. If this row does not exist or if $a \ge d$, then $h^{\uparrow n}(x)$ is undefined. Otherwise, we define $h^{\uparrow n}(x) = \max \left\{ c, ~ \dfrac{a + \min\{b,d\} }{2} \right\}$.
Let $h_{\downarrow n}(x)$ be defined similarly, but to be a $y$-value $n$ notches below $\big( x, h(x) \big)$ with respect to ${\mathcal{C}}$ such that $c < y \le d$.
*$w$-Fracturing Strategy*\
Let $w$ be a given positive integer. In a game of $A_{m,3}$, follow the steps below as player B. Player B will decide whether to play on the line $y=x$ to extend a central up-run ${\mathcal{C}}$ or to play above (*fracturing left*) or below (*fracturing right*) ${\mathcal{C}}$. When doing the latter, player B creates *wastebins* ${\mathcal{W}}_j$ and corresponding intervals $W_j$. The sets ${\mathcal{B}}_j$ are the *banked up-runs*, the portions of the central up-run ${\mathcal{C}}$ which become fixed when player B fractures left or right. The $x$-values $a \le a' \le a'' \le b'' \le b' \le b$ are used to decide whether to add to ${\mathcal{C}}$, create a new wastebin, or add to an existing wastebin. The $y$-values $c<d$ are used to ensure the wastebins form an up-run. The function $\varphi$ encodes the decision that player B makes.
1. (Initialize) Define $z_j = \frac{(w-j)(w-j+1)}{2} +1$ for $1 \le j \le w-1$ and $z_j = 1$ for $j \ge w$. Let $a = a' = a'' = 0 = c$, let $b'' = b' = b = 1 = d$, and let ${\mathcal{C}}= {\mathcal{C}}_1 = \varnothing$. Define $\varphi(x) := x$ for all $x \in (0,1)$, and let $i = 1$.
2. \[Fstep:nextmove\] (Choose $y$) Assuming the game is not over, player A chooses an $x$-value, say $\hat{x}$. Choose the $y$-value $\varphi(\hat{x})$ to make the point $q$. If $\hat{x} < a$ or $b < \hat{x}$, then go to Step \[Fstep:previousfracture\]. If $a < \hat{x} < a''$, then go to Step \[Fstep:leftfracture\]. If $b'' < \hat{x} < b$, then go to Step \[Fstep:rightfracture\]. Else, $a'' < \hat{x} < b''$; add $q$ to ${\mathcal{C}}$ and ${\mathcal{C}}_i$ and go to Step \[Fstep:update2\].
3. \[Fstep:previousfracture\] (Previous fracture) In this case, we must have $\hat{x} \in W_j$ for some $j \in [i-1]$. Add $q$ to ${\mathcal{W}}_j$ and return to Step \[Fstep:nextmove\].
4. \[Fstep:leftfracture\] (Fracture left) Let $\ell = \min_{p \in \operatorname{SE}(q)} x(p)$ and $r = \max_{p \in \operatorname{SE}(q)} x(p)$. Let $W_i = (a, \ell )$, let ${\mathcal{W}}_i = \{q\}$, and let ${\mathcal{B}}_i = \{ p \in {\mathcal{C}}_i : x(p) \le r \}$. Then redefine $a = \ell$, $a' = r$, and $c = y(q)$. Go to Step \[Fstep:update1\].
5. \[Fstep:rightfracture\] (Fracture right) Let $\ell = \min_{p \in \operatorname{NW}(q)} x(p)$ and $r = \max_{p \in \operatorname{NW}(q)} x(p)$. Let $W_i = (r, b )$, let ${\mathcal{W}}_i = \{q\}$, and let ${\mathcal{B}}_i = \{ p \in {\mathcal{C}}_i : \ell \le x(p) \}$. Then redefine $b' = \ell$, $b = r$, and $d = y(q)$. Go to Step \[Fstep:update1\].
6. \[Fstep:update1\] (Begin updating) Increment $i$. Define ${\mathcal{C}}_i = {\mathcal{C}}_{i-1} \setminus {\mathcal{B}}_{i-1}$. Go to Step \[Fstep:update2\].
7. \[Fstep:update2\] (Finish updating) Redefine $a''$ to be the maximum value of $x(p)$ from $p \in {\mathcal{C}}_i$ such that $|\operatorname{NE}(p) \cap {\mathcal{C}}_i| \ge z_i -1$, or to be $a'$ if there is no such $p$. Similarly, redefine $b''$ to be the minimum value of $x(p)$ from $p \in {\mathcal{C}}_i$ such that $|\operatorname{SW}(p) \cap {\mathcal{C}}_i| \ge z_i -1$, or to be $b'$ if there is no such $p$. Now, we redefine $\varphi$ on $(a,b)$: $$\varphi(x) =
\begin{cases}
h^{\uparrow (w-i+1)}(x), & a < x < a''\\
h(x), & a'' < x < b''\\
h_{\downarrow (w-i+1)}(x), & b'' < x < b.
\end{cases}$$ The definition of $\varphi$ on $(0,a) \cup (b,1)$ remains unchanged. Return to Step \[Fstep:nextmove\].
\[lem:fracturing\] If player B uses the $w$-Fracturing Strategy in a game of $A_{m,k}$, then we have the following:
1. \[lem:fracturing:CUB\] $|{\mathcal{C}}_j| \le 2z_j -1$ always for all $j$.
2. \[lem:fracturing:BUB1\] If $j<w$ and ${\mathcal{B}}_j$ is defined when $a < \hat{x} < a'$ or $b' < \hat{x} < b$, then $|{\mathcal{B}}_j| \le w-j$.
3. \[lem:fracturing:BUB2\] If $j<w$ and ${\mathcal{B}}_j$ is defined when $a' < \hat{x} < a''$ or $b'' < \hat{x} < b'$, then $|{\mathcal{B}}_j| \le z_j +w -j$.
4. \[lem:fracturing:CLB\] After redefining $a''$ and $b''$ in Step \[Fstep:update2\], if $i \le w$ and $a < a''$ or $b'' < b$ then $|{\mathcal{C}}_i| \ge z_i$.
5. \[lem:fracturing:WellDefined\] The $w$-Fracturing Strategy is well-defined.
6. \[lem:fracturing:No3down\] The game does not end with a 3-down-run.
(\[lem:fracturing:CUB\]) Every turn, ${\mathcal{C}}_i$ is newly defined or increases by at most one. If $|{\mathcal{C}}_i| = 2z_i-1$, then $a'' = b''$ and no more points can be added to ${\mathcal{C}}_i$. Once ${\mathcal{C}}_{j+1}$ is defined, the set ${\mathcal{C}}_j$ never changes.
(\[lem:fracturing:BUB1\]) Suppose without loss of generality that ${\mathcal{B}}_j$ is defined when $a < \hat{x} < a'$. Since $\hat{x} < a'$, all points in ${\mathcal{B}}_j$ are to the right of $q$, and there is at least one point in $\operatorname{SE}(q) \cap {\mathcal{C}}$ in ${\mathcal{B}}_{j'}$ for some $j' < j$. Since $q$ was placed $w-j+1$ notches above $\big( x, h(x) \big)$, we have $|{\mathcal{B}}_j| = |\operatorname{SE}(q) \cap {\mathcal{C}}_j| \le (w-j+1)-1 = w-j$.
(\[lem:fracturing:BUB2\]) Suppose without loss of generality that ${\mathcal{B}}_j$ is defined when $a' < \hat{x} < a''$. Since $a' < \hat{x}$, there may be some points in ${\mathcal{B}}_j$ to the left of $q$; by (\[lem:fracturing:CUB\]) there are at most $z_j -1$ of these. Together with the $w-j+1$ points in ${\mathcal{B}}_j$ to the right of $q$, we have $|{\mathcal{B}}_j| \le (z_j -1) + (w-j+1) = z_j +w-j$.
(\[lem:fracturing:CLB\]) If $a' < a''$ or $b'' < b'$, then $|{\mathcal{C}}_i| \ge z_i$ by redefinition of $a''$ and $b''$. We now proceed by induction on $i$. When $i=1$, we must have $a' < a''$ or $b'' < b'$ since $a=a'=0$ and $b'=b=1$. So suppose $2 \le i \le w$ and that $|{\mathcal{C}}_j| \ge z_j$ for all $j <i$. Since the size of ${\mathcal{C}}_i$ never decreases while $i$ is constant, suppose that ${\mathcal{C}}_i$ has just been defined in Step \[Fstep:update1\] after the point $q$ has been played. If $a' < x(q) < a''$ or $b'' < x(q) < b'$, then we are done; suppose that $a < x(q) < a'$ or $b' < x(q) < b$. Then by (\[lem:fracturing:BUB1\]) we have $|{\mathcal{C}}_i| = |{\mathcal{C}}_{i-1} \setminus {\mathcal{B}}_{i-1}| \ge z_{i-1} - [w-(i-1)] = z_i$.
(\[lem:fracturing:WellDefined\]) It suffices to show that in Step \[Fstep:update2\], $h^{\uparrow(w-i+1)} (x)$ is defined for all $a < x < a''$. If $a<x<a'$, then $h^{\uparrow(w-i+1)} (x)$ is defined if $|{\mathcal{C}}_i| \ge w-i$. Since $z_j > w-j$ for all $j$, this follows from (\[lem:fracturing:CLB\]). If $a'<x<a''$, then $h^{\uparrow(w-i+1)} (x)$ is defined if $|\operatorname{NE}\big(x, h(x)\big) \cap {\mathcal{C}}_i| \ge w-i+1$. Since $z_j \ge w-j+1$ for all $j$, this follows from the definition of $a''$.
(\[lem:fracturing:No3down\]) We show that ${\mathcal{P}}$ can be partitioned into two up-runs ${\mathcal{C}}$ and $\bigcup_{j=1}^i {\mathcal{W}}_j$. Since $\varphi$ is nondecreasing on $W_j$ when $W_j$ is defined and is not redefined on $W_j$, each wastebin ${\mathcal{W}}_j$ is an up-run. The use of $c$ and $d$ in the definitions of $h^{\uparrow n} (x)$ and $h_{\downarrow n} (x)$, including the fact that $c < d$ always, guarantees that $\bigcup_{j=1}^i {\mathcal{W}}_j$ is an up-run.
\[prop:lowerbound\] If player B adopts the $w$-Fracturing Strategy in a game of $A_{m,3}$ where $(w+1)^3 -(w+1) < 6m$, then at least $m+w$ total points will be played.
Suppose we are at the end of such a game. Then by Lemma \[lem:fracturing\](\[lem:fracturing:No3down\]), there is some $m$-up-run ${\mathcal{U}}$. Either ${\mathcal{U}}$ intersects some wastebin set ${\mathcal{W}}_j$, or else ${\mathcal{U}}= {\mathcal{C}}$.
**Case 1:** ${\mathcal{U}}$ intersects ${\mathcal{W}}_j$ for some $j \in [i-1]$. Let $q_j \in {\mathcal{U}}\cap {\mathcal{W}}_j$. None of the points in $\operatorname{NW}(q_j) \cup \operatorname{SE}(q_j) \subseteq {\mathcal{C}}$ can be in ${\mathcal{U}}$; by virtue of $q_j$ being added to ${\mathcal{W}}_j$, we know that $|\operatorname{NW}(q_j) \cup \operatorname{SE}(q_j)| \ge w-j+1$. Now, for each $\alpha \in [j-1]$, select some $q_\alpha \in {\mathcal{W}}_\alpha$ and let $p_\alpha = \operatorname*{argmin}_{p \in \operatorname{NW}(q_\alpha) \cup \operatorname{SE}(q_\alpha)} |x(p) - x(q_\alpha)|$. From each pair $\{q_\alpha, p_\alpha\}$, at most one point can be in ${\mathcal{U}}$. Observe that for each pair of sets ${\mathcal{W}}_\alpha, {\mathcal{W}}_{\alpha'}$ with $\alpha, \alpha' \in [i-1]$, there is a point $\hat{p} \in {\mathcal{C}}$ such that $x(q) < x(\hat{p}) < x(q')$ or $x(q') < x(\hat{p}) < x(q)$ for all $q \in W_\alpha$ and $q' \in W_{\alpha'}$. Hence $p_1, \dots, p_{j-1}$ are distinct points in ${\mathcal{C}}$ (not necessarily ordered by $x$-values), and none of them are in $\operatorname{NW}(q_j) \cup \operatorname{SE}(q_j)$. Thus there are at least $j-1$ points in $\{q_1,p_1, \dots, q_{j-1},p_{j-1} \}$ that are not in ${\mathcal{U}}$. Counting the $m$ points in ${\mathcal{U}}$ and at least $(w-j+1)+(j-1)$ points we know not to be in ${\mathcal{U}}$, we have at least $m+w$ points total.
**Case 2:** ${\mathcal{U}}= {\mathcal{C}}$. Since the wastebin sets ${\mathcal{W}}_j$ have been defined for all $j \in [i-1]$, are nonempty, and do not intersect ${\mathcal{C}}$, we show that $i-1 \ge w$. Since in this case the game cannot be over until $m = |{\mathcal{C}}| = \left|\left(\bigcup_{j \in [i-1]} {\mathcal{B}}_j \right) \cup {\mathcal{C}}_i\right|$, we suppose $i=w$ and show that $\left|\left(\bigcup_{j \in [w-1]} {\mathcal{B}}_j \right) \cup {\mathcal{C}}_w\right| < m$. This implies $i > w$. By Lemma \[lem:fracturing\](\[lem:fracturing:BUB2\]) we have $|{\mathcal{B}}_j| \le z_j + w -j$ for all $j \in [w-1]$. Also, by Lemma \[lem:fracturing\](\[lem:fracturing:CUB\]) we have that $|{\mathcal{C}}_w| \le 2z_w -1$. Hence $\left|\left(\bigcup_{j \in [w-1]} {\mathcal{B}}_j \right) \cup {\mathcal{C}}_w\right| \le \sum_{j=1}^{w-1} (z_j + w - j) + 2z_w -1 = \frac16 \left[(w+1)^3 -(w+1)\right] <m$.
Applying $w = \lfloor \sqrt[3]{6m} \rfloor -1$ to Proposition \[prop:lowerbound\] yields $$\operatorname{ESO}(m,3) \ge m + \lfloor \sqrt[3]{6m} \rfloor -1 > m + \sqrt[3]{6m} -2.$$
Concluding Remarks {#sec:conclusion}
==================
As already shown in Section \[sec:mkLB\], one may imagine variations on the Erdős-Szekeres on-line game. For example, the game $B_{m,k}$ could be generalized so that player B plays from the set $[s]$. Another variation in which player B plays from $[s]$ would restrict the game to playing $s$ points total and player B cannot repeat previous choices from $[s]$. What is the maximum length up-run or down-run that player A can force in the $s$ turns?
Alternatively, one could consider additional target patterns. For example, consider the $(s,t)$-${\mathcal{V}}$ configuration: a point $p$ which is the left-most point of an $s$-up-run and a $t$-down-run, or the right-most point of an $s$-up-run and a $t$-down-run. What is the minimum number of moves in which player A can force an $m$-up-run, a $k$-down run, or an $(s,t)$-${\mathcal{V}}$? When $s$ and $t$ are relatively smaller than $m$ and $k$, respectively, then this additional target pattern is likely to disrupt strategies which would be optimal for player B in a game of $A_{m,k}$.
Another possibility is to generalize to more than two dimensions for the board space. In $R^n$, say that two points $(a_1, \dots, a_n)$ and $(b_1, \dots, b_n)$ are increasing if $a_i \le b_i$ for all $i \in [n]$ or $b_i \le a_i$ for all $i \in [n]$, and say that they are decreasing otherwise. Let a set of $m$ pairwise increasing points be called a chain and a set of $k$ pairwise decreasing points be called a $k$-anti-chain. For $s<n$, let player A choose the first $s$ coordinates of a point, and then player B choose the last $n-s$ coordinates of the point. In how many turns can player A force an $m$-chain or a $k$-anti-chain?
Variations of the Erdős-Szekeres on-line game might not have symmetry. For example, $B(m,k)$ and $B(k,m)$ are not necessarily equal. While $B(m,2) = m$, Proposition \[prop:BgameUB\] implies $B(2,m) \le \lfloor \frac{m}{2}\rfloor +1$.
Nevertheless, we believe that $\operatorname{ESO}(m,k)$ and $B(m,k)$ are very closely related, encapsulated in the following conjecture, which is stronger than Conjecture \[con\] mentioned in the introduction.
For all $m$ and $k$, $|\operatorname{ESO}(m,k)-B(m,k)|=o(mk)$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This collaboration began as part of the 2016 Rocky Mountain–Great Plains Graduate Research Workshop in Combinatorics, supported in part by NSF-DMS Grants \#1604458, \#1604773, \#1604697 and \#1603823. The authors are grateful for the workshop, which provided the opportunity for fruitful conversations. The authors thank Gavin King, a workshop participant, for posing the “Online Happy Ending Problem” which inspired this project.
Appendix {#appendix .unnumbered}
========
**Example 1**
Now we give an example of how a game of $A_{m,3}$ might proceed when player A uses the $f$-Middling strategy mode, with $f = (3,2,1,0,0,0,\dots)$. In the figures that follow, the labels of each point indicate the order in which they were played.
Until player B deviates from ${\mathcal{S}}$, we have ${\texttt{t}}=0$ and the points look something as follows:
(0,0) rectangle (12,10); at (1,9.5) [${\texttt{t}}= 0$]{};
\(1) at (2,1) ; (4) at (3,2) ; (5) at (4,3) ; (7) at (5,4) ; (9) at (6,5) ; (8) at (7,6) ; (6) at (8,7) ; (3) at (9,8) ; (2) at (10,9) ;
at (\[shift=[(120:.5)]{}\]1) [1]{}; at (\[shift=[(-60:.5)]{}\]2) [2]{}; at (\[shift=[(-60:.5)]{}\]3) [3]{}; at (\[shift=[(120:.5)]{}\]4) [4]{}; at (\[shift=[(120:.5)]{}\]5) [5]{}; at (\[shift=[(-60:.5)]{}\]6) [6]{}; at (\[shift=[(120:.5)]{}\]7) [7]{}; at (\[shift=[(-60:.5)]{}\]8) [8]{}; at (\[shift=[(120:.5)]{}\]9) [9]{};
\
At this point we have $(a_x,b_x) \times (a_y,b_y) = (0,1) \times (0,1)$, ${\mathcal{S}}= \{1,2,3,4,5,6,7,8,9\}$, and ${\mathcal{N}}= {\mathcal{W}}= \varnothing$.
Suppose that on the next turn, player B deviates from ${\mathcal{S}}$ to form the point 10:
(0,0) rectangle (12,10); at (2,9.5) [${\texttt{t}}= 1$, $f_{\texttt{t}}= 3$]{}; at (9.5,1.8) [Active segment:]{}; at (9.5,1) [$\big( 0,x(9) \big) \times \big( 0,y(9) \big)$]{};
\(1) at (2,1) ; (4) at (3,2) ; (5) at (4,3) ; (7) at (5,4) ; (9) at (6,5) ; (10) at (6.5,4.5) ; (8) at (7,6) ; (6) at (8,7) ; (3) at (9,8) ; (2) at (10,9) ;
(0,0) rectangle (9);
at (\[shift=[(120:.5)]{}\]1) [1]{}; at (\[shift=[(-60:.5)]{}\]2) [2]{}; at (\[shift=[(-60:.5)]{}\]3) [3]{}; at (\[shift=[(120:.5)]{}\]4) [4]{}; at (\[shift=[(120:.5)]{}\]5) [5]{}; at (\[shift=[(-60:.5)]{}\]6) [6]{}; at (\[shift=[(120:.5)]{}\]7) [7]{}; at (\[shift=[(-60:.5)]{}\]8) [8]{}; at (\[shift=[(120:.5)]{}\]9) [9]{}; at (\[shift=[(0:.6)]{}\]10) [10]{};
(6.5,0) edge node (6.5,10);
\
Now increment ${\texttt{t}}$ to 1; since $|\operatorname{SW}(10)\cap{\mathcal{S}}| > f_{\texttt{t}}$ and $|\operatorname{NE}(10)\cap{\mathcal{S}}| > f_{\texttt{t}}$, player A keeps playing the $f$-Middling strategy mode. At this point we have $(a_x,b_x) \times (a_y,b_y) = (0,x(9)) \times (0,y(9))$, ${\mathcal{S}}= \{1,4,5,7\}$, ${\mathcal{N}}= \{9,8,6,3,2\}$, and ${\mathcal{W}}= \{10\}$.
Player A keeps playing in a middlemost column of ${\mathcal{S}}$ until player B deviates again, say on turn 16:
(0,0) rectangle (12,10); at (2,9.5) [${\texttt{t}}= 2$, $f_{\texttt{t}}= 2$]{}; at (6.35,1.8) [Active segment:]{}; at (6.35,1) [$\big( x(13),x(9) \big) \times \big( y(13),y(9) \big)$]{};
\(1) at (0.5,1) ; (4) at (1.25,1.5) ; (11) at (2,2) ; (14) at (3,3) ; (16) at (3.2,5.5) ; (15) at (4,4) ; (13) at (5,5) ; (12) at (6,6) ; (5) at (7.25,6.75) ; (7) at (8.5,7.5) ;
\(9) at (9.5,8.5) ; (10) at (9.75,8) ; (8) at (10,8.75) ; (6) at (10.5,9) ; (3) at (11,9.25) ; (2) at (11.5,9.5) ;
(0,0) rectangle (9); (13) rectangle (9);
at (\[shift=[(120:.5)]{}\]1) [1]{}; at (\[shift=[(-60:.4)]{}\]2) [2]{}; at (\[shift=[(-60:.4)]{}\]3) [3]{}; at (\[shift=[(120:.5)]{}\]4) [4]{}; at (\[shift=[(-60:.5)]{}\]5) [5]{}; at (\[shift=[(-60:.4)]{}\]6) [6]{}; at (\[shift=[(-60:.5)]{}\]7) [7]{}; at (\[shift=[(-60:.4)]{}\]8) [8]{}; at (\[shift=[(120:.4)]{}\]9) [9]{}; at (\[shift=[(-30:.4)]{}\]10) [10]{}; at (\[shift=[(120:.6)]{}\]11) [11]{}; at (\[shift=[(-30:.6)]{}\]12) [12]{}; at (\[shift=[(-30:.6)]{}\]13) [13]{}; at (\[shift=[(120:.6)]{}\]14) [14]{}; at (\[shift=[(-30:.6)]{}\]15) [15]{}; at (\[shift=[(120:.6)]{}\]16) [16]{};
(3.2,0) edge node (3.2,8.5);
\
Now increment ${\texttt{t}}$ to 2; since $|\operatorname{SW}(16)\cap{\mathcal{S}}| > f_{\texttt{t}}$ and $|\operatorname{NE}(16)\cap{\mathcal{S}}| > f_{\texttt{t}}$, player A keeps playing the $f$-Middling strategy mode. At this point we have $(a_x,b_x) \times (a_y,b_y) = (x(13),x(9)) \times (y(13),y(9))$, ${\mathcal{S}}= \{5,7,12\}$, ${\mathcal{N}}= \{9,8,6,3,2,1,4,11,14,15,13\}$, and ${\mathcal{W}}= \{10,16\}$.
Player A keeps playing in a middlemost column of ${\mathcal{S}}$ until player B deviates again, say on turn 19:
(0,0) rectangle (12,10); at (2,9.5) [${\texttt{t}}= 3$, $f_{\texttt{t}}= 1$]{}; at (6,1) [(Exit strategy mode.)]{};
\(1) at (0.5,0.5) ; (4) at (1,1) ; (11) at (1.5,1.5) ; (14) at (2,2) ; (16) at (2.25,3.5) ; (15) at (2.5,2.5) ; (13) at (3,3) ;
\(12) at (4,4) ; (5) at (5,4.8) ; (18) at (6,5.6) ; (19) at (6.75,7) ; (17) at (7.5,6.4) ; (7) at (8.5,7.5) ;
\(9) at (9.5,8.5) ; (10) at (9.75,8) ; (8) at (10,8.75) ; (6) at (10.5,9) ; (3) at (11,9.25) ; (2) at (11.5,9.5) ;
\(13) rectangle (9);
at (\[shift=[(120:.4)]{}\]1) [1]{}; at (\[shift=[(-60:.4)]{}\]2) [2]{}; at (\[shift=[(-60:.4)]{}\]3) [3]{}; at (\[shift=[(120:.4)]{}\]4) [4]{}; at (\[shift=[(-60:.5)]{}\]5) [5]{}; at (\[shift=[(-60:.4)]{}\]6) [6]{}; at (\[shift=[(-60:.5)]{}\]7) [7]{}; at (\[shift=[(-60:.4)]{}\]8) [8]{}; at (\[shift=[(120:.4)]{}\]9) [9]{}; at (\[shift=[(-30:.4)]{}\]10) [10]{}; at (\[shift=[(120:.4)]{}\]11) [11]{}; at (\[shift=[(-30:.6)]{}\]12) [12]{}; at (\[shift=[(-30:.4)]{}\]13) [13]{}; at (\[shift=[(120:.4)]{}\]14) [14]{}; at (\[shift=[(-30:.4)]{}\]15) [15]{}; at (\[shift=[(120:.4)]{}\]16) [16]{}; at (\[shift=[(-30:.6)]{}\]17) [17]{}; at (\[shift=[(-300:.6)]{}\]18) [18]{}; at (\[shift=[(120:.6)]{}\]19) [19]{};
(6.75,3) edge node (6.75,8.5);
\
Now increment ${\texttt{t}}$ to 3; since $|\operatorname{NE}(19)\cap{\mathcal{S}}| \le f_{\texttt{t}}$, player A now exits the $f$-Middling strategy mode. At this point we have $(a_y,b_y) = (x(13),x(9)) \times (y(13),y(9))$, ${\mathcal{S}}= \{5,7,12,17\}$, ${\mathcal{N}}= \{9,8,6,3,2,1,4,11,14,15,13\}$, and ${\mathcal{W}}= \{10,16\}$.
**Example 2**
Now we illustrate how part of a game of $A_{m,3}$ might proceed when player A uses the $w$-Barb strategy mode, with $w = 4$.
Suppose the game is underway with the following point set. Then player A chooses the column between $r_1$ and $\hat{r}_1$:
(0,0) rectangle (12,10);
\(1) at (1,1) ; (2) at (1.5,2) ; (3) at (2,3) ; (4) at (2.5,4) ; (5) at (3,5) ; (6) at (3.7,6) ; (7) at (4.5,6.7) ;
\(8) at (6,7.5) ; (9) at (7,7.8) ; (10) at (8,8.1) ; (11) at (9,8.4) ; (12) at (10,8.7) ; (13) at (11,9) ; (q1) at (8.5,7.1) ;
at (\[shift=[(90:.5)]{}\]8) [$r_1$]{}; at (\[shift=[(90:.5)]{}\]9) [$\hat{r}_1$]{}; at (\[shift=[(0:.5)]{}\]q1) [$q_1$]{};
at (11,8) [${\mathcal{B}}_1$]{}; at (2,5) [${\mathcal{U}}$]{};
(6.5,0) edge node (6.5,10);
(q1) edge node (8); (8) edge node (9); (9) edge node (10); (10) edge node (11); (11) edge node (12); (12) edge node (13);
\(1) edge node (2); (2) edge node (3); (3) edge node (4); (4) edge node (5); (5) edge node (6); (6) edge node (7);
\
Then suppose player B chooses a $y$-value that results in the point $q$ which is 3 notches below $r_1$ with respect to ${\mathcal{U}}$. Since $3 \ge w-1$, we go to Step \[Bstep:stepdown\] and then return to Step \[Bstep:play\] with the following point set. Then player A chooses the column between $r_2$ and $\hat{r}_2$:
(0,0) rectangle (12,10);
\(1) at (1,1) ; (2) at (1.5,2) ; (3) at (2,3) ; (4) at (2.5,4) ;
\(5) at (3,5) ; (6) at (3.7,6) ; (7) at (4.5,6.7) ;
\(8) at (6,7.5) ; (9) at (7,7.8) ; (10) at (8,8.1) ; (11) at (9,8.4) ; (12) at (10,8.7) ; (13) at (11,9) ; (q1) at (8.5,7.1) ; (q2) at (6.5,4.5) ;
at (\[shift=[(90:.5)]{}\]5) [$r_2$]{}; at (\[shift=[(90:.5)]{}\]6) [$\hat{r}_2$]{}; at (\[shift=[(0:.5)]{}\]q1) [$q_1$]{}; at (\[shift=[(0:.5)]{}\]q2) [$q_2$]{};
at (6,6) [${\mathcal{B}}_2$]{}; at (1,5) [${\mathcal{U}}$]{};
(3.35,0) edge node (3.35,10);
(q2) edge node (5); (5) edge node (6); (6) edge node (7); (7) edge node (8); (8) edge node (9); (9) edge node (10); (10) edge node (11); (11) edge node (12); (12) edge node (13);
\
Then suppose player B chooses a $y$-value that results in the point $q$ which is 1 notch below $r_2$ with respect to ${\mathcal{U}}$. Since $1 < w-2$, we go to Step \[Bstep:playbarb\] with the following point set:
(0,0) rectangle (12,10);
\(1) at (1,1) ; (2) at (1.5,2) ; (3) at (2,3) ; (4) at (2.5,4) ;
\(5) at (3,5) ; (6) at (3.7,6) ; (7) at (4.5,6.7) ;
\(8) at (6,7.5) ; (9) at (7,7.8) ; (10) at (8,8.1) ; (11) at (9,8.4) ; (12) at (10,8.7) ; (13) at (11,9) ; (q1) at (8.5,7.1) ;
(q2) at (6.5,4.5) ;
\(q) at (3.35,3.5) ;
at (\[shift=[(90:.5)]{}\]5) [$r_2$]{}; at (\[shift=[(90:.5)]{}\]6) [$\hat{r}_2$]{}; at (\[shift=[(0:.5)]{}\]q1) [$q_1$]{}; at (\[shift=[(0:.5)]{}\]q2) [$q_2$]{}; at (\[shift=[(-20:.5)]{}\]q) [$q$]{}; at (\[shift=[(150:.5)]{}\]4) [$p$]{};
at (6,3) [${\mathcal{B}}$]{};
\(1) edge node (2); (2) edge node (3); (3) edge node (q); (q) edge node (q2); (q) edge node (6);
(q2) edge node (5); (5) edge node (6); (6) edge node (7); (7) edge node (8); (8) edge node (9); (9) edge node (10); (10) edge node (11); (11) edge node (12); (12) edge node (13);
\
Then player A treats $p$ and $q_1$ as a loss and plays the barb ${\mathcal{B}}$ by choosing the column between $q$ and $\hat{r}_2$.
[^1]: Department of Mathematics, University of Denver, Denver, CO 80208; [[email protected], [email protected]]{}.
[^2]: Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO 80217; [[email protected], [email protected]]{}.
[^3]: Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218; [[email protected]]{}.
[^4]: Department of Mathematics, Vanderbilt University, Nashville, TN 37240; [[email protected]]{}.
[^5]: The Python code used to obtain these results in Table \[tab:smallresults\] can be viewed online at [math.ucdenver.edu/\~nelsenl/projects/ErdosSzekeresOnline](http://math.ucdenver.edu/~nelsenl/projects/ErdosSzekeresOnline).
[^6]: One can replace the rules for going to Steps \[Bstep:stepdown\] and \[Bstep:playbarb\] with rules that compare $d$ to $|\operatorname{SW}(q)|$ rather than comparing $d$ to $w-i$. This would not improve the result of Lemma \[lemma:m3barbbound\], but would be a better strategy for player A when player B does not play optimally.
|
---
author:
- Barbara Betz
- and Miklos Gyulassy
title: 'Erratum: Constraints on the Path-Length Dependence of Jet Quenching in Nuclear Collisions at RHIC and LHC'
---
The Figures 4 (b1) and (b2) and Figures 9 (b1) and (b2) were calculated incorrectly. See replacement Figures \[Fig1\] and \[Fig2\] below. The corrected figures imply that:
- The nuclear modification factor $R_{AA}$ at LHC energies for pure elastic energy loss \[with $(a,b,c,q)=(0,0,2,-1)$\] in the new Fig. \[Fig1\] is now found to be compatible with both RHIC and LHC energies for $\kappa_{\rm RHIC}=\kappa_{\rm LHC}$. The jet $v_2$-asymmetry is, however, still a factor of $\sim 2$ too low.
- The SLTc scenario assuming a radiative jet-energy loss coupling $\kappa(T)$ that is enhanced by a factor of three in the transition range of $113 < T< 173$ MeV [@Liao:2008dk] does in fact describe the LHC $R_{AA}$-data but is sensitive to the bulk hydrodynamic background temperature field. For this $\kappa(T)$ model the RL viscous hydro field [@Luzum:2008cw] is prefered by both $R_{AA}$ and $v_2$.
![Corrected Figure 4 (b1) and (b2) of the original publication. Panel (a) shows data for the pion nuclear modification factor $R_{AA}$ from ALICE and CMS , while panel (b) depicts the high-$p_T$ elliptic flow as extracted from ALICE , CMS , and ATLAS . The model calculations assume elastic energy loss, $dE/dx=\kappa T^2$, with [*no*]{} energy-loss fluctuations using different bulk hydro temperature flow fields at LHC energies: viscous $\eta/s=0.08$ VISH2+1 (solid), viscous $\eta/s=0.08$ RL Hydro (dashed-dotted), and the $v_\perp=0.6$ blast wave model (dotted). []{data-label="Fig1"}](Fig14.pdf){width="6in"}
![Corrected Figure 9 (b1) and (b2) of the original publication. The pion nuclear modification factor and the high-$p_T$ elliptic flow are compared to the SLTc energy loss model, $dE/dx=\kappa(T) x T^3$, with enhanced coupling near $T_c$ , no energy-loss fluctuations, and different bulk QGP flow fields at LHC energies .[]{data-label="Fig2"}](Fig15.pdf){width="6in"}
[00]{}
C. Shen, U. Heinz, P. Huovinen and H. Song, Phys. Rev. C [**84**]{}, 044903 (2011); Z. Qiu, C. Shen and U. Heinz, Phys. Lett. B [**707**]{}, 151 (2012).
M. Luzum and P. Romatschke, Phys. Rev. C [**78**]{}, 034915 (2008); \[Erratum-ibid. C [**79**]{}, 039903 (2009)\]; Phys. Rev. Lett. [**103**]{}, 262302 (2009).
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B. Abelev [*et al.*]{} \[ALICE Collaboration\], Phys. Lett. B [**720**]{}, 52 (2013).
B. Abelev [*et al.*]{} \[ALICE Collaboration\], Phys. Lett. B [**719**]{}, 18 (2013).
S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Eur. Phys. J. C [**72**]{}, 1945 (2012).
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|
---
address:
- |
Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany\
email: [email protected]
- |
Korteweg de Vries Institute for Mathematics\
University of Amsterdam\
Plantage Muidergracht 24\
1018TV Amsterdam\
The Netherlands\
email: [email protected]
author:
- Bernhard Krötz
- Eric Opdam
title: Analysis on the crown domain
---
[^1]
Introduction
============
Our concern is with harmonic analysis on a Riemannian symmetric space $$X_{\mathbb{R}}=G_{\mathbb{R}}/K_{\mathbb{R}}$$ of the noncompact type. Here $G_{\mathbb{R}}$ denotes a connected reductive algebraic group and $K_{\mathbb{R}}$ is a maximal compact subgroup thereof.
Given a $K_{\mathbb{R}}$-spherical irreducible unitary representation $(\pi, \H_\pi)$ of $G_{\mathbb{R}}$ with $K_{\mathbb{R}}$-fixed ray $\H_\pi^{K_{\mathbb{R}}-{\rm fix}}={\mathbb{C}}v_K$, we obtain an $G_{\mathbb{R}}$-equivariant continuous map
$$i_\pi: X_{\mathbb{R}}\to \H_\pi, \ \ gK_{\mathbb{R}}\mapsto \pi(g) v_K .$$ We assume that $\pi\neq {\bf 1}$ is non-trivial and then $i_\pi$ is injective. The map $i_\pi$ is analytic, hence admits holomorphic extension to a maximal $G_{\mathbb{R}}$-neighborhood $\Xi_\pi$ of $X_{\mathbb{R}}$ in $X_{\mathbb{C}}=G_{\mathbb{C}}/K_{\mathbb{C}}$. It is a remarkable fact that $\Xi_\pi$ is independent of the choice of $\pi\neq {\bf 1}$ ([@KSI],[@KSII], [@KOS]) and hence defines a natural domain $\Xi$ in $X_{\mathbb{C}}$, referred to as the [*crown domain*]{}. A result in this paper determines the precise growth rate of $\|i_\pi\|$ when approaching the boundary of $\Xi$.
We have to clarify what we understand by the term “approaching the boundary”. The crown domain admits a natural Shilov-type boundary [@GKI], referred to as the distinguished boundary $\partial_d\Xi$ of $\Xi$. In a first step we give a simple description of $\partial_d\Xi$ in terms of the affine Weyl group, hereby extending and unifying results from [@GKI]. At this point it is relevant that the $G_{\mathbb{R}}$-equivalence classes in $\partial_d\Xi$ are described by a finite union of Weyl group orbits.
Given a distinguished boundary point $z\in \partial_d\Xi$ and $(z_n)\subset \Xi$ a sequence converging radially to $z$ we are interested in the growth of $\|i_\pi (z_n)\|$ in terms of ${\rm dist}(z_n,z)$. We determine
- For fixed $z_n$, sufficiently close to $z$, optimal lower exponential bounds for $\|i_\pi(z_n)\|$ in terms of the parameter of $\pi$ and ${\rm dist}(z_n,z)$;
- For fixed $\pi$, the precise blow up rate of $\|i_\pi(z_n)\|$ for $z_n\to z$ in terms of ${\rm dist}(z_n,z)$.
We use these results to prove estimates for Maaß automorphic forms. For example, a theorem of Langlands asserts that cuspidal automorphic forms are of rapid decay [@La], [@HC]. An unpublished theorem of J. Bernstein goes beyond and asserts exponential decay. In this paper we establish precise exponential decay rates. The basic idea of proof goes back to J. Bernstein and our contribution lies in a incorporation of geometric methods and hard estimates.[^2] In particular, we show that the crown domain admits a natural parameterization by unipotent $G_{\mathbb{R}}$-orbits which makes Bernstein’s idea work out efficiently.
Finally we wish to point out that we make a detailed study of proper actions of $G_{\mathbb{R}}$ on $X_{\mathbb{C}}$ in Section 4. As a byproduct of these investigations we obtain a complex geometric classification of the different series of representations of the group $G=\Sl(2,{\mathbb{R}})$ (cf. Theorem \[th=sl2r\] below).
[*Acknowledgment:*]{} We express our gratitude to Joseph Bernstein for generously sharing his insights with us. We thank the number theorists Erez Lapid and Andre Reznikov for pointing out interesting questions and their useful hints to the literature. We are indebted to Philip Foth who did an elegant matrix computation for us. Finally we thank Joachim Hilgert for asking a good question during one of our talks.
The complex crown and its distinguished boundary
================================================
This section is divided into two parts. First we recall the definition and basic properties of the complex crown $\Xi$ of a Riemannian symmetric space $X$ (see [@KSII] for a comprehensive account). Second we shall unify and extend results from [@GKI] on the distinguished boundary of $\Xi$.
The complex crown
-----------------
Let $G$ be a connected, real semisimple, noncompact Lie group. Write $\gf$ for the Lie algebra of $G$ and denote by $\gf_{\mathbb{C}}$ its complexification. We fix a maximal compact subalgebra $\kf\subset \gf$ and set $K=\exp(\kf)$.
Let us denote by $G_{\mathbb{C}}$ the universal complexification of $G$ and by $\iota: G \to G_{\mathbb{C}}$ the homomorphism sitting over the injection $\gf\hookrightarrow \gf_{\mathbb{C}}$. Write $K_{\mathbb{C}}$ for the analytic subgroup of $G_{\mathbb{C}}$ corresponding to $\kf_{\mathbb{C}}$.
Our concern is with the Riemannian symmetric space $X=G/K$. The complex symmetric space $X_{\mathbb{C}}=G_{\mathbb{C}}/K_{\mathbb{C}}$ naturally acts as a complexification of $X$ and the assignment $gK\mapsto \iota(g)K_{\mathbb{C}}$ identifies $X$ as a totally real submanifold of $X_{\mathbb{C}}$ in a $G$-equivariant way. We denote the base point $eK_{\mathbb{C}}$ of $X_{\mathbb{C}}$ by $x_0$.
\[rem=split\] Let $\gf=\gf_1+\ldots + \gf_l$ be the factorization of $\gf$ in simple Lie algebras and let $\kf=\kf_1+ \ldots+\kf_l$ be the associated splitting for $\kf$. Denote by $G_j$, $K_j$ the analytic subgroups of $G$ corresponding to $\gf_j$, $\kf_j$. Then with $X_j=G_j/ K_j$ there is the equivariant isomorphism $$X\simeq X_1\times \ldots \times X_l\, .$$ In a similar manner (and obvious notation) $$X_{\mathbb{C}}\simeq X_{1,{\mathbb{C}}}\times\ldots \times X_{l,{\mathbb{C}}}\ .$$
In the light of the discussion in Remark \[rem=split\] it is no loss of generality to assume henceforth that $\gf$ is simple.
Let $\gf=\kf\oplus \pf$ be the Cartan decomposition associated to the choice of $\kf$, and choose $\af$ a maximal abelian subspace in $\pf$. The [*complex crown*]{} $\Xi$ of $X$ is by definition $$\Xi=G\exp(i\pi \Omega/ 2). x_0\subset
X_{\mathbb{C}},$$ where $\Omega\subset\af$ is given by $$\Omega=\{Y\in \af\mid {\rm spec}({\rm ad} Y)\subset ]-1, 1
[\}.$$ According to [@AG], $\Xi$ is a $G$-invariant open subdomain of $X_{\mathbb{C}}$ with the $G$-action proper. Actually $\Xi$ is Stein (see [@KSII] and the references therein). Let us point out that $\Xi$ is independent of the choice of the flat $\af$ and therefore naturally attached to $X$.
The set $\Omega$ can be described in terms of the restricted root system $\Sigma=\Sigma(\gf,\af)$ as follows: $$\label{eq:omega}
\Omega=\{Y\in\af\mid|\a(Y)|<1\,\forall\a\in\Sigma\}.$$ In particular we see that $\overline \Omega$ is a compact $W$-invariant polyhedron. Here $W$, as usual, denotes the Weyl group of $\Sigma$.
\[rem=tan\](Realization in the tangent bundle) Set $\Omega^K={\pi\over 2}\Ad (K)\Omega$. As $\Omega$ is an open $W$-invariant convex subset of $\af$, Kostant’s linear convexity theorem implies that $\Omega^K \subset \pf$ is an open $K$-invariant convex subset of $\pf$. Write $TX=G\times_K \pf$ for the tangent bundle of $X$. Notice that $G$ acts properly on $TX$ and that $G\times_K\Omega^K$ is a contractible $G$-equivariant subset of $TX$ (base and fiber are contractible). In [@AG] it was shown that the map $$\label{eq=tan} G\times_K \Omega^K\to \Xi, \ \ [g,Y]\mapsto
g\exp(iY). x_0$$ is a $G$-equivariant diffeomorphism. In particular, $G$ acts properly on $\Xi$ and $\Xi$ is contractible.
In the sequel we write $\tf=i\af$ and let $T=\exp \tf$ be the corresponding torus in $G_{\mathbb{C}}$. Notice that $T_{\mathbb{C}}= A_{\mathbb{C}}=AT$ with $A=\exp \af$. We will also use the notation $T_\Omega=\exp(i\pi \Omega/2)$.
\[rem=bd\] (The boundary of $\Xi$)
[(i)]{} (Semisimple boundary part) The topological boundary $\partial\Xi$ is a complicated union of $G$-orbits. This is because not all $G$-orbits in $\partial\Xi$ meet $T. x_0$. Those which do make up the semisimple (or elliptic) part of the boundary $\partial_s\Xi=
G\exp(i\pi \partial\Omega/ 2). x_0$ of $\Xi$ (see [@AG; @KSII]). Equivalently, $\partial_s\Xi$ describes the closed $G$-orbits in $\partial \Xi$. One knows that each $G$-orbit in $\partial\Xi$ has a a unique semisimple orbit in its closure [@FH], but a satisfactory general description of $\partial \Xi$ is still missing.
[(ii)]{} (Properness) The polyhedron $\Omega$ is maximal with regard to proper $G$-action, i.e. there does not exists a larger connected subset $\tilde \Omega\supset \Omega$ such that $G$ would act properly on $G\exp(i\pi\tilde\Omega/2). x_0$ (cf. [@AG]). We mention that $G$-stabilizers of points in $\exp(i\pi\partial \Omega/2). x_0$ are noncompact subgroups [@AG].
[(iii)]{} (Dependence on isogenies) It follows from (\[eq=tan\]) that $\Xi$ is homeomorphic to $\pf\times \Omega^K$. It means in particular that $\Xi$ only depends on the isogeny class of the connected group $G$. However, the situation becomes different once we start to consider the boundary $\partial \Xi$ of the crown in $X_{\mathbb{C}}$. It turns out that $\partial
\Xi$ is sensitive with regard to the choice of the connected group $G$; for instance $\partial \Xi$ is different for $\SO(n,{\mathbb{C}})$ and its simply connected cover $\SO(n,{\mathbb{C}})^\sim $. We will comment more on that when we will discuss collapsing of boundary orbits below (Example \[collapse\]).
Distinguished and minuscule boundary of the crown {#sub:distbdy}
-------------------------------------------------
The [*distinguished boundary*]{} $\partial_d\Xi$ of the crown, introduced in [@GKI], is defined by $$\partial_d\Xi=G\exp(i\pi\partial_e\Omega/2). x_0\subset \partial\Xi$$ where $\partial_e\Omega$ is the (finite set) of extreme points of the compact polyhedron $\overline \Omega$. In view of this definition and the results of the previous subsection we may and will assume that $\mathfrak{g}$ is simple in this subsection. Let us recall that the distinguished boundary plays the rôle of a noncompact Shilov-type boundary of $\Xi$; one has the following elementary result.
\[prop:sup\]([@GKI]) Let $f$ be a holomorphic function on $\Xi$ which extends to a bounded continuous function on $\overline{\Xi}$. Then $$\operatorname{sup}_{x\in \Xi}(|f(x)|)=
\operatorname{sup}_{x\in \partial_d(\Xi)}(|f(x)|).$$
In [@GKI] a complete characterization of those crowns $\Xi$ was given which admit symmetric spaces as components in $\partial_d\Xi$. Cases relevant for [@GKI] are those $\Sigma$ which are not of type $E_8, G_2$ or $F_4$.
The objective of this section is to give a uniform approach to $\partial_d\Xi$ in the general case. Our first result is a description of $\partial_e\Omega$ in terms of structure theory which is stunningly simple (cf. Theorem \[th=omega\] below). We will define the [*minuscule part*]{} of the distinguished boundary and tie it with the results of [@GKI]. After that we classify the non-symmetric boundary components of $\partial_d\Xi$. Finally we discuss collapsing of distinguished boundary orbits.
Write $\Sigma^l=\{ \alpha: 2\alpha\not\in \Sigma\}$ for the irreducible reduced subsystem of unmultipliable roots in $\Sigma=\Sigma(\mathfrak{g},\mathfrak{a})$. It is clear that $\Sigma^l$ completely describes $\Omega$, i.e., $$\Omega=\{ Y\in \af\mid |\alpha(Y)|<1, \forall \alpha\in \Sigma^l\}
\, .$$ Fix a basis for $\Sigma^l$, say $\Pi=\{ \alpha_1,\ldots, \alpha_n\}$, and write $C\subset \af$ for the closure of the associated Weyl chamber. Let $\beta$ be the highest root corresponding to $\Pi$ and $$\beta=k_1 \alpha_1+ \ldots + k_n \alpha_n$$ its expansion in the simple roots (hence $k_i\in {\mathbb{Z}}_{>0}$). We record the obvious relation $$\label{eq=aff} \overline \Omega\cap C=\{ Y\in
C : \beta(Y)\leq 1\}\, .$$ It means that $ \overline \Omega\cap C$ is a fundamental domain for the affine Weyl group $W^{\rm aff}=W\ltimes Q^\vee$ with $Q^\vee={\rm span}_{\mathbb{Z}}\Sigma^\vee$ the coroot lattice in $\af$ (observe that ${\rm span}_{\mathbb{Z}}\Sigma^\vee={\rm span}_{\mathbb{Z}}(\Sigma^l)^\vee$).
Define $\omega_i\in \af$ by $\alpha_j(\omega_i)=\delta_{ij}$. It is straightforward from (\[eq=aff\]) that $$\label{eq=inc}
\partial_e\Omega\cap C\subset\left \{ \omega_1/ k_1, \ldots,
\omega_n/ k_n\right \}$$ and so $\partial_e \Omega\subset W. \left \{ \omega_1/k_1, \ldots,
\omega_n/ k_n\right\}$ (cf. [@GKI], Lemma 3.17).
In general the inclusion in (\[eq=inc\]) is proper and we have to determine which $\omega_i/ k_i$ actually occur. The key observation is contained in Lemma \[lem=cone\] below.
We need some terminology. Let $(V, (\cdot, \cdot))$ be an Euclidean space and $W\subset {\rm O}(V)$ be a Weyl group of finite type associated to a root system with root basis $F$. We shall assume that the action is effective, or equivalently that $V^*={\rm span}_{\mathbb{R}}F$. For a subset $P\subset F$ let $W_P<W$ be the corresponding parabolic subgroup. As before $$C=\{ v\in V: \alpha(v)\geq 0 \forall \alpha\in F\}$$ denotes the closure of the Weyl chamber. A closed convex cone $\Gamma\subset V$ will be called non-degenerate if its edge $E(\Gamma)=\Gamma\cap -\Gamma$ is equal to $\{0\}$. Clearly $\Gamma$ is non-degenerate iff there exists a linear functional $\omega\in V^*$ such that $\omega|_{\Gamma\backslash\{0\}}>0$.
\[lem=cone\] Let $W$ be a Weyl group of finite type acting effectively on an Euclidean space $V$. Let $C$ be the closure of the corresponding Weyl chamber. Then the following statements are equivalent:
1. $W$ is irreducible.
2. $W_P.C$ is non-degenerate for all proper subsets $P\subsetneq F$.
3. $W_P.C$ is non-degenerate for a maximal proper subset $P\subsetneq F$.
$(i)\Rightarrow (ii)$: If $P=\emptyset$, then $W_P.C=C$ is non-degenerate. So let us henceforth assume that $P\neq \emptyset$. Denote by $V_{\rm fix}=\{ v\in V\mid (\forall w\in W_P)\ w(v)=v \}$ the space of $W_P$-fixed points. Then $$V=V_{\rm fix}\oplus V_{\rm eff}$$ with $V_{\rm eff} ={\rm span}_{\mathbb{R}}P =V_{\rm fix}^\bot$ the effective part for the $W_P$-action. We note that $C\cap V_{\rm fix}\neq \{0\}$ and fix a non-zero element $u$ in this intersection.
Assume that $W$ is irreducible. According to [@Hump], Ch. IV, Exc. 8, one has $(x,y)>0$ for all $x,y\in C\backslash\{0\}$. In particular if $\omega\in V^*$ is defined by $\omega(v):=(u,v)$ then $\omega|_{C\backslash\{0\}}>0$. As $u$ is $W_P$-fixed, it follows that $\omega|_{W_P. C\backslash\{0\}}>0$ and consequently $W_P. C$ is non-degenerate.
$(ii)\Rightarrow(iii)$ is clear, moving on to $(iii)\Rightarrow(i)$: We argue by contradiction and assume that $W$ is reducible. Then there exist splittings $W=W_1\times W_2$, $V=V_1\oplus V_2$, $F=F_1\amalg F_2$ with $W_1$ irreducible, $F_1\subset P$ and $V_1, V_2\neq \{0\}$. But then $V_1\subset W_P. C$ and $W_P. C$ is degenerate.
Let us now return to our initial setting with the irreducible restricted root system $\Sigma=\Sigma(\gf,\af)$ (then the reduced root system $\Sigma^l$ of unmultipliable roots is irreducible as well). We write $D$ for the Dynkin diagram associated with the bases $\Pi$ of $\Sigma^l$, and $D^*=D(W^{\rm aff})$ for its affine extension. Let $\Pi_0=\{\alpha_0,\alpha_1,\dots,\alpha_n\}$ denote the underlying set of affine simple roots.
\[th=omega\] Let $\gf$ be a simple Lie algebra and $\Sigma=\Sigma(\gf,\af)$ the associated irreducible root system. Then for all $1\leq i\leq n$ the following statements are equivalent:
1. $\omega_i/k_i\in \partial_e\Omega$.
2. $D^*-\{\alpha_i\}$ is connected.
Fix $1\leq i\leq n$ and denote the stabilizer of $\omega_i/k_i$ in $W^{\rm aff}$ by $W^{(i)}$. Notice that $W^{(i)}\simeq W(D^*-\{\alpha_i\})$ is a Weyl group of finite type, and that $\Pi^{(i)}=\{\alpha_0,\dots,\alpha_{i-1},\alpha_{i+1},\dots,\alpha_n\}$ is a set of simple roots for its root system. Let us denote by $C^{(i)}$ the associated closed Weyl chamber.
Let $U$ denote an open ball around $\omega_i/ k_i$ such that for $w\in W$ one has $w(U)\cap C\not=\emptyset$ iff $w$ fixes $\omega_i/k_i$. The isotropy group of $\omega_i/k_i$ in $W$ is $W_P$ where $P=\Pi\cap \Pi^{(i)}$. Observe that $P$ is a maximal proper subset both of $\Pi$ and of $\Pi^{(i)}$.
Observe that $C \cap \overline \Omega$ is the fundamental alcove of $W^{\rm aff}$ with respect to $\Pi_0$. Hence $(C \cap \overline \Omega)\cap U=C^{(i)}\cap U$. Moreover $\overline \Omega\cap U=W(C \cap \overline \Omega)\cap U
=W_P(C \cap \overline \Omega)\cap U=W_P(C^{(i)}\cap U)=
W^{(i)}_{P}(C^{(i)}\cap U)=W^{(i)}_{P}C^{(i)}\cap U$. Hence $\omega_i/k_i$ is an extremal point of $\overline\Omega$ iff the convex cone $W^{(i)}_{P}C^{(i)}$ (with vertex $\omega_i/k_i$) is non-degenerate. Apply Lemma \[lem=cone\].
Let us call $\omega_i$ [*minuscule*]{} if $k_i=1$. Notice that $\omega_i\in \partial\Omega$. Let us denote the union of all $W$-orbits through minuscule $\omega_i$ by $\partial_m \Omega$ and refer to it as the [*minuscule part*]{} of $\partial\Omega$. Similarly we define the [*minuscule boundary*]{} of $\Xi$ by $$\partial_m\Xi=G\exp(i\pi \partial_m \Omega/2). x_0\, .$$
\[prop:incl\] One has $$\partial_m\Omega\subset \partial_e\Omega$$ and in particular $\partial_m\Xi\subset\partial_d\Xi$.
Let $A=(a_{ij})$ (with $i,j\in\{0,1,\dots,n\}$) be the generalized Cartan matrix associated with the extended Dynkin diagram $D^*$. We consider $A$ as a matrix with respect to the bases $\{\a_0,\a_1,\dots,\a_n\}$ of affine simple roots. By elementary theory of generalized Cartan matrices (cf. [@KacFirstEdition Theorem 4.8]) the one-dimensional kernel of $A$ is generated by a unique positive, primitive element $\delta$ in the affine root lattice, namely $\delta=\a_0+\beta$. In other words, if we put $k_0=1$ then for each $j$: $2k_j+\sum_{i\not=j} a_{ji}k_i=0$. Hence if $\omega_j$ is minuscule (i.e. $k_j=1$) then either $\a_j$ is an end point in $D^*$ (i.e. has only one neighbor in $D^*$) or else $\a_j$ has precisely two neighbors $\a_{i}$, $\a_{l}$ with $k_{i}=k_{l}=1$. But in this last case $D^*$ must be (by an easy inductive argument) a circular graph. We conclude in both cases that $D^*-\{\a_j\}$ is connected as desired.
For later reference and convenience to the reader we list $\partial_e\Omega$ and $\partial_m\Omega$. Theorem \[th=omega\] and the tables of [@B] yield:
Table 1
(Correcting literature) The first named author would like to take the opportunity to point out an error in [@GKI] regarding $\partial_e\Omega$ for the $E_7$-case. Due to a computational mistake the $W$-orbit through $\omega_2/2$ was missed.
For a point $\omega_j/k_j\in \partial\Omega$ set $$z_j=\exp(i\pi\omega_j/2k_j). x_0\in \partial_d\Xi$$ and denote by $H_j$ the stabilizer of $G$ in $z_j$. We already remarked earlier that $H_j$ is a noncompact subgroup. Let us denote by $\h_j$ its Lie algebra. Our next objective is to classify the stabilizer algebras $\h_j$ for those $z_j$ which appear in $\partial_d\Xi$.
Write $F=A_{\mathbb{C}}\cap K_{\mathbb{C}}=T \cap K$ and notice that $F$ is a finite two group. We will often identify $A_{\mathbb{C}}. x_0$ with $A_{\mathbb{C}}/F$ and remark that elements $z\in A_{\mathbb{C}}.x_0$ have well defined squares $z^2 \in A_{\mathbb{C}}$. For each $1\leq j\leq n$ let us define the centralizer subgroup $$G_j:=Z(z_j^4)=\{ g\in G\mid z_j^4 gz_j^{-4}=g\}$$ and denote by $\gf_j$ its Lie algebra.
\[lem=m\]Let $1\leq j\leq n$. Then the following assertions hold:
1. $\gf_j=\gf$ if and only if $\omega_j$ is minuscule.
2. $\gf_j$ is a $3$-graded reductive Lie algebra $$\gf_j=\gf_{j,-} \oplus \gf_{j,0} \oplus \gf_{j,+}$$ where $\gf_{j,\pm}=\{ Y\in \gf\mid
[\omega_j,Y]=\pm k_j Y\}$ and $\gf_{j,0}=\{ Y\in \gf\mid
[\omega_j,Y]=0\}$.
Associated to $\omega_j$ is the standard grading $$\label{eq=grad} \gf=\sum_{l=-k_j}^{k_j} \gf_l$$ with $\gf_l=\{ Y\in \gf\mid [\omega_j,Y]=lY\}$. Notice that $\Ad(z_j^4)$ acts on $\gf_l$ as the scalar $e^{2il\pi/k_j}$. The assertions of the lemma follow with $\gf_{j,\pm}=\gf_{\pm k_j}$.
Let us denote by $\theta$ the Cartan involution of $\gf=\kf\oplus \pf$. Observe that $Y\in \gf$ belongs to $\h_j$ if and only if $\Ad(z_j^{-1})(Y)\in \kf_{\mathbb{C}}$, in other words $$\h_j=\{ Y\in \gf\mid \Ad(z_j^2) (\theta Y)=Y\}\,,$$ (cf. [@GKI], Lemma 3.4). We reveal the structure of $\h_j$.
\[lem=h\] Let $1\leq j\leq n$. Then $\h_j$ is $\theta$-stable subalgebra of $\gf_j$. Moreover, its Cartan decomposition is given by $$\h_j=\gf_{0,j}^\theta \oplus (\gf_{j,-}\oplus \gf_{j,+})^{-\theta}\, .$$
Recall the grading $\gf=\sum_{l=-k_j}^{k_j} \gf_l$ from (\[eq=grad\]). Then for each $0\leq l\leq k_j$ the operator $\Ad(z_j^2) \circ \theta $ leaves $(\gf_l \oplus \gf_{-l})_{\mathbb{C}}$ stable; explicitly $$(Y_l,Y_{-l})\mapsto (e^{il\pi/k_j}\theta (Y_{-l}), e^{-il\pi/k_j}\theta(Y_l))\qquad (Y_l,Y_{-l})\in
(\gf_l \oplus \gf_{-l})_{\mathbb{C}}\, .$$ Hence $\left (\Ad(z_j^2)\circ \theta \right)(\gf_l \oplus \gf_{-l}) \cap \gf\neq\{0\}$ precisely for $l=0,k_j$. The assertions of the lemma follow.
\[cor=min\] Let $1\leq j\leq n$. Then $\dim \h_j\leq \dim \kf$ with equality precisely if $\omega_j$ is minuscule.
As a consequence of Lemma \[lem=m\] and Lemma \[lem=h\] we can extend [@GKI], Theorem 3.26 (2).
For a boundary orbit $G.z_j\subset \partial\Xi$ the following statements are equivalent.
1. $\omega_j$ is minuscule.
2. $\dim \h_j=\dim \kf$.
3. $\Ad(z_j^{-1})(\h_j)_{\mathbb{C}}=\kf_{\mathbb{C}}$.
4. $\h_j$ is a symmetric subalgebra of $\gf$.
5. $G.z_j$ is a totally real submanifold of $X_{\mathbb{C}}$.
6. $G.z_j$ is a totally real submanifold of $X_{\mathbb{C}}$ of maximal dimension.
(i)$\iff$(ii): Corollary \[cor=min\].
(ii)$\iff$(iii): $\Ad(z_j^{-1}) \h_j\subset \kf_{\mathbb{C}}$ holds for all $1\leq j\leq n$ by the definition of $\h_j$.
(i)$\Leftarrow$(iv): If $\omega_j$ is minuscule, then $\gf_j=\gf$ by Lemma \[lem=m\](i). In particular $\tau_j=\Ad(z_j^2)\circ \theta$ defines an involution and $\h_j$ being the $\tau_j$-fixed point set is symmetric.
(iv)$\Rightarrow$(i): Notice that $\gf_j$ is a reductive subalgebra properly containing $\h_j$. Now if $\h_j$ is symmetric, then it is a maximally reductive proper subalgebra of $\gf$. Thus $\gf=\gf_j$, i.e. $\omega_j$ is minuscule.
(v)$\Rightarrow$(ii): If $G.z_j$ is totally real, then $\dim_{\mathbb{R}}G.z_j\leq \dim_{\mathbb{R}}X$. The latter inequality rewrites as $\dim \h_j \geq \dim \kf$. Because of $\dim \h_j\leq \dim \kf$ in all cases, it follows that $\dim \h_j=\dim \kf$.
(vi)$\Rightarrow$(v) is clear.
\(ii) $\Rightarrow$ (vi): (ii) implies that $\dim_{\mathbb{R}}G.z_j =\dim_{\mathbb{R}}X$. It remains to show that $G.z_j$ is totally real. By $G$-homogeneity, it is sufficient to show that $T_{z_j} (G.z_j)$ is totally real in $T_{z_j} (X_{\mathbb{C}})$. The assignment $Y\mapsto {d\over dt}\Big|_{t=0} \exp(tY).z_j$ identifies $\gf_{\mathbb{C}}/ \Ad(z_j) \kf_{\mathbb{C}}$ with $T_{z_j} (X_{\mathbb{C}})$. Now observe that $\gf_{\mathbb{C}}/ \Ad(z_j) \kf_{\mathbb{C}}=\gf_{\mathbb{C}}/\h_{\mathbb{C}}$ by the equivalence of (ii) and (iii). Thus all we have to show is that $\gf +\h_{\mathbb{C}}/ \h_{\mathbb{C}}$ is totally real in $\gf_{\mathbb{C}}/ \h_{\mathbb{C}}$, which is apparent.
Suppose that $\omega_j$ is minuscule. Then $\gf=\gf_{j, -}\oplus \gf_{j,0}\oplus \gf_{j,+}$ is a $3$-graduation and $\tau_j=\Ad(z_j^2)\circ \theta$ is an involution with fixed point algebra $\h_j$. In other words $(\gf, \h_j)$ is a noncompactly causal (NCC) symmetric pair. Moreover all (NCC) symmetric pairs arise in this fashion. For a proof of all this we refer to the paper [@Kan2] of Professor Soji Kaneyuki (specifically Th. 3.1).
For the concrete classification of the $\h_j$ in the minuscule case we allow ourselves to refer alternatively to [@GKI], Th. 3.25.
We wish to complete the classification of $\partial_d\Xi$ by listing all the non-minuscule cases. The most degenerate situation might deserve special attention.
(The distinguished boundary of $G_2$) Let us consider the case of $\gf=G_2$. We use the terminology of [@B]. With $\Pi=\{\alpha_1, \alpha_2\}$ the positive roots list as $$\alpha_1,\alpha_2, \alpha_1+\alpha_2, 2\alpha_1+\alpha_2, 3\alpha_1+\alpha_2, 3\alpha_1+
2\alpha_2\, .$$ We have $\partial_e \Omega= W. {\omega_1/3}$. Hence $$\begin{aligned}
\gf_{1,0}&=&\af\oplus \gf^{\alpha_2} \oplus \gf^{-\alpha_2}\simeq \sl(2,{\mathbb{R}})\times {\mathbb{R}}\\
\gf_{1,1} &=& \gf^{3\alpha_1+\alpha_2} \oplus \gf^{3\alpha_1+2\alpha_2}\simeq {\mathbb{R}}^2\\
\gf_{1,-1} &=& \gf^{-3\alpha_1-\alpha_2} \oplus \gf^{-3\alpha_1-2\alpha_2}\simeq {\mathbb{R}}^2\end{aligned}$$ and so $\gf_1\simeq \sl(3,{\mathbb{R}})$. Finally Lemma \[lem=h\] implies $\h_1\simeq \sl(2,{\mathbb{R}})$.
Let $z_j$ be a non-minuscule boundary points. A glance at Table 1 above shows that $k_j=2$ except for $G_2$ and one case in $E_8$. Thus $\gf=\sum_{j=-k_j}^{k_j} \gf_j$ is a $5$-grading for most of the cases.
Combining Lemma \[lem=h\] and Lemma \[lem=m\] with Kaneyuki’s classification of the even part of $5$-graded Lie algebras [@Kan] we arrive at the following two lists. For the exceptional cases we use [@Kan], Table I, II.
Table 2
For the classical cases we apply [@Kan], Th. 3.2, and note that the first two cases below were already contained in [@GKI], Th. 3.25.
Table 3
We conclude this section with a discussion of collapsing of boundary orbits. Let $z_j, z_l\in \partial_d\Xi$ with $j\neq l$. If $G_{\mathbb{C}}$ is simply connected and $G\subset G_{\mathbb{C}}$, then $G.z_j\neq G.z_l$, i.e $G.z_j\cap G.z_l=\emptyset$ (cf. [@GKI], Th. 3.6). In the general case it might happen that $G.z_j=G.z_k$ and we say that $\omega_j/k_j$ and $\omega_l/k_l$ [*collapse*]{} in $\partial_d\Xi$. Collapsing appears when there exist outer automorphisms. We refrain from complete results but would like to mention some important examples.
\[collapse\] (a) Let $G={\rm PSl}(n,{\Bbb K})$ for ${\Bbb K}={\mathbb{R}}, {\mathbb{C}},
{\Bbb H}$. Then $\omega_j$ and $\omega_l$ collapse precisely for $j+l=n$.
\(b) Let $G={\rm SO}(n,n)$ for $n\geq 4$. Then $\omega_{n-1}$ and $\omega_n$ collapse.
New features of $G=\Sl(2,{\mathbb{R}})$ {#sec=nf}
=======================================
This section is devoted to the crown domain associated to the basic group $G=\Sl(2,{\mathbb{R}})$. It is divided into two parts. In the first half we give a description of the full boundary $\partial\Xi$ as a cone bundle over the affine symmetric space $G/H=\Sl(2,{\mathbb{R}})/
{\rm SO}(1,1)$. In the second part we give a novel description of the crown as a union of unipotent $G$-orbits. Later, via appropriate $\Sl(2,{\mathbb{R}})$-reduction, we will use the material collected there for our discussion of cusp forms and proper action.
Corner view
-----------
We change perspective. Instead of regarding the crown from the base point $x_0$ as a thickening of $X$, we may view $\Xi$ from a corner point $z_j$ as a domain bordered by the homogeneous space $G/H_j$. The advantage of this perspective is that it leads to a simple characterization of the full boundary $\partial \Xi$ of $\Xi$.
We will give a detailed discussion of the boundary of the complex crown when $G=\Sl(2,{\mathbb{R}})$. As $\Xi$ is attached to $X$, and so independent of the specific global structure of $G$, we may replace $\Sl(2,{\mathbb{R}})$ by $G={\rm SO}_e(1,2)$. We regard $K={\rm SO}(2,{\mathbb{R}})$ as a maximal compact subgroup of $G$ under the standard lower right corner embedding.
Let us define a quadratic form $Q$ on ${\mathbb{C}}^3$ by
$$Q ({\bf z})=z_0^2-z_1^2 -z_2^2, \qquad
{\bf z}=(z_0, z_1, z_2)^T\in {\mathbb{C}}^3\, .$$ With $Q$ we declare real and complex hyperboloids by $$X=\{ {\bf x}=(x_0, x_1, x_2)^T\in {\mathbb{R}}^3\mid Q({\bf x})=1, x_0>0\}$$ and $$X_{\mathbb{C}}=\{ {\bf z} =(z_0,z_1,z_2)^T\in {\mathbb{C}}^3\mid
Q({\bf z})=1\}\ .$$ We notice that mapping $$G_{\mathbb{C}}/K_{\mathbb{C}}\to X_{\mathbb{C}}, \ \ gK_{\mathbb{C}}\mapsto g.{\bf x}_0 \qquad ({\bf x}_0=(1,0,0))$$ is diffeomorphic and that $X$ is identified with $G/K$.
At this point it is useful to introduce coordinates on $\gf=\so(1,2)$. We set
$${\bf e_1}=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0\\ 1 & 0 & 0\end{pmatrix},
\quad
{\bf e_2}=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 0\end{pmatrix},
\quad
{\bf e_3}=
\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1\\ 0 & -1 & 0\end{pmatrix}\, .$$
We notice that $\kf={\mathbb{R}}{\bf e_3}$, $\pf= {\mathbb{R}}{\bf e_1} \oplus {\mathbb{R}}{\bf e_2}$ and make our choice of the flat piece $\af={\mathbb{R}}{\bf e_1}$. Then $\Omega=(-1, 1) {\bf e_1}$, $\Xi=G\exp (i(-\pi/ 2,\pi/2){\bf e_1}).{\bf x_0}$ and we obtain Gindikin’s favorite model of the crown $$\Xi=\{ {\bf z}={\bf x} +i{\bf y}\in X_{\mathbb{C}}\mid
x_0>0, Q({\bf x}) >0\}\, .$$ It follows that the boundary of $\Xi$ is given by
$$\label{be1}
\partial \Xi=\partial_s \Xi \amalg \partial_n \Xi$$
with semisimple part $$\label{be2}
\partial_s \Xi=\{ i{\bf y}\in i{\mathbb{R}}^3\mid
Q({\bf y})= -1\}$$ and nilpotent part $$\label{be3}
\partial_n \Xi=\{ {\bf z}={\bf x}+i{\bf y}
\in X_{\mathbb{C}}\mid x_0>0, Q{(\bf x})=0\}\, .$$
Notice that ${\bf z_1}=\exp(i\pi/2 {\bf e_1}).{\bf x_0}=(0,0,i)^T$ and that the stabilizer of ${\bf z}_1$ in $G$ is the symmetric subgroup $H={\rm SO}_e(1,1)$, sitting inside of $G$ as the upper left corner block. Hence $$\partial_s\Xi=\partial_d \Xi =G.{\bf z_1}\simeq G/H$$ Write $\tau$ for the involution on $G$ with fixed point set $H$ and let $\gf=\h \oplus \qf$ the corresponding $\tau$-eigenspace decomposition. Clearly, $\h={\mathbb{R}}{\bf e_2}$ and $\qf=\af \oplus \kf={\mathbb{R}}{\bf e_1} \oplus {\mathbb{R}}{\bf e_3}$. Notice that $\qf$ breaks as an $\h$-module into two pieces $$\qf=\qf^+ \oplus \qf^-$$ with $$\qf^\pm=\{ Y\in \qf\mid [e_2, Y]=\pm Y\} ={\mathbb{R}}({\bf e_1}\pm {\bf e_2})
\, .$$ Let us define the $H$-stable pair of half lines $$\Cc={\mathbb{R}}_{\geq 0}({\bf e_1}\oplus {\bf e_3})\cup {\mathbb{R}}_{\geq 0}
({\bf e_1}- {\bf e_3})$$ in $\qf=\qf^+ \oplus \qf^-$. We remark that $\Cc$ is the boundary of the $H$-invariant open cone $$\W=\Ad(H)({\mathbb{R}}_{> 0}{\bf e_1})= {\mathbb{R}}_{> 0}({\bf e_1}+ {\bf e_3})\oplus {\mathbb{R}}_{> 0}
({\bf e_1}- {\bf e_3})\, .$$ Recall that the tangent bundle $T(G/H)$ naturally identifies with $G\times_H \qf$ and let us mention that $G\times_H \Cc$ is a $G$-invariant subset thereof.
\[th=bd\] For $G={\rm SO}_e(1,2)$, the mapping $$b: G\times_H \Cc\to \partial \Xi, \ \ [g,Y]\mapsto g\exp(-iY).{\bf z_1}$$ is a $G$-equivariant homeomorphism.
It is of course clear that the map is equivariant and continuous. We move to surjectivity. For $s\in {\mathbb{R}}$,
$$\begin{aligned}
\label{m-co}
\exp(is({\bf e_1} + {\bf e_3})) &=& \begin{pmatrix}
1 -s^2/2 & s^2/2 & is \\ -s^2/2 & 1+ s^2/2 & is \\
is & -is & 1 \end{pmatrix}, \\
\exp(is({\bf e_1} - {\bf e_3})) &=& \begin{pmatrix}
1 -s^2/2 & -s^2/2 & is \\ s^2/2 & 1+ s^2/2 & -is \\
is & is & 1 \end{pmatrix}\, . \end{aligned}$$
Therefore, $$\label{be4} b([{\bf 1}, s({\bf e_1}\pm {\bf e_3})])=
\exp(-is({\bf e_1} \pm {\bf e_3})).{\bf z_1}=(s, \pm s, i)^T
\, .$$ From (\[be1\]) - (\[be3\]), $$\partial \Xi = G.\{ (s, \pm s, i)^T\mid
s\geq 0\}$$ and surjectivity is forced by (\[be4\]) and $G$-equivariance.
Next, we prove that $b$ is one-to-one. By $G$-equivariance, all we have to show is that $$\label{be5}
b([g, s({\bf e_1}\pm {\bf e_3})])=b([{\bf 1}, t({\bf e_1}\pm {\bf e_3})])$$ for some $g\in G$ and $s,t\geq 0$, forces $g\in H$ and $\Ad(g)(s({\bf e_1}\pm {\bf e_3}))=t({\bf e_1}\pm {\bf e_3})$. We write (\[be5\]) out and see $$\label{be6}
g.(s, \pm s, i)^T = (t, \pm t, i)^T\, .$$ We take imaginary parts of this identity and deduce that $g(0,0,i)^T=(0,0,i)^T={\bf z_1}$, i.e. $g\in H$. With this information we go back in (\[be6\]), take the real part and get $g(s,\pm s,0)^T= (t,\pm t, 0)^T$. We observe that the latter means $\Ad(g)(s({\bf e_1}\pm {\bf e_3}))=t({\bf e_1}\pm {\bf e_3})$ and end the proof of injectivity.
Finally we mention that $b$ is an open mapping and this finishes the proof.
For $G={\rm SO}_e(1,2)$ one has $$\pi_1(\partial\Xi)=\pi_1(G/H)={\mathbb{Z}}\, .$$
Unipotent parameterization
--------------------------
We give now a novel description of the crown as a union of unipotent $G$-orbits.
If not stated otherwise, $G=\Sl(2,{\mathbb{R}})$. The standard choices of coordinates are $$\af={\mathbb{R}}\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}
\qquad\hbox{and} \qquad \nf={\mathbb{R}}\begin{pmatrix} 0 & 1\\ 0 & 0\end{pmatrix}$$ and we observe that $\Omega=(-1/2, 1/2)\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}$.
The key observation is contained in the following lemma.
\[lem=or\] Let $G=\Sl(2,{\mathbb{R}})$. For all $0\leq |t|<{\pi/4}$, $t\in{\mathbb{R}}$, one has the identity $$\label{eq=or} G\begin{pmatrix} 1 & i \sin 2t \\ 0 & 1
\end{pmatrix}.x_0=G\begin{pmatrix}e^{it} & 0 \\ 0 & e^{-it}
\end{pmatrix}.x_0\, .$$
For the proof it is convenient to switch to the hyperbolic model and replace $G$ by ${\rm SO}_e(1,2)$ (we recall that $X=G/K$ and $\Xi$ are independent of the globalization $G$ of $\gf$; Remark \[rem=bd\](c) ). As before, we choose $\af={\mathbb{R}}{\bf e_1}$. We come to our choice of $\nf$. For $z\in {\mathbb{C}}$ let
$$n_z=\left( \begin{array}{ccc} 1+\frac{1}{2} z^2 & z &
-{1\over 2} z^2 \\ z & 1 & -z \\
\frac{1}{2} z^2 & z & 1-\frac{1}{2}
z^2 \end{array} \right)$$ and $$N_{\mathbb{C}}=\{ n_z \mid z\in{\mathbb{C}}\}\, .$$
Further for $t\in {\mathbb{R}}$ with $|t|<{\pi\over 2}$ we set
$$a_t=\left( \begin{array}{ccc} \cos t & 0 & -i\sin t \\ 0 &
1 & 0 \\ -i \sin t & 0 & \cos t \end{array} \right) \in
\exp (i\Omega)\, .$$
The statement of the lemma translates into the assertion $$Gn_{i\sin t }.{\bf x}_0=G a_t.{\bf x}_0\, .$$
Clearly, it suffices to prove that $$a_t.{\bf x}_0=(\cos t, 0, -i\sin t)^T\in Gn_{i \sin t}.{\bf x}_0\, .$$
Now let $k\in K$ and $b\in A$ be elements which we write as
$$k= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 &
\cos \theta & \sin\theta \\ 0 & -\sin \theta & \cos \theta
\end{array} \right) \quad \hbox{and}\quad
b=\left( \begin{array}{ccc} \cosh r & 0 & \sinh r \\ 0 &
1& 0 \\ \sinh r & 0 & \cosh r \end{array} \right)$$ for real numbers $r,\theta$. For $y\in {\mathbb{R}}$, a simple computation yields that
$$kbn_{iy}.{\bf x}_0= \left(\begin{array}{c}
\cosh r (1-{1\over 2} y^2) - {1\over 2}
y^2\sinh r \\
iy\cos\theta + \sin\theta( \sinh r (1-{1\over 2} y^2) -
{1\over 2}y^2
\cosh r)\\ -iy\sin\theta +
\cos\theta( \sinh r (1-{1\over 2}y^2) -{1\over 2}y^2
\cosh r)
\end{array} \right)\, .$$ Now we make the choice of $\theta={\pi \over 2}$ which gives us that
$$kb n_{iy}.{\bf x}_0= \left(\begin{array}{c}
\cosh r (1-{1\over 2} y^2) - {1\over 2}
y^2\sinh r \\
\sinh r (1-{1\over 2}y^2) -{1\over 2}y^2
\cosh r\\ -iy
\end{array} \right)\, .$$ As $y=\sin t $ we only have to verify that we can choose $r$ such that $\sinh r (1-{1\over 2}y^2) -{1\over 2}y^2
\cosh r=0$. But this is equivalent to $$\tanh r = {{1\over 2} y^2 \over 1 -{1\over 2}y^2}\, .$$ In view of $-1< y=\sin t <1 $, the right hand side is smaller than one and we can solve for $r$.
Let us define a domain in $\nf$ $$\Lambda=\left\{\begin{pmatrix} 0 & x \\ 0 & 0
\end{pmatrix}\in \nf\mid \, x \in {\mathbb{R}}, |x|<1 \right\}\, .$$ A remarkable consequence of the preceding Lemma is the following result which we will establish in full generality later on.
\[th=sl2\] For $G=\Sl(2,{\mathbb{R}})$ one has $$\Xi=G\exp(i\Lambda).x_0\, .$$
\[rem=fb\] (a) (Relation to $KNK$) As observed by Kostant, for any semisimple Lie group $G$ one has $G=KNK$. As one referee pointed out, a more careful study of the $KNK$-decomposition of $G$ was undertaken by H. Lee Michelson in [@Mi]. In particular, Prop. 3.1 in [@Mi] applied to $G=\Sl(2,{\mathbb{R}})$ states that $$\label{mm} K \begin{pmatrix} e^t & 0 \\ 0 & e^{-t}\end{pmatrix}K =
K \begin{pmatrix} 1 & 2 \sinh t \\ 0 & 1 \end{pmatrix}K$$ for all $t\in {\mathbb{R}}$. It is tempting to believe that Lemma \[lem=or\] would follow from some sort of analytic continuation of (\[mm\]). However, we observe a subtle difference in this matter: the location of the even prime. This is surprising and we thank this referee of having raised the question.
\(b) It is not a priori clear that $G\exp(i\Lambda).x_0$ is open in $X_{\mathbb{C}}$. This is because of the fact that the natural map $$G\times \nf \to X_{\mathbb{C}}, \ \ (g, Y)\mapsto g\exp(iY).x_0$$ has singular differential at $(g, 0)$, $g\in G$.
\(c) Lemma \[lem=or\] allows us to give a characterization of $\Xi$ as a fiber bundle related to the nilcone. Write $\Nc\subset \gf$ for the cone of nilpotent elements in $\gf$ and note that $\Nc=\Ad(K)\nf$. Define a subset of $\Lambda$ by $$\Lambda^+=\left\{\begin{pmatrix} 0 & x \\ 0 & 0 \end{pmatrix}\mid
x \in \nf, \, 0\leq x <1 \right\}\, .$$ and put $\Nc^+=\Ad(K)\Lambda^+$. Then it follows from Lemma \[lem=or\] that the mapping $$G\times_K \Nc^+\to \Xi, \ \ [g,Y]\mapsto g\exp(iY).x_0$$ is a homeomorphism.
While it is not possible to enlarge $\Xi$ to a larger domain in hyperbolic directions, i.e. beyond $\Omega$, the situation is quite different for unipotent elements.
\[lem=np\] The differential of the mapping $$G\times \nf \to X_{\mathbb{C}}, \ \ (g, Y)\mapsto g\exp(iY).x_0$$ is invertible at all points $(g,Y)\in G\times \nf$ with $Y\neq 0$.
By $G$-equivariance of the map, it will be the sufficient to show that the map is submersive at all points $({\bf 1},Y)$ with $Y\neq 0$. This assertion in turn translates into the identity $$e^{-i {\rm ad} Y}\gf +i \nf + \kf_{\mathbb{C}}=\gf_{\mathbb{C}}$$ which is satisfied whenever $Y\neq 0$.
For $a<b$ we define an open subset of $\nf$ by $$\Lambda_{a,b}=\left\{\begin{pmatrix} 0 & x \\ 0 & 0
\end{pmatrix}\in \nf\mid x \in {\mathbb{R}}, a < x<b \right\}\,$$ and declare $G$-invariant connected subsets of $X_{\mathbb{C}}$ by $$\Xi_{a,b}=G \exp (i\Lambda_{a,b}).x_0\, .$$ Of further interest for us is the limiting object for $a\to -\infty , b \to \infty$,
$$\Xi_N= G\exp(i\nf).x_0=GN_{\mathbb{C}}.x_0\, .$$
For all $a<b$, the sets $\Xi_{a,b}$ are open. In particular $\Xi_N$ is open.
If $0\not \in (a,b)$, then the assertion follows from Lemma \[lem=np\]. Thus we may assume that $0\in (a,b)$. Suppose first that $|a|, |b|\leq 1$. Then by Lemma \[lem=or\] there exists a symmetric, i.e. $W={\mathbb{Z}}_2$-invariant, interval $\Omega_{a,b}\subset \Omega$ such that $\Xi_{a,b}=G\exp(i\Omega_{a,b}).x_0$. The latter set is open by Remark \[rem=tan\]. Finally assume that $b>1$ or $a<-1$. Then $$\Xi_{a,b} =\Xi_{\max\{a, -1\}, \min\{1,b\}}
\cup G\exp( i \Lambda_{a,b} \setminus\{0\}).x_0$$ is the union of two open sets (use Lemma \[lem=np\] for the second term) and we conclude the proof of the lemma.
We exhibit the structure of the domain $\Xi_N$. For that it is useful to move to the hyperboloid picture with $G={\rm SO}_e(1,2)$. Define the horocycle space of $X$ by
$${\rm Hor} (X)=\{ \xi\in {\mathbb{R}}^3\mid Q (\xi)=0, \xi_0>0\}$$ and notice that the map $$G/N\to {\rm Hor}(X), \ \ gN\mapsto g.\xi_0$$ with $\xi_0=(1, 0, 1)^T$ is a diffeomorphism. Let us denote by $${\bf z}\cdot {\bf w}=z_0 w_0- z_1 w_1 - z_2 w_2$$ the complex bilinear form obtained from polarizing $Q$.
\[prop=xic\] For the domain $\Xi_N$ the following assertions hold:
1. $\Xi_N=\{ {\bf z}\in X_{\mathbb{C}}\mid {\bf z}\cdot \xi =1\ \hbox{for some}
\ \xi \in {\rm Hor}(X)\}$.
2. $\Xi_N= X_{\mathbb{C}}- \partial_d\Xi - \{{\bf z}\in X_{\mathbb{C}}\mid Q({\bf x})>0,
x_0<0\} $.
3. For all $z\in \Xi_N$, the $G$-stabilizer $$G_z=\{ g\in G\mid g.z=z\}$$ is a compact subgroup of $G$.
\(i) We only have to notice that $$N_{\mathbb{C}}.x_0=\{{\bf z}\in X_{\mathbb{C}}\mid {\bf z}\cdot \xi_0=1\}\, .$$
\(ii) We use the characterization of $\Xi_N$ from (i). We have to show that
$$\label{eq=xin}
\Xi_N=\{{\bf z}={\bf x}+i {\bf y}\in X_{\mathbb{C}}\mid {\bf x}\neq 0\}\, .$$
For elements $z\in X_{\mathbb{C}}$ we will distinguish three cases: $Q ({\bf x})>0$, $Q ({\bf x})<0$ and $Q ({\bf x})=0$. Before we do our case by case analysis let us mention the fact that elements ${\bf z}={\bf x}+i{\bf y} \in {\mathbb{C}}^3$ belong to $X_{\mathbb{C}}$ precisely when $$Q({\bf x})- Q({\bf y})=1 \qquad\hbox{and}
\qquad {\bf x}\cdot {\bf y}=0\, .$$
Case 1: $Q ({\bf x})>0$ and $x_0<0$. We claim that the $G$-orbit through ${\bf z}$ has a representative of the type ${\bf z}=(x_0, iy_1, 0)^T$. In fact, as $Q({\bf x})>0$, the $G$-orbit through ${\bf x}$ has a representative $(x_0, 0, 0)^T$ with $x_0<0$. From $ {\bf x}\cdot {\bf y}=0$ we then conclude that ${\bf y}=(0, y_1, y_2)^T$. Further we may alter ${\bf y}$ by the stabilizer $K={\rm SO}(2, {\mathbb{R}})$ of ${\bf x}$. Thus we may assume that ${\bf y}=(0, y_1, 0)^T$. But then ${\bf z}\cdot \xi=1$ for $\xi=(1/x_0, 0, 1/x_0)^T\in {\rm Hor}(X)$. In particular $$\{z\in X_{\mathbb{C}}\mid Q({\bf x})>0\}\subset \Xi_N\, .$$
Case 2: $Q ({\bf x})<0$. We claim that the $G$-orbit through ${\bf z}$ has a representative of the type ${\bf z}=(0, iy_1, x_2)^T$. Indeed, as $Q({\bf x})<0$, we may assume that ${\bf x}=(0, 0, x_2)^T$ with $x_2>0$. Orthogonality $ {\bf x}\cdot {\bf y}=0$ then implies that ${\bf y}=(y_0, y_1, 0)^T$. Notice that $$Q({\bf y})=y_0^2 -y_1^2 = -1 - x_2^2 <0\, .$$ It is allowed to change ${\bf y}$ by displacements of $H={\rm SO}(1,1)$, the stabilizer of ${\bf x}$. As $H$ acts transitively on all connected component of the level sets of $y_0^2 -y_1^2$, it is no loss of generality to assume that ${\bf y}=(0,y_1, 0)^T$. But then ${\bf z}\cdot \xi =1$ for $\xi=(1/x_2, 0, -1/x_2)^T\in {\rm Hor}(X)$ and we conclude that $$\{z\in X_{\mathbb{C}}\mid Q({\bf x})<0\}\subset \Xi_N\, .$$
Case 3: $Q ({\bf x})=0$ and ${\bf x}\neq 0$. We assert that the $G$-orbit through ${\bf z}$ has a representative of the type ${\bf z}=(1+iy_0,1+iy_0,\pm 1)^T$. Namely, as $Q({\bf x})=0$ and ${\bf x}\neq 0$, the $G$-orbit through ${\bf x}$ contains the element ${\bf x}=(1,1,0)^T$. Then ${\bf x}\cdot {\bf y}=0$ and $Q({\bf y})=-1$ force $y=(y_0, y_0, \pm 1)$. Hence we can choose ${\bf z}$ of the asserted form. But then $\xi=( {y_0^2+1\over 2} , {y_0^2- 1\over 2}, \mp y_0)^T
\in {\rm Hor }(X)$ with ${\bf z}\cdot \xi=1$.
Finally, we observe that elements of the type ${\bf z}= i{\bf y}$ cannot belong to $\Xi_N$ by (i).
\(iii) Notice that $g.{\bf z}={\bf z}$ means that $g.{\bf x}={\bf x}$ and $g.{\bf y}={\bf y}$. We analyze the three cases in (ii). If $Q({\bf x})>0$, then $G.{\bf x}\simeq X$ and $G_{\bf x}$ is compact. If $Q({\bf x})<0$, then we may assume that ${\bf z}=(0, iy_1, x_2)^T$. Hence $g.{\bf x}={\bf x}$ forces $g\in H$ and then $g.{\bf y}={\bf y}$ yields $g={\bf 1}$. Finally if $Q({\bf x})=0$ and ${\bf x}\neq 0$, then our choice of ${\bf z}$ can be ${\bf z}=(1+iy_0,1+iy_0,\pm 1)^T$. Then $g.{\bf x}={\bf x}$ implies that $g$ is unipotent while $g.{\bf y}={\bf y}$ forces $g$ to be hyperbolic. Hence $g={\bf 1}$ in this case also.
The statement in Proposition \[prop=xic\] (iii) suggest that the $G$-action on $\Xi_{a,b}$ should be proper. However, this is not always the case as our next result shows. For that
\[prop=ab\]Suppose that $(a,b)\cap (-1,1)\neq \emptyset$. Then the following assertions hold:
1. If $\max\{ |a|, |b|\}\leq 1$, then the $G$-action on $\Xi_{a,b}$ is proper.
2. If $\min\{ |a|, |b|\} >1$, then the $G$-action on $\Xi_{a,b}$ is not proper.
If $|a|, |b|\leq 1$, then $\Xi_{a,b}\subset \Xi$ and as the $G$-action is proper on $\Xi$, the same holds for $\Xi_{a,b}$. We move to (ii). Assume now that $|a|>1$ and $|b|>1$. Then $\Xi_{a,b}$ contains both elements ${\bf w}_+=n_i.{\bf x}_0$ and ${\bf w}_-=n_{-i}.{\bf x}_0$. We note that $${\bf w}_+:=\begin{pmatrix} 1/2 \\ i\\ -1/2\end{pmatrix}=
n_i.{\bf x}_0=
\left( \begin{array}{ccc} 1/2 & i & 1/2 \\ i & 1 & -i \\
-1/2 & i & 1/2
\end{array} \right)\cdot \begin{pmatrix} 1 \\ 0\\ 0\end{pmatrix}\,.$$ For $n\in {\mathbb{N}}$ we define elements ${\bf z}_n\in X_{\mathbb{C}}$ by $${\bf z}_n =(1/2, 0, -1/2 + e^{-n})^T + i( 0, \sqrt{{3/4} + (e^{-n} -1/2)^2}, 0)
^T\, .$$ Notice that $\lim_{n\to \infty} {\bf z}_n={\bf w}_+$. Hence there exists an $n_0\in {\mathbb{N}}$ such that ${\bf z}_n\in \Xi_{a,b}$ for all $n\geq n_0$. Now set
$$b_n=\left( \begin{array}{ccc} \cosh n & 0 & \sinh n \\ 0 &
1& 0 \\ \sinh n & 0 & \cosh n \end{array} \right)\in A\, .$$ Note that eigenvectors of $b_n$ are $${\bf f}_1=\begin{pmatrix} 1 \\ 0 \\1\end{pmatrix},
\quad {\bf f}_2=\begin{pmatrix} 0 \\ 1 \\0\end{pmatrix},
\quad {\bf f}_3=\begin{pmatrix} 1 \\ 0 \\-1\end{pmatrix}$$ with eigenvalues $e^n$, $1$ and $e^{-n}$ respectively. Thus $$\begin{aligned}
b_n.{\bf z}_n = & b_n.{\bf x_n} + i {\bf y}_n \\
= & e^n\cdot {x_{0,n} +x_{2,n}\over 2}\cdot {\bf f}_1 +
e^{-n} \cdot {x_{0,n} -x_{2,n}\over 2}\cdot {\bf f}_2 + \\
& + i \sqrt{{3/4}
+ (e^{-n} -1/2)^2}\cdot {\bf f}_3 \\
=& 1/2 \cdot{\bf f}_1 + e^{-n} (1-e^{-n}/2)\cdot {\bf f}_2 +
i \sqrt{{3/4} + (e^{-n} -1/2)^2}\cdot {\bf f}_3\end{aligned}$$ and thus $\lim_{n\to \infty} b_n.{\bf z}_n ={\bf w}_-\in \Xi_{a,b}$. Hence $(b_n.{\bf z}_n)_{n\geq n_0}$ stays in a compact subset of $\Xi_{a,b}$ but with $(b_n)_{n\geq n_0}$ an unbounded sequence. Thus the action of $G$ on $\Xi_{a,b}$ is not proper.
We conclude this section with a final result for proper $G$-action.
\[prop=sl2\] Let $G=\Sl(2,{\mathbb{R}})$ and let $D\subset X_{\mathbb{C}}$ be a $G$-invariant domain with $X\subset D$. If the action of $G$ on $D$ is proper, then:
1. $\partial_s\Xi\cap D=\emptyset$.
2. $\partial_n\Xi\not\subseteq D$.
In particular, if $\partial_n\Xi\cap D =\emptyset$, then $D\subseteq \Xi$.
Let $X\subset D\subset X_{\mathbb{C}}$ be an open $G$-invariant domain with proper $G$-action. Suppose that $D\cap \partial \Xi\neq \emptyset$ and let $z$ be a point thereof. Then $z\not \in \partial_d\Xi=G/H$ as $H$ is noncompact and the $G$-action on $D$ is proper. Hence $z\in \partial_n\Xi$. It follows from Theorem \[th=bd\] that $$\partial_n\Xi= Gn_i.{\bf x}_0 \amalg Gn_{-i}.{\bf x}_0\, .$$ Thus $D\cap \Xi_{a,b}\neq \emptyset $ for some $a,b$ with $\max\{|a|, |b|\}>1$ – the assertion now follows from the previous proposition.
\[remidemmi\] There exist larger $G$-domains $D\supsetneq\Xi$ with the $G$-action proper. We provide the recipe for their construction in case of $G=\Sl(2,{\mathbb{R}})$. Recall that $X$ identifies with the upper halfplane and henceforth we view $X$ in the projective space $\PP^1({\mathbb{C}})$. Notice that $G_{\mathbb{C}}$ acts on $\PP^1({\mathbb{C}})$ by fractional linear transformation. Denote by $\overline X$ the lower half plane and notice that $\Xi$ is $G$-isomorphic to $X\times \overline X$. In this realization $X$ sits in $\Xi=X\times \overline X$ via $z\mapsto (z,\overline z) $. We view $\Xi\in \PP^1({\mathbb{C}})\times \PP^1({\mathbb{C}})$ and note that $$X_{\mathbb{C}}=\{ (z,w)\in \PP^1({\mathbb{C}})\times \PP^1({\mathbb{C}})\mid z\neq w\}\, .$$ Furthermore $$\partial_s\Xi=\{(x,y)\in \PP^1({\mathbb{R}})\times \PP^1({\mathbb{R}})\mid
x\neq y\}\,$$ and $$\partial_n\Xi= X\times \PP^1({\mathbb{R}})\amalg \PP^1({\mathbb{R}})\times \overline X\, .$$ In particular we see that $$D=(X\times \PP^1({\mathbb{C}}))\cap X_{\mathbb{C}}$$ provides a $G$-domain in $X_{\mathbb{C}}$ such that
- $\partial_s\Xi\cap D=\emptyset$,
- $\partial_n\Xi\not\subset D$,
- $G$ acts properly on $D$.
With this picture of $\Xi$ one can easily sharpen Proposition \[prop=ab\] to: $G$ acts properly on $\Xi_{a,b}$ if and only if $\min\{|a|,|b|\}\leq 1$.
Properness and maximality of holomorphic extension
==================================================
The first part of this section is valid for general $G$; the subsection after for $G=\Sl(2,{\mathbb{R}})$ only.
As we mentioned earlier in Remark \[rem=bd\](ii), it was proved in [@AG], that $\Omega$ is maximal with respect to proper $G$-action. We will refine this result in Theorem \[th=p\] below. This new geometric fact translates into a maximality assertion for holomorphic extension of representations.
\[th=p\] Let $X\subset D$ be a $G$-domain in $X_{\mathbb{C}}$ with the $G$-action proper. Then the following assertions hold:
1. $\partial_s\Xi\cap D=\emptyset$.
2. $\partial_n\Xi\not\subset D$. In particular, if $\partial_n\Xi \cap D=\emptyset$, then $D\subset \Xi$.
\(i) It was shown in [@AG] that $G$-stabilizers on $\partial_s\Xi$ are noncompact. Hence the assertion.
\(ii) Suppose that $\partial_n\Xi\subset D$ and let $z$ be a point of $\partial_n\Xi$. As $D$ is open we may assume that $z$ is generic in the sense of [@FH], Section 4.2. It follows from [@FH], Th. 4.3.5., that there is a subgroup $G_0\subset G$ which is locally isomorphic to $\Sl(2,{\mathbb{R}})$ such that the crown $\Xi_0$ associated to $G_0$ embeds $G_0$-equivariantly into $\Xi$ with $z\in \partial_n \Xi_0$ in addition. As $\partial_n\Xi_0\subset \partial_n\Xi$ we obtain a contradiction to Proposition \[prop=sl2\].
We turn to applications in representation theory. For that it is convenient to look at the preimage $$\tilde \Xi = G\exp(i\pi/2\Omega)K_{\mathbb{C}}$$ of $\Xi$ in $G_{\mathbb{C}}$.
We let $(\pi, \H)$ be a unitary irreducible representation of $G$ and write $\H_K$ for the associated Harish-Chandra modul of $K$-finite vectors. Then, for $v\in \H_K$, it was shown in [@KSI] that the orbit map $$F_v: G\to \H, \ \ g\mapsto \pi(g) v$$ extends to a $G$-equivariant holomorphic map $\tilde \Xi\to \H$, also denoted by $F_v$ in the sequel. We wish to show that $\tilde \Xi$ is maximal and want to relate this to the properness of the action of $G$ on $\Xi$. The link is established through the following fact.
\[lem=hm\]Let $(\pi, \H)$ be a unitary representation of a reductive group $G$ which does not contain the trivial representation. Then $G$ acts properly on $\H -\{0\}$.
Let $C\subset \H -\{0\}$ be a compact subset and $C_G=\{ g\in G\mid \pi(g)C\cap C\neq \emptyset\}$. Suppose that $C_G$ is not compact. Then there exists a sequence $(g_n)_{n\in {\mathbb{N}}}$ in $C_G$ and a sequence $(v_n)_{n\in {\mathbb{N}}}$ in $C$ such that $\pi(g_n)v_n \in C$ and $\lim_{n\to \infty} g_n =\infty$. As $C$ is compact we may assume that $\lim_{n\to\infty} v_n =v$ and $\lim_{n\to\infty} \pi(g_n)v_n =w$ with $v,w\in C$. We claim that $$\label{hm}
\lim_{n\to\infty} \langle \pi(g_n)v, w\rangle \neq 0\, .$$
In fact $\|\pi(g_n) v_n -\pi(g_n)v\|=\| v_n -v\|\to 0$ and thus $\pi(g_n)v\to w$ as well. As $w\in C$, it follows that $w\neq 0$ and our claim is established.
Finally we observe that (\[hm\]) contradicts the Riemann-Lebesgue lemma for representations which asserts that the matrix coefficient vanishes at infinity.
From Lemma \[lem=hm\] we deduce the following result.
\[t1\] Let $(\pi, \H)$ be an irreducible unitary representation of $G$ which is not trivial. Let $v\in \H_K$, $v\neq 0$, be a $K$-finite vector. Let $\tilde D$ be a maximal $G\times K_{\mathbb{C}}$-invariant domain in $G_{\mathbb{C}}$ with respect to the property that the orbit map $F_v: G\to \H, \ \ g\mapsto \pi(g)v$ extends to a $G$-equivariant holomorphic map $\tilde \Xi\to \H$. Then $G$ acts properly on $\tilde D/ K_{\mathbb{C}}\subset X_{\mathbb{C}}$.
We argue by contradiction and assume that $G$ does not act properly on $D=\tilde D/K_{\mathbb{C}}$. We obtain sequences $(z_n')_{n\in {\mathbb{N}}}\subset D$ and $(g_n)_{n\in {\mathbb{N}}} \subset G$ such that $\lim_{n\to\infty} z_n' =z'\in D$, $\lim_{n\to\infty} g_n z_n' =w'\in D$ and $\lim_{n\to\infty} g_n=\infty$. We select preimages $z_n$, $z$ and $w$ of $z_n'$, $z'$ and $w'$ in $\tilde D$. We may assume that $\lim_{n\to\infty} z_n=z$ and find a sequence $(k_n)_{n\in {\mathbb{N}}}$ in $K_{\mathbb{C}}$ such that $\lim_{n\to\infty} g_n z_n k_n = w$.
Before we continue we claim that $$\label{eq=nz} (\forall z\in \tilde D)\qquad \pi(z)v\neq 0$$ In fact assume $\pi(z)v=0$ for some $z\in \tilde D$. Then $\pi(g)\pi(z)v=0$ for all $g\in G$. In particular the map $G\to \H, \ \ g\mapsto \pi(g)v$ is constantly zero. However this map extends to a holomorphic map to a $G$-invariant neighborhood in $G_{\mathbb{C}}$. By the identity theorem for holomorphic functions this map has to be zero as well. We obtain a contradiction to $v\neq 0$ and our claim is established.
Write $V= {\rm span} \{\pi(K)v\}$ for the finite dimensional space spanned by the $K$-translates of $v$. In our next step we claim that
$$\label{eq=bou} (\exists c_1, c_2>0) \qquad c_1 < \|\pi(k_n)v\|< c_2\, .$$
In fact from
$$\lim_{n\to\infty} \pi(g_nz_nk_n)v = \pi(w)v\quad \hbox{and}\quad
\|\pi(g_n z_n k_n)v\|=\|\pi (z_n) \pi(k_n)v\|$$ we conclude with (\[eq=nz\]) that there are positive constants $c_1',c_2'>0$ such that $c_1'<\|\pi (z_n) \pi(k_n)v\|<c_2'$ for all $n$. We use that $\lim_{n\to\infty} z_n =z\in \tilde D$ to obtain $\pi(z_n)|_V-\pi(z)|_V\to 0$ and our claim follows.
We define $C$ to be the closure of the sequences $(\pi(z_nk_n)v)_{n\in {\mathbb{N}}}$ and $(\pi(g_nz_nk_n)v)_{n\in {\mathbb{N}}}$ in $\H$. With our previous claims (\[eq=nz\]) and (\[eq=bou\]) we obtain that $C\subset \H -\{0\}$ is a compact subset. But $C_G=\{g\in G\mid \pi(g) C\cap C\neq \emptyset\}$ contains the unbounded sequence $(g_n)_{n\in {\mathbb{N}}}$ and hence is not compact - a contradiction to Lemma \[lem=hm\].
Let $(\pi, \H)$ and $v\in \H$ be as in the theorem. Then we might ask whether the stronger statement $$\lim_{z\to \partial \tilde\Xi} \|\pi(z)v\| =\infty$$ holds true. For the special case of $v=v_K\in \H_K$ a $K$-fixed vector this was established in [@KSII], Th. 2.4.
Domains of holomorphy for the unitary dual of $G=\Sl(2,{\mathbb{R}})$
---------------------------------------------------------------------
Let now $G=\Sl(2,{\mathbb{R}})$. With the coordinates of Remark \[remidemmi\] we have $$X_{\mathbb{C}}=\left (\PP^1({\mathbb{C}})\times \PP^1({\mathbb{C}})\right \backslash \diag(\PP^1({\mathbb{C}})),$$ $$\Xi=X\times \overline X,$$ where $X$ denotes the upper and $\overline X$ the lower halfplane. Then there are two interesting $G$-domains in $X_{\mathbb{C}}$ which contain $\Xi$. These are:
- $S^+=(\PP^1({\mathbb{C}})\times\overline X)\cap X_{\mathbb{C}}$,
- $S^-=(X\times \PP^1({\mathbb{C}}))\cap X_{\mathbb{C}}$.
The following assertions hold:
1. $S^+=G\exp(i\Lambda_{(-1,\infty)}).x_0$,
2. $S^-=G\exp(i\Lambda_{(-\infty,1)}).x_0$.
In particular, $S^\pm$ are maximal $G$-domains in $X_{\mathbb{C}}$ on which $G$ acts properly.
The first two assertions come down to a very elementary computation; the last one follows from Proposition \[prop=sl2\].
As $\gf$ is of Hermitian type, the $\kf_{\mathbb{C}}$-module $\pf_{\mathbb{C}}$ splits into two inequivalent subspaces $\pf_{\mathbb{C}}=\pf^+ \oplus \pf^-$ with $$\pf^\pm ={\mathbb{C}}\cdot \begin{pmatrix} 1 & \pm i \\ \pm i & 1 \end{pmatrix}\, .$$ Set $P^\pm =\exp (\pf^\pm)$. Then the preimages $\tilde S^\pm$ of $S^\pm$ in $G_{\mathbb{C}}$ are given by
- $\tilde S^+=G K_{\mathbb{C}}P^+ $,
- $\tilde S^-= G K_{\mathbb{C}}P^-$.
We obtain the following result.
\[th=sl2r\] Let $(\pi, \H)$ be an irreducible unitary representation of $G=\Sl(2,{\mathbb{R}})$. Let $v\in \H_K$ be a non-zero vector and $f_v: G\to \H, \ g\mapsto \pi(g)v$ the corresponding orbit map. Then the domains of holomorphy of $f_v$ are given by:
1. $G_{\mathbb{C}}$, if $\pi$ is trivial.
2. $\tilde S^+$, if $\pi$ is a non-trivial highest weight representation.
3. $\tilde S^-$, if $\pi$ is a non-trivial lowest weight representation.
4. $\tilde \Xi$, if $\pi$ is none of the above, i.e. a unitarizable principal series.
\(i) is clear.
\(ii) If $\H_K$ is a highest weight module, then all its vectors are $K_{\mathbb{C}}P^+$-finite. Hence $\tilde S^+=GK_{\mathbb{C}}P^+$ lies in the domain of holomorphy of $f_v$. By the preceding Proposition $\tilde S^+$ is maximal for proper action and the assertion follows from Theorem \[t1\].
\(iii) Analogous to (ii).
\(iv) For $\tilde \Xi$ to be contained in the domain of holomorphy we refer to the general result of [@KSII], Th. 1.1. If $\pi$ is $K$-spherical, then $\tilde \Xi$ is indeed maximal as it follows from [@KOS], Th. 5.1 and Remark (\[rem=corr\]) below. Finally, the case of non-spherical principal series is similar to the spherical case (the same proof as in [@KOS] applies).
\[rem=corr\] (Correcting literature) In the proof of Th. 5.1 in [@KOS] there is an inaccuracy which we wish to correct here. Actually we have to address the proof of the key result Th. 5.4 in [@KOS]: it asserts for $G=\Sl(2,{\mathbb{R}})$ that a spherical function with imaginary parameter blows up at the boundary of $\Xi$. Now $\partial\Xi=\partial_s\Xi\amalg \partial_n\Xi$. The arguments given for the blow-up at the semisimple boundary $\partial_s\Xi$ are fine; the ones for the blow-up at $\partial_n\Xi$ are not correct and should be modified. With the notation of [@KOS] we have for $a_r=\begin{pmatrix} r & 0 \\ 0 & {1\over r}\end{pmatrix}\in A$, $r>0$, and $-1< t< 1$ that $$P\left(a_r\begin{pmatrix} 1& it \\ 0 & 1\end{pmatrix}.x_0\right)=r^2 +{1\over r^2} - t^2 r^2\, .$$ In particular, if $|t|>1$, then there would exist a sequence $r_n\to r_0$ such that $P\left(a_{r_t}\begin{pmatrix} 1& it \\
0 & 1\end{pmatrix}\right)\to -2^+$. Now we can use the argument given in the proof of Th. 5.4 in [@KOS].
Secondly, let us mention that Th. 5.1 and Th. 5.4 are true for all positive definite spherical functions $\neq {\bf 1}$, not only for those with imaginary parameters as stated – the argument is literally the same.
Theorem \[th=sl2r\] says that there are different domains of holomorphy for different series of representations. We expect an analogous result for arbitrary semisimple Lie groups and intend to return to this topic elsewhere.[^3] Let us mention that the crown is maximal for the class of non-trivial spherical unitary representations of $G$ by [@KOS], Th. 5.4 and the remark above.
Holomorphic extension of spherical functions
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This section is a short essay on spherical functions on $X$ which highlights their natural holomorphic extension to the crown $\Xi$. Here $G$ is arbitrary semisimple (within our self-imposed restrictions).
As always some notations upfront. For $\alpha\in \Sigma$ write $\gf^\alpha$ for the corresponding root space. Choose a positive system $\Sigma^+$ and define $\nf=\sum_{\alpha\in \Sigma^+} \gf^\alpha$. Set $N=\exp \nf$. The Iwasawa decomposition $G=NAK$ yields the analytic diffeomorphism $$\label{eq=iwa}
N\times A\mapright{\simeq} X,\ \ (n,a)\to na.x_0\, .$$ In particular, every $x\in X$ can be uniquely written as $x=n(x) a(x).x_0$ with $n(x)\in N$ and $a(x)\in A$ both depending analytically on $x$.
Let $N_{\mathbb{C}}=\exp\nf_{\mathbb{C}}$. If we complexify the Iwasawa decomposition of $X$ we obtain a Zariski open subset $N_{\mathbb{C}}A_{\mathbb{C}}.x_0\subsetneq X_{\mathbb{C}}$ which contains the crown, i.e. $$\label{eq=cin} \Xi\subset N_{\mathbb{C}}A_{\mathbb{C}}.x_0$$ (see [@KSI] for classical groups and [@Hu][^4] as well as [@Ma] in general). Let us mention that $\Omega\subset \af$ is a maximal domain for the inclusion (\[eq=cin\]) to hold, i.e. $G\exp(i\pi \tilde \Omega/2)\not\subset N_{\mathbb{C}}A_{\mathbb{C}}.x_0$ for any domain $\tilde \Omega\subset \af$ strictly containing $\Omega$ (cf. [@Ba] and [@KSII], Th. 2.4 with proof).
Define the finite $2$-group $F=A_{\mathbb{C}}\cap K_{\mathbb{C}}=T\cap K$ and record that the map $$N_{\mathbb{C}}\times A_{\mathbb{C}}/ F\mapright{\simeq} N_{\mathbb{C}}A_{\mathbb{C}}.x_0, \ \
(n,aF)\mapsto na.x_0$$ is biholomorphic. It follows that each element $z\in N_{\mathbb{C}}A_{\mathbb{C}}.x_0$ can be uniquely expressed as $z=n_{\mathbb{C}}(z) a_{\mathbb{C}}(z).x_0$ with $n_{\mathbb{C}}(z)\in N_{\mathbb{C}}$ and $a_{\mathbb{C}}(z)\in A_{\mathbb{C}}/F$ both holomorphic in $z$. One obtains an $N$-invariant holomorphic assignment $$a_{\mathbb{C}}: \Xi \to A_{\mathbb{C}}/ F$$ We have already remarked that $\Xi$ is contractible and this yields that $a_{\mathbb{C}}$ lifts to a holomorphic map $\Xi \to A_{\mathbb{C}}$, as well denoted by $a_{\mathbb{C}}$, such that $a_{\mathbb{C}}(x_0)=e$. Likewise there is a holomorphic logarithm $\log a_{\mathbb{C}}: \Xi \to \af_{\mathbb{C}}$ extending $\log a: X\to \af$. In particular, for all $\lambda\in \af_{\mathbb{C}}^*$ we can define the holomorphic $\lambda$-power of $a_{\mathbb{C}}$ by
$$a_{\mathbb{C}}(z)^\lambda=e^{\lambda (\log a_{\mathbb{C}}(z))} \qquad (z\in \Xi)$$
We would like to mention the complex convexity theorem ([@GKII], [@KOt]) which states that $$\label{cc}
{\mbox{\rm Im}\,}\log a_{\mathbb{C}}(G\exp(iY).x_0)={\rm co} (W.Y) \qquad (Y\in \pi\Omega/ 2)$$ with ${\rm co}(\cdot)$ denoting the convex hull of $(\cdot)$. As a consequence we obtain a refinement of the inclusion (\[eq=cin\]): $$\Xi \subset N_{\mathbb{C}}AT_\Omega.x_0$$
For $\alpha\in\Sigma$ let us define $m_\alpha=\dim \gf^\alpha$ and note that the multiplicity assignment $\alpha\mapsto m_\alpha$ is $W$-invariant. As usual we set $\rho={1\over 2}\sum_{\alpha\in \Sigma^+} m_\alpha \alpha$. Motivated by our previous discussion we define the spherical function with parameter $\lambda\in \af_{\mathbb{C}}^*$ ab initio as a holomorphic function on $\Xi$:
$$\phi_\lambda(z)=\int_K a_{\mathbb{C}}(kz)^{\rho +i\lambda} \, dk \qquad (z\in \Xi)$$
Of later relevance for us will be the doubling formula for spherical functions ([@KSI], Th. 4.2). For the convenience of the reader we briefly recall the short argument. We translate the inclusion (\[eq=cin\]) into representation theory: Using the compact realization of a spherical minimal principal series module $(\pi_\lambda,\H_\lambda)$ one shows that the orbit map of a spherical vector $v_{\lambda}\in\H_\lambda$ $$\label{or1}
F: X\to \H_\lambda,\ \ x\to\pi_\lambda(x)v_\lambda$$ extends to a holomorphic map $$\label{or2}
F: \Xi\to \H_\lambda\ \ z\to\pi_\lambda(z)v_\lambda$$ see [@KSI], Prop. 4.1. This allows us to express the spherical function $\phi_\lambda$ as a holomorphic matrix coefficient $\phi_\lambda(z)=\langle v_\lambda,\pi_\lambda(z)v_\lambda \rangle$ for $z\in \Xi$ (where we have adopted the physicist’s convention that sesquilinear pairings are linear on the right hand side, and anti-linear on the left hand side).
Now let $z\in AT_\Omega$ such that $z^2\in AT_\Omega$ and observe that $$\phi_\lambda(z^2)=\langle \pi_\lambda^*(\overline{z^{-1}})
v_\lambda,\pi_\lambda(z)v_\lambda\rangle$$ with $\pi_\lambda^*$ the conjugate contragredient representation. It follows that $\phi_\lambda|_{AT_\Omega}$ extends to a holomorphic function on $AT_\Omega^2$. In particular we see from this formula in the case of the unitary spherical minimal principal series $\lambda\in \af^*$ that the function $\phi_{\l}$ is positive on $T_\Omega^2$, and that for all $x=gt.x_0\in \Xi$ (recall the notation $v_{\l}^x:=\pi_\lambda(x)v_{\l}$): $$\begin{aligned}
\label{eq:l2norm}
\langle v_{\l}^x,v_{\l}^x\rangle&=\phi_{\l}(t^2)\\
\nonumber &=\int_K |a_{\mathbb{C}}(kt)^{2(\rho+i\lambda)}|dk.\end{aligned}$$
Sharp uniform lower bound for holomorphically extended orbit maps of spherical representations
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Given a non-trivial unitary spherical representation $(\pi, \H)$ of $G$ with normalized $K$-spherical vector $v_K$ we wish to control the norm of the holomorphically extended orbit map $$F_\pi: \Xi\to {\mathbb{C}}, \ \ z\mapsto \pi(z)v_K$$ in two aspects:
- For $z\in \Xi$ sufficiently close to $\partial_d\Xi$ we are aiming to give optimal lower bounds for $\|F_\pi(z)\|$ uniform in the representation parameter $\lambda(\pi)\in \af_{\mathbb{C}}^*$;
- For fixed $\pi$ we are looking for optimal upper bounds of $\|F_\pi(z)\|$ for $z$ approaching the distinguished boundary.
In view of the fundamental identity (\[eq:l2norm\]) we can translate the problems above into growth behavior of analytically continued spherical functions. In this section and the next we will address these two aspects. We begin with the uniform lower bounds.
Fix a distinguished boundary point $t=z_j=\exp(i\pi\omega_j/2k_j).x_0$ of the crown domain. For $0<\e<1$ set $$t_\e =\exp(i(1-\e)\pi\omega_j/2k_j).x_0\, .$$ The objective of this section is to provide sharp lower estimates for $\phi_\lambda(t_\e^2)$ which are uniform in $\e$ and $\lambda\in \af^*$. Our approach is based on the doubling identity (\[eq:l2norm\]) which implies that $$\label{ble} \phi_\lambda(t_\e^2)\geq \int_U
|a_{\mathbb{C}}(kt_\e)^{2(\rho +i\lambda)}|
\ dk$$ where $U$ is any neighborhood of $e\in K$. It turns out that the desired estimate will depend on the nature of the distinguished boundary point $t=z_j$, in particular whether $z_j$ is minuscule or not. We will treat the minuscule case first and later reduce the general case to the minuscule situation.
If $t=z_j$ is minuscule, then $G=Z(t^4)$ by Lemma \[lem=m\], i.e. $t^4$ is central. The following lemma, especially seen in the context of (\[cc\]), is quite remarkable.
\[lem=key\]Let $t=z_j$ be a minuscule boundary point and $U$ a connected and simply connected compact neighborhood of $e\in K$ such that $Ut\subset N_{\mathbb{C}}A_{\mathbb{C}}.x_0$. Then for all $k\in U$ the middle projection $a_{\mathbb{C}}(kt)\in A_{\mathbb{C}}$ is well defined and we have $a_{\mathbb{C}}(kt)=r(kt)t$ with $r(kt)\in A$ continuously depending on $k$.
The assertion of the lemma is local and thus it is no loss of generality to assume that $G\subset G_{\mathbb{C}}$ with $G_{\mathbb{C}}$ simply connected. In particular the Cartan involution $\theta:G\to G$ extends to a holomorphic involution on $G_{\mathbb{C}}$, again denoted by $\theta$. We notice that $G_{\mathbb{C}}^\theta=K_{\mathbb{C}}$. Likewise $G_{\mathbb{C}}$ admits a complex conjugation $g\mapsto \overline g$ with respect to $G$
Fix $k\in K$. Then $kt=nak'$ for some $n\in N_{\mathbb{C}}$, $a\in A_{\mathbb{C}}$ and $k'\in K_{\mathbb{C}}$. Define $x:=kt\theta(kt)^{-1}$ and note that
$$\label{*}
x=kt^2k^{-1}=na^2\theta(n)^{-1}\, .$$
On the other hand, as $t^4$ is central, $$\label{**}
t^{-4}x=kt^{-2}k^{-1}=\overline{kt^2k^{-1}}
=\overline{x}=\overline {n}\overline{a}^2\theta(\overline n)^{-1}\, .$$ Combining the information of (\[\*\]) and (\[\*\*\]) yields $$t^4 \overline {n} \overline {a}^2 \theta(\overline {n})^{-1}=
x=na^2 \theta(n)^{-1}\, .$$ Once more we use the fact $t^4$ is central and obtain $$t^4 \overline{a}^2 = \underbrace{(\overline {n}^{-1} n)}_{\in N_{\mathbb{C}}}
a^2
\underbrace{\theta(n^{-1}\overline {n})}_{\in \theta(N_{\mathbb{C}})}\, .$$ Bruhat implies that $\overline{n}=n$. Consequently $t^4=a^2\overline{a}^{-2}$, and this forces $a=r(tk) t$ for some $r(tk)\in A$.
Choose $0<\e_0<1$ small enough such that $Ut_\e\subset N_{\mathbb{C}}A_{\mathbb{C}}.x_0$ for all $\e\in (0,\e_0)$. In particular $a_{\mathbb{C}}(kt_\e)$ is well defined for all $k\in U$ and $\e\in (0, \e_0)$. As $a_{\mathbb{C}}(kt)\in A t$ for all $k\in U$ by the lemma, linear Taylor approximation yields that there are balls $B_r, B_{r'}$ in $\af$ centered at $0$ with radii $r, r'>0$ such that
$$a_{\mathbb{C}}(kt_\e)\in t \exp(B_r) \exp(i\e B_{r'})$$
for all $k\in U$ and $\e\in (0,\e_0)$. Thus it follows that there exists a constant $c>0$ such that for all $\lambda\in \af^*$, $k\in U$ and $\e\in (0, \e_0)$ the estimate $$\label{zztop}
|a_{\mathbb{C}}(kt_\e)^{2(\rho+ i\lambda)}|\geq c e^{\lambda(\pi\omega_j) -
r''\e |\lambda|}$$ holds for some $r''\geq r'$.
\[prop=before\] Let $t=z_j=\exp(i\pi \omega_j/2).x_0$ be a minuscule boundary boundary point of $T_\Omega$. Then there exist constants $\e_0\in(0,1)$ and $R>0$, $C>0$ such that $$\phi_\lambda(t_\e^2)\geq C\max_{w\in W}e^{ \pi\l(w\omega_j)(1 - R\e)},$$ for all $\l\in\af^*$ and $\e\in(0,\e_0)$.
According to Harish-Chandra one has $\phi_\lambda=\phi_{w\lambda}$ for all $\lambda$. Thus it is no loss of generality to assume that $\lambda(\omega_j)=\max_{w\in W} \lambda(w\omega_j)$. We notice that $||\lambda||:=\max_{w\in W} \lambda(w\omega_j)$ defines a norm on $\af^*$. Hence, by the equivalence of norms on Euclidean spaces, there exist a constant $d>0$ such that $|\cdot |\leq d||\cdot ||$. Now the the assertion follows from (\[zztop\]) and the basic lower estimate (\[ble\]).
Let us now turn to the general case where $t=z_j=\exp(i\pi\omega_j/ 2k_j).x_0$ is an arbitrary extremal boundary point of $T_\Omega$. We recall the groups $G_j=Z(t^4)$ with Lie algebra $\gf_j$. The main result of this section is:
\[lowerth\] Let $t=\exp(i\pi \omega_j/2k_j)$ be an extremal boundary point of $T_\Omega$. Then there exist constants $\e_0\in(0,1)$ and $R>0$, $C>0$ such that $$\phi_\lambda(t_\e^2)\geq
C \e^{(\operatorname{dim}\gf-\operatorname{dim}
\gf_j)/4}\max_{w\in W} e^{\pi\l(w\omega_j)(1-R\e)}$$ for all $\l\in\af$, and for all $\e\in(0,\e_0)$.
First, for $\omega_j$ minuscule one has $\gf_j=\gf$ and the assertion follows from Proposition \[prop=before\] above. The general case will be reduced to this situation.
We begin with some remarks on the reductive Lie algebra $\gf_j$. Recall that $\gf_j$ is $\theta$-stable and hence $\gf_j=\kf_j\oplus \pf_j$ with $\kf_j=\kf\cap \gf_j$ and $\pf_j=\pf\cap \gf_j$. By definition $\af\subset\gf_j$ and hence $\af$ is maximal abelian in $\pf_j$. Let $\Sigma_j=\Sigma(\gf_j,\af)$ be the corresponding reduced root system and $\Omega_j\subset \af$ the associated polyhedron. A quick look at our classification of the $\gf_j$’s shows that $\gf_j$ is simple modulo a compact ideal. Hence $\Sigma_j$ is irreducible and $\omega_j/k_j$ becomes a minuscule boundary point of $\Omega_j$.
Write $\kf_j^\perp$ for the orthogonal complement to $\kf_j$ in $\kf$ with respect to the Cartan-Killing form of $\gf$. Let $V_j,V_j'$ be small balls around $0$ in $\kf_j$,$\kf_j^\perp$ such that the map $$V_j\times V_j'\to K,\ \ (v,v')\mapsto \exp(v)\exp(v')$$ is a diffeomorphism. Set $U_j=\exp(V_j)$, $U_j'= \exp(V_j')$ and define $U=U_j U_j'$. Then $U$ is a connected and simply connected neighborhood of $e$ in $K$. We assume that $Ut\subset N_{\mathbb{C}}A_{\mathbb{C}}.x_0$ and choose $\e_0>0$ such that $Ut_\e\subset N_{\mathbb{C}}A_{\mathbb{C}}.x_0$ for $\e\in (0,\e_0)$ holds in addition.
Our previous discussion combined with Lemma \[lem=key\] implies that $$\label{eq=1} a_{\mathbb{C}}(kt)\in r(kt) t \qquad \forall k\in U_j'$$ and $r(kt)\in A$ depending continuously on $k$. Next consider the map $\psi: U\to A_{\mathbb{C}}, k\mapsto a_{\mathbb{C}}(kt)$. We claim that $$\label{eq=2}
d\psi(e)= 0\, .$$ In fact, this is well known, and follows from $\operatorname{pr}_{\af_{\mathbb{C}}}(\operatorname{Ad}(t^{-1})\kf)=\{0\}$ with $\operatorname{pr}_{\af_{\mathbb{C}}}: \gf_{\mathbb{C}}\to \af_{\mathbb{C}}$ the linear projection along $\nf_{\mathbb{C}}\oplus \kf_{\mathbb{C}}$.
Using the information of (\[eq=1\]) and (\[eq=2\]), linear Taylor approximation yields constants $r,r'>0$ such that $$\label{eq=3}
a_{\mathbb{C}}(\exp(v)\exp(v')t_\e)\in t \exp(B_r)\exp(i(\e +\|v'\|^2) B_{r'})$$ for all $(v,v')\in V_j\times V_j'$ and $\e\in(0,\e_0)$. It follows from equation (\[ble\]) that there exists a constant $c>0$ such that $$\label{ble'}\phi_\lambda(t_\e^2)\geq c
\int_{V_j} \int_{V_j'}
|a_{\mathbb{C}}(\exp(v)\exp(v')t_\e)^{2(\rho +i\lambda)}|
\ dv dv'$$ for all $\lambda\in \af^*$ and $\e\in (0,\e_0)$. Thus if we choose $V_j'$ to be ball of radius $\sqrt{\e}$, then (\[eq=3\]) and (\[ble’\]) yield constants $r'', c'>0$ such that $$\phi_\lambda(t_\e^2) \geq c' \e^{ (\operatorname{dim} \kf_j^\perp)/2}
e^{\lambda(\pi\omega_j/k_j) -r''\e|\lambda|}\,$$ for all $\lambda\in \af^*$ and $\e\in (0,\e_0)$ (note that the $\e$ -dependence of $V_j'$ is incorporated in the factor $\e^{ (\operatorname{dim} \kf_j^\perp)/2}$). We observe that $\operatorname{dim} \gf-\operatorname{dim}
\gf_j= 2 \operatorname{dim} \kf_j^\perp$ and finish the proof with the same argument for Proposition \[prop=before\] before.
We consider the lower estimate in Theorem \[lowerth\] is optimal. It is for the following reason: the crucial point in the above argumentation was the fact that $d\psi(e)=0$, to be very precise it was the fact $d ({\mbox{\rm Im}\,}\log \psi)(e)|_{\kf_j^\perp}=0$ which entered. This is actually the best one can hope for as the second derivative $d^2({\mbox{\rm Im}\,}\log \psi)(e)$ is already non-degenerate on $\kf_j^\perp\times \kf_j^\perp$. In fact, fix $\lambda \in \af^*$ regular, and set $F_\lambda =\lambda \circ {\mbox{\rm Im}\,}\log \psi$. Then [@DKV], pp. 343–346, implies $$d^2 F_\lambda (e)(Z,W)=-{1\over 2}\sum_{\alpha\in \Sigma^+}
\langle \alpha, \lambda\rangle {\mbox{\rm Im}\,}(1-t^{-2\alpha}) \langle Z_\alpha, W_\alpha
\rangle\qquad (Z,W\in \kf)$$ where $Z_\alpha$, resp. $W_\alpha$ is the orthogonal projection of $Z$, resp. $W$, onto $\kf\cap (\gf^\alpha+
\gf^{-\alpha})$. In particular if $\alpha\in \Sigma\backslash \Sigma_j$, then ${\mbox{\rm Im}\,}(1-t^{-2\alpha})\neq 0$. It follows that $d^2 F_\lambda(e)$ and hence $d^2({\mbox{\rm Im}\,}\log \psi)(e)$ is non-degenerate on $\kf_j^\perp\times\kf_j^\perp$.
Sharp upper bound for holomorphically extended orbit maps of spherical representations
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In this section we consider the problem to give an upper estimate for the square norm of the holomorphic extension of the orbit map $\Xi\ni x\to v^x\in\mathcal{H}$. Recall from (\[eq:l2norm\]) that $$\label{eq:orbnorm}
(v^{x}_{i\l},v^{x}_{i\l})=\phi^X_{i\l}(t^2.x_0)$$ if $x=gt.x_0\in\Xi$ with $g\in G$ and $t\in T_\Omega$. Here $\phi^X_\mu$ denotes the elementary spherical function on $X$ with spectral parameter $\mu\in\af_{\mathbb{C}}^*$. Therefore we concentrate on the question of estimating the singular behavior of the holomorphic extension of the restriction of the elementary spherical function $\phi^X_{\mu}|_{A}$ to $A_{\mathbb{C}}\supset A$ when we approach $t^2.x_0\in T.x_0\subset A_{\mathbb{C}}.x_0$ where $t=t(\eta)=\exp(i\pi\eta/2)$ with $\eta=\omega_j/k_j\in\af_{\mathbb{C}}$ an extremal boundary point of $\Omega$ in $\Omega\cap C$. Thus we are interested in the singular expansion in $\epsilon$ of the pull-back of (the holomorphic continuation of) spherical functions $\phi_\mu$ via the embedding $\mathbb{D}^\times\ni\epsilon\to A_\mathbb{C}.x_0$ of a small punctured disk $\mathbb{D}_r^\times
=\{\e\in{\mathbb{C}}\mid 0<|\e|< r\}$ given by $\epsilon\to t_\e^2.x_0$. For $\mu$ fixed the restriction of $\phi^X_\mu$ has a convergent logarithmic singular expansion at $\epsilon=0$. This means that there exists a finite set $S\subset{\mathbb{C}}\times\mathbb{Z}_{\geq0}$ such that if $(s,l),(s^\prime,l^\prime)\in S$ then $s-s^\prime\not\in\mathbb{Z}\backslash\{0\}$ and such that we have a unique decomposition (for $\epsilon$ varying in any sector $S_{r,\theta_1,\theta_2}=\{\e\in\mathbb{D}_r^\times\mid \theta_1<
\operatorname{arg}(\e)<\theta_2\}$ of $\mathbb{D}_r^\times$) of the form $$\label{eq:leadexp}
\phi^X_{\mu}(t^2_\e.x_0)=\sum_{(s,l)\in S}
\e^s\log^l\e f_{s,l}(\e)$$ where each $f_{s,l}$ is holomorphic on $\mathbb{D}_r$ and such that $f_{s,l}(0)\not=0$.
The projection of the set $S$ on the first factor ${\mathbb{C}}$ is called the set of exponents of the pull back of $\phi^X_\mu$ to $\mathbb{D}^\times_r$. In our case this set will always belong to $\mathbb{R}$. The minimum of this set is denoted by $s^X_{\eta,\mu}$ and is called the *leading exponent* of the singular expansion of the pull back of $\phi^X_\mu$ to $\mathbb{D}^\times_r$. We call the largest $l\in\mathbb{Z}_{\geq0}$ such that the $(s^X_{\eta,\mu},l)\in S$ the *logarithmic degeneracy* of the leading exponent.
So our problem boils down to the determination of the leading exponent $s^X_{\eta,\mu}$ at $\e=0$ of the pull back of $\phi^X_\mu$ on the distinguished embedded punctured disk given above, and its logarithmic degeneracy. In the Appendix \[app:exp\] we define an appropriate notion of the exponent of a regular holonomic system of differential equations and using the basic properties of these exponents we compute the exponents of $\phi_\mu$ at the extremal boundary points $\eta$ of $\Omega\cap C$ for $\phi_\mu$ a solution of a more general system of differential equations, namely the system of hypergeometric equations associated with the root system $\Sigma$. This system of equations is a parameter deformation of the system of equations for the restriction for the elementary spherical functions $\phi^X_\mu$ to $A_{\mathbb{C}}.x_0$. This deformation is an essential ingredient for the computations of the exponents. These results imply the following:
\[thm:leadexp\] We use the notations as introduced above. Consider the functions $s_\eta(m)$ and $d_\eta(m)$ of the multiplicity parameters $m=(m_\a)$ as listed in the table in Theorem \[thm:table\]. Suppose that the Riemannian symmetric space $X$ has root system $\Sigma$ and root multiplicity parameters $m^X=(m^X_\a)$, then we put $s^X_\eta:=s_\eta(m^X)$ and $d_\eta^X=d_\eta(m^X)$.
For all $\mu\in\af_{\mathbb{C}}^*$ we have $s^X_{\eta,\mu}\geq s^X_\eta$, and if $s^X_{\eta,\mu}= s^X_\eta$ then $d^X_\eta$ is an upper bound for the logarithmic degeneracy of $s^X_{\eta,\mu}$.
We postpone the proof of this theorem in the general case to the Appendix \[app:exp\]. For the complex cases (i.e. when $X$ is a Riemannian symmetric space of type IV) the proof will be given below.
As an immediate consequence of theorem \[thm:leadexp\] we have:
\[thm:esteta\] We use the notations as introduced above. Given an extremal boundary point $\eta=\omega_j/k_j$ of $\Omega$ we consider $t_\epsilon=\exp(i\pi\eta/2)\in A_{\mathbb{C}}$. Fix $-\pi<\theta_1<\theta_2<\pi$. Let $\mu\in\af_{\mathbb{C}}^*$, then there exist constants $r>0$, $C>0$ such that for all $\e\in S_{r,\theta_1,\theta_2}$: $$\label{eq:mainest}
|\phi^X_\mu(t^2_\epsilon.x_0)|\leq C\e^{s^X_\eta}|\log(\e)|^{d^X_\eta}$$ where $s_\eta^X=s_\eta(m^X)$ and $d_\eta^X=d_\eta(m^X)$ for the functions $s_\eta$ and $d_\eta$ listed in Theorem \[thm:table\].
For later applications it is useful to have a slightly weaker but more handy version of the estimate above. Let us define
$$\label{def=ds} s^X:=\max_\eta s_\eta^X\quad\hbox{and}
\quad
d^X:= \max_{\eta:s_\eta^X=s^X} d_\eta^X\, .$$
The theorem above combined with the maximum principle of holomorphic functions then yields:
\[thm=se\] For each $\mu\in \af_{\mathbb{C}}^*$ there exists a constant $C=C(\mu)>0$ such that for all $Y\in \partial\Omega$ and $0<\e<1$ $$|\phi^X_\mu(\exp(i(1-\e) \pi Y).x_0)|\leq C\e^{s^X}|\log(\e)|^{d^X}\, .$$
Since $\Xi$ only depends on the isogeny class of $G$ (Remark \[rem=bd\](c)) it suffices to do the analysis in the situation where $G_{\mathbb{C}}$ is simply connected. In addition we assume the restricted root system $\Sigma$ of $X$ to be irreducible. Recall that twice the character lattice of $A_{\mathbb{C}}$ is equal to the weight lattice of the restricted root system $\Sigma^l$. The categorical quotient $W\backslash A_{\mathbb{C}}$ (as well as $W\backslash A_{\mathbb{C}}/F$) is affine space.
There are two special cases which can be treated by direct methods, the real rank one case and the complex case. It is both instructive and useful to consider these cases first before going to the general case which is treated in the Appendix \[app:exp\].
### The real rank one case
This case was treated in detail in [@KSI Theorem 5.1] but it is useful to discuss the difference between the approach used in [@KSI] and the approach in the present paper.
Let $G$ be a real semisimple group with real rank one. Then $\Sigma=\Sigma(\mathfrak{g},\mathfrak{a})$ is of the form $\Sigma=\{\pm\alpha\}$ (reduced case) or $\Sigma=\{\pm\alpha/2,\pm\alpha\}$ (non-reduced case). Let $\eta\in\mathfrak{a}$ be such that $\alpha(\eta)=1$, so that $\eta$ is at the boundary of $\Omega$. We put $q:=\dim\mathfrak{g}^\alpha$ and $p:=\dim\mathfrak{g}^{\alpha/2}$ (so that $q\geq 1$, with $p=0$ (reduced case) or $p\geq q$ (non-reduced case); this is convenient but admittedly a bit unconventional in the reduced case).
In [@KSI] the Harish-Chandra integral representation for the spherical function $\phi_\lambda$ is analyzed directly to obtain the precise asymptotic behaviour of the holomorphic extension of $\phi_\lambda(\exp(i\pi(1-\epsilon)\eta))$ as $\epsilon\to 0$ if $\lambda\in i\mathfrak{a}^*$. The result of [@KSI] says that for $\lambda(\alpha^\vee)\in i\mathbb{R}$ one has $$\label{eq:asexplog}
\phi_\lambda(\exp(i\pi(1-\epsilon)\eta))\asymp |\log\epsilon|$$ if $q=1$, and $$\label{eq:asexp}
\phi_\lambda(\exp(i\pi(1-\epsilon)\eta))\asymp \epsilon^{1-q}$$ if $q>1$.
The method of the present paper is not based on the analysis of the integral representation of the spherical function $\phi_\lambda$ but rather on the analysis of the radial system of eigenfunction equations for $\phi_\lambda$ with respect to the commutative algebra of $G_\mathbb{C}$-invariant differential operators on $\Xi$. In the case at hand, this amounts to the well known fact that $$\phi_\lambda(\exp(i\pi(1-\epsilon)\eta))=F(a,b,c;z)$$ with $$\begin{aligned}
a&=\lambda(\alpha^\vee)+p/4+q/2\\
b&=-\lambda(\alpha^\vee)+p/4+q/2\\
c&=1/2+p/2+q/2\\
z&=(1-\cos(\pi(1-\epsilon)))/2=1-\pi^2\epsilon^2/4+\dots\end{aligned}$$ where $F(a,b,c;z)$ (with $a,b,c\in\mathbb{C}, |z|<1$) denotes the Gauss hypergeometric function. This function is the unique holomorphic solution of the hypergeometric differential equation $$\label{eq:Ghyp}
z(1-z)\frac{d^2F}{d^2z}+(c-(a+b+1)z)\frac{dF}{dz}-abF=0$$ on the unit disc, normalized by $F(a,b,c;0)=1$. The hypergeometric differential equation is a second order Fuchsian equation with regular singularities at $z=0, z=1$ and $z=\infty$. The exponents of (\[eq:Ghyp\]) at $z=1$ are easily seen to be equal to $0$ and $c-a-b=1/2(1-q)$. Hence the problem of finding the singular expansion of $\phi_\lambda(\exp(i\pi(1-\epsilon)\eta))$ at $\epsilon=0$ is now transformed to a simple exercise on the analytic continuation of the solutions of (\[eq:Ghyp\]). If $\phi_\lambda(\exp(i\pi(1-\epsilon)\eta))=F(a,b,c;z)$ is holomorphic at $z=1$ (i.e. at $\epsilon=0$) then $F(a,b,c;z)$ extends holomorphically to $\mathbb{C}$. Since $\infty$ is a regular singular point of (\[eq:Ghyp\]) this implies that $F(a,b,c;z)$ is a polynomial in $z$ in this case (essentially a so-called Jacobi polynomial). The hypergeometric series terminates iff $a$ or $b$ is a non-positive integer (since $c$ is not a negative integer in our situation), i.e. iff $\lambda(\alpha^\vee)=\pm(p/4+q/2+n)$ with $n\in\mathbb{Z}_{\geq 0}$. Hence for these values of $\lambda$ the function $\phi_\lambda(\exp(i\pi(1-\epsilon)\eta))$ is asymptotic to a constant (for $\epsilon\to 0$), and for all other values of $\lambda$ we find $(\ref{eq:asexplog})$ or $(\ref{eq:asexp})$ (whichever is relevant).
In the general case we need to deal with Harish-Chandra’s radial system of eigenfunction equations for the $G$-invariant differential operators, which is a holonomic system of differential equations on $W\backslash A_\mathbb{C}/F$ with regular singularities along the discriminant locus. This gives rise to various questions and problems which are addressed in the Appendix \[app:exp\]. First of all it is not clear in general how to define a set of singular exponents at a point of the singular locus of a regular singular holonomic system. In Subsection \[sub:expAR\] we give an appropriate definition of the notion of exponents in the special case of a holonomic system of differential equations with regular singularities along an affine hyperplane arrangement. We apply this definition to the pull back to $\mathfrak{a}_\mathbb{C}$ of the Harish-Chandra system of equations. The next problem is the actual computation of the set of exponents at $\eta$. We do not know how to do this directly in an algebraic way in general; instead we use the relation between the exponents and monodromy of the solutions of the system of equations and, in a crucial way, the parameter deformation of the Harish-Chandra system provided by the so-called hypergeometric system of differential equations for root systems. Via the monodromy representation affine Hecke algebras play a role in dealing with these computations. A final problem in the higher rank case is the selection of those exponents among the full set of exponents at $\eta$ which may be involved in the expansion of $\phi_\lambda$ at $\eta$, and among those, the leading exponent. Again some basic representation theory of the affine Hecke algebra plays an important role at this stage. The outcome is remarkable; the leading exponent is attached to a specific irreducible character of the isotropy group $W^a_\eta$ of $\eta$ in the affine Weyl group $W^a$. We call this irreducible character the *leading character* at $\eta$. It turns out that the leading character (for generic $\lambda$) depends on the local geometry of $\Omega$ at $\eta$, but not on the root multiplicities.
### The complex case
We consider the complex case $X=G/U$ where $G$ is the connected simply connected complex simple group and $U$ its maximal compact subgroup. As is well known by the work of Harish-Chandra [@HC0] the spherical functions are of an elementary nature in this case, allowing us to analyze the desired exponents directly. The restricted root system $\Sigma$ of $X$ is equal to twice the root system of $G$, and all root multiplicities are equal to $2$. We introduce the Weyl denominator $\delta$ on $A_{\mathbb{C}}/F$ by $$\delta(a)=\prod_{\a\in\Sigma_+}(\a(a)-\a(a)^{-1})$$ and we denote by $A_{\mathbb{C}}^{\operatorname{reg}}/F$ the complement of the set $\d=0$ in $A_{\mathbb{C}}/F$. The algebra $\Ri_X$ of radial parts of invariant differential operators on $X$ consists of the differential operators on $A_{\mathbb{C}}^{\operatorname{reg}}/F$ of the form $$\Ri_X=\{\delta^{-1}\circ\partial(p)\circ\delta\mid p\in{\mathbb{C}}[\af^*]^W\}$$ The Harish-Chandra isomorphism $\gamma_X:\Ri_X\to{\mathbb{C}}[\af^*]^W$ is given by $\gamma_X(\delta^{-1}\circ\partial(p)\circ\delta)=\partial(p)$. Now $\phi^X_\mu|_{A_{\mathbb{C}}}$ satisfies the following system of eigenfunction equations $$\label{eq:cpx}
D\phi=\gamma_X(D)(\mu)\phi,\ \forall D\in\Ri_X$$ This is a $W$-equivariant system of differential equations on $A_{\mathbb{C}}^{\operatorname{reg}}/F$. It is also equivariant for the action of the $2$-group $F=A_{\mathbb{C}}\cap U$. Therefore we can view (\[eq:cpx\]) as a system of differential equations on $W\backslash A_{\mathbb{C}}/F-\{d=0\}$, where $d=\delta^2$ is the discriminant of $W$, viewed as a polynomial on $W\backslash
A_{\mathbb{C}}/F$. We call $\L_X$ the sheaf of local solutions of (\[eq:cpx\]).
A general local solution to this set of equations is of the form $$\label{eq:factor}
\phi=\d^{-1}\psi$$ where $\psi$ is a local solution of the constant coefficient system $$\label{eq:stein}
\partial(p)\psi=p(\mu)\psi,\ \forall p\in{\mathbb{C}}[\af^*]^W$$ Let $\exp:\af_{\mathbb{C}}\to A_{\mathbb{C}}$ denote the exponential map such that $\exp(2\pi iX)=1$ iff $X\in
Q(\Sigma^\vee)$. Consider the covering map $$\label{eq:cov}
\pi:\af_{\mathbb{C}}^{\operatorname{reg}}\to
W\backslash A_{\mathbb{C}}^{\operatorname{reg}}/F$$ where $\af_{\mathbb{C}}^{\operatorname{reg}}$ is the complement in $\af_{\mathbb{C}}$ of the set of affine root hyperplanes, the zero sets of the affine roots $a=\a-n$ (with $\a\in\Sigma$ and $n\in\mathbb{Z}$), and $\pi$ is given by $\pi(X)=W\exp(\pi i X)F$. The following proposition is well known.
\[prop:dim\] The space of solutions of (\[eq:cpx\]) on a nonempty open ball $U\subset W\backslash A_{\mathbb{C}}^{\operatorname{reg}}/F$ consists of holomorphic functions and has dimension $|W|$ (independent of $\mu$). Let $V\subset\pi^{-1}(U)$ be a connected component. The pull back of a local solution of (\[eq:cpx\]) on $U$ via $\pi|_V$ extends to a global holomorphic function on $\af_{\mathbb{C}}^{\operatorname{reg}}$.
We use the general form (\[eq:factor\]) of the local solutions. By a well known result of Steinberg [@S] the global solution space of (\[eq:stein\]) on $\af_{\mathbb{C}}$ has dimension equal to $|W|$ (independent of $\mu$) and consists of entire functions. On the other hand the left ideal in the ring of differential operators with holomorphic coefficients on $\af_{\mathbb{C}}$ generated by the operators $\partial(p)-p(\lambda)$ (with $p\in{\mathbb{C}}[\af^*]^W$) is cofinite, a complement being generated by constant coefficient operators $\partial(q)$ where $q$ is running over a set of polynomials representing a basis of the coinvariant algebra. Hence the local solution space is at most of dimension $|W|$. The proposition follows.
The system (\[eq:cpx\]) is holonomic of rank $|W|$. Upon choosing a base point $p\in\af_{\mathbb{C}}^{\operatorname{reg}}$ we may view the monodromy representation as a representation of the group of deck transformations of the covering map $\pi$, which is the affine Weyl group $W^a=W\ltimes Q(\Sigma^\vee)$ acting on $\af_{\mathbb{C}}$. The restriction of the monodromy representation to $W$ is equivalent to the regular representation.
All is clear except for the last assertion. By equation (\[eq:factor\]) it is enough to know this for the space of solutions of (\[eq:stein\]). This is well known, and follows from the case $\mu\in\af_{\mathbb{C}}^{\operatorname{reg}}$ by rigidity of characters of a finite group.
Notice that the center of the group ring of $W^a$ is ${\mathbb{C}}[Q(\Sigma^\vee)]^W$. Thus the central characters of irreducible representations of $W^a$ correspond to $W$-orbits of points of the complex algebraic torus $$T^L=\af_{\mathbb{C}}^*/P(\Sigma)$$ whose exponential map we will denote by $\exp^L$ (i.e. ($\exp^L(\mu)=1$ iff $\mu\in P(\Sigma)$). Then $T^L$ is the dual torus of $A_{\mathbb{C}}$.
The monodromy representation of (\[eq:cpx\]) has central character $W\operatorname{exp}^L(\mu)\in W\backslash T^L$ (observe that this central character is unitary iff $\mu\in\af^*$ is real). The monodromy representation is irreducible iff $\exp^L(\mu)$ has trivial isotropy for the action of $W$ on $T^L$.
The monodromy representation clearly has central character $W\exp^L{\mu}$, and dimension $|W|$ by Proposition \[prop:dim\]. The last assertion follows easily from the Mackey induction procedure.
The spherical function $\phi^X_\mu|_{A_{\mathbb{C}}}$ is a special solution of (\[eq:cpx\]), which can be characterized by saying that it is a nonzero $W$-fixed vector in the monodromy representation. By the above corollary the space of $W$-fixed vectors is one-dimensional. Looking at (\[eq:factor\]) we see that $\phi^X_\mu|_{A_{\mathbb{C}}}$ is of the form $\d^{-1}\psi$ where $\psi$ is a $W$-skew solution (say on $\af_{\mathbb{C}}$) of (\[eq:stein\]). Then $\psi$ is divisible by $\prod_{\a\in\Sigma_+}\a$, and thus $\phi^X_\mu|_{A_{\mathbb{C}}}$ extends to a holomorphic solution on $AT_\Omega^2$.
We continue the discussion by considering $\phi^X_\mu|_{A_{\mathbb{C}}}$ via $\pi$ as a holomorphic function on $i\af+\Omega$ of the form $\d^{-1}\psi$ with $\psi$ a $W$-skew solution of (\[eq:stein\]). Let $\eta=\omega_i/k_i$ be as before. We denote by $W_{\eta}$ the isotropy group of ${\eta}$ in $W$, and by $W_{\eta}^a$ the isotropy group of ${\eta}$ in the affine Weyl group $W^a$.
The natural homomorphism $W^a\to W$ restricts to an isomorphism from $W^a_{\eta}$ onto the isotropy group $W_{t(\eta)^2F}\subset W$ of $t(\eta)^2F=\exp(i\pi\eta)F$ for the action of $W$ on $A_{\mathbb{C}}/F$. In particular $W_{t(\eta)^2F}=W$ if $\eta\in\partial(\Omega)$ is minuscule.
The injectivity is clear. The isomorphism of complex algebraic tori $A_{\mathbb{C}}/F\approx A_{\mathbb{C}}$ given by $aF\to a^2$ is $W$-equivariant. Therefore $W_{t(\eta)^2F}$ is equal to the isotropy group of $W_{t(\eta)^4}$ for the action of $W$ on $A_{\mathbb{C}}$. Since $A_{\mathbb{C}}$ has the weight lattice $P(\Sigma)$ as its character lattice the group $W_{t(\eta)^4}$ is generated by reflections (by a well known result of Steinberg [@S2]). Now suppose that $s=s_\a\in W_{t(\eta)^4}$ is a reflection. But $s\in W_{t(\eta)^4}\Longleftrightarrow
s(\eta)-\eta\in Q(\Sigma^\vee)\Longleftrightarrow
\a(\eta)=n\in{\mathbb{Z}}$. Hence the affine reflection $s_{\a-n}$ satisfies $s_{\a-n}\in W^a_\eta$ and is mapped to $s$, proving the surjectivity.
Let $\Sigma_{+,{\eta}}^a$ be the set of positive affine roots which vanish on ${\eta}$, and let $\Sigma_{+,{\eta}}=\Sigma_{+,{\eta}}^a\cap\Sigma$. Then the Dynkin diagram of $\Sigma_{+,\eta}^a$ is $D^*-\{\a_\eta\}$ where $\a_\eta$ is the unique simple root of $\Sigma$ such that $\a_\eta(\eta)\not=0$, and $\Sigma_{\eta}\subset\Sigma_{\eta}^a$ is the maximal standard parabolic subsystem in which we delete $\a_0$ from the set of simple roots of $\Sigma_{\eta}^a$.
In a small neighborhood of the extremal boundary point $\eta$ of $\Omega$ we consider the Taylor expansion of $\psi$. The lowest homogeneous term $h_{\eta,\mu}$ of $\psi$ at $\eta$ is a $W$-harmonic polynomial which is $W_{\eta}$ skew, and $\phi^X_\mu|_{A_{\mathbb{C}}}\circ\pi$ can be uniquely expressed in the form $$\phi^X_\mu|_{A_{\mathbb{C}}}\circ\pi=
(\prod_{a\in\Sigma_{\eta,+}^a}a)^{-1}(h_{\eta,\mu}+
\mathrm{higher\ order\ terms\ at\ } \eta)$$ Since $h_{\eta,\mu}$ is divisible by $\prod_{\a\in\Sigma_{\eta,+}}\a$ we see that in this case the leading exponent $s_{\eta,\mu}^X$ satisfies $$s_{\eta,\mu}^X\geq s_\eta^X:=-|\Sigma_{\eta,+}^a-\Sigma_{\eta,+}|$$ For $\mu$ in an open, dense subset of $\af_{\mathbb{C}}^*$ this bound is sharp. The bound is sharp if $\mu=i\lambda$ with $\lambda\in\af^*$. It is not so easy to describe the function $\mu\to s_{\eta,\mu}^X$ exactly. We observe that this function is upper semi-continuous.
The above analysis can not be used directly in general, since the spherical functions do not have a simple factorization formula like (\[eq:factor\]) in general. For our later use it is helpful to describe the above result in terms of the monodromy representation. By the above we see that $h_{\eta,\mu}$ belongs to space of $W$-harmonic polynomials which transform by the sign representation under the action of $W_{\eta}$. This means that $h_\eta$ is a $W$-harmonic polynomial in the direct sum of the isotypical components of the irreducible characters of $W^a_{\eta}$ which are induced from the sign representation $\operatorname{det}_{\eta}$ of $W_{\eta}$. Therefore the homogeneous degree of $h_\eta$ at $\eta$ is at least equal to the harmonic birthday of the irreducible character (*the leading character*) $\tilde\sigma_\eta\in\operatorname{Irr}(W_{\eta}^a)$ given by the *truncated induction* $$\tilde\sigma_\eta\in\operatorname{Irr}(W_{\eta}^a)=
j_{W_{\eta}}^{W_{\eta}^a}(\operatorname{det_{\eta}})$$ It follows by truncated induction (see [@Ca Section 11.2]) that the harmonic birthday of this irreducible character is equal to $|\Sigma_{\eta,+}|$, and that this representation has multiplicity $1$ in this degree. Moreover, the same is true in the space of $W$-harmonic polynomials.
\[cpx\] Assume we are in the complex case, so $X$ is a Riemannian symmetric space of type IV with restricted root system $\Sigma$. Let $\eta\in\partial_e{\Omega}\cap{\mathbb{C}}$ be an extremal boundary point of $\Omega$ as before, and define $-s_\eta^X:=|\Sigma_{\eta,+}^a-\Sigma_{\eta,+}|$ as the number of roots $\a$ in $\Sigma_+$ with $\a(\eta)=1$. Let $s^X_{\eta,\mu}$ be the leading exponent at $z=t(\eta)^2.x_0$ in the sense of (\[eq:leadexp\]). For all $\mu\in\af_{\mathbb{C}}^*$ the logarithmic degeneracy of the leading exponent $s^X_{\eta,\mu}$ is $0$. For all $\mu\in\af_{\mathbb{C}}^*$ we have $s_{\eta,\mu}^X\geq s_\eta^X$. For $\mu$ in a dense open set of $\af_{\mathbb{C}}^*$ containing $i\af^*$ this inequality is an equality (cf. Theorem \[thm:table\]).
\[thm:table\] In Table 4 below we have used the numbering of the extremal boundary points $\eta_j=\omega_j/k_j\in\partial\Omega$ corresponding to the distinguished boundary orbits as in Table 1. Table 4 displays lower bounds for the leading exponents of the holomorphically extended elementary spherical functions at the extremal points $\eta_j$ in the sense of (\[eq:leadexp\]). In the case where this lower bound is attained the table displays the corresponding logarithmic degeneracy and a leading character (leading characters are explained in Appendix \[app:exp\]).
The convention for the root multiplicities is as follows. We use $m_1\geq 1$ for the root multiplicity of a long root $\a$ (or simply $m$ if $\Sigma$ is reduced and simply laced). The multiplicity of half a long root in $\Sigma$ is denoted by $m_{1/2}\geq 0$ (i.e. we view $C_n$ as the special case of $BC_n$ where $m_{1/2}=0$). The multiplicity of unmultipliable roots $\b\in\Sigma$ (i.e. $2\b\not\in\Sigma$) which are not long roots is denoted by $m_2\geq 1$ (if such roots exist).
Table 4
The proof Table 4 and these facts is given in the Appendix Section \[app:exp\]. In the parameter family of hypergeometric functions $\phi_{\mu,m}$ (cf. Appendix \[app:exp\]) with real multiplicities $m$ the indicated lower bounds are sharp generically in $m$, provided that $m$ satisfies the inequalities $1\leq m_1\leq m_2$. The leading character is independent of $m$ in this cone (hence only depends on the geometry of $\Omega$ at the extremal point $\eta$).
Unipotent model for the crown domain
====================================
In this section we give a new geometrical characterization of the crown by unipotent $G$-orbits. Fot the beginning there is no restriction on $G$ and we define a connected $G$-subset of $X_{\mathbb{C}}$ by
$$\Xi_N=G\exp(i\nf).x_0=GN_{\mathbb{C}}.x_0\, .$$ For $G=\Sl(2,{\mathbb{R}})$ we have shown that $\Xi_N$ is an open subset, but in the general case this is not clear to us. The next lemma contains the crucial information.
$\Xi\subset\Xi_N$.
Let $ Y\in \pi\Omega/ 2$. We recall the complex convexity theorem (\[cc\]) $${\mbox{\rm Im}\,}\log a_{\mathbb{C}}(K\exp(iY).x_0)={\rm co} (W.Y) \, .$$ In particular, there exists a $k\in K$ such that ${\mbox{\rm Im}\,}\log a_{\mathbb{C}}(k\exp(iY).x_0)=0$, or, in other words, $$k\exp(iY) \in N_{\mathbb{C}}A K_{\mathbb{C}}= A N_{\mathbb{C}}K_{\mathbb{C}}\, .$$ We conclude that $G\exp(iY)\subset GN_{\mathbb{C}}K_{\mathbb{C}}$ and then $G\exp(i\Omega)\subset GN_{\mathbb{C}}K_{\mathbb{C}}$, i.e. $\Xi\subset \Xi_N$.
Let us define a domain $\Lambda\subset \nf$ by
$$\Lambda=\{ Y\in \nf\mid \exp(iY).x_0\in \Xi\}_0$$ where $\{\cdot \}_0$ stands for the connected component of $\{\cdot \}$ containing $0$. As $\Xi$ is open, it is clear that $\Lambda$ is open as well.
\[lem=lb\]Suppose that $\Omega_c\subset \Omega$ is a compact subset. Then the set $$\Lambda_c=\{ Y\in \nf\mid \exp(iY).x_0\in G\exp(i\Omega_c).x_0\}$$ is compact in $\nf$.
First we observe that $\Lambda_c$ is closed as $G\exp(i\Omega_c).x_0$ is closed in $X_{\mathbb{C}}$.
Let $Y\in \Lambda_c$. Then $\exp(iY)=g\exp(iZ).x_0$ for some $g\in G$ and $Z\in \Omega_c$. With $g=n^{-1} a^{-1} k$ for $n\in N$, $a\in A$ and $k\in K$ we obtain that $$\label{eq=ii}k\exp(iZ).x_0= an\exp(iY).x_0\, .$$
We recall that $\Xi\subset A_{\mathbb{C}}N_{\mathbb{C}}.x_0$ and that there is a well defined holomorphic projection $$\tilde n: A_{\mathbb{C}}N_{\mathbb{C}}.x_0\to N_{\mathbb{C}}\, .$$ Further we note that the map $$N\times \nf\to N_{\mathbb{C}}, \ \ (n, Y)\mapsto n\exp(iY)$$ is a diffeomorphism. In particular $N\backslash N_{\mathbb{C}}\simeq \nf$ under a homeomorphic map $\psi$. Consider the continuous map $$f=\psi\circ \tilde n: A_{\mathbb{C}}N_{\mathbb{C}}.x_0\to \nf$$ and note that (\[eq=ii\]) shows that $f(k\exp(iZ).x_0)= Y$. Therefore $$\Lambda_c\subset f(K\exp(i\Omega_c).x_0)\, .$$ Since $K\exp(i\Omega_c).x_0$ is compact and $f$ is continuous, the assertion of the lemma follows.
We arrive at the main result of this section.
$\Xi=G\exp(i\Lambda).x_0$.
We argue by contradiction. Suppose that the assertion is false. Then there exists $Z\in \Omega$ such that $\exp(iZ).x_0\not\in G\exp(i\Lambda).x_0$ and sequences $(Z_n)_{n\in {\mathbb{N}}}\subset \Omega$, $(g_n)_{n\in {\mathbb{N}}}\subset G$ and $(Y_n)_{n\in {\mathbb{N}}}\subset \Lambda$ such that $Z_n\to Z$ and $\exp(iZ_n).x_0=g_n \exp(iY_n).x_0$. Let $\Omega_c\subset \Omega$ be a compact subset of $\Omega$ with $(Z_n)_{n\in {\mathbb{N}}}\subset \Omega_c $. Then $\exp(iY_n).x_0\in G\exp(i\Omega_c).x_0$ and we conclude from Lemma \[lem=lb\] that $(Y_n)_{n\in {\mathbb{N}}}$ is a bounded sequence in $\nf$. W.l.o.g. we may assume that $Y_n\to Y\in \nf$. As $\Omega_c$ is compact, the set $G\exp(i\Omega_c).x_0$ is closed in $X_{\mathbb{C}}$ (cf. Remark \[rem=tan\]) and thus $\exp(iY).x_0\in G\exp(i\Omega_c).x_0
\subset \Xi$. Hence $Y\in \Lambda$. Because $G$ acts properly on $\Xi$ we conclude that $(g_n)_{n\in {\mathbb{N}}}$ is bounded and it is no loss of generality to assume that $\lim_{n\to\infty} g_n =g$. But then $\exp(iZ).x_0=g\exp(iY).x_0\in
G\exp(i\Lambda).x_0$, a contradiction.
The determination of the precise shape of $\Lambda$ is a difficult problem, especially for higher rank groups. Generally one might ask: Is $\Lambda$ always bounded? Is $\Lambda$ convex?
Recall the fact that $G\exp(iZ).x_0=G\exp(iZ').x_0$ for $Z,Z'\in \pi\Omega/ 2$ means that $W.Z=W.Z'$. Thus we obtain a well defined map $$p: \Lambda\to \Omega/ W$$ via $G\exp(iY).x_0=G\exp(i\pi p(Y)/2).x_0$ for $Y\in \Lambda$. The following would be interesting to know: What are the fibers of the map $p$? What are the preimages of the extreme points? Is there an expressable relationship between $Y$ and $p(Y)$?
The case of real rank one
-------------------------
In this subsection we will determine the precise shape of $\Lambda$ for groups $G$ with real rank one. We begin with a criterion which will allow us explicit computations.
\[lem=r1\] Suppose that $G$ has real rank one. Then $$\Lambda=\{ Y\in \nf\mid N\exp(iY).x_0\subset \overline N_{\mathbb{C}}A_{\mathbb{C}}.x_0\}_0$$ with $\{ \cdot \}_0$ denoting the connected component of $\{ \cdot \}$ containing $0$.
Set $$\Lambda_1=\{ Y\in \nf\mid N\exp(iY).x_0\subset \overline N_{\mathbb{C}}A_{\mathbb{C}}.x_0\}_0$$ and note that $$\label{eq=l1}
\Lambda_1=\{ Y\in \nf\mid \exp(iY).x_0\subset\bigcap_{n\in N} n \overline N_{\mathbb{C}}A_{\mathbb{C}}.x_0\}_0\, .$$ We recall the fundamental fact on (general) complex crowns that $$\Xi=\left[\bigcap_{g\in G} g\overline N_{\mathbb{C}}A_{\mathbb{C}}. x_0\right]_0$$ with $[\cdot]_0$ denoting the connected component of $[, ]$ containing $x_0$. We are now going to use the fact that $G$ has real rank one. In particular $W=\{ 1, w\}={\mathbb{Z}}_2$ and the Bruhat decomposition of $G$ reads $G=N MA\overline N \cup w MA\overline N$. Hence $$\Xi=\left[\bigcap_{n\in N} n\overline N_{\mathbb{C}}A_{\mathbb{C}}. x_0\cap N_{\mathbb{C}}A_{\mathbb{C}}.x_0\right]_0\,.$$ As a result (\[eq=l1\]) translates into $$\Lambda_1=\{ Y\in \nf \mid \exp(iY)\in \Xi\}_0 =\Lambda\,$$ and the proof is complete.
We introduce coordinates on $\nf=\gf^\alpha + \gf^{2\alpha}$. As usual we write $p=\dim \gf^\alpha$ and $q=\dim \gf^{2\alpha}$ and let $c={1\over 4(p+4q)}$. We endow $\nf$ with the inner product $\langle Y_1, Y_2\rangle = - \kappa (Y_1, \theta(Y_2))$ where $\kappa$ denotes the Cartan-Killing form of $\gf$. For $z\in \Xi$ we write in the sequel $a_{\mathbb{C}}(z)$ for the $A_{\mathbb{C}}$-part of $z$ in the Iwasawa decomposition $\overline N_{\mathbb{C}}A_{\mathbb{C}}.x_0$. For $Y\in \gf^\alpha, Z\in \gf^{2\alpha}$ we recall the formula
$$\label{eq=e1} a_{\mathbb{C}}(\exp(Y+Z).x_0)^\rho =\left[(1+c\|Y\|^2)^2 + 4c \|Z\|^2\right]^{p+2q\over 4}\,.$$
The complex linear extension of $\langle\cdot,\cdot \rangle$ to $\nf_{\mathbb{C}}$ shall be denoted by the same symbol. We obtain the following criterion for $\Lambda$.
\[lem=exc1\] If $G$ has real rank one, then $$\begin{aligned}
\Lambda =&\{ (Y,Z)\in \gf^\alpha \oplus\gf^{2\alpha}\mid
(\forall (Y',Z')\in \nf) \quad (1+c\langle Y'+iY, Y'+iY\rangle )^2 \\
&\quad +4c \langle Z'+iZ +i{1/2} [Y',Y],
Z'+iZ +i{1/2} [Y',Y]\rangle \neq 0\}_0\, .\end{aligned}$$
A standard argument (see [@KSI], Lemma 1.6) combined with Lemma \[lem=r1\] yields that $$\Lambda=\{ Y\in \nf\mid (\forall n\in N)\ a_{\mathbb{C}}(n\exp(iY)) \hbox{ is defined}\}_0\, .$$ Now for $n=\exp(Y'+Z')\in N$ with $(Y',Z')\in\nf$ and $(Y,Z)\in \Lambda$ one has $$n\exp(i(Y+Z))=\exp(Y'+iY + Z'+i(Z +1/2 [Y',Y]))$$ and the assertion follows in view of the explicit formula (\[eq=e1\]).
We use the criterion in Lemma \[lem=exc1\] to determine $\Lambda$ explicitly. However, this is not so easy as it looks in the beginning. We shall begin with two important special cases and start with the Lorentz groups $G={\rm SO}_e(1,p+1)$ where $q=0$.
\[lem=lo\] Assume that $G$ is locally ${\rm SO}_e(1,p+1)$. Then $c={1\over 4p}$ and $$\Lambda=\{ Y\in \nf={\mathbb{R}}^p\mid c
\|Y\|^2 < 1\}\, .$$
In view of the previous lemma we have to look at the connected component of those $Y\in \nf$ such that $$1+c\langle Y'+iY, Y'+iY\rangle= 1 +c ( \|Y'\|^2 -\|Y\|^2 - 2i \langle Y, Y'\rangle) \neq 0$$ for all $Y'\in \nf$. The assertion follows.
Next we consider the case of the group $G={\rm SU}(2,1)$. Here $p=2$ and $q=1$ and so $c={1\over 24}$. Define matrix elements
$$X_\alpha=\begin{pmatrix} 0 & 1 & 0\\ -1 & 0 & 1\\ 0 & 1 & 0\end{pmatrix},
\quad Y_\alpha=\begin{pmatrix} 0 & i & 0\\ i & 0 & -i\\ 0 & i & 0\end{pmatrix}
, \quad X_{2\alpha}={1\over 2}
\begin{pmatrix} i & 0 & -i\\ 0 & 0 & 0\\ i & 0 & -i\end{pmatrix}\,$$ and note that $$\gf^\alpha= {\mathbb{R}}X_\alpha \oplus {\mathbb{R}}Y_\alpha\quad \hbox{and}\qquad \gf^{2\alpha}={\mathbb{R}}X_{2\alpha}\, .$$ We record the commutator relation $[X_\alpha, Y_\alpha]=4 X_{2\alpha}$ and the orthogonality relation $\langle X_\alpha, Y_\alpha\rangle =0$. Finally we need that $\|X_\alpha\|^2=\|Y_\alpha\|^2={1\over c}$ and $\|X_{2\alpha}\|^2 ={1\over 4c}$.
\[lem=su\] For $G$ locally ${\rm SU}(2,1)$ one has $$\begin{aligned}
\Lambda & =\{ xX_\alpha+ yY_\alpha +zX_{2\alpha}\in \nf \mid 2(x^2+y^2) + |z|< 1\}\\
&=\{(Y,Z)\in \nf\mid 2c \|Y\|^2 + 2\sqrt{c} \|Z\|<1\}\, .\end{aligned}$$
We want to determine those $Y\in \nf$ which belong to $\Lambda$. By the $M$-invariance of $\Lambda$ we may restrict our attention to elements of the form $Y=xX_\alpha+ zX_{2\alpha}\in \Lambda$. We have to find the connected component of those $x,z$ such that $$\begin{aligned}
& (1+c\langle(u+ix)X_\alpha + vY_\alpha, (u+ix)X_\alpha + vY_\alpha\rangle)^2\\
& \quad +4c \langle (w+iz) X_{2\alpha} + {1\over 2} ixv[Y_\alpha, X_\alpha],
(w+iz) X_{2\alpha} + {1\over 2} ixv[Y_\alpha, X_\alpha]\rangle =0\end{aligned}$$ has no solution for $u,v,w\in {\mathbb{R}}$. We employ the precedingly collected material on commutators, orthogonality and norms and obtain the equivalent version
$$(1+(u+ix)^2 + v^2)^2 + (w+i(z+ 2xv))^2 =0$$ for $u,v,w\in {\mathbb{R}}$. However, this is equivalent to $$1+(u+ix)^2 + v^2 = \pm i(w+i(z+2xv))= \pm i w \mp (z+2xv)\,.$$ Comparing real and imaginary part yields the system of equations
$$(1 -x^2 \pm 2z) + u^2 +v^2 = \mp 2xv$$ $$2ux =\pm w\, .$$ We can always choose $w$ so that the second equation is satisfied. Hence we look for $x,z$ such that the quadratic equation in $v$
$$v^2 -2xv + (1 -x^2 \pm z + u^2)=0\,.$$ has no solution for all $u$. Clearly we can take $u=0$ and assume $\pm z= -|z|$. We are left with analyzing the discriminant $$4x^2 - 4( 1-x^2 -|z|)<0\,.$$ This inequality translates into $2x^2+ |z|<1$ and concludes the proof of the lemma.
Consider the domain $$\tilde\Lambda=\{Y\in \nf\mid \exp(iY).x_0\in \overline N_{\mathbb{C}}A_{\mathbb{C}}.x_0\}_0\,.$$ It is clear that $\Lambda\subset\tilde\Lambda$ and it is easy to determine $\tilde \Lambda$ explicitly: $$\begin{aligned}
\tilde\Lambda& =\{ (Y,Z)\in \nf\mid (1 - c\|Y\|^2)^2 -4c\|Z\|^2 >0\}\\
&=\{ (Y,Z)\in \nf\mid c\|Y\|^2 +2\sqrt{c}\|Z\| <1\}\, .\end{aligned}$$ Now for $q=0$ we have seen that $\tilde\Lambda=\Lambda$. However, as our previous analysis of the ${\rm SU}(2,1)$-case shows, one has $\Lambda\neq \tilde\Lambda$ in general.
Before we come to the determination of $\Lambda$ for all rank one cases some remarks concerning the nature of the constant $c={1\over 4(p +4q)}$ are appropriate.
\[rem=norm\]Let $\gf$ be of real rank one. Let $E\in
\gf^{\alpha}$, resp. $E\in \gf^{2\alpha}$ and set $F=\theta(E)$, $H=[E,F]$. If we assume that $\{H, E, F\}$ is an $\sl(2)$-triple, i.e. $[H,E]=2E$ and $[H,F]=-2F$, then elementary $\sl(2)$-representation theory gives $\|E\|^2={1\over c}$ if $E\in
\gf^\alpha$ and $\|E\|^2={1\over 4c}$ if $E\in \gf^{2\alpha}$.
\[th=r1\] Let $G$ be a simple Lie group of real rank one.
1. If $q=0$, then $$\Lambda=\left\{ Y\in \nf\mid c
\|Y\|^2 < 1\right\}\, .$$
2. If $q>0$, then $$\Lambda=\{ (Y,Z)\in \nf\mid 2c\|Y\|^2 + 2\sqrt{c}\|Z\|< 1\} \, .$$
\(i) Lemma \[lem=lo\].
\(ii) Set $$\tilde \Lambda=\{ (Y,Z)\in \nf\mid 2c\|Y\|^2 + 2\sqrt{c}\|Z\|<
1\} \, .$$ We first show that $\tilde \Lambda \subset \Lambda$. Let $(Y,Z)\in \tilde \Lambda$, $Y\neq 0$ and $Z\neq 0$. Consider the Lie algebra $\gf_0$ generated by $Y, \theta(Y), Z, \theta
(Z)$. Standard structure theory says that $\gf_0\simeq \su(2,1)$, see [@Hel], Ch. IX, §3. Choose $E_\alpha\in {\mathbb{R}}Y $ and $E_{2\alpha}\in {\mathbb{R}}Z$ such that $\{
[E_\alpha, \theta (E_\alpha)], E_\alpha, \theta(E_\alpha)\} $ as well as $\{ [E_{2\alpha}, \theta (E_{2\alpha})], E_{2\alpha},
\theta(E_{2\alpha})\} $ are $\sl(2)$-triples. Let $y,z\in {\mathbb{R}}$ such that $Y=yE_\alpha$ and $Z=zE_{2\alpha}$. In view of the previous Remark \[rem=norm\] the condition $ 2c\|Y\|^2 + 2\sqrt{c}\|Z\|<
1$ is equivalent to $2y^2 + |z|< 1$. But this is just the condition for $\exp(i(Y+Z)).x_0$ to be contained in the crown domain $\Xi_0$ for the group $G_0=\langle \exp \gf_0\rangle < G$ (see Lemma \[lem=su\]). Now, with the obvious notation, we have $\Omega=\Omega_0$ and so $\Xi_0\subset \Xi$. This concludes the proof of $\tilde \Lambda \subset \Lambda$.
It remains to verify that $\Lambda\subset \tilde \Lambda$. For that it is sufficient to show the following: if $(Y,Z)\in \partial \tilde \Lambda$, then $(Y,Z)\not\in \Lambda$. Let also $(Y,Z)\in \partial \tilde \Lambda$ with $(Y,Z)\in \Lambda$ and let $\gf_0$ as before. Note that $\exp(i(Y+Z)).x_0\in \partial\Xi_0$. As $\Xi_0\subset \Xi$ is closed and $\exp(i(Y+Z)).x_0\in \Xi$, there would exist $G_0$-domain $\Xi_0'\subset X_{0,{\mathbb{C}}}$, properly containing $\Xi_0$, and on which $G_0$ acts properly. But this contradicts Theorem \[th=p\].
The case where $\Xi$ is a Hermitian symmetric space
---------------------------------------------------
It can happen that $\Xi$ allows additional symmetries, i.e. the group of holomorphic automorphisms is strictly larger then $G$. For example, when $X=G/K$ is a Hermitian symmetric space, then $\Xi$ is biholomorphic to $X\times \overline X$ where $\overline
X$ denotes $X$ endowed with the opposite complex structure (see [@KSII], Th. 7.7). In this example $\Xi=X\times \overline X$ is again a Hermitian symmetric space and $\Aut(\Xi)= G\times G$ is twice the size of $G=\Aut(X)$. In [@KSII], Th. 7.8, one can find a classification of all those cases where $\Xi$ is a Hermitian symmetric space for a larger group $S$. For all these cases it turns out that there is an interesting subset $\Lambda^+\subset \Lambda$ such that $\Xi=G\exp(i\Lambda^+).x_0$, and moreover, it is possible to give a precise relation between unipotent $G$-orbits through $\exp(i\Lambda^+).x_0$ and the elliptic $G$-orbits through $\exp(i\Omega).x_0$. As the case of the symplectic group is of special interest, in particular for later applications to automorphic forms, and as it is always good to have a illustrating example, we shall begin with a discussion for this group.
### The symplectic group
In this section $G={\rm Sp}(n,{\mathbb{R}})$ for $n\geq 1$. Let us denote by ${\rm Sym}(n,{\mathbb{R}})$, resp. $M_+(n,{\mathbb{R}})$, the symmetric, resp. strictly upper triangular, matrices in $M(n,{\mathbb{R}})$. Our choices of $\af$ and $\nf$ shall be
$$\af=\left\{ {\rm diag}(t_1, \ldots, t_n, -t_1, \ldots, -t_n)\mid t_i\in {\mathbb{R}}\right\}$$ and $$\nf=\left\{ \begin{pmatrix} Y & Z\\ 0 & -Y^T\end{pmatrix}\mid Z\in {\rm Sym}(n,{\mathbb{R}}), Y\in M_+(n,{\mathbb{R}})\right\}\, .$$ Of special interest is an abelian subalgebra of $\nf$ $$\nf^+=\left \{ \begin{pmatrix} 0& Z\\ 0 & 0\end{pmatrix}\mid Z\in {\rm Sym}(n,{\mathbb{R}})\right\}, .$$ We recall that the maximal compact subgroup $K<G$ is isomorphic to $U(n)$ and that $X=G/K$ admits a natural realization as a Siegel upper halfplane:
$$X={\rm Sym}(n,{\mathbb{R}})+ i {\rm Sym}^+(n,{\mathbb{R}})\subset {\rm Sym}(n,{\mathbb{C}})$$ where ${\rm Sym}^+(n,{\mathbb{R}})$ denotes the positive definite symmetric matrices. The action of $G$ on $X$ is given by generalized fractional transformations: if $g=\begin{pmatrix} A& B\\ C & D\end{pmatrix}\in G$ and $Z\in X$, then $$g(Z)=(AZ +B) (CZ+D)^{-1}\, .$$ Notice that the base point $x_0$ with stabilizer $K$ becomes $x_0=i I_n$ with $I_n$ the identity matrix. The natural realization of $\overline X$ is the lower half plane $$\overline X = {\rm Sym}(n,{\mathbb{R}}) - i {\rm Sym}^+(n,{\mathbb{R}})\, .$$ In the sequel we view $X$ inside of $X\times \overline X$ as a totally real submanifold via the embedding $$X\hookrightarrow X\times \overline X, \ \ Z\mapsto (Z, \overline Z)$$ where $\overline Z$ denotes the complex conjugation in ${\rm Sym}(n,{\mathbb{C}})$ with respect to the real form ${\rm Sym}(n,{\mathbb{R}})$. As we remarked earlier, $\Xi$ is naturally biholomorphic to $X\times \overline X$. Let us now consider a domain in $\nf^+$
$$\Lambda^+=\{ Y\in \nf^+\mid \exp(iY).x_0\in \Xi\}_0\, .$$
\[th=sp\]For $G={\rm Sp}(n,{\mathbb{R}})$ the following assertions hold:
1. If $\|\cdot \|$ denotes the operator norm on $M(n,{\mathbb{R}})$, then $$\Lambda^+=\left\{ \begin{pmatrix} 0& Z\\ 0 & 0\end{pmatrix}\in \nf^+ \mid \|Z\|< 1\right\}\, .$$
2. With $\Lambda^{++}=\Lambda\cap {\rm diag}(n,{\mathbb{R}})$ one has
1. $\Lambda^{++}=\{ {\rm diag}(t_1, \ldots, t_n)\in \nf^+\mid |t_i|<1\}$
2. $\Lambda^+=\Ad K_0(\Lambda^{++})$ with $K_0={\rm SO}(n,{\mathbb{R}})<K $.
3. $\Xi=G\exp(i\Lambda^+).x_0=G\exp(i\Lambda^{++}).x_0$.
\(i) Let $Z\in {\rm Sym}(n,{\mathbb{R}})$ and $\tilde Z= \begin{pmatrix} 0& Z\\ 0 & 0\end{pmatrix}$ the corresponding element in $\nf^+$. Then $$\exp(i\tilde Z)=\begin{pmatrix} I_n& iZ\\ 0 & I_n\end{pmatrix}$$ and accordingly $$\exp(i\tilde Z).x_0=\exp(i\tilde Z) (iI_n, -iI_n)=
(i(I_n+Z), -i(I_n-Z))\, .$$ Therefore $\exp(i\tilde Z).x_0\in X\times \overline X$ if and only if $I_n+Z \in {\rm Sym}^+(n,{\mathbb{R}})$ and $I_n-Z \in {\rm Sym}^+(n,{\mathbb{R}})$. Clearly, this is equivalent to $\|Z\|<1$ and the proof of (i) is finished.
\(ii) This is immediate from (i).
\(iii) It is enough to show that $\Xi=G\exp(i\Lambda^{++}).x_0$ and for that it suffices to verify that $\exp(i\Omega).x_0\subset G\exp(i\Lambda^{++}).x_0$. Now, as $\Sigma$ is of type $C_n$, the domain $\Omega$ is a cube $$\Omega =\{{\rm diag}(t_1, \ldots, t_n, -t_1, \ldots, -t_n)\mid |t_i|<{\pi \over 4}\}\, .$$ Let us write $E_{ij}$ for the elementary matrices in $M(2n,{\mathbb{R}})$. Then for each $1\leq j\leq n$ we define an $\sl(2)$-subalgebra $\gf_j$ by $$\gf_j={\rm span}_{\mathbb{R}}\{ E_{jj}-E_{j+n, j+n}, E_{j, j+n}, E_{n+j, j}\}\, .$$ We note that the $\gf_j$ pairwise commute and so $\gf_0=\gf_1\oplus\ldots \oplus
\gf_n$ is a subalgebra of $\gf$ which contains $\af$. Now we are in the situation to use $\sl(2)$-reduction and the assertion becomes a consequence of Lemma \[lem=or\].
For later reference we wish to make the last part of the above theorem more precise. For ${\bf z}=(z_1, \ldots, z_n) \in {\mathbb{C}}^n$ let us define a matrix in $N_{\mathbb{C}}^+$
$$n_{\bf z}=\begin{pmatrix} I_n & {\rm diag}({\bf z})\\ 0& I_n\end{pmatrix}\, .$$ Moreover if ${\bf z}\in ({\mathbb{C}}^*)^n$, then we set $$a_{{\bf z}}={\rm diag}(z_1,\ldots, z_n, z_1^{-1}, \ldots, z_n^{-1})\in A_{\mathbb{C}}\, .$$ In the course of the proof of Theorem \[th=sp\] (iii) we have shown the following:
\[lem=orn\] Let ${\bf t}=(t_1,\ldots, t_n)\in {\mathbb{R}}^n$ with $|t_i|<{\pi\over 4}$. Set ${\bf e}^{i\bf t}=(e^{it_1}, \ldots, e^{it_n})$ and ${\bf sin}(2 {\bf t})=(\sin 2t_1, \ldots, \sin 2t_n)$. Then $$G n_{{\bf sin}(2{\bf t})}.x_0= G a_{{\bf e}^{i{\bf t}}}.x_0\, .$$
### The general case of Hermitian $\Xi$ {#ssh}
In this subsection we will assume that $\Xi$ is a Hermitian symmetric space for an overgroup $S\supset G$. From a technical point of view it is however better to work with an alternative characterization, namely (cf. [@KSII], Th. 7.8) $\Sigma$ is of type $C_n$ or $BC_n$ for $n\geq 2$ or $\gf=\so(1,k)$ with $k\geq 2$ for the rank one cases.
If $\Sigma$ is of type $C_n$ or $BC_n$, then
$$\Sigma=\{ \pm \gamma_i\pm \gamma_j\mid 1\leq i,j\leq n\}\backslash \{0\}
\cup\{\pm {1\over 2}\gamma_i: 1\leq i\leq n\}$$ with the second set on the right to be considered not present in the $C_n$-case. As a positive system of $\Sigma$ we choose $$\Sigma^+=\{ \gamma_i\pm \gamma_j\mid 1\leq i\leq j\leq n\}\backslash \{0\}
\cup\{{1\over 2}\gamma_i: 1\leq i\leq n\}\, .$$ Further we consider the $A_{n-1}$-subsystem $$\Sigma_0=\{ \pm \gamma_i \mp \gamma_j\mid 1\leq i\neq j\leq n\}\,$$ and set $$\Sigma^{++}=\Sigma^+\cap (\Sigma^+\backslash\Sigma_0) \quad \hbox{and}\quad
\Sigma^{--}=-\Sigma^{++}\, .$$ Next we define subalgebras of $\gf$ by
$$\nf^+=\bigoplus_{\alpha\in \Sigma^{++}} \gf^\alpha, \quad
\nf^-=\bigoplus_{\alpha\in \Sigma^{--}} \gf^\alpha \quad\hbox{and}
\quad \gf(0)=\af \oplus\mathfrak{m} \oplus \bigoplus_{\alpha\in \Sigma_0} \gf^\alpha\, .$$ We note that $\nf^+$ is a subalgebra of $\nf$ and that $$\gf=\nf^- \oplus \gf(0) \oplus \nf^+$$ is a direct decomposition with $[\gf(0),\nf^{\pm}]\subset \nf^\pm$. Define elements $T_j\in \af$ by the requirement $\gamma_i(T_j)=\delta_{ij}$ and note that $$\Omega=\bigoplus_{j=1}^n \left ]-{\pi\over 4}, {\pi\over 4}\right [ T_j \, .$$
Now for an element $Y_j\in \gf^{2\gamma_j}$ we find $E_j\in {\mathbb{R}}Y_j$, unique up to sign, such that $\{ T_j, E_j, \theta(E_j)\}$ form an $\sl(2)$-triple. Define $y_j\in {\mathbb{R}}$ by $Y_j= y_j E_j$. With that we can define an open ball in $\bigoplus_{j=1}^n \gf^{2\gamma_j}$ by
$$\Lambda^{++}=\left\{ Y=\sum_{j=1}^n Y_j \in \bigoplus_{j=1}^n \gf^{2\gamma_j}\mid |y_j|< 1\right \}$$ Further write $\kf(0)=\gf(0)\cap \kf$ and set $K(0)=\exp \kf(0)$. Finally we define a subset of $\nf^+$ by $$\Lambda^+=\Ad K_0 (\Lambda^{++})\, .$$
Define an abelian subspace of $\nf^+$ by $\nf^{++}=\bigoplus_{\alpha\in \Sigma^{++}\cap C_n} \gf^\alpha$. Then $\Lambda^+$ is a bounded convex domain in $\nf^{++}$.
\[ts\] If $\Xi$ is a Hermitian symmetric space for an overgroup $S\supset G$, then the following assertions hold.
1. If $T =\sum_{j=1}^n t_j T_j\in \Omega$, and $\{ T_j, E_j, \theta(E_j)\}$ is any $\sl(2)$-triple with $E_j\in \gf^{2\gamma_j}$, then $$G\exp\left(i\sum_{j=1}^n \sin (2t_j)E_j\right).x_0= G \exp\left(i\sum_{j=1}^n t_j T_j\right).x_0\, .$$
2. $\Xi =G\exp(i\Lambda^+).x_0=G\exp(i\Lambda^{++}).x_0$.
\(i) We define subalgebras of $\gf$ which are isomorphic to $\sl(2,{\mathbb{R}})$ by $\gf_j={\mathbb{R}}T_j \oplus {\mathbb{R}}E_j \oplus {\mathbb{R}}\theta(E_j)$. The $\gf_j$’s commute in $\gf$ and so $\gf_0=\gf_1\oplus\ldots\oplus \gf_n$ defines a subalgebra of $\gf$. In view of Lemma \[lem=or\], the assertion now follows by $\sl(2)$-reduction.
\(ii) This is a consequence of (i).
If $\gf$ is Hermitian and of tube type, then $\Lambda^+$ is a bounded open convex set in $\nf^+$ and $$\Lambda^+=\{ Z\in \nf^+\mid \exp(iZ).x_0\in \Xi\}_0\, .$$ This can be proved as in the ${\rm Sp}(n)$-case by employing the machinery of Jordan algebras.
Some partial results for the special linear groups {#sssl}
--------------------------------------------------
In this subsection we exclusively deal with $G=\Sl(n,{\mathbb{R}})$. To determine the exact shape of $\Lambda$ for $n\geq 3$ seems to be very challenging; already the case of $n=3$ appears to be very intricate. Instead we will exhibit a fairly large cube-domain inside of $\Lambda$; further we will estimate the corresponding hyperbolic parameterization.
In order to perform reasonably efficient computations we use the matrix model for $X_{\mathbb{C}}$. Let us denote by ${\rm Sym} (n,{\mathbb{C}})_{{\rm det}=1}$ the affine variety of complex symmetric matrices with unit determinant. The map $$X_{\mathbb{C}}=\Sl(n,{\mathbb{C}})/{\rm SO}(n,{\mathbb{C}})\to {\rm Sym} (n,{\mathbb{C}})_{{\rm det}=1}, \ \ gK_{\mathbb{C}}\mapsto
gg^t$$ is an isomorphism. Within this model for $X_{\mathbb{C}}$, the Riemmannian symmetric space $X$ identifies with ${\rm Sym} (n,{\mathbb{R}})_{{\rm det}=1}^+$, the determinant one section in the cone of positive definite symmetric matrices. Now the crown domain $\Xi$ contains the determinant one cut $\Xi_0$ of the tube domain, i.e.
$$\Xi_0=\{ Z\in X_{\mathbb{C}}\mid {\mbox{\rm Re}\,}Z \gg 0\}\, .$$
As usual we write $E_{ij}=(\delta_{ki}\delta_{ij})_{lj}$ for the elementary matrices. We choose $N$ to be the group of unipotent upper triangular matrices and consider the mapping $$m: {\mathbb{C}}^{n-1}\to N_{\mathbb{C}}, \ \ (z_1,\ldots, z_{n-1})
\mapsto \exp(z_1 E_{12})\cdot \ldots \cdot
\exp(z_{n-1}E_{n-1\, n})\, .$$ In matrix notation $m$ is given by
$$m(z_1,\ldots, z_{n-1})=\begin{pmatrix} 1 & z_1 & & &\\
& 1 & z_2 & & \\
& & \ddots &\ddots &\\
& & & 1 & z_{n-1} \\
& & & & 1 \end{pmatrix}\, .$$
We define a subset of $N_{\mathbb{C}}$ by
$$\NN^+=m\left (i\prod_{j=1}^{n-1} (-1,1)\right)$$ and claim:
$\NN^+\cdot x_0\subset \Xi_0$. In particular $G\NN^+\cdot x_0\subset \Xi_0\subset \Xi$.
This is an elementary matrix computation. Let $(t_1,\ldots, t_{n-1})\in {\mathbb{R}}^{n-1}$, $|t_i|<1$ and set $$n=\begin{pmatrix} 1 & it_1 & & &\\
& 1 & it_2 & & \\
& & \ddots &\ddots &\\
& & & 1 & it_{n-1} \\
& & & & 1 \end{pmatrix}\, .$$
One has to verify that ${\mbox{\rm Re}\,}(nn^t)\gg 0$. A straightforward calculation yields
$$nn^t=\begin{pmatrix} 1 & it_1 & & &\\
it_1 & 1-t_1^2 & it_2 & & \\
& it_2 & \ddots &\ddots &\\
& & \ddots & 1-t_{n-2}^2 & it_{n-1} \\
& & & it_{n-1} & 1-t_{n-1}^2 \end{pmatrix}\, .$$ Therefore
$${\mbox{\rm Re}\,}(nn^t)=\begin{pmatrix} 1 & & &\\
& 1-t_1^2 & & \\
& &\ddots & \\
& & & 1-t_{n-1}^2\end{pmatrix}\gg 0$$
Next we discuss hyperbolic parameterization for elements in $\NN^+$. Here our results are somewhat partial but perhaps still interesting. For what follows we are indebted to Philip Foth. For $t\in {\mathbb{R}}$ with $|t|<1$ we consider the element $$z(t)= m(i(t, \ldots, t))= \begin{pmatrix} 1 & it & & &\\
& 1 & it & & \\
& & \ddots &\ddots &\\
& & & 1 & it \\
& & & & 1 \end{pmatrix}\, .$$
We wish to estimate the element $a(t)\in \exp(i\pi \Omega/2)$ for which $$Gz(t)\cdot x_0=Ga(t)\cdot x_0$$ holds. The result is as follows.
\[prost\] Let $G=\Sl(n,{\mathbb{R}})$. Fix $t\in{\mathbb{R}}$, $|t|<1$ and set $z(t)=m(i(t, \ldots, t))$. Then $Gz(t)\cdot x_0=Ga(t)\cdot x_0$ with $$a(t)=\diag (e^{i\phi_1(t)}, \ldots,
e^{i\phi_n(t)}),\qquad \diag(\phi_1(t), \ldots, \phi_n(t))\in \pi\Omega/ 2$$ and $$\label{es1}
|\phi_j(t)|\leq \left|{1\over 2} \tan^{-1} \left({2t\over 1-t^2}\right)\right| \qquad (1\leq j\leq n)\, .$$
We proceed indirectly and use the complex convexity theorem (\[cc\]). For $k\in K$ we have to show that the components of ${\mbox{\rm Im}\,}\log a_{\mathbb{C}}(kz(t))$ satisfy the estimate (\[es1\]). To compute $a_{\mathbb{C}}(kz(t))$ we write the corresponding matrix identity out:
$$\underbrace{\begin{pmatrix} 1 & * & \ldots & \ldots &*\\
& 1 & * &\ldots & * \\
& & \ddots &\ddots &\vdots\\
& & & 1 & * \\
& & & & 1 \end{pmatrix}}_{\in N_{\mathbb{C}}}
\cdot
\underbrace{\begin{pmatrix} * &\ldots &*\\
* &\ldots &*\\
\vdots &\vdots &\vdots\\
* &\ldots &*\\
k_1 & \ldots & k_n\end{pmatrix}}_{\in K}
\cdot \underbrace{\begin{pmatrix} 1 & it & & &\\
& 1 & it & & \\
& & \ddots &\ddots &\\
& & & 1 & it \\
& & & & 1 \end{pmatrix}}_{=z(t)} =$$
$$=
\underbrace{\begin{pmatrix} a_{{\mathbb{C}},1}(t)& & & & \\
& a_{{\mathbb{C}},2}(t) & & \\
& & \ddots & \\
& & & a_{{\mathbb{C}},n}(t)
\end{pmatrix}}_{\in A\exp(i\pi\Omega/2)}
\cdot
\underbrace{\begin{pmatrix} * & \ldots & * \\
* & \ldots & * \\
\vdots & \vdots & \vdots \\
k_1' & \ldots & k_n'
\end{pmatrix}}_{\in K_{\mathbb{C}}}\, .$$
We match the bottom rows and arrive at:
$$(k_1, it k_1 +k_2, itk_2+k_3, \ldots, it k_{n_1} +k_n)= a_{{\mathbb{C}},n}(t)(k_1', \ldots, k_n')\, .$$ We square the entries and sum them up: $$1-t^2(k_1^2+\ldots+ k_{n-1}^2) +2it(k_1 k_2+\ldots+ k_{n-1}k_n)= a_{{\mathbb{C}},n}(t)^2\, .$$ The result is
$$|\phi_n(t)| = \left| {1\over 2} \tan^{-1}\left ({ 2t(k_1k_2+\ldots+ k_{n-1} k_n)\over
1-t^2(k_1^2+\ldots +k_{n-1}^2)}\right)\right|\, .$$ Finally we use the estimates $k_1^2+\ldots +k_{n-1}^2\leq 1$ and $k_1k_2+\ldots+ k_{n-1} k_n\leq 1$ and obtain that
$$|\phi_n(t)| = \left| {1\over 2} \tan^{-1}\left({ 2t\over
1-t^2}\right)\right|\, .$$ This proves (\[es1\]) for the last entry. The general case follows by Weyl group invariance.
One can show that $z(t)\cdot x_0\in \Xi$ precisely for $|t|<1$.
Exponential decay of Maaß cusp forms I: The example of $G=\Sl(2,{\mathbb{R}})$
==============================================================================
It is a result obtained by Langlands that cuspidal automorphic forms are of rapid decay. But actually more is true and the decay is of exponential type. The purpose of this section is to give an introduction to this circle of problems with a solid discussion of the case of $G=\Sl(2,{\mathbb{R}})$. We will restrict our attention to Maaß cusp forms and to the modular group $\Gamma=\Sl(2,{\mathbb{Z}})$ in order to keep the exposition basic. It is possible to verify the exponential decay by our explicit knowledge of the Whittaker functions in this case. This will be presented first. In general however, concrete knowledge of the Whittaker functions is not available and an alternative approach is needed. It was Joseph Bernstein who came up with the idea to use analytic continuation to obtain exponential decay. We shall present his ideas in the geometric framework which we developed in the preceding section.
Concrete approach
-----------------
For the rest of this section we let $G=\Sl(2,{\mathbb{R}})$ and keep our choices including notation from Section \[sec=nf\]. In the sequel we will identify $X=G/K$ with the upper half plane, i.e. $X=\{ z\in {\mathbb{C}}\mid {\mbox{\rm Im}\,}z >0\}$ and $G$ acting by fractional linear transformations:
$$g(z)={az+ b\over cz+d} \qquad \hbox{for}\quad g=\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in G, \ z\in X\, .$$ In these coordinates the base point $x_0$ is the imaginary unit $x_0=i$ and the Iwasawa decomposition states that the map
$$N\times A \to X, \ \ (n_x, a_y)\mapsto n_xa_y(i)=x+iy\, ,$$ where
$$n_x=\begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix} \in N \quad \hbox{and}\quad
a_y=\begin{pmatrix} \sqrt{y} & 0 \\ 0 & {1\over \sqrt{y}}\end{pmatrix} \in A$$ with $x\in {\mathbb{R}}, y>0$, is a diffeomorphism. The Laplace-Beltrami operator of $X$ is given by $\Delta=-y^2(\partial_x^2 + \partial_y^2)$ and we note that $\DD(X)={\mathbb{C}}[\Delta]$. We make a simpler choice for a lattice $\Gamma<G$, namely $\Gamma=\Sl(2,{\mathbb{Z}})$ the modular group. Then by a [*Maaß automorphic form*]{} we understand an analytic function $\phi: X\to{\mathbb{C}}$ such that
- $\phi$ is $\Gamma$-invariant
- $\phi$ is an eigenfunction for $\DD(X)$, i.e. there exists $\lambda\in {\mathbb{C}}$ such that $\Delta \phi =\lambda(1-\lambda)\phi$.
- $\phi$ is of moderate growth, i.e. there exists $\alpha\in{\mathbb{R}}$ such that $$|\phi(x+iy)|\ll y^\alpha \qquad (y>1)\, .$$
Moreover, a Maaß automorphic form is called a [*cusp form*]{} if $$\int_{N\cap \Gamma\backslash N } \phi(nz) \ dn=0 \qquad \hbox {for all $z\in X$}\, .$$ Note that $N\cap \Gamma=\begin{pmatrix} 1 & {\mathbb{Z}}\\ 0 & 1\end{pmatrix}$ so that $N\cap \Gamma\backslash N \simeq {\mathbb{Z}}\backslash {\mathbb{R}}$ is a circle. In our special case the results of Langlands reads as follows.
Maaß cusp forms $\phi$ are of rapid decay, i.e. $$|\phi(x+iy)| \ll y ^\alpha \qquad (y>1)$$ for any $\alpha\in {\mathbb{R}}$.
However, more is true and we can state:
\[th=mcf\] Maaß cusp forms $\phi$ are of exponential decay, i.e. there is a constant $C>0$ such that $$|\phi(x+iy)| \leq C e^{-2\pi y}\qquad (y>1) \, .$$
Before we prove this theorem, we will recall the Whittaker expansion of a Maaß cusp form: If $\phi$ is a Maaß cusp form with $\Delta \phi=\lambda(1-\lambda)\phi$, then
$$\label{eq=FB}
\phi(x+iy)=\sum_{n\in {\mathbb{Z}}^\times} a_n \sqrt{y} K_\nu(2\pi |n| y) e^{2\pi i nx}\,$$
where $K_\nu$ is the McDonald Bessel function
$$K_\nu(y)= {1\over 2} \int_0^\infty e^{-y(t+{1\over t})/2} t^\nu {dt\over t} \qquad (y>0)\,$$ with parameter $\lambda={1\over 2} + \nu $ and the $a_n$ are complex numbers satisfying the Hecke bound
$$\label{eq=hb} |a_n| \ll |n|^{1\over 2} \, .$$
As a final piece of information we need the asymptotic expansion of the Bessel function
$$\label{eq=Kas} K_\nu(y)\sim \left({\pi\over 2y}\right)^{1\over 2}
e^{-y} \cos (\nu \pi)\, .$$
We can now prove Theorem \[th=mcf\].
We plug the estimates (\[eq=hb\]) and (\[eq=Kas\]) in the Fourier expansion (\[eq=FB\]) and use the convention that $C$ denotes a positive constant whose actual value may change from line to line: for $y$ large we obtain
$$\begin{aligned}
|\phi(x+iy)| &\leq \sum_{n\neq 0} |a_n| \sqrt{y} \cdot |K_\nu(2\pi |n|y)| \\
&\leq C \sum_{n\neq 0} |n|^{1\over2} \sqrt{y} \left({\pi\over 2\pi |n|y}\right)^{1\over 2}
e^{-2\pi|n|y} \\
&\leq C \sum_{n\neq 0} e^{-2\pi|n|y} \\
&\leq C {e^{-2\pi y}\over 1- e^{-2\pi y }}\\
&\leq C e^{-2\pi y} \,.\end{aligned}$$
The method of analytic continuation
-----------------------------------
We now present an alternative approach to Theorem \[th=mcf\], essentially due to J. Bernstein, which uses the method of analytic continuation. The final result is slightly weaker than the optimal estimate in Theorem \[th=mcf\], but this will be balanced by the conceptionality of the approach.
Let $\phi$ be a Maaß cusp form. Let us fix $y>0$ and consider the $1$-periodic function
$$F_y: {\mathbb{R}}\to {\mathbb{C}}, \ \ u\mapsto \phi(n_ua_y(i))=\phi(u+iy)\, .$$ This function being smooth and periodic admits a Fourier expansion
$$F_y(u)=\sum_{n\neq 0} A_n(y) e^{2\pi i n x}\, .$$ Here, $A_n(y)$ are complex numbers depending on $y$. Now observe that $$n_ua_y=a_y a_y^{-1}n_u a_y= a_y n_{u/y}$$ and so $$F_y(u)=\phi(a_y n_{u/ y}.x_0)\, .$$ As $\phi$ is a $\DD(X)$-eigenfunction, it admits a holomorphic continuation to $\Xi$ and thus it follows from Lemma \[lem=or\] and Theorem \[th=sl2\] that $F_y$ admits a holomorphic continuation to the strip domain $$S_y=\{ w=u+iv \in {\mathbb{C}}\mid |v|< y\}\, .$$ Let now $\e>0$, $\e$ small. Then, for $n>0$, we proceed with Cauchy
$$\begin{aligned}
A_n(y) & =\int_0^1 F_y( u -i(1-\e)y) e^{-2\pi i n (u-i(1-\e)y)}\ du \\
&= e^{-2\pi n (1-\e)y} \int_0^1 F_y( u-i(1-\e)y) e^{-2\pi i n u} \ du \\
&= e^{-2\pi n (1-\e)y} \int_0^1 \phi ( a_y n_{u/y} n _{-i(1-\e)}.x_0) e^{-2\pi i n u} \ du \, .\end{aligned}$$
Thus we get, for all $\e>0$ and $n\neq 0$ the inequality
$$\label{eq=ineq}
|A_n(y)|\leq e^{-2\pi |n| y(1-\e)} \sup_{\Gamma g\in \Gamma\backslash G} |\phi(\Gamma g n_{\pm i(1-\e)}.x_0)|$$
We need an estimate.
\[lem=esti\]Let $\phi$ be a Maaß cusp form. Then there exists a constant $C$ only depending on $\lambda$ such that for all $0< \e < 1$ $$\sup_{\Gamma g\in \Gamma\backslash G} |\phi(\Gamma g n_{i(1-\e)}.x_0)|\leq C |\log \e|^{1\over 2}$$
Let $-\pi/4< t_\e <\pi/4$ be such that $\pm (1-\e)=\sin 2t_\e$. Then, by Lemma \[lem=or\] we have $G n_{\pm i(1-\e)}.x_0 =G a_{\e}.x_0$ with $a_\e=\begin{pmatrix} e^{it_\e} & 0 \\ 0& e^{-it_\e}\end{pmatrix}$. Now note that $t_\e \approx \pi/4 -\sqrt{2\e}$ and thus [@KSI], Th. 5.1 and Th. 6.17 , give that $$\sup_{\Gamma g\in \Gamma\backslash G} |\phi(ga_\e.x_0)|\leq C |\log \e|^{1\over 2}\, .$$ This concludes the proof of the lemma.
We use the estimates in Lemma \[lem=esti\] in (\[eq=ineq\]) and get
$$\label{eq=ineq1}
|A_n(y)|\leq C e^{-2\pi |n| y(1-\e)} |\log \e|^{1\over 2}\, ,$$
and specializing to $\e=1/y$ gives that $$\label{eq=ineq2}
|A_n(y)|\leq C e^{-2\pi |n| (y-1)} (\log y)^{1\over 2} \, .$$ This in turn yields for $y>2$ that
$$\begin{aligned}
|\phi(iy)| & = |F_y(0)|\leq \sum_{n\neq 0} |A_n(y)|\\
&\leq C (\log y)^{1\over 2}\sum_{n\neq 0} e^{-2\pi |n| (y-1)}\\
&\leq C (\log y)^{1\over 2} \cdot e^{-2\pi y}\end{aligned}$$
It is clear, that we can replace $F_y$ by $F_y(\cdot +x)$ for any $x\in {\mathbb{R}}$ without altering the estimate. Thus we have proved:
Let $\phi$ be a Maaß cusp form. Then there exists a constant $C>0$, only depending on $\lambda$, such that $$|\phi(x+iy)|\leq C (\log y)^{1\over 2} \cdot e^{-2\pi y} \qquad (y>2)\, .$$
It is not too hard to make the constant in the theorem precise. We will do this in the next section when we give a general discussion of the rank one cases.
Exponential decay of Maaß cusp forms II: the rank one cases
===========================================================
The example of $G=\Sl(2,{\mathbb{R}})$ admits a straightforward generalization to all rank one cases and this will be outlined below. Throughout this section we let $G$ be of real rank one, i.e. $\dim \af=1$. We fix a noncocompact lattice $\Gamma<G$ and call a parabolic subgroup $MAN$ [*cuspidal*]{} for $\Gamma$ if $\Gamma\cap N$ is a lattice in $N\cap \Gamma$. Notice that this implies that $\Gamma\cap Z(N)$ is a lattice in $Z(N)$ where $Z(N)$ is the center of $N$. Recall the constant $c={1\over 4 (p+4q)}$ and let $d={1\over \sqrt c}$ if $q=0$ and $d={1\over 2\sqrt{c}}$ otherwise. We define the [*period* ]{} $r_\Gamma$ of $\Gamma$ to be the positive number
$$r_\Gamma={1\over d} \min\{ \|\log \gamma\|: \gamma\in Z(N)\cap \Gamma, \gamma\neq {\bf 1},
N \ \hbox{cuspidal}\}\, .$$
We fix now $MAN$ and $E'\in \log (Z(N)\cap \Gamma)$, $E'\neq 0$, such that $\|E'\|$ is minimal for all possible choices of $N$. Then $\|E'\|= d r_\Gamma$.
Next let $E\in {\mathbb{R}}^+ E'$ be such that for $F=\theta(E)$ and $H=[E,F]$ the set $\{ H, E, F\}$ forms an $\sl(2)$-triple. Recall from Remark \[rem=norm\] that $\|E\|=d$ so that $$\label{eq=period} E' =r_\Gamma E\, .$$ For $y>1$ we set $a_y=\exp(\log y\cdot H/2)\in A$.
We fix a Maaß cusp form $\phi$ for $\Gamma$, fix $n\in N$ and $y>1$ and consider the function
$$F_{n,y}: {\mathbb{R}}\to {\mathbb{C}}, \ \ u\mapsto \phi(\exp(uE)na_y.x_0)\, .$$ From the relation (\[eq=period\]), it follows that $F_{n,y}$ is periodic with period $r_\Gamma$. Thus $F_{n,y}$ admits a Fourier expansion
$$F_{n,y}(u)=\sum_{k\in {\mathbb{Z}}^\times} A_k(n,y) e^{{2\pi i k\over r_\Gamma} u}\, .$$ As $\exp (uE)\in Z(N)$ we notice next that $$\exp(uE)na_y=na_y \exp(u/y E)\,$$ and we conclude with Theorem \[th=r1\] that $F_{n,y}$ extends a holomorphic function on the strip domain $S_y=\{ u+iv\in {\mathbb{C}}\mid |v|<y\}$. For $0<\e<1$ we obtain, as in the previous section, the coefficient estimate
$$\label{esti=c1}
|A_k(n,y)| \leq e^{-{2\pi y(1-\e)\over r_\Gamma}} \sup_{\Gamma g\in \Gamma\backslash G}
|\phi(\Gamma g \exp(i(1-\e)E).x_0)|\, .$$
From now on we make the slightly restrictive assumption that $\phi$ corresponds to a spherical principal series representation $\pi_\lambda$ with $\lambda\in i\af^*$. Often we will identify $i\af^*$ with $i{\mathbb{R}}$ via $\lambda=\lambda\cdot \rho$.
\[lem=esti1\] Let $\phi$ be a Maaß cusp form associated to $\pi_\lambda$. Then for all $0<\e<1$ the following estimate holds $$\sup_{\Gamma g\in \Gamma\backslash G} |\phi (\Gamma g\exp(i(1-\e)E).x_0)|
\leq C(\lambda) \begin{cases} |\log \e|^{1\over 2} & \text{if $p=1$ and $q=0$} \\
\e^{{1-p\over 4}} & \text{if $p>1$ and $q=0$}\\
|\log \e|^{1\over 2} & \text{if $q=1$} \\
\e^{ {1-q\over 4}} & \text{if $q>1$}\end{cases}$$ where $$C(\lambda)=C \cdot e^{{\pi\over 2}|\lambda|} (|\lambda|+1)^{1+[\dim X/2]}$$ and $C>0$ a constant independent of $\lambda$.
In first order approximation we have $G\exp(i(1-\e)E).x_0=G\exp(i (\pi/4 -2\sqrt\e)H).x_0$ as in the proof of Lemma \[lem=esti\]. Now for fixed $\lambda$, the assertion follows from [@KSI], Th. 5.1 (or alternatively from our table in Theorem \[thm:table\]) and Th. 6.17. The precise shape of the constant $C(\lambda)$ is found by tracing the proofs in [@KSI].
Finally, specializing to $y={1\over \e}$ in (\[esti=c1\]) we obtain from Lemma (\[lem=esti1\]) the following result:
Let $\phi$ be a Maaß cusp form associated to $\pi_\lambda$ with $\lambda\in i\af^*$. Then, for all $n\in N$ and $y>2$ the following estimate holds: $$\sup_{n\in N} |\phi (na_y.x_0)|
\leq C(\lambda) e^{-{2\pi y\over r_\Gamma}} \begin{cases} (\log y)^{1\over 2} & \text{if $p=1$ and $q=0$} \\
y^{{p-1\over 4}} & \text{if $p>1$ and $q=0$}\\
(\log y)^{1\over 2} & \text{if $q=1$} \\
y^{{q-1\over 4}} & \text{if $q>1$}\end{cases}\, .$$
Exponential decay of Maaß cusp forms III: the higher rank cases
===============================================================
Throughout this section we denote by $G$ a simple Lie group and by $\Gamma<G$ a noncocompact lattice. We say that a parabolic subgroup $P=MAN$ is [*cuspidal*]{} if $\Gamma_N=\Gamma\cap N$ is a lattice in $N$.
A $\Gamma$-invariant smooth ${\mathbb D}(X)$-eigenfunction on $X$ is called a [*weak Maaß automorphic form*]{}. A weekly automorphic Maaß form is called a [*Maaß cusp form*]{} if it is of moderate growth and $$\int_{\Gamma_N\backslash N} f(\Gamma_N n g)\ d(\Gamma_Nn)=0$$ for all $g\in G$ and all proper cuspidal parabolic subgroups $P=MAN$.
The crucial fact for us is that all ${\mathbb D}(X)$-eigenfunctions on $X$ extend holomorphically to $\Xi$ (cf. [@KSII], Th. 1.1). Hence all weak Maaß automorphic forms extend to holomorphic functions on $\Xi$. If moreover $\Gamma$ is torsion free, then $\Gamma$ acts properly on $\Xi$ (as the $G$-action is proper) and we can form the quotient $\Gamma\backslash \Xi$ in the category of complex manifolds. Thus Maaß forms have $\Gamma\backslash \Xi$ as their natural domain of definition.
For the rest of this section we let $P=MAN$ be a minimal parabolic subgroup which is cuspidal. In addition we make the following
[**assumption**]{}: For each root $\alpha\in \Sigma^+$ the group $\Gamma\cap \exp(\gf^\alpha)$ is a lattice in $\exp (\gf^\alpha)$.
For each $\alpha\in \Sigma^+$ and $E_\alpha\in \gf^\alpha$ we set $F_\alpha=\theta(E_\alpha)$ and $H_\alpha=[E_\alpha,F_\alpha]$. We always normalize $E_\alpha$ in such a way that $\{ E_\alpha, F_\alpha, H_\alpha\}$ forms an $\sl(2)$-triplet.
For $\alpha\in \Pi$ we define an ideal in $\Sigma$ by
$$\Sigma_\alpha=\left\{ \beta=\sum_{\gamma\in\Pi} n_\gamma \gamma\mid
n_\alpha>0\right\}\subset \Sigma^+,$$ and write $\u_\alpha=\bigoplus_{\beta\in \Sigma_\alpha} \gf^\beta$ for the corresponding ideal in $\nf$. We set $U_\alpha=\exp(\u_\alpha)$ and notice that $U_\alpha$ is the nilradical of the cuspidal parabolic subgroup $P_\alpha$ attached to $\alpha\in\Pi$.
Associated to $\alpha\in\Pi $ we define positive constants
$$r_{\alpha, \Gamma}=\max_{\beta\in\Sigma_\alpha}
\min\{ c>0\mid \exp(cE_\beta)\in \Gamma, \
E_\beta\in \gf^\beta \ \hbox{normalized}\}\, ,$$ and
$$c_\alpha=\min\left\{ c>0\mid {c\over 2} H_\beta\in \partial\Omega
,\ \beta\in\Sigma_\alpha\right\}\, .$$
The relevance of the number $c_\alpha$ is the following: it is the maximal number such that $\exp(itE_\beta).x_0\in \Xi$ for all $0\leq t<c_\alpha$ and $\beta\in\Sigma_\alpha$.
For a subgroup $U<N$ with $\Gamma_U\backslash U$ and compact we define the [*constant term*]{} of a function $f\in C^\infty(\Gamma_N \backslash G)$ with respect to $U$ as $$\pi_U f (Ug)=\int_{U_N\backslash U} f(\Gamma_N ug) \ d(U_N u)\, .$$ Note that $\pi_U f \in C^\infty (\Gamma_N U\backslash G )$. For $U=U_\alpha$ we use the simplifying notation $\pi_\alpha=\pi_{U_\alpha}$ for the constant term with respect to $U_\alpha$.
We can now state the holomorphic analog of the Main Lemma (Lemma 10) in [@HC].
\[mlem\] [(Main Lemma)]{} Let $\alpha\in \Pi$ and $0<\e<1$. Let $f\in C^\infty(\Gamma_N\backslash G)$ such that $f$ admits a holomorphic continuation to $\Gamma_N\backslash \tilde \Xi$. Then there exists a constant $C_\alpha>0$, only depending on $\alpha$, such that $$\label{ce}
\left |(f-\pi_\alpha f)(\Gamma_N a)\right|\leq
C_\alpha e^{-{a^\alpha(1-\e)\over r_{\alpha, \Gamma}}} \cdot
\sup_{g\in G\atop \beta\in\Sigma_\alpha} \left|f(\Gamma_N
g\exp(i (1-\e)c_\alpha E_\beta))\right|$$ for all $a\in A^+$.
We follow [@HC], Ch. I, $\S$ 7. We order the roots of $\Sigma_\alpha$, say
$$\beta_1> \beta_2>\ldots > \beta_s\, ,$$ and form ideals of $\nf$ by $$\nf_i=\bigoplus_{j=1}^i \gf^{\beta_j}\qquad (0\leq i \leq s)\, .$$ Set $U_i=\exp(\nf_i)$. Note that $\Gamma\cap U_i$ is cocompact in $U_i$ by our assumption on $\Gamma$. Thus $\pi_{U_i}$ is defined. Note that $\pi_{U_0}=\mathrm{id}$ and $\pi_{U_s}=\pi_\alpha$. We now verify the stronger statement (cf. [@HC], Lemma 19):
$$\label{rce}
f- \pi_{U_j} f \quad \hbox{satisfies (\ref{ce})}\, .$$
We prove (\[rce\]) by induction, following the arguments for the proof of [@HC], Lemma 19. The case $i=0$ is clear. Notice that $\u_i=\u_{i-1} \oplus \gf^{\beta_i}$. Choose a basis $E_1, \ldots , E_p$ of $\gf^{\beta_i}$ of normalized elements. We require in addition that $\exp(r_{\alpha, \Gamma}E_j)
\in \Gamma_N$. For $0\leq j\leq p$ we set
$$\v_j=\u_i \oplus\bigoplus_{k=1}^j {\mathbb{R}}E_k\quad\hbox{and}\quad
V_j=\exp (\v_j)\, .$$ Observe that each $V_j$ is a normal subgroup of $N$ with $\Gamma\cap V_j <V_j$ cocompact. In particular $\pi_{V_j}$ is defined. Set $\phi_{f,j}=\pi_{V_j} f$ for $0\leq j\leq p$ and $$\psi_{f,j}=\phi_{f,j-1}-\phi_{f,j}\qquad (1\leq j\leq p)\, .$$ Then $$\pi_{U_{i-1}} f -\pi_{U_i} f = \phi_{f,0}-\phi_{f,p}
=\sum_{j=1}^p \psi_{f,j}$$ and $$f-\pi_{U_j}f =\sum_{i=1}^{j} \pi_{U_{i-1}}f -\pi_{U_i}f$$ imply that it is sufficient to establish for all $1\leq j\leq p$ (cf. [@HC], Lemma 20):
$$\label{rce2}
\psi_{f,j} \quad \hbox{satisfies (\ref{ce})}\, .$$
For that let us fix $j$ and write $\phi_f=\phi_{f,j-1}$, $V=V_j$. So $\phi_f=\pi_V f $. Consider the mapping
$$V_{j-1}\backslash V_j\to{\mathbb{C}}, \ \ v\mapsto \phi_f(va)$$ and note that this function is left invariant under $\Gamma\cap V_j$. As $\exp(r_{\alpha,\Gamma}E_j)\in\Gamma\cap
V_j$ we obtain that
$$\phi_f (a)=\sum_{q\in{\mathbb{Z}}} \vartheta_{f,q} (a)
\quad\hbox{where}
\quad \vartheta_{f,q}(a)={1\over r_{\alpha, \Gamma}}\int_0^{r_{\alpha,\Gamma} }
\phi_f(\exp(t E_j)a)
e^{-{2\pi i q t\over r_{\alpha,\Gamma}}} \ dt\, .$$ We fix $a\in A_+$, $q\in{\mathbb{Z}}$ and consider the function $$F_{f,q}(z)=\phi_f(\exp(zE_j) a)
=\phi_f(a\exp(a^{-\beta_j} z E_j))$$ for $z\in{\mathbb{R}}$. We conclude that $F_{f,q}$ admits a holomorphic continuation to the strip domain
$$S=\{ z\in {\mathbb{C}}\mid |{\mbox{\rm Im}\,}z|< c_\alpha\cdot a^{\beta_j}\}\, .$$ Thus, as in the preceding two sections, we obtain for $0<\e<1$ the estimate
$$|\theta_{f,q}(a)| \leq M_\e \cdot e^{-2\pi q a^{\beta_j} (1-\e) c_\alpha\over {r_{\alpha,\Gamma}}}$$ where $$M_\e=\sup_{g\in G} |f(\Gamma_N g\exp(i(1-\e) c_\alpha E_j)|\, .$$ We sum up the geometric series and note that $\vartheta_{f,0}=\phi_{f,j}$ and obtain the desired estimate for $\psi_{f,j}=\phi_f-\phi_{f,j}$. This proves the lemma.
This lemma has an an immediate consequence the following important result (compare to [@HC], Corollary to Lemma 10).
[(Main Estimate)]{}\[cor=M\] Suppose that $f$ is Maaß[ ]{} cusp form. Then there exist a constant $C>0$, independent from $f$, such that for all $a\in A^+$ $$\label{ME1}|f(\Gamma a)|\leq C\cdot \min_{\alpha\in \Pi} e^{ - {2\pi (1-\e) a^\alpha c_\alpha\over
r_{\alpha, \Gamma}}}\cdot M_\e$$ with $$\label{M1} M_\e=\sup_{g\in G} \sup_{\beta\in\Sigma_\alpha\atop \alpha\in\Pi}
|f(\Gamma g \exp(i(1-\e)c_\alpha E_\beta)|\, .$$
It is instructive to consider the following example $$G=\Sl(n,{\mathbb{R}})\qquad \hbox{and} \quad \Gamma=\Sl(n,{\mathbb{Z}})\, .$$ In this situation we have $r_{\alpha,\Gamma}=c_\alpha=1$ for all $\alpha$ and the estimate in the Corollary becomes $$|f(\Gamma a)|\leq C\cdot e^{ - 2\pi (1-\e) \cdot \max_{1\leq i\leq n-1} {a_{i}\over a_{i+1}}}\cdot M_\e$$ with $$M_\e=\sup_{g\in G} \sup_{1\leq i<j\leq n} |f(\Gamma g \exp(i(1-\e)E_{ij}))|\, .$$
Refinements of the Main Estimate {#ss=me}
--------------------------------
It is possible to do a little bit better as in the Main Estimate once we apply the more refined geometric results from Subsections \[ssh\] and \[sssl\]. To state the inequalities in a more compact form we define
$$r_\Gamma=\sup_\alpha r_{\alpha,\Gamma}\,.$$
We begin with the Hermitian cases, i.e. where $\Sigma$ is of type $C_n$. We restrict our attention to the maximal Siegel parabolic with abelian nilradical and proceed as in Lemma \[mlem\] going simultaneously in the direction of the strongly orthogonal $E_j$ (cf. the notation in Subsection \[ssh\]). Then Theorem \[ts\] gives the following result.
\[mlemref1\] [(Main Estimate refined – the Hermitian case)]{} Suppose that $\Xi$ is a Hermitian symmetric space and $f$ is a Maaß cusp form on $X$. Then, with the notation of Subsection \[ssh\], there exists a constant $C>0$, independent from $f$, such that for all $t_i\geq 0$ $$\label{ME2}|f(\Gamma \exp(\sum_{j=1}^n t_jT_j))|\leq C \cdot e^{-{2\pi(1-\e)(\sum_{j=1}^n t_j)\over r_\Gamma}}\cdot M_\e$$ with $$\label{M2} M_\e=\sup_{g\in G} \left|f\left(\Gamma g \exp(i(1-\e)\sum_{j=1}^n E_j))\right)\right
|\, .$$
Finally we draw our attention to the case of $G=\Sl(n,{\mathbb{R}})$ and our fine geometric results in Subsection \[sssl\]. We will state our result for the Whittaker functionals of a Maaß cusp form. It is no loss of generality to assume that $N$, the group of unipotent upper triangular matrices, is cuspidal for the lattice $\Gamma$. Let us fix a unitary character $\chi: N\to {\mathbb S}^1$. As $\chi$ is necessarily trivial on $[N,N]$, it is clear that $\chi$ is given by a parameter ${\bf m}=(m_1, \ldots, m_{n-1})\in {\mathbb{R}}^{n-1}$, namely
$$\chi\begin{pmatrix} 1 & t_1 & * & \ldots & *\\
& \ddots& \ddots & * & *\\
& & 1 & t_{n-1}& *\\
& & & & 1\end{pmatrix}
=e^{2\pi i \sum_{j=1}^{n-1} t_j m_j}\, .$$ In the sequel we assume that $\chi$ is trivial on $\Gamma_N$.
For a cusp form $f$ we then define the [*Whittaker function with respect to $\chi$*]{} by
$$W(f,\chi)(g)=\int_{\Gamma_N \backslash N} f(\Gamma n g) \chi(n) \ d(\Gamma_N n)
\qquad (g\in G)$$ and note that $W(f,\chi)\in C^\infty (G/N, \chi)$. The obvious application of our standard technique yields:
\[mlemref2\] [(Main Estimate refined – Whittaker functionals for the special linear group)]{} Let $G=\Sl(n,{\mathbb{R}})$ and $f$ be a Maaß cusp form on $X$. Then, with the notation of Subsection \[sssl\], there exists a constant $C>0$, independent from $f$, such that for all $a=\diag(a_1,\ldots, a_n)\in A^+$ $$\label{ME3}
|W(f,\chi) (a)|\leq C \cdot e^{-{2\pi(1-\e)\over r_\Gamma}\sum_{j=1}^{n-1} |m_j|\cdot {a_j\over a_{j+1}}}
\cdot M_\e$$ with $$\label{M3} M_\e=\sup_{g\in G} \left|f(\Gamma g z(1-\e))\right
|\, .$$
A Bergman estimate on the local crown domains
---------------------------------------------
To proceed with our estimates on Maaß cusp forms we need to control the quantitities
$$M_\e=\sup_{g\in G} |f(\gamma gn_\e)|$$ for certain $n_\e\in N_{\mathbb{C}}$. In order to do so we estimate $M_\e$ against an $L^2$-norm, which can be controlled in terms of representation theory.
We state the result.
\[prop=berg\] Let $\Gamma$ be a lattice in the semi-simple group $G$. Fix an element $Z_0\in \partial \pi \Omega/2$ and a constant $0<\e<1$. Let $f$ be a $\Gamma$-invariant holomorphic function on $\Xi$. Then there exists a constant $C>0$, independent from $f$, such that for all $g\in G$ $$\begin{aligned}
|f(\Gamma &g \exp(i(1-\e)Z_0).x_0)| \leq C \cdot
\e^{-\dim X +{1\over 2} \mathrm{rank} X +{1\over 2}}\cdot \\
& \cdot \sup_{\{Z\in\Omega\mid \|Z-(1-\e)Z_0\|<\e/2\}}
\left(\int_{\Gamma\backslash G} |f(\Gamma g\exp(iZ).x_0)|^2 \ d(\Gamma g) \right)^{1\over 2}
\, .\end{aligned}$$
Before we start with the proof let us recall the basic Bergman estimate for polydiscs in ${\mathbb{C}}^n$. Fix $z_0\in {\mathbb{C}}^n$. For $r>0$ let us define the polyydisc centered at $z_0$ with radius $r$ by $$P(z_0, r)=\{ z\in {\mathbb{C}}^n \mid \|z-z_0\|_\infty <r\}\, .$$ One expands a holomorphic function $f\in \O(P(z_0,r))$ in a power series at $z_0$, and uses orthogonality of the monomials; the result is the [*Bergman estimate*]{}
$$\label{besti}
|f(z_0)|\leq {1\over \pi^n\cdot r ^n} \cdot \|f\|_{L^2 (P(z_0, r))}\, .$$
We turn to the proof of the proposition.
We normalize the Killing norm $\|\cdot\|$ on $\pf$ such that $\|Z_0 \|=1$. Let $Z_\e=(1-\e) Z_0$. We define various balls in $\pf$ and $\pf_{\mathbb{C}}$:
$$\begin{aligned}
B_1 & =\{ U\in \pf\mid \|U-Z_\e\|<\e/2\},\\
B_2 & =\{ V\in \pf \mid \|V\|<\e/2\},\\
B & =\{Z=U+iV\mid U\in B_1, V\in B_2\}.\end{aligned}$$
If necessary we may replace $\e$ by $c\e$ for some positive constant in the definition of $B_1$ and henceforth assume that $B_1\subset \pi\hat\Omega/2$. Then it is clear that $\exp(B_2)\exp(B_1).x_0\subset\Xi$, but what about $\exp(B).x_0$ ? This is not clear, but after some controlled shrinking we are in good shape:
There exists $c>0$ such that for all $0<\e < 1$ $$\exp(B_2)\exp(iB_1).x_0\supset \exp(\{ Z=U+iV\in \pf_{\mathbb{C}}: \|Z-Z_\e\|< c\e/2\})\, .$$
We remark that $\pi \overline {\hat \Omega}/2$ is compact and that
$$d\exp(iZ): \pf_{\mathbb{C}}\to T_{\exp(iZ).x_0} X_{\mathbb{C}}$$ is invertible for all $Z\in \pi \overline {\hat \Omega}/2$. In fact, the Jacobian of $\exp$ at $iZ$ is given by
$$|\det d\exp(iZ)|=\left | \prod_{\alpha\in\Sigma^+}
{\sinh \alpha(iZ)\over \alpha(iZ)}\right|\, .$$ The assertion follows from the implicit function theorem.
It is no loss of generality to assume that the constant $c$ in the previous lemma is $1$. At any rate the previous lemma combined with the Bergman estimate yields
$$\label{besti2}
|\phi(\exp(iZ_\e).x_0)|\leq C \cdot{1\over \e^{\dim X}} \left(\int_{\exp(B_2)\exp(iB_1).x_0} |\phi(z)|^2 \ dz\right)
^{1\over 2}$$
for a constant $C>0$ and all functions $\phi\in \O(\Xi)$. Here, $dz$ denotes the Haar measure on $X_{\mathbb{C}}$.
Next we set $B_{\af,1}=B_1\cap \af$ and note that $$\Ad (K) B_{\af,1}\supseteq B_1'\, .$$ with $B_1'= \{ U\in \pf\mid \|U-Z_\e\|<c\e/2\}$ for some constant $c>0$. Again it is no loss of generality to assume that $B_1'=B_1$. As a consequence we derive from (\[besti2\]) that $$\label{besti3}
|\phi(\exp(iZ_\e).x_0)|\leq C\cdot {1\over \e^{\dim X}} \left(\int_{\Gamma\backslash G \exp(iB_1).x_0}
|\phi(z)|^2 \ dz\right)
^{1\over 2}$$ for all $\Gamma$-invariant holomorphic functions $\phi$ on $\Xi$. Finally we use the integration formula [@KSII], Prop. 4.6, and obtain with $$J(Y)=\prod_{\alpha\in\Sigma^+} |\sin 2\alpha(Y)|^{m_\alpha} \qquad (Y\in\af)$$ that
$$\label{besti4}
\int_{\Gamma\backslash G \exp(iB_1).x_0}
|\phi(z)|^2 \ dz =\int_{\Gamma\backslash G}\int_{B_{\af,1}}
|\phi(\Gamma g\exp(iY).x_0)|^2 \cdot J(Y) \ d(\Gamma g) \ dY$$
Notice that $J(Y)\leq 2 \e$ as at least one root is going to vanish on $Z_\e$ for $\e\to 0$. Thus after combining (\[besti3\]) and (\[besti4\]) we obtain for all $g\in G$ that
$$\begin{aligned}
|f(g\exp(iZ_\e).x_0)|& \leq C \cdot \e^{-\dim X+ {1\over 2}(1+ \mathrm{rank} X)}
\cdot \\
& \cdot \sup_{Y\in B_{\af,1}} \left(\int_{\Gamma\backslash G}
f(\Gamma g\exp(iY).x_0|^2 \ dz\right)
^{1\over 2}\end{aligned}$$
for all $\Gamma$-invariant $f\in\O(\Xi)$.
Main estimates in final form
----------------------------
In this concluding subsection we put our previously obtained results together in order to obtain final version of our main estimates Corollary \[cor=M\], Lemma \[mlemref1\] and Lemma \[mlemref2\]. The main task is to obtain estimates for the quantities $M_\e$ in (\[M1\]), (\[M2\]) and (\[M3\]). We give details for the main case in (\[M1\]), and confine ourselves with stating the analogous results for the remaining two cases. So we wish to control the behavior of $$M_\e=\sup_{g\in G} \sup_{\beta\in\Sigma_\alpha\atop \alpha\in\Pi}
|f(\Gamma g \exp(i(1-\e)c_\alpha E_\beta)|\, .$$ First we deduce from Lemma \[lem=or\] that $$\label{m1}
M_\e\leq
\sup_{g\in G}\sup_{Y\in \partial\Omega}
|f(\Gamma g \exp(i(1-2\sqrt{\e})\pi Y/2)|\, .$$ Set $$r^X:=-\dim X + {1\over 2} \mathrm{rank} X + {1\over 2}\, .$$ Then it follows from Proposition \[prop=berg\] and (\[m1\]) that $$\label{m2}
M_\e\leq C \cdot \e^{{r^X}/2}
\sup_{Y\in \partial\Omega}
\cdot
\left(\int_{\Gamma\backslash G} |f(\Gamma g\exp(i(1-c \sqrt{\e})\pi Y/2).x_0)|^2 \ d(\Gamma g) \right)^{1\over 2}$$ for constants $C,c>0$ only depending on $X$. Assume that $f$ corresponds to the spherical representation $\pi_\mu$. Recall the exponents $s^X$ and $d^X$ from (\[def=ds\]). Now, Theorem \[thm=se\] applies and we arrive at $$\label{m3}
M_\e\leq C(\mu) \cdot \e^{{r^X}/2+ s^X/4}| \log \e|^{d^X/2}$$ for a constant $C=C(\mu)$ depending on $\mu$ and the geometry of $X$. If we specialize in Corollary \[cor=M\] to $\e=\min_{\alpha\in \Pi} a^{-\alpha}$ we get from (\[m3\]) the following
\[mef1\][(Main Estimate)]{} Suppose that $f$ is Maaß[ ]{} cusp form corresponding to $\pi_\mu$. Then there exist a constant $C=C(\mu)>0$ such that for all $a\in A^+$ $$\label{ME11} |f(\Gamma a)|\leq C\cdot \min_{\alpha\in \Pi} e^{ - {2\pi a^\alpha c_\alpha\over
r_{\alpha, \Gamma}}}\cdot \max_{\alpha\in \Pi} a^{-\alpha(r^X/2+ s^X/4)}\cdot |\alpha(\log a)|^{d^X/2}\, .$$
In similar manner we obtain a more concrete version of Lemma \[mlemref1\]:
\[mef2\] [(Main Estimate refined – the Hermitian case)]{} Suppose that $\Xi$ is a Hermitian symmetric space and $f$ is a Maaß cusp form on $X$ corresponding to $\pi_\mu$. Then, with the notation of Subsection \[ssh\], there exists a constant $C=C(\mu)>0$ such that for all $t_i\geq 0$ $$\label{ME22}|f(\Gamma \exp(\sum_{j=1}^n t_jT_j))|\leq
C \cdot e^{-{2\pi(\sum_{j=1}^n t_j)\over r_\Gamma}}\cdot
\left(\sum_{j=1}^n t_j\right)^{ -r^X/2 - s^X/4} \, .$$
Finally we state a new version of Lemma \[mlemref2\] (for which one also needs to employ the estimate in Proposition \[prost\]) :
\[mef3\] [(Main Estimate refined – Whittaker functionals for the special linear group)]{} Let $G=\Sl(n,{\mathbb{R}})$ and $f$ be a Maaß cusp form on $X$ corresponding to $\pi_\mu$. Then there exists a constant $C=C(\mu)$ such that for all $a=\diag(a_1,\ldots, a_n)\in A^+$ $$\begin{aligned}
|W(f,\chi) (a)|\leq & C \cdot e^{-{2\pi\over r_\Gamma} (\sum_{j=1}^{n-1} |m_j|\cdot {a_j\over a_{j+1}})}\cdot
\left(\sum_{j=1}^{n-1} |m_j|\cdot {a_j\over a_{j+1}}\right)^{-r^X - s^X/2}\\
& \cdot
\left|\log \left(\sum_{j=1}^{n-1} |m_j|\cdot {a_j\over a_{j+1}}\right)\right|
\, .\end{aligned}$$
[(Some generalizations of the Main Estimates)]{} Let $F$ either denote a Maaß cusp form $f$ or a Whittaker function $W(f,\chi)$ in case $G=\Sl(n,{\mathbb{R}})$. The general form of our estimates in Theorems \[mef1\], \[mef2\], \[mef3\], then is
$$\label{111}
(\forall a\in A^+) \qquad |F(\Gamma a)|\leq C e^{-\phi(a)} \cdot P(a)$$
where $C>0$ is a constant only depending on the representation $\pi_\mu$ associated to $F$, $$\phi(a)=\sum_{\alpha\in\Sigma^+} c_\alpha a^\alpha \qquad (c_\alpha\geq 0)$$ is a “positive” linear functional and $P(a)$ is a polynomial in the variables $a^\alpha, \alpha(\log a)$ with $\alpha\in\Sigma^+$.
\(a) [(Extension to a Siegel domain)]{} We restricted ourselves to estimates on $A^+$. However, for certain applications in number theory one needs estimates which are uniform on a Siegel domain $${\mathfrak S}_t=\omega A_t K \qquad (t>1)$$ where $\omega\subset N$ is a fixed compact and $A_t=\{ a\in A\mid
a^\alpha> t\ \forall \alpha \in \Pi\}$. The version of (\[111\]) on the whole Siegel domain ${\mathfrak S}_t$ is $$\label{222}
(\forall n\in\omega) (\forall a\in A_t) \qquad |F(\Gamma na)|\leq C_t e^{-c_t\phi(a)} \cdot P(a)$$ where $c_t,C_t>0$ are such that $c_t\to 1^- $ for $t\to \infty$. Let us explain how this is derived from (\[111\]). For the proof of (\[111\]) we use certain subsets $\Lambda_0\subset \Lambda$ and applied the fact that $F$ extends to a function on $\Gamma \exp(i\Ad(a) \Lambda_0) a.x_0$; recall, the precise rate of exponential decay was directly linked to the geometry of $\Lambda_0$. If one wants estimates on ${\mathfrak S} _t$ one needs to bring in $\omega$-variables, i.e. we look for maximal subsets $\Lambda_t\subset\Lambda_0$ such that $F$ extends to $\Gamma \exp(i \Ad(a)\Lambda_t) \omega a.x_0$. This is equivalent to the requirement of $$\exp(i\Lambda_t) a\omega a^{-1}.x_0\subset \Xi$$ Now $a\omega a^{-1}$ shrinks to $\{\bf 1\}$ for $t\to \infty$, meaning $\bigcup_{t>0} \Lambda_t=\Lambda_0$.
\(b) [(Extension to other $K$-types)]{} We only considered Maaß cusp forms, i.e. cusp forms associated to a trivial $K$-type. However, it is possible to extend to other $K$-types $\sigma\in\hat K$. For that one needs uniform quantitative control of the projection $\kappa: N_{\mathbb{C}}A_{\mathbb{C}}K_{\mathbb{C}}\to K_{\mathbb{C}}$ on $K$-orbits through $\exp(i\pi \Omega/2)$. This will be defered to another paper. In any case, the result then is
$$\label{333}
(\forall nak\in{\mathfrak S}_t) \qquad |F(\Gamma nak)|\leq C_t e^{-c_t\phi(a)} \cdot P_\sigma (a)$$
with $C_t=C_t(\mu,\sigma)$ and $P=P_\sigma$ now depending on $\sigma$. Further explanation is given in item (c) below.
\(c) [(Extension to non-spherical representations)]{} So far we only considered spherical representations $\pi=\pi_\mu$. But estimate (\[333\]) remains true for an arbitrary irreducible unitary representation $\pi$ and arbitrary $K$-types $\sigma$. Here is the reason. We can embed $\pi$ into a principal series representation induced off a minimal parabolic (subrepresentation theorem). One uses the fact that the smooth structures are unique (Casselman-Wallach). Now for a principal series one can look at the corresponding Eisenstein integrals for the $K$-types (Harish-Chandra) and everything boils down to estimate the spherical function and the sup-norm of a holomorphically extended $K$-type (for that one needs the quantative control of $\kappa$). The other details can be found in the proof of [@KSI], Th. 3.1.
\[rem11\] [(Quality of the estimates)]{} The rate of exponential decay given in above three Theorems are sharp. We provide some evidence in the section below. Concerning the polynomial part, one could likely replace $r^X$ by zero. However that would require to prove Conjecture C in [@KSI]; something which is out of reach with currently available techniques.
[(Applications to automorphic forms)]{}
\(a) [($L$-functions)]{} For various reasons on wants to know whether certain automorphic $L$-functions are meromorphic of finite order. For instance this information is required if one wants to exhibit zero-free regions (in the spirit of de la Vallée Poussin) for those $L$-functions. We refer to [@GLS], [@GL], [@GS] for results in this direction. We wish to point out that our estimates help to establish that $L$-functions with appropriate integral representations are in fact of finite order.
\(b) [(Voronoi summation)]{} Recently Voronoi summation was established for ${\mathrm {Gl}}(3,{\mathbb{R}})$, cf. [@MS]. Shortly after it was extended to ${\mathrm {Gl}}(n,{\mathbb{R}})$ in [@GLi]. In the approach of [@GLi] it becomes visible that exponential decay is an important analytical ingredient to establish Voronoi summation.
Final Remarks
=============
Estimates on Whittaker functionals for $\Gl(n)$ are sharp
---------------------------------------------------------
We show that the rate of exponential decay for Whittaker functionals for $G=\Gl(n,{\mathbb{R}})$ proved in Theorem \[mef3\] is optimal.
To begin with we recall the Whittaker expansion of Piatetski-Shapiro and Shalika for a cuspidal Maaß form of the group $\Gl(n,{\mathbb{R}})$. For simplicity let us restrict ourselves to the case $n=3$. Our arithmetic subgroup of choice will be $\Gamma=\Gl(3,{\mathbb{Z}})$. Let us define subgroups of $\Gamma$ by
$$\Gamma^2=\begin{pmatrix} \Gl(2,{\mathbb{Z}}) & 0\\ 0 & {\bf 1}
\end{pmatrix}\quad\hbox{and}\quad \Gamma^2_N=\Gamma^2\cap N\, .$$
In the sequel we use the notation introduced in Subsection \[ss=me\]. For $\chi$ corresponding to ${\bf m}=(1,1)$ and $f$ a Maaß cusp form we set $W(f)=W(f,\chi)$. For $n_1,n_2\in{\mathbb{N}}$ we put
$$W_{n_1,n_2}(f)(z)=W(f)\left( \begin{pmatrix} n_1 n_2 & & \\ & n_1 & \\
& & 1\end{pmatrix} z\right)$$ where $z\in X=\Gl(3,{\mathbb{R}})/ {\rm O}(3,{\mathbb{R}})$. The Whittaker expansion of $f$ reads as
$$f(z)=\sum_{\gamma\in \Gamma_N^2\backslash \Gamma^2}\sum_{n,1,n_2\in{\mathbb{N}}}
{a_{n_1,n_2}\over n_1 n_2} \cdot W_{n_1,n_2}(f)(z)$$ for complex coefficients $a_{n_1, n_2}$, [@Sh], Th. 5.9. We normalize $f$ such that $a_{1,1}=1$ and draw our attention to the main result in [@Bu], (10.1), which gives a formula for the Mellin transform of $W(f)$:
$$\begin{aligned}
\label{eq=bump}& \int_0^\infty \int_0^\infty W \begin{pmatrix}y_1 y_2 & & \\
& y_1 &\\ & & 1\end{pmatrix}\cdot y_1^{s_1-1}\cdot y_2^{s_2-1}
\ {dy_1\over y_1} {dy_2\over y_2}=\\
\nonumber&\quad = {1\over 4} \pi^{-s_1-s_2} \cdot{\Gamma\left({s_1+\alpha\over 2}\right)
\Gamma\left({s_1+\beta\over 2}\right)\Gamma\left({s_1+\gamma\over 2}\right)
\cdot\Gamma\left({s_2-\alpha\over 2}\right)
\Gamma\left({s_2-\beta\over 2}\right)\Gamma\left({s_2-\gamma\over 2}\right)
\over \Gamma\left({s_1+s_2\over 2}\right)}\, .\end{aligned}$$
Here $s_1,s_2$ are sufficiently large real numbers and $\alpha,\beta,\gamma=-\alpha-\beta$ are complex numbers related to the parameter of the principal series representation associated to $f$ (see [@Bu], p. 161). We perform a Stirling approximation of the right hand side (RHS) of (\[eq=bump\]) and obtain
$$(RHS)(s_1,s_2)\sim {1\over 4} \pi^{-s_1-s_2} \cdot \sqrt{2\pi}^5 \cdot
e^{-(s_1+s_2)}\cdot { \left({s_1\over 2}\right)^{3({s_1\over 2} -{1\over 2})}
\left({s_2\over 2}\right)^{3({s_2\over 2} -{1\over 2})} \over
\left({{s_1+s_2}\over 2}\right)^{({{s_1+s_2}\over 2} -{1\over 2})}}\, .$$ We specialize to $s_1=s_2=s$ and get the simpler expression
$$\label{eq=a1}
(RHS)(s,s)\sim {1\over 2} (2\pi)^{-2s+5/2} \cdot e^{-2s}\cdot {s^{2s-1/2} \over 2^s}\, .$$
Similarly, if we keep one variable fixed to be zero we get
$$\label{eq=a2}
(RHS)(s,0)= RHS(0, s)\sim C
(2\pi)^{-s} \cdot e^{-s}\cdot {s^{s-1}}\, .$$
We wish to compare these asymptotics with what we obtain by applying the estimate for $W(f)$ from Theorem \[mef3\] With $N>3$ we get $$\label{eq=a3}
\left|W(f)\begin{pmatrix} y_1y_2 & & \\ & y_1 & \\ & & 1\end{pmatrix}\right|
\leq C (1+ y_1^N + y_2^N ) e^{-2\pi (y_1+y_2)} \, .$$
We insert the estimate (\[eq=a3\]) into the left hand side (LHS) of \[eq=bump\] and arrive at the inequality $$\label{eq=a4}
LHS(s,s)\leq C (2\pi)^{-2 s} e^{-2s} (s+N-1)^{2s+2N-3}$$ and likewise $$\label{eq=a5}
LHS(s,0)\leq C (2\pi)^{- s} e^{-s} (s+N-1)^{s+N-3/2}\, .$$
Conjecturally we could even take any $N>0$ (cf. Remark \[rem11\]). In any case, if we compare (\[eq=a5\]) with (\[eq=a2\]) we see that the exponential decay for the Whittaker functional established in Theorem \[mef3\] is optimal. In fact, any better exponential decay rater would lead to decrease of $(2\pi)^{-2 s}$ to $(2\pi +a)^{-2s}$ for some $a>0$ on the right of (\[eq=a5\]); this would contradict the asymptotics in (\[eq=a2\]).
Automorphic holomorphic triple products
---------------------------------------
We introduce a holomorphic version of triple products and raise some natural questions. The setting here is: $G$ a semisimple noncompact Lie group and $\Gamma<G$ a cocompact lattice. For three automorphic forms $\phi_1, \phi_2,\phi_3$ one $\Gamma\backslash G$ one forms the automorphic triple product, or automorphic trilinear functional in the terminology of J. Bernstein and A. Reznikov,
$$\ell_{\rm aut}(\phi_1, \phi_2,\phi_3)=\int_{\Gamma\backslash G} \phi_1(\Gamma g) \phi_2(\Gamma g )
\phi_3(\Gamma g) \ d(\Gamma g)\, .$$
Assume now that the $\phi_i$ are Maaß forms so that the integral defining $\ell_{\rm aut}$ is effectively over the locally symmetric space $\Gamma\backslash X$. From the general theory we know that the $\phi_i$ extend to holomorphic functions $\tilde\phi_i$ on the local crown domain $\Gamma\backslash \Xi$. For the moment we restrict ourselves to the basic case of $G=\Sl(2,{\mathbb{R}})$ with comments on the general situation thereafter.
We form the [*holomorphic automorphic triple product*]{} by
$$\ell_{\rm aut}^{\rm hol}(\phi_1, \phi_2,\phi_3)=\int_{\Gamma\backslash \Xi} \tilde \phi_1(\Gamma z)
\tilde \phi_2(\Gamma z ) \tilde \phi_3(\Gamma z) \ d(\Gamma z)\,$$ where $d(\Gamma z)$ is the measure on $\Gamma\backslash \Xi$ induced from the Haar measure on $X_{\mathbb{C}}$. That $\ell_{\rm
aut}^{\rm hol}$ is actually defined is content of the next lemma.
Let $G=\Sl(2,{\mathbb{R}})$ and $\Gamma<G$ be a cocompact lattice. Let $\phi_1,\phi_2,\phi_3$ be Maaß automorphic forms. Then the intergral defining $\ell_{\rm aut}^{\rm hol}$ converges absolutely.
In view of [@KSI], Th. 5.1 and Th. 6.17, there exists a constant $C>0$ such that we have for all $Y\in\Omega$
$$\sup_{g\in G} |\tilde \phi_i(\Gamma g \exp(i\pi Y/2).x_0)|\leq C \left |\log \cos \alpha(\pi Y/2)\right|$$ Thus in view of the polar decomposition of the measure $\Xi$ (see [@KSII], Prop. 4.6), we get $$|\ell_{\rm aut}^{\rm hol}(\phi_1, \phi_2,\phi_3)|\leq C^3 \int_0^1 \left |\log \cos \alpha(\pi Y/2)\right|^3
\cdot \sin \alpha(\pi Y) \ dY <\infty$$ and this proves the lemma.
Determine the relation between $\ell_{\rm aut}$ and $\ell_{\rm aut}^{\rm hol}$ and explain its significance.
For a general semisimple Lie group the integrals defining $\ell_{\rm aut}^{\rm hol}$ are not absolutely convergent. However, we have some freedom in the choice of the $G$-invariant measure on $\Xi$. Under the parameterization map $p: G/M\times \Omega^+\to \Xi$ the pull back of the Haar measure $dz$ on $\Xi$ is given by $$p^*(dz)=d(gM)\times J(Z)dZ$$ with $J(Z)=\prod_{\alpha\in\Sigma^+} [\sin \alpha(\pi Z)]^{m_\alpha}$ (see [@KSII], Prop. 4.6). For example if we replace $J$ by sufficiently high power $J^k$, then $\ell_{\rm aut}^{\rm hol}$ is defined with regard to this measure. It is possible to carry out the details using the results obtained in this article.
Exponential decay of automorphic triple products
------------------------------------------------
With the methods of analytic continuation one can prove exponential decay of automorphic triple products (see [@Sa] and [@BR] for the first results). To be a more precise, consider a compact locally symmetric space $\Gamma\backslash X$ and fix a Maaß form $\phi$. Then for a Maaß form $\phi_\pi$ corresponding to an automorphic representation $\pi$ there is interest in finding the precise exponential decay of
$$|\ell_{\rm aut}(\phi,\phi_\pi,\overline{\phi_\pi})|$$ in terms of the parameter $\lambda(\pi)$ of $\pi$. This was first determined by Sarnak for $G=\Sl(2,{\mathbb{C}})$ [@Sa] and then by Petridis [@Pet] and Bernstein-Reznikov [@BR] for $G=\Sl(2,{\mathbb{R}})$. Optimal bounds for all rank one groups were established in [@KSI]. For higher rank groups , such as $\Sl(n,{\mathbb{R}})$, partial results were obtained in [@KSI]. These bounds however fail to be optimal in general. The results in this paper combined with the methods of [@BR] and [@KSI] allow to establish non-trivial (although) non-optimal bounds for the exponential decay of automorphic triple products.
Appendix: Leading exponents of holomorphically extended elementary spherical functions {#app:exp}
======================================================================================
In this appendix we prove Theorem \[thm:esteta\] and the table of Theorem \[thm:table\] for the asymptotic behavior of norm of the holomorphic extension of the orbit map $G/K\ni gK\to\pi(g)v$ of a spherical vector $v\in\H$ in an irreducible spherical representation $(H,\pi)$ of $G$ when the argument approaches the distinguished boundary of the crown domain $\Xi$. The key property equation (\[eq:orbnorm\]) translates this to the problem of finding the asymptotic behavior of certain solutions of a system of differential equations when approaching the singular locus of the system. In the theory of ordinary Fuchsian differential equations this boils down to the study of characteristic exponents at its singular points of $\mathbb{P}^1$ and their relation to the monodromy of the system. A beautiful application (closely related to our problem in fact, via (\[eq:l2norm\])) of this classical theory is the study of the asymptotic behavior of certain classes of oscillatory integrals via the monodromy of the Gauß-Manin connection of the Milnor fibration of the phase function [@Malg].
In the case of “Fuchsian systems” of differential equations in several complex variables we first need to develop some fundamental facts on exponents and their first properties. In the case of a regular holonomic $\mathcal{D}$-module of the form $\mathcal{D}/\mathcal{J}$ on ${\mathbb{C}}^n$ which is $\mathcal{O}$-coherent on the complement of a *hyperplane arrangement* in ${\mathbb{C}}^n$ we propose a definition of the set of exponents of local solutions of the system of equations $D\phi=0,\ \forall D\in\mathcal{J}$ at any point $\eta\in{\mathbb{C}}^n$. This translates our original problem to that of determining the set of exponents of a special solution to Harish-Chandra’s radial system of differential equations on $A_C.x_0$ (namely the holomorphic extension of the restriction of the elementary spherical function to $A.x_0$) at the extremal boundary points $t(\eta)^2 x_0$ of $T_\Omega$.
What turns this into a successful method is the fact that there exists a well behaved parameter deformation of Harish-Chandra’s radial system of differential equations for which we have rather explicit knowledge of the monodromy representation of its solutions *for generic parameters*. This deformation is the hypergeometric system of differential equations [@HO1],[@HS],[@Op4]. Its monodromy factors through an affine Hecke algebra, thus bringing the representation theory of affine Hecke algebras into play. In the spirit of the study of the Bessel function equations [@Op2] this leads to the description of the set of all exponents of the hypergeometric system at $t(\eta)^2.x_0$. Using that and the relation between exponents and monodromy (which we will carefully establish below) we can compute the leading exponents of the holomorphically extended hypergeometric function at the points $t(\eta)^2 x_0$.
Specialization of the parameters then leads to the desired lower bounds for the leading exponents of holomorphically extended elementary spherical functions on a Riemannian symmetric space $X$, leading to the proof of Theorem \[thm:esteta\] and Theorem \[thm:table\].
One remarkable phenomenon that comes out of these considerations is that the leading exponent of the hypergeometric function at an extremal point $t(\eta)^2.x_0$ is related to a *leading character* $\s_\eta$ of the isotropy group $W^a_\eta\subset W$ of $t(\eta)^2.x_0$ which depends only on the geometry of $\Omega$ locally at the extremal point $\eta$, but not on the multiplicity $m$ (if $m$ is real and satisfies certain inequalities which hold for the multiplicity functions of Riemannian symmetric spaces).
Exponents and hyperplane arrangements {#sub:expAR}
-------------------------------------
In this subsection we propose a definition of exponents of local Nilsson class functions [@Bj Chapter 6.4] on the complement of a hyperplane arrangement of ${\mathbb{C}}^n$ at points $\eta\in{\mathbb{C}}^n$. The main results are that the exponents at $\eta$ are invariant for local monodromy at $\eta$ and the relation between exponents and monodromy.
Let $\eta\in{\mathbb{C}}^n$ and let $\phi$ be a local Nilsson class function at $\eta$. By this we mean a multivalued holomorphic function $\phi$ on the complement $N\backslash Y:=N^{\operatorname{reg}}$ of an analytic hypersurface $Y\subset {\mathbb{C}}^n$ inside a small open ball $N\subset {\mathbb{C}}^n$ centered at $\eta$ such that
1. $\phi$ has finite determination order in $N^{\operatorname{reg}}$.
2. The pull back of any branch of $\phi$ via any holomorphic map $j:\mathbb{D}\to N$ with the property that $j^{-1}(Y)\subset\{0\}$ has moderate growth at $0\in\mathbb{D}$.
Suppose that $j$ as in LN2 is an embedding such that $j(0)=\eta$. Then the pull back of (any branch of) $\phi$ via $j$ has a singular expansion at $\e=0$ (where $\e$ denotes the standard coordinate in the unit disk $\mathbb{D}$) of the form $$\label{eq:singexp}
\phi(j(\e))=\sum_{s,l}\e^s\log^l(\e)f_{s,l}(\e),$$ a sum over a finite set of pairs $(s,l)$ with $s\in{\mathbb{C}}$ and $l\in\mathbb{Z}_{\geq 0}$, such that for each pair $(s,l)$ in this sum the function $f_{s,l}(\e)$ is holomorphic on $\mathbb{D}^\times$ with at most a pole at $\e=0$. This expansion is obviously not unique, and even if one tries to make it unique by imposing additional requirements one will find that the set $S$ which enters in (\[eq:zsingexp\]) will in general depend on the chosen embedding $j$ and of the chosen branch of $\phi$ (with respect to local monodromy in $N^{\operatorname{reg}}$) in an essential way. In order to define exponents of $\phi$ at $\eta$ we assume from now on the following.
1. For sufficiently small $N$ we may take $Y=Y^\eta$ to be a linear hyperplane arrangement centered at $\eta$.
We call a holomorphic map $i:\mathbb{D}^{n}\to N$ a standard coordinate map if
1. $i(0,z)=\eta$ for all $z\in\mathbb{D}^{n-1}$.
2. $i(\mathbb{D}^\times\times\mathbb{D}^{n-1})
\subset N^{\operatorname{reg}}$.
3. The lift of the map $i:\mathbb{D}^\times\times\mathbb{D}^{n-1}\to N^{\operatorname{reg}}$ to the blow-up $X_\eta\to {\mathbb{C}}^n$ of $N$ at the point $\eta$ extends to a coordinate map $i:\mathbb{D}^n\to X_\eta$ such that $i(\mathbb{D}^n)\cap Z=\emptyset$, where $Z$ denotes the strict transform of $Y\cap N$.
Let $i$ be a standard coordinate map. Choose a base point $p=i(P)\in i(\mathbb{D}^\times\times\mathbb{D}^{n-1})$ and fix a germ $\phi_p$ of a branch of $\phi$ at $p$. Let $C\subset\mathbb{D}^\times$ be a cut disk (the complement in $\mathbb{D}$ of a ray emerging from $0$) such that $P\in C\times\mathbb{D}^{n-1}$. The pull back of $\phi_p$ via $i$ to $P\in C\times\mathbb{D}^{n-1}$ is the germ of a Nilsson class function on $\mathbb{D}^\times\times\mathbb{D}^{n-1}$. Hence we have the following standard result [@Bj Proposition 4.4.2]:
There exists a finite set $S$ of pairs $(s,l)$ with $s\in{\mathbb{C}}$ and $l\in\mathbb{Z}_{\geq0}$ such that the unique analytic continuation of $i^*(\phi_p)$ to $C\times\mathbb{D}^{n-1}$ admits an expansion of the form $$\label{eq:zsingexp}
\phi_p(i(\e,z^\prime))=\sum_{(s,l)\in S}
\e^s\log^l(\e)f_{s,l}(\e,z^\prime),$$ where each $f_{s,l}$ extends meromorphically to $\mathbb{D}^\times\times\mathbb{D}^{n-1}$.
As before this expansion is not unique for obvious reasons but we can rearrange (\[eq:zsingexp\]) in such a way that
1. all $f_{s,l}$ extend holomorphically on $\mathbb{D}^n$,
2. if the pairs $(s,l)$ and $(s^\prime,l^\prime)$ occur in (\[eq:zsingexp\]) then $s-s^\prime$ is not equal to a nonzero integer, and
3. if the pair $(s,l)$ occurs in (\[eq:zsingexp\]) then there exists an $l^\prime\in{\mathbb{Z}}_{\geq 0}$ such that $f_{s,l^\prime}(0,\cdot)\not\equiv 0$.
That makes the expansion unique.
\[dfn:expphi\] Let $\phi$ be a local Nilsson class function at $\eta\in{\mathbb{C}}^n$ and let $i:\mathbb{D}^\times\times\mathbb{D}^{n-1}\to
N^{\operatorname{reg}}$ be a standard coordinate map. Choose a base point $p$ in the image of $i$, and choose a germ $\phi_p$ of a branch of $\phi$ at $p$. We define the finite set $E^{\eta,i,p}(\phi_p)\subset{\mathbb{C}}\cup\{\infty\}$ of exponents of $\phi_p$ at $\eta$ as the projection of the finite set $S\subset{\mathbb{C}}\times\mathbb{Z}_{\geq0}$ defined above to the first component if $\phi\not=0$. We put $E^{\eta,i,p}(0)=\{\infty\}$.
\[prop:indep\] The set $E^{\eta,i,p}(\phi_p)$ is independent of the choice of $i$ (satisfying the requirements (i),(ii) and (iii) above) and is independent of analytic continuation of $\phi_p$ within $N^{\operatorname{reg}}$. Hence we may speak about the set of exponents $E^{\eta}(\phi)$ without referring to a specific branch of $\phi$ and coordinate map $i$. If $w:N\to N$ is a linear automorphism of the hyperplane arrangement $Y^\eta$ then $E^\eta(\phi^w)=E^\eta(\phi)$.
By equation (\[eq:zsingexp\]) it is clear that $E^{\eta,i,p}(\phi_p)$ is independent of analytic continuation of $\phi_p$ along paths inside $i(\mathbb{D}^\times\times\mathbb{D}^{n-1})$. Suppose that $i,i^\prime$ both satisfy the requirements above, and suppose that $i(\{0\}\times\mathbb{D}^{n-1})\cap
i^\prime(\{0\}\times\mathbb{D}^{n-1})\not=\emptyset$. Let $V\subset i(\{0\}\times\mathbb{D}^{n-1})\cap
i^\prime(\{0\}\times\mathbb{D}^{n-1})\subset E$ be a connected contractible open set (where $E$ denotes the exceptional divisor). By the properties of $i$ and $i^\prime$ we have $(i^\prime)^{-1}(i(\e,z^\prime))=(\e^\prime,w^\prime)$ with $\e^\prime(\e,z^\prime)=\e h(\e,z^\prime)$ where $h$ is holomorphic and nonzero on $i^{-1}(V)$. If we plug this in the expansion (\[eq:zsingexp\]) we see that the exponents defined by $i$ and by $i^\prime$ are equal if we use branches of $\phi$ on the image of $i$ and on the image of $i^\prime$ which are related by analytic continuation via the connected component of the intersection of the images of $i$ and $i^\prime$ which contains $V$. By AR we see that that any path in $N^{\operatorname{reg}}$ is homotopic to a path which is contained in a finite union of coordinate patches of the form $i(\mathbb{D}^\times\times\mathbb{D}^{n-1})$. With the above this shows at once that the set of exponents does not depend on the choice of the coordinate map $i$ and is independent for analytic continuation of $\phi_p$ within $N^{\operatorname{reg}}$. For the last assertion we remark that $i_w=w\circ i$ is also a standard coordinate map, hence $E^\eta(\phi^w)=
E^{\eta,i_w,w(p)}((\phi^{w})^{w(p)})=E^{\eta,i,p}(\phi_p)=E^\eta(\phi)$.
What lies behind this notion of exponents is the well known “decone construction” on a central hyperplane arrangement. This elementary construction implies that if $Y^\eta$ is nonempty then $N^{\operatorname{reg}}$ is a isomorphic to a product $$N^{\operatorname{reg}}\simeq\mathbb{D}^\times\times E^{\operatorname{reg}}$$ Indeed, the restriction of the Hopf fibration $p:{\mathbb{C}}^n\backslash\{0\}\to E=\mathbb{P}({\mathbb{C}}^n)$ to the complement of one of the hyperplanes $H$ of $Y$ is a trivial fibration since $E\backslash\mathbb{P}(H)\simeq {\mathbb{C}}^{n-1}$ is contractible. Hence the further restriction of this fibration to $N^{\operatorname{reg}}$ is a fortiori trivial. Thus we have a decomposition $$\label{eq:funddec}
\pi_1(N^{\operatorname{reg}},p)\simeq{\mathbb{Z}}\times
\pi_1(E^{\operatorname{reg}},[p])$$ Now let $\L\subset\O(N^{\operatorname{reg}})$ be a local system of finite rank $r$ of germs of Nilsson class function on $N^{\operatorname{reg}}$. We remark that, as a result of LN1, the germs of any local Nilsson class function $\phi$ on $N^{\operatorname{reg}}$ are contained in such a local system.
\[dfn:gamma\] We denote by $T_p^\eta$ the monodromy map on $\L_p\simeq{\mathbb{C}}^r$ which corresponds to analytic continuation along the loop $\g_p^\eta:t\to \exp(2i\pi t)p$. Observe that $[\g_p^\eta]$ is a generator of ${\mathbb{Z}}$ in (\[eq:funddec\]).
In view of (\[eq:funddec\]) we may use $T_p^\eta$ to split the sheaf $\L$ as a direct sum $$\label{eq:block}
\L=\bigoplus_{t\in{\mathbb{C}}^\times}\L^\eta(t)$$ of generalized eigensheaves of $T^\eta$. In view of (\[eq:zsingexp\]) it is clear that if $\phi\not=0$ then $\phi\in\L_p^\eta(t)$ iff $E^\eta(\phi)=\{s\}$ for some exponent $s\in{\mathbb{C}}$ such that $t=\exp(2i\pi s)$.
Given $s\in{\mathbb{C}}\cup\{\infty\}$ we define a subsheaf $F_s^\eta(\L)\subset \L^\eta(\exp(2i\pi s))$ of $\L$ by setting for each $p\in N^{\operatorname{reg}}$: $$F_s^\eta(\L)_p=\{\phi\in\L_p\mid E^\eta(\phi)
=\{\kappa\}\mathrm{\ with\ }
\kappa-s\in\mathbb{Z}_{\geq 0}\cup\{\infty\}\}$$ One checks easily that this is a linear subspace of $\L_p$. By Proposition \[prop:indep\] it is invariant for the parallel transport in the local system $\L$, hence it defines a subsheaf. Moreover, these subsheaves of $\L$ are invariant for the action of the group of automorphisms of $\L$ which are induced by linear automorphisms of the arrangement $Y^\eta$. For each $t\in{\mathbb{C}}^\times$ the subsheaves $F^\eta_s(\L)\subset\L$ with $s\in{\mathbb{C}}$ such that $\exp(2i\pi s)=t$ define a descending filtration $$\dots\supset F^\eta_s(\L)\supset F^\eta_{s+1}(\L)\supset\dots$$ of the direct summand $\L^\eta(t)$ of $\L$.
We define a local system $\operatorname{Gr}^\eta(\L)$ by $$\operatorname{Gr}^\eta(\L)=\bigoplus_{s\in{\mathbb{C}}}
\operatorname{Gr}^\eta_s(\L),\mathrm{\ with\ }
\operatorname{Gr}^\eta_s(\L)=
F^\eta_s(\L)/F^\eta_{s+1}(\L)$$
For each $s\in{\mathbb{C}}$ we define the multiplicity $\operatorname{mult}^\eta(\L,s)$ of $s$ as an exponent at $\eta$ of the system of $\L$ by $$\operatorname{mult}^\eta(\L,s)=\dim(\operatorname{Gr}^\eta_s(\L))$$
The (multi-)set $E^\eta\subset{\mathbb{C}}$ of exponents of $\L$ at $\eta$ are the complex numbers $s\in{\mathbb{C}}$ such that $\operatorname{mult}^\eta(\L,s)>0$.
\[cor:expT\] The (multi-)set $\exp(2i\pi E^\eta)\subset{\mathbb{C}}^\times$ is the generalized eigenvalue spectrum of $T^\eta$ acting on $\L$.
\[ex:bench\] Consider for $\mu\in\af_{\mathbb{C}}^*$ the sheaf $\L$ of local solutions of the set of equations $$\partial(p)\phi=p(\mu)\phi,\ \forall p\in{\mathbb{C}}[\af_{\mathbb{C}}^*]^W$$ and let $\eta\in\af_{\mathbb{C}}$ be any point. Any local solution $\phi$ is holomorphic at $\eta$ and is completely determined by its harmonic derivatives $\partial(q)(\phi)(\eta)$ at $\eta$. Hence the set of exponents of $\L$ at $\eta$ is independent of $\eta$ and $\mu$, and is equal to the set $0,1,\dots,|\Sigma^l_+|$ where $\operatorname{mult}^\eta(\L,s)
=\dim{\operatorname{Harm}_s}(W)$, the dimension of the space of $W$-harmonic polynomials of homogeneous degree $s$.
Consider the sheaf $\L$ of local solutions of (\[eq:cpx\]). Suppose that $\eta\in\af_{\mathbb{C}}^{\operatorname{reg}}$ is a regular point. Again a local solution $\phi$ of (\[eq:cpx\]) near $\eta$ is holomorphic at $\eta$ and is completely determined by its harmonic derivatives $\partial(q)(\phi)(\eta)$ at $\eta$. Hence the answer is the same as in the previous example.
\[ex:cpx\] Let $\L$ be as in the previous example, but now we take $\eta=i\pi\omega_j/k_j$ as in subsection \[sub:distbdy\]. The exponents of $\L$ at $\eta$ are equal to $-|\Sigma^a_{\eta,+}|,\dots,|\Sigma^l_+|-|\Sigma^a_{\eta,+}|$, and if $\phi_\mu$ denotes the holomorphic extension to $\af+i\pi\Omega$ of the spherical function, then for generic $\mu$ we have $E^{\eta}(\phi_\mu)=\{|\Sigma_{\eta,+}|-|\Sigma^a_{\eta,+}|\}$ (see Proposition \[cpx\]).
Harish-Chandra’s radial system of differential equations
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In this subsection we describe the system of differential equation we are mainly interested in, the radial differential equations for an elementary spherical functions $\phi_\mu^X$ on a Riemannian symmetric space $X=G/K$ restricted to a maximal flat, totally geodesic subspace $A_X=A.x_0\subset X$.
The elementary spherical function $\phi_\mu^X$ (with $\mu\in\af_{\mathbb{C}}^*$) on $X=G/K$ is a $K$-invariant solution of the $G$-invariant system of differential equations $$(\Delta-\gamma_X(\Delta)(\mu))\phi=0\ \forall\Delta\in\mathbf{D}(X)$$ where $\mathbf{D}(X)$ denotes the ring of $G$-invariant differential operators on $X$, and where $\gamma_X:\mathbf{D}(X)\to\mathbb{C}[\af^*]^{W}$ is the Harish-Chandra isomorphism.
By separation of variables we see that the restriction of $\phi^X_\mu$ to $A_X$ is a $W$-invariant solution of the system of differential equations $$\label{eq:sphsys}
(D-\gamma_X(D)(\mu))\phi=0\ \forall D\in\Ri_X$$ on $A_X$ or on its complexification $A_{X,{\mathbb{C}}}=A_{\mathbb{C}}.x_0$, where $\Ri_X\simeq \mathbf{D}(X)$ is the algebra of radial parts of the operators $\Delta\in\mathbf{D}(X)$. Notice that we use the same notation $\g_X$ for the Harish-Chandra isomorphism defined on $\Ri_X$.
Let $T_X=T.x_0\subset A_{X,{\mathbb{C}}}$ be the compact form of $A_{X,{\mathbb{C}}}$. It is a maximal flat totally geodesic subspace of a compact dual symmetric space $U/K$ (which is by our choices simply connected). The restrictions to $T_X$ of the zonal spherical functions of $U/K$ are $W$-invariant simultaneous eigenfunctions of $\Ri_X$. Since these zonal polynomials constitute a linear basis of the space of $W$-invariant Laurent polynomials on $A_{X,{\mathbb{C}}}=A_{\mathbb{C}}/F$ this implies that the operators in $\Ri_X$ descend to *polynomial* differential operators on the complex affine quotient space $W\backslash A_{\mathbb{C}}/F$.
The hypergeometric system of differential equations
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In this subsection we describe a parameter deformation of the Harish-Chandra system (\[eq:sphsys\]) of differential equations that we will use to study properties of solutions of (\[eq:sphsys\]). This parameter family of systems of differential equations is called the system of hypergeometric equations associated with root systems. As was explained, this deformation is an essential ingredient for the computation of the leading exponents (\[eq:leadexp\]) of the spherical functions at extremal points of $T_{\Omega}$.
We need to introduce some notations. Let $\Sigma$ be a (not necessarily reduced) irreducible root system in $\af^*$. We consider indeterminates $\mathbf{m}_{\a}$ which are labeled by the *$W$-orbits* of the roots $\a\in\Sigma$ (in other words, $\mathbf{m}_\a=\mathbf{m}_\b$ if $W\a=W\b$). Let ${\mathbb{C}}[\mathbf{m}_\a]$ be the complex polynomial algebra over these indeterminates $\mathbf{m}_\a$. If $X$ is a Riemannian symmetric space with restricted root system isomorphic to $\Sigma$ then $m_\a^X\in\mathbb{N}$ denote the root multiplicities of $X$.
The following result is one of the cornerstones of the theory of hypergeometric functions for root systems.
Let $\mathbb{A}$ denote the Weyl algebra of polynomial differential operators on the complex affine space $W\backslash
A_\mathbb{C}/F\simeq {\mathbb{C}}^n$ with coefficients in the polynomial ring ${\mathbb{C}}[\mathbf{m}_\a]$. There exists a unique subalgebra $\Ri\subset{\mathbb{C}}[\mathbf{m}_\a]\otimes\mathbb{A}$ with the following properties:
\(1) The algebra $\Ri$ is isomorphic to the polynomial ring ${\mathbb{C}}[{\mathbf{m}}_\a][\af^*_{\mathbb{C}}]^W$ via a Harish-Chandra isomorphism $\g$ of algebras. This isomorphism $\g$ has the characterizing property that any element $D\in\Ri$ is asymptotically equal to the constant coefficient operator $\g(D)(\cdot-\rho(\mathbf{m}))$ on $A_\mathbb{C}$ (viewed as an element of the symmetric algebra on $\af_{\mathbb{C}}$) along regular directions towards infinity in $A_+$.
\(2) If we specialize $\mathbf{m}$ at the multiplicity function $m^X$ for a Riemannian symmetric space $X=G/K$ with restricted root system $\Sigma_X\subset\af^*$ such that $\Sigma^l=\Sigma^l_X$, such that $A_{\mathbb{C}}$ is the maximal torus of $G_{\mathbb{C}}$, then $\Ri$ specializes to $\Ri_X$ and $\g$ to the Harish-Chandra isomorphism $\g_X$.
It is remarkable that the theory of Dunkl operators provides a proof of this theorem which is both elementary and simple [@H].
\[prop:RS\] Let $\mathbb{D}^{\operatorname{reg}}$ be the ring of algebraic differential operators on the affine variety $A^{\operatorname{reg}}_{\mathbb{C}}/F$. For each multiplicity parameter $m=(m_\a)$ and $\mu\in\af^*_{\mathbb{C}}$ let $\mathcal{I}_{m,\mu}\subset\mathbb{D}^{\operatorname{reg}}$ denote the $W$-invariant left ideal $$\label{eq:ideal}
\mathcal{I}_{m,\mu}:=\sum_{D\in\Ri_m}
\mathbb{D}^{\operatorname{reg}}(D-\g_m(D)(\mu))$$ Here $\Ri_m$ is the specialization of $\Ri$ at $m$, and $\g_m$ the corresponding Harish-Chandra homomorphism.
Consider the $\mathbb{D}^{\operatorname{reg}}$-module $\mathcal{M}_{\mu,m}=\mathbb{D}^{\operatorname{reg}}/
\mathcal{I}_{m,\mu}$ on $A_{\mathbb{C}}^{\operatorname{reg}}/F$. Then in the terminology of Chapter IV, section 7 of [@Bo], $\mathcal{M}_{\mu,m}$ is an algebraic connection on $A_{\mathbb{C}}^{\operatorname{reg}}/F=
A_{\mathbb{C}}/F-\{\delta=0\}$ of rank $|W|$ which is regular. Moreover $\mathcal{M}_{\mu,m}$ is $W$-equivariant.
The elements of $\Ri_m$ are algebraic and the coefficients are known to be regular on $A_{\mathbb{C}}^{\operatorname{reg}}/F$ (the simplest way to see this is to use Dunkl-Cherednik operators [@H], [@Op3]). It is known that $\mathcal{M}_{\mu,m}$ is $\mathcal{O}(A_{\mathbb{C}}^{\operatorname{reg}}/F)$- free of rank $|W|$ by [@HO1], and it is clear that $\mathcal{M}_{\mu,m}$ is $W$-equivariant. It remains to prove the regularity.
The elements of $\Ri_m$ descend to the regular part of the adjoint torus $A_{\mathbb{C}}^{\operatorname{adj,reg}}/F$ with character lattice $Q=\mathbb{Z}\Sigma\subset\af^*$. We view this as an open subset of the toric completion of $A_{\mathbb{C}}^{\operatorname{adj}}/F$ associated with the decomposition of $\af^*$ in Weyl chambers. This is a projective variety. It clearly suffices to prove the regularity on $A_{\mathbb{C}}^{\operatorname{adj,reg}}/F$.
On $A_{\mathbb{C}}^{\operatorname{adj,reg}}/F$ one can explicitly rewrite the module $\mathcal{M}_{\mu,m}$ as a connection of rank $|W|$ with logarithmic singularities at infinity (see [@HO1]). According to [@HO1] the connection matrix depends polynomially on the parameters $\mu$ and $m$. It remains to show that the connection is also regular singular at the components of the discriminant locus $\delta=0$. Since the connection depends polynomially on the parameters $\mu,m$ it is easy to see that the set of parameters $\mu,m$ for which the connection is regular singular is a Zariski-closed set. If $m=0$ the system is trivially regular singular. If $\mu$ is sufficiently generic then the theory of shift operators gives equivalences between the modules $\mathcal{M}_{\mu,m}$ and $\mathcal{M}_{\mu,m^\prime}$ if $m-m^\prime$ belong to the “lattice of integral shifts” (see e.g. [@Op0] or [@HS]) in the space of multiplicity parameters. The result follows.
\[rem:cycl\] The element $u=1\in \M_{\mu,m}$ is a cyclic vector. Via $u$ the complex vector space of $D$-module homomorphisms of $\M_{\mu,m}$ to $\mathcal{O}_p$ correspond to the space $\L_p(\mu,m)$ of solutions in $\mathcal{O}_p$ of the $W$-invariant system of differential equations $$\label{eq:hypsys}
(D-\gamma_m(D)(\mu))\phi=0\ \forall D\in\Ri_m$$ on $A_{\mathbb{C}}^{\operatorname{reg}}/F$.
The local system $\L(\mu,m)$ of germs of solutions of (\[eq:hypsys\]) is a local system of germs of Nilsson class functions on $\af_{\mathbb{C}}^{\operatorname{reg}}$. Hence the results of Subsection \[sub:expAR\] are applicable to $\L(\mu,m)$.
By [@Bj Proposition 4.6.6] it is sufficient to check the moderate growth conditions for solutions of (\[eq:hypsys\]) on the dense open set of subregular points of $\d=0$. Since we can rewrite the system (\[eq:hypsys\]) as a meromorphic connection on $\af_{\mathbb{C}}^{\operatorname{reg}}$ which is regular singular along $\d=0$ according to Proposition \[prop:RS\] this follows from [@Bo Remark (5.9)] (see also [@D]).
Let $X$ be a Riemannian symmetric space with maximal flat geodesic subspace $A.x_0$. The holomorphic extension of the restriction to $A.x_0$ of the spherical function $\phi^X_\mu$ to $AT_\Omega^2 x_0$ is a holomorphic $W$-invariant solution of (\[eq:sphsys\]) on $AT_\Omega^2.x_0$. This function is the specialization of a holomorphic family (in the parameter $m$) of solutions of (\[eq:hypsys\]) by virtue of the following theorem:
\[thm:hypfun\]([@HO1],[@HS],[@Op4]) There exists an $\e>0$ such that for all multiplicity parameters $m\in\Qc(-\e)$, the space of multiplicity parameters such that $\operatorname{Re}(m_\a)\geq -\e\
\forall\a\in\Sigma$, the hypergeometric system (\[eq:hypsys\]) has a unique solution $\phi_{\mu,m}$, the hypergeometric function, which extends to a $W$-invariant and holomorphic function on $AT_\Omega^2.x_0$. The function $(t,\mu,m)\to\phi_{\mu,m}(t)$ is holomorphic on $(AT_\Omega^2.x_0)\times\Qc(-\e)\times\af_{\mathbb{C}}^*$.
Recall the covering map $\pi:\af_{\mathbb{C}}\to A_{\mathbb{C}}/F\simeq A_{\mathbb{C}}.x_0$ of (\[eq:cov\]) which is given by the exponential map $\pi(X)=\exp(\pi i X)F$. Via this map we will lift the differential equations (\[eq:hypsys\]) to $\af_{\mathbb{C}}^{\operatorname{reg}}$ and work on $\af_{\mathbb{C}}$ rather than $A_{\mathbb{C}}/F$. On this space the system of differential equations (\[eq:hypsys\]) is invariant for the action of the affine Weyl group $W^a=W\ltimes {Q^\vee}$. In particular, we will work on the tube domain $i\af+\Omega\subset \af_{\mathbb{C}}$ instead of $AT_\Omega^2/F\subset A_{\mathbb{C}}/F$ (recall that the logarithm is well defined on $AT_\Omega^2$). It is well known [@HO1] that the spherical system of eigenfunction equations can be cast in the form of an integrable connection on $\af_{\mathbb{C}}$ with singularities along the collection of affine hyperplanes $\a(H)\in{\mathbb{Z}}$ (not $\in\pi
i {\mathbb{Z}}$ as in [@HO1], since we have multiplied everything by $(\pi i)^{-1}$).
The indicial equation {#sub:ind}
---------------------
We will show in this subsection that the exponents of the hypergeometric equations (\[eq:hypsys\]) at $\eta\in\af_{\mathbb{C}}$ coincide with the eigenvalues of the residue matrix of a specially chosen integrable connection with simple poles which is equivalent to (\[eq:hypsys\]). The characteristic equation of the residue matrix has coefficients which are polynomials in the parameters $m_\a$. This equation is called the indicial equation of (\[eq:hypsys\]) at $\eta$.
Let us first construct an explicit standard coordinate map $i$ as used in the definition of the set of exponents. Consider a parameterized line $x\to \eta+xV_1$ through $\eta$, where $V_1$ is small and chosen in such a way that this line is not contained in the union of the singular affine hyperplanes. We choose coordinates $(z_1=\e,z_2,\dots z_n)$ (with $z_1=\e\in\mathbb{D}^\times$ and for $i>1$: $z_i\in \mathbb{D}$), which we will often write as $z=(\e,z^\prime)\in\mathbb{D}^\times
\times\mathbb{D}^{n-1}$ with $z^\prime=(z_2,\dots,z_n)$. First we choose $V_2,\dots,V_n$ in $\af$ such that $\Vert
V_i\Vert$ is small for all $i$, and such that $(V_1,V_2,\dots,V_n)$ is a basis of the real vector space $\af$. Then our coordinate map $i$ is given by $$i(\e,z^\prime)=\eta+\e(V_1+\sum_{i\geq 2} z_iV_i)\in \af.$$ If we lift this coordinate map to the blow-up of $\af_{\mathbb{C}}$ at $\eta$ then the coordinates can be naturally extended to the polydisk $\mathbb{D}^n$, and this is then a coordinate neighborhood of a regular point of the exceptional divisor $E$. The intersection of this neighborhood with $E$ is described by the equation $z_1=0$. The complement of $z_1=0$ in $\mathbb{D}^n$ is $\mathbb{D}^\times\times\mathbb{D}^{n-1}$, the “punctured polydisk”. The Euler vector field $\E^\eta$ is given in these coordinates by $z_1\partial/\partial{z_1}=\e\partial/\partial{\e}$.
Let $p$ be a point in the punctured polydisk $\mathbb{D}^\times\times\mathbb{D}^{n-1}$ and let $\O_p$ denote the ring of holomorphic germs at $p$. Consider a subspace $U^*$ of dimension $|W|$ of the ring of holomorphic linear partial differential operators on $\mathbb{D}^\times\times\mathbb{D}^{n-1}$ such that at all points $p\in
\mathbb{D}^\times\times\mathbb{D}^{n-1}$, the free $\O_p$-module $\O_p\otimes U^*$ is a complement for the left ideal $\mathcal{I}_{\mu,m}$. We require further that the elements of $U^*$ commute with the Euler vector field $\E^\eta$ (in other words, they are homogeneous of degree $0$), and that $1\in U^*$. Such linear subspaces $U^*$ exist, for instance one could take as a basis $b_i=
\e^{\operatorname{deg}(q_i)}\partial(q_i)$, where $q_i$ runs over a homogeneous basis of $W$-harmonic polynomials on $\af^*_{\mathbb{C}}$ with $q_1=1$.
We rewrite the differential equations (\[eq:hypsys\]) (with $\mu\in\af_{\mathbb{C}}^*$) in connection form with respect to the above basis $\{b_i\}$ and coordinates $\{z_i\}$. We define matrices $A^i_{\mu,m}\in\operatorname{End}_{\O_p}(\O_p\otimes U)$ (where $U$ denotes the dual of $U^*$, with dual basis $b_i^*$) which are characterized by the requirement that $$\label{eq:A}
\frac{\partial}{\partial z_i}\circ b_k\in\sum_{j}
(A^{i}_{\mu,m})^{\mathrm{tr}}_{jk}b_j+\I_{\mu,m}.$$ As an $\O_p$-module, the cyclic $D$-module $(M_{\mu,m},u)$ is equal to $\O_p\otimes U^*u$, with basis $\overline{b_i}=b_i.u$. Then the desired (flat) connection form of (\[eq:hypsys\]) is defined on the free $\O_p$-module $\O_p\otimes U$ by $$\label{con}
\frac{\partial \Phi}{\partial z_i}=A^i_{\mu,m}\Phi\ \ (\Phi\in\O_p\otimes U).$$ By construction, if $\phi$ is a solution of (\[eq:hypsys\]) then $$\label{eq:iso}
\Phi(\phi):=\sum_i b_i(\phi)b_i^*$$ is a solution vector of (\[con\]). Conversely, if $\Phi$ is a solution vector of (\[con\]) then the first coordinate $\phi=\langle b_1,\Phi\rangle$ is a solution of (\[eq:hypsys\]). Since the linear map $\phi\to\Phi=\sum_i b_i(\phi)b_i^*$ is clearly injective we see by a dimension count that these linear maps are inverse isomorphisms between the solution spaces of these two systems of differential equation.
\[rem:loc\] Since the local solution space of an integrable connection at a regular point $p$ can be identified with the fiber of the underlying vector bundle at $p$, the above gives an isomorphism (depending on $p$) between the local solution space $\L_p(\mu,m)$ of (\[eq:hypsys\]) at $p$ and the complex vector space $U$.
We claim that the system (\[con\]) has simple singularities at $\e=0$. The basis vectors $b_i=\e^{\operatorname{deg}(q_i)}\partial(q_i)$ have homogeneous degree zero and thus belong to the ring $\mathcal{D}_0$ of holomorphic differential operators on $\mathbb{D}^{n}$ generated by vector fields tangent to $\e=0$ (i.e. by $\partial/\partial{z_i}$ with $i>1$ and by $\e\partial/\partial\e$). Therefore our claim is easily implied by (also compare to [@HO1 Proposition 3.2]):
Given $B\in\mathcal{D}_0$ there exists a unique section $u(B)_{\mu,m}=\sum_{j}u(B)^j_{\mu,m}b_j\in
\mathcal{O}(\mathbb{D}^n)\otimes U^*$ such that $$\label{eq:Bmat}
B\in u(B)_{\mu,m}+\I_{\mu,m}$$ The map $\mathcal{D}_0\ni B\to u(B)_{\mu,m}$ is an $\mathcal{O}(\mathbb{D}^n)$-module morphism which depends polynomially on $\mu$ and $m$. For all $B\in\mathcal{D}_0$, $u(B)_{\mu,m}|_{\{0\}\times \mathbb{D}^{n-1}}$ is independent of $\mu$.
We use induction on the order $d$ of $B$. Using the well known theorem that ${\mathbb{C}}[\af_{\mathbb{C}}^*]$ is the free ${\mathbb{C}}[\af_{\mathbb{C}}^*]^W$-module generated by the $W$-harmonic polynomials, we have a unique decomposition $$B=\sum_{i,j} f_{i,j}(\e,z^\prime)b_i \e^{d_{i,j}}\partial(p_{i,j})$$ with $p_{i,j}\in{\mathbb{C}}[\af_{\mathbb{C}}]^W$ a homogeneous polynomial of degree $d_{i,j}$ such that $d_{i,j}+
\operatorname{deg}(b_i)\leq d$, and where $f_{i,j}(\e,z^\prime)$ is holomorphic for all $i,j$. Now $\e^{d_{i,j}}\partial(p_{i,j})=
\e^{d_{i,j}}(D_{p_{i,j}}-\g(D_{p_{i,j}})(\mu))$ modulo lower order operators in $\mathcal{D}_0$, where we have used the fact that for $p\in{\mathbb{C}}[\af_{\mathbb{C}}]^W$ homogeneous, the lowest homogeneous part $h^\eta(D_{p})$ at $\eta$ of $D_{p}\in\Ri_m$ contains the highest order term $\partial(p)$ of $D_p$ (see [@Op0]). By the induction hypothesis we conclude the existence $u(B)_{\mu,m}$. Using the independence of the $W$-harmonic polynomials over the ring ${\mathbb{C}}[\af_{\mathbb{C}}^*]^W$ and the induction hypothesis the uniqueness of $u(B)_{\mu,m}$ follows too. By induction and using the fact that the operators $D_p$ depend polynomially on $\mu$ and $m$ we conclude that $u(B)_{\mu,m}$ is holomorphic on $\mathbb{D}^n$ and polynomial in $\mu$ and $m$. Since in the induction step $\mu$ only occurs via the terms of the form $\e^{d_{i,j}}\g(D_{p_{i,j}})(\mu)$ we see that $\mu$ does not influence the evaluation at $\e=0$ of $u(B)_{\mu,m}$.
Let $R_m$ be the residue matrix of $A^1_{\mu,m}$ at $z_1=0$. By the previous lemma $R_m$ is independent of $\mu$ and is polynomial in $m$. As is well known (cf. [@Bo], Chapter IV, section 4, or [@D]) $R_m$ is independent of the coordinate map $i$. Moreover, let us consider on $V=i(\{0\}\times\mathbb{D}^{n-1})$ the integrable connection defined by the restrictions $B^i_{\mu,m}:={A^i_{\mu,m}}|_{V}$ for $i>1$. Then the residue $R_m$ is known to be flat for this integrable connection on $V$. In particular, its characteristic equation is independent of $z^\prime$.
\[lem:eigen\] The exponents of (\[eq:hypsys\]) at $\eta$ are the eigenvalues of the residue matrix $R_m$ of $A^1_{\mu,m}$ at $z_1=0$. The characteristic polynomial of $R_m$ is independent of $\mu$ and of $z^\prime$ and has polynomial coefficients in the $m_\a$.
By changing the basis of the trivial vector bundle (with fiber $U$) on $i(\mathbb{D}^n)$ by a suitable invertible matrix depending on $z^\prime$ only we may assume that $B^i_m=0$ on $V$ for all $i>1$. We denote the finite dimensional complex vector space of sections spanned by this basis of flat sections $\mathcal{U}$. By the flatness of $R_m$ for the restricted connection on $V$ as above, $R_m$ is constant in this new basis (i.e. independent of $z^\prime$). Let $s$ be an eigenvalue of $R_m$, and let $v$ be a generalized $R_m$-eigenvector with eigenvalue $s$. Put $u(\e)=\exp(\log(\e)R_m)v=\e^s\exp(\log(\e)(R_m-s\operatorname{Id}))v$, and observe that $$\label{eq:q0}
q_0^{^{(s,v)}}(\e,z^\prime):=\exp(\log(\e)(R_m-s\operatorname{Id}_\mathcal{U}))v$$ is a $\mathcal{U}$-valued polynomial in $\log(\e)$. We denote the series expansion of $\e A^1_{\mu,m}$ in $\e$ with respect to a fixed basis of $\mathcal{U}$ by $$\e A^1_{\mu,m}(\e,z^\prime)=R_m+\sum_{k>1}\e^k
A^1_{\mu,m,k}(z^\prime)$$ with $A_{\mu,m,k}^1(z^\prime)$ holomorphic for $z^\prime\in\mathbb{D}^{n-1}$. Now we use the following relative version of [@Wa Ch. IV, §24, Hilfssatz XI]: If $q_i(\e,z^\prime)$ ($i<k$) are $\mathcal{U}$-valued polynomials in $\log(\e)$ of degree $\leq N$ with coefficients in the ring of holomorphic functions on $\mathbb{D}^{n-1}$, then the equation $$\label{eq:rec}
\e\frac{\partial{q_k}}{\partial{\e}}+((s+k)\operatorname{Id}_\mathcal{U}-R_m)q_k=
\sum_{i=0}^{k-1}A_{\mu,m,k-i}^1(z^\prime)q_i$$ has at least one solution $q_k$ which is polynomial in $\log(\e)$ and has coefficients in the ring of holomorphic functions in $z^\prime\in\mathbb{D}^{n-1}$. The solution $q_k$ is unique and has degree $\leq N$ if $(s+k)$ is not an eigenvalue of $R_m$. In general there exist several solutions $q_k$ which are polynomial in $\log(\e)$ and these solutions are all of degree $\leq N+r$ in $\log(\e)$, where $r$ is the maximal length of a Jordan block of $R_m$ with eigenvalue $(s+k)$.
Given a set $\{q_k^{(s,v)}\}$ of solutions of the recurrence relations (\[eq:rec\]) (with $q_k^{(s,v)}$ polynomial in $\log(\e)$ for all $k$, and $q_0^{(s,v)}$ given by (\[eq:q0\])) there exists a convergent (but multivalued) series solution $\Phi^{(s,v)}$ of (\[con\]) on $i(\mathbb{D}^\times\times\mathbb{D}^{n-1})$ of the form $$\label{eq:series}
\Phi^{(s,v)}(\e,z^\prime)=\e^s\sum_{k\geq 0}\e^k q_k^{(s,v)}(\e,z^\prime)$$ (see e.g. [@Wa Ch. IV, §24, XII]). Notice that the degree of $q_k^{(s,v)}(\e,z^\prime)$ ($k\geq 0$) as a polynomial in $\log(\e)$ with coefficients in the ring of holomorphic functions in $z^\prime\in\mathbb{D}^{n-1}$ is uniformly bounded.
Such series expansion is not necessarily unique, but by choosing such a series solutions $\Phi^{(s,v)}$ for a set of pairs $(s,v)$ where $s$ runs through the set of eigenvalues of $R_m$ and for each $s$, $v$ runs through a basis of the generalized $s$-eigenspace of $R_m$ then the collection of multivalued solutions $\Phi^{(s,v)}$ on $i(\mathbb{D}^\times\times\mathbb{D}^{n-1})$ constitutes a basis for the space of multivalued solutions of (\[con\]). On the other hand we have seen above that the flat sections on $i(\mathbb{D}^\times\times\mathbb{D}^{n-1})$ all are of the form $\Phi=\sum_i b_i(\phi)b_i^*$ where $\phi=\langle b_1,\Phi\rangle$ is a solution of (\[eq:hypsys\]). Hence the set of exponents of (\[eq:hypsys\]) must coincide with the set of eigenvalues of $R_m$, counted with multiplicity.
As a result of the above theorem the following definition makes sense.
Let $R_{\bf{m}}=R_{\bf{m}}^\eta$ denote the $|W|\times|W|$-matrix with coefficients in the ring ${\mathbb{C}}[{\bf{m}}_\a]\otimes\O(\mathbb{D}^{n-1})$ such that $R_m=R^\eta_m$ is the specialization of $R_{\bf{m}}^\eta$ at ${\bf{m}}=m$ (this matrix depends on the coordinate map $i$). We call the characteristic polynomial $I_{\bf{m}}^\eta\in{\mathbb{C}}[{\bf{m}_\a}][X]$ of $R_{\bf{m}}^\eta$ the “indicial polynomial” of (\[eq:hypsys\]) at $\eta$.
\[cor:ind\] The (multi-)set $E^\eta$ of exponents of (\[eq:hypsys\]) at $\eta$ is equal to the (multi-)set of roots of the indicial polynomial $I_m^\eta$ of (\[eq:hypsys\]) at $\eta$.
Hecke algebras and exponents
----------------------------
We now bring into play well known results on the monodromy of the system of hypergeometric differential equations. We have quite good control for generic parameters as a consequence of the main result, the fact that this representation of the affine braid group factors through an affine Hecke algebra. We apply these results to prove that the indicial polynomial $I^\eta$ at $\eta$ factors completely over the ring of rational polynomials in the indeterminates $\bf{m}_\a$ with roots that are affine linear functions in the $\bf{m}_\a$ with half integral coefficients.
By affine Weyl group symmetry we may assume without loss of generality that $\eta\in \overline{\Omega}\cap C$, the fundamental alcove. From now on we will make this assumption.
By Corollary \[cor:expT\] the generalized eigenvalue spectrum of $\L_p(\mu,m)$ under the action of $T^\eta_p$ contains information on the set $E^\eta$ of exponents of (\[eq:hypsys\]). Since by \[eq:funddec\] $T^\eta_p$ is certainly central in $\Pi_1(N^{\operatorname{reg}},p)$, the decomposition of $\L_p(\mu,m)$ in indecomposable blocks for the monodromy action of $\Pi_1(N^{\operatorname{reg}},p)$ on $\L_p(\mu,m)$ refines the decomposition in generalized $T^\eta_p$ eigenspaces (by virtue of Schur’s Lemma).
Therefore we now recall some fundamental facts on the monodromy representation of the fundamental group $\Pi_1(W^a\backslash\af_{\mathbb{C}}^{\operatorname{reg}},p)$ (at a regular base point $p\in\af_{\mathbb{C}}^{\operatorname{reg}}$ in the fundamental alcove $\overline{\Omega}\cap C$) on the local solution space $\L_p=\L_p(\mu,m)$ of (\[eq:hypsys\]). By a well known result of Looijenga and Van der Lek ([@L], also see [@HO1], [@HS], [@Op4]) the group $\Pi_1(W^a\backslash\af_{\mathbb{C}}^{\operatorname{reg}},p)$ is isomorphic to the affine braid group $B^a$ of $W^a=W\ltimes Q(\Sigma^\vee)$, the affine Weyl group of the affine root system $\Sigma^a=\Sigma^l\times\mathbb{Z}$. In order to formulate the result we need to define an affine root multiplicity function $m^a$ on the affine roots in $\Sigma^a$ as follows. For the affine simple roots $a_0=1-\theta,a_1=\a_1,\dots,a_n=\a_n$ we define $$\begin{aligned}
\label{eq:maff}
m_{a_0}^a&=m_\theta\\ \nonumber m_{a_i}^a&=m_{\a_i}+m_{\a_i/2}\end{aligned}$$ and then we extend this to $\Sigma^a$ by $W^a$-invariance.
(cf. [@HO1], [@HS], [@Op4]) The monodromy action on the $W^a$-equivariant local system $\L_p(\mu,m)$ on $\af_{\mathbb{C}}^{\operatorname{reg}}$ factors through an affine Hecke algebra $H(W^a,q^a)$ in the following sense.
Let $q^a$ be the label function *on the affine root system* $\Sigma^a=\Sigma^l\times\mathbb{Z}$ defined by $q_b^a=\exp(-i\pi(m^a_b)$ for all $b\in\Sigma^a$. For the simple affine roots $a_i$ we write $q^a_{a_i}:=q^a_i$. The monodromy matrices $M_{\mu,m}(b_i)$ ($i=0,\dots,n$) of the generators $b_i$ of $B^a$ satisfy $(M_{\mu,m}(b_i)-1)(M_{\mu,m}(b_i)+q^a_i)=0$. The monodromy representation $M_{\mu,m}$ of $\Pi_1(W^a\backslash\af_{\mathbb{C}}^{\operatorname{reg}},p)$ depends analytically on the parameters $m$ and $\mu$.
Recall that $W_\eta^a$ is the isotropy subgroup of $\eta$ in $W^a$, which is a finite reflection group, and let $\Sigma_\eta^a$ be the corresponding root system. There is a natural monomorphism $W_\eta^a\to W$ with image $\tilde{W}^a_\eta\subset W$. We put $N_\eta=[W:\tilde{W}_\eta^a]$ for the index of this subgroup.
Let us denote by $B_\eta^a\subset B^a$ the braid group of $W_\eta^a$, which we can identify, by Brieskorn’s theorem on the fundamental group of the regular orbit space of a finite reflection group, with the fundamental group of the “local regular orbit space” at $\eta$, namely $\Pi_1(W^a_\eta\backslash N^{\operatorname{reg}},p)$.
Let $m_\eta^a$ be the restriction of $m^a$ to $\Sigma_\eta^a$, and let $q_\eta^a$ be corresponding the corresponding root multiplicity function on $\Sigma_\eta^a$. Let $\Qc$ denote the finite dimensional complex vector space of complex multiplicity functions $m$ on (the possibly non-reduced) root system $\Sigma$. In a dense, open set $\Qc_\eta^{\operatorname{reg}}\subset\Qc$ of values of the parameter $m$, the finite dimensional Hecke algebra $H(W_\eta^a,q_\eta^a)$ (with $q_\eta^a=q(m_\eta^a)$) is a semisimple algebra. If we assume that $m\in \Qc_\eta^{\operatorname{reg}}$ then, by Tits’ deformation lemma, we can index its set of irreducible modules by $\widehat{W_\eta^a}$, the set of irreducible representations of $W_\eta^a$. Given $\tau\in
\widehat{W_\eta^a}$ and $m\in \Qc_\eta^{\operatorname{reg}}$ we will write $\pi_\tau^\eta(q_\eta^a)$ for the corresponding irreducible $H(W_\eta^a,q_\eta^a)$-module. Upon restriction of the monodromy action of $B^a$ on $\L_p(\mu,m)$ to $B_\eta^a$ we have:
\[cor:monfin\] Let $q=q(m)$ and $q_\eta^a=q(m_\eta^a)$ for $m\in
\Qc_\eta^{\operatorname{reg}}$. The monodromy action of $\Pi_1(W^a_\eta\backslash N^{\operatorname{reg}},p)$ on $\L_p(\mu,m)$ factors through the semisimple finite type Hecke algebra $H(W_\eta^a,q_\eta^a)$, and the local solution space $\L_p(\mu,m)$ decomposes under this action in isotypical components $$\label{eq:mondec}
\L_p(\mu,m)=\bigoplus_{\tau\in\widehat{W_\eta^a}}
\L_p(\mu,m)(\tau)$$ such that for each $\tau\in\widehat{W_\eta^a}$, $\L_p(\mu,m)(\tau)\simeq K(\tau,m)\otimes\pi_\tau^\eta(q_\eta^a)$ with $\dim(K(\tau,m))=N_\eta\operatorname{deg}_\tau$ (independent of $m\in\Qc_\eta^{\operatorname{reg}}$).
Using the rigidity of semisimple finite dimensional algebras (Tits’ deformation lemma, [@Ca], Proposition 10.11.4) the multiplicity of $\pi_\tau^\eta(q_\eta^a)$ is constant in $(\mu,m)\in\af_{\mathbb{C}}^*\times\Qc_\eta^{\operatorname{reg}}$. We may therefore compute the multiplicity by evaluating at $(\mu,m)=(0,0)$. Hence it is equal to the multiplicity of $\tau$ in the restriction of the regular representation of $W$ to $\tilde{W}_\eta^a$, which is $N_\eta\operatorname{deg}_\tau$.
The following topological observation due to Deligne [@D1] is crucial for our purpose:
\[lem:delcent\] Let $\b^\eta_p\in B_\eta^a$ denote the local braid in $\Pi_1(W^a_\eta\backslash N^{\operatorname{reg}},p)$ which corresponds to a reduced expression of the longest element of $W_\eta^a$. Then $(\b^\eta_p)^2=[\g^\eta_p]$ (see Definition \[dfn:gamma\]). In particular this element is central in $B_\eta^a$.
Given $\tau\in\widehat{W_\eta^a}$ we denote by $p^i_{\eta,\tau}$ (with $i=1,\dots,N_\eta\operatorname{deg}_\tau$) the embedding degrees of $\tau$ in the graded vector space of $W$-harmonic polynomials. We choose these $W$-harmonic embedding degrees so that $i\to p^i_{\eta,\tau}$ is a non-decreasing sequence. In particular, $p^1_{\eta,\tau}$ is the “harmonic birthday” of $\tau$ in the $W$-harmonic polynomials.
\[prop:ev\] Let $\tau\in\widehat{W_\eta^a}$ and let $m\in\Qc_\eta^{\operatorname{reg}}$. The multiset $E^\eta(\tau,m)$ of exponents of $\L_p(\mu,m)(\tau)$ consists of the complex numbers $$s^i_{\eta,\tau}(m)=
p^i_{\eta,\tau}-\frac{1}{2}c_{\eta,\tau}(m),$$ where $i$ runs from $1$ to $N_\eta\operatorname{deg}_\tau$, each $s^i_{\eta,\tau}(m)$ occurring with multiplicity $\operatorname{deg}_\tau$. Here $c_{\eta,\tau}(m)$ is the affine linear function of the multiplicity parameters $m_\a$ with nonnegative integral coefficients defined by (cf. (\[eq:maff\]) for the definition of $m^a_{\eta,b}$): $$c_{\eta,\tau}(m)=\sum_{b\in\Sigma_{\eta,+}^a}
(1-\frac{\chi_\tau(s_b)}{\operatorname{deg}_\tau})m_{\eta,b}^a$$
Since $T_p^\eta$ is the monodromy action of the (locally) central braid $(b^\eta_p)^2$ (by Lemma \[lem:delcent\]) we see that $T_p^\eta$ acts trivially in the multiplicity space $K(\tau,m)$ and acts by scalar multiplication in the irreducible representation $\pi^\eta_\tau(q_\eta^a)$ of the Hecke algebra $H(W^a_\eta,q^a_\eta)$ by some scalar $C$. This $C$ is an element of the ring of Laurent polynomial in the Hecke algebra labels $(q_{\eta,b}^a)^{1/2}$ (with $b\in\Sigma^a_\eta$) since this is the splitting ring of the Hecke algebra. By taking the determinant of $T_p^\eta$ in $\pi^\eta_\tau(q_\eta^a)$ we find easily that $C^{\operatorname{deg}_\tau}=
\exp(-i\pi\operatorname{deg}_\tau c_{\eta,\tau}(m))$. This implies that $C$ is a root of $1$ times a monomial in the $(q^a_{\eta,b})^{\pm 1/2}$. For $(q^a_{\eta,b})^{1/2}=1$ we have $C=1$, hence $$C=\exp(-i\pi c(m))$$ Let $\mathcal{N}$ denote the collection of functions $\nu$ on the set $\tau\in\widehat{W^a_\eta}$ which associate to each $\tau$ a finite multiset $\nu(\tau)=\{\nu_{\tau,j}\mid j=1,\dots,N_\eta\operatorname{deg}_\tau^2\}$ of $N_\eta\operatorname{deg}_\tau^2$ integers $\nu_{\tau,j}\in{\mathbb{Z}}$. By Corollary \[cor:expT\] and Corollary \[cor:ind\] it follows that for each $m\in\Qc_\eta^{\operatorname{reg}}$ the set of roots of the indicial polynomial $I^\eta_m$ is a multiset of the form $\rho_{\tau,\nu,j}(m)=\nu_{\tau,j}-1/2c_{\eta,\tau}(m)$ for some $\nu\in\mathcal{N}$. For each $\nu\in\mathcal{N}$ the set $\Qc(\nu)\subset\Qc$ of multiplicity parameters $m\in\Qc$ for which the multiset of roots of $I^\eta_m$ is equal to the multiset $\{\rho_{\tau,\nu,j}(m)\}$ is Zariski-closed (since $I^\eta_m$ is a polynomial in $m$, by Corollary \[cor:ind\]). Moreover the union of these sets contains $\Qc_\eta^{\operatorname{reg}}$. Since $\mathcal{N}$ is countable, Baire’s category theorem implies that there must exist at least one $\nu_0\in\mathcal{N}$ such that the interior (in the analytic topology) of $\Qc(\nu_0)$ is nonempty, and hence such that $\Qc=\Qc(\nu_0)$.
In the situation $m\in\Qc_\eta^{\operatorname{reg}}$ we have the splitting in the isotypical components (\[eq:mondec\]). It follows that the set $E^\eta(\tau,m)$ consists of the subset $\rho_{\tau,\nu_0,j}(m)$ ($j=1,\dots,N_\eta\operatorname{deg}_\tau^2$) of roots of the indicial equation. Finally we need to determine $\nu_0(\tau)$. This is resolved by taking $m=0\in\Qc_\eta^{\operatorname{reg}}$ and comparing to Example \[ex:bench\], after making the additional remark that the monodromy representation $\pi^\eta_\tau(q^a_\eta)$ is by definition equal to $\tau$ if $q^a_\eta=1$.
\[cor:nolog\] For $m\in\Qc^{\operatorname{reg}}_\eta$ the action of $T^\eta_p$ is semisimple. In particular, there are no logarithmic terms in the decomposition (\[eq:zsingexp\]) if $m\in\Qc^{\operatorname{reg}}_\eta$ and if $\phi_p\in\L_p(\mu,m)$.
So we conclude this subsection with the following remarkable result:
The indicial polynomial factorizes as $$I_m(X)=\prod_{\tau\in\widehat{W^a_\eta}}
\prod_{i=1}^{N_\eta\operatorname{deg}_\tau}
(X-s^i_{\eta,\tau}(m))^{\operatorname{deg}_\tau}$$ For $m\in\Qc^{\operatorname{reg}}$ this factorization is compatible with the decomposition of $\L_p(\mu,m)$ in blocks of the form $\L_p(\mu,m)(\tau)$ as in Corollary \[cor:monfin\].
Computation of the leading exponents
------------------------------------
Let $\phi_{\mu,m}\in\L_p(\mu,m)$ with $p\in\overline{\Omega}\cap C$ denote the hypergeometric function, the solution of (\[eq:hypsys\]) whose germ at points of the fundamental alcove $\overline{\Omega}\cap C$ we define by analytic continuation along a path in $\overline{\Omega}\cap C$ of the unique normalized $W$-invariant solution of (\[eq:hypsys\]) which extends holomorphically to a neighborhood of $0\in \af_{\mathbb{C}}$.
\[cor:cont\] By definition, $\phi_{\mu,m}$ extends holomorphically over all finite walls, the walls of $C$, to a $W$-invariant function on $\Omega$, the interior of $WC$. In particular, if $\eta\in
\overline{\Omega}\cap C$ and $\theta(\eta)\not=1$ then $E^\eta(\phi_{\mu,m})=\{\kappa\}$ with $\kappa\in{\mathbb{Z}}_{\geq 0}$ (generically $\kappa=0$, of course).
We will be interested in this section in the case where $\eta=\omega_j/k_k$ as in Theorem \[th=omega\]. In this case we know that $\Sigma^a_\eta$ is an irreducible root system. From the definition of $\phi_{\mu,m}$ we see that
\[cor:trivind\] Let $m\in\Qc_\eta^{\operatorname{reg}}$. Let $W_\eta\subset W_\eta^a$ be the maximal parabolic subgroup of $W_\eta^a$ generated by the simple reflections $s_i$ of $W$ which fix $\eta$. Then $\phi_{\mu,m}\in\L_p(\mu,m)$ belongs to the subspace $\L_p^\eta(\mu,m)$ defined by $$\L_p^\eta(\mu,m):=
\bigoplus_{\tau\in J_\eta}\L_p(\mu,m)(\tau)$$ where $J_\eta\subset\widehat{W^a_\eta}$ is the subset consisting of irreducible representations which occur in the induction of the trivial representation of $W_\eta$ to $W_\eta^a$.
The above fact restricts the $T^\eta_p$-spectrum of $\phi_{\mu,m}$, and thus the set of exponents $E^\eta(\phi_{\mu,m})$, drastically for $m\in\Qc^{\operatorname{reg}}$. We assume from now on that $m$ is real valued, which we denote by $m\in\Qc({\mathbb{R}})$. By Theorem \[prop:ev\] the multiset $E^\eta(\phi_{\mu,m})$ consists of real numbers now.
Let $m\in\Qc({\mathbb{R}})$. We call the smallest element in the the multiset $E^\eta(\phi_{\mu,m})$ the *leading exponent* of $\phi_{\mu,m}$ at $\eta$. The irreducible characters $\tau\in J_\eta\subset W^a_\eta$ affording the leading exponent are called *leading characters*.
If $\Sigma^a_\eta$ is reduced and simply laced we denote the root multiplicity by $m=m_1\geq 1$. In general $m_1$ denotes the root multiplicity of the longest roots. The multiplicity of half a long root is denoted by $m_{1/2}\geq 0$ (i.e. we consider $C_n$ as the special case of $BC_n$ where $m_{1/2}=0$; since the geometry of $\Omega$ depends on $\Sigma^l$ only this is allowed). Let $\eta\in\partial{\Omega}\cap C$ be an extremal boundary point of $\Omega$ and assume that $m\in \Qc({\mathbb{R}})$ is in the cone $\mathcal{C}\in\Qc({\mathbb{R}})$ defined by the inequalities $$\begin{aligned}
\label{eq:condm}
1&\leq m_1\leq m_2\end{aligned}$$ (these inequalities are obviously satisfied by the multiplicity function $m^X$ of a Riemannian symmetric space $X$ with restricted root system $\Sigma_X$ such that $\Sigma_X^l=\Sigma^l$). The leading exponent $s_\eta(m)$ of $\phi_{\mu,m}$ at $\eta$ satisfies $$\label{eq:ineq}
s_\eta(m)\geq s^1_{\eta,\tau}(m)$$ where $$\tau=\sigma_\eta=\operatorname{det}_\eta^a\otimes
j_{W_\eta}^{W_\eta^a}(\operatorname{det}_\eta)$$ Here $\operatorname{det}_\eta^a$ is the determinant representation of $W_\eta^a$, and $\operatorname{det}_\eta$ its restriction to $W_\eta$. Moreover, for generic $m\in\mathcal{C}$ the inequality (\[eq:ineq\]) is an equality and $\sigma_\eta$ is a leading character.
This is based on a case-by-case analysis. We first assume that $m\in\Qc^{\operatorname{reg}}_\eta({\mathbb{R}})$ is regular. We compute in all cases the set $J_\eta$ of irreducible components $\tau$ of the induction of the trivial representation of $W_\eta$ to $W^a_\eta$ (which is relatively easy, as $W_\eta$ is a rather large subgroup of $W^a_\eta$). In the classical cases we use the Littlewood-Richardson rule, and in the exceptional cases we refer to the character tables in the computer algebra packet CHEVIE. We use below the notations for the irreducible characters as used in [@Ca]).
We have luck: if we consider for each $\pi\in J_\eta$ the smallest associated exponent $s^1_{\eta,\tau}(m)$ (using Theorem \[prop:ev\]) we can simply check that these are indeed all greater than or equal to $s^1_{\eta,\tau}(m)$, where $\tau=\s_\eta$ and if $m\in\mathcal{C}$. Recall that $\s_\eta\in J_\eta$ was the term which gave the unique leading exponent in the complex case (see Example \[ex:cpx\], Proposition \[cpx\]), which corresponds only to one interior point $m^{X_{\mathbb{C}}}\in\mathcal{C}$ of $\mathcal{C}$. In any case, this surprising fact is enough to prove that for generic $m\in\mathcal{C}$ the value $s^1_{\eta,\tau}(m)$ really is the leading exponent of $\phi_{\mu,m}$ at $\eta$ by the fact that $\phi_{\mu,m}$ is holomorphic in the parameter $m$ (see Theorem \[thm:hypfun\]).
Below will now show these claims in a case-by-case analysis:
*Type $A_{l-1}(l\geq3)$:* For $\Sigma=A_{l-1}$ all the nodes of the Dynkin diagram are minuscule and thus extremal according to Theorem \[th=omega\]. Let $\omega_j$ be the $j$-th node of the Dynkin diagram. By symmetry we may assume without loss of generality that $1\leq j\leq l/2$. Recall that the irreducible characters $\chi_\l$ of $S_l$ are parameterized by the partitions $\l$ of $l$ in such a way that $\chi_l=1$ and $\chi_{1^l}=\e$ (the determinant representation). We denote the i-th exponent corresponding to $\chi_{\l}$ by $\s_\l^i(m)$.
By the Littlewood-Richardson rule [@M Section I.9] we have $$\label{eq:LR}
\operatorname{Ind}_{S_j\times S_{l-j}}^{S_l}(\chi_j\times \chi_{(l-j)})=
\bigoplus_{0\leq i\leq j}\chi_{(l-i,i)}$$ and we have that (see [@Ca Sections 11.2, 11.4]): $$\s_{\omega_j}:=\e\otimes
j_{S_{l-j}\times S_j}^{S_l}(\e_{l-j}\times\e_j)
=\chi_{(l-j,j)}$$ Using Theorem \[prop:ev\] and standard facts on representations of $S^l$ we find that: $$\label{eq:formexp}
s_{(l-i,i)}^1(m)=i(1-(l+1-i)m/2)$$ Under the condition (\[eq:condm\]) (namely $m\geq 1$) we see that among the exponents $s_{(l-i,i)}(m)$ at $\omega_j$ (thus with $i\leq j$) indeed $$s_{\omega_j}(m):=s_{(l-j,j)}^1(m)=j(1-(l+1-j)m/2)$$ is the unique minimal one, unless $l$ is even, $m=1$ and $j=l/2$. In this last case the two components $(l/2,l/2)$ and $(l/2+1,l/2-1)$ of (\[eq:LR\]) both have the same exponent $l(l-2)/8$.
*Type $B_l(l\geq3)$ ($\eta=\omega_1$):* Recall that the irreducible characters $\chi_{(\l,\mu)}$ of $B_l$ are parameterized by ordered pairs $(\lambda,\mu)$ of partitions of total weight $l$. Here $\chi_{(l,0)}=1$ and $\chi_{(0,1^l)}=\e$. We have (using the LR rule again for wreath products, see [@M I.Appendix B]): $$\begin{aligned}
\label{eq:bn}
\operatorname{Ind}_{B_{l-1}}^{B_l}&(\chi_{(l-1,0)})\\
\nonumber&=\operatorname{Ind}_{B_{l-1}\times B_1}^{B_l}(\chi_{(l-1,0)}\times\chi_{(1,0)})+
\operatorname{Ind}_{B_{l-1}\times B_1}^{B_l}(\chi_{(l-1,0)}\times\chi_{(0,1)})=\\
\nonumber&=\chi_{(l,0)}+\chi_{(l-1,1)}+\chi_{((l-1,1),-)}\end{aligned}$$ and thus $$J_{\omega_1}=\{\chi_{(l,0)},\chi_{(l-1,1)},\chi_{((l-1,1),-)}\}$$ From [@Ca Proposition 11.4.2] we find $$\s_{\omega_1}:=
\e_l\otimes j_{B_{l-1}}^{B_l}(\chi_{\e_{l-1}})=
\e_l\otimes\chi_{(1,1^{l-1})}=\chi_{(l-1,1)}$$ and the birthday of $\chi_{(1,1^{l-1})}$ is $|\Sigma(B_{l-1})_+|=(l-1)^2$.
Using Theorem \[prop:ev\] and standard results on representations of $W(B_l)$ (e.g. [@Ca Chapter 11]) we find $$\begin{aligned}
\label{eq:bnj}
s_{(l-1,1)}^1(m_2,m_1)&=1-m_2-(l-1)m_1\\
\nonumber s_{((l-1,1),-)}^1(m_2,m_1)&=2-lm_1\end{aligned}$$ Under the condition (\[eq:condm\]) (i.e. if $1\leq m_1\leq m_2$) then we see that the first one is indeed always smaller than the second one.
*Type $B_l(l\geq3)$ ($\eta=\omega_l/2$):* Not minuscule, with $W^a_\eta=W(D_l)$ and $W_\eta=S_l$, so this reduces to the minuscule case $D_l, \eta=\omega_l$ (with $m=m_1$) if $l\geq4$, or to $A_3, \eta=\omega_1$ if $l=3$.
*Types $BC_l(l\geq 1)$ and $C_l(l\geq2)$:* We treat these cases together, since the geometry of $\Omega$ is the same.
We have one boundary orbit to consider, namely $\eta=\omega_l$, a minuscule case. We have $W^a_\eta=W(C_l)$ and $W_\eta=W(A_{l-1})=S_l$, with root multiplicities $m^a_{a_0}=m_1$ for the long roots of $C_l$, and $m_2$ for the short roots of $C_l$.
In the construction of [@Ca Proposition 11.4.2] it is easy to see that the irreducible character $\chi_{(i,l-i)}$ of $W(C_l)$ is realized on the space of polynomials in ${\mathbb{C}}[x_1,\dots,x_n]$ by the action of $W(C_l)$ on the monomial $x_1\dots x_{l-i}$ ($i=0,\dots,l$). Hence this character contains the trivial character of $S_l$ and has dimension $\operatorname{binomial}(l,i)$, and has its birthday in degree $l-i$. By dimension count we find that $$\label{eq:cl}
\operatorname{Ind}_{S_l}^{W(C_l)}(\chi_l)=\oplus_{i=0}^l\chi_{(i,l-i)}$$ and so $$J_{\omega_l}=\{(\chi_{(i,l-i)}\}_{i=0}^l$$ We also see easily from the above realization that $$\label{eq:clno}
s_{(i,l-i)}^1(m)=(l-i)(1-m_1-im_2)$$ The characters $\e\otimes\chi_{(i,l-i)}=
\chi_{(1^{l-i},1^i)}$ all contain the sign representation of $S_l$ ($i=0,\dots,l$); thus together they fill up (multiplicity free) the character of $W(C_l)$ induced from the sign representation of $S_l$. According to [@Ca Proposition 11.4.2] the birthday of $\chi_{(1^{l-i},1^i)}$ is in degree $|\Sigma(D_{l-1})_+|+|\Sigma(C_i)_+|=l(l-1)+i((2l-1)-2i)$. We see that the minimum is attained for $i=r$ if $l=2r$ or if $l=2r+1$. Hence if $l=2r$ we get $$\s_{\omega_{2r}}=\chi_{(r,r)}$$ whereas in the case $l=2r+1$ we have $$\s_{\omega_{2r+1}}=\chi_{(r,r+1)}$$
Case $l=2r(r\geq1)$ even: One checks that $$s_{(i,2r-i)}^1(m)-s_{(r,r)}^1(m)=(r-i)((r-i)m_2-m_1+1)$$ which is strictly positive on $\mathcal{C}$ for $0\leq i\leq 2r$ and $i\not=r$.
Case $l=2r+1(r\geq0)$ odd: One checks that $$s_{(i,2r-i+1)}^1(m)-s_{(r,r+1)}^1(m)=(r-i)((r-i+1)m_2-m_1+1)$$ For $0\leq i\leq 2r+1$ and $i\not=r$ this is nonnegative on $\mathcal{C}$, and it is zero precisely when $m_1=1$ and $i=r+1$. Observe that this is also true if $r=0$.
*Type $D_l(l\geq4)$, $\eta=\omega_1$:* This is a minuscule case. Recall that the irreducible characters $\chi_{(\l,\mu)}$ of $D_l$ are parameterized by unordered pairs $(\lambda,\mu)$ of partitions of total weight $l$ where $\l\not=\mu$, and characters $\chi_{(\l,\l)}^\prime,\chi_{(\l,\l)}^{\prime\prime}$ if $l$ is even (weight of $\l$ is $l/2$). The character $\chi_{(\l,\mu)}$ is the restriction of the character of $W(B_l)$ with the same label $(\l,\mu)$ to $W(D_l)$. This restriction stays irreducible unless $\l=\mu$, in which case the character splits as a sum of two irreducible characters which we distinguish by ${}^\prime$ and ${}^{\prime\prime}$. Thus $\chi_{(l,0)}=1$ and $\chi_{(1^l,0)}=\e$.
By restriction of (\[eq:bn\]) to $W(D_l)$ we find $$J_{\omega_1}=\{\chi_{(l,0)},\chi_{(l-1,1)},\chi_{((l-1,1),-)}\}$$ From [@Ca Proposition 11.4.2] we find $$\s_{\omega_1}:=
\e_l\otimes j_{D_{l-1}}^{D_l}(\chi_{\e_{l-1}})=
\e_l\otimes\chi_{((2,1^{l-1}),-)}=\chi_{((l-1,1),-)}$$ where the birthday of $\chi_{((2,1^{l-1}),-)}$ is in $|\Sigma(D_{l-1})_+|=(l-1)(l-2)$.
Using Theorem \[prop:ev\] and standard results on representations of $W(D_l)$ (e.g. [@Ca Chapter 11]) we find (one should compare this to (\[eq:bnj\])) $$\begin{aligned}
s_{(l-1,1)}^1(m)&=1-(l-1)m\\
s_{((l-1,1),-)}^1(m)&=2-lm\end{aligned}$$ Under the condition (\[eq:condm\]) (i.e. if $1\leq m$) then we see indeed that the second one is smaller than the first one, except in the case $m=1$ when they coincide.
*Type $D_l(l\geq4)$, $\eta=\omega_l$:* This is minuscule too. For the computation of $J_{\omega_l}$ we recall the realizations for the characters $\chi_{(l-i,i)}$ as described in the text above (\[eq:cl\]). We introduce an intertwining operator $\J$ for the restriction of these representations to $W(D_l)$. If $\Omega\subset\{1,\dots,l\}$ we denote by $x_\Omega$ the product of the $x_i$ with $i\in\Omega$. We now define $\J(x_{\Omega})=x_{\Omega^c}$ and extend by linearity. Then $\J$ is an intertwining isomorphism $\J:\pi_{(\a,\b)}|_{W(D_l)}\to\pi_{(\b,\a)}|_{W(D_l)}$, and if $\a=\b$ then $\J$ splits $\pi_{(\a,\a)}|_{W(D_l)}$ in $\pi_{(\a,\a)}^\prime$ (the $+1$-eigenspace of $\J$) and $\pi_{(\a,\a)}^{\prime\prime}$ (the $-1$-eigenspace of $\J$). Thus $\pi_{(\a,\a)}^\prime$ contains the $S_l$-spherical vector with this convention. Hence if $l=2r$ then $$J_{\omega_l}=\{\chi_{(r,r)}^{\prime}\}\cup\{\chi_{(i,2r-i)}\}_{i=r+1}^{2r}$$ and if $l=2r+1$ then $$J_{\omega_l}=\{\chi_{(i,2r-i+1)}\}_{i=r+1}^{2r+1}$$ As in the text below (\[eq:cl\]) we find that if $l=2r+1$ then $$\s_{\omega_{2r+1}}=\chi_{(r+1,r)}$$ whereas if $l=2r$ then $$\label{eq:Dodd}
\s_{\omega_{2r}}=\chi_{(r,r)}^\prime$$
In the odd case $l=2r+1$ we thus get the specialization of the result (\[eq:cl\]) for $C_l$ at $m_1=0$, namely: $$s_{\chi_{(2r+1-i,i)}}^1(m)=(2r+1-i)(1-im)$$ but this time this has a unique minimal value among $J_{\omega_l}$ at $i=r+1$ (which proves our claim in this case, in view of (\[eq:Dodd\])). Hence $s_{\omega_{2r+1}}=r(1-(r+1)m)$.
In the even case we need to look more closely at our model for $\chi_{(r,r)}^\prime$ first. The degree of this representation is $\operatorname{binomial}(2r,r)/2=\operatorname{binomial}(2r-1,r-1)$. The dimension of the $-1$ eigenspace of a reflection is equal to $\operatorname{binomial}(2r-2,r-1)/2=\operatorname{binomial}(2r-3,r-2)$. This leads to $$s_{\chi_{(r,r)}}^1(m)=r(1-rm)$$ which is still same the same answer as we had in $C_{2r}$ when substituting $m_1=0$ (cf. (\[eq:clno\])). Therefore this exponent indeed represents the unique minimal exponent among those associated with the characters in $J_{\omega_1}$ proving the claim in this case as well.
*Type $E_6$, $\eta=\omega_1$:* This is minuscule. By the character tables in “CHEVIE” we find that $$J_{\omega_1}=\{\chi_{1,0},\chi_{6,1},\chi_{20,2}\}$$ and $$\begin{aligned}
s^1_{6,1}(m)&=1-6m\\
s^1_{20,2}(m)&=2-9m\end{aligned}$$ The second one is the unique minimal exponent, and we check that $s^1_{20,2}(2)=-16=|\Sigma(D_5)_+|-|\Sigma(E_6)_+|$. In view of Proposition \[cpx\] this proves the claims in this case.
*Type $E_7$, $\eta=\omega_7$:* This is minuscule. By the character tables in “CHEVIE” we find that $$J_{\omega_1}=\{\chi_{1,0},\chi_{7,1},\chi_{27,2},\chi_{21,3}\}$$ and $$\begin{aligned}
s^1_{7,1}(m)&=1-9m\\
s^1_{27,2}(m)&=2-14m\\
s^1_{21,3}(m)&=3-15m\end{aligned}$$ The last one is the unique minimal exponent, except when $m=1$ when it coincides with the second one. We check that $s^1_{21,3}(2)=-27=|\Sigma(E_6)_+|-|\Sigma(E_7)_+|$. In view of Proposition \[cpx\] this proves the claims.
*Type $E_7$, $\eta=\omega_2/2$:* This is not minuscule, and reduces to the case ($A_7$, $\eta=\omega_1$).
*Type $E_8$, $\eta=\omega_1/2$:* This is not minuscule, and reduces to the case ($D_8$, $\omega_1$).
*Type $E_8$, $\eta=\omega_2/3$:* This is not minuscule, and reduces to the case ($A_8$, $\omega_1$).
*Type $F_4$:* This is not minuscule, and reduces to the case ($B_4$, $\omega_1$).
*Type $G_2$:* This is not minuscule, and reduces to the case ($A_2$, $\omega_1$).
(of the proof of the previous Theorem) For $m\in\partial{\mathcal{C}}$ (the boundary of $\mathcal{C}$ there are at most two inequivalent irreducibles $\tau,\pi\in J^\eta$ such that $s^1_{\eta,\tau}(m)=
s^1_{\eta,\pi}(m)$. The cases where this occurs are indicated in the last column of the table of Theorem \[thm:table\].
Let $m\in\partial{\mathcal{C}}$ be such that there are two inequivalent irreducible representations $\tau,\pi\in J^\eta$ with coinciding exponents $s^1_{\eta,\tau}(m)=s^1_{\eta,\pi}(m)$. Then the term of (\[eq:zsingexp\]) corresponding to the leading exponent $s^1_{\eta,\tau}(m)$ contains possibly a $\log(\e)$ term of degree at most one. Otherwise the leading term in (\[eq:zsingexp\]) has no logarithmic term.
This is an easy consequence of Corollary \[cor:nolog\] and the fact that $\phi_{\mu,m}$ is holomorphic in $m$. Indeed, suppose that an expression of the form (with $s\in\mathbb{R}\backslash\{0\}$ fixed) $$f(m,\e)=a(m,\e)+b(m,\e)\e^{sm}$$ is a local Nilsson class function of $(m,\e)\in\mathbb{D}\times\mathbb{D}^\times$ with $a(m,\e)$, $b(m,\e)$ both holomorphic for $\e\in\mathbb{D}$ for all fixed $m\in\mathbb{D}^\times$. Then analytic continuation around $\e=0$ implies that $$f^\prime(m,\e)=a(m,\e)+\exp(2i\pi sm)b(m,\e)\e^{sm}$$ is in the local Nilsson class on $\mathbb{D}\times\mathbb{D}^\times$ too. Hence $a$ and $b$ have poles in $m$ of order at most $1$ and their residues at $m=0$ cancel. Now use $\log(\e)=s^{-1}\lim_{m\to 0}(m^{-1}(\e^{sm}-1))$.
This finishes the proofs of Theorem \[thm:esteta\] and Theorem \[thm:table\].
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[^1]: During the preparation of this paper the second named author was partially supported by a Pionier grant of the Netherlands Organization for Scientific Research (NWO). Part of this research was carried out in the fall of 2004, during which period both authors enjoyed the hospitality of the Research Institute for the Mathematical Sciences in Kyoto, Japan. It is our pleasure to thank the RIMS for its hospitality and for the stimulating environment it offers.
[^2]: JB explained to us the case of $G=\Sl(2,{\mathbb{R}})$, cf. the first half of Subsection 9.2.
[^3]: Added in proof: This is now established, see [@K].
[^4]: We would like to caution the reader that the proof in [@Hu] is severely wrong; a correct proof – unfortunately not emphasized – appeared later in [@FH].
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abstract: 'Following the idea of Aganagic–Okounkov [@AOelliptic], we study vertex functions for hypertoric varieties, defined by $K$-theoretic counting of quasimaps from $\PP^1$. We prove the 3d mirror symmetry statement that the two sets of $q$-difference equations of a 3d hypertoric mirror pairs are equivalent to each other, with Kähler and equivariant parameters exchanged, and the opposite choice of polarization. Vertex functions of a 3d mirror pair, as solutions to the $q$-difference equations, satisfying particular asymptotic conditions, are related by the elliptic stable envelopes. Various notions of quantum $K$-theory for hypertoric varieties are also discussed.'
author:
- Andrey Smirnov and Zijun Zhou
bibliography:
- 'reference.bib'
title: '3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties'
---
Introduction
============
Motivation and background in physics
------------------------------------
Let $G_\RR$ be a compact Lie group, and $\mathbf{M}$ be a quaternionic representation of $G$. The pair $(G_\RR, \mathbf{M})$ defines a 3d ${\mathcal{N}}= 4$ supersymmetric gauge theory in physics, where $G_\RR$ is the gauge group, and the representation $\mathbf{M}$ describes the collection of matter fields. There are two interesting components of the moduli space of vacua associated with such theories, called the *Higgs branch* and *Coulomb branch* respectively, which recently received plenty of attention in mathematics. Given a 3d ${\mathcal{N}}= 4$ theory ${\mathcal{T}}$, its Higgs branch ${\mathcal{M}}_H({\mathcal{T}})$ is mathematically the hyper-Kähler quotient associated with the $G_\RR$-representation $\mathbf{M}$, while its Coulomb branch ${\mathcal{M}}_C ({\mathcal{T}})$, also admits a mathematical construction recently by Bravermann–Finkelberg–Nakajima [@Nak1; @BFN2], in the case where $\mathbf{M} = \mathbf{N} \oplus \mathbf{N}^*$ is of cotangent type.
3d mirror symmetry [@PhysMir3; @PhysMir2; @PhysMir1; @Ga-Wit; @HW; @BDGH] predicts a duality phenomenon between certain pairs of 3d ${\mathcal{N}}= 4$ supersymmetric gauge theories, ${\mathcal{T}}$ and ${\mathcal{T}}'$, which are called mirror pairs. Given explicitly in terms of lagrangian descriptions, these two theories are expected to be different presentations of the same physical theory, and hence admit the same, or equivalent correlation functions. Moreover, a particular property of the duality is that, the Higgs and Coulomb branches of a mirror pair is expected to be exchanged: $${\mathcal{M}}_H ({\mathcal{T}}) = {\mathcal{M}}_C ({\mathcal{T}}'), \qquad {\mathcal{M}}_C ({\mathcal{T}}) = {\mathcal{M}}_H ({\mathcal{T}}'),$$ as well as the FI parameters and mass parameters, which are often added to deform the theories and result in resolutions of the branches.
There are several aspects of the 3d mirror symmetry, from which one can extract interesting mathematical conjectures.
1. As in [@BDGH], one can introduce boundary conditions in the Omega background, which implies an equivalence of categories of modules over the quantized Higgs and Coulomb branches. Mathematically, this is realized as Kozsul duality for the category ${\mathcal{O}}$’s, or symplectic duality, for symplectic resolutions on the categorical level [@BLPW; @matdu].
2. One can consider geometric interpretations of the identification ${\mathcal{M}}_H ({\mathcal{T}}) = {\mathcal{M}}_C ({\mathcal{T}}')$ between Higgs and Coulomb branch of a mirror pair. For example, the Hikita conjecture [@Hik] and related works [@KMP].
3. There should be some interplay between the Coulomb branch ${\mathcal{M}}_C({\mathcal{T}})$ and Higgs branch ${\mathcal{M}}_H ({\mathcal{T}})$ of the same theory. In particular, the (quantized) Coulomb branch could be related to quasimap counting on the Higgs branch. There are several attempts in this direction, for example [@BDGHK] and recent papers [@MSY; @HKW].
4. Correlation functions or partition functions of a mirror pair should be equated or related, for example [@BFK; @CDZ]. The approach of this paper is also of this kind, where we follow the idea of Aganagic–Okounkov [@Oko; @AOelliptic; @AOBethe]. Physically, the invariants we consider are the *vortex partition functions* with domain $S^1 \times_q D^2$, or the 3d holomorphic blocks, in the sense of [@BDP]. Mathematically, these are the generating functions of quasimaps to the Higgs branch, or quantum $K$-theory, as we will explain in the paper, which also implies symmetries among geometric objects called elliptic stable envelopes. Attempts in this direction are such as [@GaKor; @Kor; @RSVZ; @RSVZ2]. We also notice the recent work [@Hik], related to hypertoric elliptic stable envelopes.
Vertex functions, $q$-difference equations, and elliptic stable envelopes
-------------------------------------------------------------------------
From now on, in the algebraic-geometric language, we consider a complex reductive group $G$ (which is considered as the complexification of $G_\RR$), a $G$-representation $\mathbf{N}$. The quaternionic representation, with a fixed chosen complex structure, is considered as $\mathbf{M} = \mathbf{N} \oplus \mathbf{N}^*$. The hyperkähler quotient is then equivalent to the holomorphic symplectic reduction. More precisely, the Higgs branch of the associated 3d ${\mathcal{N}}= 4$ theory is the GIT quotient $X := \mu^{-1} (0) /\!/_\theta G$, where $\mu: \mathbf{M} \to \mathfrak{g}^*$ is the complex moment map, and $\theta$ is a chosen character of $G$, serving as the stability condition.
A quasimap from $\PP^1$ to the holomorphic symplectic quotient $X$ is defined to be a morphism from $\PP^1$ to the stacky quotient $[\mu^{-1} (0) / G]$, which generically maps into the stable locus $X$. In [@Oko], A. Okounkov introduced the *vertex function* $V(q, z, a)$, defined as generating functions for the $K$-theoretic equivariant counting of those quasimaps, which satisfy an extra requirement that the point $\infty \in \PP^1$ (or the $\infty$ of the last bubble component, in the relative version) is not a base point. Here, $q$ is a fixed complex number such that $|q|<1$; $z$ and $a$ stand respectively for the collections of Kähler parameters (which records the degrees of the quasimaps) and equivariant parameters.
In the case that $X$ is a Nakajima quiver variety, the vertex function is shown to satisfy two sets of $q$-difference equations: either by $q$-shifts of $z$-variables, or by $q$-shifts of $a$-variables. The analytic property of such $q$-difference equations as studied in [@Oko; @OS] shows that the associated $q$-difference modules are holonomic and admit regular singularities with respect to the variables $z$ and $a$ *separately*, but not simultaneously. The vertex function (scaled by an appropriate prefactor), by definition, happens to generate the $z$-solutions, i.e., (multi-valued) solutions that are holomorphic in $z$-variables in a punctured neighborhood of the limit point $z\to 0$, but have infinitely many poles in any punctured neighborhood of the chosen limit point $a\to 0$. It is then natural to look for the monodromy transformation that relates the two kind of solutions: $z$-solutions and $a$-solutions, which is done by Aganagic–Okounkov [@AOelliptic]. The monodromy matrix is found to be the *elliptic stable envelopes*, an elliptic analogue of the cohomological and $K$-theoretic stable envelopes [@MO].
Motivated by 3d mirror symmetry, it is naturally conjectured by Aganagic–Okounkov [@AOelliptic] that in cases where the 3d mirror $X'$ of $X$ exists, the $a$-solutions are indeed the vertex functions defined for $X'$, and the two sets of $q$-difference equations for $X'$ are the same as those for $X$, with Kähler parameters $z$ and equivariant parameters $a$ exchanged with each other. As a corollary, one can also deduce a conjecture that the elliptic stable envelopes for $X$ and $X'$, properly renormalized, are transpose to each other. The 3d mirror symmetry for elliptic stable envelopes is proved for $T^* Gr (n, k)$, $n\geq 2k$ in [@RSVZ], and cotangent bundle of a complete flag variety in [@RSVZ2]. In this paper, we prove the 3d mirror symmetry for both the vertex functions and elliptic stable envelopes, in the hypertoric case.
Hypertoric 3d mirror symmetry
-----------------------------
In the special case of *abelian* gauge theories, in other words, the gauge group is a torus, the mirror theory always exists and admits very explict descriptions [@PhysMir1; @KS]. Mathematically, their Higgs branches are abelian hyper-Kähler quotients of a quaternionic representation, which are called *toric hyperkähler varieties* or *hypertoric varieties*. To define a hypertoric variety, one starts with a short exact sequence $$\xymatrix{
0 \ar[r] & \ZZ^k \ar[r]^\iota & \ZZ^n \ar[r]^\beta & \ZZ^d \ar[r] & 0.
}$$ The map $\iota$ then describes a action of the torus ${\mathsf{K}}:= (\CC^*)^k$ on $\mathbf{N} = \CC^n$, and then on the symplectic vector space $T^*\CC^n$. The hypertoric variety is then defined as $$X := \mu^{-1} (0) /\!/_\theta {\mathsf{K}},$$ where $\mu: T^*\CC^n \to (\CC^k)^\vee$ is the complex moment map. $\theta$ here is a character of $(\CC^*)^k$, which we identify with an element in $(\ZZ^k)^\vee$. We will always assume that $\theta$ is chosen generically, in which case $X$ is smooth. We will consider the group action on $X$ by $\TT := {\mathsf{T}}\times \CC^*_\hbar$, where ${\mathsf{T}}:= (\CC^*)^n$ acts on $\CC^n$, which descends to $X$, and $\CC^*_\hbar$ scales the symplectic form, whose equivariant parameter we denote by $\hbar$.
We denote by $a_1, \cdots, a_n$ the equivariant parameters of ${\mathsf{T}}$, considered as coordinates on the torus ${\mathsf{A}}:= {\mathsf{T}}/ {\mathsf{K}}= (\CC^*)^d$. We also take $z_1, \cdots, z_n$ as *Kähler parameters*, treated as coordinates on ${\mathsf{K}}^\vee$. Both collections of parameters are redundant and subject to certain relations (see Section \[sec-para\]).
Denote by $p_1 = 0 \in \PP^1$ and $p_2:= \infty \in \PP^1$. Let ${\operatorname{QM}}(X, \beta)$ be the moduli space of quasimaps from $\PP^1$ to $X$, with degree $\beta \in H_2 (X, \ZZ)$. It is equipped with the natural perfect obstruction theory, and hence the associated virtual structure sheaf ${\mathcal{O}}_{\mathrm{vir}}\in K_\TT ({\operatorname{QM}}(X, \beta))$. Let ${\operatorname{QM}}(X, \beta)_{{\mathrm{ns}}\ p_2}$ be the open substack consisting of those quasimaps where $p_2$ is *nonsingular*, i.e., not a base point. The (bare) *vertex function* with descendent insertion $\tau$, is defined as $$V^{(\tau)}(q,z, a) := \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta {\operatorname{ev}}_{2, *} \left( {\operatorname{QM}}(X, \beta)_{{\mathrm{ns}}\ p_2}, \widehat{\mathcal{O}}_{\mathrm{vir}}\cdot {\operatorname{ev}}_1^* \tau \right) \quad \in K_{\TT\times \CC^*_q} (X)_{\mathrm{loc}}[[ z^{{\operatorname{Eff}}(X)} ]] ,$$ where $\tau\in K_\TT ({\mathfrak{X}})$ is a Kirwan lift of $K$-theory class to the stacky quotient ${\mathfrak{X}}:= [\mu^{-1} (0) / {\mathsf{K}}]$, ${\operatorname{ev}}_1$ and ${\operatorname{ev}}_2$ are evaluation maps at $p_1$ or $p_2$ from the moduli stack to ${\mathfrak{X}}$ or $X$, depending on whether the point is assumed to be nonsingular, and $\widehat{\mathcal{O}}_{\mathrm{vir}}$ is the twist of ${\mathcal{O}}_{\mathrm{vir}}$ by a square root of the virtual canonical bundle and a chosen polarization $T^{1/2}_X$, i.e., a “half" of the tangent bundle $T_X$. Moreover, $q$ is the character of $T_{p_1} \PP^1$ under the action of the torus $\CC^*_q$ on $\PP^1$. The invariants have to lie in $K_{\TT \times \CC_q^*} (X)_{\mathrm{loc}}$, which means to apply $\CC_q^*$-localization and pass to the fraction field $\CC(q)$, because the map ${\operatorname{ev}}_2$ is not proper, and its push-forward has to be defined via such localization.
The geometry of hypertoric varieties can be described very nicely in combinatorics, using the language of hyperplane arrangements, which makes it convenient to apply $\TT \times \CC_q^*$-localization computations. , The vertex functions can then be calculated explicitly, and written in the form of a contour integral of Barnes–Mellin type. One can then realize them, appropriately renormalized by some prefactors and denoted by $\widetilde V$, as solutions of $q$-difference systems. Let $Z_i$ (resp $A_i$) be the operators that shifts $z_i \mapsto q z_i$ (resp. $a_i \mapsto q a_i$) and keeps other variables unchanged.
\[Theorem \[q-diff-eqn\]\]
1) The modified vertex function $\widetilde V^{({\mathbf{1}})} (q,z, a) \big|_{\mathbf{p}}$ is annihilated by the following $q$-difference operators: $$ \prod_{i\in S^+} ( 1 - Z_i ) \prod_{i\in S^-} ( 1 - \hbar Z_i ) - z_\sharp^\beta \prod_{i\in S^+} ( 1 - \hbar Z_i ) \prod_{i\in S^-} ( 1 - Z_i ) , \qquad S = S^+ \sqcup S^- : \text{circuit},$$ where $z_{\sharp, i}:= z_i (-\hbar^{-1/2})$, $\beta$ is the curve class corresponding to $S$, and $z_\sharp^\beta := \prod_{i\in S^+} z_{\sharp, i} \prod_{i\in S^-} z_{\sharp, i}^{-1}$.
2) The modified vertex function $\widetilde V^{({\mathbf{1}})} (q,z, a) \big|_{\mathbf{p}}\cdot e^{-\sum_{i=1}^n \frac{\ln z_{\sharp, i} \ln a_i}{\ln q} }$ is annihilated by the following $q$-difference operators: $$ \prod_{i\in R^+} (1- A_i) \prod_{i\in R^-} (1 - q\hbar^{-1} A_i ) - (\hbar a)^\alpha \prod_{i\in R^+} (1 - q\hbar^{-1} A_i ) \prod_{i\in R^-} (1 - A_i ) , \qquad R = R^+ \sqcup R^- : \text{cocircuit},$$ where $\alpha$ is the root corresponding to $R$, and $(\hbar a)^\alpha := \prod_{i\in R^+} (\hbar a_i) \prod_{i\in R^-} (\hbar a_i)^{-1}$.
Here for the equations 2), as in [@MO; @Oko], one also needs to choose a cocharacter $\sigma: \CC^* \to {\mathsf{A}}$, which we identify with an element in $\ZZ^d$. The choice of $\sigma$ determines a chamber in the space of equivariant parameters, as well as an ordering of the fixed point set $X^\TT$.
The 3d mirror of a hypertoric variety $X$ is still a hypertoric variety $X'$, constructed by dualizing the defining short exact sequence $$\xymatrix{
0 \ar[r] & (\ZZ^d)^\vee \ar[r]^{\beta^\vee} & (\ZZ^n)^\vee \ar[r]^{\iota^\vee} & (\ZZ^k)^\vee \ar[r] & 0.
}$$ The stability condition and cocharacter of $X'$ is chosen as $\theta' = -\sigma$, $\sigma' = -\theta$. There are natural identifications of the spaces of parameters ${\mathsf{K}}' = {\mathsf{A}}^\vee$, ${\mathsf{A}}' = {\mathsf{K}}^\vee$, and also bijections between the fixed point sets $X^\TT$ and $(X')^{\TT'}$. We consider the vertex function $V' (q, z', a')$ for $X'$, but defined for an *opposite* choice of polarization $T^{1/2}_{X'}$. We finally have the following main result.
\[Theorem \[main-theorem\]\] Under the identification of parameters $$\kappa_{\mathrm{vtx}}: \qquad {\mathsf{K}}^\vee \times {\mathsf{A}}\times \CC^*_\hbar \xrightarrow{\sim} {\mathsf{A}}' \times ({\mathsf{K}}')^\vee \times \CC^*_\hbar, \qquad ( z_{\sharp, i} , a_i, \hbar) \mapsto ( (a'_i)^{-1}, z'_{\sharp, i}, q \hbar^{-1}),$$ the product $$V'(q, z', a') = {\mathfrak{P}}\cdot V (q, z, a) \quad \in \quad K_\TT (X^\TT)$$ forms a global class in $K_{\TT'} (X')$, and coincides with the vertex function $V'(q, z', a')$ of the 3d-mirror $X'$, with the opposite polarization $T^{1/2}_{X'}$.
Here the matrix ${\mathfrak{P}}$ is defined via the elliptic stable envelope matrix $ \dfrac{{\operatorname{Stab}}_\sigma ({\mathbf{q}}) |_{\mathbf{p}}}{\Theta (T^{1/2}_X |_{\mathbf{p}}) }$, ${\mathbf{p}}, {\mathbf{q}}\in X^\TT$, appropriately renormalized by other factors contributed from the fixed point ${\mathbf{p}}$. For any fixed point ${\mathbf{q}}\in X^\TT$, the elliptic stable envelope ${\operatorname{Stab}}_\sigma ({\mathbf{q}})$ is defined as a particular section of a line bundle over the equivairant elliptic cohomology scheme ${\operatorname{Ell}}_\TT (X) \times {\operatorname{Ell}}_{{\mathsf{T}}^\vee} ({\operatorname{pt}})$, and admits an explicit expression as a monomial of theta functions. They also satisfy a 3d mirror symmetry correspondence as follows.
\[Theorem \[Thm-Stab\]\]
Under the isomorphism of parameters $$\kappa_{{\operatorname{Stab}}} : \ {\mathsf{K}}^\vee \times {\mathsf{A}}\times \CC^*_\hbar \xrightarrow{\sim} {\mathsf{A}}' \times ({\mathsf{K}}')^\vee \times \CC^*_\hbar, \qquad ( z_i , a_i, \hbar) \mapsto ( a'_i, z'_i, \hbar^{-1}),$$ we have:
1) There is a line bundle $\mathfrak{M}$ on ${\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*} (X \times X')$ such that $$(i_{{\mathbf{p}}'}^*)^* \mathfrak{M} = \mathfrak{M} ({\mathbf{p}}) , \qquad (i_{\mathbf{p}}^*)^* \mathfrak{M} = \mathfrak{M} ({\mathbf{p}}').$$
2) There is a section $\mathfrak{m}$ of $\mathfrak{M}$, called the “duality interface", such that $$(i_{{\mathbf{p}}'}^*)^* \mathfrak{m} = {\mathbf{Stab}}_{\sigma} ({\mathbf{p}}) , \qquad (i_{\mathbf{p}}^*)^* \mathfrak{m} = {\mathbf{Stab}}'_{\sigma'} (p_{{\mathbf{p}}'}) .$$
3) In the hypertoric case, the duality interface $\mathfrak{m}$ admits a simple explicit form: $$\mathfrak{m} = \prod_{i=1}^n \vartheta (x_i x'_i) .$$ In particular, it comes from a section of a universal line bundle on the prequotient ${\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^* \times {\mathsf{T}}^\vee \times ({\mathsf{T}}')^\vee} ({\operatorname{pt}})$, and does not depend on the choices of $\theta$ or $\sigma$.
\[Corollary \[Cor-Stab\]\] We have the following symmetry between elliptic stable envelopes: $$\frac{{\operatorname{Stab}}_\sigma ({\mathbf{p}}) |_{\mathbf{q}}}{{\operatorname{Stab}}_\sigma ({\mathbf{q}}) |_{\mathbf{q}}} = \frac{{\operatorname{Stab}}'_{\sigma'} ({\mathbf{q}}') |_{{\mathbf{p}}'}}{{\operatorname{Stab}}'_{\sigma'} ({\mathbf{p}}') |_{{\mathbf{p}}'}},$$ where ${\mathbf{p}}, {\mathbf{q}}\in X^\TT$, and ${\mathbf{p}}', {\mathbf{q}}' \in (X')^{\TT'}$ are fixed points corresponding to each other.
Quantum $K$-theory
------------------
From the enumerative geometric point of view, quasimaps play a crucial role in Gromov–Witten type theories. Using the notion of *relative quasimaps* introduced in [@Oko], where one allow the domain to “bubble" at $\infty$, in [@PSZ; @KPSZ] a version of quantum $K$-theory (which we call PSZ quantum $K$-theory to avoid confusion) is defined, as a deformation of the usual ring structure $K_\TT(X)$. The $q$-difference system above satiesfied by the vertex functions, with respect to the $z$-variables, actually determines the PSZ quantum $K$-theory ring structure.
\[Theorem \[PSZ-relations\]\] We have the following presentations of ring structures (which are equivalent to each other):
1) The PSZ quantum $K$-theory ring of $X$ is generated by the quantum tautological line bundles $\widehat L_i (z)$, $1\leq i\leq n$, up to the relations $$\prod_{i\in S^+} ( 1 - \widehat L_i (z) ) * \prod_{i\in S^-} ( 1 - \hbar \widehat L_i (z) ) - z_\sharp^\beta \prod_{i\in S^+} ( 1 - \hbar \widehat L_i (z) ) * \prod_{i\in S^-} ( 1 - \widehat L_i (z) ) , \qquad S = S^+ \sqcup S^- : \text{circuit},$$ where $z_{\sharp, i}:= z_i (-\hbar^{-1/2})$, $\beta$ is the curve class corresponding to $S$, $z_\sharp^\beta := \prod_{i\in S^+} z_{\sharp, i} \prod_{i\in S^-} z_{\sharp, i}^{-1}$, and all products $\prod$ are quantum products $*$.
2) The divisorial quantum $K$-theory ring of $X$ is generated by the line bundles $L_i$, $1\leq i\leq n$, up to the relations $$\prod_{i\in S^+} ( 1 - L_i ) \prod_{i\in S^-} ( 1 - \hbar L_i ) - z_\sharp^\beta \prod_{i\in S^+} ( 1 - \hbar L_i ) \prod_{i\in S^-} ( 1 - L_i ) , \qquad S = S^+ \sqcup S^- : \text{circuit},$$ where $z_{\sharp, i}:= z_i (-\hbar^{-1/2})$, $\beta$ is the curve class corresponding to $S$, and $z_\sharp^\beta := \prod_{i\in S^+} z_{\sharp, i} \prod_{i\in S^-} z_{\sharp, i}^{-1}$.
The quasimap we considered here is actually a special case of the $\epsilon=0+$ quasimaps with a parametrized domain component defined by [@CKM]. It is then natural to ask how it is related to the quantum $K$-theory defined via counting stable maps by Lee and Givental [@Lee; @GL]. In general, it is expected to be studied via certain $\epsilon$-wall-crossing techniques for stability conditions over the moduli stacks. But for us, thanks to the explicit computations available, we are able to extract information directly from the vertex functions.
\[Coroallary \[V-tau\]\] Let $\tau$ be a Laurent polynomial in $q$ with coefficients in $K_{\mathsf{T}}(X) \otimes \Lambda$. The descendent bare vertex function $$(1-q) V^{(\tau)} (q^{-1} , z) \big|_{z_\sharp = Q}$$ lies in the range of permutation-equivariant big $J$-function of $X$.
Here one has to consider Givental’s *permutation-equivariant* quantum $K$-theory [@Giv1], which behaves better with localizations and twisted theories [@GV]. We then introduce a Givental type quantum $K$-theory, based on the the $K$-theoretic Gromov–Witten potential with both permutation-equivariant and ordinary inputs. Following the idea in [@IMT], there are some operators $B_{i, {\mathrm{com}}} \in {\operatorname{End}}K_\TT(X) \otimes \Lambda$, introduced via $q$-difference operators. Let $t_0 \in K_\TT (X) \otimes \Lambda$ be the point where $(1-q) V^{({\mathbf{1}})} (q^{-1} , z) \big|_{z_\sharp = Q} = J(t_0, Q)$.
\[Theorem \[GivQK-relation\]\]
Let $X$ be a hypertoric variety, and $t_0 \in K(X) \otimes \Lambda$ be as in Theorem \[V\]. We fix the insertions $x = 0$ and $t = t_0$.
1) For any circuit $S = S^+\sqcup S^-$, and the corresponding curve class $\beta$, the identity class ${\mathbf{1}}\in K_{\mathsf{T}}(X)$ is annihilated by the following operator $$\prod_{i\in S^+} ( 1 - B_{i, {\mathrm{com}}}) \prod_{i\in S^-} (\hbar - B_{i, {\mathrm{com}}} ) - Q^\beta \prod_{i\in S^+} (\hbar - B_{i, {\mathrm{com}}} ) \prod_{i\in S^-} (1 - B_{i, {\mathrm{com}}}),$$ where $Q^\beta := \prod_{i\in S^+} Q_i \prod_{i\in S^-} Q_i^{-1}$.
2) The Givental quantum $K$-theory ring of $X$ is generated by the classes $B_{i, {\mathrm{com}}} {\mathbf{1}}$, $1\leq i\leq n$, up to the following relations: for any circuit $S = S^+\sqcup S^-$, and the corresponding curve class $\beta$ $$\prod_{i\in S^+} ( 1 - B_{i, {\mathrm{com}}} {\mathbf{1}}) \bullet \prod_{i\in S^-} (\hbar - B_{i, {\mathrm{com}}} {\mathbf{1}}) = Q^\beta \prod_{i\in S^+} (\hbar - B_{i, {\mathrm{com}}} {\mathbf{1}}) \bullet \prod_{i\in S^-} (1 - B_{i, {\mathrm{com}}} {\mathbf{1}}),$$ where all the products are the quantum product $\bullet$.
Structure of the paper
----------------------
The paper is organized as follows. In Section 2, we review basic constructions and the equivariant geometry of hypertoric varieties. We explicitly described the $K$-theory $K_\TT (X)$, the tautological line bundles $L_i$ and the characters one obtains when restricting them to a fixed point ${\mathbf{p}}$. In Section 2.7, we introduce the redundant and global equivariant and Kähler parameters. In Section 3, we recall the definition of elliptic stable envelopes and their characterization via theta functions in the hypertoric case. We define the vertex function and PSZ quantum $K$-theory in Section 4, and then compute them in Section 5 via $\TT$-localization. In Section 6, we prove our main theorems on the 3d mirror symmetry for vertex functions and elliptic stable envelopes. The relationship to Givental’s permutation-equivariant quantum $K$-theory is studied in Section 7.
Acknowlegements
---------------
The authors would like to thank Mina Aganagic and Andrei Okounkov for their extraordinary work [@AOelliptic], where we learn the idea and start this project. The second author would also like to thank Ming Zhang and Yaoxiong Wen for discussions on Givental’s quantum $K$-theory. The work of A.S. is partly supported by the Russian Science Foundation under grant 19-11-00062. The work of Z.Z. is supported by FRG grant 1564500.
Geometry of hypertoric varieties
================================
Basic construction {#Sec-basic}
------------------
In this section we review the definition and geometric properties of hypertoric varieties. For the details, there are plenty of references, such as [@BD; @HH; @HP; @HS; @Kon; @Kon2]. As hyperkähler analogues of toric varieties, which can be constructed as symplectic reductions of complex vector spaces, hypertoric varieties are hyperkähler reductions of quarternion vector spaces.
Let $k,n$ be nonnegative integers, with $k\leq n$. Consider a short exact sequence of free $\ZZ$-modules: $$\label{knd-seq}
\xymatrix{
0 \ar[r] & \ZZ^k \ar[r]^\iota & \ZZ^n \ar[r]^\beta & \ZZ^d \ar[r] & 0.
}$$ Denote the complex tori by ${\mathsf{K}}:= (\CC^*)^k$, ${\mathsf{T}}:= (\CC^*)^n$ and ${\mathsf{A}}:= (\CC^*)^d$. When tensoring with $\CC$, the embedding $\iota$ specifies an embedding of the complex torus ${\mathsf{K}}$ into ${\mathsf{T}}$, and hence defines an action of ${\mathsf{K}}$ on the affine space $\CC^n$. The abelian groups $\ZZ^k$ and $\ZZ^n$ above are viewed as lattices of cocharacters of the tori.
The ${\mathsf{K}}$-action naturally extends to the cotangent space $T^* \CC^n \cong \CC^n \oplus \CC^n$, preserving the canonical holomorphic symplectic form. Let $\mu: T^* \CC^n \to {\operatorname{Lie}}({\mathsf{K}})^\vee$ be the moment map of this action. With a choice of a stability parameter $\theta \in {\operatorname{Lie}}({\mathsf{K}})^\vee$, the hypertoric variety $X$ is defined as the GIT quotient $$X := \mu^{-1} (0) /\!/_\theta {\mathsf{K}}.$$ We will also consider the stacky quotient, denoted by $\mathfrak{X} := [\mu^{-1} (0) / {\mathsf{K}}]$.
Later we will see that the quasimap theory actually depend on the presentation of the GIT quotient, i.e. the sequence (\[knd-seq\]), rather than the resulting quotient variety $X$ itself. More precisely, we will consider hypertoric data *up to automorphisms of $\ZZ^k$ and $\ZZ^d$*. In other words, two sets of hypertoric data as (\[knd-seq\]) will be considered equivalent, if they can be related by change of bases for $\ZZ^k$ and $\ZZ^d$. The corresponding Kähler and equivariant parameters will also be understood up to those changes of bases.
Hyperplane arrangement and circuits
-----------------------------------
The information of a hypertoric variety can be conveniently organized in the combinatoric data of a collection ${\mathcal{H}}$ of affine hyperplanes in $\RR^d$, called a *hyperplane arrangement*.
In the sequence (\[knd-seq\]), let $e_i\in \ZZ^n$, $1\leq i\leq n$ be the standard basis. Consider the dual exact sequence $$\xymatrix{
0 \ar[r] & (\ZZ^d)^\vee \ar[r]^{\beta^\vee} & (\ZZ^n)^\vee \ar[r]^{\iota^\vee} & (\ZZ^k)^\vee \ar[r] & 0.
}$$
Let $\widetilde\theta \in (\ZZ^n)^\vee$ be a lift of $\theta$ along $\iota^\vee$. Then the hyperplane arrangement ${\mathcal{H}}= \{ H_i \mid 1\leq i \leq n\}$ is defined as the collection of the following (affine) hyperplanes $$H_i := \{x\in (\RR^d)^\vee \mid \langle x, \beta (e_i) \rangle = - \langle \widetilde\theta, e_i \rangle \}.$$ The hypertoric variety $X$ can be recovered from such an ${\mathcal{H}}$. A different choice of the lift $\widetilde\theta$ corresponds to translating the hyperplanes simultaneously by an element in $(\ZZ^d)^\vee$, which does not affect the associated $X$.
Each hyperplane $H_i$ divides $(\RR^d)^\vee$ into two half-spaces: $$H_i^+ := \{x\in (\RR^d)^\vee \mid \langle x, \beta (e_i) \rangle \geq - \langle \widetilde\theta, e_i \rangle \}, \qquad H_i^- := \{x\in (\RR^d)^\vee \mid \langle x, \beta (e_i) \rangle \leq - \langle \widetilde\theta, e_i \rangle \}.$$
A hypertoric variety $X$ is smooth if and only if the hyperplane arrangement ${\mathcal{H}}$ satisfies the following two conditions:
1) simple, i.e., for any $0\leq m \leq n$, every $m$ hyperplanes in ${\mathcal{H}}$ intersect, if nonempty, in codimension $m$;
2) unimodular, i.e., any collection of $d$ linearly independent vectors in the conormals $\{\beta (e_1), \cdots, \beta (e_n) \}$ form a basis of $\ZZ^d$ over $\ZZ$.
Unless otherwise specified, we will always assume that our hypertoric variety $X$ is *smooth*, which is always the case when the stability condition $\theta$ is chosen generically.
A subset $S \subset \{1, \cdots, n\}$ is called a *circuit*, if it is a minimal subset such that $\bigcap_{i\in S} H_i = \emptyset$. In other words, it gives a minimal relations among the images of $e_i$’s for $i\in S$.
Each circuit has a unique splitting $S = S^+ \sqcup S^-$, determined as follows. The smoothness assumption on $X$ implies that in the relation among $e_i$’s for $i\in S$, all coefficients of $e_i$’s must be $\pm 1$. The subsets $S^\pm$ are determined by the presentation of the relation $$\beta_S := \sum_{i\in S^+} e_i - \sum_{i\in S^-} e_i \in \ker \beta,$$ such that $\langle \beta_S, \widetilde\theta \rangle \geq 0$.
Equivariant geometry and line bundles
-------------------------------------
Analogous to toric varieties, the hypertoric variety $X$ admits a torus action, naturally inherited from the standard torus action by $(\CC^*)^n$ on $\CC^n$. Moreover, there is a 1-dimensional torus $\CC^*_\hbar$ that scales the cotangent fiber of $T^* \CC^n$, which also descends to $X$. Hence $X$ admits a torus action by $\TT := {\mathsf{T}}\times \CC^*_\hbar$, whose equivariant parameters are denoted by $a_1, \cdots, a_n, \hbar \in K_\TT ({\operatorname{pt}})$.
Each character in $(\ZZ^n)^\vee$ defines a natural $\TT$-equivariant line bundle on $X$. In particular, for the standard dual basis $e_i^* \in (\ZZ^n)^\vee$, $1\leq i\leq n$, we have line bundles $$L_i := \mu^{-1}(0)^s \times_{\mathsf{K}}\CC_{e_i^*},$$ where $\mu^{-1}(0)^s \subset \mu^{-1}(0)$ is the stable locus for the ${\mathsf{K}}$-action, and $\CC_{e_i^*}$ denotes the 1-dimensional ${\mathsf{K}}$-representation defined by the character $\iota^\vee e_i^*$.
Similarly, each character in $(\ZZ^k)^\vee$ defines a (non-equivariant) line bundle on $X$. Let $\{f_j^* \mid 1\leq j\leq k\}$ be the standard dual basis for $(\ZZ^k)^\vee$. We have the tautological line bundles $$N_j := \mu^{-1}(0)^s \times_{\mathsf{K}}\CC_{f_j^*}.$$
The relationship between $L_i$ and $N_j$’s is encoded in the map $\iota^\vee$: $$\label{LL}
L_i = \CC_{e_i^*} \otimes \bigotimes_{j=1}^k N_j^{\otimes \iota_{ij}},$$ where $(\iota_{ij})$ is the $n\times k$ matrix given by $\iota$.
We are interested in the $\TT$-equivariant $K$-theory of $X$. Recall that [@HH] $K_\TT (X)$ satisfies the *Kirwan surjectivity* [^1], i.e. the following surjection $$K_\TT ({\operatorname{pt}}) [s_1^{\pm 1}, \cdots, s_k^{\pm 1} ] \twoheadrightarrow K_\TT (X),$$ where the image of $s_j$ is the $K$-theory class of $N_j$.
More precisely, the kernel of the surjection can be described explicitly. We have $$\label{K(X)}
K_\TT (X) \cong \CC [ a_1^{\pm 1}, \cdots, a_n^{\pm 1}, \hbar^{\pm 1}, s_1^{\pm 1}, \cdots, s_k^{\pm 1} ] / \langle \prod_{i\in S^+} (1-x_i) \prod_{i\in S^-} (1- \hbar x_i) \mid S \text{ is a circuit} \rangle ,$$ where $x_i$ is the class of $L_i$ in $K_\TT (X)$, which can be expressed in $s_j$’s through (\[LL\]), i.e., $x_i = a_i \prod_{j=1}^k s_j^{\iota_{ij}}$.
In particular, if we view ${\operatorname{Spec}}K_\TT (X)$ as an affine scheme embedded in an algebraic torus $(\CC^*)^{n+k+1}$ with coordinates $a_i$, $\hbar$, $s_j$, it is given by the intersection of certain hypersurfaces, each defined by a circuit $S$ as a union of hyperplanes in the torus. Moreover, one can see that this intersection is indeed *transversal*, reflecting the fact that $X$, equipped with the $\TT$-action, is a *GKM variety* (i.e., admits finitely many fixed points and finitely many 1-dimensional orbits).
In later sections of the paper, we will sometimes choose particular bases of $\ZZ^k$ and $\ZZ^d$ to make computations more convenient. The corresponding constructions, such as $N_j$, $\theta$, will change according to the change of bases. However, we will always fix the basis of $\ZZ^n$. In other words, the line bundle $L_i$ and the stability parameter $\widetilde\theta$ will stay the same for all choices.
Restriction to ${\mathsf{T}}$-fixed points
------------------------------------------
To conclude this section, we would like to specify the restriction of line bundles $N_j$ and $L_i$ to each fixed point in $X^\TT$. The $\TT$-invariant loci of $X$ can be described by the hyperplane arrangement ${\mathcal{H}}$: a $\TT$-fixed point of $X$ corresponds to a vertex in ${\mathcal{H}}$; a $\TT$-invariant 1-dimensional orbit corresponds to an edge, etc.
Let ${\mathbf{p}}= \{{\mathbf{p}}_1, \cdots, {\mathbf{p}}_d \} \subset \{1, \cdots, n\}$, with ${\mathbf{p}}_1 < \cdots < {\mathbf{p}}_d$, be a subset with $d$ elements such that $$\bigcap_{i\in {\mathbf{p}}} H_i \neq \emptyset.$$ By our smooth assumption, the above intersection of $H_i$’s is a vertex in ${\mathcal{H}}$, and therefore corresponds to a fixed point ${\mathbf{p}}\in X^\TT$. From now on, we will abuse the notation ${\mathbf{p}}$ (and also ${\mathbf{q}}$) for the followings:
\(i) the subset ${\mathbf{p}}$,
\(ii) the bijection from $\{1, \cdots, d\}$ to the set ${\mathbf{p}}$,
\(iii) the vertex in the hyperplane arrangement,
\(iv) the fixed point ${\mathbf{p}}\in X^\TT$.
Let ${\mathcal{A}}_{\mathbf{p}}:= \{1, \cdots, n \} \backslash {\mathbf{p}}$ be the complement subset. Denote that ${\mathcal{A}}_{\mathbf{p}}= \{ {\mathcal{A}}_{{\mathbf{p}},1} , \cdots, {\mathcal{A}}_{{\mathbf{p}}, k} \}$, where ${\mathcal{A}}_{{\mathbf{p}}, 1} < \cdots < {\mathcal{A}}_{{\mathbf{p}}, k}$. We will also abuse the notation ${\mathcal{A}}_{\mathbf{p}}$ for the bijection from $\{1, \cdots, k\}$ to ${\mathcal{A}}_{\mathbf{p}}$.
In particular, for (ii) above, we mean that if $i = {\mathbf{p}}_I$ for some $1\leq I\leq d$, we denote by $I = {\mathbf{p}}^{-1} (i)$; similarly if $j = {\mathcal{A}}_{{\mathbf{p}}, J}$ for some $1\leq J\leq k$, denote $J = ({\mathcal{A}}_{\mathbf{p}})^{-1} (j)$.
The orientations given by the conormal vectors $\beta (e_i)$ of $H_i$’s, for $i\not\in {\mathbf{p}}$, determines a splitting ${\mathcal{A}}_{\mathbf{p}}= {\mathcal{A}}_{\mathbf{p}}^+ \sqcup {\mathcal{A}}_{\mathbf{p}}^-$, where $${\mathcal{A}}_{\mathbf{p}}^+ := \{i \not\in {\mathbf{p}}\mid {\mathbf{p}}\in H_i^+ \}, \qquad {\mathcal{A}}_{\mathbf{p}}^- := \{i \not\in {\mathbf{p}}\mid {\mathbf{p}}\in H_i^- \}.$$
The following system of equations $$a_m s_1^{\iota_{m 1}} \cdots s_k^{\iota_{mk}} = \left\{ \begin{aligned}
& 1, \qquad && m \in {\mathcal{A}}_{\mathbf{p}}^+ \\
& \hbar^{-1}, \qquad && m \in {\mathcal{A}}_{\mathbf{p}}^-
\end{aligned}\right.$$ admits a unique set of solutions $(s_1 ({\mathbf{p}}), \cdots, s_k ({\mathbf{p}}))$. The restriction of the line bundle $N_j$ to ${\mathbf{p}}$ is $$\left. N_j \right|_{{\mathbf{p}}} = s_j ({\mathbf{p}}).$$ The restriction of the line bundle $L_i$ to ${\mathbf{p}}$ is $$\label{restriction}
\left. L_i \right|_{\mathbf{p}}= \left\{ \begin{aligned}
& 1 , \qquad && i \in {\mathcal{A}}_{\mathbf{p}}^+ \\
& \hbar^{-1}, \qquad && i \in {\mathcal{A}}_{\mathbf{p}}^- \\
& a_i s_1 ({\mathbf{p}})^{\iota_{i1}} \cdots s_k ({\mathbf{p}})^{\iota_{ik}}, \qquad && i \in {\mathbf{p}}.
\end{aligned} \right.$$
This is essentially Theorem 3.5 in [@HH].
We choose the basis of $\ZZ^k$ such that the matrix $\iota$ is of the following special form: $$\label{V-frame-i}
\iota_{mj} ({\mathbf{p}}) := \delta_{mj}, \qquad \iota_{ij} ({\mathbf{p}}) := C_{ij} ({\mathbf{p}}), \qquad m, j \not\in {\mathbf{p}}, \ i \in {\mathbf{p}}$$ where $C_{ij} ({\mathbf{p}})$ is a $d\times k$-matrix, with indices taken in ${\mathbf{p}}\times {\mathcal{A}}_{\mathbf{p}}$. Moreover, one can choose a basis of $\ZZ^d$ such that the matrix $\beta$ is also of a special form: $$\label{V-frame-b}
\beta_{ij} ({\mathbf{p}}) = -C_{ij} ({\mathbf{p}}), \qquad \beta_{il} ({\mathbf{p}}) = \delta_{il}, \qquad j \not\in {\mathbf{p}}, \ i, l \in {\mathbf{p}}.$$ Here the ${\mathbf{p}}$ in parentheses is to emphasize the dependence on ${\mathbf{p}}$. We call this choice of bases the *standard ${\mathbf{p}}$-frame*.
For the standard ${\mathbf{p}}$-frame, the relationship between $x_i = L_i$, $1\leq i\leq n$, and $s_J = N_J$, $1\leq J\leq k$, is $$x_i = \left\{ \begin{aligned}
& a_i s_I, \qquad && i = {\mathcal{A}}_{{\mathbf{p}}, I} \in {\mathcal{A}}_{\mathbf{p}}\\
& a_i s_1^{C_{i, {\mathbf{p}}^c_1}} \cdots s_k^{C_{i, {\mathbf{p}}^c_k}} , \qquad && i \in {\mathbf{p}}.
\end{aligned} \right.$$ The restriction of the line bundle $N_J$, $1\leq J\leq k$ to ${\mathbf{p}}$ is $$\label{restriction-s(V)}
\left. N_J \right|_{\mathbf{p}}= s_J ({\mathbf{p}}) = \left\{ \begin{aligned}
& a_j^{-1}, \qquad && j = {\mathcal{A}}_{{\mathbf{p}}, J} \in {\mathcal{A}}_{\mathbf{p}}^+ \\
& \hbar^{-1} a_j^{-1} , \qquad && j = {\mathcal{A}}_{{\mathbf{p}}, J} \in {\mathcal{A}}_{\mathbf{p}}^-.
\end{aligned}\right.$$ The restriction of the line bundle $L_i$, $1\leq i\leq n$ to ${\mathbf{p}}$ is $$\label{restriction-V}
\left. L_i \right|_{\mathbf{p}}= \left\{ \begin{aligned}
& 1 , \qquad && i \in {\mathcal{A}}_{\mathbf{p}}^+ \\
& \hbar^{-1}, \qquad && i \in {\mathcal{A}}_{\mathbf{p}}^- \\
& a_i \prod_{j\not\in {\mathbf{p}}} a_j^{-C_{ij} ({\mathbf{p}})} \cdot \hbar^{- \sum_{j \in {\mathbf{p}}^{c-}} C_{ij} ({\mathbf{p}}) } . \qquad && i \in {\mathbf{p}},
\end{aligned} \right.$$
We would like to rewrite this formula in a more intrinsic way. The presentation of $K_\TT (X)$ (\[K(X)\]) can be expressed as $$\CC [ a_1^{\pm 1}, \cdots, a_n^{\pm 1}, \hbar^{\pm 1}, x_1^{\pm 1}, \cdots, x_n^{\pm 1} ] / \langle \prod_{i=1}^n (x_i / a_i)^{\beta_{ji}} - 1, 1\leq j\leq d; \prod_{i\in S^+} (1-x_i) \prod_{i\in S^-} (1- \hbar x_i) , S: \text{circuits} \rangle.$$ The picture of the affine scheme ${\operatorname{Spec}}K_\TT (X)$ is clear (view $\hbar$ as a constant): the first set of relations cuts out a codimension-$d$ subspace in the ambient torus $(\CC^*)^{2n}$, and the second furthermore cuts out a union of subspaces, each isomorphic to $(\CC^*)^n$, intersecting transversally. Each fixed point ${\mathbf{p}}\in X^\TT$ corresponds to an irreducible component ${\operatorname{Spec}}K_\TT ({\mathbf{p}}) \cong (\CC^*)^n$ of ${\operatorname{Spec}}K_\TT (X)$.
View $x_i$ as a function on ${\operatorname{Spec}}K_\TT (X)$. The restriction formula (\[restriction-V\]) can then be written as the residue of the function $x_i$ along one of the components: $$x_i |_{\mathbf{p}}:= L_i |_{{\mathbf{p}}} = \int_{\gamma({\mathbf{p}})} x_i \cdot \frac{d\ln x_1 \wedge \cdots \wedge d\ln x_n}{\bigwedge_{m=1}^d \Big( \sum_{i=1}^n \beta_{mi} d\ln x_i \Big) } ,$$ where $\gamma({\mathbf{p}})$ is the compact real $k$-cycle around the irreducible component ${\operatorname{Spec}}K_\TT ({\mathbf{p}})$, specified by $x_j = 1$, $j\in {\mathcal{A}}_{\mathbf{p}}^+$ and $x_j = \hbar^{-1}$, $j\in {\mathcal{A}}_{\mathbf{p}}^-$.
Effective curves, walls, and chambers {#section-eff}
-------------------------------------
There is a bijection [@Kon] between circuits and primitive effective curves in $X$. We will abuse the notation and denote also by $\beta_S$ the primitive effective curve corresponding to the circuit $S$.
All irreducible $\TT$-invariant curves $C$ in $X$ are of the following form. Let ${\mathbf{p}}$ and ${\mathbf{q}}$ be two vertices in the hyperplane arrangement, such that ${\mathbf{p}}= ({\mathbf{q}}\backslash \{j\} ) \sqcup \{i\}$ and ${\mathbf{q}}= ({\mathbf{p}}\backslash \{i\} ) \sqcup \{j\}$, for some $1 \leq i\neq j\leq n$. There is a unique $\TT$-invariant curve $C$ connecting the fixed points ${\mathbf{p}}$ and ${\mathbf{q}}$. It is clear that the circuit that defines $C$ is $$S_{{\mathbf{p}}{\mathbf{q}}} := {\mathbf{p}}\sqcup \{j\} = {\mathbf{q}}\sqcup \{ i \}.$$
\[bridge\]
(i) $\deg L_m \big|_C = 0$, for $m \not\in S_{{\mathbf{p}}{\mathbf{q}}}$; $\deg L_i \big|_C = \pm 1$, if $i\in {\mathcal{A}}_{\mathbf{q}}^\pm$.
(ii) The character $$T_{\mathbf{p}}C = \left\{ \begin{aligned}
& x_i |_{\mathbf{p}}, && \quad i\in {\mathcal{A}}_{\mathbf{q}}^+ \\
& \hbar^{-1} x_i^{-1} |_{\mathbf{p}}, && \quad i\in {\mathcal{A}}_{\mathbf{q}}^- .
\end{aligned} \right.$$
\(i) is true by direct computation. (ii) follows from the fact that $x_i |_{\mathbf{p}}= x_i |_{\mathbf{q}}\cdot (T_{\mathbf{p}}C)^{\deg L_i |_C}$, and $x_i |_{\mathbf{q}}= 1$ or $\hbar^{-1}$, depending on $i\in {\mathcal{A}}_{\mathbf{q}}^\pm$.
Recall that as in [@Kon], the space $\RR^k$ of stability parameters of $X$ admits a wall-and-chamber structure. For each circuit $S = S^+ \sqcup S^-$, there is a codimension-1 hyperplane $$P_S := {\operatorname{Span}}_\RR \{ \iota^\vee e_i^* \mid i\not\in S \}\subset \RR^k,$$ which we call a *wall* in $\RR^k$. A connected component of the complement of walls $$\mathfrak{K} \subset \RR^k \backslash \bigcup_{S: \text{circuit}} P_S$$ is called a *chamber*.
Constructed by the real moment map in the hyperKähler definition, there is a family of (possibly singular) hypertoric varieties over $\RR^k$. For each $\theta \in \RR^k$, the fiber over $\theta$ is the hypertoric variety $X$ defined with the choice of stability condition $\theta$, which is smooth if $\theta$ is away from all the walls. The geometry of $X$ stays the same for all $\theta$’s in a given chamber $\mathfrak{K}$, and admits a symplectic flop phenomenon when $\theta$ crosses a wall.
With the space $\RR^k$ identified with $H^2 (X, \RR)$, the real moment map can also be understood as a real period map, whose image is the real Kähler class. In this sense, the chamber $\mathfrak{K}$ is nothing but the *Kähler cone* of $X$. The effective cone ${\operatorname{Eff}}(X) \otimes \RR$ can be described as the dual of $\mathfrak{K}$.
Roots, walls and chambers {#section-root}
-------------------------
Following Maulik–Okounkov [@MO], the space of equivariant parameters $\RR^d = \RR^n / \RR^k$ also admits a wall-and-chamber structure. Choose a cocharacter of the torus $\sigma : \CC^* \to (\CC^*)^d$, which we view as an element in $\ZZ^d$. Let $X^\sigma \subset X$ be the fixed loci of $X$ under the action of the 1-dimensional subtorus defined by $\sigma$. We also choose a lift $\widetilde \sigma \in \ZZ^n$ of $\sigma$ along $\beta$, i.e. $\beta (\widetilde\sigma) = \sigma$.
The subset in $\RR^d$ where $\sigma$ is “generic", i.e., $X^\sigma = X^{(\CC^*)^d} = X^\TT$, is the complement of a union of hyperplanes, which we call *walls*. Each wall is of the form $W_\alpha := \{ \lambda \in \RR^d \mid \langle \lambda, \alpha \rangle = 0 \}$, for some primitive $\alpha \in (\ZZ^d)^\vee$, which we call a *root*.
Alternatively, if we identify $\alpha$ with its image $\beta^* \alpha$ in $(\ZZ^n)^\vee$, then a root is a minimal relation among images of the standard basis vectors $\iota^\vee e_i^*$. Therefore, we may also identify $\alpha$ as a *cocircuit*, i.e. a subset $R \subset \{1, \cdots, n\}$, with unique splitting $R = R^+ \sqcup R^-$, such that $$\alpha_R := \sum_{i\in R^+} e_i^* - \sum_{i\in R^-} e_i^* \in \ker \iota^\vee,$$ and $\langle \alpha_R, \widetilde\sigma \rangle \geq 0$.
A connected component of the complement of the union of all root hyperplanes is also called a *chamber*: $${\mathfrak{C}}\subset \RR^d \backslash \bigcup_{\alpha: \text{cocircuit}} W_\alpha.$$ For a fixed choice of $\widetilde\sigma$, a root $\alpha$ is called positive if it is nonzero and $\langle \widetilde\sigma, \alpha \rangle \geq 0$.
Equivariant and Kähler parameters {#sec-para}
---------------------------------
To end this section, we would like to elaborate more on the parameters which our vertex functions and elliptic stable envelopes will depend on. Recall the multiplicative version of the short exact sequence (\[knd-seq\]): $$\xymatrix{
1 \ar[r] & {\mathsf{K}}\ar[r]^-{\exp \iota} & {\mathsf{T}}\ar[r]^-{\exp \beta} & {\mathsf{A}}\ar[r] & 1,
}$$ where ${\mathsf{K}}= (\CC^*)^k$, ${\mathsf{T}}= (\CC^*)^n$ and ${\mathsf{A}}= (\CC^*)^d$. We have used the action by $(\CC^*)^n$ in the previous section to define the equivariant $K$-theory. The coordinates $a_1, \cdots, a_n$ on ${\mathsf{T}}$ are called the *equivariant parameters*. They correpond to the standard basis on $\ZZ^n$, which we fix once and for all.
However, this torus action is actually redundant: it acts by the factorization through the morphism $\exp \beta: {\mathsf{T}}\to {\mathsf{A}}$. The actual non-redundant equivariant parameters are functions on the quotient torus ${\mathsf{A}}$, or in other words, monomials in $a_1, \cdots, a_n$ that vanish on the kernel ${\mathsf{K}}$.
Every choice of basis on $\ZZ^d$, or equivalently, every presentation of the map $\beta$, defines a particular choice of coordinates on ${\mathsf{A}}$. For example, let ${\mathbf{p}}$ be a vertex in the hyperplane arrangement ${\mathcal{H}}$. If we choose the presentation $\beta = (-C , I)$ in the standard ${\mathbf{p}}$-frame, the corresponding choice of coordinates would be $$\alpha_i ({\mathbf{p}}) := a_i \prod_{j\not\in {\mathbf{p}}} a_j^{-C_{ij}}, \qquad i\in {\mathbf{p}}.$$
The same happens for the Kähler parameters. Consider the dual exact sequence of tori $$\xymatrix{
1 \ar[r] & {\mathsf{A}}^\vee \ar[r]^-{\exp \beta^\vee} & {\mathsf{T}}^\vee \ar[r]^-{\exp \iota^\vee} & {\mathsf{K}}^\vee \ar[r] & 1.
}$$ We fix coordinates on $z_1, \cdots, z_n$ on ${\mathsf{T}}^\vee$ once and for all, as the “redundant" Kähler parameters. Then the non-redundant Kähler parameters are coordinates on the quotient torus ${\mathsf{K}}^\vee$, or equivalently, monomials in $z_1, \cdots, z_n$, vanishing on the dual kernal ${\mathsf{A}}^\vee$.
Every choice of basis on $\ZZ^k$, or equivalently, every presentation of the map $\iota$, defines a particular choice of coordinates on ${\mathsf{K}}^\vee$. For the standerd ${\mathbf{p}}$-frame, $\iota = \begin{pmatrix}
I \\
C
\end{pmatrix}$, there is a particular choice of Kähler parameters, or in other words, a choice of representatives for the Kähler parameters $$\zeta_j ({\mathbf{p}}) := z_j \prod_{i\in {\mathbf{p}}} z_i^{C_{ij}}, \qquad j\not\in {\mathbf{p}}.$$
The vertex function for $X$, once defined, will be a $K_\TT (X)$-valued function, over the product of the dual Kähler torus ${\mathsf{K}}^\vee$ and the equivariant torus ${\mathsf{A}}$ (and furthermore, certain partial compactifications of them). In particular, for any effective curve $\beta$, with associated circuit $S = S^+ \sqcup S^-$, the monomial $$z^\beta := \prod_{i\in S^+} z_i \prod_{i\in S^-} z_i^{-1}$$ is a well-defined function on the quotient torus ${\mathsf{K}}^\vee$. Similarly, for any root $\alpha$ [^2], with associated cocircuit $R = R^+ \sqcup R^-$, the monomial $$a^\alpha := \prod_{i\in R^+} a_i \prod_{i\in R^-} a_i^{-1}$$ is a well-defined function on the quotient torus ${\mathsf{A}}$.
Polarization
------------
The notion of polarization, although not essentially, is important for our formation of elliptic stable envelopes and vertex functions.
\[Defn-pol\] Let $X$ be a hypertoric variety.
1) A collection of $K$-theoretic classes $T^{1/2}_{{\mathbf{p}}} \in K_\TT ({\mathbf{p}})$, for ${\mathbf{p}}\in X^\TT$, is called a *localized polarization*, if it satisfies $$T^{1/2}_{\mathbf{p}}+ \hbar^{-1} (T_{\mathbf{p}}^{1/2})^\vee = T_X |_{\mathbf{p}}\in K_\TT ({\mathbf{p}}), \qquad {\mathbf{p}}\in X^\TT.$$
2) A localized polarization is called a *global polarization*, or simply a polarization, if it comes from a global $K$-class, i.e., there exists $T_X^{1/2} \in K_\TT (X)$, such that $T^{1/2}_{\mathbf{p}}= T_X^{1/2} |_{\mathbf{p}}$.
Usually a polarization will be given as a Kirwan lift in the $K_\TT ({\mathfrak{X}})$, i.e., as a Laurent polynomial in the Chern roots $x_i$’s.
Elliptic cohomology and stable envelopes
========================================
Equivariant elliptic cohomology for hypertoric varieties
--------------------------------------------------------
Let $q\in \CC^*$ be a complex number, with $|q|<1$, and let $E := \CC^* / q^\ZZ$ be the elliptic curve, with modular parameter $q$. Equivariant elliptic cohomology is a covariant functor that associates to every $\TT$-variety a scheme ${\operatorname{Ell}}_\TT (X)$. In this section, we describe explicitly the $\TT$-equivariant elliptic cohomology and its extended version of the hypertoric variety $X$. For general definitions of equivariant elliptic cohomology, we refer the readers to [@ell1; @ell2; @ell3; @ell4; @ell5; @ell6].
For $X = {\operatorname{pt}}$, the equivariant elliptic cohomology is the abelian variety $${\mathscr{E}}_\TT := {\operatorname{Ell}}_\TT ({\operatorname{pt}}) = E^{\dim \TT},$$ the coordinates on which we refer to as the *elliptic equivariant parameters*, and by abuse of notation, still denote by $a_1, \cdots, a_n$ and $\hbar$. Let ${\mathsf{S}}(X) := E^k$, whose coordinates we call *elliptic Chern roots*, and still denote by $s_1, \cdots, s_k$.
Let $X$ be a hypertoric variety, which is in particular, a GKM variety. The explicit description of equivariant $K$-theory $K_\TT (X)$ can be generalized to the elliptic setting, hence the following diagram $$\xymatrix{
{\operatorname{Ell}}_\TT(X) \ar[d] \ar@{^{(}->}[r] & {\mathsf{S}}(X) \times {\mathscr{E}}_\TT \\
{\mathscr{E}}_\TT .
}$$ Again, ${\operatorname{Ell}}_\TT (X)$ is a closed subvariety in the ambient space ${\mathsf{S}}(X) \times {\mathscr{E}}_\TT$, finite over ${\mathscr{E}}_\TT$, with simple normal crossing singularities.
By $\TT$-localization, the irreducible components of ${\operatorname{Ell}}_\TT (X)$ are parameterized by the fixed point set $X^\TT$, and each of them is isomorphic to the base ${\mathscr{E}}_\TT$. We denote by ${\mathsf{O}}_{\mathbf{p}}$ the irreducible component corresponding to a fixed point ${\mathbf{p}}\in X^\TT$, and call it an *orbit*. The fact that $X$ is a GKM variety implies that orbits are glued in a very nice way to form the scheme ${\operatorname{Ell}}_\TT (X)$:
We have $${\operatorname{Ell}}_{\TT}(X)=\Big(\coprod\limits_{{\mathbf{p}}\in X^{\TT}}\, {\mathsf{O}}_{\mathbf{p}}\Big) /\Delta,$$ where $/\Delta$ denotes the intersections of $\TT$-orbits ${\mathsf{O}}_{\mathbf{p}}$ and ${\mathsf{O}}_{\mathbf{q}}$ along the hyperplanes $${\mathsf{O}}_{\mathbf{p}}\supset \chi^{\perp}_{C} \subset {\mathsf{O}}_{\mathbf{q}},$$ for all ${\mathbf{p}}$ and ${\mathbf{q}}$ connected by an equivariant curve $C$, and $\chi_{C}$ is the $\TT$-character of the tangent space $T_{\mathbf{p}}C$.
The intersections of the orbits ${\mathsf{O}}_{\mathbf{p}}$ and ${\mathsf{O}}_{\mathbf{q}}$ are transversal and hence the scheme ${\operatorname{Ell}}_\TT (X)$ is a variety with simple normal crossing singularities.
Let ${\mathscr{E}}_{{\mathsf{T}}^\vee} := E^n$ [^3] be the space whose coordinates we call the *elliptic Kähler parameters* and still denote by $z_1, \cdots, z_n$. The *extended equivariant elliptic cohomology* of $X$ is defined to be the product $$\textsf{E}_\TT (X) := {\operatorname{Ell}}_\TT (X) \times {\mathscr{E}}_{{\mathsf{T}}^\vee}.$$ It admits the same structure as above $\textsf{E}_\TT (X) = (\coprod\limits_{{\mathbf{p}}\in X^{\TT}}\, \widehat{\mathsf{O}}_{\mathbf{p}}) /\Delta$, with $\widehat{\mathsf{O}}_{\mathbf{p}}:= {\mathsf{O}}_{\mathbf{p}}\times {\mathscr{E}}_{{\mathsf{T}}^\vee}$ and $\Delta$ the same as above.
Recall that in (\[V-frame-i\]) and (\[V-frame-b\]), we can choose different presenation of $\iota$ and $\beta$, for different fixed points ${\mathbf{p}}$. That means for different orbits ${\mathsf{O}}_{\mathbf{p}}$, we have chosen different coordinates $s_1, \cdots, s_k$ on the ambient space ${\mathsf{S}}(X)$.
Elliptic functions
------------------
We define the theta function associated with $E$ explicitly by: $$\vartheta (x) := (x^{1/2} - x^{-1/2} ) \prod_{d=1}^\infty (1 - q^d x) \prod_{d=1}^\infty ( 1- q^d x^{-1} ), \qquad x \in \CC^*.$$ Note that $\vartheta (1) = 0$ and $\vartheta (qx) = - q^{-1/2} x^{-1} \vartheta (x)$, which means that $\vartheta (x)$ defines a section of a line bundle of degree one on the elliptic curve $E$. It will be convenient to describe sections for line bundles on product of elliptic curves in terms of theta functions.
For a sum of variables $\sum_i x_i$, we denote $$\Theta \Big( \sum_i x_i \Big) := \prod_i \vartheta (x_i).$$
Elliptic stable envelopes
-------------------------
Recall that ${\mathsf{A}}= (\CC^*)^n / (\CC^*)^k$ is the quotient torus. Since ${\mathsf{K}}= (\CC^*)^k$ acts on $X$ trivially, ${\mathsf{A}}$ is the actual non-redundant torus acting on $X$. Equivariant parameters on ${\mathsf{A}}$ can be viewed as functions on $(\CC^*)^n$ that vanish on $(\CC^*)^k$. As in Section \[sec-para\], For a given fixed point ${\mathbf{p}}\in X^\TT$ and the correponding standard ${\mathbf{p}}$-frame, there is a convenient choice of coordinates of equivariant parameters: $$\alpha_i ({\mathbf{p}}) = a_i \prod_{j\not\in {\mathbf{p}}} a_j^{-C_{ij}}, \qquad i\in {\mathbf{p}}.$$
The elliptic stable envelope depends on the choice of a polarization (see Definition \[Defn-pol\]). In this section, we choose the polarization as $T^{1/2}_X = \sum_{i=1}^n L_i - {\mathcal{O}}^{\oplus k}$, or $\sum_{i=1}^n x_i - k$, written in terms of Chern roots.
One also needs to choose an element $\sigma \in \ZZ^d$, which determines a cocharacter $\sigma: \CC^* \to {\mathsf{A}}$. Let $\widetilde\sigma \in \ZZ^n$ be a lift of $\sigma$. We assume that $\sigma$ is chosen generically, such that the fixed point set $X^\sigma$ under the 1-dimension subtorus action by $\sigma$ is the same as $X^\TT$.
The choice of $\sigma$ determines a splitting ${\mathbf{p}}= {\mathbf{p}}^+ \sqcup {\mathbf{p}}^-$, where $$\label{split-V}
{\mathbf{p}}^+ := \{ i \in {\mathbf{p}}\mid \langle \alpha_i ({\mathbf{p}}) , \sigma \rangle >0 \},$$ and similar with ${\mathbf{p}}^-$.
We denote by $${\operatorname{Attr}}_\sigma ({\mathbf{p}}) := \{ x\in X \mid \lim_{t\to \infty} \sigma (t) \cdot x = {\mathbf{p}}\}$$ the attracting set of the fixed point ${\mathbf{p}}$. The full attracting set ${\operatorname{Attr}}^f_\sigma ({\mathbf{p}})$ is defined to be the minimal closed subset of $X$ which contains ${\mathbf{p}}$ and is closed under taking ${\operatorname{Attr}}_\sigma(\cdot)$
The attracting set has the following description, originally given [@She]. Recall that the hyperplane arrangement ${\mathcal{A}}$ associated with $X$ gives a collection of affine hyperplanes $H_i$ in $\RR^d$. The space $\RR^d$ is then divided into polytopes by $H_i$’s, where each polytope corresponds to a $\TT$-invariant lagrangian submanifold in $X$. Each fixed point ${\mathbf{p}}$ is identified with a vertex, which is the intersection $\bigcap_{i\in {\mathbf{p}}} H_i$. Given the chosen cocharacter $\sigma$, consider the cone defined as $$\bigcap_{i\in {\mathbf{p}}^+} H_i^+ \cap \bigcap_{i\in {\mathbf{p}}^-} H_i^-.$$ The attacting set ${\operatorname{Attr}}_\sigma ({\mathbf{p}})$ is then the union of lagrangians corresponding to the polytopes in this cone. In particular, we have:
\[Attr\] Given ${\mathbf{p}}, {\mathbf{q}}\in X^\TT$, ${\mathbf{q}}\in \overline{{\operatorname{Attr}}_\sigma ({\mathbf{p}})}$ if and only if ${\mathbf{q}}\in \bigcap_{i\in {\mathbf{p}}^+} H_i^+ \cap \bigcap_{i\in {\mathbf{p}}^-} H_i^-$, which is also equivalent to ${\mathcal{A}}_{\mathbf{q}}^- \cap {\mathbf{p}}^+ = {\mathcal{A}}_{\mathbf{q}}^+ \cap {\mathbf{p}}^- = \emptyset$.
The polarization, when restricted to a fixed point, decomposes into three parts $T^{1/2}_X |_{\mathbf{p}}= T^{1/2}_X |_{\mathbf{p}}^+ + T^{1/2}_X |_{\mathbf{p}}^{\mathsf{A}}+ T^{1/2}_X |_{\mathbf{p}}^-$, whose characters are positive, zero and negative with respect to $\sigma$. For our choice of polarization $T^{1/2}_X = \sum_{i=1}^n x_i - k$, we see that $T^{1/2}_X |_{\mathbf{p}}^A = \sum_{j\not\in {\mathbf{p}}} x_j |_{\mathbf{p}}$, $T^{1/2}_X |_{\mathbf{p}}^\pm = \sum_{i\in {\mathbf{p}}^\pm} x_i |_{\mathbf{p}}$.
The elliptic stable envelope [@AOelliptic] associates each fixed point ${\mathbf{p}}$ with a section ${\operatorname{Stab}}_\sigma ({\mathbf{p}})$ of a certain explicit line bundle $\mathcal{T} ({\mathbf{p}})$ on the orbit $\widehat{\mathsf{O}}_{{\mathbf{p}}} \subset \mathsf{E}_{\mathsf{T}}(X)$, which is the pull-back of a line bundle ${\mathcal{T}}^{os} ({\mathbf{p}})$ on the ambient abelian variety $\mathsf{S}(X) \times {\mathscr{E}}_\TT \times {\mathscr{E}}_{{\mathsf{T}}^\vee}$ along the elliptic Chern class map $\mathsf{E}_\TT (X) \to \mathsf{S}(X) \times {\mathscr{E}}_\TT \times {\mathscr{E}}_{{\mathsf{T}}^\vee}$. The line bundle is uniquely determined by the $q$-quasi-periods of its sections, which can be read off from the explicit formula for ${\operatorname{Stab}}_\sigma ({\mathbf{p}})$ we give in Theorem \[Stab-formula\].
There exists a unique section ${\operatorname{Stab}}_\sigma ({\mathbf{p}})$ of ${\mathcal{T}}({\mathbf{p}})$, holomorphic over ${\mathscr{E}}_\TT$, meromorphic over ${\mathscr{E}}_{{\mathsf{T}}^\vee}$ and in $\hbar$, satisfying the following properties:
(i) ${\operatorname{Supp}}({\operatorname{Stab}}_\sigma ({\mathbf{p}})) \subset {\operatorname{Attr}}^f_\sigma ({\mathbf{p}})$;
(ii) its restriction to ${\mathbf{p}}$ is $${\operatorname{Stab}}_\sigma ({\mathbf{p}}) |_{\mathbf{p}}= (-1)^{|{\mathbf{p}}^+|} \Theta (N_{\mathbf{p}}^-) = \prod_{i\in {\mathbf{p}}^+} \vartheta (\hbar x_i |_{\mathbf{p}}) \prod_{i\in {\mathbf{p}}^-} \vartheta (x_i |_{\mathbf{p}}) ,$$ where $N_{\mathbf{p}}^-$ is the negative half in the decomposition $T_{\mathbf{p}}X = N_{\mathbf{p}}^+ + N_{\mathbf{p}}^-$ which pairs with $\sigma$ negatively.
We have the following explicit formula for the elliptic stable envelope. Note that in our hypertoric case, ${\operatorname{Stab}}_\sigma({\mathbf{p}})$ is actually supported on $\overline{{\operatorname{Attr}}_\sigma ({\mathbf{p}})}$, which is stronger than the general definition.
\[Stab-formula\] The stable envelope ${\operatorname{Stab}}_\sigma ({\mathbf{p}})$ is the pull-back along $\mathsf{S}(X) \times {\mathscr{E}}_\TT \times {\mathscr{E}}_{{\mathsf{T}}^\vee}$ of the following section $$\prod_{i \in {\mathbf{p}}^+} \vartheta ( \hbar x_i ) \prod_{i \in {\mathbf{p}}^-} \vartheta (x_i ) \prod_{j \in {\mathcal{A}}_{\mathbf{p}}^+} \dfrac{\vartheta \Big( x_j \zeta_j ({\mathbf{p}}) \hbar^{- \sum_{i \in {\mathbf{p}}^+} C_{ij}} \Big) }{\vartheta \Big( \zeta_j ({\mathbf{p}}) \hbar^{- \sum_{i \in {\mathbf{p}}^+} C_{ij}} \Big) } \prod_{j \in {\mathcal{A}}_{\mathbf{p}}^-} \dfrac{\vartheta \Big( x_j \zeta_j ({\mathbf{p}}) \hbar^{- \sum_{i \in {\mathbf{p}}^+} C_{ij}} \Big) }{\vartheta \Big( \hbar^{-1} \zeta_j ({\mathbf{p}}) \hbar^{- \sum_{i \in {\mathbf{p}}^+} C_{ij} } \Big) }$$ of the line bundle ${\mathcal{T}}^{os} ({\mathbf{p}})$.
The shift of $\hbar$-factors here is exactly determined by the line bundle ${\mathcal{T}}^{os} ({\mathbf{p}})$. It suffices to check that the given section satisfies (i) and (ii), which follow from Lemma \[Attr\] and direct computation.
Vertex function and quantum product
===================================
Quasimaps and bare vertex
-------------------------
Recall that the hypertoric variety $X$ is a subscheme of the quotient stack ${\mathfrak{X}}= [\mu^{-1} (0) / {\mathsf{K}}] $. Consider the following definition from [@Oko].
A quasimap from $\PP^1$ to $X$ is a morphism $f: \PP^1 \to {\mathfrak{X}}$, such that away from a $0$-dimensional subscheme in $\PP^1$, the morphism $f$ maps into the stable locus $X\subset {\mathfrak{X}}$.
Let $L_i$, $1\leq i\leq n$ and $N_j$, $1\leq j \leq k$ be the tautological line bundles[^4] on ${\mathfrak{X}}$, defined similarly as in previous sections. The datum of a quasimap $f$ is equivalent to the collection of line bundles $f^* N_j$, and sections of $\bigoplus_{i=1}^n L_i \oplus \hbar^{-1} \bigoplus_{i=1}^n L_i^{-1}$, satisfying the moment map equations and stability condition. A point on $\PP^1$ is called a base point for $f$ if it is mapped into the unstable locus ${\mathfrak{X}}\backslash X$.
For a quasimap $f$, taking the degrees of $f^* N_j$ defines a homomorphism $H^2 (X, \ZZ) \to \ZZ$, or equivalently, a curve class $\deg f \in H_2 (X, \ZZ)$, which we call the degree class of $f$. We will call $\deg f$ *effective*, if it lies in ${\operatorname{Eff}}(X)$, the monoid of effective curve classes in $X$.
Now we consider the moduli of quasimaps. Let ${\operatorname{Hom}}(\PP^1, {\mathfrak{X}})$ be the moduli stack parameterizing all representable morphisms from $\PP^1$ to ${\mathfrak{X}}$, which is an Artin stack locally of finite type. Since the domain of quasimaps is fixed, the universal curve is a trivial family over the moduli stack, fitting in the following universal diagram $$\xymatrix{
{\operatorname{Hom}}(\PP^1, {\mathfrak{X}}) \times \PP^1 \ar[r]^-f \ar[d]_\pi & {\mathfrak{X}}\\
{\operatorname{Hom}}(\PP^1, {\mathfrak{X}}).
}$$ There is a perfect obstruction theory, with virtual tangent complex given by $$T_{\mathrm{vir}}= R^\bullet \pi_* f^* T_{\mathfrak{X}}.$$ One observes that the obstruction part actually vanishes. Hence ${\operatorname{Hom}}(\PP^1, {\mathfrak{X}})$ is a smooth Artin stack.
Fix $\beta \in {\operatorname{Eff}}(X)$, and $p_1=0$, $p_2=\infty \in \PP^1$. Let ${\operatorname{QM}}(X, \beta)$ be the stack parameterizing quasimaps from $\PP^1$ to ${\mathfrak{X}}$, which lies in ${\operatorname{Hom}}(\PP^1, {\mathfrak{X}})$ as an open substack, and is of finite type since we fix the degree. Hence ${\operatorname{QM}}(X, \beta)$ is a Deligne–Mumford stack, equipped with the inherited perfect obstruction theory. The standard construction in [@BF; @Lee] defines a virtual structure sheaf $${\mathcal{O}}_{\mathrm{vir}}\in K_\TT ({\operatorname{QM}}(X, \beta)).$$
By Okounkov [@Oko], it is more natural to twist the obstruction theory by a certain square root line bundle and form a modified virtual structure sheaf.
Let $T^{1/2}_X$ be a fixed global polarization. The modified virtual structure sheaf is defined as: $$\label{defn-cO_vir}
\widehat{\mathcal{O}}_{\mathrm{vir}}:= {\mathcal{O}}_{\mathrm{vir}}\otimes \left( K_{\mathrm{vir}}\frac{\det {\mathcal{T}}^{1/2} \big|_{p_2}}{\det {\mathcal{T}}^{1/2} \big|_{p_1}} \right)^{1/2} \in K_\TT ({\operatorname{QM}}(X, \beta)),$$ where $K_{\mathrm{vir}}:= \det T_{\mathrm{vir}}^\vee$, and ${\mathcal{T}}^{1/2}$ is the tautological bundle associated with a lift of the polarization $T^{1/2}_X$ to $K_\TT ({\mathfrak{X}})$. The existence of the square root line bundle $K_{\mathrm{vir}}^{1/2}$ relies on the existence of a polarization on the target $X$, and a spin structure $K_{\PP^1}^{1/2}$ on the domain, which only makes sense if the target is symplectic and the domain is fixed. This crucial modification has the effect of making the obstruction theory equivariantly symmetric.
In order to define invariants, one has to be able to insert $K$-theoretic classes from $X$. Consider the following open substack $${\operatorname{QM}}(X, \beta)_{{\mathrm{ns}}\ p_2} \subset {\operatorname{QM}}(X, \beta)$$ consisting of those quasimaps for which $p_2\in \PP^1$ is nonsingular, or in other words, *not* a base point. There are evaluation maps $${\operatorname{ev}}_2: {\operatorname{QM}}(X, \beta)_{{\mathrm{ns}}\ p_2} \to X, \qquad {\operatorname{ev}}_1: {\operatorname{QM}}(X, \beta)_{{\mathrm{ns}}\ p_2} \to {\mathfrak{X}},$$ where a quasimap $f$ is mapped to the image $f(p_2)$ or $f(p_1)$. Let $\CC^*_q$ be the torus on $\PP^1$, where $q\in K_{\CC_q^*}({\operatorname{pt}})$ is the character defined by $T_{p_1} \PP^1$.
A general $K$-theoretic invariant one can define is to pair the modified virtual structure sheaf $\widehat{\mathcal{O}}_{\mathrm{vir}}$ with classes pulled back from $X$ or ${\mathfrak{X}}$ via ${\operatorname{ev}}_2$ or ${\operatorname{ev}}_1$, and then push forward to $X$ or alternatively, to ${\operatorname{pt}}$. As the stack ${\operatorname{QM}}(X, \beta)_{{\mathrm{ns}}\ p_2}$ is not proper, but admits proper fixed loci under the action of $\TT \times \CC^*_q$. the push-forward in the last step is only well-defined if we work in the *localized* $\CC^*_q$-equivariant theory, i.e., to work in $K_{\TT \times \CC^*_q} (X)_{\mathrm{loc}}:= K_{\TT \times \CC^*_q} (X) \otimes \CC(q)$.
Given $\tau\in K_\TT({\mathfrak{X}})$, the *bare vertex function with decendent insertion $\tau$* is defined as $$V^{(\tau)}(q,z) := \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta {\operatorname{ev}}_{2, *} \left( {\operatorname{QM}}(X, \beta)_{{\mathrm{ns}}\ p_2}, \widehat{\mathcal{O}}_{\mathrm{vir}}\cdot {\operatorname{ev}}_1^* \tau \right) \quad \in K_{\TT\times \CC^*_q} (X)_{\mathrm{loc}}[[ z^{{\operatorname{Eff}}(X)} ]] .$$
General definitions of stable quasimaps are given in [@CKM]. Our definition of quasimaps with fixed domain $\PP^1$ is the special case of Definition 7.2.1 there, as stable quasimaps of genus $0$ to $X$, with one parameterized domain component, and without any marked points.
Relative quasimaps and capped vertex
------------------------------------
Another version of the vertex function comes from counting the relative quasimaps, motivated from the relative DT/PT theory. Let $l\geq 0$ be an integer. Consider the following $$\PP^1[l]:= \PP^1 \cup \PP^1 \cup \cdots \cup \PP^1,$$ constructed by attaching a chain of $l$ $\PP^1$’s to the domain $\PP^1$ at the point $\infty$. The newly attached rational curves are called bubbles or rubber components. Let $p_2$ be the point $\infty$ on the last bubble.
A *relative* quasimap to $X$ is a map $f: \PP^1 [l] \to {\mathfrak{X}}$, for some integer $l\geq 0$, such that it generically maps into the stable locus $X\subset {\mathfrak{X}}$, and moreover, the point $p_2$ is not a base point.
Rather than quasimaps with fixed domain $\PP^1$ in the previous section, relative quasimaps admit nontrivial automorphisms from scaling the bubbles. To describe this construction, we adopt the language from the relative GW/DT/PT theory. For more detailed definitions, see for example [@Li; @ACFJ; @OP; @Oko; @Zhou].
By constructions in [@Li; @ACFJ], there exists a smooth Artin stack ${\mathcal{B}}$, together with a universal famity ${\mathcal{C}}$ over ${\mathcal{B}}$, parameterizing all possible *extended pairs* of the form $(\PP^1[l], p_2)$, in the following sense. For any geometric point ${\operatorname{Spec}}\CC \to {\mathcal{B}}$, the fiber of the family ${\mathcal{C}}\to {\mathcal{B}}$ over that point is of the form $\PP^1 [l]$, for some integer $l\geq 0$. Moreover, the automorphism group for this point is $(\CC^*)^l$, acting on the fiber $\PP^1 [l]$ by scaling the $l$ bubbles.
Let $S$ be a scheme. A family of expanded pairs of $(\PP^1, \infty)$ over $S$ is the family ${\mathcal{C}}_S \to S$, arising from a Cartesian diagram of the following form $$\xymatrix{
{\mathcal{C}}_S \ar[d] \ar[r] & {\mathcal{C}}\ar[d] \\
S \ar[r] & {\mathcal{B}}.
}$$ A *family of relative quasimaps* (with respect to the divisor $\infty \in \PP^1$) is a family of expanded pairs $\pi: {\mathcal{C}}_S \to S$, together with a map $f: {\mathcal{C}}_S \to {\mathfrak{X}}$, such that over each geometric point $s\in S$, the fiber gives a relative quasimap.
A relative quasimap $f: \PP^1 \to {\mathfrak{X}}$ is called *stable*, if its automorphism group is finite. Equivalently, it means that the degree of $f$ on each bubble is nontrivial.
Let $X_0$ be the affine quotient, defined by $\mu^{-1}(0) /\!/_{\theta = 0} {\mathsf{K}}$. Let ${\operatorname{QM}}(X, \beta)_{{\mathrm{rel}}\ p_2}$ be the stack parameterizing all stable relative quasimaps to $X$. Standard argument as in relative DT/PT theory shows that it is a DM stack of finite type, proper over $X_0$, and it admits a perfect obstruction theory, relative over the smooth Artin stack parameterizing principal ${\mathsf{K}}$-bundles over the fibers of ${\mathcal{C}}\to {\mathcal{B}}$. After the same twisting as in (\[defn-cO\_vir\]), we have the modified virtual structure sheaf $\widehat{\mathcal{O}}_{\mathrm{vir}}$.
Similarly we have the evaluation maps $${\operatorname{ev}}_2: {\operatorname{QM}}(X, \beta)_{{\mathrm{rel}}\ p_2} \to X, \qquad {\operatorname{ev}}_1: {\operatorname{QM}}(X, \beta)_{{\mathrm{rel}}\ p_1} \to {\mathfrak{X}}.$$ However, this time ${\operatorname{ev}}_2$ is *proper*, and one can work in the *non-localized* $\CC^*_q$-equivariant theory or the non-equivariant theory.
Given $\tau\in K_\TT({\mathfrak{X}})$, the *capped vertex function with descendent insertion $\tau$* is defined as $$\widehat V^{(\tau)} (q,z) := \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta {\operatorname{ev}}_{2, *} \left( {\operatorname{QM}}(X, \beta)_{{\mathrm{rel}}\ p_2}, \widehat{\mathcal{O}}_{\mathrm{vir}}\cdot \tau\big|_{p_1} \right) \quad \in K_{\TT\times \CC^*_q} (X) [[ z^{{\operatorname{Eff}}(X)} ]].$$ Its non-equivariant counterpart with respect to $\CC^*_q$, defined by the non-equivariant pushforward along the evaluation map, is called the *quantum tautological class*: $$\widehat\tau(z):= \widehat V^{(\tau)}(z) \big|_{q=1} \quad \in K_\TT (X) [[ z^{{\operatorname{Eff}}(X)} ]].$$
The notion of relative quasimaps is also a special case of Definition 7.2.1 in [@CKM]. For each domain $\PP^1[l]$ of the relative quasimaps, there is a contraction map $\PP^1[l] \to \PP^1$, mapping all bubble components to the node where the bubbles are attached to the rigid $\PP^1$. The contraction map can also be defined in families. Let ${\operatorname{QM}}_{0, 1}^{CKM} (X, \beta)_{para}$ be the moduli stack of stable quasimaps of genus $0$ to $X$, with one parameterized domain component, and $1$ marked point $p$. The moduli space of relative quasimaps ${\operatorname{QM}}(X, \beta)_{{\mathrm{rel}}\ p_2}$ is the closed substack in ${\operatorname{QM}}_{0, 1}^{CKM} (X, \beta)_{para}$, consisting of quasimaps where the marked point $p$ is mapped to $p_2 \in \PP^1$ under the contraction map.
Gluing operator, degeneration and PSZ quantum $K$-theory {#QK}
--------------------------------------------------------
More generally, if we choose $N$ points $p_1, \cdots, p_N$ on $\PP^1$, one can specify a “nonsingular" or “relative" condition at each point, and insert appropriate descendent insertions at other points. Pushforward by the evaluation map ${\operatorname{ev}}_1\times \cdots \times {\operatorname{ev}}_N$ would define a $K$-theory class in $K(X)^{\otimes N}$, i.e. an $N$-point class. However, one has to work with the $\CC_q^*$-equivariant theory or non-equivariant theory, depending on the following specific cases.
\(i) If $N\leq 2$, and all points $p_i$ (identified with either $0$ or $\infty\in \PP^1$) are equipped with the “relative" condition, this $N$-point class can be defined to lie in the non-localized ring $K_{\TT\times \CC_q^*}(X)^{\otimes N} [[ z^{{\operatorname{Eff}}(X)} ]]$.
\(ii) If $N\leq 2$ and some of the points $p_i$ (identified with either $0$ or $\infty \in \PP^1$) are equipped with the “nonsingular" condition, then it lies in the localized ring $K_{\TT\times \CC_q^*}(X)_{\mathrm{loc}}^{\otimes N} [[ z^{{\operatorname{Eff}}(X)} ]]$.
\(iii) Finally, if $N\geq 3$, then $\CC_q^*$ does not preserves the points $p_i$’s, and all $p_i$’s have to be assigned the “relative" condition. The push-forward map only makes sense $\CC^*_q$-non-equivariantly, producing an $N$-point class in $K_{\TT}(X)^{\otimes N} [[ z^{{\operatorname{Eff}}(X)} ]]$.
Let $K_X$ denote the canonical line bundle. A natural nondegenerate bilinear form on $K_\TT(X)$ is $$({\mathcal{F}}, {\mathcal{G}}):= \chi \Big( X, {\mathcal{F}}\otimes {\mathcal{G}}\otimes K_X^{-1/2} \Big), \qquad {\mathcal{F}}, {\mathcal{G}}\in K_\TT (X).$$ By the Fourier–Mukai philosophy, an $(N+M)$-point class in $K_\TT (X)^{\otimes (N+M)}$ can be viewed as an operator $K_\TT (X)^{\otimes N} \to K_\TT (X)^{\otimes M}$, where we are free to choose $N$ points as the input and $M$ points as the output. In particular, a class ${\mathsf{O}}\in K_\TT (X)^{\otimes 2}$ defines an operator ${\mathsf{O}}\in {\operatorname{End}}K_\TT (X)$ by ${\mathsf{O}}{\mathcal{G}}:= {\operatorname{pr}}_{1*} \left( {\mathsf{O}}\otimes {\operatorname{pr}}_2^* {\mathcal{G}}\right)$. We will often abuse the same notation for both the class in $K_\TT (X)^{\otimes 2}$ and the operator in ${\operatorname{End}}K_\TT (X)$.
The *gluing operator* is defined as $$G(q,z) := \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta ({\operatorname{ev}}_{p_1}\times {\operatorname{ev}}_{p_2})_* \left( {\operatorname{QM}}(X, \beta )_{{\mathrm{rel}}\ p_1, {\mathrm{rel}}\ p_2}, \widehat{\mathcal{O}}_{\mathrm{vir}}\right) \in K_{\TT \times \CC_q^*} (X)^{\otimes 2} [[ z^{{\operatorname{Eff}}(X)} ]].$$
The gluing operator plays a crucial role in the degeneration of quasimap theory. Note that the $\beta =0$ part has only contribution from the constant maps, and therefore $G = (\Delta_X)_* K_X^{1/2} + O(z) \in K_{\TT \times \CC_q^*} (X)^{\otimes 2} [[ z^{{\operatorname{Eff}}(X)} ]]$. In particular, $G = K_X^{1/2} \otimes + O(z)$ as an operator, and admits an inverse $G^{-1}$.
We introduce the following $N$-to-$M$ operator $$C(p_{N+1}, \cdots, p_{N+M} \mid p_1, \cdots, p_N ) : K_{\mathsf{T}}(X)^{\otimes N} \to K_{\mathsf{T}}(X)^{\otimes M} [[z^{{\operatorname{Eff}}(X)} ]],$$ defined by quasimap counting. Let $p_1, \cdots, p_{N+M}$ be points on $C = \PP^1$.
For any $N, M \geq 0$, define && C(p\_[N+1]{}, , p\_[N+M]{} p\_1, , p\_N ) (\_1, , \_N)\
&:=& \_[(X)]{} z\^( \_[p\_[N+1]{}, , p\_[N+M]{}]{} )\_\* ( (X, )\_[p\_1, , p\_[N+M]{}]{} , \_\_[i=1]{}\^N \_[p\_i]{}\^\* (G\^[-1]{} |\_[q= 1]{} \_N) ), where the push-forward ${\operatorname{ev}}_{p_{N+1}, \cdots, p_{N+M}}$ is taken to be $\CC^*_q$-non-equivariant.
Now suppose that the domain $C=\PP^1$ degenerates into the union of two rational curves $C_0 = C_+ \cup C_-$. Degeration formula relates the quasimap theory with domain $C$ to the relative quasimap theories with domains $C_\pm$, with respect to the new relative points introduced by the node. The degeneration formula [@Oko] states that the operators satisfy the identity $$C(p_{N+1}, \cdots, p_{N+M} \mid p_1, \cdots, p_N ) = C_+( p_{N+1}, \cdots, p_{N+M} \mid \bullet ) \circ G^{-1} |_{q=1} \circ C_-( \bullet \mid p_1, \cdots, p_N ),$$ where $\bullet$ denotes the new relative point introduced by the node. Moreover, if $N+M \leq 2$, the degeneration formula also works $\CC_q^*$-equivariantly, and the non-equivariant gluing $G^{-1} |_{q=1}$ should be replaced by the equivariant version $G^{-1}$.
In particular, the quantum tautulogical class associated to the identity can be recovered as the $1$-point function: $\hat{\mathbf{1}}(z) = C(p_1 \mid \ )$, and the specialization of the inverse gluing operator can be recovered as the $2$-point function: $\chi(X, {\mathcal{F}}\otimes ( G(q, z)^{-1} |_{q=1} \cdot {\mathcal{G}}) ) = C( \ | p_1, p_2) ({\mathcal{F}}, {\mathcal{G}})$.
We are particularly interested in the case $(N,M) = (2,0)$ and $(2,1)$. The following definition is made by Pushkar–Smirnov–Zeitlin [@PSZ].
The *PSZ quantum K-theory ring* of $X$ is defined as the vector space $K_\TT(X) [[ z^{{\operatorname{Eff}}(X)} ]]$, equipped with the *quantum pairing* $\langle\ , \ \rangle_{PSZ}$, the *quantum product* $*$, and the *quantum identity* $\hat{\mathbf{1}}(z)$, defined as follows $$\langle {\mathcal{F}}, {\mathcal{G}}\rangle_{PSZ} := C(\ \mid p_1, p_2) ({\mathcal{F}}, {\mathcal{G}}), \qquad {\mathcal{F}}* {\mathcal{G}}:= C(p_3 \mid p_1, p_2) ({\mathcal{F}}, {\mathcal{G}}), \qquad \hat{\mathbf{1}}(z) := C(p_1 \mid \ ).$$
As in [@PSZ], one can prove that this is actually what we expect:
The quantum $K$-theory ring defined as above is a Frobenius algebra.
Capping operator and $q$-difference equation
--------------------------------------------
The two versions of vertex functions are related by the following *capping operator* [^5] $$\Psi(q,z) := \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta ({\operatorname{ev}}_{p_1}\times {\operatorname{ev}}_{p_2})_* \left( {\operatorname{QM}}(X, \beta )_{{\mathrm{ns}}\ p_1, \ {\mathrm{rel}}\ p_2}, \widehat{\mathcal{O}}_{\mathrm{vir}}\right) \in K_{\TT \times \CC_q^*} (X)_{\mathrm{loc}}^{\otimes 2} [[ z^{{\operatorname{Eff}}(X)} ]],$$ where the notation means we consider quasimaps that are required to be nonsingular at $p_1\in \PP^1$ and allowed to bubble out at $p_2\in\PP^1$. The relationship between the bare and capped vertices is the following, which is proved in [@Oko] by relative localization and degeneration.
\[capping-eqn\] View $\Psi (q, z)$ as in ${\operatorname{End}}K_{\TT \times \CC^*_q} (X)_{\mathrm{loc}}$. We have $$\hat V^{(\tau)}(q,z) = \Psi(q,z) \, V^{(\tau)}(q,z) .$$
Note that both factors on the right hand side lie in the localized K-group, while the left hand side is a non-localized $K$-class. This would be clear in a more precise way as we consider the $q\to 1$ asymptotic of this equality.
Analogous to the quantum differential equation in the cohomological theory, the capping operator satisfies a certain $q$-difference equation. Consider the operator $$\begin{aligned}
M_i(q,z) &:=& \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta ({\operatorname{ev}}_{p_1}\times {\operatorname{ev}}_{p_2})_* \left( {\operatorname{QM}}(X, \beta )_{{\mathrm{rel}}\ p_1, \ {\mathrm{rel}}\ p_2}, \widehat{\mathcal{O}}_{\mathrm{vir}}\cdot \det H^\bullet (L_i \otimes \pi^* {\mathcal{O}}_{p_2}) \right) \circ G^{-1} \\
&\in& K_{\TT \times \CC_q^*} (X)^{\otimes 2} [[ z^{{\operatorname{Eff}}(X)} ]],\end{aligned}$$ where $\pi: \PP^1 [l]\to \PP^1$ is the projection, and the term $H^\bullet(-)$ here means the (virtual) tautological bundle on the moduli space ${\operatorname{QM}}(X, \beta)$ whose fiber at a quasimap is represented by the (virtual) space $H^\bullet(L_i \otimes \pi^* {\mathcal{O}}_{p_2})$. It is important that the operator $M_i(z)$ here lies in the non-localized K-group. The following equation is proved by considering the degeneration of capping operators with descendents [@Oko; @OS].
\[Psi\] View $\Psi (q, z)$ as in ${\operatorname{End}}K_{\TT \times \CC^*_q} (X)_{\mathrm{loc}}$. The capping operator $\Psi(z)$ satisfies $$Z_i \Psi(q, z) = (L_i^{-1} \otimes ) \circ \Psi(q,z) \circ M_i (q^{-1} ,z),$$ where $Z_i$ is the operator which shifts $z_i \mapsto q z_i$ and keep the other Kähler parameters.
Moreover, the non-equivariant limit (with respect to $\CC_q^*$) of $M_i$ is the quantum multiplication by the quantum tautological line bundle: $$M_i(1, z) = \widehat L_i (z) * .$$
Divisorial quantum $K$-theory
-----------------------------
In this subsection we would like to slightly modify the PSZ ring structure, and obtain one that can be compared with Givental’s quantum $K$-theory. Recall that ${\mathfrak{X}}= [\mu^{-1} (0) / {\mathsf{K}}]$ is the stacky quotient, and $K_\TT ({\mathfrak{X}}) \cong K_\TT (B {\mathsf{K}})$ consists of Laurent polynomials in $x_1, \cdots, x_n$, up to the linear relations: $$K_\TT ({\mathfrak{X}}) \cong \CC [ a_1^{\pm 1}, \cdots, a_n^{\pm 1}, \hbar^{\pm 1}, x_1^{\pm 1}, \cdots, x_n^{\pm 1} ] / \langle \prod_{i=1}^n (x_i / a_i)^{\beta_{ji}} - 1, 1\leq j\leq d \rangle.$$ The ordinary $K$-theory ring $K_\TT (X)$ is the quotient of $K_\TT ({\mathfrak{X}})$ with relations parameterized by the circuits of $X$. We will define a new ring structure, also as a quotient of $K_\TT ({\mathfrak{X}})$, which deforms $K_\TT (X)$. On the other hand, it is essentially “the same" as the PSZ ring.
\[quantum-prod\] Given $\tau, \eta \in K_\TT ({\mathfrak{X}})$, one has $$\widehat{\tau \eta} (z) = \widehat\tau (z) * \widehat\eta (z).$$
By definition, $\widehat{\tau \eta} (z)$ is the evaluation at $q = 1$ of $\widehat V^{(\tau\eta)} (q, z)$. The descendent insertion is $(\tau \eta) |_{p_1}$, which is equivalent to $\tau |_{p_2} \cdot \eta |_{p_3}$ under the $q=1$ evaluation, where $p_2$, $p_3$ are two arbitrary points on $\PP^1$. By the degeneration formula, this is the same as $\widehat\tau (z) * \widehat\eta (z)$.
The lemma implies that the following $${\mathcal{I}}:= \{ \tau \in K_\TT ({\mathfrak{X}}) [[z^{{\operatorname{Eff}}(X)}]] \mid \hat\tau (z) = 0 \},$$ is an ideal in $K_\TT ({\mathfrak{X}}) [[z^{{\operatorname{Eff}}(X)}]] $. Furthermore, the bilinear pairing $$\langle \tau, \eta \rangle := \langle \hat\tau (z), \hat\eta (z) \rangle_{PSZ}$$ vanishes for any $\tau$ or $\eta\in {\mathcal{I}}$, and hence descends to a pairing $\langle \ , \ \rangle_{div}$ on the quotient. The map $$\label{isom}
K_\TT ({\mathfrak{X}}) [[z^{{\operatorname{Eff}}(X)}]] / I \to (K_\TT (X) [[z^{{\operatorname{Eff}}(X)} ]], * ) , \qquad \tau \mapsto \hat\tau (z)$$ is an isomorphism of $K_\TT ({\operatorname{pt}}) [[z^{{\operatorname{Eff}}(x)}]]$-algebras, preserving the pairings. It is therefore an isomorphism of Frobenius algebras.
We define the quotient ring $K_\TT ({\mathfrak{X}}) [[z^{{\operatorname{Eff}}(X)}]] / I$ as the *divisorial quantum $K$-theory* of $X$, with *quantum identity* ${\mathbf{1}}$, and *quantum pairing* $\langle \ , \ \rangle_{div}$. In particular, it is a Frobenius algebra, whose specialization at $z = 0$ recovers the ordinary $K$-theory ring $K_\TT (X)$.
Localization computations
=========================
Bare vertex and integral presentation
-------------------------------------
In this section we compute the explicit formula for bare vertex functions of the hypertoric variety $X$, by equivariant localization. Let ${\mathbf{p}}\subset \{1, \cdots, n\}$ denote a vertex in the hyperplane arrangement ${\mathcal{H}}$, and also the corresponding $\TT$-fixed point.
For this particular ${\mathbf{p}}$, we choose the standard ${\mathbf{p}}$-frame as in (\[V-frame-i\]) and (\[V-frame-b\]). For example, if ${\mathbf{p}}= \{ k+1, \cdots, n \}$, then the matrices $\beta$ and $\iota$ take the form $$\beta = \begin{pmatrix}
-C & I
\end{pmatrix}, \qquad \iota = \begin{pmatrix}
I \\
C
\end{pmatrix}.$$
The prequotient vector space $M = \CC^n$, as a $\TT$-representation, splits as $$M = \sum_{i=1}^n \CC_{e_i^*} = \sum_{i\not\in {\mathbf{p}}} \CC_{e_i^*} + \sum_{i\in {\mathbf{p}}} \CC_{e^*_i},$$ whose associated tautological bundle (by (\[restriction-V\])) can be written as $${\mathcal{M}}= \sum_{i=1}^n L_i = \sum_{i\in {\mathcal{A}}_{\mathbf{p}}^+ } L_i + \sum_{i\in {\mathcal{A}}_{\mathbf{p}}^-} \hbar^{-1} \otimes L_i + \sum_{i\in {\mathbf{p}}} a_i \prod_{j\not\in {\mathbf{p}}} a_j^{-C_{ij}} \cdot \hbar^{- \sum_{j \in {\mathbf{p}}^{c-} } C_{ij} } \otimes \prod_{j\not\in {\mathbf{p}}} L_j^{\otimes C_{ij}}.$$
Let $f: \PP^1 \to {\mathfrak{X}}$ be a $\TT$-equivariant quasimap whose image lies in the fixed point ${\mathbf{p}}$. Then $$\deg f^* L_i \geq 0, \quad \text{for any } i\in {\mathcal{A}}_{\mathbf{p}}^+; \qquad \deg f^* L_i \leq 0, \quad \text{for any } i\in {\mathcal{A}}_{\mathbf{p}}^-.$$
Let $(z, w) \in T^* \CC^n$ be a representative of ${\mathbf{p}}$. We know from [@HP] that $z_i \neq 0$, $w_i = 0$ for $i\in {\mathcal{A}}_{\mathbf{p}}^+$, and $z_i = 0$, $w_i\neq 0$ for $i\in {\mathcal{A}}_{\mathbf{p}}^-$. Since $f$ generically maps to ${\mathbf{p}}$, the section of $f^* L_i^{\pm 1}$ defined by $f$ is nonzero, for $i\in {\mathcal{A}}_{\mathbf{p}}^\pm$. The lemma follows.
The lemma states that all quasimaps mapping into ${\mathbf{p}}$ lie in the cone $$\prod_{i\in {\mathcal{A}}_{\mathbf{p}}^+} \RR_{\geq 0} \times \prod_{i\in {\mathcal{A}}_{\mathbf{p}}^- } \RR_{\leq 0} \subset H_2 (X, \RR).$$
\[subcone\] $\prod_{i\in {\mathcal{A}}_{\mathbf{p}}^+ } \RR_{\geq 0} \times \prod_{i\in {\mathcal{A}}_{\mathbf{p}}^- } \RR_{\leq 0}$ is a subcone of ${\operatorname{Eff}}(X) \otimes \RR$. We denote it by ${\operatorname{Eff}}_{\mathbf{p}}(X)$.
As $\iota^\vee = (I, C^T)$, given stability condition $\theta \in (\ZZ^k)^\vee$, we can choose an equivalent lift $\widetilde\eta \in (\ZZ^n)^\vee$ as $\widetilde\eta_j = \theta_j$ for $j\not\in {\mathbf{p}}$, and $\widetilde\eta_i = 0$ for $i\in {\mathbf{p}}$. By definition we have ${\mathcal{A}}_{\mathbf{p}}^+ = \{j\not\in {\mathbf{p}}\mid \theta_j >0 \}$, ${\mathcal{A}}_{\mathbf{p}}^- = \{ j \not\in {\mathbf{p}}\mid \theta_j <0 \}$. In other words, $\theta$ lies in the cone ${\operatorname{Eff}}_{\mathbf{p}}(X)^\vee = \prod_{i\in {\mathcal{A}}_{\mathbf{p}}^+ } \RR_{\geq 0} \times \prod_{i\in {\mathcal{A}}_{\mathbf{p}}^- } \RR_{\leq 0} \subset H^2 (X, \RR)$. On the other hand, the Kähler cone ${\mathfrak{K}}= {\operatorname{Eff}}(X)^\vee \otimes \RR$ is the smallest $k$-dimensional cone generated by $\iota^\vee e_i^*$, which contains $\theta$. Therefore, we have ${\mathfrak{K}}\subset {\operatorname{Eff}}_{\mathbf{p}}(X)^\vee$. Taking the dual proves the lemma.
Let $f: \PP^1 \to {\mathfrak{X}}$ be a $\TT$-equivariant quasimap whose image lies in the fixed point ${\mathbf{p}}$. The virtual tangent sheaf at the moduli point $[f]$ is $$T_{\mathrm{vir}}= H^\bullet({\mathcal{M}}\oplus \hbar^{-1}{\mathcal{M}}^*) - (1+\hbar^{-1}) \Big( \sum_{i\not\in {\mathbf{p}}} {\mathcal{O}}\Big).$$
By the definition (\[defn-cO\_vir\]), i.e. $${\mathcal{O}}_{\mathrm{vir}}\big|_f = \frac{1}{\Lambda^\bullet(T_{\mathrm{vir}}^\vee \big|_f)} = S^\bullet(T_{\mathrm{vir}}^\vee \big|_f), \qquad \widehat{\mathcal{O}}_{\mathrm{vir}}= q^{- \frac{1}{2} \deg {\mathcal{T}}^{1/2}} {\mathcal{O}}_{\mathrm{vir}}\otimes K_{\mathrm{vir}}^{1/2},$$ we see that the contribution to ${\mathcal{O}}_{\mathrm{vir}}$ from a term $$a L + \hbar^{-1} a^{-1} L^{-1}, \qquad \text{with} \ \deg L = d$$ in $T_{\mathrm{vir}}\big|_f$ is $$\{a\}_d := (-q^{1/2}\hbar^{-1/2})^d \frac{(\hbar a)_d}{(q a)_d} = (-q^{1/2}\hbar^{-1/2})^{-d} \frac{(a^{-1} )_{-d} }{(q \hbar^{-1} a^{-1} )_{-d} } , \qquad d\in \ZZ,$$ where $$(x)_d := \frac{\phi(x)}{\phi (q^d x)}, \qquad \phi(x) := \prod_{l=0}^\infty (1-q^l x) .$$ Note that $\phi (x)$ is convergent if we view $q$ as a complex number with $|q|<1$, and $\{a\}_d$ being rational, is well-defined for general $q$.
\[Prop-vertex\] The bare vertex function is $$\label{vertex}
V^{(\tau)}(q,z) = \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta q^{-\frac{1}{2} (\beta, \det {\mathcal{T}}^{1/2}) } \prod_{i=1}^n \{ x_i \}_{D_i} \cdot \tau ( x_1 q^{D_1}, \cdots, x_n q^{D_n} ),$$ where $x_i = L_i \in K_\TT (X)$, $D_i := \deg L_i$, for $1\leq i\leq n$, $\tau = \tau (x_1, \cdots, x_n) \in K_\TT ({\mathfrak{X}})$ is a Laurent polynomial in $n$ variables, and $z^\beta := z_1^{D_1} \cdots z_n^{D_n}$.
Note that the expression on the RHS of (\[vertex\]) is independent of the choice of basis for $\ZZ^k$. The proposition follows from direct computations of the vertex function, restricted to each fixed point.
Note that by our definition of Kähler parameters in Section \[sec-para\], we have $z^\beta = z_1^{D_1} \cdots z_n^{D_n} = \prod_{j\not\in {\mathbf{p}}} \zeta_j ({\mathbf{p}})^{D_j}$. Moreover, the $z^\beta$ term of the restriction $V^{(\tau)}(q,z) \big|_{\mathbf{p}}$ of (\[vertex\]) to the fixed point ${\mathbf{p}}$ vanishes unless $\beta \in {\operatorname{Eff}}_{\mathbf{p}}(X)$.
Note that different choices of the global polarization $T^{1/2}_X$, and hence the associated ${\mathcal{T}}^{1/2}$, result in different vertex functions. But they are related by $q$-shifts of the Kähler parameters. As observed in [@AOelliptic], with some extra factors, the bare vertex function can be written in the form of an integral over the Chern roots, along some appropriately chosen contours.
\[loc-pol\]
1) Choose the *global polarization* as $$\label{Polar-X}
T^{1/2}_X = \sum_{i=1}^n L_i - {\mathcal{O}}^{\oplus k} = \sum_{i=1}^n x_i - k \ \in \ K_\TT (X).$$ Define the shifted Kähler parameters $$z_\sharp^\beta := z^\beta \cdot (-\hbar^{ \frac{1}{2} \beta \cdot \det T^{1/2}_X }), \qquad \text{i.e.} \qquad z_{\sharp, i} := z_i \cdot (-\hbar^{-1/2}), \qquad 1\leq i\leq n.$$
2) Define the *localized polarization* as $$\label{polarization}
{\mathsf{Pol}}_{\mathbf{p}}(x) := \sum_{i\in {\mathcal{A}}_{\mathbf{p}}^+} \hbar^{-1} x_i^{-1} + \sum_{i\in {\mathcal{A}}_{\mathbf{p}}^- \cup {\mathbf{p}}} x_i - k \hbar^{-1} \ \in \ K_\TT (X).$$ In particular, the restriction ${\mathsf{Pol}}_{\mathbf{p}}(x) |_{\mathbf{p}}= \sum_{i\in {\mathbf{p}}} x_i |_{\mathbf{p}}= T_X^{1/2} |_{\mathbf{p}}$ is well-defined. Define another version of shifted Kähler parameters $$z_\epsilon^\beta := z_\sharp^\beta \cdot (-q^{1/2} \hbar^{-1/2})^{\frac{1}{2} (-1+\beta \cdot \det {\mathsf{Pol}}_{\mathbf{p}}(x) )}, \qquad \text{i.e.} \qquad z_{\epsilon, i} ({\mathbf{p}}) := z_{\sharp, i} \cdot (q\hbar^{-1})^{-\epsilon (i)},$$ where $\epsilon (i) = 1$ if $i\in {\mathcal{A}}_{\mathbf{p}}^+$, and $\epsilon (i) = 0$ if $i\in {\mathcal{A}}_{\mathbf{p}}^- \cup {\mathbf{p}}$.
3) Define the *modified bare vertex functions* as $$\widetilde V^{(\tau)} (q,z) |_{\mathbf{p}}:= V^{(\tau)}(q,z) \big|_{\mathbf{p}}\cdot e^{\sum_{i=1}^n \frac{\ln z_{\epsilon,i} ({\mathbf{p}}) \ln x_i |_{\mathbf{p}}}{\ln q} } \cdot \Phi ((q-\hbar) T_X^{1/2} |_{\mathbf{p}}),$$ where $\Phi$ be the multiplicative function determined by $\Phi(\sum_i x_i) := \prod_i \phi(x_i)$.
Recall that the bare vertex function $V^{(\tau)}$, lives in $K_\TT (X)$, and depends only on the choice of the global polarization $T_X^{1/2}$. However, the modified bare vertex function $\widetilde V^{(\tau)} |_{\mathbf{p}}$ depends also on the localized polarization ${\mathsf{Pol}}_{\mathbf{p}}$, which is only defined for each fixed point ${\mathbf{p}}$, and does not necessarily lift to a global class in $K_\TT (X)$. In general, one is allowed to choose a different localized polarization such as ${\mathsf{Pol}}_{\mathbf{p}}(x) = \sum_{i\in {\mathcal{A}}_{\mathbf{p}}^+ \cup {\mathbf{p}}'} \hbar^{-1} x_i^{-1} + \sum_{i\in {\mathcal{A}}_{\mathbf{p}}^- \cup {\mathbf{p}}''} x_i - k \hbar^{-1}$, and the corresponding $z_\epsilon$ such as $\epsilon (i) = 1$ if $i\in {\mathcal{A}}_{\mathbf{p}}^+ \cup {\mathbf{p}}'$, and $\epsilon (i) = 0$ if $i\in {\mathcal{A}}_{\mathbf{p}}^- \cup {\mathbf{p}}''$. We will see such a different choice on the mirror side.
Direct computation yields the following.
$$\widetilde V^{(\tau)} (q,z) \big|_{\mathbf{p}}= \frac{1}{(2\pi i)^k} \int_{q \gamma({\mathbf{p}})} \frac{d\ln x_1 \wedge \cdots d\ln x_n}{\bigwedge_{m=1}^d \Big( \sum_{i=1}^n \beta_{mi} d\ln x_i \Big) } \cdot e^{\sum_{i=1}^n \frac{\ln z_{\epsilon,i} ({\mathbf{p}}) \ln x_i }{\ln q} } \cdot \Phi'((q - \hbar) \mathsf{Pol}_{\mathbf{p}}(x)) \cdot \tau( x_1 , \cdots, x_n),$$ where $q\gamma (V)$ is a noncompact real $k$-cycle in $(\CC^*)^k = \left\{ \prod_{i=1}^n (x_i / a_i)^{\beta_{ji}} = 1 \right\} \subset (\CC^*)^n$, enclosing the following $q$-shifts of poles: $$\label{poles}
x_i = \left\{ \begin{aligned}
& q^{d_i}, \qquad && d_i \geq 0, \ i \in {\mathcal{A}}_{\mathbf{p}}^+ \\
& \hbar^{-1} q^{d_i}, \qquad && d_i \leq 0, \ i\in {\mathcal{A}}_{\mathbf{p}}^- .
\end{aligned} \right.$$
\[limit-point\] If we use $z_{\epsilon, i}$, $i\not\in {\mathbf{p}}$ as the coordinates, then up to an exponential factor, the integral is convergent in the region described as follows: $$|z^\beta| \ll 1,$$ for any $\beta \in {\operatorname{Eff}}(X) - \{0\}$. In particular, we have $|\zeta_i ({\mathbf{p}})|\ll 1$ for $i\in {\mathcal{A}}_{\mathbf{p}}^+$ and $|\zeta_i ({\mathbf{p}})| \gg 1$ for $i\in {\mathcal{A}}_{\mathbf{p}}^- $, and there is a limit point $$\zeta_i ({\mathbf{p}}) \to 0, \qquad i \in {\mathcal{A}}_{\mathbf{p}}^+; \qquad \zeta_i({\mathbf{p}}) \to \infty , \qquad i\in {\mathcal{A}}_{\mathbf{p}}^-.$$ We will denote by $\zeta ({\mathbf{p}}) \xrightarrow{\theta} 0$ the process of taking the limit of $\zeta$, or equivalently $z$, in the manner as above, in the well-defined $\theta$-dependent region. In particular, under this limit, $z^\beta \to 0$ or $\infty$ if and only if $\beta \cdot \theta >0$ or $<0$.
Asymptotics
-----------
In this subsection, we would like to prove a rigidification result which particularly holds for hypertoric varieties, which follows from an estimate of the behavior of the are vertex function as $q\to 0$ or $\infty$. The result does not hold in general for non-abelian holomorphic symplectic quotients.
\[rig\] Let $X$ be a hypertoric variety. The capped vertex function $\widehat V^{({\mathbf{1}})} (q,z)$ is independent of $q$, and hence equal to the PSZ quantum identity class $\widehat{\mathbf{1}}(z)$. Moreover, one has the limit $$V^{({\mathbf{1}})} (q , z) \to \left\{ \begin{aligned}
& \widehat {\mathbf{1}}(z) , && \qquad q \to 0 \\
& G(q, z) \big|_{q=1}^{-1} \cdot \widehat {\mathbf{1}}(z) , && \qquad q \to \infty.
\end{aligned} \right.$$
Recall that by definition $\widehat {\mathbf{1}}(z) = \lim_{q\to 1} \widehat V^{({\mathbf{1}})} (q,z)$, and by Proposition \[capping-eqn\] the capped and bare vertices are related by the capping operator $$\hat V^{({\mathbf{1}})}(q,z) = \Psi(q,z) \cdot V^{({\mathbf{1}})}(q,z).$$ Let’s study the $q\to 1$ asymptotic behavior of this equation.
\[capping-limit\] The capping operator $\Psi (q,z)$ satisfies $$\Psi (q, z) \to \left\{ \begin{aligned}
& {\operatorname{Id}}, && \qquad q \to 0 \\
& G(q,z)|_{q=1} , && \qquad q \to \infty
\end{aligned} \right.$$
This is Lemma 7.1.11 in [@Oko].
\[bounded\] The bare vertex function $V^{({\mathbf{1}})} (q,z)$ admits finite limits when $q \to 0$ or $\infty$.
We have the explicit formula $$\left. V^{({\mathbf{1}})} (q, z) \right|_{\mathbf{p}}= \sum_{\beta \in {\operatorname{Eff}}(X)} z^\beta q^{- \frac{1}{2} \sum_{i=1}^n D_i } \prod_{i=1}^n \{ x_i |_{\mathbf{p}}\}_{D_i}.$$ By definition, we have for any $x$ and $D\in \ZZ$, $$\{ x \}_D \sim \left\{ \begin{aligned}
& \mathrm{const} \cdot q^{|D|/2}, && \qquad q \to 0 \\
& \mathrm{const} \cdot q^{-|D| / 2} , && \qquad q \to \infty.
\end{aligned} \right.$$ We see that the factor $q^{ - \frac{1}{2} \sum_{i=1}^n D_i } $ is completely controlled by the term $q^{\pm \sum_{i=1}^n |D_i| / 2}$ and hence $V^{({\mathbf{1}})} (q, z)$ is bounded as $q \to 0$ or $\infty$.
As a class in the *non-localized* $K$-theory ring $K_{\TT \times \CC_q^*} (X) [[ z^{{\operatorname{Eff}}(X)} ]]$, each $z^\beta$-term of the capped vertex $\widehat V^{({\mathbf{1}})} (q, z)$ is a Laurent polynomial in $q$. Moreover, by the previous two lemmas, it admits no poles at $q = 0$ and $\infty$, which implies that it is actually constant in $q$. Therefore, $$\lim_{q \to 1} \widehat V^{({\mathbf{1}})} (q , z) = \lim_{q\to 0} \widehat V^{({\mathbf{1}})} (q , z) .$$ The last statement follows from Lemma \[capping-limit\].
For some special targets $X$, one can explicitly compute the limit of the vertex function, and hence the PSZ quantum identity class. Consider the following two assumptions:
$\mathrm{(A+)}$ for any circuit $\beta\neq 0$, there exists some $i$, such that $D_i >0$;
$\mathrm{(A-)}$ for any circuit $\beta\neq 0$, there exists some $i$, such that $D_i <0$.
For example, $T^*\PP^n$ satisfies $\mathrm{(A+)}$ but not $\mathrm{(A-)}$; ${\mathcal{A}}_n$ satisfies both $\mathrm{(A+)}$ and $\mathrm{(A-)}$.
1) If $X$ satisfies $\mathrm{(A+)}$, then $\widehat {\mathbf{1}}(z) = G(q,z) \big|_{q=1} \cdot {\mathbf{1}}$.
2) If $X$ satiesfies $\mathrm{(A-)}$, then $\widehat {\mathbf{1}}(z) = {\mathbf{1}}$.
$q$-difference equations
------------------------
For any function $f(a, z)$ depending on the (redundant) equivariant parameters $a_i$ and Kähler parameters $z_i$, $1\leq i\leq n$, consider the following $q$-shift operators: (A\_i f) (a\_1, , a\_i, a\_n, z ) &:=& f (a\_1, , q a\_i, a\_n, z )\
(Z\_i f) (a, z\_1, , z\_i, , z\_n ) &:=& f (a, z\_[1]{}, , q z\_[i]{}, , z\_[n]{} ). The effect of the inverse operators $A_i^{-1}$ and $Z_i^{-1}$ are to shift the variables by $q^{-1}$.
We would like to apply those shift operators to the vertex function V\^[()]{} (q,z) |\_&=& \_[q ()]{} e\^[\_[i=1]{}\^n ]{}\
&& \_[i\_\^+]{} \_[i\_\^- ]{}\
&=:& \_[q ()]{} e\^[W(x)]{} . Recall in Definition \[loc-pol\] 2) that $z_{\sharp, i} = z_i \cdot (-\hbar^{-1/2}) = z_{\epsilon, i} ({\mathbf{p}}) \cdot (q\hbar^{-1})^{\epsilon (i)}$, where $\epsilon (i) = 1$ or $0$ for $i\in {\mathcal{A}}_{\mathbf{p}}^+$ or ${\mathcal{A}}^- \cup {\mathbf{p}}$ respectively. Note that unlike $z_\epsilon$, $z_\sharp$ is independent of ${\mathbf{p}}$.
\[q-diff-operator\] The $q$-shift operators act as: $$Z_i \widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}=
\int_{q \gamma ({\mathbf{p}})} x_i \cdot e^{W(x)} , \qquad
A_i^{-1} \widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}=
(q\hbar^{-1}) z_{\sharp, i}^{-1} \int_{q \gamma ({\mathbf{p}})} \Big( \frac{1 - x_i^{-1} }{1 - q \hbar^{-1} x_i^{-1} } \Big) \cdot e^{W(x)} ,$$ $$A_i \widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}=
z_{\sharp, i} \int_{q \gamma ({\mathbf{p}})} \Big( \frac{1 - \hbar x_i }{1 - q x_i } \Big) \cdot e^{W(x)} .$$ In particular, all these $q$-shift operators commute with each other.
The only nontrivial actions are those of $A_i$, $i\not\in {\mathbf{p}}$, since they not only act on the integrand, but also shift the contour $q \gamma({\mathbf{p}})$. Let’s consider the case $i\in {\mathcal{A}}_{\mathbf{p}}^+$; the case $i\in {\mathcal{A}}_{\mathbf{p}}^-$ is similar.
Recall that the contour $q \gamma ({\mathbf{p}})$ encloses poles of the integrand, described as in (\[poles\]). The operator $A_i^{-1}$ for $i\in {\mathcal{A}}_{\mathbf{p}}^+$ shifts to a contour $A_i^{-1} (q \gamma ({\mathbf{p}}) )$. The poles enclosed by $A_i^{-1} (q \gamma ({\mathbf{p}}))$ sasitsfy the same conditions as in (\[poles\]), except for $i$: $x_i = q^{1 + d_i}$, $d_i \geq 0$. Note that the points with $x_i = q$ are no longer poles of the integrand. Thus it does no harm to change the contour back to $q \gamma ({\mathbf{p}})$. We then have $$A_i^{-1} \widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}= z_{\epsilon, i} ({\mathbf{p}})^{-1} \int_{q \gamma ({\mathbf{p}})} \Big( \frac{1 - x_i^{-1}}{1 - q \hbar^{-1} x_i^{-1}} \Big) \cdot e^{W(x)} .$$ The action of $A_i$ follows directly.
\[q-diff-eqn\]
1) The modified vertex function $\widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}$ is annihilated by the following $q$-difference operators: $$\label{q-diff-Z}
\prod_{i\in S^+} ( 1 - Z_i ) \prod_{i\in S^-} ( 1 - \hbar Z_i ) - z_\sharp^\beta \prod_{i\in S^+} ( 1 - \hbar Z_i ) \prod_{i\in S^-} ( 1 - Z_i ) , \qquad S = S^+ \sqcup S^- : \text{circuit},$$ where $z_{\sharp, i}:= z_i (-\hbar^{-1/2})$, $\beta$ is the curve class corresponding to $S$, and $z_\sharp^\beta := \prod_{i\in S^+} z_{\sharp, i} \prod_{i\in S^-} z_{\sharp, i}^{-1}$.
2) The modified vertex function $\widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}\cdot e^{-\sum_{i=1}^n \frac{\ln z_{\sharp, i} \ln a_i}{\ln q} }$ is annihilated by the following $q$-difference operators: $$\label{q-diff-A}
\prod_{i\in R^+} (1- A_i) \prod_{i\in R^-} (1 - q\hbar^{-1} A_i ) - (\hbar a)^\alpha \prod_{i\in R^+} (1 - q\hbar^{-1} A_i ) \prod_{i\in R^-} (1 - A_i ) , \qquad R = R^+ \sqcup R^- : \text{cocircuit},$$ where $\alpha$ is the root corresponding to $R$, and $(\hbar a)^\alpha := \prod_{i\in R^+} (\hbar a_i) \prod_{i\in R^-} (\hbar a_i)^{-1}$.
Let ${\mathcal{M}}_1$ be the left ${\mathcal{D}}_q$-module generated by $\widetilde V \big|_{\mathbf{p}}$. We know that $\widetilde V \big|_{\mathbf{p}}$ only depends on the non-redundant equivariant parameters $\alpha_i ({\mathbf{p}}) := a_i \prod_{j\not\in {\mathbf{p}}} a_j^{-C_{ij}}$, $i\in {\mathbf{p}}$. So for each circuit $S = S^+ \sqcup S^-$, the operator $\prod_{i\in S^+} A_i \prod_{i\in S^-} A_i^{-1}$ acts as identity in ${\mathcal{M}}_1$. In other words, on ${\mathcal{M}}_1$ we have $$\prod_{i\in S^+} A_i = \prod_{i\in S^-} A_i , \qquad \forall j\not\in V.$$ On the other hand, for any $i$, the relation between $Z_i$ and $A_i$’s (in ${\mathcal{M}}_1$) is $( 1 - q Z_i ) z_{\sharp, i}^{-1} A_i = 1 - \hbar Z_i$, or equivalently (using $q Z_i z_{\sharp, i}^{-1} = z_{\sharp, i}^{-1} Z$) $$\label{ZA}
z_{\sharp, i}^{-1} (1 - Z_i) A_i = 1 - \hbar Z_i ,$$ Therefore, we have in ${\mathcal{M}}_1$, \_[iS\^+]{} ( 1 - Z\_i ) \_[iS\^-]{} ( 1 - Z\_i ) &=& \_[iS\^-]{} z\_[, i]{}\^[-1]{} \_[iS\^+ S\^-]{} ( 1 - Z\_i ) \_[iS\^-]{} A\_i\
&=& \_[iS\^-]{} z\_[,i]{}\^[-1]{} \_[iS\^+ S\^-]{} ( 1 - Z\_i ) \_[iS\^+]{} A\_i\
&=& z\_\^\_[iS\^+]{} ( 1 - Z\_i ) \_[iS\^-]{} ( 1 - Z\_i ) , We obtain 1).
For 2), let ${\mathcal{M}}_2$ be the ${\mathcal{D}}_q$-module generated by $\widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}\cdot e^{-\sum_{i=1}^n \frac{\ln z_{\sharp, i} \ln a_i}{\ln q} }$. There is an isomorphism ${\mathcal{M}}_1 \cong {\mathcal{M}}_2$, sending $Z_i \mapsto a_i Z_i$, $A_i \mapsto z_{\sharp, i} A_i$. One can check that $\sum_{i=1}^n \ln z_{\epsilon, i} ({\mathbf{p}}) \ln x_i |_{\mathbf{p}}- \sum_{i=1}^n \ln z_{\epsilon, i} ({\mathbf{p}}) \ln a_i = -\sum_{j\in {\mathcal{A}}_{\mathbf{p}}^+ } \ln \zeta_{\epsilon, j} ({\mathbf{p}}) \ln a_j - \sum_{j\in {\mathcal{A}}_{\mathbf{p}}^- } \ln \zeta_{\epsilon, j} ({\mathbf{p}}) \ln (\hbar a_j)$, and hence for each cocircuit $R$, the operators $\prod_{i\in R^+} Z_i \prod_{i\in R^-} Z_i^{-1}$ act as identity in ${\mathcal{M}}_2$. On the other hand, by (\[ZA\]), we have $1- A_i = (\hbar a_i) (1 - q\hbar^{-1} A_i ) Z_i$. By similar arguments, we obtain 2).
The vertex functions can be uniquely characterized as solutions of the $q$-difference equations (with respect to either Káhler or equivariant parameters), with prescribed asymptotes.
\[uniqueness\] Given ${\mathbf{p}}\in X^\TT$, the function $\widetilde V^{({\mathbf{1}})} (q, z) \big|_{\mathbf{p}}$ is the unique solution of the $q$-difference system (\[q-diff-Z\]), with asymptotic behavior $$\widetilde V^{({\mathbf{1}})} (q, z) \big|_{\mathbf{p}}\sim e^{\sum_{i=1}^n \frac{\ln z_{\epsilon, i} ({\mathbf{p}}) \ln x_i |_{\mathbf{p}}}{\ln q} } \cdot \prod_{i\in {\mathbf{p}}} \frac{\phi (q x_i |_{\mathbf{p}})}{\phi ( \hbar x_i |_{\mathbf{p}})} \cdot (1 + o(\zeta ({\mathbf{p}}) ) ) ,$$ as $\zeta \xrightarrow{\theta} 0$, where the limit is explained in Remark \[limit-point\].
By the standard approach, the higher order $q$-difference system (\[q-diff-Z\]) can be written as a first-order holonomic $q$-difference system, of rank ${\operatorname{rk}}K_\TT (X)$. The existence and uniqueness of the solution follows from discussions in [@Aom; @FR; @AOelliptic].
Relations for PSZ and divisorial quantum $K$-theory
---------------------------------------------------
The PSZ quantum $K$-theory ring we introduced in Section \[QK\] can be explicitly determined by the $q$-difference equations \[q-diff-Z\] with respect to Kähler parameters.
\[PSZ-relations\] We have the following presentations of ring structures (which are equivalent to each other):
1) The PSZ quantum $K$-theory ring of $X$ is generated by the quantum tautological line bundles $\widehat L_i (z)$, $1\leq i\leq n$, up to the relations $$\prod_{i\in S^+} ( 1 - \widehat L_i (z) ) * \prod_{i\in S^-} ( 1 - \hbar \widehat L_i (z) ) - z_\sharp^\beta \prod_{i\in S^+} ( 1 - \hbar \widehat L_i (z) ) * \prod_{i\in S^-} ( 1 - \widehat L_i (z) ) , \qquad S = S^+ \sqcup S^- : \text{circuit},$$ where $z_{\sharp, i}:= z_i (-\hbar^{-1/2})$, $\beta$ is the curve class corresponding to $S$, $z_\sharp^\beta := \prod_{i\in S^+} z_{\sharp, i} \prod_{i\in S^-} z_{\sharp, i}^{-1}$, and all products $\prod$ are quantum products $*$.
2) The divisorial quantum $K$-theory ring of $X$ is generated by the line bundles $L_i$, $1\leq i\leq n$, up to the relations $$\prod_{i\in S^+} ( 1 - L_i ) \prod_{i\in S^-} ( 1 - \hbar L_i ) - z_\sharp^\beta \prod_{i\in S^+} ( 1 - \hbar L_i ) \prod_{i\in S^-} ( 1 - L_i ) , \qquad S = S^+ \sqcup S^- : \text{circuit},$$ where $z_{\sharp, i}:= z_i (-\hbar^{-1/2})$, $\beta$ is the curve class corresponding to $S$, and $z_\sharp^\beta := \prod_{i\in S^+} z_{\sharp, i} \prod_{i\in S^-} z_{\sharp, i}^{-1}$.
Recall that in Lemma \[q-diff-operator\], the action of the $q$-difference operator $Z_i$ on the bare vertex function is $$Z_i \widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}=
\int_{q \gamma ({\mathbf{p}})} x_i \cdot e^{W(x)} = \widetilde V^{(x_i)} (q,z) \big|_{\mathbf{p}}.$$ In general, let $\tau (x_1, \cdots, x_n)$ be a Laurent polynomial in $x_1, \cdots, x_n$, with coefficients in $K_{{\mathsf{T}}\times \CC_q^*} ({\operatorname{pt}})$. We have $$\tau (Z_1, \cdots, Z_n) \widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}= \widetilde V^{(\tau)} (q,z) \big|_{\mathbf{p}}.$$ Now suppose that for some $\tau$, and for any ${\mathbf{p}}\in X^{\mathsf{T}}$, we have $\tau (Z_1, \cdots, Z_n) \widetilde V^{({\mathbf{1}})} (q,z) \big|_{\mathbf{p}}= \widetilde V^{(\tau)} (q,z) \big|_{\mathbf{p}}= 0$. It follows that $V^{(\tau)} (q,z) = 0$, and hence $\widehat V^{(\tau)}(q,z) = \Psi(q,z) \cdot V^{(\tau)}(q,z) = 0$ by Proposition \[capping-eqn\]. Evaluating at $q=1$, we have $$\tau (\widehat L_1 (z), \cdots, \widehat L_n (z) ) = \widehat\tau (z) = 0,$$ by Lemma \[quantum-prod\], where for products in $\tau$ on the LHS we take the quantum product $*$. The theorem then follows from equation (\[q-diff-Z\]).
The result can also be obtained following the approach in [@PSZ]. The quantum $K$-theory relations here can be interpreted as Bethe-ansatz equations.
3d mirror symmetry for hypertorics
==================================
Abelian mirror construction
---------------------------
To construct the dual of the hypertoric variety $X$, consider the dual of the sequence (\[knd-seq\]): $$\label{knd-seq-dual}
\xymatrix{
0 \ar[r] & \ZZ^d \ar[r]^{\iota'} & \ZZ^n \ar[r]^{\beta'} & \ZZ^k \ar[r] & 0,
}$$ where $\iota' = \beta^\vee$, and $\beta' = \iota^\vee$.
With given stability parameter $\widetilde\theta$ and chamber parameter $\widetilde\sigma$ of $X$, we choose the stability and chamber parameter for the mirror as $$\label{mirror-theta-sigma}
\widetilde\theta' = - \widetilde\sigma, \qquad \widetilde\sigma' = - \widetilde\theta.$$
We now view the dual sequence (\[knd-seq-dual\]) as the defining sequence for a new hypertoric variety, denote by $X'$. We define $X'$ as the *3d mirror* of the hypertoric variety $X$. Denote by ${\mathsf{K}}'$, ${\mathsf{T}}'$, ${\mathsf{A}}'$ the corresponding tori, and by ${\mathcal{H}}'$ the hyperplane arrangment. Let $L'_i$ be the tautological line bundle defined by the $i$-th standard basis vector in $\ZZ^n$, and let $x'_i$ be its $K$-theory class. Let $a'_i$, $z'_i$, $s'_i$ be the equivariant parameters, Kähler parameters and Chern roots for $X'$ respectively.
Let ${\mathbf{p}}\subset \{1, \cdots, n\}$ be the subset corresponding to a vertex in ${\mathcal{H}}$. Take ${\mathbf{p}}' := {\mathcal{A}}_{\mathbf{p}}= \{ 1, \cdots, n \} \backslash {\mathbf{p}}$.
${\mathbf{p}}'$ is a vertex in the dual hyperplane arrangement ${\mathcal{H}}'$.
Recall that the hyperplane $H_i$ has normal vector $a_i = \beta (e_i) \in \RR^d$. The fact that ${\mathbf{p}}= \bigcap_{i\in {\mathbf{p}}} H_i \neq \emptyset$ is equivalent to the linear independence of the vectors $\{ a_i \mid i\in {\mathbf{p}}\}$. In particular, in the standard ${\mathbf{p}}$-frame, this is equivalent to the fact that the matrix $\beta$ is of the form $(-C, I)$, where the identity submatrix $I$ is for the columns $\{i \in {\mathbf{p}}\}$. Also, the matrix $\iota$ is of the form $\begin{pmatrix}
I \\
C
\end{pmatrix}$, where $I$ is for the rows $\{ j \not\in {\mathbf{p}}\}$.
Now we look at the dual picture. The matrix $\beta' = \iota^\vee$ is of the form $(I, C^T)$, whose columns indexed by $\{j \in {\mathbf{p}}' \} = \{j\not\in {\mathbf{p}}\}$ are linearly independent. Therefore, ${\mathbf{p}}'$ is a vertex in ${\mathcal{H}}'$.
As a result, we have the following natural bijection between the fixed point sets: $$\label{bj-fixed}
\textsf{bj}: X^\TT \xrightarrow{\sim} (X')^{\TT'}, \qquad {\mathbf{p}}\mapsto {\mathbf{p}}'.$$
For a given ${\mathbf{p}}$, if we choose the standard ${\mathbf{p}}$-frame (\[V-frame-i\]) (\[V-frame-b\]), the dual variety $X'$ will also be in the standard ${\mathbf{p}}'$-frame. More precisely, this means $$\iota'_{li} ({\mathbf{p}}') = \delta_{li}, \qquad \iota'_{ji} ({\mathbf{p}}') = - C_{ij} ({\mathbf{p}}), \qquad l, i \not\in {\mathbf{p}}', \ j \in {\mathbf{p}}',$$ $$\beta'_{ji} ({\mathbf{p}}') = C_{ij} ({\mathbf{p}}), \qquad \beta'_{jm} ({\mathbf{p}}') = \delta_{jm}, \qquad i \not\in {\mathbf{p}}', \ j, m \in {\mathbf{p}}'.$$
Recall that the choice of $\widetilde\theta$ determines the splitting ${\mathcal{A}}_{\mathbf{p}}= {\mathcal{A}}_{\mathbf{p}}^+ \sqcup {\mathcal{A}}_{\mathbf{p}}^-$, and the choice of $\widetilde\sigma$ determines the splitting ${\mathbf{p}}= {\mathbf{p}}^+ \sqcup {\mathbf{p}}^-$. The stability and chamber parameters on the mirror side, specified by (\[mirror-theta-sigma\]), also determines the similar splittings of ${\mathcal{A}}_{{\mathbf{p}}'} = {\mathbf{p}}$ and ${\mathbf{p}}' = {\mathcal{A}}_{\mathbf{p}}$.
\[V-V’\] We have $${\mathcal{A}}_{{\mathbf{p}}'}^{\pm} = {\mathbf{p}}^\mp, \qquad ({\mathbf{p}}')^\pm = {\mathcal{A}}_{\mathbf{p}}^\mp .$$
The vertex ${\mathbf{p}}= (v_i)_{i\in {\mathbf{p}}} \in \RR^d$ is the unique solution to the equations $\langle {\mathbf{p}}, \beta (e_i) \rangle = - \langle \widetilde\theta, e_i \rangle$, $i\in {\mathbf{p}}$. In particular, in the standard ${\mathbf{p}}$-frames, the unique solution is $v_i = -\widetilde\theta_i$, $i\in {\mathbf{p}}$.
Recall that by definition, ${\mathcal{A}}_{\mathbf{p}}^+ \subset {\mathbf{p}}'$ consists of those $j\in {\mathbf{p}}'$ for which ${\mathbf{p}}\in H_j^+$. This means that $\langle \beta^* ({\mathbf{p}}), e_j \rangle = \langle {\mathbf{p}}, \beta (e_j) \rangle > - \widetilde\theta_j$. Therefore, in the standard ${\mathbf{p}}$-frame, the inequalities become $$\sum_{i\in {\mathbf{p}}} (-C_{ij}) v_i = \sum_{i\in {\mathbf{p}}} C_{ij} \widetilde\theta_i > - \widetilde\theta_j.$$ Applying the mirror construction $\widetilde\sigma' = - \widetilde\theta$, we have $$\langle a'_j \prod_{i\in {\mathbf{p}}} (a'_i)^{C_{ij}} , \widetilde\sigma' \rangle < 0,$$ which is exactly the condition (\[split-V\]) characterizing $({\mathbf{p}}')^-$. Hence we’ve proved that ${\mathcal{A}}_{\mathbf{p}}^+ = ({\mathbf{p}}')^-$, and the others are similar.
The restriction formula of tautological line bundles to a fixed point ${\mathbf{p}}'$ (under the standard ${\mathbf{p}}'$-frame) is \[restriction-V’\] x’\_j |\_[’]{} := . L’\_j |\_[’]{} &=& {
& 1 , && j \_[’]{}\^+\
& \^[-1]{}, && j \_[’]{}\^-\
& a’\_j \_[i’]{} (a’\_j)\^[-C’\_[ji]{}]{} \^[- \_[j \_[’]{}\^-]{} C’\_[ji]{} ]{} . && j ’,
.\
&=& {
& 1 , && j \^-\
& \^[-1]{}, && j \^+\
& a’\_j \_[i]{} (a’\_j)\^[C\_[ij]{}]{} \^[ \_[i \^+]{} C\_[ij]{} ]{} . && j ,
.
Duality of wall-and-chamber structures
--------------------------------------
Recall that in Section \[section-eff\], the space $\RR^k$ of stability conditions of $X$ admits a wall-and-chamber structure. Each circuit $S$ defines a wall $P_S$, and the stability condition $\widetilde\theta$, or essentially its image $\theta = \iota^\vee \widetilde\theta$, specifies the Kähler cone ${\mathfrak{K}}$ in the complement of all walls. The boundary walls of ${\mathfrak{K}}$ form a basis of the effective cone ${\operatorname{Eff}}(X)$.
Similarly in Section \[section-root\], the space $\RR^d$ also admits a wall-and-chamber structure, where each wall $W_\alpha$ is indexed by a cocircuit, or equivalently a root $\alpha$. A generic choice of the chamber parameter $\widetilde\sigma$, or essentially its image $\sigma = \beta \widetilde\sigma$, singles out a chamber ${\mathfrak{C}}$, whose boundary walls give the positive simple roots.
Now let’s consider the dual hypertoric variety $X'$, with the choice of $\widetilde\theta'$ and $\widetilde\sigma'$ specified in (\[mirror-theta-sigma\]). We denote ${\mathfrak{K}}'$ and ${\mathfrak{C}}'$ the chambers determined by the choice. The following duality between the wall-and-chamber structures follows from the definition of circuits and cocircuits.
The wall-and-chamber structures of $X$ and $X'$ are dual to each other. More precisely, this means that
- There is a bijection between the circuits of $X$ and the (negative) cocircuits of $X'$, and vice versa.
- ${\mathfrak{K}}= {\mathfrak{C}}'$, ${\mathfrak{C}}= {\mathfrak{K}}'$. In particular, indecomposable effective curves of $X$ can be identified with negative simple roots of $X'$, and vice versa.
Duality interface and elliptic stable envelopes
-----------------------------------------------
Recall the definition of equivariant and Kähler parameters in Section \[sec-para\]. For a hypertoric variety $X$, equivariant parameters are defined as coordinates on the quotient torus ${\mathsf{A}}= (\CC^*)^d$. On the other hand, by our mirror construction, ${\mathsf{A}}^\vee = {\mathsf{K}}'$ for $X'$. In other words, there is a canonical isomorphism between the equivariant parameters of $X$ and the Kähler parameters of $X'$. We also include the $\CC^*_\hbar$ factor and denote it by $$\label{id-para}
\kappa_{{\operatorname{Stab}}}: \ {\mathsf{K}}^\vee \times {\mathsf{A}}\times \CC^*_\hbar \xrightarrow{\sim} {\mathsf{A}}' \times ({\mathsf{K}}')^\vee \times \CC^*_\hbar, \qquad ( z_i , a_i, \hbar) \mapsto ( a'_i, z'_i, \hbar^{-1}).$$ We see that $\kappa_{{\operatorname{Stab}}}$ is actually induced by the identity map of ${\mathsf{T}}^\vee \times {\mathsf{T}}= {\mathsf{T}}' \times ({\mathsf{T}}')^\vee$, together with the natural interpretation of ${\mathsf{K}}$, ${\mathsf{A}}$, ${\mathsf{K}}^\vee$ and ${\mathsf{A}}^\vee$ as sub-tori or quotient tori of ${\mathsf{T}}$ or ${\mathsf{T}}^\vee$. In particular, in the standard ${\mathbf{p}}$-frame, the isomorphism $\kappa$ can also be written explicitly as $$\zeta_j ({\mathbf{p}}) \mapsto \alpha'_j ({\mathbf{p}}'), \qquad \alpha_i ({\mathbf{p}}) \mapsto \zeta'_i ({\mathbf{p}}'), \qquad \hbar\mapsto \hbar^{-1},$$ where $j\not\in {\mathbf{p}}$, $i\in {\mathbf{p}}$.
By the explicit formula of elliptic stable envelopes, we can write down the formula on the dual side, which by Lemma \[V-V’\] is ’\_[’]{} (’) &=& \_[j (’)\^+]{} ( x’\_j ) \_[j (’)\^-]{} ( x’\_j ) \_[i \_[’]{}\^+]{} \_[i \_[’]{}\^-]{}\
&=& \_[j \_\^-]{} ( x’\_j ) \_[j \_\^+]{} ( x’\_j ) \_[i \^-]{} \_[i \^+]{} . The diagonal elements are $$\left. {\operatorname{Stab}}'_{\sigma'} ({\mathbf{p}}') \right|_{{\mathbf{p}}'} = \prod_{j \in {\mathcal{A}}_{\mathbf{p}}^+} \theta ( x'_j |_{{\mathbf{p}}'} ) \prod_{j \in {\mathcal{A}}_{\mathbf{p}}^-} \theta ( \hbar x'_j |_{{\mathbf{p}}'} )
= (-1)^{|{\mathcal{A}}_{\mathbf{p}}^-|} \Theta (N_{{\mathbf{p}}'}^{'-}) ,$$ whose image under $\kappa^{-1}$ are exactly the denominators in ${\operatorname{Stab}}_\sigma ({\mathbf{p}})$. In other words, we can consider the following normalized version of elliptic stable envelopes \_() &:=& \_() . ’\_[’]{} (’) |\_[’]{}\
&=& \_[i \^+]{} ( x\_i ) \_[i \^-]{} (x\_i ) \_[j ]{} ( x\_j \_j () \^[- \_[i \^+]{} C\_[ij]{}]{} ) .
Consider the product $X \times X'$, viewed as a ${\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*$-variety, and the following equivariant embeddings: $$\xymatrix{
X = X \times \{ {\mathbf{p}}' \} \ar@{^{(}->}[r]^-{i_{{\mathbf{p}}'}} & X \times X' & \{ {\mathbf{p}}\} \times X' = X' \ar@{_{(}->}[l]_-{i_{\mathbf{p}}} .
}$$ We view $X\times \{{\mathbf{p}}\}$ as a ${\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*$-variety with trivial action on the second factor. Then $${\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*} (X \times \{ {\mathbf{p}}' \} ) = {\operatorname{Ell}}_\TT (X) \times {\mathscr{E}}_{{\mathsf{T}}'} = \textsf{E}_\TT (X),$$ where we apply the identification (\[id-para\]) ${\mathscr{E}}_{{\mathsf{T}}'} \cong {\mathscr{E}}_{{\mathsf{T}}^\vee}$. Similarly, ${\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*} (\{ {\mathbf{p}}\} \times X' ) = \textsf{E}_{\TT'} (X')$.
Let $\mathfrak{X} := [\mu^{-1} (0) / {\mathsf{K}}]$ be the stacky quotient, and $\mathfrak{X}'$ similarly. The diagram above hence induces the following $$\xymatrix{
\textsf{E}_\TT (X) \ar[r]^-{i_{{\mathbf{p}}'}^*} & {\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*} (X \times X') & \textsf{E}_{\TT'} (X') \ar[l]_-{i_{{\mathbf{p}}}^*} \\
{\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^* \times {\mathsf{T}}^\vee \times ({\mathsf{T}}')^\vee } ({\operatorname{pt}}) \ar[r] & {\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*} (\mathfrak{X} \times \mathfrak{X}' ) \ar[u] & .
}$$
We have the following main theorem.
\[Thm-Stab\]
Under the isomorphism of parameters $$\kappa_{{\operatorname{Stab}}}: \ {\mathsf{K}}^\vee \times {\mathsf{A}}\times \CC^*_\hbar \xrightarrow{\sim} {\mathsf{A}}' \times ({\mathsf{K}}')^\vee \times \CC^*_\hbar, \qquad ( z_i , a_i, \hbar) \mapsto ( a'_i, z'_i, \hbar^{-1}),$$ we have:
1) There is a line bundle $\mathfrak{M}$ on ${\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^*} (X \times X')$ such that $$(i_{{\mathbf{p}}'}^*)^* \mathfrak{M} = \mathfrak{M} ({\mathbf{p}}) , \qquad (i_{\mathbf{p}}^*)^* \mathfrak{M} = \mathfrak{M} ({\mathbf{p}}').$$
2) There is a section $\mathfrak{m}$ of $\mathfrak{M}$, called the “duality interface", such that $$(i_{{\mathbf{p}}'}^*)^* \mathfrak{m} = {\mathbf{Stab}}_{\sigma} ({\mathbf{p}}) , \qquad (i_{\mathbf{p}}^*)^* \mathfrak{m} = {\mathbf{Stab}}'_{\sigma'} (p_{{\mathbf{p}}'}) .$$
3) In the hypertoric case, the duality interface $\mathfrak{m}$ admits a simple explicit form: $$\mathfrak{m} = \prod_{i=1}^n \theta (x_i x'_i) .$$ In particular, it comes from a section of a universal line bundle on the prequotient ${\operatorname{Ell}}_{{\mathsf{T}}\times {\mathsf{T}}' \times \CC_\hbar^* \times {\mathsf{T}}^\vee \times ({\mathsf{T}}')^\vee} ({\operatorname{pt}})$, and does not depend on the choices of $\theta$ or $\sigma$.
It suffices to check 3). We compute $(i_{{\mathbf{p}}'}^*)^* \mathfrak{m}$ by restricting $x'_j$ to the fixed point ${\mathbf{p}}' \in X'$, followed by the change of variables $\kappa$. By (\[restriction-V’\]) we have (i\_[’]{}\^\*)\^\* &=& \_[j\^+]{} (\^[-1]{} x\_j) \_[j\^-]{} (x\_j) \_[j]{} ( a’\_j \_[i ]{} (a’\_i)\^[C\_[ij]{}]{} \^[\_[i\^+]{} C\_[ij]{} ]{} x\_j )\
&& \_[j\^+]{} (x\_j) \_[j\^-]{} ( x\_j) \_[j]{} ( \_j () \^[ - \_[i\^+]{} C\_[ij]{} ]{} x\_j )\
&=& \_(). The computation for $(i_{\mathbf{p}}^*) \mathfrak{m}$ is similar.
\[Cor-Stab\] We have the following symmetry between elliptic stable envelopes: $$\frac{{\operatorname{Stab}}_\sigma ({\mathbf{p}}) |_{\mathbf{q}}}{{\operatorname{Stab}}_\sigma ({\mathbf{q}}) |_{\mathbf{q}}} = \frac{{\operatorname{Stab}}'_{\sigma'} ({\mathbf{q}}') |_{{\mathbf{p}}'}}{{\operatorname{Stab}}'_{\sigma'} ({\mathbf{p}}') |_{{\mathbf{p}}'}},$$ where ${\mathbf{p}}, {\mathbf{q}}\in X^\TT$, and ${\mathbf{p}}', {\mathbf{q}}' \in (X')^{\TT'}$ are fixed points corresponding to each other.
Opposite polarization and vertex function for mirror {#oppo}
----------------------------------------------------
Recall that to define the vertex function, we need to choose a global polarization; to add prefactors and form modified vertex functions, we need to choose a localized polarization. We choose them for the mirror $X'$ as follows.
(i) The global polarization $T_{X'}^{1/2}$ in the definition of vertex functions would be $$T_{X'}^{1/2} = \sum_{i=1}^n \hbar^{-1} L_i^{-1} - \hbar^{-1} {\mathcal{O}}^{\oplus d} = \sum_{i=1}^n \hbar^{-1} x_i^{-1} - d \hbar^{-1} ,$$ which is *opposite*, compared to the choice $T_X^{1/2}$ in (\[Polar-X\]). As a result, the shifted Kähler parameters $z'_\sharp$ are $$z'_{\sharp, i} = z'_i \cdot (-\hbar^{1/2}).$$
(ii) The localized polarization ${\mathsf{Pol}}'_{{\mathbf{p}}'}$ is chosen as $${\mathsf{Pol}}'_{{\mathbf{p}}'} (x') = \sum_{{\mathcal{A}}_{{\mathbf{p}}'}^+ \cup ({\mathbf{p}}')^-} \hbar^{-1} (x'_i)^{-1} + \sum_{{\mathcal{A}}_{{\mathbf{p}}'}^- \cup ({\mathbf{p}}')^+} x'_i - d \hbar^{-1}.$$ The shifted Kähler parameters $z'_\epsilon$ are $$z'_{\epsilon, i} = z'_{\sharp, i} \cdot (q\hbar^{-1})^{- \epsilon' (i)},$$ where $\epsilon' (i) = 0$ for $i\in {\mathcal{A}}_{{\mathbf{p}}'}^+ \cup ({\mathbf{p}}')^-$ , and $\epsilon' (i) = -1$ for $i\in {\mathcal{A}}_{{\mathbf{p}}'}^- \cup ({\mathbf{p}}')^+$.
Under these choices, we restate the characterization for vertex functions of $X'$.
\[uniqueness-X’\] Given ${\mathbf{p}}' \in (X')^{\TT'}$, the modified bare vertex function $( \widetilde V')^{({\mathbf{1}})} (q, z') |_{{\mathbf{p}}'}$ is uniquely characterized by the following.
(i) It is annihilated by the following $q$-difference operators $$\label{q-diff-Z'}
\prod_{i\in (S')^+} ( 1 - Z'_i ) \prod_{i\in (S')^-} ( 1 - \hbar Z'_i ) - (q \hbar^{-1} z'_\sharp)^\beta \prod_{i\in (S')^+} ( 1 - \hbar Z'_i ) \prod_{i\in (S')^-} ( 1 - Z'_i ) ,$$ where $Z'_i$ is the operater $z'_i \mapsto q z'_i$, $S' = (S')^+ \sqcup (S')^-$ runs through all circuits of $X'$.
(ii) It admits the asymptotic behavior $$( \widetilde V')^{({\mathbf{1}})} (q, z') \big|_{{\mathbf{p}}'} \sim e^{\sum_{i=1}^n \frac{\ln z'_{\epsilon, i} ({\mathbf{p}}') \ln x'_i |_{{\mathbf{p}}'}}{\ln q} } \cdot \prod_{i\in {\mathcal{A}}_{{\mathbf{p}}'}^+ \cup ({\mathbf{p}}')^-} \frac{\phi (q\hbar^{-1} (x'_i)^{-1} |_{{\mathbf{p}}'} )}{\phi ( (x'_i)^{-1} |_{{\mathbf{p}}'} )} \prod_{i\in {\mathcal{A}}_{{\mathbf{p}}'}^- \cup {\mathbf{p}}'} \frac{\phi (q x'_i |_{{\mathbf{p}}'} )}{\phi ( \hbar x'_i |_{{\mathbf{p}}'} )} \cdot (1 + o(\zeta' ({\mathbf{p}}') ) ) ,$$ as $\zeta' \xrightarrow{\theta'} 0$, where the limit is explained in Remark \[limit-point\].
Mirror symmetry for $q$-difference equations and vertex functions
-----------------------------------------------------------------
In this subsection, we study the mirror symmetry statement for $q$-difference equations and vertex functions.
Consider the matrix ${\mathfrak{P}}$ whose entries are defined by [^6] $${\mathfrak{P}}_{{\mathbf{q}}, {\mathbf{p}}} := \frac{{\operatorname{Stab}}^\sharp_{\sigma} ({\mathbf{q}}) |_{\mathbf{p}}}{\Theta (T^{1/2}_X \big|_{\mathbf{p}})} \cdot \frac{\Phi ((q-\hbar) T^{1/2}_X \big|_{\mathbf{p}}) }{\kappa_{\mathrm{vtx}} (\Phi ((q-\hbar) {\mathsf{Pol}}'_{{\mathbf{q}}'} |_{{\mathbf{q}}'} ) )} \cdot (-\hbar^{1/2})^{|{\mathbf{q}}^+|} \prod_{i\in {\mathbf{q}}^+} x_i |_{\mathbf{q}},$$ for ${\mathbf{q}}, {\mathbf{p}}\in X^{\mathsf{T}}$, where ${\operatorname{Stab}}^\sharp_{\sigma} ({\mathbf{q}})$ is defined as the stable envelope ${\operatorname{Stab}}_\sigma ({\mathbf{q}})$ with change of variables $\zeta_j ({\mathbf{q}}) \mapsto \zeta_{\sharp, j} ({\mathbf{q}})^{-1}$, and $\kappa_{\mathrm{vtx}}$ is the identification of parameters in the following main theorem.
\[main-theorem\] Under the identification of parameters $$\kappa_{\mathrm{vtx}}: \qquad {\mathsf{K}}^\vee \times {\mathsf{A}}\times \CC^*_\hbar \xrightarrow{\sim} {\mathsf{A}}' \times ({\mathsf{K}}')^\vee \times \CC^*_\hbar, \qquad ( z_{\sharp, i} , a_i, \hbar) \mapsto ( (a'_i)^{-1}, z'_{\sharp, i}, q \hbar^{-1}),$$ the product $$V'(q, z', a') = {\mathfrak{P}}\cdot V (q, z, a) \quad \in \quad K_\TT (X^\TT)$$ forms a global class in $K_{\TT'} (X')$, and coincides with the vertex function $V'(q, z', a')$ of the 3d-mirror $X'$, with the opposite polarization $T^{1/2}_{X'}$.
The following two lemmas provide the key incredients in the proof of this main theorem.
\[Criterion-1\] Given ${\mathbf{q}}' \in (X')^{\TT'}$, under the change of variables $\kappa_{\mathrm{vtx}}$, the functions $$\kappa_{\mathrm{vtx}} \Big( e^{\sum_{i=1}^n \frac{\ln z'_{\epsilon, i} ({\mathbf{q}}') \ln x'_i |_{{\mathbf{q}}'}}{\ln q}} \Phi ((q-\hbar) {\mathsf{Pol}}'_{{\mathbf{q}}'} |_{{\mathbf{q}}'} ) \Big) \cdot ( {\mathfrak{P}}\cdot V(q,z,a) ) \big|_{\mathbf{q}}$$ are annihilated by the $q$-difference operators (\[q-diff-Z’\]) for the mirror $X'$.
Explicit computation shows that the $q$-shifts of $$\dfrac{{\operatorname{Stab}}^\sharp_{\sigma} ({\mathbf{q}}) }{\Theta ( {\mathsf{Pol}}_{\mathbf{p}})} = \prod_{i \in {\mathbf{q}}^+} \frac{\theta ( \hbar x_i )}{ \theta ( x_i )} \prod_{j \in {\mathcal{A}}_{\mathbf{q}}^+} \dfrac{\theta \Big( x_j \zeta_{\sharp, j} ({\mathbf{q}})^{-1} \hbar^{-\sum_{i \in {\mathbf{q}}^+} C_{ij}} \Big) }{ \theta ( x_j ) \theta \Big( \zeta_{\sharp, j} ({\mathbf{q}})^{-1} \hbar^{- \sum_{i \in {\mathbf{q}}^+} C_{ij}} \Big) } \prod_{j \in {\mathcal{A}}_{\mathbf{q}}^-} \dfrac{\theta \Big( x_j \zeta_{\sharp, j} ({\mathbf{q}})^{-1} \hbar^{- \sum_{i \in {\mathbf{q}}^+} C_{ij}} \Big) }{ \theta ( x_j ) \theta \Big( \hbar^{-1} \zeta_{\sharp, j} ({\mathbf{q}})^{-1} \hbar^{ -\sum_{i \in {\mathbf{q}}^+} C_{ij} } \Big) } \prod_{i\in {\mathcal{A}}_{\mathbf{p}}^+} \frac{\theta (x_i) }{ \theta (\hbar^{-1} x_i^{-1} ) } ,$$ with respect to variables $a_i$, $z_i$ and $s_i$ are the same as $$\exp \Big( \sum_{i=1}^n \frac{\ln z_{\sharp, i} \ln x_i}{\ln q} - \sum_{i=1}^n \frac{\ln z_{\sharp, i} \ln x_i |_{\mathbf{q}}}{\ln q} - \sum_{i\in {\mathbf{q}}^+} \frac{\ln x_i |_{\mathbf{q}}\ln \hbar}{\ln q} + \sum_{i\in {\mathcal{A}}_{\mathbf{p}}^+} \frac{\ln x_i \ln \hbar}{\ln q} \Big).$$ Therefore ${\mathfrak{P}}\cdot V(q,z, a)$ satisfies the same $q$-difference equations (with respect to $z_i$’s and $a_i$’s) as $$\sum_{\mathbf{p}}e^{ \sum_{i=1}^n \frac{\ln z_{\epsilon, i} ({\mathbf{p}}) \ln x_i |_{\mathbf{p}}}{\ln q} - \sum_{i=1}^n \frac{\ln z_{\epsilon, i} ({\mathbf{q}}) \ln x_i |_{\mathbf{q}}}{\ln q} - \sum_{i\in {\mathbf{q}}^+} \frac{\ln x_i |_{\mathbf{q}}\ln \hbar}{\ln q} } \cdot \prod_{i\in {\mathbf{q}}^+} x_i |_{\mathbf{q}}\cdot \prod_{i\in {\mathbf{p}}} \frac{\phi ( q x_i |_{\mathbf{p}})}{\phi ( \hbar x_i |_{\mathbf{p}})} \cdot V (q,z,a) \big|_{\mathbf{p}}$$ where we have used the observation that $\sum_{i=1}^n \ln z_{\epsilon, i} ({\mathbf{p}}) \ln x_i |_{\mathbf{p}}= \sum_{i=1}^n \ln z_{\sharp, i} \ln x_i |_{\mathbf{p}}$. Hence, by Lemma \[id-prime\], we have $$\kappa_{\mathrm{vtx}} \Big( e^{\sum_{i=1}^n \frac{\ln z'_{\epsilon, i} ({\mathbf{q}}') \ln x'_i |_{{\mathbf{q}}'}}{\ln q}} \Phi ((q-\hbar) {\mathsf{Pol}}'_{{\mathbf{q}}'} |_{{\mathbf{q}}'} ) \Big) \cdot ( {\mathfrak{P}}\cdot V(q,z,a) ) \big|_{\mathbf{q}}$$ satisfies the same $q$-difference equations as && e\^[\_[i=1]{}\^n - \_[i=1]{}\^n - \_[i\^+]{} ]{} \_[i\^+]{} x\_i |\_V (q, z, a)\
&=& e\^[- \_[i=1]{}\^n - \_[i\^+, j\_\^-]{} C\_[ij]{} ]{} V (q, z, a). By Theorem \[q-diff-eqn\], it satisfies the equation (\[q-diff-A\]), which under the change of variables $\kappa_{\mathrm{vtx}}$, is the same as the equation (\[q-diff-Z’\]).
\[Criterion-2\] Under the change of variables $\kappa_{\mathrm{vtx}}$, the functions $( {\mathfrak{P}}\cdot V(q,z,a) ) \big|_{\mathbf{q}}$ admit the following asymptotic behavior $$( {\mathfrak{P}}\cdot V(q,z,a) ) \big|_{\mathbf{q}}\sim (1 + o(\alpha({\mathbf{q}})))$$ as $\alpha ({\mathbf{p}}) \xrightarrow{\sigma} 0$ (see Appendix \[limit-point-a\]).
We first consider the asymptotic behavior of the function along a generic *$q$-geometric progression* of the form $$\alpha_i ({\mathbf{q}}) = w_i q^{\pm N}, \qquad i\in {\mathbf{q}}^\pm, \qquad N\to \infty .$$ Consider $\kappa_{\mathrm{vtx}} \Big( \Phi ((q-\hbar) {\mathsf{Pol}}'_{{\mathbf{q}}'} |_{{\mathbf{q}}'} ) \Big) \cdot ({\mathfrak{P}}\cdot V(q,z,a) ) \big|_{\mathbf{q}}$, which is $$\sum_{{\mathbf{p}}\in X^\TT} {\operatorname{Cont}}_{\mathbf{p}}:= (-\hbar^{1/2})^{|{\mathbf{q}}^+|} \prod_{i\in {\mathbf{q}}^+} x_i |_{\mathbf{q}}\cdot \sum_{{\mathbf{p}}\in X^\TT} \frac{ {\operatorname{Stab}}^\sharp_{\sigma} ({\mathbf{q}}) |_{\mathbf{p}}}{\Theta (T_X^{1/2} |_{\mathbf{p}}) } \cdot \Phi ((q-\hbar) T^{1/2}_X |_{\mathbf{p}}) \cdot V(q,z,a) \big|_{\mathbf{p}}.$$ We claim that along any generic $q$-geometric progression of the above form, the summation over ${\mathbf{p}}$ is *dominated by the diagonal term*. Let’s analyze the contributions from each fixed point ${\mathbf{p}}$ in more details.
**Case 1**: ${\mathbf{p}}\neq {\mathbf{q}}$.
Let ${\mathbf{p}}\in X^\TT$ be such that ${\mathbf{q}}\neq {\mathbf{p}}$. It’s clear that the matrix ${\mathfrak{P}}_{{\mathbf{q}}, {\mathbf{p}}}$ vanishes unless ${\mathbf{q}}\in \overline{{\operatorname{Attr}}_\sigma ({\mathbf{p}})}$. The contribution from ${\mathbf{p}}$, viewed as a function in terms of $a_i$’s and $z_i$’s, is a product of factors of the following three types (using the identity $\theta (x) = x^{1/2} \phi (qx) \phi (x^{-1})$):
(i) $x_i^{\pm 1} |_{\mathbf{p}}$ or $x_i^{\pm 1} |_{\mathbf{q}}$, $1\leq i\leq n$;
(ii) $\dfrac{\phi (u x_i^{\pm 1} |_{\mathbf{p}}) }{\phi (v x_i^{\pm 1} |_{\mathbf{p}})}$, $1 \leq i\leq n$, where $u$ and $v$ are some monomials of $q$ and $\hbar$;
(iii) $\dfrac{\phi (u \zeta_{\sharp, j} ({\mathbf{q}})^{\mp 1} x_j^{\pm 1} |_{\mathbf{p}}) }{ \phi (x_j^{\pm 1} |_{\mathbf{p}}) \phi (v \zeta_{\sharp, j} ({\mathbf{q}})^{\mp 1} )}$, $j\not\in {\mathbf{q}}$, where $u$ and $v$ are some monomials $q$ and $\hbar$, and $x_j |_{\mathbf{p}}\neq 1$;
(iv) $V(q, z, a) |_{\mathbf{p}}$.
We see that along a generic $q$-geometric progression $\{\alpha_i ({\mathbf{q}}) = w_i q^{\pm N} \}$, as $N\to \infty$, factors of type (i) and (ii), either have finite limits, or have asymptotes of the form $\text{const}\cdot (u/v)^N$. Factor (iv) admits a finite limit by Proposition \[limit-V\]. However, factors of type (iii) have the asymptotic form $( u \zeta_{\sharp, j} ({\mathbf{q}})^{\mp 1} )^N$. Therefore, the contribution from ${\mathbf{p}}$ is asymptotically of the form $${\operatorname{Cont}}_{\mathbf{p}}\sim f(q, \hbar)^N \cdot g(q, \hbar, \zeta_\sharp) \cdot \Big( \prod_{j\in {\mathcal{A}}_{\mathbf{q}}\cap {\mathbf{p}}^+ } \zeta_{\sharp, j} ({\mathbf{p}}) \prod_{j \in {\mathcal{A}}_{\mathbf{q}}\cap {\mathbf{p}}^-} \zeta^{-1}_{\sharp, j} ({\mathbf{q}}) \Big)^N$$ where $f$ is a monomial in $q$ and $\hbar$, only depending on $q$ and $\hbar$, and independent of the initial point $w_i$, and $g$ is a function depending only on $q$, $\hbar$ and $\zeta_\sharp$. Now as ${\mathbf{p}}\neq {\mathbf{q}}$, we always have ${\mathcal{A}}_{\mathbf{q}}\cap {\mathbf{p}}\neq \emptyset$, and one can uniformly choose $\zeta_{\sharp, j} ({\mathbf{p}})$ sufficiently small (or large, depending on $j\in {\mathcal{A}}_{\mathbf{q}}\cap {\mathbf{p}}^\pm$), such that the asymptote above goes to $0$, as $N\to \infty$.
In other words, for ${\mathbf{q}}\neq {\mathbf{p}}$, locally in some open subset $U({\mathbf{q}})$ of the domain of $\zeta_\sharp$, the limit of ${\operatorname{Cont}}_{\mathbf{p}}({\mathfrak{P}}\cdot V(q,z,a) ) \big|_{\mathbf{q}}$ along any generic $q$-geometric progression is $0$. Note that as ${\mathbf{q}}\in \overline{{\operatorname{Attr}}_\sigma ({\mathbf{p}})}$, ${\mathcal{A}}_{\mathbf{q}}\cap {\mathbf{p}}^\pm = {\mathcal{A}}_{\mathbf{q}}^\pm \cap {\mathbf{p}}^\pm$ by Lemma \[Attr\], and one can always take the open subset $U({\mathbf{q}})$ such that it does not depend on ${\mathbf{p}}$.
**Case 2**: ${\mathbf{p}}= {\mathbf{q}}$.
We can compute the diagonal contribution explicitly. Recall that ${\operatorname{Stab}}^\sharp_\sigma ({\mathbf{q}}) |_{\mathbf{q}}= (-1)^{|{\mathbf{q}}^+|} \Theta (N_{\mathbf{q}}^-)$. Hence \_&=& (-1)\^[|\^+|]{} (-\^[1/2]{})\^[|\^+|]{} \_[i\^+]{} x\_i |\_ ((q-) T\^[1/2]{}\_X |\_) V(q,z,a) |\_\
&=& (-\^[1/2]{})\^[|\^+|]{} \_[i\^+]{} x\_i \_[i\^+]{} \_[i]{} V(q, z, a) |\_\
&=& \_[i\^+]{} \_[i\^-]{} V(q, z, a) |\_. As $\alpha({\mathbf{q}}) \xrightarrow{\sigma} 0$, by Proposition \[limit-V\], it has the limit $\lim_{\alpha({\mathbf{q}}) \xrightarrow{\sigma} 0} V(q,z,a) |_{\mathbf{q}}= \kappa_{\mathrm{vtx}} \Big( \Phi ((q-\hbar) {\mathsf{Pol}}'_{{\mathbf{q}}'} |_{{\mathbf{q}}'} ) \Big)$.
Now let’s go back to the lemma, and consider $\sum_{{\mathbf{p}}\in X^\TT} {\operatorname{Cont}}_{\mathbf{q}}$. We will apply the holomorphicity results from Aganagic–Okounkov [@AOelliptic]. By Theorem 5 of [@AOelliptic], the function $( {\mathfrak{P}}\cdot V(q, z, a) ) |_{\mathbf{q}}$ is holomorphic with respect to $a_i$’s in a *punctured* neighborhood of the limit point $\alpha ({\mathbf{p}}) \xrightarrow{\sigma} 0$. On the other hand, within the region $\zeta_\sharp \in U({\mathbf{q}})$, comibing Case 1 and 2, we see that it has a finite limit $\kappa_{\mathrm{vtx}} \Big( \Phi ((q-\hbar) {\mathsf{Pol}}'_{{\mathbf{q}}'} |_{{\mathbf{q}}'} ) \Big)$ along any $q$-geometric progression. The lemma then follows from Riemann extension theorem and analytic continuation.
To prove the theorem, it suffices to check that $e^{\sum_{i=1}^n \frac{\ln z'_{\epsilon, i} ({\mathbf{p}}') \ln x'_i |_{{\mathbf{p}}'}}{\ln q} } \cdot ({\mathfrak{P}}\cdot V(q,z,a) ) \big|_{\mathbf{q}}$ satisfies the uniqueness criteria for $\widetilde V'$ in Lemma \[uniqueness-X’\]. Now Criterion (i) and (ii) there are respectively checked in Lemma \[Criterion-1\] and \[Criterion-2\]. The theorem follows.
Relationship to Givental’s quantum $K$-theory
=============================================
$K$-theoretic Gromov–Witten theory and $J$-function
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As in the usual Gromov–Witten theory, the $K$-theoretic analogue, introduced by Givental [@Giv-WDVV] and Lee [@Lee], considers the moduli space of stable maps $\overline{\mathcal{M}}_{0, N} (X, \beta)$, parameterizing genus-$0$ stable maps into $X$. The usual GW perfect obstruction theory defines a virtual structure sheaf ${\mathcal{O}}_{\mathrm{vir}}$. Most of the properties of cohomological GW theory can be generalized to $K$-theory, although in an essentially nontrivial way.
In [@Giv1], Givental introduced another variant of $K$-theoretic GW theory, called the *permutation-equivariant* quantum $K$-theory. The idea is to consider the invariants not merely as numbers, but as $S_N$-modules, keeping track on the permutations of marked points. The permutation-equivariant theory turns out to behave much better than the ordinary version of $K$-theory.
Let $X$ be a quasiprojective variety. Let $\Lambda$ be a $\lambda$-algebra [^7] over $\QQ$, which contains the ring of symmetric functions on a certain number of variables, and the Kähler parameters $Q_i$. Let ${\mathbf{t}}(q)$ be a Laurent polynomial in $q$ with coefficients in $K(X) \otimes \Lambda$. The moduli space of stable maps $\overline{\mathcal{M}}_{0, N} (X, \beta)$ admits an action of the group $S_N$, permuting the marked points. The virtual structure sheaf ${\mathcal{O}}_{\mathrm{vir}}$ is equivariant under the $S_N$-action, and hence defines a $K$-theory class on the quotient.
The correlation functions are defined as $$\left\langle {\mathbf{t}}(L), \cdots, {\mathbf{t}}(L) \right\rangle_{0, N, \beta}^{S_N} := \chi \Big( \left[ \overline{\mathcal{M}}_{0, N} (X, \beta) / S_N \right] , {\mathcal{O}}_{\mathrm{vir}}\otimes \prod_{i=1}^N {\mathbf{t}}(L_i) \Big),$$ where $L_i$ is the tautological cotangent line bundle at the $i$-th marked point.
The genus-$0$ invariants are encoded in the *big $J$-function*, defined as $$J ({\mathbf{t}}(q), Q) := 1-q + {\mathbf{t}}(q) + \sum_{i, N\geq 2, \beta} \phi_i \Big\langle {\mathbf{t}}(L), \cdots, {\mathbf{t}}(L) , \frac{\phi^i}{1 - qL} \Big\rangle_{0, N+1}^{S_N} Q^\beta,$$ where $S_N$ only acts on the first $N$ points, $\{\phi_i\}$ is a basis of $K(X)$ and $\{\phi^i\}$ is the dual basis with respect to the Mukai pairing.
The language of the loop space is introduced to describe the range of the big $J$-function. Consider the space ${\mathcal{K}}:= K(X)\otimes \Lambda (q)$, consisting of rational functions in $q$ with coefficients in $K(X) \otimes \Lambda$. ${\mathcal{K}}$ admits a symplectic form $\Omega (f, g) := \left( {\operatorname{Res}}_{q = 0} + {\operatorname{Res}}_{q = \infty} \right) (f (q), g (q^{-1}) ) \frac{dq}{q}$, and a decomposition into Lagrangian subspaces ${\mathcal{K}}= {\mathcal{K}}_+ \oplus {\mathcal{K}}_- = T^* {\mathcal{K}}_+$. Here ${\mathcal{K}}_+$ is the subspace $K(X)\otimes \Lambda [q, q^{-1}]$, and ${\mathcal{K}}_-$ is the subspaces of *reduced rational functions*, i.e., rational functions in $q$, regular as $q \to 0$, and tends to $0$ as $q\to \infty$. The big $J$-function can be viewed naturally as the graph of a function from ${\mathcal{K}}_+$ to ${\mathcal{K}}_-$, with ${\mathbf{t}}(q) \in {\mathcal{K}}_+$, and $1-q$ the *dilaton shift* of the origin. When $X$ admits the action by a torus $\TT$, as before, everything can be made $\TT$-equivariantly.
The aim of this subsection, is to prove the vertex functions $V^{(1)} (q^{-1}, z)$, defined in previous sections for hypertoric varieties, multiplied by $(1-q)$, represents a value of the big $J$-function, up to a certain $q$-shift of Kähler parameters. To show that, we will apply a criteria by Givental which characterize the range of a big $J$-function.
Let $X$ be a GKM variety, with the torus action by $\TT$. We take the $\lambda$-algebra to be $$\Lambda := K_\TT ({\operatorname{pt}}) [[Q^{{\operatorname{Eff}}(X)}]],$$ and let $\Lambda_+$ to be the ideal generated by $1 - a_i^{\pm 1}$, $1 - \hbar^{\pm 1}$ and $Q$. We assume that all coefficients ${\mathbf{t}}_k$ of ${\mathbf{t}}(q)$ are taken in $K_\TT (X) \otimes \Lambda_+$.
When $\Lambda$ does not contain the ring of symmetric functions, following Example 4 of [@Giv1], we will refer to the permutation-equivariant quantum $K$-theory as *symmetrized* quantum $K$-theory. The full permutation-equivariant invariants, when $\Lambda$ includes the ring of symmetric functions, actually contain all information about the pushforward of ${\mathcal{O}}_{\mathrm{vir}}$ along the projection $\left[ \overline{{\mathcal{M}}}_{0,N}(X, \beta) / S_N \right] \to B S_N$.
Let $\{ f^{({\bf p})} \in \Lambda (q) \mid {\bf p} \in X^\TT \}$ be a set of elements. For ${\bf p, q} \in X^\TT$ connected by a 1-dimensional $\TT$-invariant curve $C$, we denote by $\lambda_{\bf p,q}$ the $\TT$-character $T_{\bf p} C$. The proof of the following proposition is the same as in [@Giv2].
\[Givental-cri\] Suppose that $\{ f^{({\bf p})} \in \Lambda (q) \mid {\bf p} \in X^\TT \}$ satiesfy the following two criteria.
(i) For each ${\bf p} \in X^\TT$, considered as a meromorphic function in $q$ with only poles at roots of unity, $f^{({\bf p})}$ represents a value of the big $J$-function $J_{{\operatorname{pt}}}$ of a point target space.
(ii) Outside $q = 0, \infty$ and roots of unity, $f^{({\bf p})}$ may have poles only at $q = \lambda_{\bf p,q}^{1/m}$, $m = 1, 2, \cdots$, with residues $${\operatorname{Res}}\limits_{ q = \lambda_{\bf p,q}^{ 1/m} } f^{({\bf p})} (q) \frac{dq}{q} = - \frac{ Q^{m [C]}}{m E_{\bf p,q} (m)} \cdot f^{( {\bf q} )} ( \lambda_{\bf p, q}^{1/m} ),$$ where $[C]$ is the curve class of the curve $C$, and $$E_{\bf p, q} (m) = \bigwedge^\bullet \left( T_\varphi \overline{\mathcal{M}}_{0,2} (X, m[C]) - T_{\bf p} X \right)^\vee$$ where $\varphi: \PP^1 \to C \subset X$ is the deg-$m$ covering map over $C$, ramified at $0$ and $\infty$.
Then there exists ${\mathbf{t}}(q) \in K_\TT (X) [q, q^{-1}]$, such that $f^{({\bf p})} = \left. J_X ({\mathbf{t}}(q) ) \right|_{\bf p}$, for each ${\bf p} \in X^\TT$. In other words, $f^{({\bf p})}$’s represents a value of the permutation-equivariant big $J$-function $J_X$ of $X$.
\[V\] Let $X$ be a hypertoric variety, and $z_\sharp$ be the $q$-shifted Kähler parameters defined as $z_{\sharp, i} := z_i \cdot (-\hbar^{-1/2})$. The vertex function $$(1-q) V^{({\mathbf{1}})} (q^{-1} , z) \big|_{z_\sharp = Q}$$ represents a value $J(t_0, Q)$ of the big $J$-function of $X$, for some $t_0 \in K_\TT (X) \otimes \Lambda$.
Take ${\mathbf{p}}\in X^\TT$. Explicitly, one can compute that $$V^{({\mathbf{1}})}(q,z) \big|_{\mathbf{p}}= \sum_{\beta \in {\operatorname{Eff}}(X)} z_\sharp^\beta \prod_{i=1}^n \frac{ ( \hbar x_i |_{\mathbf{p}})_{D_i} }{ ( q x_i |_{\mathbf{p}})_{D_i} } ,$$ where $D_i = \beta \cdot L_i$. By similar arguments as in Corollary 1 of [@Giv4], we see that Criterion (i) holds. Indeed, it follows from Theorem in [@Giv2], which describes the big $J$-function of a point, and Lemma in [@Giv4], which gives a set of difference operators that preserves the range of the big $J$-function of a point.
We now check that $V^{({\mathbf{1}})} (q , z)$ satiesfies Criterion (ii), with $q$ replaced by $q^{-1}$. Let ${\mathbf{q}}$ be another vertex in the hyperplane arrangement, such that ${\mathbf{p}}= ({\mathbf{q}}\backslash \{j\} ) \sqcup \{i\}$ and ${\mathbf{q}}= ({\mathbf{p}}\backslash \{i\} ) \sqcup \{j\}$, for some $1\leq i\neq j\leq n$. Let $C$ be the $\TT$-invariant curve connecting ${\mathbf{p}}$ and ${\mathbf{q}}$. Then we have two cases as in Lemma \[bridge\] (ii). For simplicity, we assume that $i\in {\mathcal{A}}_{\mathbf{q}}^+$; the other case $i\in {\mathcal{A}}_{\mathbf{q}}^-$ is similar. We have $\lambda:= \lambda_{{\mathbf{p}}, {\mathbf{q}}} = x_i |_{\mathbf{p}}$.
Let $m\geq 0$ be an integer. Consider the coefficent of $Q^\beta$ in $V^{({\mathbf{1}})} (q,z) |_{\mathbf{p}}$. Its residue at $q = \lambda^{-1/m}$ vanishes unless $D_i \geq m$. In that case, for $\beta \in {\operatorname{Eff}}_{\mathbf{p}}(X)$, we claim that the curve class $\beta - m[C] \in {\operatorname{Eff}}_{\mathbf{q}}(X)$. In fact, for any $l\not\in {\mathbf{p}}\cup {\mathbf{q}}$, we know that in $\RR^d$ the vertices ${\mathbf{p}}$ and ${\mathbf{q}}$ lie on the same side of the hyperplane $H_l$, which implies that ${\mathcal{A}}_{\mathbf{p}}^\pm$ and ${\mathcal{A}}_{\mathbf{q}}^\pm$ only differ by the indices $i$ or $j$. On the other hand, the curve class $[C]$ is defined by the circuit $S_{{\mathbf{p}}{\mathbf{q}}} = {\mathbf{p}}\cup {\mathbf{q}}$; in other words, it is of the form $[C] = e_i + \sum_{l\in {\mathbf{q}}} \epsilon_l e_l$, where $\epsilon_l = \pm 1$, depending on $l \in S_{{\mathbf{p}}{\mathbf{q}}}^\pm$. Therefore, we have $$(\beta - m[C], L_l) = \left\{ \begin{aligned}
& (\beta, L_l ) , && \qquad l \in {\mathcal{A}}_{\mathbf{p}}^\pm \backslash \{j\} = {\mathcal{A}}_{\mathbf{q}}^\pm \backslash \{i\} \\
& D_i - m , && \qquad l = i .
\end{aligned} \right.$$ We see that $\beta - m [ C] \in {\operatorname{Eff}}_{\mathbf{q}}(X)$. The claim holds.
Direct computation shows that $${\operatorname{Res}}_{q = \lambda^{-1/m}} \frac{ ( \hbar x_i |_{\mathbf{p}})_{D_i} }{ ( q x_i |_{\mathbf{p}})_{D_i} } \frac{dq}{q} = \frac{1}{m} \left. \frac{ ( \hbar x_i |_{\mathbf{p}})_{m} }{ ( q x_i |_{\mathbf{p}})_{m} } \right|_{q = \lambda^{-1/m}} \cdot \left. \frac{ ( \hbar x_i |_{\mathbf{q}})_{D_i - m} }{ ( q x_i |_{\mathbf{q}})_{D_i - m} } \right|_{q = \lambda^{-1/m}} ,$$ $$\left. \frac{ ( \hbar x_l |_{\mathbf{p}})_{D_l} }{ ( q x_l |_{\mathbf{p}})_{D_l} } \right|_{q = \lambda^{-1/m}} = \left. \frac{ ( \hbar x_l |_{\mathbf{q}})_{D_l\mp m} }{ ( q x_l |_{\mathbf{q}})_{D_l \mp m} } \right|_{q = \lambda^{-1/m}} \cdot \left. \frac{ ( \hbar x_l |_{\mathbf{p}})_{\pm m} }{ ( q x_l |_{\mathbf{p}})_{\pm m} } \right|_{q = \lambda^{-1/m}}, \qquad l\in S_{{\mathbf{p}}{\mathbf{q}}}^\pm \backslash \{i\},$$ and on the other hand, $$E_{{\mathbf{p}}, {\mathbf{q}}}^{-1} (m) = \prod_{l\in S_{{\mathbf{p}}{\mathbf{q}}}^+} \left. \frac{ ( \hbar x_l |_{\mathbf{p}})_{m} }{ ( q x_l |_{\mathbf{p}})_{m} } \right|_{q = \lambda^{-1/m}} \cdot \prod_{l\in S_{{\mathbf{p}}{\mathbf{q}}}^-} \left. \frac{ ( \hbar x_l |_{\mathbf{p}})_{-m} }{ ( q x_l |_{\mathbf{p}})_{-m} } \right|_{q = \lambda^{-1/m}}.$$ We see that Criterion (ii) is satisfied. The theorem follows from Proposition \[Givental-cri\].
\[V-tau\] Let $\tau$ be a Laurent polynomial in $q$ with coefficients in $K_{\mathsf{T}}(X) \otimes \Lambda$. The descendent bare vertex function $$(1-q) V^{(\tau)} (q^{-1} , z) \big|_{z_\sharp = Q}$$ lies in the range of big $J$-function of $X$.
By the Theorem 2 of explicit reconstruction in [@Giv8], given a point $\sum_\beta I_\beta Q^\beta$ in the rangle ${\mathcal{L}}$ of big $J$-function, the point $$\sum_\beta I_\beta Q^\beta \tau (x_1 q^{D_1}, \cdots, x_n q^{D_n})$$ also lies in ${\mathcal{L}}$. The corollary then follows.
Quantum $K$-theory (in the sense of Givental)
---------------------------------------------
In Section \[QK\], we defined a quantum $K$-theory ring in terms of virtual counting of parameterized quasimaps from $\PP^1$ to $X$. On the other hand, it is standard to define quantum cohomology or $K$-theory ring [@Lee], in terms of genus-zero stable maps into $X$. It is natural to ask whether these two versions of quantum $K$-theories coincide. The question is somehow complicated due to the lack of divisor axiom in $K$-theory, as opposed to the cohomological theory. We will see later that the PSZ quantum $K$-theory is essentially the same as the algebra generated by the $A_i$ operators, introduced in [@IMT], but in general different from Givental’s quantum $K$-theory.
Let’s review the definition of Givental’s quantum $K$-theory ring, which we denote by $\bullet$. Let $X$, $\Lambda$ as in the previous subsection. In [@Giv7], Givental introduced a genus-zero $K$-theoretic GW potential with *mixed inputs*: $${\mathcal{F}}({\mathbf{x}}, {\mathbf{t}}) = \sum_{\beta \in {\operatorname{Eff}}(X)} \sum_{M, N=0}^\infty \left\langle {\mathbf{x}}(L) , \cdots, {\mathbf{x}}(L); {\mathbf{t}}(L) , \cdots, {\mathbf{t}}(L) \right\rangle_{0, M+N, \beta}^{S_N} \frac{Q^\beta}{M!},$$ where ${\mathbf{x}}(q)$, ${\mathbf{t}}(q)$ are Laurent polynomials with coefficients in $K(X) \otimes \Lambda$; only ${\mathbf{t}}$’s are considered as *permutation-equivariant* inputs, and ${\mathbf{x}}$’s are considered as *ordinary* inputs.
The graph of the potential ${\mathcal{F}}({\mathbf{x}}, {\mathbf{t}})$ (up to dilaton shift) therefore defines a *mixed* $J$-function $J({\mathbf{x}}(q), {\mathbf{t}}(q), Q)$. In particular, the permutation-equivariant $J$-function can be recovered by setting ${\mathbf{x}}= 0$.
Let $\{\phi_\alpha\}$ be a basis of $K(X)$. We take constant Laurent polynomial as inputs: ${\mathbf{x}}(q) = x = \sum_\alpha x^\alpha \phi_\alpha$, ${\mathbf{t}}(q) = t \in K(X) \otimes \Lambda$. Given basis elements $\phi_\alpha, \phi_\beta, \phi_\gamma \in K (X)$, the *quantum pairing* is defined as $$G (\phi_\alpha, \phi_\beta) := \frac{\partial^2}{\partial x^\alpha \partial x^\beta} {\mathcal{F}}(x, t).$$ The *quantum product* is determined by the 3-point function of three basis elements $\phi_\alpha, \phi_\beta, \phi_\gamma$ $$G (\phi_\alpha \bullet \phi_\beta, \phi_\gamma ) := \frac{\partial^3}{\partial x^\alpha \partial x^\beta \partial x^\gamma} {\mathcal{F}}(x, t) .$$ By the WDVV equation in [@Giv7], the ring is equipped with a structure of Frobenius algebra, with pairing $G$, product $\bullet$ (both dependent on $x$ and $t$), and identity ${\mathbf{1}}\in K_{\mathsf{T}}(X)$. Similar as in the comological theory, one can define a quantum connection using the quantum product $\bullet$ and then the quantum $K$-ring structure can be packaged in the language of ${\mathcal{D}}$-modules.
For given constant Laurent polynomials $x$ and $t$ as above, there is an operator $S (x, t, q)^{-1}: K(X) \otimes \Lambda \to {\mathcal{K}}_-$, whose inverse is defined as $$S (x, t, q)^{-1} \phi := \phi + \sum_{i, M, N, \beta} \phi_i \Big\langle \phi, x, \cdots, x, t, \cdots, t, \frac{\phi^i}{1 - qL} \Big\rangle_{0, M+N+2}^{S_N} \frac{Q^\beta}{M!}.$$ $S$ is a symplectomorphism, i.e., satisfying $S(q) = S^* (q^{-1})$. In particular, the $J$-function $J(x, t, Q) = (1-q) S (x, t, q)^{-1} {\mathbf{1}}$. The image $S(x, t, q)^{-1} {\mathcal{K}}_+$ is called the *ruling space*, which satisfies the following properties [@Giv7; @Giv8]:
(i) The range ${\mathcal{L}}_{\mathrm{perm}}$ of permutation-equivariant big $J$-functions ${\mathbf{t}}(q) \mapsto J(0, {\mathbf{t}}(q), Q)$ of $X$ is swept by the images of $S(t,q)^{-1}$: $${\mathcal{L}}_{\mathrm{perm}} = \bigcup_{t \in K(X) \otimes \Lambda_+} (1-q) S(0, t, q)^{-1} {\mathcal{K}}_+.$$
(ii) For each fixed $t$, the tangent space of the Lagrangian cone ${\mathcal{L}}_t$ for the ordinary big $J$-function ${\mathbf{x}}(q) \mapsto J ({\mathbf{x}}(q), t, Q)$ is $T_t := S({\mathbf{x}}(q), t, Q)^{-1} {\mathcal{K}}_+$, and tangent to ${\mathcal{L}}_t$ exactly along the subspace $(1-q) T_t$. In particular, ${\mathcal{L}}_t$ is also swept by the union of those ruling spaces.
(iii) Let $Q_i$ be the Kähler parameter with respect to $\widetilde L_i$. Each ruling space admits a ${\mathcal{D}}_q$-module structure under the $q$-difference operators $q^{Q_i \frac{\partial}{\partial Q_i}}$.
The key observation of [@IMT] is that the operator $S(x, t, q)^{-1}$ serves simultaneously as the fundamental solution to a $q$-difference system with respect to $q^{Q_i \partial_{Q_i}}$, and the fundamental solution to a $q$-differential system with respect to the variables $\frac{\partial}{\partial x^\alpha}$. They take the following forms: $$\label{2-systems}
(1-q) \frac{\partial}{\partial x^\alpha} S(x, t, q)^{-1} = S(x, t, q)^{-1} \circ (\phi_\alpha \bullet - ), \qquad L_i^{-1} q^{Q_i \frac{\partial}{\partial Q_i}} \circ S(x, t, q)^{-1} = S(x, t, q)^{-1} \circ B_i q^{Q_i \frac{\partial}{\partial Q_i}} ,$$ where the first equation is by the definition of the quantum product $\bullet$, and the second is obtained from the $q$-difference module structure. Here $B_i \in {\operatorname{End}}K(X) \otimes \Lambda [q, q^{-1}]$ are uniquely characterized by the above equation. The two systems above are compatible to each other, in the sense that the quantum connection and $q$-difference operators commute. In particular, one can define the operators $$B_{i, {\mathrm{com}}} := B_i \big|_{q = 1} \in {\operatorname{End}}K(X) \otimes \Lambda.$$ Now let $X$ be a hypertoric variety. Using the result on the vertex function and $J$-functions in the previous section, together with the $q$-difference equations (\[q-diff-Z\]), we obtain the following result.
\[GivQK-relation\] Let $X$ be a hypertoric variety, and $t_0 \in K_\TT (X) \otimes \Lambda$ be as in Theorem \[V\]. We fix the insertions $x = 0$ and $t = t_0$.
1) For any circuit $S = S^+\sqcup S^-$, and the corresponding curve class $\beta$, the identity class ${\mathbf{1}}\in K_{\mathsf{T}}(X)$ is annihilated by the following operator $$\prod_{i\in S^+} ( 1 - B_{i, {\mathrm{com}}}) \prod_{i\in S^-} (\hbar - B_{i, {\mathrm{com}}} ) - Q^\beta \prod_{i\in S^+} (\hbar - B_{i, {\mathrm{com}}} ) \prod_{i\in S^-} (1 - B_{i, {\mathrm{com}}}),$$ where $Q^\beta := \prod_{i\in S^+} Q_i \prod_{i\in S^-} Q_i^{-1}$.
2) The Givental quantum $K$-theory ring of $X$ is generated by the classes $B_{i, {\mathrm{com}}} {\mathbf{1}}$, $1\leq i\leq n$, up to the following relations: for any circuit $S = S^+\sqcup S^-$, and the corresponding curve class $\beta$ $$\prod_{i\in S^+} ( 1 - B_{i, {\mathrm{com}}} {\mathbf{1}}) \bullet \prod_{i\in S^-} (\hbar - B_{i, {\mathrm{com}}} {\mathbf{1}}) = Q^\beta \prod_{i\in S^+} (\hbar - B_{i, {\mathrm{com}}} {\mathbf{1}}) \bullet \prod_{i\in S^-} (1 - B_{i, {\mathrm{com}}} {\mathbf{1}}),$$ where all the products are the quantum product $\bullet$.
By Theorem \[V\], the $J$-function $J(t_0, Q) = S(0, t_0, q)^{-1} {\mathbf{1}}$ satisfies the $q$-difference equations (\[q-diff-Z\]), with $Z_i$ replaced by $\widetilde L_i q^{-Q_i \partial_{Q_i}}$. By the 2nd equation in (\[2-systems\]), for any Laurent polynomial $f(X_1, \cdots, X_n)$, we have $$S \circ f ( L_1^{-1} q^{Q_1 \partial_{Q_1}} , \cdots, L_n^{-1} q^{Q_n \partial_{Q_n}} ) \circ S^{-1} = f(B_1 q^{Q_1 \partial_{Q_1}}, \cdots, B_n q^{Q_n \partial_{Q_n}} ).$$ Apply operators on both sides to the identity ${\mathbf{1}}$, and take $q\to 1$. We obtain 1).
For 2), it suffices to notice that by compatibility, for any $\alpha$, $i$, one has $B_{i, {\mathrm{com}}} \phi_\alpha = B_{i, {\mathrm{com}}} (\phi_\alpha \bullet {\mathbf{1}}) = \phi_\alpha \bullet (B_{i, {\mathrm{com}}} {\mathbf{1}})$. Hence the operator $B_{i, {\mathrm{com}}}$ is the same as the quantum multiplication $(B_{i, {\mathrm{com}}} {\mathbf{1}}) \bullet (-)$. 2) then follows from 1).
Some identities of Kähler and equivariant parameters
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We list here some computations for Kähler and equivariant parameters. Let ${\mathbf{p}}\in X^\TT$ be a fixed point, and $\iota = \begin{pmatrix}
I \\
C
\end{pmatrix}$ be the matrix in the standard ${\mathbf{p}}$-frame.
$$\sum_{i=1}^n \ln z_i \ln x_i |_{\mathbf{p}}- \sum_{i=1}^n \ln z_i \ln a_i = -\sum_{j\in {\mathcal{A}}_{\mathbf{p}}^+ } \ln \zeta_j ({\mathbf{p}}) \ln a_j - \sum_{j\in {\mathcal{A}}_{\mathbf{p}}^- } \ln \zeta_j ({\mathbf{p}}) \ln (\hbar a_j)$$
Recall the variable $\zeta_j ({\mathbf{p}})$, $j\not\in {\mathbf{p}}$ is defined as $\zeta_j ({\mathbf{p}}) := z_j \prod_{i\in {\mathbf{p}}} z_i^{C_{ij}}$. Then by the restriction formula (\[restriction-V\]), LHS &=& \_[j\_\^-]{} z\_i \^[-1]{} + \_[i]{} z\_i ( a\_i - \_[j\_\^+]{} C\_[ij]{} a\_j - \_[j\_\^-]{} C\_[ij]{} (a\_j) ) - \_[i=1]{}\^n z\_i a\_i\
&=& -\_[j\_\^+]{} a\_j ( z\_j + \_[i]{} C\_[ij]{} z\_i ) - \_[j\_\^-]{} (a\_j) ( z\_j + \_[i]{} C\_[ij]{} z\_i )\
&=& RHS.
Let ${\mathbf{p}}' \in (X')^\TT$ be the corresponding fixed point in the mirror, and $z_\epsilon$, $z'_\epsilon$ be defined as in Definition \[loc-pol\] 2) and Section \[oppo\]. More precisely, $$z_{\epsilon, i} = \left\{ \begin{aligned}
& z_{\sharp, i} \cdot (q\hbar^{-1} )^{-1} , && \ i \in {\mathcal{A}}_{\mathbf{p}}^+ \\
& z_{\sharp, i} , && \ i \in {\mathcal{A}}_{\mathbf{p}}^- \cup {\mathbf{p}}\end{aligned}\right. , \qquad z'_{\epsilon, i} = \left\{ \begin{aligned}
& z'_{\sharp, i} , && \ i \in {\mathcal{A}}_{{\mathbf{p}}'}^+ \cup ({\mathbf{p}}')^- \\
& z'_{\sharp, i} \cdot (q\hbar^{-1} ) , && \ i \in {\mathcal{A}}_{{\mathbf{p}}'}^- \cup ({\mathbf{p}}')^+ .
\end{aligned}\right.$$
\[id-prime\] Under the change of variables $\kappa_{\mathrm{vtx}}$, $$\sum_{i=1}^n \ln z'_{\epsilon, i} ({\mathbf{p}}') \ln x'_i |_{{\mathbf{p}}'} - \sum_{i=1}^n \ln z_{\epsilon, i} ({\mathbf{p}}) \ln x_i |_{\mathbf{p}}= - \sum_{i=1}^n \ln z_{\sharp, i} \ln a_i - \sum_{i\in {\mathbf{p}}^+} \ln (q\hbar^{-1}) \ln x_i |_{\mathbf{p}}+ \sum_{j \in {\mathcal{A}}_{\mathbf{p}}^-, i\in {\mathbf{p}}^+} C_{ij} \ln (q\hbar^{-1} ) \ln \hbar .$$
Compute \_[i=1]{}\^n z\_[, j]{} () x\_i |\_&=& - \_[j\_\^-]{} z\_[, i]{} + \_[i]{} z\_[, i]{} ( a\_i \_[j]{} a\_j\^[-C\_[ij]{}]{} \^[- \_[j\_\^-]{} C\_[ij]{}]{} )\
&=& - \_[j\_\^-]{} z\_[, j]{} + \_[i]{} z\_[, i]{} a\_i - \_[i, j]{} C\_[ij]{} z\_[, i]{} a\_j - \_[i, j\_\^-]{} C\_[ij]{} z\_[, i]{} . On the other hand, since ${\mathcal{A}}_{{\mathbf{p}}'}^\pm = {\mathbf{p}}^\mp$, $({\mathbf{p}}')^\pm = {\mathcal{A}}_{\mathbf{p}}^\mp$, and $C'_{ji} = - C_{ij}$, \_[i=1]{}\^n z’\_[, i]{} () x’\_i |\_[’]{} &=& - \_[i \_[’]{}\^-]{} z’\_[, i]{} + \_[j ’]{} z’\_[, j]{} ( a’\_j \_[i ’ ]{} (a’\_i)\^[-C’\_[ji]{}]{} \^[- \_[i\_[’]{}\^-]{} C’\_[ji]{}]{} )\
&& + \_[j(’)\^+]{} (q\^[-1]{}) ( a’\_j \_[i ’ ]{} (a’\_i)\^[-C’\_[ji]{}]{} \^[- \_[i\_[’]{}\^-]{} C’\_[ji]{}]{} )\
&=& - \_[i\^+]{} z’\_[, i]{} + \_[j]{} z’\_[, j]{} a’\_j + \_[j, i]{} C\_[ij]{} z’\_[, j]{} a’\_i + \_[j, i(’)\^+]{} C\_[ij]{} z’\_[, j]{}\
&& + \_[j\_\^-]{} (q\^[-1]{}) a’\_j + \_[j\_\^-, i]{} C\_[ij]{} (q\^[-1]{}) a’\_i + \_[j \_\^-, i(’)\^+]{} C\_[ij]{} (q\^[-1]{} ) , which under the change of variables $\kappa_{\mathrm{vtx}}$ is && - \_[i \^+]{} a\_i (q\^[-1]{} ) - \_[j]{} a\_j z\_[, j]{} - \_[j, i]{} C\_[ij]{} a\_j z\_[, i]{} + \_[j , i\^+]{} C\_[ij]{} a\_j (q\^[-1]{} )\
&& - \_[j \_\^-]{} z\_[, j]{} - \_[j \_\^-, i ]{} C\_[ij]{} z\_[, i]{} + \_[j \_\^-, i\^+]{} C\_[ij]{} (q\^[-1]{} ) . The lemma follows by direct comparison.
Limit of bare vertex functions {#limit-point-a}
==============================
Let ${\mathbf{p}}\in X^\TT$ be a fixed point, and $\iota = \begin{pmatrix}
I \\
C
\end{pmatrix}$ be the matrix in the standard ${\mathbf{p}}$-frame. By the explicit formula (\[vertex\]), we have the bare vertex function $$V^{({\mathbf{1}})} (q, z, a) |_{\mathbf{p}}= \sum_{\substack{d_j \geq 0, j\in {\mathcal{A}}_{\mathbf{p}}^+ \\ d_j \leq 0, j \in {\mathcal{A}}_{\mathbf{p}}^-} } \zeta ({\mathbf{p}})^d q^{ - \frac{1}{2} \sum_{j\not\in {\mathbf{p}}} d_j - \frac{1}{2} \sum_{i\in {\mathbf{p}}, j\not\in {\mathbf{p}}} C_{ij} d_j } \prod_{j\in {\mathcal{A}}_{\mathbf{p}}^+} \{1\}_{d_j} \prod_{j\in {\mathcal{A}}_{\mathbf{p}}^-} \{\hbar^{-1} \}_{d_j} \prod_{i\in {\mathbf{p}}} \{ x_i |_{\mathbf{p}}\}_{\sum_{j\not\in {\mathbf{p}}} C_{ij} d_j}$$ where $\zeta ({\mathbf{p}})^d := \prod_{j\not\in {\mathbf{p}}} \zeta_j ({\mathbf{p}})^{d_j}$.
Now we consider $V^{({\mathbf{1}})} (q, z, a) |_{\mathbf{p}}$ as a meromorphic function in terms of equivariant parameters $a_i$’s, in the region specified by the following condition: $$| a^\alpha | \ll 1,$$ for any *positive* root $\alpha$. In particular, we have $|\alpha_i ({\mathbf{p}})| \ll 1$ for $i\in {\mathbf{p}}^+$, and $|\alpha_i ({\mathbf{p}})| \gg 1$ for $i\in {\mathbf{p}}^-$.
We denote by the notation $\alpha ({\mathbf{p}}) \xrightarrow{\sigma} 0$ the following process of taking limit: $$\alpha_i ({\mathbf{p}}) \to \infty, \qquad i \in {\mathbf{p}}^+; \qquad \alpha_i ({\mathbf{p}}) \to 0 \qquad i \in {\mathbf{p}}^-.$$ Note that under the change of variables $\kappa_{\mathrm{vtx}}$, this limit is the as the one as described in Remark \[limit-point\], applied to $X'$ and $\theta'$.
\[limit-V\] In the region of $a_i$’s described above, we have $$\lim_{\alpha ({\mathbf{p}}) \xrightarrow{\sigma} 0} V^{(1)} (q, z, a) = \prod_{j\in {\mathcal{A}}_{\mathbf{p}}^+} \frac{\phi \left( \hbar \zeta_{\sharp, j} ({\mathbf{p}}) (q\hbar^{-1} )^{-\sum_{i\in {\mathbf{p}}^+} C_{ij} } \right) }{\phi \left( \zeta_{\sharp, j} ({\mathbf{p}}) (q\hbar^{-1} )^{-\sum_{i\in {\mathbf{p}}^+} C_{ij} } \right)} \prod_{j\in {\mathcal{A}}_{\mathbf{p}}^-} \frac{\phi \left( q \zeta_{\sharp, j} ({\mathbf{p}})^{-1} (q\hbar^{-1} )^{\sum_{i\in {\mathbf{p}}^+} C_{ij} } \right) }{\phi \left( q\hbar^{-1} \zeta_{\sharp, j} ({\mathbf{p}})^{-1} (q\hbar^{-1} )^{\sum_{i\in {\mathbf{p}}^+} C_{ij} } \right)} .$$ In particular, under the change of variables $\kappa_{\mathrm{vtx}}$, it is $$\prod_{j\in ({\mathbf{p}}')^-} \frac{\phi \left( q\hbar^{-1} (x'_j)^{-1} |_{{\mathbf{p}}'} \right) }{\phi \left( (x'_j)^{-1} |_{{\mathbf{p}}'} \right)} \prod_{j\in ({\mathbf{p}}')^+} \frac{\phi \left( q x'_j |_{{\mathbf{p}}'} \right) }{\phi \left( \hbar x'_j |_{{\mathbf{p}}'} \right)} = \Phi ((q - \hbar) {\mathsf{Pol}}'_{{\mathbf{p}}'} (x') |_{{\mathbf{p}}'} ).$$
It is easy to see that, for any $a$ and $d\in \ZZ$, $$\{a \}_d \to \left\{ \begin{aligned}
& (- q^{1/2} \hbar^{-1/2} )^d , && \qquad a \to 0 ; \\
& (- q^{1/2} \hbar^{-1/2} )^{- d} , && \qquad a \to \infty .
\end{aligned} \right.$$ Therefore, \_[() 0]{} V\^[(1)]{} (q, z, a) &=& \_[ ]{} ()\^d q\^[ - \_[j]{} d\_j - \_[i, j]{} C\_[ij]{} d\_j ]{} \_[j\_\^+]{} (- q\^[1/2]{} \^[-1/2]{} )\^[d\_j]{}\
&& \_[j\_\^-]{} ( - q\^[1/2]{} \^[-1/2]{} )\^[-d\_j]{} \_[i\^+]{} (- q\^[1/2]{} \^[-1/2]{})\^[ - \_[j]{} C\_[ij]{} d\_j ]{} \_[i\^-]{} (- q\^[1/2]{} \^[-1/2]{})\^[ \_[j]{} C\_[ij]{} d\_j ]{}\
&=& \_[ ]{} \_[j\_\^+]{} \_[, j]{}\^[d\_j]{} \_[j\_\^-]{} (q \^[-1]{} \_[, j]{}\^[-1]{} )\^[- d\_j]{} \_[i\^+]{} (q \^[-1]{} )\^[ - \_[j]{} C\_[ij]{} d\_j ]{}\
&=& \_[ ]{} \_[j\_\^+]{} ( \_[, j]{} (q\^[-1]{} )\^[-\_[i\^+]{} C\_[ij]{} ]{} )\^[d\_j]{} \_[j\_\^-]{} ( q \^[-1]{} \_[, j]{}\^[-1]{} (q\^[-1]{} )\^[\_[i\^+]{} C\_[ij]{} ]{} )\^[- d\_j]{} The lemma then follows from the $q$-binomial formula $\dfrac{\phi (xz)}{\phi (z)} = \sum_{d\geq 0} \dfrac{(x)_d}{(q)_d} z^d$.
Andrey Smirnov\
Department of Mathematics,\
University of North Carolina at Chapel Hill,\
Chapel Hill, NC 27599-3250, USA;\
[email protected]
Zijun Zhou\
Department of Mathematics,\
Stanford University,\
450 Serra Mall, Stanford, CA 94305, USA\
[email protected]
[^1]: The statement in [@HH] is only on the Kirwan surjectivity for cohomology. But the GKM method adopted there applies also to the $K$-theory, and allows one to obtain the explicit presentation (\[K(X)\])
[^2]: Unfortunately, we use $\alpha$ for both roots and equivariant parameters. To distinguish them, an equivariant parameter will always be followed with a dependence on its associated fixed ${\mathbf{p}}$, e.g., $\alpha_i ({\mathbf{p}})$.
[^3]: In [@AOelliptic], this is called ${\mathscr{E}}_{{\operatorname{Pic}}_{\mathsf{T}}(X)}$.
[^4]: Here we use the same notations for tautological line bundles on ${\mathfrak{X}}$ and $X$, since they are defined in the same manner and compatible to each other via the inclusion $X \subset {\mathfrak{X}}$.
[^5]: Our convention here is different from [@Oko] and [@PSZ]: $p_1$ and $p_2$ are exchanged. Under this convention we always view $K$-classes from $p_1$ as inputs and those from $p_2$ as outputs. Our $\Psi$ is the conjugate of the original one, with $q \mapsto q^{-1}$.
[^6]: This matrix ${\mathfrak{P}}$ differs from the one in [@AOelliptic] by an exponential factor.
[^7]: The Adams operation of $\Lambda$ naturally includes the usual $\Psi^k$ operators on symmetric functions, and also on the Kähler parameters $\Psi^k (Q_i) = Q_i^k$.
|
---
address:
- |
${}^a$Joseph Henry Laboratories\
Princeton University, Princeton, NJ 08544, USA
- '${}^b$Center of Mathematical Sciences and Applications, Harvard University, Cambridge, 02138, USA'
- '${}^c$Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA'
author:
- 'Yifan Wang$^a$ and Dan Xie$^{b,c}$'
bibliography:
- 'ref.bib'
title: ' Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6d $(2,0)$ theories '
---
\
\
Introduction
============
The six dimensional $(2,0)$ superconformal theories (SCFT) are mysterious quantum field theories that arise either as low energy descriptions of five-branes in M-theory or in a decoupling limit of type IIB string probing ADE singularities [@Witten:1995zh; @Strominger:1995ac; @Witten:1995em]. They are rigid strongly coupled fixed points in six dimensions that are believed to be determined by an ADE Lie algebra and have no relevant deformations that preserve the $(2,0)$ supersymmetry [@Henningson:2004dh; @Cordova:2015vwa; @Cordova:2016xhm]. Residing in a highly constrained structure, the richness of the $(2,0)$ theories lies in the collection of extended defects and their dynamics [@Witten:2009at]. In particular, a plethora of lower dimensional supersymmetric theories have been constructed by compactifications of the $(2,0)$ theory on manifolds with defect insertions. The sheer existence of the 6d parent has lead to many highly nontrivial predictions for the physics of the lower dimensional theories as well as dualities between ostensibly different field-theoretic descriptions. For many cases, these predictions are verified by techniques that are accessible in the lower dimensions. In this way, even though the $(2,0)$ SCFT itself does not have a simple field-theoretic construction that allows direct access to its dynamics,[^1] we can gain valuable insights by studying its daughter theories.
Among all defects in $(2,0)$ SCFTs, the half-BPS codimension-two defects are one of the central focuses of investigation in recent years. They play a crucial role in the class S construction of 4d $\cN=2$ SCFTs by compactifying $(2,0)$ SCFT on a Riemann surface with punctures [@Witten:1997sc; @Gaiotto:2009we; @Gaiotto:2009hg]. In this setup, the codimension-two defects that extend in the 4d spacetime directions produce the punctures.[^2] They often give rise to global symmetries in the 4d theory and supply degrees of freedom that carry symmetry charges. As we review in section \[sec:rev\], these codimension-two defects in the (2,0) SCFT can be described by the singularities of a Lie algebra valued one-form field, the Higgs field $\Phi$, on the Riemann surface. Furthermore, these defects come in two families: the regular (tame) defects corresponding to simple poles for $\Phi$, and the irregular (wild) defects associated to higher order singularities. Classification of the regular defects was given in [@Gaiotto:2009we; @Nanopoulos:2009uw; @Chacaltana:2012zy] and the irregular defects were studied in [@Xie:2012hs; @Wang:2015mra]. In particular the regular defects carry flavor symmetries that are subgroups of ADE Lie groups.
$ \mf j $ $A_{2N}$ $A_{2N-1}$ $D_{N+1}$ $E_6$ $D_4$
----------------------------------- ---------- ------------ ----------- --------- ---------
Outer-automorphism $o$ $\mZ_2$ $\mZ_2$ $\mZ_2$ $\mZ_2$ $\mZ_3$
Invariant subalgebra $\mf g^\vee$ $B_N$ $C_N$ $B_{N}$ $F_4$ $G_2$
Flavor symmetry $\mf g$ $C_N$ $B_N$ $C_{N}$ $F_4$ $G_2$
: Outer-automorphisms of simple Lie algebras $\mfj$, its invariant subalgebra $\mfg^\vee$ and flavor symmetry $\mfg$ from the Langlands dual.[]{data-label="table:outm"}
The $(2,0)$ SCFT also has codimension-one defects that correspond to the discrete global symmetries associated to the outer-automorphism group of the ADE Lie algebra (see Table \[table:outm\]).[^3] Thus we can consider [*twisted*]{} codimension-two defects that live at the ends of these symmetry defects.[^4] On the Riemann surface in a class S setup, the codimension-one defect is represented by a twist line that either wraps a nontrivial cycle or connects a pair of twisted punctures (see Figure \[fig:defects\]).
![Local configurations of the codimension-one symmetry defects (red line) and codimension-two twisted defects (blue dot) on the Riemann surface in a class S setup.[]{data-label="fig:defects"}](twhand.pdf){width="100.00000%"}
$$\begin{aligned}
\times~ \mR^{3,1} \\ \end{aligned}$$
![Local configurations of the codimension-one symmetry defects (red line) and codimension-two twisted defects (blue dot) on the Riemann surface in a class S setup.[]{data-label="fig:defects"}](twpair.pdf){width="100.00000%"}
$$\begin{aligned}
\times~ \mR^{3,1} \\ \end{aligned}$$
Twisted regular defects carry (subgroups of) non-simply-laced BCFG flavor symmetry groups and they were studied extensively in [@Tachikawa:2009rb; @Tachikawa:2010vg; @Chacaltana:2012zy; @Chacaltana:2015bna; @Chacaltana:2014nya; @Chacaltana:2012ch]. Note that the maximal $\mZ_2$ twisted regular defects in type $A_{2N}$ and $D_{N+1}$ $(2,0)$ SCFTs share the same flavor symmetry $USp(2N)$ but differ in Witten’s global anomaly for the said symmetry [@Tachikawa:2011ch; @Tachikawa:2018rgw].
$\mfj$ Singularity Symmetry Action of generators
------------ -------------------- ---------- -----------------------------------------------
$A_{2N}$ $x^{2N+1}+yz=0$ $\mZ_4$ $(x,y,z)\rightarrow(-x,z,-y)$
$A_{2N-1}$ $x^{2N}+yz=0$ $\mZ_2$ $(x,y,z)\rightarrow(-x,z,y)$
$D_{N+1}$ $x^{N}+xy^2+z^2=0$ $\mZ_2$ $(x,y,z)\rightarrow(x,-y,-z)$
$E_6$ $x^4+y^3+z^2=0$ $\mZ_2$ $(x,y,z)\rightarrow(-x,y,-z)$
$D_4$ $x^3+y^3+z^2=0$ $S_3$ $(x,y,z)\rightarrow(y,x,-z)$
$(x,y,z)\rightarrow(\omega x, \omega^2 y, z)$
: Discrete symmetries of ADE two-fold singularities.[]{data-label="table:sym"}
One of main purposes of this paper is to classify twisted irregular defects in $(2,0)$ SCFTs. Before we summarize the results, let us briefly recall the classification of untwisted irregular defects performed in [@Xie:2012hs; @Wang:2015mra] since the method we pursue here is a direct generalization.
The classification of codimension-two defects in [@Xie:2012hs; @Wang:2015mra] was based on analyzing consistency conditions on the higher order singularities of the Higgs field $\Phi$ on a Riemann surface with a local holomorphic coordinate $z$. After the singularity is put into the convenient semisimple form by a gauge transformation, \_z(z)= [Tz\^[2+[kb]{}]{}]{}+…, b\^+, k [and]{} k>-b. \[utphi\] where $T$ is a semisimple element of $\mfj$, the consistency condition on $\Phi$ simply says (e\^[2i]{} z)= \_g (z) \[utcon\] for some inner automorphism $\sigma_g$ of the Lie algebra $\mfj$. This puts constraints on the defining data $(T,k,b)$ of the singularity, which are then solved systematically by Kac’s classification of finite order (torsion) inner automorphisms of simple Lie algebras [@kac2013infinite]. Generic 4d $\cN=2$ SCFTs engineered by such irregular defects are of the Argyres-Douglas (AD) type which have fractional scaling dimensions in the half-BPS Coulomb branch spectrum and are intrinsically strongly-coupled [@Argyres:1995jj; @Argyres:1995wt; @Argyres:1995xn].
$ \mfj$ $b$ Singularity
----------- -------- -------------------------------------- --
$A_{N-1}$ $N$ $x_1^2+x_2^2+x_3^N+z^k=0$
$~$ $N-1$ $x_1^2+x_2^2+x_3^N+x_3 z^k=0$
$D_N$ $2N-2$ $x_1^2+x_2^{N-1}+x_2x_3^2+z^k=0$
$~$ $N$ $x_1^2+x_2^{N-1}+x_2x_3^2+z^k x_3=0$
$E_6$ 12 $x_1^2+x_2^3+x_3^4+z^k=0$
$~$ 9 $x_1^2+x_2^3+x_3^4+z^k x_3=0$
$~$ 8 $x_1^2+x_2^3+x_3^4+z^k x_2=0$
$E_7$ 18 $x_1^2+x_2^3+x_2x_3^3+z^k=0$
$~$ 14 $x_1^2+x_2^3+x_2x_3^3+z^kx_3=0$
$E_8$ 30 $x_1^2+x_2^3+x_3^5+z^k=0$
$~$ 24 $x_1^2+x_2^3+x_3^5+z^k x_3=0$
$~$ 20 $x_1^2+x_2^3+x_3^5+z^k x_2=0$
: Three-fold isolated quasi-homogeneous singularities of the cDV type corresponding to the $J^{(b)}[k]$ irregular punctures of the regular-semisimple type in [@Wang:2015mra]. []{data-label="table:sing"}
A distinguished class of solutions to and , known as the [*regular-semisimple*]{} type[^5] gives rise to irregular codimension-two defects that are in one-to-one correspondence with three-fold quasi-homogeneous isolated singularities of the compound Du Val (cDV) type (see Table \[table:sing\]).[^6] This connection between the two very different types of singularities is established by the observation that identical 4d $\cN=2$ SCFTs are engineered by i) compactifying $(2,0)$ SCFT on $\mP^1$ with such a irregular defect inserted; ii) the decoupling limit of IIB string probing a cDV singularity. We review the general untwisted irregular defects, the resulting 4d $\cN=2$ SCFTs as well as their physical data in section \[sec:rev\].
In this paper, we incorporate outer-automorphism twists into the configurations of codimension-two irregular defects in the $(2,0)$ SCFT. The consistency condition for the Higgs field singularity at $z=0$ on the Riemann surface is modified to (e\^[2i]{} z)=\_g o (z) where $\sigma_g o$ labels an outer-automorphism of $\mfj$ with $o$ generating automorphisms of the Dynkin diagrams (see Table \[table:outm\]) and the parameter $b$ in is replaced by $b_t$. As we explain in section \[sec:tip\], this constraint can be solved by invoking the classification of finite order (torsion) outer-automorphisms of simple Lie algebras, which is also given in [@kac2013infinite]. Restricting the polar matrix $T$ in to be regular semisimple again gives rise to a distinguished class of twisted irregular defects, which can be put into the three-fold form in Table \[table:SW\].[^7] We also explicitly identify the continuous free parameters of these defects with flavor symmetry masses and exactly marginal couplings.
-------------------------- -------- --------------------------------------------------- -------------------------
$\mfj$ with twist $b_t$ SW geometry at SCFT point $\Delta[z]$
\[1pt\] $A_{2N}/\mZ_2$ $4N+2$ $x_1^2+x_2^2+x^{2N+1}+z^{k'+{1\over2}}=0$ ${4N+2\over 4N+2k'+3}$
\[3pt\] $2N$ $x_1^2+x_2^2+x^{2N+1}+xz^{k'}=0$ ${2N\over k'+2N}$
\[3pt\] $A_{2N-1}/\mZ_2$ $4N-2$ $x_1^2+x_2^2+x^{2N}+xz^{k'+{1\over2}}=0$ ${4N-2\over 4N+2k'-1}$
\[3pt\] $2N$ $x_1^2+x_2^2+x^{2N}+z^{k}=0$ ${2N\over 2N+k'}$
\[3pt\] $D_{N+1}/\mZ_2$ $2N+2$ $x_1^2+x_2^{N}+x_2x_3^2+x_3z^{k'+{1\over2}}=0$ ${2N+2\over 2k' +2N+3}$
\[3pt\] $2N$ $x_1^2+x_2^{N}+x_2x_3^2+z^{k'}=0$ ${2N\over k'+2N}$
\[3pt\] $D_4/\mZ_3$ $12$ $x_1^2+x_2^{3}+x_2x_3^2+x_3z^{k'\pm {1\over3}}=0$ ${12\over 12+3k'\pm1}$
\[3pt\] $6$ $x_1^2+x_2^{3}+x_2x_3^2+z^{k'}=0$ ${6\over 6+k'}$
\[3pt\] $E_6/\mZ_2$ $18$ $x_1^2+x_2^{3}+x_3^4+x_3z^{k'+{1\over2}}=0$ ${18\over 18+2k'+1}$
\[3pt\] $12$ $x_1^2+x_2^{3}+x_3^4+z^{k'}=0$ ${12\over 12+k'}$
\[3pt\] $8$ $x_1^2+x_2^{3}+x_3^4+x_2z^{k'}=0$ ${8\over 12+k'}$
\[3pt\]
-------------------------- -------- --------------------------------------------------- -------------------------
: Seiberg-Witten geometry of twisted theories at the SCFT point.[]{data-label="table:SW"}
Moving onto section \[sec:tt\], we classify 4d $\cN=2$ SCFTs that are engineered by such twisted irregular defects in class S setups which we refer to as [*twisted theories*]{}. The twisted theories come in infinite families (labelled by $b_t$) for each choice of the simple Lie algebra $\mfj$ (see Table \[table:SW\]). We spell out a simple procedure to extract physical data of such theories from our descriptions and propose formulae for the conformal and flavor central charges. Most of the theories we construct here are new but we identify in our setups several (sequences of) known constructions that only involve regular defects. They provide a nontrivial consistency check of our construction and central charge formulae.
A general 4d $\cN=2$ SCFT is known to have a nontrivial protected sector described by a 2d chiral algebra [@Beem:2013sza]. Recent developments indicate that it captures information of both Coulomb branch and Higgs branch physics of the 4d theory [@Cordova:2015nma; @Buican:2015ina; @Cecotti:2015lab; @Xie:2016evu; @Creutzig:2017qyf; @Fredrickson:2017yka; @Song:2017oew; @Beem:2017ooy; @Kozcaz:2018usv]. In particular, the Higgs branch of the 4d theory is identified with the associated variety of the 2d chiral algebra [@arakawa2015associated]. In section \[sec:voa\], we identify the associated 2d chiral algebra or vertex operator algebra (VOA) for a subclass of the twisted theories and determine the Higgs branch of these theories from the associated variety of the VOA. We end in section \[sec:sum\]. with a summary and future directions.
Background review {#sec:rev}
=================
In this section, we review the description of codimension-two BPS defects in 6d $(2,0)$ SCFTs in terms of the Higgs field and explain the class S construction of 4d $\cN=2$ SCFTs using the Hitchin system on a Riemann surface with defect insertions. We also summarize the classification of untwisted defects in the last subsection. Readers familiar with these topics may safely skip this section.
Hitchin system and 4d $\cN=2$ SCFTs
-----------------------------------
A large class of 4d $\cN=2$ superconformal field theories can be engineered by the twist compactification of the 6d $(2,0)$ type $J=ADE$ theories on a Riemann surface $\cC$, usually referred to as the UV curve or Gaiotto curve [@Witten:1997sc; @Gaiotto:2009we]. The Riemann surface $\cC$ can come with punctures at isolated points $\{p_i\}$, which correspond to codimensional-two BPS defects in the 6d SCFT. In M-theory, the $A_N$ type (2,0) SCFT captures the low energy dynamics of a stack of fivebranes and the codimensional-two defect is described by another stack of fivebranes which share four longitudinal directions with the former. To produce $4d$ theories with $\cN=2$ supersymmetry, we have the first stack of fivebranes wrapping $\cC$, whereas the defect fivebranes extend along the cotangent fibers of $\cC$ at the singularities $p_i$.
[ccccccc]{} [ ]{} & \^[2,1]{} &S\^1 && T\^\*\_[p\_i]{}& &&&\
& & & &\
& \^[2,1]{} &S\^1 & & T\^\* & &\
& & & &\
& \^[2,1]{} &S\^1 & & &\
& &&\
& & & & & &\
& &&\
& & &&\
\[m5setup\] To decode information about the 4d $\cN=2$ SCFT from this construction where the 6d parent has no explicit description (e.g. in terms of a Lagrangian), it is useful to consider the alternate compactification of the 6d theory on a circle transverse to the Riemann surface $\cC$. The resulting 5d theory is believed to be the $\cN=2$ super Yang-Mills (MSYM) with gauge group $J$ (up to higher derivative corrections) [@Seiberg:1996bd; @Seiberg:1997ax; @Douglas:2010iu; @Lambert:2010iw].
Upon twisted compactification of the 5d theory on the Riemann surface $\cal C$ with holomorphic coordinate $z$, there’s a natural principal $J$-bundle $E$ over $\cal C$ with gauge connection $A=A_z dz+A_{\bar z}d\bar z$ and two of the five 5d scalars combine into a $(1,0)$-form $\Phi=\Phi_z dz$ valued in the adjoint bundle ${\rm ad}(E)$. The supersymmetric configurations of the twisted theory are governed by the Hitchin equations &F+ \[,|\]=0,\
&|\_A =d|z(\_[|z]{}+\[A\_[|z]{},\])=0, \[hitchin\] where $F$ denotes the curvature two-form of $A$. The pair $(E,\Phi)$ subject to is referred to as the Higgs bundle and $\Phi$ is the Higgs field. In particular, the second line of implies $\Phi$ is a holomorphic section of $\cK \otimes {\rm ad}(E)$ where $\cK$ denotes the canonical bundle on $\cC$. For a fixed structure group $J$, the moduli space of Higgs bundles over $\cC$ (solutions to Hitchin equations modulo gauge redundancy) corresponds to the Hitchin moduli space $\cM_{\rm H}(\cC)$.
The twisted compactification of 5d MSYM leads to a particular description of the 3d $\cN=4$ SCFT in the IR, with a Higgs branch identical to $\cM_{\rm H}(\cC)$ thanks to supersymmetry. Meanwhile the other order of compactifications in (first on $\cC$ then $S^1$) provides another description of the same 3d SCFT related by mirror symmetry, where $\cM_{\rm H}(\cC)$ now describes the Coulomb branch.
[rcl]{} &[ @arrow 9999[@]{}[ ]{}]{} &\
[ ]{}& & [ ]{}\
&
. \[\]
$\cM_{\rm H}(\cC)$ is a hyper-Kähler manifold of rich structure that encodes dynamics of the 4d theory $\cT_4[\cC]$ as well as its 3d descendant. In one complex structure, $\cM_{\rm H}(\cC)$ is equivalent to the moduli space of $\mf j_{\mC}$-valued flat connections $\cA=A+i\Phi+i\bar{\Phi}$ on $\cC$. For example when $\cC$ has genus $g$ with no punctures, $\cM_{\rm H}(\cC)$ is parametrized by the holonomies around the $2g$ cycles labelled by elements $U_{1\leq i\leq g}$ and $V_{1\leq i\leq g}$ in $J_{\mC}$ that are subject to the constraint 1=U\_1V\_1 U\_1\^[-1]{} V\_1\^[-1]{}…U\_gV\_g U\_g\^[-1]{} V\_g\^[-1]{}, \[cs1\] and modulo $J_{\mC}$ gauge transformations. In particular, $\cM_{\rm H}(\cC)$ has complex dimension \_[H]{}()=(2g-2)J. In another complex structure,[^8] $\cM_{\rm H}(\cC)$ exhibits a natural fibration structure :\_[H]{}() \[cs2\] by taking Casimirs of $\Phi$ (gauge invariant differentials), known as the Hitchin fibration [@Hitchin:1986vp; @Hitchin:1987mz] The base manifold (a complex affine space) $\cB$ is special Kähler and identified with the Coulomb branch of the 4d theory $\cT_4[\cC]$, while the fiber is (generically) a complex torus of dimension ${1\over 2}\dim \cM_{\rm H}(\cC)$ and corresponds to the electric and magnetic holonomies[^9] on $S^1$. In this complex structure, there is a holomorphic symplectic form $\Omega_I$ which defines a Poisson bracket for functions on $\cM_{\rm H}(\cC)$. Hence the Hitchin moduli space $\cM_{\rm H}(\cC)$ becomes a complex integrable system with commuting Hamiltonians given by the Casimirs of $\Phi$ and the fibers of $\cM_{\rm H}(\cC)$ are identified with the orbits of the Hamiltonian flows which are special Lagrangian with respect to $\Omega_I$ [@Donagi:1995cf].
We can recover the usual Seiberg-Witten (SW) description [@Seiberg:1994rs; @Seiberg:1994aj] of the low energy dynamics of the 4d theory $\cT_4[\cC]$ as follows. The SW curve $\Sigma$ governing the Coulomb branch dynamics is equivalent to the spectral curve of the Hitchin system,[^10] (x dz-)=0, and the SW differential is identified with $\lambda=xdz$. More explicitly, the SW curves can be put in the following forms & A\_[N-1]{}: x\^N+\_[i=2]{}\^N \_i(z) x\^[N-i]{}=0,\
& D\_[N]{}: x\^[2N]{}+\_[i=1]{}\^[N-1]{} \_[2i]{}(z)x\^[2N-2i]{}+(\_N)\^2=0,\
& E\_6: \_2(z), \_5(z), \_6(z), \_8(z), \_9(z), \_[12]{}(z),\
& E\_7: \_2(z), \_6(z), \_8(z), \_[10]{}(z), \_[12]{}(z), \_[14]{}(z), \_[18]{}(z),\
& E\_8: \_2(z), \_8(z), \_[12]{}(z), \_[14]{}(z), \_[18]{}(z), \_[20]{}(z), \_[24]{}(z), \_[30]{}(z). Here $\phi_i(z)$ is a degree $i$ differential on $\cC$ and they generate the ring of fundamental invariants (Casimirs) of the Lie algebra. For $E_N$ case, we only list the independent differentials. The coefficients $u_{i,j}$ of these differentials in the $z$ expansion \_i(z)=\_[j]{} u\_[i,j]{} z\^j, \[epsw\] encode the Coulomb branch parameters of the theory.[^11]
Since the integral of the SW differential along a one-cycle gives the mass of a BPS particle, the SW differential $\lambda=x dz$ has scaling dimension one and consequently &+=1. This allows us to determine the scaling dimensions of $u_{i,j}$ by demanding each term of to have the same scaling dimensions[^12]: relevant chiral couplings are given by those with $\Delta [u_{i,j}]<1$, Coulomb branch operators if $\Delta [u_{i,j}] > 1$ and masses if $ \Delta [u_{i,j}]=1$.[^13]
Review of untwisted (ir)regular defects
---------------------------------------
The relevant codimension-two defects in the 6d $(2,0)$ theory of type $\mf j$ can be characterized by singular boundary conditions for the Higgs field $\Phi$ on $\cC$. Supposing the defect is located at $z=0$ on $\cC$, it is convenient to perform a gauge transformation (that may involve fractional powers of $z$) to put $\Phi_z$ in the following form: \_z= \_[k-b]{} [T\_z\^[2+/b]{}]{}+…, b\^+, k\[uts\] where each $T_\ell$ is a semisimple element of ${\mf j}$. In other words, by a gauge transformation, we have $T_\ell\in \mfh$, a Cartan subalgebra of $\mfj$, for all $\ell$ [@Witten:2007td]. We have suppressed the non-divergent terms in .[^14] To ensure the Higgs field $\Phi$ is well-defined on $\cC$, we require (e\^[2i]{} z)= g (z) g\^[-1]{} g T\_g\^[-1]{}= \^T\_ \[utgi\] for some $g\in J$ and $\omega=e^{2\pi i\over b}$.
The usual regular (tame) punctures correspond to $b=-k$ in which case $\Phi$ has a simple pole (go to a branch cover if necessary) \_z= [T z ]{}+ …\[ruts\] and the constraint is trivialized by taking $g=1$. The regular punctures are thus classified by the conjugacy classes of $T$. When $\mfj$ is a classical Lie algebra, these conjugacy classes are labelled by the Hitchin partitions which are related to the Nahm partitions by the Spaltenstein map. The generalization to exceptional Lie algebras involves the Bala-Carter labels [@Chacaltana:2012zy]. The flavor symmetry associated to the puncture can be directly read off from the Nahm label and the entries of $T$ correspond to the mass parameters. The local contributions to the conformal and flavor central charges can also be computed systematically [@Chacaltana:2012zy].
The irregular (wild) punctures arise when $k>-b$ where $\Phi$ has a higher order pole.[^15] Now puts nontrivial constraints on $T_\ell$ and $b$ which were solved by Kac [@kac_1990].[^16] One can associate to $g$ an inner automorphism of order $b$ (torsion automorphism) $\sigma_g$ of $\mf j$ which then introduces a grading on $\mf j$ =\_[m /b]{} j\^m. In particular $T_\ell$ is a semisimple element in $\mf j^\ell$. Such finite-order inner automorphisms are classified by Kac (see §8 of the textbook [@kac_1990]).[^17]
Since we are interested in 4d $\cN=2$ SCFTs, the configurations of defects on the Riemann surface $\cC$ must be consistent with a $U(1)_R$ symmetry. In the absence of irregular punctures on $\cC$, the $U(1)_R$ generator is identified with the rotation generator $R_{45}$ for the $SO(5)$ R-symmetry group of the 5d MSYM that acts on the Higgs field as $[R_{45},\Phi]=\Phi$. The regular codimension-two defects are conformal[^18] thus the number and positions of the regular punctures on $\cC$ as well as the topology of $\cC$ are unconstrained.[^19] On the other hand, in the presence of an irregular defect defined by , the potential 4d $U(1)_R$ generator involves a combination of $R_{45}$ and $U(1)_z$ that acts as U(1)\_R SO(2)\_[45]{} U(1)\_z: e\^[i]{},z e\^[i ]{} z. so that the leading polar matrix $T_k$ in is preserved. For this to be globally defined on $\cC$, we are restricted to consider $\cC=\mP^1$ with either a single irregular puncture, or an irregular puncture accompanied by a regular puncture, located at the two fixed points $z=0,\infty$ of $U(1)_z$ (see Figure \[fig:utsp\]).
![The class S setup that involves one irregular defect (star) with or without one regular defect (dot) on a sphere. They engineer $J^{(b)}[k]$ and $(J^{(b)}[k],Y)$ theories in [@Wang:2015mra] respectively. Here $J^{(b)}[k]$ labels the irregular defect and $Y$ labels the regular defect.[]{data-label="fig:utsp"}](utsp1.pdf){width=".8\textwidth"}
![The class S setup that involves one irregular defect (star) with or without one regular defect (dot) on a sphere. They engineer $J^{(b)}[k]$ and $(J^{(b)}[k],Y)$ theories in [@Wang:2015mra] respectively. Here $J^{(b)}[k]$ labels the irregular defect and $Y$ labels the regular defect.[]{data-label="fig:utsp"}](utsp.pdf){width=".8\textwidth"}
A distinguished class of irregular punctures of the [*regular semisimple*]{} type in general 6d (2,0) SCFTs were classified in [@Wang:2015mra], generalizing earlier work for $A$-type (2,0) theories [@Xie:2012hs]. The Hitchin pole in these cases is characterized by a [regular]{} semi-simple element $T_k$ in $\mf j$. The commutant of $T_k$ in $\mfj$ is a Cartan subalgebra $\mfh \ni T_k$. Restricted to $\mfh$, the inner automorphism $\sigma_g$ with a regular eigenvector $T_k$ corresponds to a regular element in the Weyl group $W(\mfh)$ in Springer’s classification of regular elements for complex reflection groups [@springer1974regular]. The orders of these regular elements (known as regular numbers $d$ in [@springer1974regular]) are listed in Table \[table:bv1\] and each order is associated to an unique regular element up to conjugation by $W(\mfh)$.
$\mfj $ $d$ (is divisor of)
----------- ---------------------
$A_{N-1}$ $N,~N-1$
$D_N$ $2N-2,~N$
$E_6$ $12,~9,~8 $
$E_7$ $18,~14$
$E_8$ 30, 24, 20
: Regular numbers (orders of regular semisimple elements) of Weyl groups.[]{data-label="table:bv1"}
The corresponding torsion automorphisms have the same orders except for $\mfj=A_{2N}$ and $d=2N-1$ in which case $\sigma_g$ has order $4N-2$.[^20] Each regular element induces a grading of its commutant Cartan subalgebra =\_[m /d]{} h\^m. where $\mfh^1$ contains the regular semisimple element. The corresponding irregular Hitchin pole is determined by [^21]
This class of irregular punctures and the resulting 4d theories were labelled by $J^{(b)}[k]$ in [@Wang:2015mra] where $b$ takes the values of the regular numbers in Table \[table:bv1\] including their divisors.[^22] The Coulomb branch spectrum and conformal central charges were computed in [@Wang:2015mra], where it was shown that these theories are in one to one correspondence with those constructed by IIB string probing three-fold isolated quasi-homogeneous singularities of compound Du Val (cDV) type (see Table \[table:sing\]). The case of including a regular puncture with nontrivial flavor symmetry was also considered in [@Wang:2015mra]. In particular a class of theories denoted by $(J^{(b)}[k],F)$ with nonabelian ADE flavor symmetry were constructed by including an additional maximal (full) regular puncture. A nice feature of these $J^{(b)}[k]$ and $(J^{(b)}[k],F)$ theories is that many of them have simple 2d chiral algebras that correspond to either W-algebra minimal model or affine Kac-Moody algebra.
$ \mfj$ Singularity $b$
----------- -------------------------------------- -------- --
$A_{N-1}$ $x_1^2+x_2^2+x_3^N+z^k=0$ $N$
$~$ $x_1^2+x_2^2+x_3^N+x_3 z^k=0$ $N-1$
$D_N$ $x_1^2+x_2^{N-1}+x_2x_3^2+z^k=0$ $2N-2$
$~$ $x_1^2+x_2^{N-1}+x_2x_3^2+z^k x_3=0$ $N$
$E_6$ $x_1^2+x_2^3+x_3^4+z^k=0$ 12
$~$ $x_1^2+x_2^3+x_3^4+z^k x_3=0$ 9
$~$ $x_1^2+x_2^3+x_3^4+z^k x_2=0$ 8
$E_7$ $x_1^2+x_2^3+x_2x_3^3+z^k=0$ 18
$~$ $x_1^2+x_2^3+x_2x_3^3+z^kx_3=0$ 14
$E_8$ $x_1^2+x_2^3+x_3^5+z^k=0$ 30
$~$ $x_1^2+x_2^3+x_3^5+z^k x_3=0$ 24
$~$ $x_1^2+x_2^3+x_3^5+z^k x_2=0$ 20
: Three-fold isolated quasi-homogeneous singularities of cDV type corresponding to the $J^{(b)}[k]$ irregular punctures of the regular semisimple type in [@Wang:2015mra] . []{data-label="table:sing"}
In particular the flavor central charge of the $(J^{(b)}[k],F)$ theory [@Xie:2013jc; @Xie:2016evu; @Xie:2017vaf; @Xie:2017aqx]: k\_F=h-[bb+k]{} was identified with the level of the affine Kac-Moody algebra $\widehat \mfj$ by $k_{2d}=-k_F$. In section \[sec:voa\], we will see how the twisted theories realize the other types of affine Kac-Moody algebra and W-algebra associated to non-simply-laced simple Lie algebras.
Twisted irregular defects {#sec:tip}
=========================
In this section, we study the classification of general twisted codimension-two BPS defects in 6d $(2,0)$ SCFTs. Along the way, we make connection to known results in the literature. We then describe in detail the classification of a distinguished class of twisted irregular defects of the [*regular-semisimple*]{} type. The physical interpretation for the parameters of the twisted irregular defect is explained in the last subsection.
Classification of general twisted defects
-----------------------------------------
When $\mf j$ has a nontrivial outer-automorphism group $\rm{Out}(\mf j)$, we can decorate the puncture (defect) with a monodromy twist by $o \in {\rm Out}(\mf j)$ around the singularity (see Table \[table:outm\]). In the IIB realization of the $(2,0)$ SCFT from the decoupling limit of string probing an ADE singularity, the above outer-automorphism group arises from the symmetry of the singularity [@slodowy2001simple] (see Footnote \[foot:iibsym\] and Table \[table:sym\]).
Locally the Higgs field behaves as \_z= \_[k]{} [T\_z\^[2+/b\_t]{}]{}+…, b\_t\^+, k-b\_t\[ts\] and the twist amounts to modifying the requirement to (e\^[2i]{} z)= g\[o((z))\]g\^[-1]{} g \[o(T\_)\] g\^[-1]{}= \^T\_ \[tgi\] for some $g\in J/{ G}^\vee$ where ${G}^\vee$ is the invariant subgroup of $J$ with respect to $o$ [@Chacaltana:2012zy]. Globally the twisted punctures must come in pairs connected by twist lines.
Compared to the untwisted case, the gauge transformation required for the Higgs field to be well-defined is now $\sigma_g o $ which is a [*twisted torsion automorphism*]{} of $\mfj$ of order $b_t$ that projects to a nontrivial element in ${\rm Out}(\mfj)$.[^23] It induces a grading on the Lie algebra =\_[m /b\_t]{} j\^m. \[tg\] such that $T_\ell$ is a semisimple element of $\mfj^\ell$ which has eigenvalue $\omega^\ell$ under $\sigma_g o$.[^24] The twisted torsion automorphisms for simple Lie algebras were classified in [@kac_1990].
A subclass of twisted punctures with leading simple pole are called regular twisted punctures in which case the constraint is solved by $\sigma_g=1$ and \_z=[T\_[-2]{}z]{}+[T\_[-1]{} z\^[1/2]{}]{}+T\_0+…, T\_[-2]{},T\_[0]{}\^0 [and]{} T\_[-1]{} \^1 for $o$ of order 2 in the case of $\mfj=A_n,D_n,E_6$ and \_z=[T\_[-3]{}z]{}+[T\_[-2]{} z\^[2/3]{}]{}+[T\_[-1]{} z\^[1/3]{}]{}+T\_0+…, T\_[-3]{},T\_[0]{}\^0, T\_[-2]{} \^2 [and]{} T\_[-1]{} \^1 for $o$ of order 3 in the case of $\mfj=D_4$. Various physical information associated to these twisted punctures were studied in [@Chacaltana:2012zy] including the local contributions to the Coulomb branch and Higgs branch, flavor symmetry and central charges. The 4d class S theories constructed from these punctures were studied extensively [@Chacaltana:2012ch; @Chacaltana:2013oka; @Chacaltana:2014nya; @Chacaltana:2015bna; @Chacaltana:2016shw].
More generally, we have an twisted irregular puncture from . In the presence of outer-automorphism twist, to construct 4d $\cN=2$ SCFT with one twisted irregular puncture, we must pair it with a regular twisted puncture on $\mP^1$.
Twisted irregular defects of regular semisimple type {#sec:rst}
----------------------------------------------------
Similar to the $J^{(b)}[k]$ type of untwisted irregular punctures (defects) introduced in [@Wang:2015mra], there is a distinguished class of twisted irregular punctures of the [*regular semisimple*]{} type. This is achieved when $T_k$ is a regular semisimple element satisfying (\_g o) T\_k=\^ k T\_k. \[tpr\] for $\omega=e^{2\pi i\over b_t}$ and a twisted torsion automorphism $\sigma_g o$ of $\mfj$.
Restricted to the commutant Cartan subalgebra $\mfh$ of $T_k$, $\sigma_g o$ induces an element in the twisted Weyl group $W_t(\mfh)\equiv W(\mfh) \rtimes{\rm Out}(\mfj)$. In particular, it corresponds to a twisted regular element of $W_t(\mfh)$ in Springer’s classification that generalizes the (untwisted) regular elements of $W(\mfh)$ which are associated to inner torsion automorphisms of $\mfj$ (for all simple Lie algebras) with regular eigenvectors [@springer1974regular]. Since we always work with semisimple elements of $\mfh$ in this paper, we will abuse the notation and use $\sigma_g o$ to denote both the twisted torsion automorphism and the corresponding twisted regular element of $W_t(\mfh)$, similarly for $\sigma_g$ in the untwisted case.[^25] As in the untwisted case, each twisted regular element is determined uniquely (up to conjugation) by a positive integer $d_t$ such that its regular eigenvector in $\mfh$ has eigenvalue $e^{2\pi i\over d_t}$. The order of the twisted regular element is $d'_t=\operatorname{lcm}(d_t,|o|)$.[^26] In particular, the twisted regular element $\sigma_g o$ from is associated with The twisted regular numbers $d_t$ were classified by Springer [@springer1974regular] and summarized in Table \[table:dt\].
---------------- ----------------------------------------------------- ------------------------ --
$d_t \equiv 0 \,({\rm mod\,} |o|) $ is a divisor of $d_t $ is a divisor of
${}^2A_{2N}$ $4N+2$ $2N$
${}^2A_{2N-1}$ $4N-2$ $2N$
${}^2D_N$ $2N$ $2N-2$
${}^2E_6$ $18$ $12,~8$
${}^3D_4$ $12$ $6$
---------------- ----------------------------------------------------- ------------------------ --
: Twisted regular numbers $d_t$ for twisted Weyl groups.[]{data-label="table:dt"}
In particular the maximal values for $d_t$ correspond to the orders of twisted Coxeter elements and are called twisted Coxeter numbers $h_t$ of $\mfj$ in [@springer1974regular] (see the second column Table \[table:dt\]). In general, a twisted regular element induces a grading on the Cartan subalgebra =\_[m /d’\_t]{} \^m,\[tgh\] such that $\mfh^{d'_t/d_t}$ contains a regular semisimple element. The corresponding twisted irregular Hitchin pole is specified by \[bkT\]
We group these Hitchin poles into class I and II for ${}^2A_N, {}^2D_N, {}^3D_4$, and class I, II and III for ${}^2E_6$ according to the values of $d_t$ listed in the Table \[table:dt\], with the understanding that their allowed divisors fall into the corresponding classes.[^27] Below we give explicitly the Hitchin pole for $A_{2N-1},A_{2N}$ and $D_N$ when the associated twisted torsion automorphism $\sigma_g o$ has the maximal order in each class.
For example, consider the case $\mf j=A_{2N-1}$. There are two classes of twisted irregular punctures. Denoting the standard basis of $\mR^{2N}$ by $\{e_1,e_2,\dots,e_{2N}\}$ and the Cartan subalgebra of $A_{2N-1}$ by $\mR^{2N}/\{\sum_{i=1}^{2N}e_{i}\}$, the generator of ${\rm Out}(A_{2N-1})=\mZ_2$ is defined by e\_i -e\_[2N+1-i]{} while the Weyl group $W(A_{2N-1})$ acts by permutation on $e_i$.
### $A_{2N-1}$ Class I {#a_2n-1-class-i .unnumbered}
=[ T z\^[2+[k4N-2]{}]{}]{}+…\[hp1\] for $\gcd(k,4N-2)=1$, where T=
0 & & & & &\
& 1 & & & &\
& & \^[2]{} & & &\
& & & & &\
\
& & & & & \^[ 4N-4 ]{}
,\^[4N-2]{}=1 The required gauge transformation $\sigma_g$ corresponds to a permutation in the $\mZ_{2N-1}$ subgroup of $W(A_{2N-1})$ acting the lower right $2N-1$ diagonal entries of $T$.
### $A_{2N-1}$ Class II {#a_2n-1-class-ii .unnumbered}
=[ T z\^[2+[k2N]{}]{}]{}+…\[hp2\] for $\gcd(k,2N)=1$, where T=
1 & & & & &\
& & & & &\
& & \^2 & & &\
& & & & &\
\
& & & & & \^[ 2N-1]{}
,\^[2N]{}=1 The required $\sigma_g$ now corresponds to a permutation in the $\mZ_{2N}$ subgroup of $W(A_{2N-1})$ .
The case $\mf j=A_{2N}$ is similar and again has two classes of twisted irregular punctures. The generator of ${\rm Out}(A_{2N})=\mZ_2$ defined by[^28] e\_i -e\_[2N+2-i]{}. while the Weyl group $W(A_{2N+1})$ acts by permutation on $e_i$ with $1\leq i\leq 2N+1$.
### $A_{2N}$ Class I {#a_2n-class-i .unnumbered}
=[ T z\^[2+[k4N+2]{}]{}]{}+…\[hp1\] for $\gcd(k,4N+2)=1$, where T=
1 & & & & &\
& \^2 & & & &\
& & \^[4]{} & & &\
& & & & &\
\
& & & & & \^[ 4N ]{}
,\^[4N+2]{}=1
### $A_{2N}$ Class II {#a_2n-class-ii .unnumbered}
=[ T z\^[2+[k2N]{}]{}]{}+…\[hp2\] for $\gcd(k,2N)=1$, where T=
0 & & & & &\
& 1 & & & &\
& & & & &\
& & & & &\
\
& & & & & \^[ 2N-1]{}
,\^[2N]{}=1
For $\mf j=D_{N}$ there are two classes of twisted irregular punctures. Identifying the Cartan subalgebra of $D_{N}$ with $\mR^N$ in the standard basis $\{e_1,e_2,\dots, e_N\}$, ${\rm Out}(D_{2N})=\mZ_2$ is generated (up to conjugation) by e\_1- e\_1,e\_i e\_i [for]{} 2iN while the Weyl group $W(D_{2N})$ acts by permutation and even number of sign flips on $e_i$.
### $D_N$ Class I {#d_n-class-i .unnumbered}
=[ T z\^[2+[k2N]{}]{}]{}+…\[hp1\] for $\gcd(k,2N)=1$, where T=
0& 1 & & & & & &\
-1& 0 & & & & & &\
& & 0 & & & & &\
& & -& 0 & & & &\
& & & & & & &\
& & & & & & 0 & \^[N-1]{}\
& & & & & & -\^[N-1]{} & 0\
,\^[2N]{}=1
### $D_N$ Class II {#d_n-class-ii .unnumbered}
=[ T z\^[2+[k2N-2]{}]{}]{}+…\[hp2\] for $\gcd(k,2N-2)=1$, where T=
0& 0 & & & & & &\
0& 0 & & & & & &\
& & 0 & 1 & & & &\
& & -1 & 0 & & & &\
& & & & & & &\
& & & & & & 0 & \^[N-1]{}\
& & & & & & -\^[N-1]{} & 0\
, \^[2N-2]{}=1.
Physical parameters from the punctures
--------------------------------------
The defining data of the punctures can be identified with the parameters of the resulting 4d theories. In particular, for SCFTs, we are interested in the masses for flavor symmetries and exactly marginal couplings. Of course one can enumerate such parameters in the SW (spectral) curve. Here we describe how these data can be extracted directly from the punctures.
The grading induces a natural conjugation action of a reductive Lie group $J_0$ associated to $\mfj^0$ on $\mfj^\ell$. Therefore $\mfj^\ell$ are $J_0$-modules of finite ranks. Furthermore, the $U(1)_R$ symmetry associated with the singularity acts by U(1)\_R SO(2)\_[45]{} U(1)\_z: e\^[i]{},z e\^[i ]{} z. such that $T_k \in \mfj^{d'_t/d_t}$ has zero $U(1)_R$ charge. In general $T_\ell$ has $U(1)_R$ charge q\_R\[T\_\]=[k-k+b\_t]{} Hence $T^{-b_t} \in \mfj^0$ has $U(1)_R$ charge $q_{R}=1$. From $\cN=2$ superconformal symmetry, we deduce that $T^{-b_t}$ contains the mass parameters of the theory whereas $T_k$ is associated to exactly marginal couplings.
The maximal number of exactly marginal couplings is determined by the rank of $\mfj^{d'_t/d_t}$ as a $J_0$-module to be n\_[marg]{}=(\^[d’\_t/ d\_t]{}|J\^0)-1= (\^[d’\_t/ d\_t]{})-1. Note that we have fixed the redundancy due to conjugation by $J_0$ as well as rescaling of the $z$ coordinate.
The maximal number of mass parameters is captured by the dimension of the intersection between the centralizer of the semisimple part of $\mfj^{d'_t/ d_t}$ and $\mfj^0$ n\_[mass]{}=(C(\^[d’\_t/ d\_t]{}\_s)\^0)= (\^0). Both $n_{\rm marg}$ and $n_{\rm mass}$ can be extracted from the grading of $\mfh$ described in [@springer1974regular]. We summarize the results in Table \[table:aedata\]-\[table:d4data\].
$d_t $ Number of marginal couplings Number of mass parameters
------------------------------------ ------------------------------ ---------------------------
$d_t=1$ $N-1$ $N$
$d_t \in 2\mZ+1$, $d_t|N$ $N/d_t-1$ $N/d_t$
$d_t \in 4\mZ$, $d_t|2N$ $2N/d_t-1$ $2N/d_t$
$d_t\in 4\mZ+2$, $d_t|2(2N) $ $4N/d_t-1$ $0$
$d_t>2 \in 4\mZ+2$, $d_t|2(2N+1) $ $2(2N+1)/d_t-1$ $0$
: Marginal couplings and mass parameters from twisted $A_{2N}$ punctures[]{data-label="table:aedata"}
$d_t$ Number of marginal couplings Number of mass parameters
---------------------------------- ------------------------------ ---------------------------
$d_t=1$ $N-1$ $N$
$d_t>1 \in 2\mZ+1$, $d_t|N$ $N/d_t-1$ $N/d_t$
$d_t \in 4\mZ$, $d_t|2N$ $2N/d_t-1$ $2N/d_t$
$d_t>2 \in 4\mZ+2$, $d_t|2(2N) $ $4N/d_t-1$ $0 $
$d_t \in 4\mZ+2$, $d_t|2(2N-1) $ $2(2N-1)/d_t-1$ $0$
: Marginal couplings and mass parameters from twisted $A_{2N-1}$ punctures[]{data-label="table:aodata"}
$d_t$ Number of marginal couplings Number of mass parameters
------------------------------------------ ------------------------------ ---------------------------
$d_t=1$ $N-2$ $N-1$
$d_t \in 2\mZ+1$, $d_t|(N-1)$ $(N-1)/d_t-1$ $(N-1)/d_t$
$d_t \in 2\mZ$, $(2(N-1))/d_t\in 2\mZ+1$ $2(N-1)/d_t$ 0
$d_t \in 2\mZ$, $(2(N-1))/d_t\in 2\mZ$ $2(N-1)/d_t-1$ 1
$d_t>2 \in 2\mZ $, $2N/d_t\in 2\mZ+1 $ $2N/d_t-1$ $0$
: Marginal couplings and mass parameters from twisted $D_{N}$ punctures[]{data-label="table:ddata"}
$d_t$ Number of marginal couplings Number of mass parameters
------- ------------------------------ ---------------------------
1 3 4
2 5 0
3 1 0
4 1 0
6 2 0
8 0 1
12 0 0
18 0 0
: Marginal couplings and mass parameters from twisted $E_6$ punctures[]{data-label="table:e6data"}
$d_t$ Number of marginal couplings Number of mass parameters
------- ------------------------------ ---------------------------
1 1 2
2 1 2
3 1 0
6 1 0
12 0 0
: Marginal couplings and mass parameters from twisted $D_4$ punctures[]{data-label="table:d4data"}
Twisted theories and central charges {#sec:tt}
====================================
Given the classification of the twisted irregular defects in the previous section, we now use them to construct 4d $\cN=2$ SCFTs and study properties of the resulting theories. In particular, we determine their flavor symmetry and Coulomb branch spectrum, and propose conjectured formulae for the flavor and conformal central charges. We later offer nontrivial checks for these conjectures.
Classification of theories from twisted irregular defects
---------------------------------------------------------
We are interested in 4d $\cN=2$ SCFTs engineered by compactifiying six dimensional $(2,0)$ SCFT of ADE type on twice-punctured $\mP^1$ with outer-automorphism twist. The twist line connects one twisted irregular singularity and one twisted regular singularity on $\mP^1$. We will refer to such 4d theories as twisted theories (see Figure \[fig:tsp\]).
![The class S setup for twisted theories: one twisted irregular defect (star) and one twisted regular defect (dot) on a sphere.[]{data-label="fig:tsp"}](tsp.pdf){width=".8\textwidth"}
One common feature of the twisted theories is their non-simply-laced flavor groups $G$ (see Table \[table:outm\]) coming from the twisted regular punctures.
As we have reviewed in the last section, the classification of twisted irregular punctures is reduced to that of torsion outer-automorphisms of the Lie algebra $\mfj$ and the associated gradings . In this section, we focus on the regular semisimple type (see Section \[sec:rst\]). The torsion outer-automorphism in this case corresponds to a regular element of the twisted Weyl group $W_t(\mfh)$ and induces a grading on the Cartan subalgebra $\mfh$, labelled by $d_t$ in Table \[table:dt\].
The corresponding twisted irregular singularity of the Higgs field takes the following forms & \_z=[Tz\^[2+[2k’+1b\_t]{}]{}]{}+…, \^[|o|]{}=\^2 A\_N, \^2 D\_N, \^2 E\_6 [and]{} b\_t=h\_t\
& \_z=[Tz\^[2+[3k’1b\_t]{}]{}]{}+…, \^[|o|]{}=\^3 D\_4 [and]{} b\_t=h\_t\
& \_z=[Tz\^[2+[k’b\_t]{}]{}]{}+…, b\_th\_t. \[thptogether\] Here $T$ is regular semisimple, $b_t$ takes the values of $d_t'$ as in and $k'$ is an arbitrary integer such that the leading pole order is larger than one. Recall that $h_t$ denotes the twisted Coxeter number.
For simplicity we take the regular twisted puncture to be maximal. Some physical properties of the resulting twisted theories can be extracted as follows:
- The SW curve at the conformal point takes the forms listed in Table \[table:SW\]. Note that now the $z$ variable admits fractional powers.
$\mfj$ Degrees of the Casimirs $d_i$ Transformation under $o$
------------------ ------------------------------- ----------------------------------------------------------------------------------------
$A_{N}$ $2,3,\ldots, N$ $\phi_{d_i}\to (-1)^{d_i}\phi_{d_i}$
\[-1em\] $D_{N}$ $2,4,\ldots, 2N-2,N$ $\tilde{\phi}_{N}\to-\tilde{\phi}_{N}$
\[-1em\] $D_{4}$ $2,4,4$ $\phi_4\to e^{2\pi i\over 3}\phi_4,~\tilde{\phi}_4\to e^{4\pi i\over 3}\tilde{\phi}_4$
\[-1em\] $E_6$ $2,5,6,8,9,12$ $\phi_{d_i}=(-1)^{d_i}\phi_{d_i}$
: Casimirs and their transformations under outer-automorphisms[]{data-label="table:transf"}
- The regular twisted puncture gives rise to non-Abelian flavor symmetry $G$ (see Table \[table:outm\]). There are two cases with $C_N$ flavor symmetry from $\mZ_2$ twisted defects in $A_{2N}$ and $D_{N+1}$ $(2,0)$ theories. We label them by $C^{\rm anom}_N$ and $C^\text{\sout{anom}}_N$ respectively.[^29] They differ by Witten’s global anomaly for $USp(2N)$ [@Tachikawa:2018rgw]: the former carries a nontrivial anomaly whereas the latter is non-anomalous.
- The number of mass parameters for $U(1)$ flavor symmetries associated to the twisted irregular puncture comes from $\Delta=1$ parameters in the Hitchin poles, as summarized in Table \[table:aedata\]-\[table:d4data\].
- The dimension of the conformal manifold can be extracted from $\Delta=0$ parameters for the Hitchin poles, as summarized in Table \[table:aedata\]-\[table:d4data\].
- The full Coulomb branch spectrum can be found by expanding the degree $d_i$ differentials $\phi_{d_i}(z)$ in $z$, where $d_i$ labels the degrees of the Casimirs of the parent ADE theory, see Table \[table:transf\] for these numbers. The novelty here is that some of the differentials are no longer holomorphic and they often have a Laurent expansion with half integral (or plus-minus one third) powers of $z$, according to their transformation rule under the outer-automorphism in Table \[table:transf\]. The spectrum of half-BPS Coulomb branch chiral primaries is then summarized in Table \[table:coulomb\].
Flavor group $G$ Coulomb branch spectrum $\Delta$
-------------------------- -----------------------------------------------------------------
$C_N^\text{{anom}}$ $d_i-{2j+1\over 2}\Delta[z]>1,~~d_i=3,5,\ldots, 2N +1,~j\geq 0$
$d_i-j\Delta[z]>1,~~d_i=2,4,\ldots, 2N,~j\geq 1$
$B_N$ $d_i-{2j+1\over 2}\Delta[z]>1,~~d_i=3,5,\ldots, 2N-1,~j\geq 0$
$d_i-j\Delta[z]>1,~~d_i=3,5,\ldots, 2N,~j\geq 1$
$C_N^\text{\sout{anom}}$ $d_i-{2j+1\over 2}\Delta[z]>1,~~d_i=N+1,~j=0,\ldots$
$d_i-{j}\Delta[z]>1,~~d_i=2,\ldots,2N,~j=1,\ldots$
$G_2$ $d_i-{3j+1\over 3}\Delta[z]>1,~~d_i=4,~j\geq 0$
$d_i-{3j+2\over 3}\Delta[z]>1,~~d_i=4,~j\geq 0$
$d_i-{j}\Delta[z]>1,~~d_i=2,~j\geq 1$
$F_4$ $d_i-{2j+1\over 2}\Delta[z]>1,~~d_i=5,9,~j\geq 0$
$d_i-{j}\Delta[z]>1,~~d_i=2,6,8,12,~j\geq 1$
: Coulomb branch spectrum of the twisted theories from a twisted irregular puncture of the regular-semisimple type and a maximal twisted regular puncture. The Coulomb branch chiral primaries are constrained to have $\Delta>1$ by unitarity.[]{data-label="table:coulomb"}
dimension $h$ $h^{\vee}$ $r^\vee$ $n$
-------------------------- ----------- -------- ------------ ---------- -----
$A_{N-1}$ $N^2-1$ $N$ $N$ 1 1
$B_N$ $(2N+1)N$ $2N$ $2N-1$ 2 2
$C_N^\text{{anom}}$ $(2N+1)N$ $2N$ $N+1$ 2 4
$C_N^\text{\sout{anom}}$ $(2N+1)N$ $2N$ $N+1$ 2 2
$D_N$ $N(2N-1)$ $2N-2$ $2N-2$ 1 1
$E_6$ 78 12 12 1 1
$E_7$ 133 18 18 1 1
$E_8$ 248 30 30 1 1
$F_4$ 52 12 9 2 2
$G_2$ 14 6 4 3 3
: Some useful Lie algebra data. $h$ is the Coxeter number and $h^{\vee}$ is the dual Coxeter number. $r^\vee$ is the lacety of the Lie algebra and $n$ is equal to $r^\vee$ except for $C_N^\text{{anom}}$. []{data-label="table:lie"}
- We present conjectures for the flavor and conformal central charges with nontrivial evidences in the next section.
Flavor and conformal central charges
------------------------------------
Before we state the conjectural formulae for the flavor and conformal central charges, let us provide some motivations for how they come about. The important observation is, in all previously known class S constructions that involve maximal regular punctures, it appears that the flavor central charge is determined by certain maximal scaling dimension among the CB spectrum contributed by the maximal puncture:[^30] k\_[G]{}=& \_[max]{}, &G=A\_N,D\_N,E\_N\
k\_[G]{}=& \_[max]{},&G=B\_N,C\_N\^,F\_4,G\_2\
k\_[G]{}=&[12]{}\_[max]{}+[12]{},&G=C\_N\^ \[fcobs\] Here $\Delta_{\rm max}$ denotes the maximal scaling dimension contributed by the maximal regular puncture which determines the flavor central charge in the untwisted theories. In twisted theories, empirical evidence suggests that one should instead take the maximal scaling dimension $\widetilde \Delta_{\rm max}$ from the [*twisted*]{} differentials (i.e. a differential that transforms non-trivially under the twist, see Table \[table:transf\]) at the maximal puncture. The $\mZ_2$ twisted maximal punctures of $A_{2N}$ type requires special attention.[^31] For example, in a $\mZ_2$ twisted $A_{2N}$ type class S construction without irregular punctures, the maximal scaling dimension contributed by the twisted differentials at the maximal twisted regular singularity is $2N+1$ from $\phi_{2N+1}$, and the $USp(2N)$ flavor central charge computed in [@Chacaltana:2014nya] is equal to (in our normalization) $h^\vee(USp(2N))=N+1$ in agreement with . The introduction of irregular punctures into the setup modifies the $U(1)_R$ symmetry of the 4d theory, but we expect continues to hold. The formula for $2a-c$ is given in [@Shapere:2008zf] while our new formula for $c$ has a close relation to the 2d chiral algebra which we will explain in Section \[sec:voa\].[^32]
The central charge of the flavor symmetry $G$ of a twisted theory defined by a twisted irregular defect of the regular semisimple type in and a maximal twisted regular defect takes the following form: \[fla\] The Lie algebra data $h^\vee$ and $n$ are listed in Table \[table:lie\].
The conformal central charges of a general twisted theory are determined by Here $f$ is the number of mass parameters contributed by the irregular singularity, $k_G$ denotes the flavor symmetry central charge listed in , and $\Delta[u_i]$ are the Coulomb branch scaling dimensions listed in Table \[table:coulomb\].
Let us provide some evidences for the above conjectures by considering twisted theories defined with integral order Hitchin poles, in which case the irregular singularity takes the form \_z=[Tz\^[2+k]{}]{}+…\[integralpole\] where $T$ is regular semisimple and the associated grading of $\mfh$ corresponds to $d_t=1$. We will use following formula 2a-c=[14]{}\_[i]{}(2\[u\_i\]-1), a-c=-[\_ 24]{} \[higgs\] to compute their central charges. The second equation above is known to hold for the untwisted irregular puncture defined by integral order Hitchin poles [@Xie:2012hs] and we assume that it is also valid for the current situation.[^33]
Here we take the regular twisted puncture to be maximal in which case the Higgs branch dimension is [@Chacaltana:2012zy; @Wang:2015mra] \_=[12]{}((G)-(G))+(G). \[hb\] The last term above comes from the twisted irregular puncture.
On the other hand, since the irregular singularity contributes $f=\rank(G)$ mass parameters (see top rows of Table \[table:aedata\]-\[table:d4data\]), the conjectural formula for central charge $c$ takes the following form c=[112]{}[k\_G (G) -k\_G+h\^]{}-[112]{}(G). \[4d2dca\] Here the conjectured flavor central charge takes the form k\_G=h\^-[1n]{}[1k+1]{}. \[4dkg\] We have verified that that the two formulae and give the same answers. Below we give some details for two instances of such checks for illustration.
Let’s consider the $C^{\text{\sout{anom}}}_{N-1}$ theory which is constructed by $\mZ_2$ twist of the $D_N$ $(2,0)$ theory. The Hitchin pole fixes the $U(1)_R$ charge (hence scaling dimension) of $z$, =[1k+1]{} and the Coulomb branch spectrum can be enumerated using Table \[table:coulomb\] ={2i-[jk+1]{}>1 | i=1,2,…,N-1, j1}{N-[2j+12(k+1)]{}>1 | j0}. Therefore & 2a-c=[14]{}\_[i]{}(2-1)= (N-1) N (4 k N-2 k+4 N-5). Along with the Higgs branch dimension from & \_= (N-1) N we obtain from $$\begin{aligned}
& a= \frac{1}{24} (N-1) N (8 k N-4 k+8 N-9),~c=\frac{1}{6} (N-1) N (2 k N-k+2 N-2)\end{aligned}$$ which is in agreement with where the flavor central charge is determined by to be k\_[C\_[N-1]{}]{}=N-[12(k+1)]{}. .
Similarly let us consider the $F_4$ theory which is engineered by $\mZ_2$ twist of the $E_6$ $(2,0)$ theory with irregular punctures. Once again =[1k+1]{} and the Coulomb branch spectrum is ={d-[jk+1]{}>1 | d=2,6,8,12 [and]{} j1}{d-[2j+12(k+1)]{}>1 | d=5,9 [and]{} j0} from Table \[table:coulomb\], which gives & 2a-c=[14]{}\_[i]{}(2\[u\_i\]-1)=78 k+71. Next the Higgs branch dimension follows from & \_=28. We thus obtain from & a= 78 k+, c=78 k+ which is in agreement with where k\_[F\_4]{}=9-[12(k+1)]{}. from .
Twisted theories with Lagrangians
---------------------------------
It turns out that many sub-families of the theories engineered from twisted irregular defects actually have Lagrangian descriptions.[^34] Since the Coulomb branch spectrum for Lagrangian theories have integral scaling dimensions, a necessary condition is $\Delta[z]\equiv 0\,({\rm mod}~2)$ for $A_{n},D_n,E_6$ theories with $\mZ_2$ twist, and $\Delta[z]\equiv 0\,({\rm mod}~3)$ for $D_4$ with $\mZ_3$ twist. This can be achieved by choosing $k$ appropriately with respect to $b_t$ in the Hitchin pole .
Since such theories have a weakly coupled frame, we can use the formulae a=[5n\_v24]{}+[n\_h24]{}, c=[n\_v6]{}+[n\_h12]{} \[weakcc\] to compute the conformal central charges. Here $n_v$ and $n_h$ count the number of vector and hypermultiplets in the quiver gauge theory description. Similarly the central charges associated to the $G_{\rm flavor}$ flavor symmetry of hypermultiplets can be obtained straightforwardly from identifying the embedding $G_{\rm flavor}\times G_{\rm gauge} \subset USp(2n_h)$. This allows us to verify the conjectured formulae and .
As we will see, often times the Lagrangian description only makes manifest a subgroup of the full flavor symmetry which is realized in our description by the single regular puncture.
\[eg:3\] Let’s take $C^\text{\sout{anom}}_N$ which comes from the $\mZ_2$ twist of $D_{N+1}$ $(2,0)$ theory. The twisted irregular puncture is specified by $$b_t=2N,\quad k=-N$$ so that we have the scaling dimension $\Delta[z]=2$. The Coulomb branch spectrum for this theory (using Table \[table:SW\] and \[table:coulomb\]) is & =\_[j=1]{}\^[N-1]{}{2i|1ij}{N-2i>1|| i0} The central charges from and are N [even]{}: & k\_[USp(2N)]{}=N, a=[124]{}N(1+N+4N\^2),c=[112]{}N\^2(1+2N).\
N [odd]{}: & k\_[USp(2N)]{}=N, a=[124]{}(-4+N+N\^2+4N\^3),c=[112]{}(-1+N\^2+2N\^3). \[kaceg3\] Note that from Table \[table:ddata\], the $N$ even case has $N$ marginal couplings and no mass parameters from the irregular puncture whereas the $N$ odd case has $N-1$ marginal couplings and an extra mass parameter. This results in the different expressions for the conformal central charges above.
On the other hand, this theory has a quiver gauge theory description by
[ccc]{}
for $N$ even, and
[ccc]{}
for $N$ odd. It is easy to see that the quiver is balanced thus all vector multiplets are conformally gauged.
The boxed nodes label the flavor symmetry of hypermultiplets. Here the symmetry is $USp(2N)$ with central charge k\_[USp(2N)]{}=N as provided by $2N$ hypermultiplets in the fundamental representation of $USp(2N)$. We can also count the number of hyper and vector multiplets from the quiver. N [even]{}: &n\_h=[13]{}N(-2+3N+2N\^2), n\_v=[13]{}(N+2N\^3)\
N [odd]{}: &n\_h=1+[23]{}N(-2+N+2N\^2), n\_v=[13]{}(-3+N+2N\^3) which gives the central charges by N [even]{}: &a=[124]{}N(1+N+4N\^2), c=[112]{}N\^2(1+2N)\
N [odd]{}: &a=[124]{}(-4+N+N\^2+4N\^3), c=[112]{}(-1+N\^2+2N\^3) Both the flavor and conformal central charges from the Lagrangian are in agreement with obtained from our conjectured formulae and .
Let’s take $G=C_N^\text{{anom}}$ which comes from the $\mZ_2$ twist of $A_{2N}$ $(2,0)$ theory. The twisted irregular puncture is specified by $$b=4N+2,~~k=-2N-1$$ so that $\Delta[z]=2$. As before, the Coulomb branch spectrum (from Table \[table:coulomb\]) and central charges (from and ) are & =\_[j=1]{}\^[N-1]{}{2i|1ij} \_[j=1]{}\^[N-1]{}{2i|1ij}{ 2i || 1iN}\
&k\_G=N+[12]{} , a=[N(8N\^2+7N+3)24]{} ,c=[N(1+2N)\^212]{}. \[kaceg4\] In particular, the irregular singularity contributes no additional mass parameters and the theory has $2N-1$ marginal couplings (see Table \[table:aedata\]). The theory has a Lagrangian description by
[ccc]{}
so we can compute the central charges using the field content n\_v=[N(4N\^2+3N+2)3]{}, n\_h=[N(4N\^2+6N-1)3]{} which gives a=[N(8N\^2+7N+3)24]{}, c=[N(1+2N)\^212]{}. Moreover the flavor central charge is supplied by $2N+1$ half hypers in the fundamental representation of $USp(2N)$ thus k=N+[12]{} Everything above is consistent with .
Let’s take $G=B_N$ which is derived from the $\mZ_2$ twist of $A_{2N-1}$ $(2,0)$ theory. The twisted irregular puncture is specified by $$b_t=2N,~~k=-N$$ and then $\Delta[z]=2$. The Coulomb branch spectrum and central charges from our general formulae &=\_[j=1]{}\^[N-1]{}{2i|1ij} \_[j=1]{}\^[N-1]{}{2i|1ij}\
&k\_[SO(2N+1)]{}=2N-2, a=[N(8N\^2-5N-3)24]{}, c=[N(2N\^2-N-1)6]{}. \[kaceg5\] From Table \[table:aodata\] the irregular singularity contributes no additional mass parameters and $2N-2$ marginal couplings.
The theory has a Lagrangian description as
[ccc]{}
so we can compute the central charges using the field content n\_v=[N(4N\^2-3N-1)3]{}, n\_h=[4N( N\^2 -1)3]{} Hence a=[N(8N\^2-5N-3)24]{}, c=[N(2N\^2-N-1)6]{}. Moreover the flavor central charge is supplied by $2N-2$ half-hypers in the fundamental representation of $SO(2n+1)$ thus k\_[SO(2N+1)]{}=2N-2 Again we see they are in agreement with ,
Half-hypermultiplet
A free half-hyper multiplet in the fundamental representation of $USp(2N)$ flavor symmetry can be constructed using $A_{2N}$ $(2,0)$ theory with $\mZ_2$ twist. The irregular puncture is specified by b\_t=4N+2, k=1-(4N+2) It is easy to see from our general formulae that the Coulomb branch is empty in this case and the central charges are k\_[USp(2N)]{}=[12]{}, 2a=c=[N12]{} as expected for a half-hyper in the fundamental representation of $USp(2N)$ (or $N$ free half-hypers).
We emphasize here that this is the only twisted theory within our construction that has an empty Coulomb branch yet nonvanishing central charge. \[eg:hh\]
Let’s take $G=C^\text{\sout{anom}}_{N-1}$ which is derived from the $\mZ_2$ twist of $D_{N}$ $(2,0)$ theory. We take $N=3n$ with $n\in \mZ^+$ and the irregular puncture is specified by b\_t=6n, k=3-6n so that $\Delta[z]=2n$. The Coulomb branch spectrum and central charges from our general formulae &={2,4,…,2n}{2,4,…,4n-2}\
&k\_[USp(6n-2)]{}=2n, a= (66 n\^2-33 n+5 ) , c=3 n\^2-+ \[kaceg7\] From Table \[table:ddata\] the irregular singularity contributes no additional mass parameters and $2$ marginal couplings.
The theory has a Lagrangian description by
[ccc]{}
We can then check the central charge by counting the multiplets n\_v=(n-1)(2n-1)+2n(4n-1),n\_h=2(n-1)2n+2n2(3n-1) which gives a= (66 n\^2-33 n+5 ) , c=3 n\^2-+. Furthermore the flavor central charge for $USP(6N-2)$ is k\_G=2N from the quiver, in perfect agreement with .
The simplest example in this sequence of theories is when $n=1$ which can be equivalently described by a cyclic quiver with two $SU(2)$ nodes,
[ccc]{}
and comes from type $A_1$ $(2,0)$ theory on $T^2$ with two punctures. This theory has $USp(4)$ enhanced flavor symmetry which is manifest in our description from type $D_3$ $(2,0)$ theory on $\mP^1$ with twisted irregular punctures.[^35]
Let’s take $G=C_N^\text{\sout{anom}}$ which is engineered from the $\mZ_2$ twist of $D_{N+1}$ $(2,0)$ theory. We take $N=3n$ and the irregular singularity specified by b\_t=6n, k=3-6n so that $\Delta[z]=2n$. The Coulomb branch spectrum and central charges from our general formulae &={2,4,…,2n}{2,4,…,4n}{2n+1}\
&k\_[USp(6n)]{}=2n+1, a= (66 n\^2+45 n+5 ) , c=3 n\^2++ \[kaceg8\] From Table \[table:ddata\] the irregular singularity contributes one mass parameter and $2$ marginal couplings.
The Lagrangian description is given by
[ccc]{}
From the number of multiplets n\_v=(2n+1)(4n+1)+n(2n+1),n\_h=3n(4n+2)+(4n+2)n+2n, we obtain the conformal central charges from a= (66 n\^2+45 n+5) , c= 3 n\^2+2 n+, as well as the flavor central charge k\_[USp(6n)]{}=2n+1, in agreement with .
Consider the $F_4$ theory constructed from $E_6$ $(2,0)$ theory with $\mZ_2$ twist and the irregular puncture is specified by b\_t=12, k=-9. so that $\Delta[z]=2$. The Coulomb branch spectrum and central charges can be read off from our general formulae &={10,8,6,4,2,8,6,4,2,6,4,2,4,2,4,2}\
&k\_[F\_4]{}=8, a=[2036]{}, c=[1043]{}. \[kaceg9\] From Table \[table:e6data\] the irregular singularity contributes no mass parameter and $5$ marginal couplings.
The theory has a Lagrangian description by
[ccc]{}
By counting multiplets we obtain the conformal central charges a=[2033]{}, c=[1043]{}, as well as the flavor central charge for the $SO(9) \subset F_4$ k\_[SO(9)]{}=8. They are consistent with since the index of embedding $I_{\mf{so}_9\hookrightarrow \mff_4 }=1$ and our description with irregular puncture makes manifest the enhanced $F_4$ flavor symmetry of the theory.[^36]
Twisted theories and non-Lagrangian conformal matter
-----------------------------------------------------
In addition to the Lagrangian examples discussed in the last section, our twisted theories also generalize many non-Lagrangian conformal matter theories constructed in class S with regular (tame) punctures. Below we provide various examples and their reincarnation in our construction with twisted irregular punctures. Various physical data of these conformal matter theories have been extracted from the SW geometry, superconformal index, and decoupling limits of certain Lagrangian theories. We will view these information as support for our construction of the much larger class of theories and nontrivial evidence for our conjectured formulae for the central charges and . As we will see in the examples, our construction often makes manifest the enhanced global symmetry which is obscure in the ordinary (regular punctures) class S setup.
$R_{2,2N}$ conformal matter
The $R_{2,2N}$ non-Lagrangian theory was constructed in [@Chacaltana:2014nya] from the $\mZ_2$ twist of type $A_{2N}$ $(2,0)$ theory with three regular punctures on a sphere: one minimal untwisted puncture, and two maximal twisted punctures.[^37] This theory also arises in the decoupling limit of $\cN=2$ $SU(2N+1)$ SYM coupled to one symmetric and one antisymmetric rank two tensor hypermultiplets in a S-dual frame [@Chacaltana:2014nya].
The Coulomb branch spectrum of $R_{2,2N}$ SCFT to be ={3,5,7,…, 2N+1} \[R22Ncoulomb\] and the conformal central charges are a=[14N\^2+19N+124]{},c=[8N\^2+10N+112]{}. \[R22Nac\] The theory has enhanced $U(1)\times USp(4N)$ flavor symmetry where only the maximal subgroup $U(1)\times USp(2N)\times USp(2N)$ is manifest from the regular punctures in $A_{2N}$. The $USp(4N)$ factor has central charge k\_[USp(4N)]{}=N+1. \[R22Nfc\]
Alternatively, the $R_{2,2N}$ theories can be constructed from type $A_{4N}$ $(2,0)$ theory with $\mZ_2$ twist and the twisted irregular puncture is specified by b\_t=4N, k=1-4N such that $\Delta[z]=4N$. One can immediately read off the Coulomb branch spectrum from Table \[table:coulomb\] and the result coincides with . The manifest $USp(4N)$ flavor symmetry comes from the regular twisted puncture and its flavor central charge is determined by to be . Our description also makes obvious Witten’s global anomaly for $USp(4N)$ [@Tachikawa:2018rgw]. From Table \[table:aedata\] we see that the irregular puncture provides the additional mass parameter responsible for the $U(1)$ factor in the flavor symmetry. It is also easy to check that the central charges computed from is consistent with the result from [@Chacaltana:2014nya].
[$USp(2N)$ conformal matter]{}
Let is consider $A_{2N}$ $(2,0)$ theory with $\mZ_2$ twist and the irregular puncture is specified by b\_t=2N, k=1-2N which gives $\Delta[z]=2N$. For $N$ even this is just the $R_{2,2N}$ theories in the last example. Here we focus on $N$ odd which has $USp(2N)$ flavor symmetry. Our general prescription for the twisted theories give &={N+1, N-1,…, 2}\
&k\_[USp(2N)]{}=[N2]{}+1, a=[(7N+5)(N+2)48]{}, c=[(2N+1)(N+2)12]{}. \[USPcmkac\] From Table \[table:aedata\] we see the irregular puncture contributes one marginal coupling but no mass parameters.
For $N=5$, this is related to the $SU(2)_4\times USp(10)_{7\over 2}$ SCFT of [@Chacaltana:2013oka] with $\Delta=\{4,6\}$ and $(n_h,n_v)=(35,18)$ where the $SU(2)_4$ flavor symmetry is gauged by an $SU(2)$ vector multiplet.
[ccc]{}
For $N=3$, this is the rank one $SU(2)_4\times USp(6)_{5\over 2}$ SCFT of [@Chacaltana:2013oka] with $\Delta=\{4 \}$ and $(n_h,n_v)=(15,7)$ where the $SU(2)$ flavor symmetry is gauged by a $SU(2)$ vector multiplet.
For $N=1$, we get the familiar $\cN=4$ $SU(2)$ SYM with $n_h=n_v=3$! The $SU(2)$ flavor symmetry with central charge $k_{SU(2)}={3\over 2}$ is now realized manifestly by the twisted regular puncture in $A_2$ $(2,0)$ theory. It also carries Witten’s global anomaly for $SU(2)$.
[$SO(2N+1)$ and $SO(2N+1)\times U(1)$ conformal matter ]{}
Let’s consider the $A_{2N-1}$ $(2,0)$ theory with the $\mZ_2$ twist and the irregular singularity is defined by b\_t=2N, k=1-2N so that $\Delta[z]=2N$. From the general formulae, we have &={N-1-2i>1|i0}\
&k\_[SO(2N+1)]{}=N-1 and the conformal central charges N [even]{}: &a=[7N\^2-5N-1048]{}, c=[2N\^2-N-212]{},\
N [odd]{}: &a=[7N\^2-5N-248]{}, c=[2N\^2-N-112]{}. From Table \[table:aodata\], in the $N$ even case the irregular puncture contributes no marginal couplings but one additional mass parameter, whereas the $N$ odd case has one marginal coupling and no extra mass parameter.
Upon closer inspection, it turns out that this $\mZ_2$ twisted class $A_{2N-1}$ setup does not make manifest the full flavor symmetry. For $N$ odd, the theory is identical to the $USp(N-1)$ SYM conformally coupled to $N+1$ fundamental flavors which has $SO(2N+2)$ symmetry. For $N$ even the theory is identified with the $R_{2,N-1}$ theories in [@Chacaltana:2010ks]. Our formulae above again give the correct conformal and current central charges as computed previously with standard methods.
[$G_2$ conformal matter]{}
Let’s consider the $D_4$ $(2,0)$ theory with $\mZ_3$ twist and choose the irregular puncture to be given by b\_t=12, k=-8. Then $\Delta [z]=3$. Our general prescription gives &={2,3,3}, k\_[G\_2]{}=3, a=[278]{}, c=[72]{}. \[G2cm\] From Table \[table:d4data\] we see the irregular puncture contributes one marginal coupling but no mass parameters.
This is identified with the $E_6$ Minahan-Nemeschansky (MN) theory [@Minahan:1996cj] with an $SU(3)$ subgroup of $E_6$ flavor symmetry gauged.
[ccc]{}
More explicitly, the relevant maximal Lie algebra embedding is $\mfe_6 \supset \mf{su}_3\oplus \mfg_2$ with embedding indices determined by the branching rule $\bf 27\to (6,1)\oplus (3,7)$ to be I\_[\_2\_6 ]{}=1,I\_[\_3\_6 ]{}=2. Therefore the gauged $SU(3)$ global symmetry has central charge $k_{SU(3)}=6$ which ensures conformal invariance. The commutant $G_2$ becomes the residue global symmetry with $k_{G_2}=3$ consistent with .
[$F_4\times U(1)$ conformal matter]{}
Let’s consider the $E_6$ $(2,0)$ theory with $\mZ_2$ twist and the irregular singularity is defined by b\_t=8, k=-7 such that $\Delta[z]=8$. The physical data from our general formulae &={4,5}\
&k\_[F\_4]{}=5, a=[143]{}, c=[163]{} The full global symmetry of the theory is $U(1)\times (F_4)_5$ where extra $U(1)$ comes from the irregular puncture (see Table \[table:e6data\]).
This theory is identified with the $SO(9)_{5}\times U(1)$ SCFT in [@Chacaltana:2015bna]. Note that our description predicts the enhancement of flavor symmetry from $SO(9)_5$ to $(F_4)_5$.
[$F_4$ conformal matter]{}
Let’s consider another irregular defect in the $E_6$ theory with $\mZ_2$ twist specified by b\_t=12, k=-10. Then $\Delta[z]=6$ and the physical data from our general formulae &={2,2,6,6}\
&k\_[F\_4]{}=6, a=[476]{}, c=[263]{} From Table \[table:e6data\], the irregular puncture contributes two marginal coupling but no mass parameters.
This theory is identified with the $(E_8)_6$ MN theory and $(D_4)_2$ SW theory[^38] with the diagonal $(G_2)_8$ flavor symmetry subgroup gauged
[ccc]{}
The relevant subalgebras are $\mfg_2 \oplus \mff_4 \subset \mf{e}_8$ and $\mfg_2 \subset \mf{so}_7 \subset \mf{so}_8$. It follows from the branching rules $\bf 248\to (14,1 )\oplus(7,26)\oplus(1,52)$ and $\bf 8 \to 7\oplus 1$ that I\_[\_2\_8 ]{}= I\_[\_2\_8]{}=1. Thus the $G_2$ diagonal subgroup of has central charge $6+2=8$ which ensures conformal invariance.
Vertex operator algebra of twisted theories {#sec:voa}
===========================================
It was shown in [@Beem:2013sza] that for any 4d $\mathcal{N}=2$ SCFT, one can associate a 2d chiral algebra or vertex operator algebra (VOA). The basic correspondence is as follows:
- The 2d Virasoro central charge $c_{2d}$ is given in terms of the conformal anomaly $c_{4d}$ of the 4d theory as c\_[2d]{} = -12 c\_[4d]{}.
- The global symmetry algebra $\mathfrak{g}$ becomes an affine Kac-Moody algebra $\hat{\mathfrak{g}}_{k_{2d}}$ and the level of affine Kac-Moody algebra $k_{2d}$ is related to the 4d the flavor central charge $k_F$ by $$k_{2d}=- k_F \ .
\label{eq:centralchargerelation}$$
- The (normalized) vacuum character of the chiral algebra/VOA is identical to the Schur index of the 4d $\cN=2$ theory: $$\begin{aligned}
\chi_0(q) = \text{I}_{\text{Schur}}(q) \ . \end{aligned}$$
We focus on the twisted theory where there is no mass parameter from the irregular singularity, and the regular singularity is labeled by a nilpotent orbit $Y$.[^39] Our proposal for the corresponding VOA is following
The VOA for the twisted theory is the W-algebra $W^{k_{2d}}(G,Y)$. Here $G$ is the flavor symmetry corresponding to the maximal twisted regular puncture, and $Y$ labels the corresponding nilpotent orbit. This W algebra is derived as the Drinfeld-Sokolov reduction of the Kac-Moody algebra $\widehat{\mfg}_{k_{2d}}$ associated to the simple Lie algebra $\mfg$ at level $k_{2d}$.
The ADE cases of these W-algebras is considered in [@Xie:2016evu; @Song:2017oew], and here we discover the correspondence for non-simply-laced simple Lie groups. We have verified the relation between the 4d central charges and the 2d central charges.
### Admissible levels of the 2d current algebra {#admissible-levels-of-the-2d-current-algebra .unnumbered}
A 2d current algebra level is called admissible if it can be written in one of the following forms: & k\_[2d]{}=-h+[pq]{}, (p,q)=1, ph, G=ADE,\
& k\_[2d]{}=-h\^+[pq]{}, (p,q)=(q,r\^)=1, ph\^, G=BCFG,\
& k\_[2d]{}=-h\^+[pr\^q]{}, (p,q)=(p,r\^)=1, ph, G=BCFG.
Recall the 2d levels in our case are given by following formula: $$k_{2d}=-h^{\vee}+{1\over n}{b_t\over k+b_t}~{\rm with }~k+b_t\geq 1$$ where $b_t$ takes the values of $d_t$ in Table \[table:dt\] and $n$ is as listed in Table \[table:lie\]. We observe that the 2d current algebra levels of the class I twisted theories are admissible (for generic $k'\in \mZ$): & B\_N [Class I]{}:k\_[2d]{}=-h\^+[2N-12k’+1+4N-2]{}, b\_t=4N-2\
& C\_N\^ [Class I]{}:k\_[2d]{}=-h\^+[12]{}[2N+14N+2+2k’+1]{}, b\_t=4N+2\
& C\_N\^ [Class I]{}:k\_[2d]{}=-h\^+[N+12N+3+2k’]{}, b\_t=2N+2\
&G\_2 [Class I]{}:k\_[2d]{}=-h\^+[412+3k’1]{}, b\_t=12\
&F\_4 [Class I]{}:k\_[2d]{}=-h\^+[918+2k’+1]{}, b\_t=18. \[admi\] Note that for these cases the relevant torsion outer-automorphism that defines the twisted irregular defect is always generated by the twisted Coxeter element.
The Schur index of the twisted theory is then identified with the vacuum character of vertex operator algebra, which is particularly simple for the boundary levels [@kac2017remark]: k\_[2d]{}=-h\^+[h\^q]{}, (h\^,n)=1, (h\^,q)=1. Comparing with the 2d levels in our list, we see that the boundary levels appear for the $B_N,~C_N^\text{\sout{anom}},~G_2,~F_4$ cases.
### Associated variety and Higgs branch {#associated-variety-and-higgs-branch .unnumbered}
The Higgs branch of the 4d $\cN=2$ SCFT is identified with the associated variety $X_{\cal V}$ of the VOA [@Song:2017oew; @Beem:2017ooy] \_[Higgs]{}=X\_[V]{}. For affine Kac-Moody algebra with an admissible level (see which corresponds to the case $Y=F$ and we have a maximal regular puncture), the associated variety $X_{\cal V}=X_M$ is found to be the closure of certain nilpotent orbits in [@arakawa2015associated] We summarize the result of [@arakawa2015associated] for the relevant orbits $\mathbb O_q$ and ${}^L\mathbb O_q$ here in Table \[table:no1\]-\[table:no4\].
For $q\geq h(\mfg)$, $\mathbb O_q$ is the same as the principal (maximal) nilpotent orbit $\mathbb O_{\rm prin}$ with quaternionic dimension \_ O\_[prin]{}=[12]{}(-). Similarly for $q\geq h^\vee(\mfg^\vee)$, ${}^L\mathbb O_q=\mathbb O_{\rm prin}$.
For other values of $q$, in Table \[table:no1\]-\[table:no2\], we give the usual labelling of a nilpotent orbit of classical Lie algebras by a partition $[n_i]$. The quaternionic dimension can be easily computed by (see §6 of [@collingwood1993nilpotent]) \_O\_[\[n\_i\]]{}=
[N(2N+1)2]{}-[14]{}\_i s\_i\^2+[14]{} \_[i[odd]{}]{} r\_i & [if]{} =B\_N\
[N(2N+1)2]{}-[14]{}\_i s\_i\^2-[14]{} \_[i[odd]{}]{} r\_i & [if]{} =C\_N
where $[s_i]$ is the transpose partition to $[n_i]$, and $r_j$ counts the number of appearances of the part $j$ in $[n_i]$. For exceptional Lie algebras, we use the Bala-Carter labels [@collingwood1993nilpotent] for the nilpotent orbits in Table \[table:no2\]-\[table:no1\]. We also include the quaternionic dimensions for the reader’s convenience.
Near the lower end of the list of nilpotent orbits, we have the minimal nilpotent orbit of the smallest nonzero dimension. It corresponds to the centered one-instanton moduli space of $\mfg$. Here the minimal nilpotent orbit of $C_N$ labelled by partition $[2,1^{2N-2}]$ shows up in Table \[table:no1\] at $q=1$, in which case the twisted theory is nothing but $N$ free hypermultiplets (see Example \[eg:hh\]). On the other hand, the minimal nilpotent orbits of $B_N$, $G_2$ and $F_4$ do not appear to be Higgs branches of our twisted theories of the regular semisimple type. These are consistent with the results of anomaly matching on the Higgs branch in [@Shimizu:2017kzs].
**Example**: For illustration we consider a class I $F_4$ theory in . We take $k'=-7$, so the 2d current algebra level $k_{2d}=-9+{9\over 5}$ and we have $q=5$. Looking at Table \[table:no4\], we conclude that the corresponding Higgs branch is the closure of nilpotent orbit with label $F_4(a_3)$ which has quaternionic dimension 20. The Coulomb branch spectrum of this theory is $\Delta=\left\{\frac{6}{5},\frac{12}{5},\frac{16}{5},\frac{18}{5},\frac{22}{5},\frac{24}{5},\frac{36}{5},\frac{42}{5}\right\}$ and the central charges are $a={247\over 15},c={52\over 3}$.
Let’s consider the more general twisted theories where the twisted irregular puncture is as before in but the twisted regular puncture is now labeled by a general nilpotent orbit $Y$ of the group $G$. The associated variety $X_{\cal V}$ is then given by X\_[V]{}=X\_M S\_Y. which describes the Higgs branch for these theories. Here $S_Y$ is the Slodowy slice defined by the nilpotent orbit $Y$ [@Song:2017oew; @Beem:2017ooy].
Lie algebra $\mfg$ $q$ odd ${}^L\mathbb{O}_q$
-------------------- -------------- --------------------------------------------------------------------------------------
$C_N$ $q\geq 2N-1$ $\mathbb O_{\rm prin}=(2N)$
$q< 2N-1$ $(q+1,\underbrace{q,\ldots, q}_{even}, s),~0\leq s \leq q-1, ~s~{\rm even}$
$(q+1,\underbrace{q,\ldots, q}_{\rm even}, q-1,s),~0\leq s \leq q-1,~ s~~{\rm even}$
: Nilpotent orbits ${}^L\mathbb{O}_q$ in $C_N$ Lie algebras[]{data-label="table:no1"}
Lie algebra $\mfg$ $q$ odd $\mathbb O_q$
-------------------- ------------- ------------------------------------------------------------------------------
$C_N$ $q\geq 2N $ $\mathbb O_{\rm prin}=(2N)$
$q< 2N$ $(\underbrace{q,\ldots, q}_{\rm even}, s),~0\leq s \leq q-1,~s~\rm even$
$(\underbrace{q,\ldots, q}_{\rm even}, q-1,s),~0\leq s \leq q-1,~s~\rm even$
$B_N$ $q\geq 2N$ $\mathbb O_{\rm prin}=(2N+1)$
$q<2N$ $(\underbrace{q,\ldots, q}_{\rm even}, s),~0\leq s \leq q, ~s~\rm odd$
$(\underbrace{q,\ldots, q}_{\rm odd}, s,1),~0\leq s \leq q-1,~ s~\rm odd$
: Nilpotent orbits $\mathbb{O}_q$ in $B_N$ and $C_N$ Lie algebras[]{data-label="table:no2"}
Lie algebra $\mfg$ $q \neq 0{\,\rm mod\,}3$ $\mathbb{O}_q$ $\dim_\mH$
-------------------- -------------------------- ---------------- ------------
$G_2$ $\geq 6$ $G_2$ 6
$ 4,5$ $G_2(a_1)$ 5
$2$ $\tilde{A}_1$ 4
$1$ $0$ 0
: Nilpotent orbits $\mathbb{O}_q$ in $G_2$ Lie algebra[]{data-label="table:no3"}
Lie algebra $\mfg$ $q$ odd $\mathbb{O}_q$ $\dim_\mH$
-------------------- ----------- ------------------- ------------
$F_4$ $\geq 12$ $F_4$ 24
$ 9 ,11$ $F_4(a_1)$ 23
$ 7$ $F_4(a_2)$ 22
$ 5$ $F_4(a_3)$ 20
$3$ $\tilde{A}_2+A_1$ 18
1 0 0
: Nilpotent orbits $\mathbb{O}_q$ in $F_4$ Lie algebra[]{data-label="table:no4"}
Conclusion {#sec:sum}
==========
We systematically studied irregular codimension-two defects twisted by outer-automorphism symmetries in 6d $(2,0)$ theories. They engineer 4d $\mathcal{N}=2$ Argyres-Douglas (AD) SCFTs that admit non-simply-laced flavor groups. We completed the classification of twisted irregular defects of the regular semisimple type, and the result was summarized in Table \[table:SW\]. Together with the classification of the ADE cases in [@Wang:2015mra], we have a large class of of Argyres-Douglas theories with arbitrary simple flavor groups. We outlined a simple procedure to extract their Coulomb branch spectrum, central charges and in some cases, the 2d chiral algebra and Higgs branch. One can also consider the degenerations of irregular singularities and regular singularities as in [@Wang:2015mra; @Xie:2017vaf; @Xie:2017aqx] which will give rise to many new AD theories.
The theories we constructed here should be thought of building blocks towards a better understanding of the full space of 4d $\cN=2$ SCFTs. On one hand, by conformally gauging the flavor symmetries of these AD theories we can form new 4d $\cN=2$ SCFTs. On the other hand, some of our theories admit exact marginal deformations, and it is interesting to study S-duality and weakly coupled gauge theory descriptions (which may involve non-Lagrangian matters) of these theories using the method in [@Xie:2017vaf; @Xie:2017aqx].
In this large space of theories, we saw various relations between the physical data, such as those between the central charges and Coulomb branch spectrum, that call for explanations. For example it would be nice to develop field-theoretic proofs for our conjectured formulae for the central charges and perhaps along the lines of [@Shapere:2008zf]. It would also be interesting to verify that the vacuum character of our proposed VOA matches with the Schur index of the 4d theory using the 2d TQFT [@Gadde:2009kb; @Gadde:2011ik; @Song:2015wta; @Buican:2017uka].
Lastly, we identified the VOA for the twisted theories defined using irregular defects which do not carry any flavor symmetry. It would be interesting to identify the VOA for the remaining theories.[^40]
We hope to address some of these directions in future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Davide Gaiotto and Yuji Tachikawa for useful discussions and correspondences. We also thank Yuji Tachikawa for reading an earlier draft of this paper and giving many helpful comments. YW would also like to thank Yuji Tachikawa and Gabi Zafrir for collaboration on related subjects. YW is grateful for the Galileo Galilei Institute for Theoretical Physics for the hospitality during the completion of this work. The work of DX is supported by Center for Mathematical Sciences and Applications at Harvard University, and in part by the Fundamental Laws Initiative of the Center for the Fundamental Laws of Nature, Harvard University. YW is supported in part by the US NSF under Grant No. PHY-1620059 and by the Simons Foundation Grant No. 488653.
[^1]: The conformal bootstrap has proven to be an efficient tool to probe the fixed point physics. See [@Poland:2018epd] for an overview and in particular [@Beem:2015aoa] for the application of bootstrap methods to the 6d $(2,0)$ SCFT.
[^2]: For this reason, we will use defect, puncture and singularity interchangeably when referring to the codimension-two defects.
[^3]: \[foot:iibsym\]In the IIB setup, these discrete global symmetries come from discrete isometries of the ADE singularity that preserve the hyper-Kähler structure (see Table \[table:sym\] and [@Tachikawa:2011ch]). A careful reader may notice some peculiarity about the $A_{2N}$ case. For $A_{2N}$ singularity, the relevant discrete isometry is an $\mZ_4$ generated by $\sigma:(x,y,z)\to (y,-x,-z)$. But since $\sigma^2:(x,y,z)\to (-x,-y,z)$ is part of the connected $U(1)$ isometry that act by $x\to e^{i\alpha} x,y\to e^{-i\alpha} y$, combined with the fact that the $(2,0)$ SCFT has no global currents (non-R symmetry), it should act trivially on the $(2,0)$ SCFT in the decoupling limit. Therefore we only expect to see the $\mZ_2$ symmetry in the $(2,0)$ $A_{2N}$ SCFT. We thank Edward Witten for helpful discussions on this point.
[^4]: This is a higher dimensional generalization of the familiar twisted sector operators in a 2d orbifold CFT. Similarly there should also be twisted codimension-two defects in 4d $\cN=4$ super-Yang Mills theories. It would be interesting to understand their roles in S-duality and the geometric Langlands program [@Witten:2007td].
[^5]: $T$ is a regular semisimple element of the Lie algebra $\mfj$.
[^6]: More explicitly, the moduli space of complex structure deformations (more precisely miniversal or semiuniversal deformations) of a cDV singularity is identified with the Hitchin moduli space of the Higgs bundle on a sphere with one irregular puncture of the regular-semisimple type. This identification can also be interpreted as a correspondence between non-compact “CY3” integrable systems and Hitchin integrable systems in the sense of [@Diaconescu:2006ry] (see also [@beck] for a review and some recent developments). It would be interesting to make this more precise.
[^7]: We emphasize here that, although suggestive, we do not yet understand the physical meaning of the singular geometries of Table \[table:SW\] in type IIB string theory as they involve branch cuts. Nevertheless, for practical purposes, we find them to be a useful mnemonic for the 4d SCFTs engineered by such twisted irregular defects (along with another twisted regular defect).
[^8]: The two complex structures relevant for and are usually referred to as the $J$ and $I$ complex structures respectively in the literature [@Kapustin:2006pk].
[^9]: In the 3d perspective, they are parametrized by the additional (dual) scalars in the dimensional reduced vector multiplet and are part of the moduli space.
[^10]: The sheets of the SW curve $\Sigma$ viewed as a branched cover of $\cC$ are labelled by eigenvalues of $\Phi_z$.
[^11]: Some of these coefficients are redundant (unphysical) and can be fixed by a coordinate change of $(x,z)$ that preserves the SW differential $\lambda$ up to an exact term.
[^12]: This is meaningful since the theory $\cT_4[\cC]$ is assumed to be conformal.
[^13]: There are two caveats to this prescription: (i) $u_{i,j}$ with integral scaling dimension $\Delta[u_{i,j}]\geq 2$ may correspond to a homogeneous polynomial (flavor Casimir) in the mass parameters; (ii) there can be constraints among $u_{i,j}$. Both subtleties can be taken care of systematically from information of the puncture(s) on $\cC$ [@Chacaltana:2012zy].
[^14]: We emphasize here that from the perspective of the resulting 4d theory, the coefficient matrices $\{T_\ell \}$ of the polar terms correspond to parameters such as chiral couplings and masses whereas the Coulomb branch moduli are encoded in the non-divergent terms of .
[^15]: The Hitchin pole in the original holomorphic form (as opposed to ) will have higher integral pole orders but the coefficient matrices are not necessarily semisimple [@Witten:2007td].
[^16]: Here we assume that the Hitchin pole is irreducible. In other words, the structure group $J$ associated to the Higgs bundle is not reduced.
[^17]: See [@Xie:2017vaf; @Xie:2017aqx] for a recent discussion relevant for 4d $\cN=2$ SCFTs.
[^18]: The more precise statement is that the boundary condition that defines a regular defect is scale-invariant. In 4d we expect this further implies conformal invariance.
[^19]: This is true as long as the Hitchin moduli space $\cM_{\rm H}(\cC)$ has a non-negative dimension. For example, one cannot have three simple punctures on a $\mP^1$ in the $A_{N>1}$ $(2,0)$ SCFT.
[^20]: \[footnote:A2norder\]In this case, since the Weyl group $W(\mfh)$ acts faithfully on $T_\ell\in \mfh$ we can effectively identify $b$ with $d$ as opposed to the order of $\sigma_g$ which is twice as big.
[^21]: Recall that $T_k=T_1$ here since the subscript takes value mod $b$. \[footnote:idip\]
[^22]: The associated torsion inner automorphism has order $b\over \gcd(k,b)$ (see around Footnote \[footnote:A2norder\] for a caveat for $A_{2N}$). We could have restricted to $\gcd(k,b)=1$ in the regular semisimple case and consider all possible values of $b$ (including the divisors of the entries in Table \[table:bv1\]) when defining the $J^{(b)}[k]$ punctures. However in writing down formulae (e.g. for central charges), we find it more convenient to allow $\gcd(k,b)\neq 1$ while keeping in mind the identification $J^{(b)}[k]=J^{(b')}[k']$ for $b'={b\over \gcd(b,k)},~k'={k\over \gcd(b,k)}$.
[^23]: In particular $b_t$ is always a multiple of $|o|$.
[^24]: In the superscript of $\mfj^\ell$, $\ell$ is understood to be mod $b_t$.
[^25]: For $\mfj=E_6$ or $E_7$ there are examples of a regular element (also twisted regular element for $E_6$) in the Weyl group that lift to two torsion automorphisms $\sigma_g,\sigma'_g$ or $\sigma_g o,\sigma'_g o$ for the twisted case (see Table 6.1, 6.2 and 6.6 in [@2012arXiv1205.0515E]). However we do not see any differences in the 4d theories engineered by punctures that are compatible with either $\sigma_g$ or $\sigma_g'$.
[^26]: We emphasize that unlike in the untwisted case, the order does not determine the twisted regular element in general.
[^27]: We emphasize here that the class I, II (and III for ${}^2 E_6$) of twisted defects are not always distinguishable. If $d_t$ is a divisor of more than one entry in Table \[table:dt\], it defines a unique defect that appears in multiple classes.
[^28]: A special feature of the $\mZ_2$ outer-automorphism of $A_{2N}$ compared to the other cases in Table \[table:outm\] is that it does not fix any simple root, instead it exchanges the pair of simple roots $\A_{N}=e_N-e_{N+1}$ and $\A_{N+1}=e_{N+1}-e_{N+2}$. This has important consequences on the 4d theories engineered by such punctures which we explain later in the paper.
[^29]: We thank Yuji Tachikawa for suggesting these names.
[^30]: Note that our normalization of the flavor central charges is related to the one in [@Chacaltana:2012zy; @Chacaltana:2014nya] by $k_G^{\rm ours}={1\over 2}k_G^{\rm theirs }$.
[^31]: Note that the ${1\over 2}$ shift in the third equation of is exactly the contribution from a half-hyper multiplet in the fundamental representation of $USp(2N)$ which would also saturate Witten’s global anomaly for these punctures [@Tachikawa:2018rgw]. Thus it’s tempting to say that this ${1\over 2}$ contribution comes from the *minimal* boundary modes of $\mZ_2$ symmetry defect. The other ${1\over 2}$ factor in should be related to the fact that the Langlands dual $SO(2N+1)$ of the $USp(2N)$ has an index of embedding equal to one in $D_{N+1}$ but [two]{} in $A_{2N}$. A better understanding of the anomaly inflow from 6d along the lines of [@Bah:2018gwc] should give a rigorous argument for .
[^32]: The essential statement here is that the 2d chiral algebra contains the affine Kac-Moody algebras associated to both the simple and $U(1)$ factors of the flavor symmetry, and the 2d stress-tensor is given by the Sugawara construction.
[^33]: The equation that relates $a-c$ to the Higgs branch dimension is a simple consequence of anomaly matching [@Shimizu:2017kzs]. It should hold whenever the Higgs branch is [*pure*]{}: the low energy theory on the Higgs branch is described just by hypermultiplets. It would be interesting to understand how it gets modified for general Argyres-Douglas type SCFTs.
[^34]: In this paper, we implicitly assume that the SW geometry together with all of its $\cN=2$ deformations fixes the $\cN=2$ SCFT uniquely. To our best knowledge there is no counter-example but it would be interesting to prove this rigorously.
[^35]: This theory also appears in the sequence considered in Example \[eg:3\] at $N=1$.
[^36]: Recall from [@Argyres:2007cn] that the Dynkin index of embedding for $G\subset J$ is computed by I\_[GJ]{}=[\_i T([**r**]{}\_i)T([**r**]{})]{} where ${\bf r}$ denotes a representation of $J$ which decomposes into $\oplus_i {\bf r}_i$ under $G$, and $T(\cdot)$ computes the quadratic index of the representation (which can be found for example in [@Yamatsu:2015npn]).
[^37]: The theory was also constructed by the circle compactification of a 5d $\cN=1$ SCFT with $\mZ_2$ twist [@Zafrir:2016wkk].
[^38]: This is the familiar Seiberg-Witten theory constructed by $SU(2)$ $\cN=2$ SYM coupled to 4 fundamental hypermultiplets [@Seiberg:1994aj]. The theory has $SO(8)$ flavor symmetry with flavor central charge $k_{D_4}=2$.
[^39]: We use the Nahm label, which is classified by the nilpotent orbits of the flavor symmetry group $G$. Note that the Hitchin label is classified by the nilpotent orbits of Langlands dual group $G^\vee$ which is generated by the invariant subalgebra of the outer-automorphism of ADE Lie algebra (see Table \[table:outm\]).
[^40]: This includes in particular the $R_{2,2N}$ theories of [@Chacaltana:2014nya].
|
---
abstract: 'Learning disentangled representation from any unlabelled data is a non-trivial problem. In this paper we propose Information Maximising Autoencoder (InfoAE) where the encoder learns powerful disentangled representation through maximizing the mutual information between the representation and given information in an unsupervised fashion. We have evaluated our model on MNIST dataset and achieved 98.9 ($\pm .1$) $\%$ test accuracy while using complete unsupervised training.'
author:
- |
Kazi Nazmul Haque[^1], Siddique Latif, and Rajib Rana\
University of Southern Queensland, Australia
title: Disentangled Representation Learning with Information Maximizing Autoencoder
---
Introduction
============
Learning disentangled representation from any unlabelled data is an active area of research [@goodfellow2016deep]. [Self supervised learning [@spyros:2018; @zhang_colorful:2016; @oord2018representation] is a way to learn representation from the unlabelled data but the supervised signal is needed to be developed manually, which usually varies depending on the problem and the dataset.]{} Generative Adversarial Neural Networks (GANs) [@goodfellow:2014] is a potential candidate for learning disentangled representation from unlabelled data ([@radford2015; @karras:2017; @Donahue2016AdversarialFL]). In particular, InfoGAN [@chen2016infogan], which is a slight modification of the GAN, can learn interpretable and disentangled representation in an unsupervised fashion. The classifier from this model can be reused for any intermediate task such as feature extraction but the representation learned by the classifier of the model is fully dependent on the generation of the model which is a major shortcoming. Because if the generator of the InfoGAN fails to generate any data manifold, the classifier is unable to perform well on any sample from that manifold. Tricks from Mutual Information Neural Estimation paper [@belghazi2018mine] might help to capture the training data distribution, yet learning all the training data manifold using GAN is a challenge for the research community [@goodfellow2016deep]. Adversarial autoencoder (AAE) [@makhzani:2016] is another successful model for learning disentangled representation. The encoder of the AAE learns representation directly from the training data but it does not utilize the sample generation power of the decoder for learning the representations. In this paper, we aim to address this challenge. We aim to build a model that utilizes both training data and the generated samples and thereby learns more accurate disentangled representation maximizing the mutual information between the random condition/information and representation space.
Information Maximizing Autoencoder {#gen_inst}
==================================
InfoAE consists of an encoder $E$, a decoder $D$ and a generator $G$. $G$ network produces latent variable space from a random latent distribution and a given condition/information. $D$ is used to generate samples from the latent variable space generated by the generator. It also maximizes the mutual information between the condition and the generated samples. $E$ is forced to learn the mapping of the train samples to the latent variable space generated by the generator. The model has three other networks for regulating the whole learning process: a classifier $C$, a discriminator $D_{i}$ and a self critic $S$. Figure \[fig:model\] shows the architecture of the model.
Encoder and Decoder
-------------------
The encoding network $E$, takes any sample $x$ $\in$ $p(x)$, where $p(x)$ is the data distribution. $E$ outputs latent variable $z_e$ = $E(x)$, where $z_e$ $\in$ $q(z)$ and $q(z)$ can be any continuous distribution learned by $E$. This $z_e$ is feed to decoder network $D$ to get sample $\hat{x_r}$ so that $\hat{x_r}$ $\in$ $p(x)$ and $\hat{x_r}$ $\approx$ $x$.
Generator and Discriminator
---------------------------
Generator network $G$ generates latent variable $z_g$ = $G($z$,c)$ $\in$ $p(z)$, from any sample $z$ and $c$ where $z$ $\in$ $u(z)$ and $c$ $\in$ $Cat(c)$. Here $p(z)$ can be any continuous distribution learned by $G$, $u(z)$ is random continuous distribution (e.g., continuous uniform distribution) and $Cat(c)$ is random categorical distribution. To validate $z_g$, decoder $D$ learns to generate sample $\hat{x_g}$ = $D(G(z_g,c))$ so that $\hat{x_g}$ $\in$ $p(x)$. The discriminator network $D_i$ forces decoder to create sample from the data distribution.
Classifier and Self Critic
--------------------------
While generator generates $z_g$ from $z$, it can easily ignore the given condition $c$. To maximise the Mutual Information (MI) between $c$ and $z_g$, we use classifier network $C$ to classify $z_g$ into $\hat{c_g}$ = $C(G(z_g,c))$ according to the given condition $c$. We also want encoder $E$ to learn encoding $\hat{z_e}$ = $E(\hat{x_g})$ so that $\hat{z_e}$ $\in$ $p(z)$ and MI$(\hat{z_e}, c)$ is maximised. To ensure MI$(\hat{z_e}, c)$ is maximised again the classifier network $C$ is utilised to classify $\hat{z_e}$ into $\hat{c_e}$ = $C(\hat{z_e})$ according to given condition $c$. To make sure $q(z)$ $\approx$ $p(z)$, we use a discriminator network $S$, which forces $E$ to encode $x$ into $p(z)$. $S$ learns through discriminating $(x,z_e)$ as fake and ($\hat{x_g}$, $z_g$) as real sample. We named this discriminator as Self Critic as it criticises two generations from the sub networks of a single model where they are jointly trained.
Training Objectives
-------------------
The InfoAE is trained based on multiple losses. The losses are : Reconstruction loss, $R_l=\sqrt{\sum(\hat{x_r} - x)^2}$ for both Encoder and Decoder; Discriminator loss, $D_{il} = \log D_{i}(x) + \log(1 - D_{i}(D(z_g))$ ; Decoder has loss $D_{lg}$, for the generated image $\hat{x_g}$ and loss $D_{le}$ for the reconstructed image $\hat{x}$, where $D_{lg} = \log(1 - D_{i}(D(z_g)))$ and $D_{le} = \log(1 - D_{i}(D(E(x))))$ ; Encoder loss $E_l = \log(1 - S(z_e, x))$ ; Self Critic loss $S_l = \log S(z_g, \hat{x_g}) + \log(1 - S(z_e, x))$ ; Two classification losses $C_{lg}$, $C_{le}$ respectively for Generator and Encoder where $C_{lg} = - \sum c \log (\hat{c_g})$ and $C_{le} = - \sum c \log (\hat{c_e})$. We get our total loss, $T_l$ in equation \[eq:1\] where $\alpha$ , $\beta$ and $\gamma$ are hyper parameters. $$T_l = \alpha * (C_{lg} + C_{le}) + \beta * (E_l + D_{le} + D_{lg}) + \gamma * R_l
\label{eq:1}$$
All the networks are trained together and the weights of the $E$, $D$, $C$, and $G$ are updated to minimise the total loss, $T_l$ while the weights of the $S$ and $D_i$ are updated to maximise the loss $S_l$, $D_{il}$, respectively. So the training objective can be express by the equation \[eq:2\] $$\min_{E,D,C,G} \max_D V(Di,S) = T_l + D_{il} + S_l
\label{eq:2}$$
Implementation Details
======================
Our model has different components as shown in Figure \[fig:model\]. We used Convolutional Neural Network (CNN) for $E$, $D_{i}$ and $S$. Batch Normalization [@ioffe2015batch] is used except for the first and the last layer. We did not use any maxpool layer and the down sampling is done through increasing the stride. For classifier $C$ and generator $G$ we used simple two layers feedforward network with hidden layer. For Decoder $D$ we used Transpose CNN.
Our experiments show that the training of the whole model is highly sensitive to $\alpha$, $\beta$ and $\gamma$. After experimenting with different values of $\alpha$, $\beta$ and $\gamma$, we received best result for $\alpha$ = 1, $\beta$ = 1 and $\gamma$ = 0.4. For $c$ variable we used random one hot encoding of size 10($c$ $\sim$ Cat($K$ = 10, $p$ = 0.1)) and $z$ $\in$ $\mathbb{R}$ $^{100}$ , which is randomly sampled from a uniform distribution $U(-1,1)$. The weights of all the networks are updated with Adam Optimizer ([@kingma2014adam]) and the learning rate of 0.0002 is used for all of them.
Results and Discussion
======================
We have evaluated the model on MNIST dataset and received outstanding results. InfoAE is trained on MNIST training data without any labels. After trainning, We encoded the test data with Encoder, $E$ and got classification label with the Classifier, $C$. Then we clustered the test data according to label and received classification accuracy of 98.9 ($\pm .05$), which is better than the popular methods as shown in Table \[errorrate\].
[ll]{} &\
\
InfoGAN ([@chen2016infogan]) &5\
Adversarial Autoencoder ([@makhzani:2016]) &4.10 ($\pm$ 1.12)\
Convolutional CatGAN ([@springenberg2015unsupervised]) &4.27\
PixelGAN Autoencoders ([@makhzani_pixelgan:2017]) &5.27 ($\pm$ 1.81)\
InfoAE &**1.1** ($\pm$ .1)\
The latent variable produced by the encoder on test data is visualized in figure \[fig:Sec\_IEMOCAP\](b). For visualization purpose we reduced the dimension of the latent vector with T-distributed Stochastic Neighbor Embedding or t-SNE [@van2008visualizing]. In the visualization, we can observe that representation of similar digits are located nearby in the 2D space while different digits. This suggests that the encoder was able to disentangle the digits category in the representation space, which has eventually resulted in the superior performance. Also, the generator was able to generate latent space according to the condition and the decoder was able to generate samples from that latent variable space, disentangling the digit category as shown in Fig. \[fig:Sec\_IEMOCAP\](a).
Let us consider two latent variables $z_{1}$ = $E(x_{1})$ and $z_{2}$ = $E(x_{2})$ where $x_{1}$, $x_{2}$ are two sample images from the test data. Now let us do a linear interpolation between $z_{1}$ toward $z_{2}$ with $z_{1} + (s/n)*(z_{2}-z_{1})$ where $s$ $\in$ $\{1,2, ....,n\}$ and $n$ is the number of steps and feed the latent variables to the Decoder for generating sample. Figure \[fig:Sec\_IEMOCAP\](c) shows the interpolation between the samples from different category. A smooth transition between different types of digits suggest the latent space is well connected.
Figure \[fig:Sec\_IEMOCAP\](d) show the interpolation between the same category of the samples and we can observe that the encoder was able to disentangle the styles of the digits in the latent space. This same category interpolation can be used as data augmentation.
[0.35]{} ![(a) Generated samples from the decoder. Rows show the samples $D(G(z,c))$ generated for different latent variables $z$ and columns show the samples generated for categorical variables $\{ c_1, c_2, . . . . . .c_{10}\}$. (b) Representation of latent variables for the MNIST test data. Different colors indicate different category of the digits. (c) Visualization of linear interpolation between two reconstructed test samples from different categories (left to right). (d) Visualization of linear interpolation between two reconstructed test samples of the same category (left to right).[]{data-label="fig:Sec_IEMOCAP"}](model2_sample.png "fig:"){width="\linewidth"}
[0.4]{} ![(a) Generated samples from the decoder. Rows show the samples $D(G(z,c))$ generated for different latent variables $z$ and columns show the samples generated for categorical variables $\{ c_1, c_2, . . . . . .c_{10}\}$. (b) Representation of latent variables for the MNIST test data. Different colors indicate different category of the digits. (c) Visualization of linear interpolation between two reconstructed test samples from different categories (left to right). (d) Visualization of linear interpolation between two reconstructed test samples of the same category (left to right).[]{data-label="fig:Sec_IEMOCAP"}](model2_viz.png "fig:"){width="\linewidth"}
\
[0.45]{} ![(a) Generated samples from the decoder. Rows show the samples $D(G(z,c))$ generated for different latent variables $z$ and columns show the samples generated for categorical variables $\{ c_1, c_2, . . . . . .c_{10}\}$. (b) Representation of latent variables for the MNIST test data. Different colors indicate different category of the digits. (c) Visualization of linear interpolation between two reconstructed test samples from different categories (left to right). (d) Visualization of linear interpolation between two reconstructed test samples of the same category (left to right).[]{data-label="fig:Sec_IEMOCAP"}](model2_interpolation.png "fig:"){width="\linewidth"}
[0.45]{} ![(a) Generated samples from the decoder. Rows show the samples $D(G(z,c))$ generated for different latent variables $z$ and columns show the samples generated for categorical variables $\{ c_1, c_2, . . . . . .c_{10}\}$. (b) Representation of latent variables for the MNIST test data. Different colors indicate different category of the digits. (c) Visualization of linear interpolation between two reconstructed test samples from different categories (left to right). (d) Visualization of linear interpolation between two reconstructed test samples of the same category (left to right).[]{data-label="fig:Sec_IEMOCAP"}](interpolation_self.png "fig:"){width="\linewidth"}
Conclusion and Future Work {#others}
==========================
In this paper we present and validate InfoAE, which learns the disentangled representation in a completely unsupervised fashion while utilizing both training and generated samples. We tested InfoAE on MNIST dataset and achieved test accuracy of 98.9 ($\pm .1$), which is a very competitive performance compared to the best reported results including InfoGAN. We observe that the encoder is able to disentangle the digit category and styles in the representation space, which results in the superior performance. InfoAE can be used to learn representation from unlabelled dataset and the learning can be utilized in a related problem where limited labeled data is available. Moreover, its power of representation learning can be exploited for data augmentation. This research is currently in progress. We are currently attempting to mathematically explain the results. We are also aiming to analyze the performance of InfoAE on large scale audio, image datasets and some of the related work [@rana1; @rana2; @rana3] on the audio space would be helpfull.
[19]{}
Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio. *Deep learning*, volume 1. MIT Press, 2016.
Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. *CoRR*, abs/1803.07728, 2018. URL <http://arxiv.org/abs/1803.07728>.
Richard Zhang, Phillip Isola, and Alexei Efros. Colorful image colorization. 9907:0 649–666, 10 2016.
Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. *arXiv preprint arXiv:1807.03748*, 2018.
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, editors, *Advances in Neural Information Processing Systems 27*, pages 2672–2680. Curran Associates, Inc., 2014. URL <http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf>.
Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. *CoRR*, abs/1511.06434, 2015.
Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. 10 2017.
Jeff Donahue, Philipp Kr[ä]{}henb[ü]{}hl, and Trevor Darrell. Adversarial feature learning. *CoRR*, abs/1605.09782, 2016.
Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In *Advances in neural information processing systems*, pages 2172–2180, 2016.
Ishmael Belghazi, Sai Rajeswar, Aristide Baratin, R Devon Hjelm, and Aaron Courville. Mine: mutual information neural estimation. *arXiv preprint arXiv:1801.04062*, 2018.
Alireza Makhzani, Jonathon Shlens, Navdeep Jaitly, and Ian Goodfellow. Adversarial autoencoders. 11 2016.
Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. *arXiv preprint arXiv:1502.03167*, 2015.
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. *arXiv preprint arXiv:1412.6980*, 2014.
Jost Tobias Springenberg. Unsupervised and semi-supervised learning with categorical generative adversarial networks. *arXiv preprint arXiv:1511.06390*, 2015.
Alireza Makhzani and Brendan Frey. Pixelgan autoencoders. 06 2017.
Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. *Journal of Machine Learning Research*, 90 (2579-2605):0 85, 2008.
Rajib Rana. Context-driven mood mining. In *MobiSys 2016 Companion-Companion Publication of the 14th Annual International Conference on Mobile Systems, Applications, and Services*, page 143. Association for Computing Machinery (ACM), 2016.
Rajib Rana, Daniel Austin, Peter G Jacobs, Mohanraj Karunanithi, and Jeffrey Kaye. Gait velocity estimation using time-interleaved between consecutive passive ir sensor activations. *IEEE Sensors Journal*, 160 (16):0 6351–6358, 2016.
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[^1]: Correspoding author email: [email protected]
|
---
abstract: 'We describe a variational approach to a notion of Hamiltonian delay equations and discuss examples.'
address:
- |
Peter Albers\
Mathematisches Institut\
Ruprecht-Karls-Universität Heidelberg
- |
Urs Frauenfelder\
Mathematisches Institut\
Universität Augsburg
- |
Felix Schlenk\
Institut de Mathématiques\
Université de Neuchâtel
author:
- 'Peter Albers, Urs Frauenfelder, Felix Schlenk'
title: 'What might a Hamiltonian delay equation be?'
---
[^1]
Introduction
============
An ordinary differential equation (ODE) on ${\mathbb{R}}^d$ is, in the simplest case, of the form $$\dot v(t) = X (v(t))$$ where $X$ is a vector field on ${\mathbb{R}}^d$. A delay differential equation (DDE) on ${\mathbb{R}}^d$ is, again in the simplest case, of the form $$\dot v(t) = X (v(t-\tau))$$ where $X$ is still a vector field on ${\mathbb{R}}^d$ and $\tau >0$ is the time delay. Delay equations therefore model systems in which the instantaneous velocity $\dot v (t)$ depends on the state of the curve $v$ at a past time. There are very many such systems in science and engineering. We refer to [@HaVL93] for a foundational text and to [@Erneux09] for a wealth of examples.
A [*Hamiltonian*]{} differential equation on ${\mathbb{R}}^{2n}$ is an ODE of the form $$\label{e:Ham}
\dot v (t) = X_H (v(t))$$ where the Hamiltonian vector field is of the special form $X_H = i\2 \nabla H$. Here $H \colon {\mathbb{R}}^{2n} \to {\mathbb{R}}$ is a smooth function and $i$ is the usual complex multiplication on ${\mathbb{R}}^{2n} \cong {\mathbb{C}}^n$. It is now tempting to define a Hamiltonian delay equation to be a DDE of the form $$\dot v (t) = X_H (v(t-\tau))$$ with $\tau >0$ and $X_H$ as before. Such systems where studied by Liu [@Liu12], who proved the existence of periodic orbits under natural assumptions on $H$.
In this paper we take a different approach to Hamiltonian delay equations, or at least to periodic orbits solving what we propose to call a Hamiltonian delay equation. Our approach is through action functionals. Let ${\mathscr{L}}= C^\infty (S^1,{\mathbb{R}}^{2n})$ be the space of smooth 1-periodic loops in ${\mathbb{R}}^{2n}$, and recall from classical mechanics that the 1-periodic solutions of are exactly the critical points of the action functional ${\mathscr{A}}\colon {\mathscr{L}}\to {\mathbb{R}}$ given by $${\mathscr{A}}(v) \,=\, \int_0^1 \bigl[ p(t) \cdot \dot q (t) - H (v(t)) \bigr] \2 dt, \qquad v(t) = (q(t),p(t)).$$ This fact, that interesting solutions can be seen as the critical points of a functional, played a key role in the creation of the modern theory of Hamiltonian dynamics and of symplectic topology, see point [**2.**]{} below. We therefore look at “delay action functionals”. If we just take $${\mathscr{A}}(v) \,=\, \int_0^1 \bigl[ p(t) \cdot \dot q (t) - H (v(t-\tau)) \bigr] \2 dt,$$ we get nothing new: The critical point equation is again $\dot v (t) = X_H(v(t))$. However, if we take two Hamiltonian functions $H,K$ on ${\mathbb{R}}^{2n}$ and the functional $${\mathscr{A}}(v) \,=\, \int_0^1 \bigl[ p(t) \cdot \dot q (t) - H(v(t)) \,K (v(t-\tau)) \bigr] \2 dt,$$ then the critical point equation is the honest delay equation $$\dot v (t) \,=\, H(v(t+\tau)) \, X_K(v(t)) + K(v(t-\tau))\, X_H(v(t)) .$$ Notice that the time shift $+\tau$ looks like “into the future”, but this does not matter along periodic orbits, along which the future can be identified with the past.
In our approach a Hamiltonian delay equation is thus a delay equation that can be obtained as critical point equation of an action functional.
In Sections \[s:2\] and \[s:4\] we compute the critical point equations of several classes of delay action functionals on the loop space of ${\mathbb{R}}^{2n}$, and more generally of exact symplectic manifolds. As a special case we shall obtain in Section \[s:3\] one instance of the delayed Lotka–Volterra equations. In fact, already in his 1928 paper [@Vol28] and in his seminal book [@Vol31] from 1931 Volterra was interested in periodic solutions of delay equations, and formulated the famous Lotka–Volterra equations with and without delay.
General properties of delay action functionals were studied for instance in Chapter VI of [@Els64] and in [@Sab69]. The problem when a delay equation on ${\mathbb{R}}^d$ is the critical point equation of a functional is analyzed in [@KPS07]. A Hamiltonian formalism for certain non-local PDEs on ${\mathbb{R}}^{2n}$, that is also based on non-local action functionals, was recently proposed in [@BaSch17].
In the rest of this introduction we further comment on why we believe that our approach to Hamiltonian delay equations is promising.
Recall that on ${\mathbb{R}}^{2n}$ one possible definition of a Hamiltonian delay equation is $$\label{e:Hamtau}
\dot v(t) = X_H(v(t-\tau)) \;.$$ On a general symplectic manifold $M$, however, this concept does not make sense, simply because $\dot v(t) \in T_{v(t)} M$ and $X_H(v(t-\tau)) \in T_{v(t-\tau)}M$ reside in different tangent spaces. On the other hand, our approach through action functionals readily extends to manifolds: Recall that a symplectic manifold is a manifold $M$ together with a non-degenerate closed 2-form $\omega$ on $M$. For simplicity we assume that $\omega$ is exact, $\omega = d \lambda$ for a 1-form $\lambda$. The Hamiltonian vector field of a smooth function $H \colon M \to {\mathbb{R}}$ is defined by $\omega (X_H, \cdot ) = -dH$, and the 1-periodic solutions of $X_H$ are exactly the critical point of the action functional $$\label{e:AH}
{\mathscr{A}}(v) \,=\, \int_0^1 \bigl[ \lambda (\dot v) - H (v(t)) \bigr] \2 dt$$ on the space of smooth 1-periodic loops in $M$. Taking $\lambda = \sum_{j=1}^n p_j\1 dq_j$ on ${\mathbb{R}}^{2n}$ we recover the case described above.
Replacing the Hamiltonian term $H(v(t))$ in by a delay term such as $H(v(t)) \,K (v(t-\tau))$ or by any of the terms described in Sections \[s:2\] and \[s:4\], we get as critical point equation a delay equation on $M$. Thus, if we start from a delay action functional and compute the critical point equation, then an “accident” as for cannot happen, and we always get a meaningful equation.
General DDEs can be readily defined on manifolds, see Section 12.1 of [@HaVL93] and [@Oli69], and there are a few results on periodic orbits of such systems, see [@BCF09] and the references therein. In contrast, there seems to be no concept of a Hamiltonian delay equation on manifolds. Our approach at least provides a natural notion of such an equation and a tool for finding periodic solutions.
While the theory of DDEs is meanwhile quite rich [@HaVL93], it is nonetheless much less developed than the theory of ODEs. One reason is that for DDEs there is no local flow on the given manifold, and so the whole theory is less geometric and more cumbersome.
In sharp contrast to general ODEs, Hamiltonian ODEs can be studied by variational methods, thanks to the action functional. For one thing, the action functional (even though neither bounded from below nor above, and strongly indefinite) can be used to do critical point theory on the loop space, as was first demonstrated by Rabinowitz [@Rab78]. At least as important, one can see symplectic topology as the geometry of the action functional. (This is almost the title and exactly the content of Viterbo’s paper [@Vi92], and also in the book [@HoZe94] the action functional is the main tool.) For instance, a selection of critical values of the action functional by min-max leads to numerical invariants of Hamiltonian systems and symplectic manifolds, that have many applications. The climactic impact of the action functional into symplectic dynamics and topology, however, is Floer homology, which is Morse theory for the action functional on the loop space.
Now, incorporating delays into an action functional, we can try to extend all these constructions to delay action functionals, and thereby create a calculus of variations for Hamiltonian delay equations, that should have many applications at least to questions on periodic orbits of such equations. For instance, a Floer theory in the delay setting should lead to the same lower bounds for the number of periodic orbits of Hamiltonian delay equations as guaranteed in the undelayed case by the solution of the Arnold conjectures. For a special class of Hamiltonian delay equations, these lower bounds are verified in [@AFS2] by means of an iterated graph construction and classical (Lagrangian) Floer homology. First steps in the construction of a delay Floer homology were taken in [@AlFr13] and [@AFS1].
Delay equations from sums and products of Hamiltonian functions {#s:2}
===============================================================
Let $(M,\omega)$ be a symplectic manifold with exact symplectic form $\omega = d \lambda$. We choose $2N+1$ autonomous Hamiltonian functions $$F,\, H_i,\, K_i \colon M \to {\mathbb{R}}, \quad i=1,\ldots,N ,$$ and define an “action functional” on the free loop space ${\mathscr{L}}\equiv {\mathscr{L}}(M) := C^\infty(S^1,M)$ by $$\label{eqn:action_fctl_with_sum}
\begin{aligned}
{\mathscr{A}}\colon {\mathscr{L}}& \to {\mathbb{R}}, \\
v &\mapsto \int_{S^1} v^*\lambda - \int_0^1 F \big( v(t) \big)dt
- \sum_{i=1}^N \int_0^1 H_i \big(v(t)\big) K_i \big(v(t-\tau) \big) dt
\end{aligned}$$ where $\tau \geq 0$ is the time delay. To find the critical point equation of ${\mathscr{A}}$ we fix $v \in {\mathscr{L}}$ and $\hat v \in T_v {\mathscr{L}}$ and compute $$\begin{aligned}
d {\mathscr{A}}(v) \hat v &=& \int_0^1 {\omega}\bigl( \hat v(t), \dot v(t) \bigr) dt
- \int_0^1 dF \bigl( v(t) \bigr) \bigl[ \hat v(t) \bigr] dt \\
&& \;\; -\sum_{i=1}^N \int_0^1 H_i \bigl( v(t) \bigr) \cdot dK_i \bigl( v(t-\tau) \bigr)
\bigl[ \hat v(t-\tau) \bigr] dt \\
&&\;\;-\sum_{i=1}^N \int_0^1 K_i \bigl( v(t-\tau) \bigr) \cdot dH_i \bigl( v(t) \bigr)
\bigl[ \hat v(t) \bigr] dt \,.\end{aligned}$$ Since $v$ and $\hat v$ are 1-periodic, $$\int_0^1 H_i \bigl( v(t) \bigr) \cdot dK_i \bigl( v(t-\tau) \bigr) \bigl[ \hat v(t-\tau) \bigr] dt
\,=\,
\int_0^1 H_i \bigl( v(t+\tau) \bigr) \cdot dK_i \bigl( v(t) \bigr) \bigl[ \hat v(t) \bigr] dt.$$ Using also the definition ${\omega}(X_F,\cdot)=-dF$ of the Hamiltonian vector field $X_F$, etc., we find $$\begin{aligned}
d {\mathscr{A}}(v) \hat v
&=& \int_0^1 {\omega}\bigl( \hat v(t), \dot v(t) \bigr) dt
- \int_0^1 {\omega}\bigl( \hat v(t), X_F(v(t)) \bigr) dt \\
&&\;\;-\sum_{i=1}^N \int_0^1 H_i \bigl( v(t+\tau) \big) \cdot
{\omega}\bigl( \hat v(t), X_{K_i} (v(t)) \bigr) dt\\
&&\;\;-\sum_{i=1}^N \int_0^1 K_i \bigl( v(t-\tau) \bigr)
\cdot {\omega}\bigl( \hat v(t), X_{H_i} (v(t)) \bigr) dt .\end{aligned}$$ The critical point equation is therefore $$\dot v(t) \,=\, X_F(v(t)) +
\sum_{i=1}^N \Bigl[ H_i (v(t+\tau)) \, X_{K_i} (v(t)) + K_i (v(t-\tau)) \, X_{H_i}(v(t)) \Bigr] .$$ We have proved the following lemma.
The critical points of ${\mathscr{A}}$ satisfy the Hamiltonian delay equation $$\label{eqn:Ham_delay_eqn}
\dot v(t) \,=\, X_F(v(t)) +
\sum_{i=1}^N \Bigl[ H_i(v(t+\tau)) \, X_{K_i} (v(t)) + K_i (v(t-\tau)) \, X_{H_i}(v(t)) \Bigr] .$$
Using that $v(t+1)=v(t)$ we obtain
For the time delay $\tau=\tfrac12$ the Hamiltonian delay equation becomes $$\dot v(t) \,=\, X_F (v(t)) +
\sum_{i=1}^N \Bigl[ H_i (v(t-\tfrac12)) \, X_{K_i}(v(t)) +
K_i(v(t-\tfrac12)) \, X_{H_i}(v(t)) \Bigr] .$$
The Lotka-Volterra equations, with and without delay {#s:3}
====================================================
In this section we extend the work of Fernandes–Oliva [@FeOl95] to positive delays. Fix a skew-symmetric $N \times N$-matrix $A=(a_{ij})$, i.e. $a_{ji}=-a_{ij}$, and $N$ real numbers $b_i$. Take $M = {\mathbb{R}}^{2N}$ with its usual exact symplectic form ${\omega}= \sum_i dq_i \wedge dp_i$, and set $$F(q,p) := \sum_{i=1}^N b_i q_i, \qquad
H_i(q,p) := -e^{p_i}, \qquad
K_i(q,p) := e^{\frac12 \sum_{j=1}^N a_{ij}q_j} .$$ The Hamiltonian vector fields are $$X_F = \sum_{i=1}^N b_i \frac{{\partial}}{{\partial}p_i}, \qquad
X_{H_i} = e^{p_i} \frac{{\partial}}{{\partial}q_i}, \qquad
X_{K_i} = \tfrac 12 \sum_{k=1}^N a_{ik} \, e^{\frac12 \sum_{j=1}^N a_{ij}q_j} \frac{{\partial}}{{\partial}p_k}.$$ Fix $\tau \geq 0$. For $v=(q,p) \in {\mathscr{L}}({\mathbb{R}}^{2N})$ the Hamiltonian (delay) equation becomes $$\dot v(t) \,=\,
\sum_{i=1}^N b_i \frac{{\partial}}{{\partial}p_i} +
\sum_{i=1}^N \Biggl[-e^{p_i(t+\tau)} \, \tfrac12 \sum_{k=1}^N a_{ik} \,
e^{\frac12 \sum_{j=1}^N a_{ij} q_j(t)} \frac{{\partial}}{{\partial}p_k} +
e^{\frac12 \sum_{j=1}^N a_{ij} q_j(t-\tau)} \, e^{p_i(t)} \frac{{\partial}}{{\partial}q_i} \Biggr]\;.$$ In other words, $$\begin{aligned}
\dot q_i(t) &=& e^{p_i(t) + \frac12 \sum_{j=1}^N a_{ij} q_j(t-\tau)} \\ [1ex]
\dot p_i(t) &=& b_i- \sum_{l=1}^N e^{p_l(t+\tau)} \, \tfrac12 \, a_{li} \, e^{\frac12\sum_{j=1}^Na_{lj}q_j(t)} \\
&=&b_i- \tfrac12\sum_{l=1}^N a_{li} \, \dot q_l(t+\tau) \\
&=&b_i+ \tfrac12\sum_{l=1}^N a_{il} \, \dot q_l(t+\tau) \end{aligned}$$ where in the last equation we have used that $A$ is skew-symmetric. Using these two equations we compute $$\begin{aligned}
\ddot q_i(t) &=&
\Bigl(\dot p_i(t) + \tfrac12 \sum_{j=1}^N a_{ij} \, \dot q_j(t-\tau) \Bigr) \,\dot q_i(t) \\
&=& \Bigl(b_i+ \tfrac12 \sum_{l=1}^N a_{il} \, \dot q_l(t+\tau) +
\tfrac12 \sum_{j=1}^N a_{ij} \, \dot q_j(t-\tau) \Bigr) \,\dot q_i(t)\;.\end{aligned}$$ Now observe that the right hand side only depends on the $\dot q_j$, but not on the $p_j$. Setting $x_i(t) := \dot q_i(t)$ we thus obtain the first order delay system $$\label{e:1oDDE}
\dot x_i(t) \,=\, b_i \, x_i(t) + \tfrac12 \sum_{j=1}^N a_{ij} \,
x_i(t) \, x_j(t+\tau) + \tfrac12 \sum_{j=1}^N a_{ij} \, x_i(t) \, x_j(t-\tau)\;.$$
[**Case $\tau =0$.**]{} Then becomes $$\dot x_i(t) \,=\, b_i \, x_i(t) + \sum_{j=1}^N a_{ij}\, x_i(t)\, x_j(t)$$ with skew-symmetric $A=(a_{ij})$. This is one instance of the Lotka–Volterra equations without delay. These equations were proposed by Lotka [@Lot10] in his studies of chemical reactions, and independently by Volterra [@Vol26] in his studies of predator-prey dynamics.
[**Case $\tau = \frac 12$.**]{} Then the equations for 1-periodic orbits become $$\dot x_i(t) \,=\,
b_i \, x_i(t) + \sum_{j=1}^N a_{ij} \, x_i(t) \, x_j(t-\tfrac 12) \,.$$ These Hamiltonian delay equations already appeared in Chapter 4 of Volterra’s book [@Vol31].
More examples {#s:4}
=============
In this section we give three rather special classes of Hamiltonian delay equations, two involving integrals. The reader may invent his own examples.
More general products of Hamiltonian functions
----------------------------------------------
In we may replace the sum by an integral and choose a double time-dependence: Consider functions $H,K \colon M \times S^1 \times S^1 \to {\mathbb{R}}$, which we write as $H_{t,\tau}(x)$ and $K_{t,\tau}(x)$ for $x \in M$ and $t,\tau \in S^1$. Then set $$\begin{aligned}
{\mathscr{A}}\colon {\mathscr{L}}& \to {\mathbb{R}}\\
v &\mapsto \int_{S^1} v^*\lambda - \int_0^1 \int_0^1 H_{t,\tau}
\bigl( v(t) \bigr) \, K_{t,\tau} \bigl( v(t-\tau) \bigr) \,d\tau dt\;.
\end{aligned}$$ For $v \in {\mathscr{L}}$ and $\hat v \in T_v {\mathscr{L}}$ we compute $$\begin{aligned}
d {\mathscr{A}}(v)\, \hat v &=& \int_0^1 {\omega}\bigl(\hat v(t), \dot v(t) \bigr) dt \\
&&\;\; -\int_0^1 \bigg[ \int_0^1 H_{t,\tau} \bigl( v(t) \bigr) \cdot
dK_{t,\tau} \bigl( v(t-\tau) \bigr) \bigl[ \hat v(t-\tau)\bigr] d\tau \\
&&\;\; - \int_0^1 K_{t,\tau} \bigl( v(t-\tau) \bigr) \cdot
dH_{t,\tau} \bigl( v(t) \bigr) \bigl[ \hat v(t) \bigr] d\tau \bigg] dt \,.\end{aligned}$$ Since $v, \hat v$ are 1-periodic and also $H,K$ are periodic in $t$, $$\int_0^1 H_{t,\tau} \bigl( v(t) \bigr) \cdot
dK_{t,\tau} \bigl( v(t-\tau) \bigr) \bigl[ \hat v(t-\tau)\bigr] d\tau
\,=\,
\int_0^1 H_{t+\tau,\tau} \bigl( v(t+\tau) \bigr) \cdot
dK_{t+\tau,\tau} \bigl( v(t) \bigr) \bigl[ \hat v(t)\bigr] d\tau .$$ Therefore, $$\begin{aligned}
d {\mathscr{A}}(v)\, \hat v &=& \int_0^1 {\omega}\bigl( \hat v(t), \dot v(t) \bigr) dt \\
&& \;\;-\int_0^1\bigg[ \int_0^1 {\omega}\Bigl( \hat v(t), H_{t+\tau,\tau} \bigl(v(t+\tau)\bigr)
\cdot X_{K_{t+\tau,\tau}} \bigl( v(t) \bigr) \Bigr) d\tau \\
&& \;\;-\int_0^1 {\omega}\Bigl( \hat v(t), K_{t,\tau} \bigl(v(t-\tau) \bigr) \cdot
X_{H_{t,\tau}} \bigl(v(t)\bigr) \Bigr) d \tau \bigg] dt \,.\end{aligned}$$ Hence the critical points of ${\mathscr{A}}$ are the solutions of the Hamiltonian delay equation $$\label{eqn:Hamiltonian_delay_from_a_product}
\dot v(t) \,=\, \int_0^1 \Bigl[ H_{t+\tau,\tau} \bigl( v(t+\tau) \bigr) \cdot
X_{K_{t+\tau,\tau}} \bigl(v(t)\bigr) + K_{t,\tau} \bigl(v(t-\tau)\bigr) \cdot
X_{H_{t,\tau}} \bigl(v(t)\bigr)\Bigr] \, d \tau\;.$$ In the special case that $H_{t,\tau}$ and $K_{t,\tau}$ are autonomous, equation simplifies to $$\dot v(t) \,=\, \int_0^1 H \bigl( v(t+\tau) \bigr) d\tau \cdot
X_K \bigl(v(t)\bigr) + \int_0^1 K \bigl( v(t-\tau) \bigr) d\tau \cdot
X_H \bigl(v(t)\bigr) \;.$$ If we define the functions $\overline H, \overline K \colon {\mathscr{L}}\to {\mathbb{R}}$ by $$\overline H(v) := \int_0^1 H \bigl( v(t) \bigr) dt \quad \text{and} \quad
\overline K(v) := \int_0^1 K \bigl(v(t)\bigr) dt$$ the above equation becomes $$\dot v(t) \,=\, \overline H(v) \, X_K(v(t)) + \overline K (v) \, X_H(v(t))\;.$$ Specializing further to $H=K$ we obtain $$\label{e:2oH}
\dot v(t) \,=\, 2 \2 \overline H(v) \,X_H(v(t))\;.$$ In this special case, preservation of energy implies that $t\mapsto H(v(t))$ is constant along solutions, and thus we may write as $$\label{eqn:Hamiltonian_delay_autonomous}
\dot v(t) \,=\, 2 \2 H(v) \,X_H(v(t)) \,=\, X_{H^2} (v(t))\;.$$
Of course, this equation can be studied by Floer theory, hence there are (in the Morse–Bott sense) multiplicity results (in terms of cup-length or Betti numbers) for periodic solutions in a certain range of Conley–Zehnder indices. Clearly, has many solutions, namely critical points of $H$. However, unless $H$ is $C^2$-small at all critical points, the Morse indices of critical points cannot all agree with their Conley–Zehnder indices, and so Floer theory implies the existence of additional non-constant solutions to . We expect that also equation admits a Floer theory. Note that typically, does not have any constant solutions, even when $K=1$ and $H$ is independent of $\tau$, hence a Floer theory should imply the existence of many interesting periodic solutions.
Exponentials of Hamiltonian functions
-------------------------------------
We consider yet another incarnation of a Hamiltonian delay equation. Take $$\begin{aligned}
{\mathscr{A}}\colon {\mathscr{L}}& \to {\mathbb{R}}\\
v&\mapsto \int_{S^1}v^*\lambda -
\int_0^1 \exp \left[ \int_0^1 H_\tau \bigl( v(t-\tau) \bigr) d\tau \right] dt
\end{aligned}$$ where $H \colon S^1\times M \to {\mathbb{R}}$ is given. We compute $$\begin{aligned}
d{\mathscr{A}}(v) \, \hat v &=& \int_0^1 {\omega}\bigl( \hat v(t), \dot v(t) \bigr) dt \\
&& \;\;-\int_0^1 \int_0^1 \exp \left[\int_0^1 H_\tau \bigl( v(t-\tau) \bigr) d\tau \right]
dH_\sigma \bigl( v(t-\sigma) \bigr) \hat v(t-\sigma) \,dt \2 d\sigma .\end{aligned}$$ Substituting $t$ by $t+\sigma$ and changing the order of integration the second summand becomes $$\begin{aligned}
&& -\int_0^1 \int_0^1 \exp \left[\int_0^1 H_\tau \bigl( v(t+\sigma-\tau) \bigr) d\tau \right]
dH_\sigma \bigl( v(t) \bigr) \2 \hat v(t) \, d\sigma \2 dt \\
&=&
-\int_0^1 {\omega}\left( \hat v(t), \int_0^1 \exp\left[\int_0^1 H_\tau (v(t+\sigma-\tau)) d\tau \right] X_{H_\sigma} \bigl(v(t)\bigr) \, d\sigma \right) dt\end{aligned}$$ The critical point equation is therefore $$\dot v(t) \,=\,
\int_0^1 \exp\left[\int_0^1 H_\tau \bigl( v(t+\sigma-\tau) \bigr) d\tau \right]
X_{H_\sigma} \bigl(v(t)\bigr) \, d\sigma \,.$$
Several inputs
--------------
We now consider a function $H \colon M \times M \to{\mathbb{R}}$ on the symplectic manifold $(M\times M,{\omega}\oplus{\omega})$, where again ${\omega}= d \lambda$, and denote by $d_1H(x,y) \colon T_xM \to{\mathbb{R}}$ the derivative of $H$ with respect to the first variable and correspondingly by $X^1_H(x,y)$ the Hamiltonian vector field of $H$ with respect to the first variable: $$d_1H(x,y)\xi \,=\, -{\omega}_x \left((X^1_H(x,y),\xi \right) \quad \forall\, \xi \in T_xM\;.$$ Further, we consider the action functional $$\begin{aligned}
{\mathscr{A}}\colon {\mathscr{L}}(M) & \to {\mathbb{R}}, \\
v&\mapsto \int_{S^1}v^*\lambda - \int_0^1 H \bigl( v(t),v(t+\tau) \bigr) dt\;.
\end{aligned}$$ Concrete examples for the function $H$ come for instance from interaction potentials (as in the $2$-body problem) or from vortex equations with delay. For $v \in {\mathscr{L}}(M)$ and $\hat v \in T_v {\mathscr{L}}(M)$ we compute $$\begin{aligned}
d{\mathscr{A}}(v) \,\dot v &=& \int_0^1 {\omega}_{v(t)} \bigl( \hat v(t), \dot v(t) \bigr) \,dt \\
&& \;\;-\int_0^1 \Bigl( d_1H \bigl(v(t),v(t+\tau)\bigr)\, \hat v(t) +
d_2H \bigl( v(t),v(t+\tau) \bigr)\, \hat v(t+\tau) \Bigr) \,dt \end{aligned}$$ The second summand is equal to $$\begin{aligned}
&&
-\int_0^1 \Bigl( d_1H \bigl(v(t),v(t+\tau)\bigr)\, \hat v(t) +
d_2H \bigl( v(t-\tau),v(t) \bigr)\, \hat v(t) \Bigr) \, dt \\
&=&
-\int_0^1 \left[ {\omega}_{v(t)} \Bigl( \hat v(t), X^1_H \bigl( v(t), v(t+\tau) \bigr) \Bigr) +
{\omega}_{v(t)} \Bigl( \hat v(t), X^2_H \bigl( v(t-\tau),v(t) \bigr) \Bigr) \right] \,dt .\end{aligned}$$ The critical point equation for ${\mathscr{A}}$ is therefore $$\label{e:critsev}
\dot v(t) \,=\, X^1_H \bigl( v(t),v(t+\tau) \bigr) + X^2_H \bigl( v(t-\tau),v(t) \bigr)\;.$$ We point out that indeed $$X^1_H \bigl(v(t),v(t+\tau)\bigr), \; X^2_H \bigl( v(t-\tau),v(t) \bigr) \in T_{v(t)}M$$ so that makes sense.
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[^1]: 2000 [*Mathematics Subject Classification.*]{} Primary 34K, Secondary 58E05,58F05, 70K42
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abstract: 'Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the normal scalar curvature. This property is conformal invariant; hence we study them in the framework of Möbius geometry, and restrict to three dimensional Wintgen ideal submanifolds in $\mathbb{S}^5$. In particular we give Möbius characterizations for minimal ones among them, which are also known as (3-dimensional) austere submanifolds (in 5-dimensional space forms).'
author:
- 'Zhenxiao Xie, Tongzhu Li, Xiang Ma, Changping Wang'
title: '**Möbius geometry of three dimensional Wintgen ideal submanifolds in $\mathbb{S}^{5}$**'
---
[**2000 Mathematics Subject Classification:**]{} 53A30, 53A55, 53C42.
[**Key words:**]{} Wintgen ideal submanifolds, DDVV inequality, Möbius geometry, austere submanifolds, complex curves\
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Introduction
============
The so-called DDVV inequality says that, given a $m$-dimensional submanifold $x:M^m\longrightarrow \mathbb{Q}^{m+p}(c)$ immersed in a real space form of dimension $m+p$ with constant sectional curvature $c$, at any point of $M$ we have $$\label{1.1}
s\leq c+||H||^2-s_N.$$ Here $s=\frac{2}{m(m-1)}\sum_{1\leq i<j\leq n}\langle R(e_i,e_j)e_j,e_i\rangle$ is the normalized scalar curvature with respect to the induced metric on $M$, $H$ is the mean curvature, and $s_N=\frac{2}{m(m-1)}||\mathbb{R}^{\perp}||$ is the normal scalar curvature. This remarkable inequality was first a conjecture due to De Smet, Dillen, Verstraelen and Vrancken [@Smet] in 1999, and proved by J. Ge, Z. Tang [@Ge] and Z. Lu [@Lu1] in 2008 independently.
As pointed out in [@Dajczer2][@Smet][@Lu1][@Lu3], it is a natural and important problem to characterize the extremal case, i.e., those submanifolds attaining the equality at every point, called *Wintgen ideal submanifolds*. In [@Ge] it was shown that the equality holds at $x\in M^m$ if and only if there exist an orthonormal basis $\{e_1,\cdots,e_m\}$ of $T_xM^m$ and an orthonormal basis $\{n_1,\cdots,n_p\}$ of $T_x^{\bot}M^m$ such that the shape operators $\{A_{n_i},i=1,\cdots,m\}$ have the form $$\label{form1}
A_{n_1}=
\begin{pmatrix}
\lambda_1 & \mu_0 & 0 & \cdots & 0\\
\mu_0 & \lambda_1 & 0 & \cdots & 0\\
0 & 0 & \lambda_1 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & \lambda_1
\end{pmatrix},~~
A_{n_2}=
\begin{pmatrix}
\lambda_2\!+\!\mu_0 & 0 & 0 & \cdots & 0\\
0 & \lambda_2\!-\!\mu_0 & 0 & \cdots & 0\\
0 & 0 & \lambda_2 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & \lambda_2
\end{pmatrix},$$ and $$A_{n_3}=\lambda_3I_p,~~~~ A_{n_r}=0, r\ge 4.$$ This is the first step towards a complete classification.
Wintgen [@wint] first proved the inequality for surfaces $M^2$ in $\mathbb{R}^4$, and that the equality holds if and only if the curvature ellipse of $M^2$ in $\mathbb{R}^4$ is a circle. Such surfaces are called *super-conformal* surfaces. They come from projection of complex curves in the twistor space $\mathbb{C}P^3$ of $\mathbb{S}^4$ [@fb]. Together with totally umbilic submanifolds (spheres and planes), they provide the first examples of Wintgen ideal submanifolds. Note that they are not necessarily minimal surfaces in space forms. In particular, being super-conformal is a conformal invariant property, whereas being minimal is not.
The conformal invariance of Wintgen ideal property in the general case was pointed out in [@Dajczer1]. Thus it is appropriate to investigate and classify Wintgen ideal submanifolds under the framework of Möbius geometry. For this purpose, the submanifold theory in Möbius geometry established by the fourth author will be briefly reviewed in Section 2.
We will always assume that the Wintgen ideal submanifolds in consideration are not totally umbilic. Note that to have the shape operators taking the form in , the distribution $\mathbb{D}=\mathrm{Span}\{e_1,e_2\}$ is well-defined. We call it *the canonical distribution*. The first Möbius classification result was obtained by us in [@Li1].\
[**Theorem A(Li-Ma-Wang[@Li1]):**]{} *Let $x:M^m\to\mathbb{S}^{m+p}(m\geq3)$ be a Wintgen ideal submanifold and it is not totally umbilic. If the canonical distribution $\mathbb{D}=\mathrm{Span}\{e_1,e_2\}$ is integrable, then locally $x$ is Möbius equivalent to either one of the following three kinds of examples described in $\mathbb{R}^{m+p}$:\
(i) a cone over a minimal Wintgen ideal surface in ${\mathbb S}^{2+p}$;\
(ii) a cylinder over a minimal Wintgen ideal surface in ${\mathbb R}^{2+p}$;\
(iii) a rotational submanifold over a minimal Wintgen ideal surface in ${\mathbb H}^{2+p}$.*\
In this paper we consider three dimensional Wintgen ideal submanifolds $x:M^3\to\mathbb{S}^5$ whose canonical distribution $\mathbb{D}$ is not integrable. There is a Möbius invariant 1-form $\omega$ associated with $x$. For its definition as well as other basic equations and invariants, see Section 3.
Our main result is stated as below, which is proved in Section 4.\
[**Theorem B:**]{} *Suppose $x:M^3\to\mathbb{S}^5$ is a Wintgen ideal submanifold whose canonical distribution $\mathbb{D}$ is not integrable. It is Möbius equivalent to a minimal Wintgen ideal submanifold in a five dimensional space form $\mathbb{Q}^5(c)$ if and only if the 1-form $\omega$ is closed.*\
Under some further conditions, in Section 5 we characterize minimal Wintgen ideal submanifolds coming from Hopf bundle over complex curves in $\mathbb{C}P^2$. We also discuss the classification of Möbius homogeneous ones among Wintgen ideal 3-dimensional submanifolds in $\mathbb{S}^5$, which include the following example: $$x: \mathrm{SO}(3)~ \longrightarrow~\mathbb{S}^5,~~~
(u, v, u\times v) \mapsto \frac{1}{\sqrt{2}}(u, v).$$
As to the geometric meaning of the 1-form $\omega$, we just mention that it could still be defined for Wintgen ideal submanifolds with dimension $m\ge 4$. In a forthcoming paper [@Li2] we will show that $d\omega=0$ is equivalent to the property that $\mathbb{D}=\mathrm{Span}\{e_1,e_2\}$ generates a 3-dimensional integrable distribution on $M^3$. Assume this is the case; then we will obtain a similar classification [@Li2] as in Theorem A. These results again demonstrate the phenomenon described by our reduction theorem [@Li0].
To understand the classification result, it is necessary to note that among Wintgen ideal submanifolds, there are a lot of minimal examples in space forms. Although they do not exhaust all possible examples, our classification demonstrates their importance as being representatives in a Möbius equivalence class of submanifolds, or as building blocks of generic examples. Those minimal Wintgen ideal surfaces are called *super-minimal* in the previous literature, including examples like complex curves in $\mathbb{C}^n$ and minimal 2-spheres in $\mathbb{S}^n$. For three dimensional submanifolds in 5-dimensional space forms $\mathbb{S}^5,\mathbb{R}^5,\mathbb{H}^5$, being minimal and Wintgen ideal is equivalent to being *austere submanifolds*, i.e. the eigenvalues of the second fundamental form with respect to any normal direction occur in oppositely signed pairs. Such submanifolds have been classified locally by Bryant [@br] for $M^3\to\mathbb{R}^5$ , by Dajczer and Florit [@Dajczer3] for $M^3\to\mathbb{S}^5$, and by Choi and Lu [@Lu] for $M^3\to\mathbb{H}^5$.
Finally we note that in [@Dajczer1], Dajczer and Florit have provided a parametric construction of Wintgen ideal submanifolds of codimension two and arbitrary dimension in terms of minimal surfaces in $\mathbb{R}^{m+2}$. Compared to our work, they had no restriction on the dimension of $M$, and the construction is explicit and valid for generic examples. On the other hand, their descriptions were not in a Möbius invariant language. In another paper [@Li3], we will give a construction of all Wintgen ideal submanifolds of codimension two and arbitrary dimension $m$ in terms of holomorphic, isotropic curves in a complex quadric $Q^{m+2}$.\
**Acknowledgement** This work is funded by the Project 10901006 and 11171004 of National Natural Science Foundation of China. We thank Professor Zizhou Tang for pointing out the homogeneous embedding of $\mathrm{SO}(3)$ in $\mathbb{S}^5$ to us. We are grateful to the referees for their helpful suggestions.
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Submanifold theory in Möbius geometry
=====================================
In this section we briefly review the theory of submanifolds in Möbius geometry. For details we refer to [@CPWang], [@liu].
Recall that in the classical light-cone model, the light-like (space-like) directions in the Lorentz space $\mathbb{R}^{m+p+2}_1$ correspond to points (hyperspheres) in the round sphere $\mathbb{S}^{m+p}$, and the Lorentz orthogonal group correspond to conformal transformation group of $\mathbb{S}^{m+p}$. The Lorentz metric is written out explicitly as $$\langle Y,Z\rangle=-Y_0Z_0+Y_1Z_1+\cdots+Y_{m+p+1}Z_{m+p+1},$$ for $Y=(Y_0,Y_1,\cdots,Y_{m+p+1}), Z=(Z_0,Z_1,\cdots,Z_{m+p+1})\in
\mathbb{R}^{m+p+2}_1$.
Let $x:M^m\rightarrow \mathbb{S}^{m+p}\subset \mathbb{R}^{m+p+1}$ be a submanifold without umbilics. Take $\{e_i|1\le i\le m\}$ as the tangent frame with respect to the induced metric $I=dx\cdot dx$, and $\{\theta_i\}$ as the dual 1-forms. Let $\{n_{r}|1\le r\le p\}$ be an orthonormal frame for the normal bundle. The second fundamental form and the mean curvature of $x$ are $$\label{2.1}
II=\sum_{ij,r}h^{r}_{ij}\theta_i\otimes\theta_j
n_{r},~~H=\frac{1}{m}\sum_{j,r}h^{r}_{jj}n_{r}=\sum_{r}H^{r}n_{r},$$ respectively. We define the Möbius position vector $Y:
M^m\rightarrow \mathbb{R}^{m+p+2}_1$ of $x$ by $$\label{2.2}
Y=\rho(1,x),~~~
~~\rho^2=\frac{m}{m-1}\left|II-\frac{1}{m} tr(II)I\right|^2~.$$ $Y$ is called *the canonical lift* of $x$ [@CPWang]. Two submanifolds $x,\bar{x}: M^m\rightarrow \mathbb{S}^{m+p}$ are Möbius equivalent if there exists $T$ in the Lorentz group $O(m+p+1,1)$ such that $\bar{Y}=YT.$ It follows immediately that $$\mathrm{g}=\langle dY,dY\rangle=\rho^2 dx\cdot dx$$ is a Möbius invariant, called the Möbius metric of $x$.
Let $\Delta$ be the Laplacian with respect to $\mathrm{g}$. Define $$N=-\frac{1}{m}\Delta Y-\frac{1}{2m^2}
\langle \Delta Y,\Delta Y\rangle Y,$$ which satisfies $$\langle Y,Y\rangle=0=\langle N,N\rangle, ~~
\langle N,Y\rangle=1~.$$ Let $\{E_1,\cdots,E_m\}$ be a local orthonormal frame for $(M^m,\mathrm{g})$ with dual 1-forms $\{\omega_1,\cdots,\omega_m\}$. Write $Y_j=E_j(Y)$. Then we have $$\langle Y_j,Y\rangle =\langle Y_j,N\rangle =0, ~\langle Y_j,Y_k\rangle =\delta_{jk}, ~~1\leq j,k\leq m.$$ We define $$\xi_r=(H^r,n_r+H^r x),~~~1\le r\le p.$$ Then $\{\xi_{1},\cdots,\xi_p\}$ form the orthonormal frame of the orthogonal complement of $\mathrm{Span}\{Y,N,Y_j|1\le j\le m\}$. And $\{Y,N,Y_j,\xi_{r}\}$ is a moving frame in $\mathbb{R}^{m+p+2}_1$ along $M^m$.
\[rem-xi\] Geometrically, at one point $x$, $\xi_r$ corresponds to the unique sphere tangent to $M^m$ with normal vector $n_r$ and the same mean curvature $H^r=\langle \xi_r,g\rangle$ where $g=(1,\vec{0})$ is a constant time-like vector. We call $\{\xi_r\}_{r=1}^p$ *the mean curvature spheres* of $M^m$.
We fix the range of indices in this section as below: $1\leq
i,j,k\leq m; 1\leq r,s\leq p$. The structure equations are: $$\label{equation}
\begin{split}
&dY=\sum_i \omega_i Y_i,\\
&dN=\sum_{ij}A_{ij}\omega_i Y_j+\sum_{i,r} C^r_i\omega_i \xi_{r},\\
&d Y_i=-\sum_j A_{ij}\omega_j Y-\omega_i N+\sum_j\omega_{ij}Y_j
+\sum_{j} B^{r}_{ij}\omega_j \xi_{r},\\
&d \xi_{r}=-\sum_i C^{r}_i\omega_i Y-\sum_{i,j}\omega_i
B^{r}_{ij}Y_j +\sum_{s} \theta_{rs}\xi_{s},
\end{split}$$ where $\omega_{ij}$ are the connection $1$-forms of the Möbius metric $\mathrm{g}$, and $\theta_{rs}$ are the normal connection $1$-forms. The tensors $${\bf A}=\sum_{i,j}A_{ij}\omega_i\otimes\omega_j,~~ {\bf
B}=\sum_{i,j,r}B^{r}_{ij}\omega_i\otimes\omega_j \xi_{r},~~
\Phi=\sum_{j,r}C^{r}_j\omega_j \xi_{r}$$ are called the Blaschke tensor, the Möbius second fundamental form and the Möbius form of $x$, respectively. The covariant derivatives $A_{ij,k}, B^{r}_{ij,k}, C^{r}_{i,j}$ are defined as usual. For example, $$\begin{aligned}
&&\sum_j C^{r}_{i,j}\omega_j=d C^{r}_i+\sum_j C^{r}_j\omega_{ji}
+\sum_{s} C^{s}_i\theta_{sr},\\
&&\sum_k B^{r}_{ij,k}\omega_k=d B^{r}_{ij}+\sum_k
B^{r}_{ik}\omega_{kj} +\sum_k B^{r}_{kj}\omega_{ki}+\sum_{s}
B^{s}_{ij}\theta_{sr}.\end{aligned}$$ The integrability conditions are given as below: $$\begin{aligned}
&&A_{ij,k}-A_{ik,j}=\sum_{r}(B^{r}_{ik}C^{r}_j
-B^{r}_{ij}C^{r}_k),\label{equa1}\\
&&C^{r}_{i,j}-C^{r}_{j,i}=\sum_k(B^{r}_{ik}A_{kj}
-B^{r}_{jk}A_{ki}),\label{equa2}\\
&&B^{r}_{ij,k}-B^{r}_{ik,j}=\delta_{ij}C^{r}_k
-\delta_{ik}C^{r}_j,\label{equa3}\\
&&R_{ijkl}=\sum_{r}(B^{r}_{ik}B^{r}_{jl}-B^{r}_{il}B^{r}_{jk}
+\delta_{ik}A_{jl}+\delta_{jl}A_{ik}
-\delta_{il}A_{jk}-\delta_{jk}A_{il}),\label{equa4}\\
&&R^{\perp}_{rs ij}=\sum_k
(B^{r}_{ik}B^{s}_{kj}-B^{s}_{ik}B^{r}_{kj}). \label{equa5}\end{aligned}$$ Here $R_{ijkl}$ denote the curvature tensor of $\mathrm{g}$. Other restrictions on tensor $\bf B$ are $$\sum_j B^{r}_{jj}=0, ~~~\sum_{i,j,r}(B^{r}_{ij})^2=\frac{m-1}{m}. \label{equa7}$$ All coefficients in the structure equations are determined by $\{\mathrm{g}, {\bf B}\}$ and the normal connection $\{\theta_{rs}\}$. Coefficients of Möbius invariants and the isometric invariants are related as below. (We omit the formula for $A_{ij}$ since it will not be used later.) $$\begin{aligned}
B^{r}_{ij}&=\rho^{-1}(h^{r}_{ij}-H^{r}\delta_{ij}),\label{2.22}\\
C^{r}_i&=-\rho^{-2}[H^{r}_{,i}+\sum_j(h^{r}_{ij}
-H^{r}\delta_{ij})e_j(\ln\rho)]. \label{2.23}\end{aligned}$$
\[rem-xi2\] For $x: M^3 \rightarrow \mathbb{R}^5$, the Möbius position vector $Y: M^3\rightarrow \mathbb{R}^7_1$ and the mean curvature sphere $\{\xi_{1},\cdots,\xi_p\}$ are given by $$Y=\rho(\frac{1+|x|^2}{2}, \frac{1-|x|^2}{2}, x),$$ $$\xi_r=\left(\frac{1+|x|^2}{2}, \frac{1-|x|^2}{2}, x\right)H^r+(x\cdot n_r,-x\cdot n_r,n_r).$$ Note that $H^r=\langle \xi_r,g\rangle$ where $g=(-1,1,\vec{0})$ is a constant light-like vector. For $x: M^3 \rightarrow \mathbb{H}^5 \subset \mathbb{R}^6_1$ (the hyperboloid model of the hyperbolic space), the corresponding formulae are $$Y=\rho(x,1),~~~\xi_r=(n_r+H^rx,H^r),~~r=1,\cdots,p.$$ In this case $H^r=\langle \xi_r,g\rangle$ where $g=(\vec{0},1)$ is a constant space-like vector. The Möbius invariants are related to the isometric invariants still by $\sim$ .
Three dimensional Wintgen ideal submanifolds in $\mathbb{S}^5$
==============================================================
From now on, we assume $x: M^{3}\to \mathbb{S}^{5}$ to be a three dimensional Wintgen ideal submanifold without umbilic points. According to and ,, that means we can choose a suitable tangent and normal frame ($\{E_1,E_2,E_3\}$ and $\{\xi_1,\xi_2\}$) such that the Möbius second fundamental form $\bf{B}$ takes the form $$\label{3.1}
B^{1}=
\begin{pmatrix}
0 & \mu & 0\\
\mu & 0 & 0\\
0 & 0 & 0
\end{pmatrix},~~~~~~
B^{2}=
\begin{pmatrix}
\mu & 0 & 0\\
0 & -\mu & 0\\
0 & 0 & 0
\end{pmatrix}, ~~~~\mu=\frac{1}{\sqrt{6}}.$$
\[rem-transform\] The distribution $\mathbb{D}=\mathrm{Span}\{E_1,E_2\}$ is well-defined. The same is true for the vector field $E_3$ up to a sign, and this sign is fixed on a connected and orientable subset of $M^3$. Notice that the tangent and normal frames still allow a simultaneous transformation $$\label{transform}
(\widetilde{E}_1,\widetilde{E}_2)=(E_1,E_2)
\begin{pmatrix} ~~\cos t &\sin t \\ -\sin t & \cos t\end{pmatrix},
~~(\widetilde{\xi}_1,\widetilde{\xi}_2)=(\xi_1,\xi_2)
\begin{pmatrix} \cos 2t &-\sin 2t \\ \sin 2t & ~~\cos 2t\end{pmatrix}$$ if we fix the induced orientation on the tangent and normal bundles and require that $B^1,B^2$ still take the form .
First we compute the covariant derivatives of $B^{r}_{ij}$. From (\[3.1\]) we get $$\label{bb1}
B^1_{33,i}=B^2_{33,i}=B^1_{12,i}=B^2_{11,i}=B^2_{22,i}=0,~~1\leq i\leq 3.$$ Other derivatives are related with the connection 1-forms $\o_{ij}$ as below: $$\label{bb2}
\begin{aligned}
\omega_{23}&=\sum_i\frac{B^1_{13,i}}{\mu}\omega_i
=-\sum_i\frac{B^2_{23,i}}{\mu}\omega_i;\\ \omega_{13}&=\sum_i\frac{B^1_{23,i}}{\mu}\omega_i
=~~\sum_i\frac{B^2_{13,i}}{\mu}\omega_i;\\
2\omega_{12}+\theta_{12}&=\sum_i\frac{-B^1_{11,i}}{\mu}\omega_i
=\sum_i\frac{B^1_{22,i}}{\mu}\omega_i
=\sum_i\frac{B^2_{12,i}}{\mu}\omega_i.
\end{aligned}$$ By and we know the symmetry property below, $$\label{bb3}
B^1_{13,2}=B^1_{23,1}=B^1_{12,3}=0,
~B^2_{13,2}=B^2_{23,1}=B^2_{12,3},$$ where we have used $B^1_{12,3}=0$ by . From this fact and comparing coefficients in , we obtain $$\label{bb4}
\mu\o_{13}(E_1)=B^2_{13,1}=B^1_{23,1}=0,~
\mu\o_{23}(E_2)=-B^2_{23,2}=B^1_{13,2}=0.$$ Similarly we know the coefficients in the following three equalities are equal to each other: $$\label{bb5}
\begin{aligned}
-\mu\o_{23}(E_1)&=-B^1_{13,1}=B^2_{23,1},\\
\mu\o_{13}(E_2)&=~~B^1_{23,2}=B^2_{13,2},\\
\mu(2\o_{12}+\theta_{12})(E_3)&=-B^1_{11,3}=B^1_{22,3}=B^2_{12,3}.
\end{aligned}$$
Next we derive the Möbius form, using and the information on $B^r_{ij,k}$: $$\label{cc1}
C^1_3=B^1_{22,3}-B^1_{23,2}=0,~C^2_3=B^2_{11,3}-B^2_{13,1}=0.$$ The other coefficients $\{C^{r}_j\}$ are obtained similarly as below: $$\label{cc2}
\begin{aligned}
&C^{1}_1=-B^{1}_{13,3}=-\mu\omega_{23}(E_3),
~~~~C^{2}_2=-B^{2}_{23,3}=~~\mu\omega_{23}(E_3), \\
&C^{1}_2=-B^{1}_{23,3}=-\mu\omega_{13}(E_3), ~~~~C^{2}_1=-B^{2}_{13,3}=-\mu\omega_{13}(E_3),\\
&C^{1}_1=B^{1}_{22,1}=~\mu(2\o_{12}+\theta_{12})(E_1)
=~B^{2}_{12,1}=-C^{2}_2, \\
&C^{1}_2=B^{1}_{11,2}=-\mu(2\o_{12}+\theta_{12})(E_2) =-B^{2}_{12,2}=~C^{2}_1.
\end{aligned}$$ For simplicity we introduce the following notations: $$\label{UVL}
\begin{aligned}
&U=~~\o_{23}(E_3)=-\frac{C^1_1}{\mu}=\frac{C^2_2}{\mu},\\
&V=-\o_{13}(E_3)=~~\frac{C^1_2}{\mu}=\frac{C^2_1}{\mu},\\
&L=\o_{13}(E_2)=-\o_{23}(E_1)=-\frac{B^1_{11,3}}{\mu}.
\end{aligned}$$ Then we summarize what we know about the connection 1-forms and the covariant derivatives $B^r_{ij,k}$ as below: $$\begin{gathered}
\o_{13}=L\o_2-V\o_3,~~~\o_{23}=-L\o_1+U\o_3;\notag\\
2\o_{12}+\theta_{12}=-U \o_1-V\o_2+L\o_3.\label{omega}\end{gathered}$$ By we have $$\label{L}
d\o_3=\o_{31}\wedge \o_1+\o_{32}\wedge\o_2\equiv 2L\o_1\wedge \o_2 ~~~mod(\o_3).$$ So the distribution $\mathbb{D}=\mathrm{Span}\{E_1, E_2\}$ is integrable if and only if $L=0$ identically.
For the information on the Blaschke tensor $\bf A$, we use . It requires to compute the covariant derivatives of $C^r_j$, which is quite straightforward: $$C^1_{1,i}=-C^2_{2,i}, ~C^1_{2,i}=C^2_{1,i},
~C^1_{3,2}=C^1_1 L=-\mu UL, ~C^1_{3,1}=-C^1_2 L=-\mu VL.$$ Now it follows that $$\begin{aligned}
\mu (A_{11}-A_{22})&=\sum_k(B^1_{2k}A_{k1}-B^1_{1k}A_{k2})
=C^1_{2,1}-C^1_{1,2},\label{A11}\\
2\mu A_{12}&=\sum_k(B^2_{1k}A_{k2}-B^2_{2k}A_{k1})
=C^2_{1,2}-C^2_{2,1}=C^1_{1,1}+C^1_{2,2},\label{A12}\\
\mu A_{13}&=\sum_k(B^1_{2k}A_{k3}-B^1_{3k}A_{k2})
=C^1_{2,3}-C^1_{3,2}=C^1_{2,3}+\mu UL,\label{A13}\\
\mu A_{23}&=\sum_k(B^1_{1k}A_{k3}-B^1_{3k}A_{k1})
=C^1_{1,3}-C^1_{3,1}=C^1_{1,3}+\mu VL.\label{A23}\end{aligned}$$
Consider a new frame $\{Y,{\hat Y},\eta_1,\eta_2,\eta_3,
\xi_1,\xi_2\}$ in ${\mathbb R}^{7}_1$ along $M^3$ as below, whose geometric meaning will be clear later (see Theorem \[thm-envelop\] and Remark \[rem-hatY\]). $$\begin{gathered}
\eta_1=Y_1+\frac{C_2^1}{\mu}Y=Y_1+VY,~~
\eta_2=Y_2+\frac{C^1_1}{\mu}Y=Y_2-UY,~~
\eta_3=Y_3-\lambda Y;\label{eta}\\
{\hat Y}=N-\frac{1}{2}(U^2+V^2+\lambda^2)Y-VY_1+U Y_2+\lambda Y_3.\label{hatY}\end{gathered}$$ Here $\lambda\in C^{\infty}(M^3)$ is an arbitrarily given smooth function at the beginning. The new frame is orthonormal except that $$\langle Y,Y\rangle=0=\langle {\hat Y},{\hat Y}\rangle,~
\langle Y,{\hat Y}\rangle=1.$$ By the original structure equations we get $$\begin{aligned}
d\xi_{1}&=-\mu\o_2\eta_1-\mu\o_1\eta_2+\theta_{12}\xi_2,\label{3.7}\\
d\xi_{2}&=-\mu \o_1\eta_1+\mu\o_2\eta_2-\theta_{12}\xi_1,\label{3.8}\\
d\eta_1&=-{\hat\o}_1Y-\o_1{\hat
Y}+\sum_k\Omega_{1k}\eta_k+\mu\o_2\xi_1+\mu\o_1\xi_2,\label{3.3}\\
d\eta_2&=-{\hat\o}_2Y-\o_2{\hat
Y}+\sum_k\Omega_{2k}\eta_k+\mu\o_1\xi_1-\mu\o_2\xi_2,\label{3.4}\\
d\eta_3&=-{\hat\o}_3Y-\o_3{\hat Y}+\sum_k\Omega_{3k}\eta_k,\label{3.5}\\
dY&=\o Y+\o_1\eta_1+\o_2\eta_2+\o_3\eta_3,\label{3.9}\\
d{\hat Y}&=-\o {\hat Y}+{\hat\o}_1\eta_1+{\hat\o}_2\eta_2+{\hat\o}_3\eta_3, \label{3.10}\end{aligned}$$ Note that and give the first motivation for the definition of $\eta_1,\eta_2$ in .
Differentiate $\sim$ . We get the following integrability equations: $$\begin{aligned}
&d\o_1=\o\wedge\o_1+\Omega_{12}\wedge\o_2+\Omega_{13}\wedge\o_3; \label{3.12}\\ &d\o_2=\o\wedge\o_2-\Omega_{12}\wedge\o_1+\Omega_{23}\wedge\o_3; \label{3.13}\\
&d\o_3=\o\wedge\o_3+\Omega_{31}\wedge\o_1+\Omega_{32}\wedge\o_2; \label{3.21}\\
&d\o_1=-(\theta_{12}+\Omega_{12})\wedge\o_2; \label{3.14}\\
&d\o_2=~~(\theta_{12}+\Omega_{12})\wedge\o_1; \label{3.15}\\
&d{\hat\o}_1=-\o\wedge{\hat\o}_1+\Omega_{12}\wedge{\hat\o}_2
+\Omega_{13}\wedge{\hat\o}_3; \label{3.16}\\
&d{\hat\o}_2=-\o\wedge{\hat\o}_2-\Omega_{12}\wedge{\hat\o}_1
+\Omega_{23}\wedge{\hat\o}_3; \label{3.17}\\
&d{\hat\o}_3=-\o\wedge{\hat \o}_3+\Omega_{31}\wedge{\hat\o}_1
+\Omega_{32}\wedge{\hat\o}_2; \label{3.22}\\
&d\Omega_{12}=\Omega_{13}\wedge\Omega_{32}-\o_1\wedge{\hat\o}_2
-{\hat\o_1}\wedge\o_2+2\mu^2\o_1\wedge\o_2; \label{3.18}\\
&d\Omega_{13}=\Omega_{12}\wedge\Omega_{23}-\o_1\wedge{\hat\o}_3
-{\hat\o}_1\wedge\o_3; \label{3.19}\\
&d\Omega_{23}=-\Omega_{12}\wedge\Omega_{13}-\o_2\wedge{\hat\o}_3
-{\hat\o}_2\wedge\o_3; \label{3.20}\\
&\Omega_{13}\wedge\o_1=\Omega_{23}\wedge\o_2;\hskip 5pt \Omega_{13}\wedge\o_2=-\Omega_{23}\wedge\o_1;\label{3.23}\\
&d\theta_{12}=2\mu^2\o_1\wedge\o_2;\label{3.24}\\
&\o_1\wedge{\hat\o_2}=-\o_2\wedge{\hat\o}_1;\hskip 5pt \o_1\wedge{\hat\o_1}=\o_2\wedge{\hat\o}_2;\label{3.25}\\
&d\o=-\o_1\wedge{\hat\o}_1-\o_2\wedge{\hat\o}_2
-\o_3\wedge{\hat\o}_3.\label{3.26}\end{aligned}$$ The 1-forms $\omega,{\hat\o}_i,\Omega_{ij}=-\Omega_{ji}$ are determined by and ,: $$\begin{aligned}
\Omega_{12}&=\langle d\eta_1,\eta_2\rangle
=\o_{12}+U\o_1+V\o_2,\label{Omega12}\\
\Omega_{13}&=\langle d\eta_1,\eta_3\rangle
=\lambda\o_1+L\o_2,\label{Omega13}\\
\Omega_{23}&=\langle d\eta_2,\eta_3\rangle
=-L\o_1+\lambda\o_2,\label{Omega23}\\
\o&=\langle dY,{\hat Y}
\rangle=-V\o_1+U\o_2+\lambda\o_3.\end{aligned}$$ It follows from that there exist some functions $\hat{F},\hat{G}$ such that $${\hat\o_1}=\hat{F}\o_1+\hat{G}\o_2,~~ {\hat\o_2}=-\hat{G}\o_1+\hat{F}\o_2.\label{FG}$$ A straightforward yet lengthy computation find $$\begin{aligned}
{\hat\o}_1&=\langle d{\hat Y},\eta_1\rangle \\
&= d\left(N-\frac{1}{2}(U^2+V^2+\lambda^2)Y-VY_1+U Y_2+\lambda Y_3\right)\cdot (Y_1+VY)\\
&=\sum_i A_{i1}\o_i-\frac{1}{2}(U^2+V^2+\lambda^2)\o_1\\
&~~~~-dV
+V^2\o_1+U\o_{21}-UV\o_2+\lambda\o_{31}-\lambda V\o_3\\
&=\left(A_{11}+\frac{1}{2}(U^2+V^2-\lambda^2)
-\frac{1}{\mu}C^1_{2,1}\right)\o_1
+\left(A_{12}-\frac{1}{\mu}C^1_{2,2}-\lambda L\right)\o_2,\end{aligned}$$ where we have used $d C^1_2=\sum_i C^1_{2,i}\o_i-C^1_1\o_{12}-C^2_2\theta_{21}$ and , . For ${\hat\o}_2$ we compute in a similar manner, using , and to verify , with the result as below: $$\begin{aligned}
\hat{F}&=A_{11}+\frac{1}{2}(U^2+V^2-\lambda^2)
-\frac{1}{\mu}C^1_{2,1},\label{hatF}\\
\hat{G}&=A_{12}-\frac{1}{\mu}C^1_{2,2}-\lambda L
=\frac{C^1_{1,1}-C^1_{2,2}}{2\mu}-\lambda L.\label{hatG}\end{aligned}$$ In particular we have the following observation.
\[lem-G\] $\hat{G}$=0 if and only if $\lambda=\frac{G}{L}$ with $L=-\frac{B^1_{11,3}}{\mu}, G=\frac{C^1_{1,1}-C^1_{2,2}}{2\mu}$.
In the end of this section we make the following important geometric observation.\
The spacelike 2-plane $\mathrm{Span}_{\mathbb{R}}\{\xi_1,\xi_2\}$ at $p\in M^3$ is well-defined, and we call it *the 3-dimensional mean curvature sphere*, because it defines a 3-sphere tangent to $M^3$ at $p$ with the same mean curvature vector. (Please compare to Remark \[rem-xi\].)
The first key observation is that in the codimension two case, $\mathrm{Span}_{\mathbb{R}}\{\xi_1,\xi_2\}$ is determined by the complex line $\mathrm{Span}_{\mathbb{C}}\{\xi_{1}-i\xi_{2}\}$ and vice versa. So $\xi_1-i\xi_2\in \mathbb{C}^7$ represents the same geometric object. It is a null vector with respect to the $\mathbb{C}$-linear extension of the Lorentz metric to $\mathbb{R}^7_1\otimes\mathbb{C}$. The complex line spanned by it corresponds to a point $$[\xi_{1}-i\xi_{2}] \in Q^5=\{[Z]\in \mathbb{C}P^6| \langle Z,Z\rangle =0\}.$$ It is similar to the conformal Gauss map of a (Willmore) surface [@br0] and to the Gauss map of a hypersurface in $\mathbb{S}^n$ [@MaOhnita].
The second key observation is that under the hypothesis of being Wintgen ideal, this 3-sphere congruence is indeed a 2-parameter family, and its envelope not only recovers $M^3$, but also extends it to a 3-manifold as a circle bundle over a Riemann surface $\overline{M}$ (a holomorphic curve). The underlying surface $\overline{M}$ comes from the quotient surface $\overline{M}=M^3/\Gamma$ (at least locally) where $\Gamma$ is the foliation of $M^3$ by the integral curves of the vector fields $E_3$. More precise statement is as below.
\[thm-envelop\] For a Wintgen ideal submanifold $x:M^3\to \mathbb{S}^5$ we have:
\(1) The complex vector-valued function $\xi_{1}-i\xi_{2}$ locally defines a complex curve $$[\xi_1-i\xi_2]:\overline{M}=M^3/\Gamma\to Q^5\subset\mathbb{C}P^6.$$ (2) *The 3-dimensional mean curvature spheres* $\mathrm{Span}\{\xi_1(p),\xi_2(p)\}$ is a two-parameter family of 3-spheres in $\mathbb{S}^5$ when $p$ runs through $M^3$.
\(3) The Lorentz 3-space $\mathrm{Span}\{Y,{\hat Y},\eta_3\}$ and the two light-like directions $[Y],[{\hat Y}]\in \mathbb{R}P^6$ correspond to a circle and two points on it. These circles are a two-parameter family. They foliate a three dimensional submanifold $\widehat{M}^3$ enveloped by the 3-dimensional mean curvature spheres $\mathrm{Span}\{\xi_{1},\xi_{2}\}$, which includes $M^3$ as part of it. On $M^3$ these circular arcs are indeed the integral curves of the vector field $E_3$.
\(4) This envelope $\widehat{M}^3\subset\mathbb{S}^5$, as a natural extension of $x:M^3\to\mathbb{S}^5$, is still a Wintgen ideal submanifold (at its regular points).
The structure equations and imply $$\label{J}
d(\xi_{1}-i\xi_{2})=i\mu(\o_1+i\o_2)(\eta_1+i\eta_2)
+i\theta_{12}(\xi_1-i\xi_2).$$ Geometrically that means $\xi=[\xi_{1}-i\xi_{2}]:M^3\to \mathbb{C}P^6$ decomposes as a quotient map $\pi:M^3\to \overline{M}=M^3/\Gamma$ composed with a holomorphic immersion $\bar\xi:\overline{M}\to \mathbb{C}P^6$. Thus conclusion (1) is proved, and (2) follows directly.
To prove (3), notice that the light-like directions in $\mathrm{Span}\{Y,{\hat Y},\eta_3\}$ represent points on a circle. Since $\{\xi_1,\xi_2,d\xi_1,d\xi_2\}$ span a 4-dimensional spacelike subspace by , the corresponding 2-parameter family of 3-dimensional mean curvature sphere congruence has an envelop $\widehat{M}$, whose points correspond to the light-like directions in the orthogonal complement $\mathrm{Span}\{Y,{\hat Y},\eta_3\}$. In particular $[Y],[{\hat Y}]$ are two points on this circle. Such circles form a 2-parameter family, with $\overline{M}$ as the parameter space. They give a foliation of $\widehat{M}^3$ which is also a circle fibration.
We assert that every integral curve $\gamma$ of $E_3$ is contained in such a circle. Because $Span\{\xi_1,\xi_2,\eta_1,\eta_2\}(p)$ is a fixed subspace along an integral curve of $E_3$ passing $p\in M^3$ by $\sim$ and ,,. Thus the integration of $Y$ along $E_3$ direction is always located in the orthogonal complement $\mathrm{Span}\{Y(p),{\hat Y}(p),\eta_3(p)\}$, which describes a circle as above. Thus $\widehat{M}^3\supset M^3$, and each circle fiber cover an integral curve of $E_3$. This verifies (3).
To prove (4), we need only to show that for arbitrarily chosen smooth function $\lambda:M^3\to \mathbb{R}$, the corresponding $[\hat{Y}]:M^3\to \mathbb{S}^5$ is a Wintgen ideal submanifold. This is because at one point $p\in M^3$, when $\lambda$ runs over $(-\infty,\infty)$, $[\hat{Y}(p)]$ given by will cover the circle fiber except $[Y(p)]$ itself; and when $\lambda: M^3\to \mathbb{R}$ is arbitrary, all such local mappings will cover $\widehat{M}$ by their images.
We have to compute out the Laplacian of $\hat{Y}$ with respect to its induced metric $\hat\o_1^2+\hat\o_2^2+\hat\o_3^2$ which is necessary to determine the normal frames (the mean curvature spheres) $\{\hat\xi_r\}$ of $\hat{Y}$. The main difficulty is that the map $\hat{Y}:M^3\to \mathbb{R}^7_1$ is generally not conformal to $Y$ . Fortunately we need only to find an orthogonal frame $\{\hat{E}_j\}_{j=1}^3$ with the same length for $\hat{Y}$, and then using the fact $
\mathrm{Span}_{\mathbb{R}}\{\hat{Y},\hat{Y}_j,\sum_{j=1}^3 \hat{E}_j\hat{E}_j(\hat{Y})\}
=\mathrm{Span}_{\mathbb{R}}\{\hat{Y},\hat{Y}_j, \hat{\Delta}\hat{Y}\}.$\
**Claim:** $\hat{Y}$ shares the same mean curvature spheres $\{\xi_1,\xi_2\}$ as $Y$.\
This requires to show $\langle\sum_{j=1}^3 \hat{E}_j\hat{E}_j(\hat{Y}),\xi_r\rangle=0.$ Since $0=\langle\hat{Y},\xi_r\rangle=\langle d\hat{Y},\xi_r\rangle=\langle \hat{Y},d\xi_r\rangle$ by , and , we need only to verify $$\langle \hat{Y},\sum_{j=1}^3 \hat{E}_j\hat{E}_j(\xi_r)\rangle=0,~~r=1,2.$$ For this purpose, suppose (keeping in mind): $${\hat\o_1}=\hat{F}\o_1+\hat{G}\o_2,~~ {\hat\o_2}=-\hat{G}\o_1+\hat{F}\o_2,~~
\hat\o_3=a\o_1+b\o_2+c\o_3.$$ Notice that we can always assume $\hat{Y}$ to be an immersion at the points where $\widehat{M}$ is regular, hence $c\ne 0$. Then one can take $$\hat{E}_1 = \hat{F} E_1+\hat{G} E_2 + a_{13}E_3,~
\hat{E}_2 = -\hat{G} E_1+\hat{F} E_2 +a_{23}E_3,~
\hat{E}_3 = a_{33}E_3,$$ where $a_{13},a_{23},a_{33}$ are uniquely determined by $\hat\o_i(\hat{E}_j)=(\hat{F}^2+\hat{G}^2)\delta_{ij}$. The explicit form of $a_{13},a_{23},a_{33}$ is not important, because when we insert the formulae above to $\sum_{j=1}^3 \hat{E}_j\hat{E}_j(\xi_r)$, the terms involving $E_3$ will always be orthogonal to $\hat{Y}$. For example, $\langle E_3(\eta_1),\hat{Y}\rangle=-\langle \eta_1,E_3(\hat{Y})\rangle=\hat\o_1(E_3)=0$ by . Thus we need only to compute the effect on $\xi_r$ of the operator below: $$(\hat{F} E_1+\hat{G} E_2)^2+(-\hat{G} E_1+\hat{F} E_2)^2
\thickapprox(\hat{F}^2+\hat{G}^2)\partial\bar{\partial}.$$ The two sides are equal up to first order differential operators like $[E_1,E_2],E_1,E_2$, whose action on $\xi_r$ must be orthogonal to $\hat{Y}$; the complex differential operators are defined as usual: $$\partial=E_1-iE_2,~\bar\partial=E_1+iE_2.$$ Since $(\o_1+i\o_2)(\bar\partial)=0$, it follows from that $$\begin{aligned}
\bar{\partial}(\xi_1-i\xi_2)&=i\theta_{12}(\bar{\partial})
(\xi_1-i\xi_2),\\
\partial\bar{\partial}(\xi_1-i\xi_2)&\in
\mathrm{Span}_{\mathbb{C}}\{\xi_1-i\xi_2,\eta_1+i\eta_2\}~~\bot~~ \hat{Y}.\end{aligned}$$ This completes the proof of the previous claim.
For $\hat{Y}$ we still take its canonical lift, whose derivatives are combinations of $\hat{Y},\eta_1,\eta_2,\eta_3$; its normal frame is still $\{\xi_1,\xi_2\}$. We read from that its Möbius second fundamental form still take the same form as . Thus conclusion (4) and the whole theorem is proved.
In the proof above, among the integrability equations from $\sim$, only and are necessary for us (to deduce the algebraic form and ,, where the explicit coefficients are not important). This is somewhat striking to the authors that the strong conclusion (4) follows from so few conditions. In a forthcoming paper [@Li3] we will give a general treatment of codimension-two Wintgen ideal submanifolds based on the observations in this theorem.
Another interesting feature is the resemblance between conclusion (4) and the duality theorem for Willmore surfaces [@br0].
Minimal Wintgen ideal submanifolds
==================================
Since we have classified all Wintgen ideal submanifolds in [@Li1] whose canonical distribution $\mathbb{D}=\mathrm{Span}\{E_1,E_2\}$ is integrable (that means $L=0$), in the rest of this paper we will only consider the case $$L\neq0.$$ From now on we take the frame and , and make the following $$\label{lambda}
\textbf{Assumption:}~~~~
\lambda=\frac{G}{L}=\frac{C^1_{1,1}-C^1_{2,2}}{-2 B^1_{11,3}}~.\qquad\qquad\qquad\qquad\qquad$$ By Lemma \[lem-G\] we have $$\label{3.27}
{\hat\o_1}=\hat{F}\o_1,~~ {\hat\o_2}=\hat{F}\o_2.$$
\[rem-hatY\] The correspondence $Y\to \hat{Y}$ describes a self-mapping of the enveloping submanifold $\widehat{M}^3$ where $\hat{Y}$ and $Y$ are located on the same circle fiber. At corresponding points they share the same normal vector fields $\{\xi_r\}$, with respect to which we can talk about principal directions. According to Theorem \[thm-envelop\] and Lemma \[lem-G\], the correspondence $Y\to \hat{Y}$ preserves the principal directions for any $\xi_r$ if and only if $\lambda=G/L$. This explains the geometric significance of the condition .
Under the assumption , $$\label{o}
\o=-V\o_1+U\o_2+\frac{G}{L}\o_3.$$ Together with and there must be $$d\o=-\o_3\wedge{\hat\o}_3.\label{3.28}$$ On the other hand, it follows from and $\lambda=G/L$ that , and now take the form $$\begin{gathered}
2\Omega_{12}+\theta_{12}=U\o_1+V\o_2+L\o_3,\label{Omega12+}\\
\Omega_{13}=\frac{G}{L}\o_1+L\o_2,~~
\Omega_{23}=-L\o_1+\frac{G}{L}\o_2.\label{Omega13+}\end{gathered}$$ Insert these into structure equations and , and simplify by , together with . We have $$\begin{aligned}
dL\wedge\o_2&=
(U\o_1+V\o_2+L\o_3)\wedge(-L\o_1+\frac{G}{L}\o_2)
-\o_1\wedge({\hat \o}_3-d\frac{G}{L})-\hat{F}\o_1\wedge\o_3,\\
dL\wedge\o_1&=
(U\o_1+V\o_2+L\o_3)\wedge(\frac{G}{L}\o_1+L\o_2)
+\o_2\wedge({\hat \o}_3-d\frac{G}{L})+\hat{F}\o_2\wedge\o_3.\end{aligned}$$ Comparing the coefficients of $\o_1\wedge\o_2,\o_1\wedge\o_3,\o_2\wedge\o_3$ in these two equations separately, we obtain $$\begin{aligned}
\hat\o_3(E_1)&=E_2(L)+UL+E_1(\frac{G}{L})-V\frac{G}{L},\label{compare1}\\
\hat\o_3(E_2)&=-E_1(L)+VL+E_2(\frac{G}{L})+U\frac{G}{L},\label{compare2}\\
\hat\o_3(E_3)&=E_3(\frac{G}{L})+L^2-\hat{F},\label{compare3}\\
E_3(L)&=G.\label{compare4}\end{aligned}$$ Similarly, inserting into and yields $$\begin{aligned}
d\hat{F}\wedge\o_1&=
(2\Omega_{12}+\theta_{12})\wedge \hat{F}\o_2
-\o\wedge \hat{F}\o_1+\Omega_{13}\wedge\hat\o_3,\\
d\hat{F}\wedge\o_2&=
-(2\Omega_{12}+\theta_{12})\wedge \hat{F}\o_1
-\o\wedge \hat{F}\o_2+\Omega_{23}\wedge\hat\o_3.\end{aligned}$$ Invoking ,, and comparing the coefficients of $\o_1\wedge\o_2,\o_1\wedge\o_3,\o_2\wedge\o_3$ in these two equations separately, one gets $$\begin{aligned}
\hat{F}&=\hat\o_3(E_3),\label{compare5}\\
E_1(\hat{F})
&=2V\hat{F}-\frac{G}{L}\hat\o_3(E_1)-L\hat\o_3(E_2),\label{compare6}\\
E_2(\hat{F})
&=-2U\hat{F}+L\hat\o_3(E_1)-\frac{G}{L}\hat\o_3(E_2),\label{compare7}\\
E_3(\hat{F})
&=-\frac{G}{L}(\hat{F}+\hat\o_3(E_3))=-2\frac{G}{L}\cdot\hat{F}.\label{compare8}\end{aligned}$$
\[rem-invariant\] We point out that $L,G,\o,\hat{F}$ are well-defined Möbius invariants. It is necessary and sufficient to verify that they are independent to the choice of the frames, or equivalently, that they are invariant under the transformation . This is obvious for $L$ by , for $G$ by , and for $\o$ by $\o=\langle dY,{\hat Y}\rangle$ where the frame vectors $\hat{Y},\eta_3$ are now canonically chosen after taking $\lambda=\frac{G}{L}$. For $\hat{F}$ we can verify the invariance under the transformation by $\hat{F}=\langle E_1({\hat Y}),\eta_1\rangle$, or just using .
On the other hand, $U,V$ correspond to $\{C^r_i\}$, components of the Möbius form, which depend on the choice of $\{E_1,E_2\}$ and $\{\xi_1,\xi_2\}$. Yet in that case we can choose the angle $t$ in suitably such that $V=0$ identically. Then the new function $U$ is well-defined and Möbius invariant. (There are other choice of the frame in a canonical way, and any of them works in the proof to Theorem \[thm-homog\] later.)
Now we can state our Möbius characterization theorem for minimal Wintgen ideal submanifolds of dimension three in five dimensional space form.
\[thm-minimal\] Let $x: M^3\to {\mathbb S}^{5}$ be a Wintgen ideal submanifold. Assume the distribution $\mathbb{D}=\mathrm{Span}\{E_1, E_2\}$ to be non-integrable, i.e., $L\ne 0$. Then the following conditions are equivalent:
\(1) $ d\o=0,$ where $\o=-V\o_1+U\o_2+\frac{G}{L}\o_3$.
\(2) The correspondence $Y\to \hat{Y}$ of the enveloping submanifold $\widehat{M}^3$ is a conformal map. ($\hat{Y}$ might be degenerate.)
\(3) $x: M^3\to {\mathbb S}^{5}$ is Möbius equivalent to a minimal Wintgen ideal submanifold in a space form. In particular, the space form is $\mathbb{S}^5$,$\mathbb{R}^5$ or $\mathbb{H}^5$ depending on whether $\hat{F}$ is positive, zero or negative.
If $Y$ and $\hat Y$ are conformal, then there exists a non-negative function $a$ so that $$\hat\o_1^2+\hat\o_2^2+\hat\o_3^2=a(\o_1^2+\o_2^2+\o_3^2).\label{3.51}$$ Combining with and we have $d\o=0$. Thus (2) implies (1).
Next we show (1) implies (2) and (3). If $d\o=0$, it follows from that $\hat \o_3=\hat{F}\o_3$. Together with , $\hat \o_j=\hat{F}\o_j$ for $j=1,2,3$. So $Y$ and $\hat Y$ are conformal, and (2) is proved.
Using $\sim$ we get $$d\hat{F}+2\hat{F}\omega = 0.\label{3.50}$$ Now the structure equations can be rewritten as below: $$\begin{aligned}
d(\hat{F}Y+{\hat Y})&=-\o (\hat{F}Y+{\hat Y})+2\hat{F}(\o_1\eta_1+\o_2\eta_2+\o_3\eta_3);\label{3.57}\\
d\eta_1&=-{\o}_1(\hat{F}Y+{\hat
Y})+\Omega_{12}\eta_2+L\o_2\eta_3+\mu\o_2\xi_1+\mu\o_1\xi_2;\\
d\eta_2&=-{\o}_2(\hat{F}Y+{\hat
Y})-\Omega_{12}\eta_1-L\o_1\eta_3+\mu\o_1\xi_1-\mu\o_2\xi_2;\\
d\eta_3&=-{\o}_3(\hat{F}Y+{\hat
Y})-L\o_2\eta_1+L\o_1\eta_2;\\
d\xi_{1}&=-\mu\o_2\eta_1-\mu\o_1\eta_2+\theta_{12}\xi_2;\\
d\xi_{2}&=-\mu \o_1\eta_1+\mu\o_2\eta_2-\theta_{12}\xi_1;\label{3.62}\\
d({\hat{F}Y-\hat Y})&=-\o (\hat{F}Y-{\hat Y}). \label{3.63}\end{aligned}$$ So $\mathrm{Span}\{\hat{F}Y-\hat Y\}$ is parallel along $M^3$, as well as its orthogonal complement $$\mathbb{V}^6=\mathrm{Span}\{\hat{F}Y+\hat Y, \eta_1, \eta_2, \eta_3, \xi_1, \xi_2\}~.$$ That means both of them are fixed subspaces of $\mathbb{R}^7_1$. The type of the inner product restricted on these subspaces depends on the sign of $\hat{F}$, which will not change on a connected open set, because $\hat{F}$ satisfies a linear PDE . We discuss them case by case.
[**Case 1:**]{} $\hat{F}>0$. This case $\mathbb{V}^6$ is a fixed space-like subspace orthogonal to a fixed time-like line $\mathbb{V}^\perp$. Define $$f=\frac{\hat{F}Y+\hat Y}{\sqrt{2\hat{F}}},\qquad g=\frac{\hat{F}Y-\hat Y}{\sqrt{2\hat{F}}},$$ which satisfy $\langle f,f\rangle=1,\langle g, g\rangle=-1$. So $g \in \mathbb{V}^\perp$ is a constant time-like vector, and $f: M^3 \rightarrow \mathbb{S}^5 \subset \mathbb{V}$ is a submanifold in the sphere.
Assume $g=(1,\vec{0})$. This is without loss of generality since we can always apply a Lorentz transformation in $\mathbb{R}^{7}_{1}$ to $Y$ and its frame at the beginning if necessary, whose effect on $x(M)\subset\mathbb{S}^5$ is a Möbius transformation. Then from the geometric meaning of the mean curvature sphere $\xi_r$ explained in Remark \[rem-xi\], we know that $x: M^3\rightarrow \mathbb \mathbb{S}^{5}$ is a minimal Wintgen ideal submanifold (up to a conformal transformation).
Note that this special minimal submanifold can now be identified with $f$ since we have $$f+g=\sqrt{2\hat{F}}Y=\sqrt{2\hat{F}}\rho (1,x).$$ Comparison shows $\sqrt{2\hat{F}}\rho=1$ and $x=f$. That $f:M^3 \rightarrow \mathbb{S}^5 \subset \mathbb{V}$ is minimal can be verified directly by $\sim$.
[**Case 2:**]{} $\hat{F}<0$. In this case, $\mathbb{V}^6$ is a fixed Lorentz subspace orthogonal to a constant space-like line $\mathbb{V}^\perp$. Define $f=(\hat{F}Y+\hat Y)/\sqrt{-2\hat{F}}, g=(\hat{F}Y-\hat Y)/\sqrt{-2\hat{F}}.$ Then similar to Case 1 we know $x$ is Möbius equivalent to $f: M^3 \rightarrow \mathbb{H}^5 \subset \mathbb{V}\cong \mathbb{R}^6_1$ which is a minimal Wintgen ideal submanifold.
[**Case 3:**]{} $\hat{F}\equiv 0$. Now $\hat \o_i \equiv0, ~i=1,2,3$. So $d\hat Y= -\o\hat{Y}$. That means $\hat Y$ determines a constant light-like direction. From $d\o=0$, we get $w=d\tau$ for some locally defined function $\tau$. Up to a Lorentz transformation one may take $e^{\tau}\hat Y=(-1, 1, \vec{0})$ which is still denoted by $g$. Since $\langle \xi_r,g\rangle=0$, from the geometric meaning of the mean curvature sphere $\xi_r$ explained in Remark \[rem-xi2\] we know that $x$ is a three dimensional minimal Wintgen ideal submanifold in $\mathbb{R}^5$ (up to a suitable conformal transformation).\
Finally we show (3) implies (1), i.e., for any minimal Wintgen ideal submanifold $x: M^3 \rightarrow \mathbb{Q}^5(c)$ whose distribution $\mathbb{D}=\mathrm{Span}\{E_1, E_2\}$ is not integrable, there is always $d\o=0$.
By assumption, for this $x$ we can always take local orthonormal frame $\{e_1, e_2, e_3\}$ and $\{n_1, n_2\}$ for the tangent and normal bundles, such that the second fundamental form is given by $$h^1=\begin{pmatrix}
0& \nu&0\\
\nu&0&0\\
0&0&0
\end{pmatrix}, \qquad
h^2=\begin{pmatrix}
\nu&0&0\\
0&-\nu&0\\
0&0&0
\end{pmatrix}.\label{h}$$ It follows that $\rho^2=6\nu^2$ by . From Remark 2.2, using we always have $$C^1_1=-C^2_2=-\frac{e_2(\nu)}{6\nu^2}, ~~~~~~~C_1^2=C^2_1=-\frac{e_1(\nu)}{6\nu^2}.$$ Consider the Möbius position vector $Y: M^3 \rightarrow \mathbb{R}^7_1$ of $x$ defined by . For the Möbius metric $$\mathrm{g}=\langle dY, dY\rangle=\rho^2 dx^2=6\nu^2 dx^2,$$ we can choose $\{E_1=\frac{e_1}{\sqrt{6}\nu}, \;E_2=\frac{e_2}{\sqrt{6}\nu}, \;E_3=\frac{e_3}{\sqrt{6}\nu}\}$ as a set of local orthonormal basis for $(M^3, \mathrm{g})$ with the dual basis $\{\o_1, \;\o_2, \;\o_3\}$. By , $$2\o_{12}+\theta_{12}=-U\o_1-V\o_2+L\o_3,$$ We have $$C^1_{1,1}=E_1(C^1_1)+C^1_2(\o_{21}+\theta_{21})(E_1)
=\frac{1}{\sqrt{6}\nu}(-E_1(E_2(\nu))+E_1(\nu)\o_{21}(E_1)),$$ $$C^1_{2,2}=E_2(C^1_2)+C^1_1(\o_{12}+\theta_{12})(E_2)
=\frac{1}{\sqrt{6}\nu}(-E_2(E_1(\nu))+E_2(\nu)\o_{12}(E_2)).$$ Using we get $$C^1_{1,1}-C^1_{2,2}=2\frac{E_3(\nu)}{\nu} \mu L.$$ So we get that $\frac{G}{L}=\frac{E_3(\nu)}{\nu}$. Since $$\o=-\sqrt{6}(C^1_2\o_1+C^1_1\o_2)+\frac{G}{L}\o_3
=\frac{E_1(\nu)}{\nu}\o_1+\frac{E_2(\nu)}{\nu}\o_2
+\frac{E_3(\nu)}{\nu}\o_3,\label{3.77}$$ it is obvious that $\o$ is an exact 1-form, and $d\o=0$. This finishes the proof.
In the proof of (1)$\Rightarrow$(3), there is always a constant vector $g$ orthogonal to $\mathrm{Span}\{\xi_1,\xi_2,\eta_1,\eta_2\}$ in either of the three cases. Thus in the foliation described in (3) of Theorem \[thm-envelop\], each leave is now a geodesic in the corresponding space form. In other words, they are ruled submanifolds. This fact is already known in the study of austere submanifolds [@br],[@Dajczer3],[@Lu].
Two Möbius characterization results
===================================
In this section we will give two characterization theorems (in terms of Möbius invariants) related with the following minimal Wintgen ideal submanifolds in $\mathbb{S}^5$.
\[ex1\] Let $\gamma:N^2\to \mathbb{C}P^2$ be a holomorphic curve, and $\pi: \mathbb{S}^5 \rightarrow \mathbb{C}P^2$ be the Hopf fibration. Then the circle bundle $M^3 \subset \mathbb{S}^5$ over $N^2$ obtained by taking the Hopf fibers over $\gamma(N^2)$ is a three dimensional minimal Wintgen ideal submanifold in $\mathbb{S}^5$ as pointed out in [@Smet] (Example 6).
Observe that $S^1$ acts by isometry on $\mathbb{S}^5$ whose orbits give the Hopf fibration. Thus for $M^3$ as above it has an induced $S^1$ symmetry. Consider the second fundamental forms given in ; the invariant $\nu$ must be a constant along every orbit of this $S^1$ action, which is exactly an integral curve of $E_3$ (see [@Smet] for details where they use $\xi$ to denote this $E_3$). So we have $G=E_3(\nu)=0$ in this special case. Since they are minimal, by Theorem \[thm-minimal\] we have $d\o=0$. These conditions characterize this class of submanifolds as below.
\[thm-Hopf\] Let $x:M^3\rightarrow\mathbb{S}^5$ be a Wintgen ideal submanifold with non-integrable distribution $\mathbb{D}=\mathrm{Span}\{E_1,E_2\}$. If it satisfies $d\o=0, G=0$, then up to a Möbius transformation on $\mathbb{S}^5$, $x$ is the Hopf lift of a holomorphic curve given in Example \[ex1\].
When $G=0$, comparing the coefficients of $\o_1\wedge\o_3$ in and using yields $2\hat{F}=L^2$. From the proof to theorem \[thm-minimal\] we know that $x$ is Möbius equivalent to a minimal Wintgen ideal submanifold $$f=\frac{\hat{F}Y+\hat Y}{\sqrt{2\hat{F}}}: M^3\rightarrow \mathbb{S}^5 \subset \mathbb{R}^6,$$ where $\mathbb{R}^6=\mathrm{Span}_{\mathbb{R}}
\{f,\eta_3,\eta_1,\eta_2,\xi_1,-\xi_2\}$. Using $\sim$ and $2\hat{F}=L^2, d\hat{F}=-2\o\hat{F}$, with respect to this frame we can write out the structure equations of $f$: $$\label{Theta1}
d\begin{pmatrix}
f\\ \eta_3\\ \eta_1\\ \eta_2\\ \xi_1\\ -\xi_2\end{pmatrix}=\begin{pmatrix}
0& L\o_3& L\o_1& L\o_2& 0& 0\\
-L\o_3& 0& -L\o_2& L\o_1& 0& 0\\
-L\o_1& L\o_2& 0& \Omega_{12}& \mu\o_2& -\mu\o_1\\
-L\o_2& -L\o_1& -\Omega_{12}& 0& \mu\o_1& \mu\o_2\\
0& 0& -\mu\o_2& -\mu\o_1& 0& -\theta_{12}\\
0& 0& \mu\o_1& -\mu\o_2& \theta_{12}& 0
\end{pmatrix}\begin{pmatrix}
f\\ \eta_3\\ \eta_1\\ \eta_2\\ \xi_1\\ -\xi_2\end{pmatrix}.$$ Denote the frame as a matrix $T:M^3\to \mathrm{SO}(6)$ with respect to a fixed basis $\{{\bf e}_k\}_{k=1}^6$ of $\mathbb{R}^6$, we can rewrite as $$\label{Theta2}
dT=\Theta T.$$ The algebraic form of $\Theta$ motivates us to introduce a complex structure ${\bf J}$ on $\mathbb{R}^6= \mathrm{Span}_\mathbb{R}\{f,\eta_3,\eta_1,\eta_2,\xi_1,\xi_2\}$ as below: $${\bf J}\begin{pmatrix}f\\
\eta_3\\ \eta_1\\ \eta_2\\ \xi_1\\ -\xi_2\end{pmatrix}
=\begin{pmatrix}
\begin{pmatrix}0& -1\\1 & 0\end{pmatrix} & & \\
& \begin{pmatrix}0& -1\\1 & 0\end{pmatrix} & \\
& &\begin{pmatrix}0& -1\\1 & 0\end{pmatrix}
\end{pmatrix}\begin{pmatrix}
f\\ \eta_3 \\ \eta_1 \\ \eta_2 \\ \xi_1 \\ -\xi_2
\end{pmatrix}.$$ Denote the diagonal matrix at the right hand side as $J_0$. Then the matrix representation of operator ${\bf J}$ under $\{{\bf e}_k\}_{k=1}^6$ is: $$J=T^{-1}J_0T.$$ Using $dT=\Theta T$ and the fact that $J_0$ commutes with $\Theta$, it is easy to verify $$dJ=-T^{-1}dT T^{-1}J_0T+T^{-1}J_0 dT
=-T^{-1}\Theta J_0T+T^{-1}J_0 \Theta T=0.$$ So ${\bf J}$ is a well-defined complex structure on this $\mathbb{R}^6$.
Another way to look at the structure equations is to consider the complex version: $$\begin{aligned}
d(f+i\eta_3)&=-iL\o_3(f+i\eta_3)+L(\o_1-i\o_2)(\eta_1+i\eta_2), \label{5.1}\\
d(\eta_1+i\eta_2)&=-L(\o_1+i\o_2)(f+i\eta_3)-i\Omega_{12}(\eta_1+i\eta_2)
+i\mu(\o_1-i\o_2)(\xi_1-i\xi_2),
\notag\\
d(\xi_1-i\xi_2)&=i\mu(\o_1+i\o_2)(\eta_1+i\eta_2)
+i\theta_{12}(\xi_1-i\xi_2).\label{5.3}\end{aligned}$$ Geometrically, this implies that $$\mathbb{C}^3=\mathrm{Span}_{\mathbb{C}}\{f+i\eta_3,
\eta_1+i\eta_2,\xi_1-i\xi_2\},$$ is a fixed three dimensional complex vector space endowed with the complex structure $i$, which is identified with $(\mathbb{R}^6,{\bf J})$ via the following isomorphism between complex linear spaces: $$v\in \mathbb{C}^3~~\mapsto~~\mathrm{Re}(v)\in \mathbb{R}^6.$$ For example, $f+i\eta_3\mapsto f,if-\eta_3\mapsto -\eta_3$ and so on.
The second geometrical conclusion is an interpretation of that $[f+i\eta_3]$ defines a holomorphic mapping from the quotient surface $\overline{M}=M^3/\Gamma$ to the projective plane $\mathbb{C}P^2$ (like the conclusion (1) in Theorem \[thm-envelop\]). Moreover, the unit circle in $$\mathrm{Span}_{\mathbb{R}}\{f,\eta_3\}=\mathrm{Span}_{\mathbb{R}}\{f,{\bf J}f\}=\mathrm{Span}_{\mathbb{C}}\{f+i\eta_3\}$$ is a fiber of the Hopf fibration of $\mathbb{S}^5\subset (\mathbb{R}^6,{\bf J})$. It corresponds to the subspace $\mathrm{Span}_{\mathbb{R}}\{Y,\hat{Y},\eta_3\}$, which is geometrically a leave of the foliation $(M^3,\Gamma)$ as described by conclusion (3) in Theorem \[thm-envelop\]. Thus the whole $M^3$ is the Hopf lift of $\overline{M}\to\mathbb{C}P^2$. In other words we have the following commutative diagram $$\begin{xy}
(30,30)*+{M^3}="v1", (60,30)*+{\mathbb{S}^5}="v2", (90,30)*+{\mathbb{C}^3}="v3";%
(30,0)*+{\overline{M}}="v4", (60,0)*+{\mathbb{C}P^2}="v5".%
{\ar@{->}^{f} "v1"; "v2"}%
{\ar@{->}^{\subset} "v2"; "v3"}%
{\ar@{->}_{M^3/\Gamma} "v1"; "v4"}%
{\ar@{->}_{[f+i\eta_3]} "v1"; "v5"}%
{\ar@{->}^{} "v4"; "v5"}%
{\ar@{->}^{\pi} "v2"; "v5"}%
{\ar@{->}_{\pi} "v3"; "v5"}%
\end{xy}$$ This finishes the proof.
Among examples given above, there is a special one coming from the lift of the famous Veronese embedding $\gamma:\mathbb{C}P^1\to \mathbb{C}P^2$ which is homogeneous. Thus the lift $M^3$ is itself a homogeneous minimal Wintgen ideal submanifold in $\mathbb{S}^5$. This special example can also be described as below.
\[ex2\] The orthogonal group $\mathrm{SO}(3)$ embedded in $\mathbb{S}^5$ homogeneously: $$\label{SO3}
x: \quad \mathrm{SO}(3)~\rightarrow~ \mathbb{S}^5, \qquad
(u, v, u\times v) \mapsto \frac{1}{\sqrt{2}}(u, v).$$ The orthonormal frames of the tangent and normal bundles can be chosen as $
e_1=(u\times v, 0), e_2=(0, u\times v), e_3=\frac{1}{\sqrt{2}}(-v, u);~~
n_1=\frac{-1}{\sqrt{2}}(v, u), n_2=\frac{-1}{\sqrt{2}}(u, -v).
$ Direct computation verifies that it is a minimal Wintgen ideal submanifold.
Consider the canonical lift $Y=\sqrt{6}(1,x): \mathrm{SO}(3)\rightarrow \mathbb{R}^7_1.$ The Möbius metric is given by $\mathrm{g}=6dx\cdot dx.$ It follows from that the Möbius form vanishes, i.e., $C^{r}_{j}=0$. Next, $\{E_j=e_j/\sqrt{6}\}$ form an orthonormal frame for $(M^3, \mathrm{g})$, with the dual 1-form $\{\o_j\}$. So the frame used in Section 3 is given by $$\eta_j=Y_j=(0,e_j), ~~\hat{Y}=N=\frac{1}{2\sqrt{6}}(-1,x), ~~\xi_r=(0,n_r).$$ The structure equations are $$d\begin{pmatrix}
Y\\ \hat{Y} \\ \eta_1 \\ \eta_2 \\ \eta_3 \\ \xi_1 \\ \xi_2
\end{pmatrix}=\begin{pmatrix}
0&0&\o_1&\o_2&\o_3&0&0\\
0&0&\frac{\o_1}{12}&\frac{\o_2}{12}&\frac{\o_3}{12}&0&0\\
\frac{-\o_1}{12}&-\o_1&0&0&\frac{\o_2}{\sqrt{6}}&
\frac{\o_2}{\sqrt{6}}&\frac{\o_1}{\sqrt{6}}\\
\frac{-\o_2}{12}&-\o_2&0&0&\frac{-\o_1}{\sqrt{6}}&
\frac{\o_1}{\sqrt{6}}&\frac{-\o_2}{\sqrt{6}}\\
\frac{-\o_3}{12}&-\o_3&\frac{-\o_2}{\sqrt{6}}&
\frac{\o_1}{\sqrt{6}}&0&0&0\\
0&0&\frac{-\o_2}{\sqrt{6}}&\frac{-\o_1}{\sqrt{6}}&0&0&
\frac{\o_3}{\sqrt{6}}\\
0&0&\frac{-\o_1}{\sqrt{6}}&\frac{\o_2}{\sqrt{6}}&0&
\frac{-\o_3}{\sqrt{6}}&0
\end{pmatrix}\begin{pmatrix}
Y\\ \hat{Y} \\ \eta_1 \\ \eta_2 \\ \eta_3 \\ \xi_1 \\ \xi_2
\end{pmatrix}.\label{4.15}$$
In [@Smet] they gave a characterization of this example as the unique Wintgen ideal submanifold $M^m\to\mathbb{Q}^{m+2}(c)$ with constant non-zero normal curvature. Here we provide another characterization of it in Möbius geometry.
In the statement below, a connected submanifold $M$ in $\mathbb{S}^5$ is said to be *locally Möbius homogenous* if for any two points $p,q\in M$, there are two neighborhoods $U_p,U_q\subset M$ of them respectively and a Möbius transformation $T$ such that $T(p)=q, T(U_p)=U_q$. An essential property of a locally (Möbius) homogenous submanifold is that any well-defined (Möbius) invariant function on it must be a constant.
\[thm-homog\] Let $x: M^3\rightarrow \mathbb{S}^5$ be a Wintgen ideal submanifold of dimension 3. If it is locally Möbius homogenous and the distribution $\mathbb{D}=\mathrm{Span}\{E_1, E_2\}$ is not integrable, then up to a Möbius transformation this is part of $x: \mathrm{SO}(3) \rightarrow \mathbb{S}^5$ given in Example \[ex2\].
According to Remark \[rem-transform\] and Remark \[rem-invariant\], the coefficients $\{\hat{F}, G, L\}$ appearing in the structure equations are geometric invariants. Thus under our assumption these functions must be constants; in particular $L$ is a non-zero constant.
By we know $G=E_3(L)=0$. From $\sim$, we have $$\qquad 2\hat{F}=L^2,~~~
\hat\o_3=LU\o_1+LV\o_2+\hat{F}\o_3.\label{5.7}$$ By one can write out explicitly that $$\begin{aligned}
\Omega_{12}&=\alpha\o_1+\beta\o_2+\gamma\o_3.\label{5.8}\\
\theta_{12}&=(U-2\alpha)\o_1+(V-2\beta)\o_2+(L-2\gamma)\o_3.
\label{5.9}\end{aligned}$$ Note that in general $\alpha, \beta, \gamma$ are not geometric invariants, because they are components of the connection 1-form $\Omega_{12}$, and when the frame $\{E_1,E_2\}$ rotate by angle $t$ in , $\Omega_{12}$ will differ by a closed 1-form $dt$.
On the other hand, by , $U^2+V^2$ is the square of the norm of the Möbius form $\Phi$ (up to a non-zero constant), hence a geometric invariant. It must also be a constant on $M^3$.\
**Claim**: The 1-form $\o=0$ identically; i.e., $U=V=0$ on $M^3$ everywhere and under any frame $\{E_1,E_2\}$. (As a consequence of this fact and the conclusion of Theorem \[thm-Hopf\], any of such examples is the Hopf lift of a complex curve in $\mathbb{C}P^2$.)\
We prove this claim by contradiction. Suppose $U^2+V^2$ is a non-zero constant. We can choose a canonical frame according to Remark \[rem-invariant\]. With respect to such a canonical frame, all coefficients $\alpha,\beta,\gamma$ in are now well-defined functions, hence be constants. From $\sim$ we have $$\begin{aligned}
d\Omega_{12}=&(\alpha^2-\alpha U+\beta^2-\beta V+2L\gamma)\o_1\wedge\o_2 \notag \\
&+[\alpha(L-\gamma)+\gamma U]\o_2\wedge\o_3
-[\beta(L-\gamma)+\gamma V]\o_1\wedge\o_3,\label{dOmega12}\\
d\theta_{12}=&-[(U-2\alpha)(U-\alpha)
+(V-2\beta)(V-\beta)-2(L-2\gamma)L]\o_1\wedge\o_2 \notag\\
&+[(U-2\alpha)(L-\gamma)+U(L-2\gamma)]\o_2\wedge\o_3 \notag\\
&-[(V-2\beta)(L-\gamma)+V(L-2\gamma)]\o_2\wedge\o_3. \label{dtheta12}\end{aligned}$$ Comparing the coefficients with and separately, we obtain $$\begin{gathered}
\alpha^2-\alpha U+\beta^2-\beta V+2L\gamma=2\mu^2-2L^2, \label{5.10}\\
\alpha(L-\gamma)+\gamma U=0,\label{5.11}\\
\beta(L-\gamma)+\gamma V=0, \label{5.12}\\
2\mu^2=-(U-2\alpha)(U-\alpha)-(V-2\beta)(V-\beta)
+2(L-2\gamma)L,\label{5.13}\\
(U-2\alpha)(L-\gamma)+U(L-2\gamma)=0,\label{5.14}\\
(V-2\beta)(L-\gamma)+V(L-2\gamma)=0. \label{5.15}\end{gathered}$$ If $\gamma=L\ne 0$, then from and we have $U=V=0$. This does not only contradict with the assumption $U^2+V^2\ne 0$, but also implies from that $\mu^2=-\alpha^2-\beta^2-L^2$, a contradiction with $L\ne 0,\mu=1/\sqrt{6}\ne 0$.
If $\gamma\ne L$, then and tell us $\alpha=-\frac{\gamma }{L-\gamma}U,\beta=-\frac{\gamma }{L-\gamma}V.$ Combined with and , we get $\gamma=2L, \alpha=2U, \beta=2V.$ Insert them into , we have $2\mu^2=-3U^2-3V^2-6L^2.$ So there must be $U=V=L=0$, which also contradicts to our assumption. Thus the claim is proved.\
Now that $U=V=\frac{G}{L}=0$, we have $\o=0$ and $2\Omega_{12}+\theta_{12}=L\o_3$. Differentiate the last equation. We get $$d(2\Omega_{12}+\theta_{12})=Ld\o_3=2L^2\o_1\wedge\o_2.$$ On the other hand, still by there is $$d(2\Omega_{12}+\theta_{12})=(6\mu^2-4L^2)\o_1\wedge\o_2.$$ Comparison shows $$L=\mu=1/\sqrt{6}$$ (assume $L>0$ without loss of generality). Then by and $2\hat{F}=L^2$ in , we obtain $$d\Omega_{12}=d\o_{12}=2(\mu^2-L^2)\o_1\wedge\o_2=0.$$ Thus $\Omega_{12}$ is a closed 1-form, which is locally an exact 1-form. Then we can use to find another frame such that $\Omega_{12}=\o_{12}=0$ and such frame is canonically chosen once it is fixed at an arbitrary point. With respect to this frame on a simply connected domain of $M^3$ we know $\alpha=\beta=\gamma=0$ in , and $\theta_{12}=\frac{1}{\sqrt{6}}\o_3$ in . The proof is finished by checking that the structure equations are the same as for $x(\mathrm{SO}(3))$.
For Möbius homogeneous Wintgen ideal submanifolds $M^3\to \mathbb{S}^5$ with $L=0$ (integrable $\mathbb{D}$), the classification will not be difficult. By the conclusion of Theorem A in the introduction, such an example comes from super-minimal surface $\overline{M}$ in four dimensional space forms. This super-conformal $\overline{M}$ must also be homogeneous by itself. According to our classification of Willmore surfaces with constant Möbius curvature [@MaWang], this $\overline{M}$ should be the Veronese surface $\mathbb{R}P^2\to \mathbb{S}^4$, and the original $M^3$ is a cone in $\mathbb{R}^5$ over this surface.
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---
abstract: 'In this paper, we analyze the quality of a large class of simple dynamic resource allocation (DRA) strategies which we name *priority planning*. Their aim is to control an undesired diffusion process by distributing resources to the contagious nodes of the network according to a predefined *priority-order*. In our analysis, we reduce the DRA problem to the *linear arrangement* of the nodes of the network. Under this perspective, we shed light on the role of a fundamental characteristic of this arrangement, the *maximum cutwidth*, for assessing the quality of any priority planning strategy. Our theoretical analysis validates the role of the maximum cutwidth by deriving bounds for the extinction time of the diffusion process. Finally, using the results of our analysis, we propose a novel and efficient DRA strategy, called *Maximum Cutwidth Minimization*, that outperforms other competing strategies in our simulations.'
author:
- |
Kevin Scaman Argyris Kalogeratos Nicolas Vayatis\
[CMLA [–]{} ENS Cachan, CNRS, France]{}\
`{scaman, kalogeratos, vayatis}@cmla.ens-cachan.fr`
bibliography:
- 'MCM2014.bib'
title: |
What Makes a Good Plan?\
An Efficient Planning Approach to Control Diffusion Processes in Networks
---
Introduction {#sec:intro}
============
Diffusion processes usually arise in systems involving agents whose behaviors depend on their close environments. Diseases, computer viruses, ideas and interests are spread through a social network by means of interactions among its users. In all these phenomena, a change in one agent may affect the actions of other neighboring agents, resulting, under certain conditions, in a massive change of behavior at the network scale.
A fair body of articles consider the problem of influence maximization, which attempts to maximize the spread of a diffusion process. On the other hand, being able to suppress or remove an undesired information or social diffusion process has received less attention, even though critical in many real-life situations. In public health, scenarios in which the spread of a virus needs to be controlled have been extensively studied by epidemiologists. Various analogues emerged in modern information networks in which a diffusion can be engineered to be viral and may cause huge negative social and economic effects. For instance, during the London riots during the summer of 2011, public opinion was obfuscated and baffled by many false rumors [@UKRIOTS2013]. During large scale natural disasters, there is also a risk to follow misinformation diffused through social media by various individuals, while trying to coordinate some rescue and volunteer teams [@MDTNHE2010; @OPCPFSDHS2014]. Another recent example arose in March 2013, when a false tweet from the account of Associated Press was headlined with “*Breaking: Two explosions in the White House and Barack Obama is injured*". The false rumor was immediately retweeted by about 2 million users causing panic and speculation for a few minutes in the US stock market. As a matter of fact, negative publicity is highly damaging for organizations and brands: even though ill-founded, a slander can dramatically affect the interests of a company due to the massive scale of the buzz-like effect.
The control of diffusion processes has been studied in various fields, including epidemiology and computer networks resilience; the respective literature can generally be divided in three complementary lines of research:
- *Static vaccination strategies*. Most of the epidemic literature focuses on static control actions such as permanently removing a set of edges or nodes of the network [@cohen2003efficient; @tong2012gelling; @wang2003epidemic; @schneider2011suppressing; @preciado2013optimal]. In this case, the available budget is considered fixed, and the effect of a control action *permanent*. Examples of static resource allocation strategies can be found in [@preciadoZEJP13; @pageRankAntidotes2009].
- *Budget optimization*. Complementary to resource allocation, the determination of the right budget of resources to be spent each time step, in order to fulfill cost and efficiency constraints, has critical impact on the resulting strategy. Several studies lay on the line of optimal budget estimation, assuming that the network administrator is capable of storing resources for later use [@klepac2012; @forster2007optimizing; @khouzani2011optimal]. These approaches usually make the simplifying assumption of *uniform mixing*, that the contagious nodes are uniformly scattered in the network. Therefore, they do not address the problem of how exactly to allocate the resources on the nodes of the network, but rather how many resources should be provided at each time to cause a desired macroscopic result.
- *Dynamic resource allocation*. A few studies consider dynamic strategies for allocating resources for dealing with epidemics. One of the most well-known such strategy is *contact-tracing* [@RSA:RSA20315] that consists in healing the neighbors of contagious nodes. In practice, this approach has been shown inefficient to contain epidemics [@RSA:RSA20315], especially when they are beyond a very initial state.
Our major contribution in this article is to introduce a particular class of strategies for suppressing an undesired diffusion. Instead of choosing a set of nodes whose behavior will be permanently modified, we allow the network administrator to change the distribution of the *resources* during the diffusion. In other words, we consider *targeted* and *temporal* action on individual nodes of the network, that can affect their behavior. Since reacting to fast spreading phenomena is difficult to achieve, we consider a simple set of dynamic strategies relying on a predefined *priority-order*. The strategy will gradually suppress the diffusion and finally remove the undesired contagion by focusing on the first contagious users according to the priority order.
In what follows, describes the model used for the diffusion process as well as the control actions available to the network administrator. presents the idea of *priority planning* as well as a natural representation of the problem connecting our analysis to *linear arrangement* problems. This anaylysis sheds light on the role of the network’s *maximum cutwidth* for the efficiency of a given strategy. By minimizing this value, we develop an efficient strategy in , and validate in that our intuition is valid by deriving theoretical results on the extinction time of the diffusion process. Finally, we present experimental results in and show that: i) The derived bounds are very close to the epidemic threshold, thus validating the fundamental role of the *maximum cutwidth* in the evaluation of such strategies. ii) The proposed strategy outperforms its competitors in a wide range of scenarios.
Diffusion and control model {#sec:model}
===========================
The *Susceptible-Infected-Susceptible* epidemic model
-----------------------------------------------------
We will consider a simple diffusion process known in the epidemiology literature as *N-Intertwined Susceptible-Infected-Susceptible* (SIS) model [@van2009virus]. According to this model, a diffusion can be spread through the edges of the network and turn the state of nodes from *susceptible* (or *healthy*) to *infected*. When a node becomes infected, it can in turn spread the contagion to its direct neighbors and, after some amount of time, return to the susceptible state without bearing any immunity. At each time, the network administrator can take control actions in order to reduce the epidemic. These actions are represented as a *budget* of *resources* (or treatments in the epidemic analog) to distribute in the network. Each resource increases the recovery rate of the receiver node. In a continuous-time framework, the distribution of control resources can be revised anytime. However, in practice the situations in which this is needed is only when there is a change in the state of the network, a new node infection or recovery, or a modification in the available resource budget.
This model is well suited to situations in which an undesired contagion affects a network, and the control action is *local* and *expensive*. Among other application examples, controlling epidemics using antidotes, limiting rumors via targeted action or allocating resources geographically to fight against a societal problem, seem valid scenarios for such a diffusion and control model.
Let $\mathcal{G} \op{=} (\mathcal{V}, \mathcal{E})$ be a network of $N$ nodes with adjacency matrix $A$, where $A_{ij} \op{=} 1$ if $i \op{\neq} j$ and edge $(i,j) \op{\in} \mathcal{E}$, else $A_{ij} \op{=} 0$. We describe the state of the diffusion process with a *state vector* $X(t) \op{\in} \real^N$ that keeps the state of each node of the network: $X_i(t) \op{=} 1$ if node $i$ is contagious at time $t$, else $X_i(t) \op{=} 0$. We also describe the control action on the network with a *resource vector* $R(t)$, where $R_i(t)$ is $1$ iff node $i$ is given a resource at time $t$. In such case, and following the epidemic analogy, we say that node $i$ is being *healed* by the resource. Using also the formalism of [@ganesh2005effect], we model the diffusion with a *continuous-time Markov process* which has the transition rates: $$\label{eq:Xt} \formulastyle
\begin{array}{l}
X_i(t) : 0\rightarrow1 \mbox{ at rate } \beta \sum_j {A_{ji} X_j(t)};\\
X_i(t) : 1\rightarrow0 \mbox{ at rate } \delta + \rho R_i(t),\\
\end{array}$$ where $\beta$, $\delta$, $\rho$ are, respectively, the transmission rate over one network edge, the recovery rate without receiving a resource, and the increase in recovery rate that a resource induces. Note that the diffusion process is continuous in time and thus $X(t)$ is a stochastic process. The action of the resource on a node increases its chances to return to the *healthy* state. Finally, we define two dimensionless parameters: $r \op{=} \frac{\beta}{\delta}$ the *effective spreading rate* of the diffusion, and $e \op{=} \frac{\rho}{\delta}$ the *resource efficiency*.
A *control strategy* takes as input the network $\mathcal{G}$, the characteristics of the diffusion ($\beta$, $\delta$, and $\rho$) and the network state $X(t)$, and returns the distribution of the resources $R(t)$ (). In the following, we refer to the problem of finding an optimal control strategy with respect to the minimization of the spread of the diffusion process as the *dynamic resource allocation* (DRA) problem. We consider as *budget* $b(t)$ the maximum number of resources that can be distributed in the network at time $t$, and that the available budget at time $t$ cannot be stored for later use. Following the epidemiology literature, we will denote as *epidemic threshold under a given strategy* the resource efficiency $e$ above which the control actions removes the contagion in *reasonable time*, that is less than exponential in the number of nodes of the network.
Priority planning: a control plan to gradually remove a contagion {#sec:priorityplanning}
-----------------------------------------------------------------
### Definition
A *priority planning* is a DRA strategy that distributes resources to the top-$b(t)$ infected nodes according to a fixed *priority ordering* of the nodes, where $b(t)$ is the available budget of resources at each time. In mathematical terms, an *$\lorder$-priority planning* is defined by a bijective mapping $\lorder\!: \mathcal{V} \op{\rightarrow} \{1, ..., N\}$ of the $N$ nodes of the network $\lorder(v)$ is the position of node $v$ in the priority order. The strategy then selects the first $b(t)$ infected nodes according to $\lorder$: $$R_i(t) = \left\{
\begin{array}{ll}
1 &\mbox{if } X_i(t) = 1 \mbox{ and } \lorder(v_i) \leq \theta(t);\\
0 &\mbox{otherwise},\\
\end{array}\right.$$ where $\theta(t)$ is an adjusted threshold set so that $\sum_i R_i(t) \op{=} b(t)$.
These strategies may be regarded as simple planning strategies for the removal of an undesired contagion: the order in which the nodes will be healed is determined prior to the beginning of the diffusion. Then, this order is respected no matter how the diffusion process evolves, and the strategy removes the contagious nodes gradually starting from the first in the priority-order to the last. Although our strategy () can handle a generic time-dependent budget, we only consider a constant budget rate $b(t) \op{=} b_{tot}$ for theoretical developments as well as focused experimental evaluation.
### Interpretation of priority planning {#sec:interpretation}
In this section, we present a novel perspective on analyzing the DRA problem which leads to efficient priority planning strategies. More specifically, we reduce the problem of determining a good priority-order to a *linear arrangement* (LA) problem.
Formally, a linear arrangement $\lorder\!: \mathcal{V} \op{\rightarrow} \{1, ..., N\}$ maps the nodes of $\mathcal{G}$ on $N$ discrete positions located on a line by assigning one position-label to each node (). LA is a class of combinatorial optimization problems whose purpose is to minimize some functional $\objf$ over the space $\mathcal{L}$ of all possible node permutations: $\lorder^* \op{=} \argmax_{\lorder \in \mathcal{L}} \objf(\mathcal{G},\lorder)$. The problems in the LA class are also referred to as *graph layout* problems (for a survey see [@SGLP2002]), and indicative applications are the graph drawing, VLSI design, and network scheduling.
Probably the most popular LA instance is the *minimum $p$-sum linear arrangement* (M$p$LA, or MLA when $p \op{=} 1$) [@OANV1964] that minimizes the following functional: $$\label{eq:p-sum}
\begin{array}{l}
\mbox{M$p$LA}\!: \ \ \objf_1(\mathcal{G}, \lorder) = \Big(\sum_{i,j} A_{ij}\left|\lorder(v_i) - \lorder(v_j)\right|^p\Big)^{1/p}.
\end{array}$$ In other words, for $p \op{=} 1$, this minimizes the weighted sum over all distances between node assignements $\ell(u)$ to the $N$ discrete equally spaced positions. Another category of LA problems is related to the minimization of some maximum value of the LA, for example the *bandwidth* (the maximum edge length), or the *workbound* (the sum of maximum edge length of each node). There is also the *minimum cutwidth linear arrangement* (MCLA) problem, where the *cutwidth* at a location $c$ lays among the node positions $c$ and $c \op{+} 1$ of the LA and is the weighted edge-cut at that position. Then, the minimized functional is the maximum of the $N \op{-} 1$ local cuts, and more precisely: $$\label{eq:cw}
\begin{array}{l}
\mbox{MCLA}\!: \ \ \objf_2(\mathcal{G}, \lorder) = \max_{c={1,...,N-1}} \sum_{i,j} A_{ij} \one_{\{\lorder(v_i) \leq c < \lorder(v_j)\}},
\end{array}$$ where $\one_{\{\cdot\}}$ is the indicator function that returns one if the input condition is true, else zero. For the simplicity of notations we also denote the value of the above functional as $\MaxCut \op{=} \objf_2(\mathcal{G}, \lorder)$. Note that symmetric LAs are evaluated as of equal quality by $\objf_1$, $\objf_2$. illustrates two LAs of a small network and an example of the cost values derived in each case by and . Among them, the arrangement of is better, indeed optimal, in terms of both objective functions $\objf_1$, $\objf_2$.
[0.5]{} ![The priority orders of two static strategies are visualized as the linear arrangements of the nodes based on two mappings $\lorder$ and $\lorder'$. Healthy nodes (white) and contagious nodes (red) form two distinct subgraphs that are connected by the *contagious edges* (dashed lines) of the *front* illustrated as a red vertical line. (a) M$p$LA cost, for $p$ 1, $\objf_1(\mathcal{G}, \lorder) \op{=} 8$ and MCLA cost $\objf_2(\mathcal{G}, \lorder) \op{=} 3$; (b) $\objf_1(\mathcal{G}, \lorder') \op{=} 4$ and $\objf_2(\mathcal{G}, \lorder') \op{=} 1$, respectively.[]{data-label="fig:static_strategy"}](network_example2.pdf "fig:"){width="0.92\linewidth"}
[0.5]{} ![The priority orders of two static strategies are visualized as the linear arrangements of the nodes based on two mappings $\lorder$ and $\lorder'$. Healthy nodes (white) and contagious nodes (red) form two distinct subgraphs that are connected by the *contagious edges* (dashed lines) of the *front* illustrated as a red vertical line. (a) M$p$LA cost, for $p$ 1, $\objf_1(\mathcal{G}, \lorder) \op{=} 8$ and MCLA cost $\objf_2(\mathcal{G}, \lorder) \op{=} 3$; (b) $\objf_1(\mathcal{G}, \lorder') \op{=} 4$ and $\objf_2(\mathcal{G}, \lorder') \op{=} 1$, respectively.[]{data-label="fig:static_strategy"}](network_example2.pdf "fig:"){width="0.92\linewidth"}
The analysis of the diffusion process under the perspective of a linear arrangement, sheds light on how to design a DRA control plan to suppress a diffusion efficiently. More specifically, knowing that the contagion can only be spread through the *contagious edges* which connect infected with healthy nodes, and that any DRA plan is based on a static priority-order for healing the nodes, we realize that it is highly critical to have as less contagious edges as possible during the whole diffusion process.
represents two diffusion instances that are being suppressed according to different plans; we consider that each priority-order is deployed from the left to the right. The current state of the plan is denoted with a red vertical line, the *front*, that roughly separates the healthy from the contagious nodes and indicates where the strategy would allocate resources. In practice, healthy nodes can also appear on the right side of the front due to self-recovery. The number of edges crossing through the front, the cutwidth at that position, indicates how vulnerable are actually the healthy nodes of the cleared part of the network laying left to the front. Therefore, we can argue that a good solution to the M$p$LA minimization would provide a smoother plan that is generally *easier to proceed from state-to-state*, while the MCLA would minimize the $\MaxCut$ which is *the most difficult state of the plan* and, thus, a determinant for its eventual success in completely removing the contagion. This implies that the minimization of the $\MaxCut$ can be a proxy for minimizing the epidemic threshold for any arbitrary network. Next, we theoretically show that these intuitive remarks are valid.
The *Maximum Cutwidth Minimization* control strategy {#sec:algorithm}
====================================================
Based on the analysis of the previous section, that uncovered a strong dependency between the epidemic threshold of the diffusion process and the *maximum cutwidth*, we propose a new DRA algorithm for arbitrary networks where the main idea is to distribute resources to contagious nodes in the order that minimizes $\MaxCut$. Specifically, given a network $\mathcal{G}$, we compute, prior to the diffusion process, the linear arrangement $\lorder_{MC}(\mathcal{G})$ of its nodes with minimum $\MaxCut$ value using any available optimization algorithm for this problem. Then, during the diffusion, the strategy distributes the budget of resources to the contagious nodes according to the order of $\lorder_{MC}(\mathcal{G})$. The pseudocode of the strategy is summarized in , while theoretical justification and results are provided in .
**Input :**network $\mathcal{G}$, infection state vector $X(t)$, budget size $b(t)$ **Output:**the resource allocation vector $R(t)$ **Prior to the diffusion:** Compute the priority-order $\lorder \op{=} \lorder_{MC}(\mathcal{G})$ by minimizing the maximum cutwidth $\MaxCut$ Order the nodes of $\mathcal{G}$ according to $\lorder$, compute the node list $(v_1,...,v_N)$ $\forall i \op{\in} \{1,..., N\}$, $\lorder(v_i) \op{=} i$\
**During the diffusion:** $X(t)$
Let $R(t)\op{=}\zero$ a zero $N$-dimensional vector, also $i \op{=} 1$ and $budget \op{=} b(t)$ $R_{v_i}(t) \leftarrow 1$ $budget \leftarrow budget - 1$ $i \leftarrow i + 1$
$R(t)$
MCLA is known to be an NP-hard problem. However, approximation heuristics do exist in the related literature [@SSCMP2012; @VFSCMP2013]. One of the difficulties of this problem is that the cost function to optimize () is extremely flat in the search space, due to the fact that slight changes in the arrangement will most probably not change $\MaxCut$. For this reason, we chose to relax the MCLA problem by optimizing the *p-sum linear arrangement* problem with $p \op{=} 1$ (MLA, see ), which is easier than MCLA and more suited to gradient descent or simulated annealing methods. Furthermore, as discussed in , MLA may produce a smooth priority-order that exhibits some desirable properties.
The scalability of the strategy is highly dependent on the chosen algorithm for finding the optimal order. For our experimental results, we applied a hierarchical approach to take advantage of the group-structure of the social network: i) first, we identified dense clusters by applying *spectral clustering* and we ordered the clusters (considered as high-level nodes) using *spectral sequencing* [@OLLEG1992], ii) then, we computed a good ordering of the nodes in each of the clusters independently using spectral sequencing followed by an iterative approach based on random node swaps, inspired by [@ETSSAAMLA2008], iii) finally, the swap-based approach was reapplied to optimize the ordering all together. The whole process achieved fairly good results (see ) in reasonable time. Since clustering and spectral sequencing depend on the computation of eigenvectors for the highest eigenvalues of an $N \op{\times} N$ sparse matrix with $|\mathcal{E}|$ non-zero entries, the overall algorithm has a complexity $O(|\mathcal{V}| \op{+} |\mathcal{E}|)$ [@Arora05fastalgorithms]. Hence, MCM is generally scalable to the size of real social networks.
Theoretical bounds for the extinction time and the epidemic threshold {#sec:theory}
=====================================================================
We now provide a justification of the design of our MCM algorithm. The following theorem gives an *upper bound* for the expected extinction time under any priority planning in the simple case of $b(t) \op{=} 1$. The general case, while much more complex to analyze, is very similar in our experiments provided that the treatment efficiency $e$ is multiplied by the available budget $b(t)$ (see ). Above a threshold value that depends on the *maximum cutwidth* $\MaxCut$ of the network under a considered priority-order, proves that the diffusion process converges in reasonable time to its absorbent state.
\[th:boundExtinctionTime\] Let $\mathcal{G} \op{=} (\mathcal{V}, \mathcal{E})$ be a network of $N$ nodes and assume a fixed budget $b(t) \op{=} 1$. Let $\lorder\!: \mathcal{V} \op{\rightarrow} \{1,..., N\}$ be an ordering of the nodes of $\mathcal{G}$, and $\MaxCut$ be the maximum cutwidth of $\lorder$, or equivalently the highest number of contagious edges during the planned removal of the contagion. Then the following upper bound holds for the expected extinction time $\Exp{\tau}$ under the $\lorder$-priority plan and starting from a total infection:
If $\rho > \beta \MaxCut \left(1 + 2\sqrt{\epsilon} + \epsilon \right) - \delta$, then\
$$\Exp{\tau} \ \leq \ \frac{N}{\rho + \delta - \beta \MaxCut \left(1 + 2\sqrt{\epsilon} + \epsilon \right)}$$ where $\epsilon = \frac{d_{max} (1 + \ln{N})}{\MaxCut}$ and $d_{max}$ is the maximum node degree of the network.
This bound relates the extinction time to the number of contagious edges in the worst step of the strategy (its *maximum cutwidth*). When $\MaxCut$ is such that $d_{max} (1 \op{+} \ln{N}) \op{\ll} \MaxCut$, this formula bounds the epidemic threshold under a given priority planning by $r \MaxCut \op{-} 1$, where $r \op{=} \beta / \delta$ is the *effective spreading rate*, thus giving a special emphasis on the role of $\MaxCut$ for assessing the quality of a given strategy.
\[th:boundThreshold\] Under the same hypotheses as , the epidemic threshold $e^*$, the lowest resource efficiency $e \op{=} \frac{\rho}{\delta}$ the diffusion converges to zero in reasonable time, is upper bounded: $$e^* \ \leq \ r \MaxCut \left(1 + 2\sqrt{\epsilon} + \epsilon \right) - 1,$$ where $\epsilon \op{=} \frac{d_{max} (1 + \ln{N})}{\MaxCut}$ and $r \op{=} \frac{\beta}{\delta}$ is the effective spreading rate.
While $\epsilon$ is necessary for the theorem to hold, we rather consider it as an artifact of the proof and not a fundamental aspect of the result. As such, we expect $r \MaxCut \op{-} 1$ to be closer to the epidemic threshold (see specifically for an experimental validation of this intuition). This result verifies that, according to the intuition provided in , removing an undesired contagion requires the resource efficiency to be *as high as needed for the worst step* of the specified plan. Equivalent to the celebrated relationship between the epidemic threshold under no control strategy and the spectral radius of the adjacency matrix, this result is fundamental for understanding the behavior of the diffusion process and designing efficient DRA strategies. The simulations in , attest that minimizing this upper bound is very efficient for dynamically controlling a diffusion process.
Experimental results {#sec:exps}
====================
Quality of the theoretical bound {#sec:qualityExps}
--------------------------------
In , we show that the relationship between the *maximum cutwidth* $\MaxCut$ and the epidemic threshold under a specific priority plan is very stable, nearly linear, and hence $\frac{r\MaxCut}{b_{tot}}$ is in fact a very good approximation. Each plotted point is the epidemic threshold under a given strategy and for a given network instance. We sampled $100$ networks of $1000$ nodes from 5 random network generators (see [@Newman:2010:NI] for details on these network types): i) , ii) Preferential attachment, iii) Small-world, iv) Geometric random [@penrose2003random] and v) 2D regular grids. We also sampled DRA control strategies at random among the set of strategies described next in , in order to cover a wide range of different scenarios.
is a summary of these results. The epidemic threshold is always below $\frac{r \MaxCut}{b_{tot}}$, but it stays very close to this value and, more importantly, shows a close to deterministic behavior with respect to this value even in the case of low infectivity (small $r$ value) where intuitively the random self-recovery of nodes become more significant. This result justifies the minimization of $\MaxCut$ as a proxy for minimizing the epidemic threshold, and thus remove the contagion more efficiently.
[0.45]{} ![Epidemic threshold the maximum cutwidth $\MaxCut$ for various random network types of $N \op{=} 1000$ nodes. For the theoretical bound, $d_{max} \op{=} 100$ was used. The dashed line indicator has slope equal to the effective spreading rate $r$.[]{data-label="fig:boundQuality"}](epidemicThresholdDynamic_delta01-eps-converted-to.pdf "fig:"){width="1\linewidth"}
[0.45]{} ![Epidemic threshold the maximum cutwidth $\MaxCut$ for various random network types of $N \op{=} 1000$ nodes. For the theoretical bound, $d_{max} \op{=} 100$ was used. The dashed line indicator has slope equal to the effective spreading rate $r$.[]{data-label="fig:boundQuality"}](epidemicThresholdDynamic_delta10-eps-converted-to.pdf "fig:"){width="1\linewidth"}
Application on a real social network {#sec:twitterexps}
------------------------------------
-------------- -------------------- ------------------ --------------------------------------------
**Strategy** **Max cutwidth** **Max cutwidth** **Expected epidemic threshold**
**** **% MCM** **($r \op{=} 0.1$, $b_{tot} \op{=} 100$)**
RAND 670,000 $\pm$ 1000 931% 670
MN 628,571 874% 629
LN 628,571 874% 629
LRSR 349,440 486% 349
**MCM** **71,956** **100%** **72**
-------------- -------------------- ------------------ --------------------------------------------
: Maximum cutwidth values ($\MaxCut$) for different strategies in the TwitterNet used in our experiments. The expected epidemic threshold is $\frac{r\mathcal{C}_{max}}{b_{tot}}$ based on and the experiments of .[]{data-label="tab:twittermaxcuts"}
[0.45]{} ![ (-) Simulation of an undesired diffusion process in the TwitterNet subset of $81,306$ nodes [@conf/nips/McAuleyL12]. $\delta \op{=} 1$, $r \op{=} 0.1$ and $b_{tot} \op{=} 100$. MCM outperforms other heuristics. () Cutwidths for LRSR and MCM. (-) Visualization of the diffusion in () at the node level (contagious nodes in black). Nodes are ordered according to LRSR and MCM linear arrangements, respectively. A closer look of () is also provided in an inset figure.[]{data-label="fig:expresults"}](final_simulation11-eps-converted-to.pdf "fig:"){width="1\linewidth"}
[0.45]{} ![ (-) Simulation of an undesired diffusion process in the TwitterNet subset of $81,306$ nodes [@conf/nips/McAuleyL12]. $\delta \op{=} 1$, $r \op{=} 0.1$ and $b_{tot} \op{=} 100$. MCM outperforms other heuristics. () Cutwidths for LRSR and MCM. (-) Visualization of the diffusion in () at the node level (contagious nodes in black). Nodes are ordered according to LRSR and MCM linear arrangements, respectively. A closer look of () is also provided in an inset figure.[]{data-label="fig:expresults"}](final_simulation4-eps-converted-to.pdf "fig:"){width="1\linewidth"}
\
[0.335]{} ![ (-) Simulation of an undesired diffusion process in the TwitterNet subset of $81,306$ nodes [@conf/nips/McAuleyL12]. $\delta \op{=} 1$, $r \op{=} 0.1$ and $b_{tot} \op{=} 100$. MCM outperforms other heuristics. () Cutwidths for LRSR and MCM. (-) Visualization of the diffusion in () at the node level (contagious nodes in black). Nodes are ordered according to LRSR and MCM linear arrangements, respectively. A closer look of () is also provided in an inset figure.[]{data-label="fig:expresults"}](LA_evaluation_cutwidths-eps-converted-to.pdf "fig:"){width="1\linewidth"}
[0.31]{} ![ (-) Simulation of an undesired diffusion process in the TwitterNet subset of $81,306$ nodes [@conf/nips/McAuleyL12]. $\delta \op{=} 1$, $r \op{=} 0.1$ and $b_{tot} \op{=} 100$. MCM outperforms other heuristics. () Cutwidths for LRSR and MCM. (-) Visualization of the diffusion in () at the node level (contagious nodes in black). Nodes are ordered according to LRSR and MCM linear arrangements, respectively. A closer look of () is also provided in an inset figure.[]{data-label="fig:expresults"}](orderLRSR.pdf "fig:"){width="1\linewidth"}
[0.31]{}
(image) at (0,0) [![ (-) Simulation of an undesired diffusion process in the TwitterNet subset of $81,306$ nodes [@conf/nips/McAuleyL12]. $\delta \op{=} 1$, $r \op{=} 0.1$ and $b_{tot} \op{=} 100$. MCM outperforms other heuristics. () Cutwidths for LRSR and MCM. (-) Visualization of the diffusion in () at the node level (contagious nodes in black). Nodes are ordered according to LRSR and MCM linear arrangements, respectively. A closer look of () is also provided in an inset figure.[]{data-label="fig:expresults"}](orderMCM.pdf "fig:"){width="1\linewidth"} ]{};
(0.32,0.45) rectangle (0.519,0.605);
For evaluating our strategy, the *maximum cutwidth minimization* (MCM), we considered a realistic scenario where an SIS epidemic is being spread in a subset of the Twitter social network, denoted as TwitterNet [^1]. TwitterNet consists of $1,000$ ego-networks extracted from the social network [@conf/nips/McAuleyL12]. In order to perform experiments in the setting considered in this article, we symmetrized this network and obtained an undirected network of $81,306$ nodes and $1,342,303$ edges. The resulting network has a single connected component and contains a rich community structure.
The diffusion control literature is mainly focused on two types of strategies: static vaccination strategies and dynamic strategies based on a uniform mixing hypothesis. We compare MCM to the first type of strategies by considering state-of-the-art vaccination algorithms and use their output as a priority-order of the nodes, and to the second by considering random allocation of resources. As discussed in , the problem of finding the optimal budget through time is complementary to our approach. From this perspective, one of the primary question we address here is whether targeting specific nodes in the network can lead to substantial gains in the efficiency of the method, compared to the random distribution of the resources to the contagious nodes. and \[fig:twitter2\] compare the MCM strategy against the following four baseline approaches for creating a priority-order:
- *Random order* (RAND): the order is a random permutation of the nodes of the network.
- *Most neighbors* (MN): this strategy gives priority to high degree nodes. Intuitively, it begins by removing the contagion from the core of the network, and gradually reaches its periphery.
- *Least neighbors* (LN): this strategy gives priority to low degree nodes. Conversely to MN, LN begins with the periphery of the network and converges to its central part.
- *Least reduction in spectral radius* (LRSR): this strategy is based on the vaccination literature and more specifically the work in [@tong2012gelling]. This strategy gives priority to nodes whose removal will lead to the maximum decrease of the spectral radius of the adjacency matrix of the resulting network.
Concerning the applicability of , the maximum node degree in the network is $d_{max} \op{=} 3383$, which leads to a small $\epsilon$ value of $0.6$ even for the MCM strategy (and $\epsilon \op{=} 0.1$ for LRSR). We can thus be relatively confident in approximating the epidemic threshold under a given priority planning by $\frac{r\MaxCut}{b_{tot}}$. shows the cuts in the two best priority-orders, LRSR and MCM, while summarizes the $\MaxCut$ values of the priority-orders produced by the different compared strategies. Note that the $\MaxCut$ of the priority-order produced by MCM is *5 times smaller* than that of LRSR which is the best among its competitors. In essence, this implies that MCM would manage to suppress the diffusion process with resource efficiencies *5 times smaller* than LRSR.
The results in and \[fig:twitter2\] illustrate that the proposed MCM strategy is more efficient than its competitors in removing the contagion from the network. In and \[fig:MCMorder\], the evolution of the diffusion process is represented as follows: each line of the figure contains the state of one node of the network throughout the whole process (black for contagious and white for healthy). Nodes are sorted in y-axis according to the considered priority-order. We can observe that the cutwidth acts as a barrier for the LRSR: the large cutwidth values at the beginning of the LRSR order ($\frac{r\mathcal{C}_{max}}{b_{tot}} \op{\approx} e \op{=} 120$ for the $5000$-th node of the linear arrangement) prevents the strategy from consistently removing the contagion from more than the first $5000$ nodes of the plan. Note that healthy nodes also appear beyond the front, however this is not due to the control actions but rather due to self-recovery.
On the contrary, MCM gradually reduces the contagion and the advancement of the front is clearly visible. These results agree with our previous analysis, and show that: i) the uniform mixing hypothesis leads to a massive drop in efficiency, since MCM substantially outperforms the random allocation strategy, ii) while efficient in the static vaccination problem, centrality-based priority-orders are suboptimal for the DRA problem, iii) a good criterion for assessing the quality of a priority-order is actually in terms of its $\MaxCut$ value.
Conclusion {#sec:conclusion}
==========
In this paper, we presented a novel type of dynamic strategies for allocating resources in a network, called *priority planning*, that aims to suppress an undesired contagion. We reduced the planning problem to that of linear arrangement of the nodes, and, based on theoretical analysis on the quality of any priority-order, i) we demonstrated the key role of the *maximum cutwidth* for assessing if a strategy would be eventually successful in removing the contagion, and ii) we derived a strategy, called *maximum cutwidth minimization* (MCM), that distributes resources to nodes in a priority-order with minimum maximum cutwidth. Our experimental results verified that, for a wide range of network types, the maximum cutwidth is indeed a good approximation of the epidemic threshold under a given strategy, and that the MCM strategy outperforms other competing strategies in a real-world social network.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is part of the SODATECH project funded by the French Government within the program of “*Investments for the Future–Big Data*”.
APPENDIX {#appendix .unnumbered}
========
Mathematical arguments {#mathematical-arguments .unnumbered}
----------------------
In order to simplify the demonstrations, and without loss of generality, we will reorder the nodes of $\mathcal{G}$ according to the priority-order $\lorder$, and thus consider that $\lorder(v_i) = i$.
#### Notations.
First, let $X \in \{0,1\}^N$ be a state vector of size $N$(describing the state of the network during the diffusion process), $\zero$ and $\one$ vectors of size $N$ that are all-zeros and all-ones, respectively, and $\bar{X} \op{=} \one \op{-} X$ . For $s \op{\subset} \{1,..., N\}$ a subset of nodes, let also $\one_s \op{=} (\one_{\{i \in s\}})_i$ be the indicator vector with ones for nodes in the set $s$. Then, we define $\tau_X$ as the extinction time of the contagion starting from the state $X$, i.e. the time needed for the Markov process to reach its absorbent state $X(t{=}\tau_X) \op{=} \zero$ when $X(t{=}0) \op{=} X$. We also denote the number of contagious nodes in network state $X$ as $N_I(X) \op{=} \one^\top X$, while $E_{I-S}(X) \op{=} X^\top A \bar{X}$ as the number of edges from a contagious to a healthy node, which edges we also refer to as *contagious edges*.
The proof of relies on the following lemmas.
\[th:increasingTau\] Under a priority plan, $X \mapsto \Exp{\tau_X}$ is monotonically increasing with respect to the natural partial order on $\{0, 1\}^N$ ($X \leq Y$ if $X_i \leq Y_i$ for all $i$).
Let $X, Y \in \{0, 1\}^N$ be two initial states of the network such that $\forall i \in \{1,..., N\}, X_i \leq Y_i$. Let also $X(t)$ be the state vector of a contagion initially in state $X$, $X(0) = X$, and $Y(t)$ be the state vector of a contagion initially in state $Y$, $Y(0) = Y$. This lemma relies on the stochastic domination of $X(t)$ by $Y(t)$. This domination is due to the fact that, under the Markov Process defined by , and when the control strategy is a priority planning, infection rates are increasing according to the natural partial order on $\{0, 1\}^N$, while recovery rates are decreasing. Thus, the initial inequality $X \leq Y$ will, in probability, grow during the contagion process.
The correct proof of this intuition relies on the strong monotonicity of the Markov Process, which we now prove. For all $X \leq Y$, the infection rate of each healthy node at state $X$ is lower than its infection rate at state $Y$, since $\sum_i A_{ji} X_i \leq \sum_i A_{ji} Y_i$. Also, the recovery rate of each infected node at state $X$ is higher than its recovery rate at state $Y$: if node $i$ is both infected in state $X$ and $Y$, and is treated in state $Y$, then $i = \min\{j\in\{1,...,N\}\!: Y_j = 1\}$ is the first infected node, and since the set of infected nodes of state $X$ is included in the set of infected nodes at state $Y$, we also have $i = \min\{j\in\{1,...,N\}\!: X_j = 1\}$ and $i$ is also treated at state $X$. We can thus apply Theorem 5.4 of [@masseyStoOrdering], and $X(t)$ is a strongly monotone Markov process.
Let $X \op{\leq} Y$. If $X(t)$, $Y(t)$ are epidemic processes such that $X(0) \op{=} X$ and $Y(0) \op{=} Y$, then the strong monotonicity of the Markov process implies that $\forall t \op{\geq} 0, \mathbb{P}(\sum_i X_i(t) \op{=} 0) \op{\geq} \mathbb{P}(\sum_i Y_i(t) \op{=} 0)$, which may be rewritten as follows: $\mathbb{P}(\tau_X \op{\leq} t) \op{\geq} \mathbb{P}(\tau_Y \op{\leq} t)$. This means that $\tau_Y$ dominates $\tau_X$ and thus $\Exp{\tau_X} \op{\leq} \Exp{\tau_Y}$.
\[th:generalBound\] Assume $b_{tot} = 1$ and let $X_n^j$ the worst state vector after $j$ additional infections from the $n^{th}$ state of the planned strategy $X_n$: $$X_n^j = \argmax_{
\begin{array}{l}
\one_s ~\st\\
\{n,..., N\} \subset s,\\
|s \cup \{1,..., n-1\}| = j
\end{array}}
{\Exp{\tau_{\one_s}}}.$$ Then the following bound for the expected extinction time under the priority planning and starting from a total infection holds:\
$\forall K \geq 1$ and $\rho > \beta \left[\sum_{n=1}^N \prod_{j=0}^K E_{I-S}(X_n^j)\right]^{\frac{1}{K+1}} - \delta$, $$\Exp{\tau_\one} \ \leq \ \frac{\sum_{k=0}^K f(k)}{(\rho + \delta)(1 - f(K+1))},$$ where $$f(k) = \sum_{n=1}^N \left(\frac{\beta}{\rho + \delta}\right)^k \prod_{j=0}^{k-1} E_{I-S}(X_n^j).$$
For every state vector $X$, we have: $$\Exp{\tau_X} = \Exp{t_1 + \tau_{X(t_1)}},$$ where $t_1$ is the time of the first change in the state vector. Three types of events can happen: either a node recovers by itself (at a rate $\delta$), or a node is healed by a resource (at a rate $\rho$ $\delta$), or a node is infected (at a rate $\beta$). The number of infected nodes is $N_I(X)$. The first contagious node, denoted as $i_X \op{=} \min\{j\in\{1,...,N\}\!: X_j \op{=} 1\}$, receives a resource, and the number of nodes that can be infected is $E_{I-S}(X)$. Thus, $$\begin{array}{ll}
\Exp{\tau_X} = &\frac{1}{\delta N_I(X) + \rho + \beta E_{I-S}(X)}\\\\
&+ \frac{\delta (N_I(X) - 1)}{\delta N_I(X) + \rho + \beta E_{I-S}(X)}\Exp{\tau_{X(t_1)} | \mbox{self-recovery of a node at } t_1}\\\\
&+ \frac{\rho + \delta}{\delta N_I(X) + \rho + \beta E_{I-S}(X)}\Exp{\tau_{X - \one_{i_X}}}\\\\
&+ \frac{\beta E_{I-S}(X)}{\delta N_I(X) + \rho + \beta E_{I-S}(X)}\Exp{\tau_{X(t_1)} | \mbox{infection at } t_1}.\\
\end{array}$$
Using , we get that $\Exp{\tau_{X(t_1)} | \mbox{self-recovery of a node at } t_1} \leq \Exp{\tau_X}$, which leads to: $$(\delta + \rho + \beta E_{I-S}(X))\Exp{\tau_X} \leq 1 + (\rho + \delta) \Exp{\tau_{X - \one_{i_X}}} + \beta E_{I-S}(X) \Exp{\tau_{X(t_1)} | \mbox{infection at } t_1}.$$
Let $u_n^j \op{=} \Exp{\tau_{X_n^j}}$. For all $j \op{\geq} 1$, by definition of $X_n^{j-1}$ and $X_n^{j+1}$, we have $\Exp{\tau_{X_n^j \op{-}\one_{i_{X_n^j}}}} \op{\leq} u_n^{j-1}$ and $\Exp{\tau_{X_n^j(t_1)} | \mbox{infection at } t_1} \op{\leq} u_n^{j+1}$. This comes from the fact that, as the order is static, the $j$ infected nodes that are among $\{1,..., n\op{-}1\}$ will receive a resource first. We thus get the following recurrence equation for the $u_n^j$: $$(\delta + \rho)(u_n^j - u_n^{j-1}) \leq 1 + \beta E_{I-S}(X_n^j)(u_n^{j+1} - u_n^j),$$ which can be iterated: $$\begin{array}{lll}
(\rho + \delta) (u_n^0 - u_{n+1}^0) &\leq &\sum_{k=0}^{K-1} (\frac{\beta}{\rho + \delta})^k \prod_{j=0}^{k-1} E_{I-S}(X_n^j)\\\\
&&+ (\rho + \delta) (u_n^{K+1} - u_n^K) (\frac{\beta}{\rho + \delta})^K \prod_{j=0}^K E_{I-S}(X_n^j)\\\\
&\leq &\sum_{k=0}^{K-1} (\frac{\beta}{\rho + \delta})^k \prod_{j=0}^{k-1} E_{I-S}(X_n^j)\\\\
&&+ (\rho + \delta) u_1^0 (\frac{\beta}{\rho + \delta})^K \prod_{j=0}^K E_{I-S}(X_n^j),\\
\end{array}$$ since $u_n^{K+1} \op{\leq} u_1^0$ using and $u_1^0 \op{=} \Exp{\tau_\one}$.
We can now derive the final formula by summing over $n$ and using the definition $f(k) = \sum_{n=1}^N (\frac{\beta}{\rho + \delta})^k \prod_{j=0}^{k-1} E_{I-S}(X_n^j)$: $$(\rho + \delta) (1 - f(K+1)) \Exp{\tau_\one} \leq \sum_{k=0}^{K-1} f(k).$$
\[th:EisBound\] For $n \op{\in} \{1,..., N\}$ and $j \op{\in} \{0,..., n\op{-}1\}$, $$E_{I-S}(X_n^j) \ \leq \ E_{I-S}(X_n) + j d_{max},$$ where $d_{max} \op{=} \max_i \sum_j A_{ij}$ is the highest degree of the network.
The contagious nodes of $X_n^j$ are the contagious nodes of $X_n$ and exactly $j$ additional nodes. Since a node can have at most $d_{max}$ neighbors, then each of the $j$ additional nodes can add at most $d_{max}$ edges to the set of contagious edges of the network.
\[th:xiBound\] Let $a \op{\geq} 0$ and $\xi$ be the (unique) positive solution to $\xi \op{-} \ln(1+\xi) \op{=} a$. The following inequality holds: $$\xi \leq a + 2 \sqrt{a}.$$
$x \op{-} \ln(1\op{+}x)$ is convex, thus always above its tangent line:\
$\forall x_0 > 0,$ $$a = \xi - \ln(1+\xi) \geq (x_0 - \ln(1+x_0)) + \frac{x_0}{1+x_0}(\xi - x_0),$$ and thus, $$\begin{array}{ll}
\xi &\leq \frac{1+x_0}{x_0}(a + \ln(1+x_0)) - 1\\\\
&\leq \frac{1+x_0}{x_0} a + x_0.\\
\end{array}$$ The final result is obtained by setting $x_0 \op{=} \sqrt{a}$.
We can now prove the using the above lemmas.
Using and , we obtain a bound on the extinction time depending on $\MaxCut \op{=} \max_n E_{I-S}(X_n)$,\
$\forall K \geq 1$ and $\rho > \beta \left[N \prod_{j=0}^K (\MaxCut + j d_{max})\right]^{\frac{1}{K+1}} - \delta$, $$\Exp{\tau_\one} \ \leq \ \frac{\sum_{k=0}^K f(k)}{(\rho + \delta)(1 - f(K+1))}$$ where $$f(k) = N (\frac{\beta}{\rho + \delta})^k \prod_{j=0}^{k-1} (\MaxCut + j d_{max}),$$ using the approximation $\sum_{n=1}^N \prod_{j=0}^{k-1} (E_{I-S}(X_n) + j d_{max}) \leq N \prod_{j=0}^{k-1} (\MaxCut + j d_{max})$.
Finally, we need to select a proper value for $K$ and derive the final result. Let $\xi$ be the unique solution of $\xi \op{-} \ln(1\op{+}\xi) \op{=} \frac{d_{max} \ln{N}}{\MaxCut}$ and $K^* \op{=} \lfloor \frac{\MaxCut}{d_{max}} \xi\rfloor$. Using the particular value of $K^*$, $$\label{eq:justaninequality}
\begin{array}{ll}
\sum_{j=0}^{K^*} \ln(1 + j \frac{d_{max}}{\MaxCut})
&\leq \int_{0}^{K^* + 1} \ln(1 + x \frac{d_{max}}{\MaxCut})dx\\
&= (K^* + 1 + \frac{\MaxCut}{d_{max}})\ln(1 + (K^* + 1) \frac{d_{max}}{\MaxCut}) - (K^* + 1)\\
&\leq (K^* + 1) \ln(1 + (K^* + 1) \frac{d_{max}}{\MaxCut}) + \frac{\MaxCut}{d_{max}}(\ln(1 + \xi) - \xi)\\
&= (K^* + 1) \ln(1 + (K^* + 1) \frac{d_{max}}{\MaxCut}) - \ln(N),\\
\end{array}$$ where the second inequality is due to $\frac{\MaxCut}{d_{max}}(K^* \op{+} 1) \op{\geq} \xi$ and the monotonic decrease of $x \op{\mapsto} \ln(1\op{+}x) \op{-} x$ for $x \op{\geq} 0$.\
From , we derive that $f(K^*\op{+}1) \op{\leq} \left[\frac{\beta}{\rho + \delta}(\MaxCut \op{+} (K^*\op{+}1) d_{max})\right]^{K^*+1}$. We thus have:\
For $\rho \op{>} \beta (\MaxCut \op{+} (K^*+1) d_{max}) \op{-} \delta$, $$\begin{array}{ll}
\Exp{\tau_\one} &\leq \frac{\sum_{k=0}^{K^*} f(k)}{(\rho + \delta)(1 - f(K^*+1))}\\\\
&\leq \frac{N\sum_{k=0}^{K^*} \left[\frac{\beta}{\rho + \delta} (\MaxCut + (K^*+1) d_{max})\right]^k}{(\rho + \delta)(1 - f(K^*+1))}\\\\
&\leq \frac{N}{(\rho + \delta)(1 - \frac{\beta}{\rho + \delta} (\MaxCut + (K^*+1) d_{max}))}\cdot\frac{1 - \left[\frac{\beta}{\rho + \delta} (\MaxCut + (K^*+1) d_{max})\right]^{K^*+1}}{1 - f(K^*+1)}\\\\
&\leq \frac{N}{\rho + \delta - \beta (\MaxCut + (K^*+1) d_{max})}.\\
\end{array}$$\
Finally, using , $d_{max} K^* \op{\leq} \MaxCut \xi \op{\leq} \MaxCut \left[\frac{d_{max} \ln{N}}{\MaxCut} \op{+} 2\sqrt{\frac{d_{max} \ln{N}}{\MaxCut}}\right]$,\
which proves the desired bound.
[^1]: Available on: `http://snap.stanford.edu/data/index.html`.
|
---
abstract: 'Global linear stability analysis of a self-similar solution describing a relativistic shell decelerated by an ambient medium is performed. The system is shown to be subject to the convective Rayleigh-Taylor instability, with a rapid growth of eigenmodes having angular scale much smaller than the causality scale. The growth rate appears to be largest at the interface separating the shocked ejecta and shocked ambient gas. The disturbances produced at the contact interface propagate in the shocked media and cause nonlinear oscillations of the forward and reverse shock fronts. It is speculated that such oscillations may affect the emission from the shocked ejecta in the early afterglow phase of GRBs, and may be the origin of the magnetic field in the shocked circum-burst medium.'
author:
- Amir Levinson
title: 'Convective Instability of a Relativistic Ejecta Decelerated by a Surrounding Medium: An Origin of Magnetic Fields in GRBs?'
---
Introduction
============
Following shock breakout and a rapid acceleration phase, the relativistic ejecta accumulated in the explosion starts decelerating, owing to its interaction with the surrounding medium. At early times a double shock structure forms, consisting of a forward shock that propagates in the ambient medium, a reverse shock crossing the ejecta and a contact interface separating the shocked ejecta and the shocked ambient medium. In the fireball scenario commonly adopted, the naive expectation has been that the crossing of the reverse shock should produce an observable optical flash. Despite considerable observational efforts, such flashes have not been detected yet. The afterglow emission, which is produced behind the forward shock, seem to indicate strong amplification of magnetic fields in the post shock region, by some yet unknown mechanism. Moreover, the lightcurve of the afterglow emission deviates at early time from that predicted by a simple blast wave model.
A question of considerable interest is the stability of the double shock system. Hydrodynamic instabilities may lead to strong distortions of the system that may generate turbulence, amplify magnetic fields, and affect the emission processes in the afterglow phase. Such effects have been studied in the non-relativistic case in connection with young supernovae remnants (SNRs). In fact, the idea that the Rayleigh-Taylor (R-T) instability may play an important role in the deceleration of a non-relativistic ejecta dates back to Gull (1973), who performed 1D simulations of young SNRs that incorporate a simple model of convection. Chevalier et al. (1992) performed a global linear stability analysis of a self-similar solution describing the interaction of a nonrelativistic ejecta with an ambient medium and found that it is subject to a convective instability. They analyzed self-similar perturbations and showed that the flow is unstable for modes having angular scale smaller than some critical value. The convective growth rate was found to be largest at the contact discontinuity surface and to increase with increasing $l$ number of the eigenmodes. They also performed 2D hydrodynamical simulations that verified the linear results and enabled them to study the nonlinear evolution of the instability. The simulation exhibits rapid growth of fingers from the contact interface that saturates, in the nonlinear state, by the Kelvin-Helmholtz instability. Strong distortion of the contact and the reverse shock was observed with little effect on the forward shock. Jun & Norman (1996) performed 2 and 3D MHD simulations of the instability to study the evolution of magnetic fields in the convection zone. They confirmed the rapid growth of small scale structure reported by Chevalier et al. (1992), and in addition found strong amplification of ambient magnetic fields in the turbulent flow around R-T fingers. On average, the magnetic field energy density reaches about 0.5% of the energy density of the turbulence, but it could well be that the magnetic field amplification was limited by numerical resolution in their simulations. The simulations of Chevalier et al. (1992) and Jun & Norman (1996) support earlier ideas, that the clumpy shell structure observed in young (pre-Sedov stage) SNRs such as Tycho, Kepler and Cas A is due to the R-T and K-H instability.
In this paper we extend the linear stability analysis of Chevalier et al. (1992) into the relativistic regime. We find that denser ejecta sweeping a lighter ambient gas are subject to the R-T instability also in the relativistic case. The stability of a double-shock system has been investigated by Xiaohu et al. (2002) using the thin shell approximation. However, this study is limited to large scale modes and neglects pressure gradients and, therefore, excludes the convective instability. Gruzinov (2000) performed a linear stability analysis of a Blandford-McKee (BMK) blast wave solution, and found that the BMK solution is stable but non-universal, in the sense that some modes decay very slowly as the system evolves. Furthermore, the onset of oscillations of an eigenmode of order $l$ has been seen in the simulation once the Lorentz factor evolved to $\Gamma <l$. The conclusion drawn based on Gruzinov’s findings is that distortion of the shock front at early times may cause significant oscillations during a large portion of its evolution. If the amplitude of these oscillations is sufficiently large, and if the same behavior holds in the nonlinear regime then this can lead to generation of vorticity in the post shock region (Milosavljevic et al. 2007; Goodman & MacFadyen 2007), and the consequent amplification of magnetic fields, as demonstrated recently by Zhang et al. (2009).
Analysis
========
We consider the interaction of a cold unmagnetized shell with a cold ambient medium having a density profile $\rho_i=br^{-k}$. The structure formed at early stages consists of a forward shock propagating in the ambient medium, a reverse shock propagating in the ejecta and a contact discontinuity separating the shocked ambient medium and the shocked ejecta. The equations governing the dynamics of the flow on each side of the contact interface can be written in the form, $$\begin{aligned}
\rho h\gamma^2\frac{d\ln\gamma}{dt}+\gamma^{2}\frac{dP}{dt}=\frac{\partial P}{\partial t},\label{momentum}\\
\frac{d}{dt}\ln\left(P/\rho^{\hat{\gamma}}\right)=0,\label{state}\\
\rho\gamma\frac{d}{dt}(h\gamma{\bf v}_T)+\nabla_T P=0,\label{tangent}\end{aligned}$$ where $\gamma=u^0$ is the Lorentz factor of the fluid, ${\bf v}_T$ is the tangential component of the 3-velocity, which we express as ${\bf v}=v_r\hat{r}+{\bf v}_T$, $\rho$, $P$ and $h$ are the proper density, pressure and specific enthalpy, $\hat{\gamma}$ is the adiabatic index, and $$\nabla_T\equiv \frac{1}{r}\frac{\partial}{\partial\theta}\hat{\theta}+\frac{1}{r\sin\theta}
\frac{\partial}{\partial\phi}\hat{\phi}.$$ Equations (\[momentum\])-(\[tangent\]) admit self-similar solutions (Nakamura & Shigeyama, 2006) in cases of a freely expanding ejecta characterized by a velocity $v_e=r/t$ at time $t$ after the explosion, and a proper density profile $$\rho_e=\frac{a}{t^3\gamma_e^n}, \label{den-ejecta}$$ where $\gamma_e=1/\sqrt{1-v_e^2}$ is the corresponding Lorentz factor of the unshocked ejecta. The Lorentz factors of the forward shock, reverse shock and the contact discontinuity surface, denoted by $\Gamma_{1}(t)$, $\Gamma_{2}(t)$ and $\Gamma_c(t)$, respectively, evolve with time as $\Gamma^2_2=At^{-m}$, $\Gamma^2_1=Bt^{-m}$, $\Gamma^2_c=Ct^{-m}$, with the constants $A,B,C$ and $m$ determined by matching the solutions in the shocked ejecta and in the shocked ambient medium at the contact discontinuity. In particular $m=(6-2k)/(n+2)$ (Nakamura & Shigeyama, 2006). The velocity of the ejecta just upstream the reverse shock is $v_e(r=r_2)=
r_2(t)/t=1-1/[2(m+1)\Gamma_2^2]$, where $r_2(t)=\int{(1-1/2\Gamma_2^2) dt}$ is trajectory of the reverse shock, implying $\gamma_e^2=(m+1)\Gamma_2^2$. Thus, the self-similar solution is applicable only to situations where the ejecta is sufficiently dense, such that the reverse shock is non or at most mildly relativistic.
We have carried out a global linear stability analysis of the self-similar solution outlined above. The details will be presented elsewhere (Levinson, in preparation). A preliminary account of the method and results is presented below.
There are total of eight independent variables, four on each side of the contact: $P,\rho, \gamma, {\bf v}_T$. The perturbations of these variables were expanded in spherical harmonics. To be more concrete, a perturbed quantity $Q$ ($Q=P, \rho$, etc.) is expressed as $Q(t,\chi,\theta,\phi)=
Q_0(t,\chi)[1+\xi_Q(\chi,t)Y_{lm}(\theta,\phi)]$, where $\chi=\{1+2(m+1)\Gamma_{1}^2\}(1-r/t)$ is the self-similarity parameter, and $Q_0$ denotes the unperturbed value. The linearized equations on each side of the contact discontinuity were then obtained upon substitution of the perturbed quantities into Eqs. (\[momentum\])-(\[tangent\]). Perturbations of the shock fronts and the contact discontinuity of the form $$\delta r_j(t,\theta,\phi)=\frac{t\delta_j(t)}{\Gamma_j^2}Y_{lm}(\theta,\phi),\label{del_rs}$$ where $j=1,2,c$ refers to the forward shock, reverse shock, and the contact discontinuity, respectively, were assumed. The perturbed shock normals are then $n_{k\mu}=n_{k\mu}^{0}+\delta n_{k\mu}$ ($k=1,2$), with $n_{k\mu}^{0}=(-\Gamma_{k}V_{k},\Gamma_{k},0)$ being the unperturbed normal and $$\delta n_{k\mu}=(-\Gamma^3_{k}\delta V_{k},\Gamma^3_{k} V_{k}\delta V_{k},
-\Gamma_{k}\nabla_T\delta r_{k}).\label{del_n}$$ Here $V_{k}$ denotes the 3-velocity of the unperturbed shock front and $\delta V_k=d\delta r_k/dt$. The linearized jump conditions at the forward and reverse shocks are given in terms of the 4-velocity $u^\mu$ and the energy-momentum tensor $T^{\mu\nu}$ as: $$\begin{aligned}
[\rho u_0^{\mu}\delta n_{k\mu}+\Delta_k (\rho u^{\mu})n^0_{k\nu}]=0,\label{jmp1}\\
\left[T_0^{\mu\nu}\delta n_{k\nu}+\Delta_k T^{\mu\nu}n^0_{k\nu}\right]=0,\label{jmp2}\end{aligned}$$ where $\Delta_k Q=\delta Q+(\partial Q_0/\partial r)\delta r_k$ denotes the Lagrange perturbation of the quantity $Q$ at the perturbed surface $k$. The relations (\[jmp1\]), (\[jmp2\]) provide 6 boundary conditions for the perturbation equations, 3 on each side of the contact. Two additional boundary conditions are obtained from pressure balance and the no flow condition, viz., $v-dr_c/dt=0$, at the contact discontinuity.
Unlike in the non-relativistic case, the boundary conditions at the shock fronts break self-similarity of the perturbations. Specifically, it can be shown that at the forward shock surface the perturbation of the tangential velocity is related to the perturbations of the radial velocity and pressure through $\xi_T=3(\xi_P-\xi_R)/[2(m+2)\Gamma_1^2]$ for $l\ne0$, with a similar relation, though somewhat more involved, at the reverse shock front. Thus, numerical simulations of the perturbation equations is needed. The only exception is the spherical mode ($l=0$) for which a self-similar solution was obtained analytically. For this solution $\delta Q\propto t^s$ with $s<0$. It can be shown that one eigenmode of order $l=0$ is associated with linear time translation of the self-similar solution. For this mode $s=-(m+1)$. We found another eigenmode of order $l=0$ that decay somewhat slower. The analytic solution for the $l=0$ mode has been used both to test the code and as initial condition for the evolution of the higher order eigenmodes. Numerical integration of the perturbation equations was performed after transforming to the so called Riemann invariants. We identified three variables that propagate from the forward shock inwards, three that propagate from the reverse shock outwards and two that propagate from the contact, one inwards and one outwards, and have chosen the boundary conditions for the Riemann invariants accordingly. It is generally found that eigenmodes having an angular scale larger than the horizon scale, specifically $l(l+1)<\Gamma^2$, are stable. Higher order modes are found to be unstable with a growth rate that increases with increasing $l$. An example is exhibited in Fig. 1. The onset of oscillations followed by a rapid growth of the initial perturbations is clearly seen. The distortion of the contact discontinuity surface becomes nonlinear very early on. The growth is algebraic in time $t$ with a growth rate of about $10$ in the example shown in Fig. 1; that is, $\delta Q\propto t^{10}$ for $Q=P,\rho, v$. As seen, the reverse shock responds quickly to the distortion of the contact. The forward shock, on the other hand, responds much later, at time when the instability near the contact already reached a nonlinear state.
Discussion
==========
The stability analysis described above seem to indicate strong convective instability at early stages of the evolution of a dense ejecta as it sweeps a lighter ambient gas. The growth rate appears to be largest at the contact discontinuity and for higher order modes. Disturbances at the interface separating the shocked ejecta and the shocked ambient medium propagate away from the contact discontinuity and cause nonlinear distortions of the shock fronts. The reverse shock responds quickly to the distortion of the contact. Propagation of the signal to the forward shock is much slower. At any rate, the instability near the contact becomes nonlinear well before the signal arrives at the forward shock, so full MHD simulations are needed to resolve the effect of the instability on the forward shock. It is naively expected that the instability will be strongly suppressed in cases where the ejecta is highly magnetized and/or if the reverse shock is highly relativistic. On the other hand, if the magnetic field strength in the unshocked ejecta is smaller than that required to suppress the instability but still much larger than that of the ambient medium, then at early stages mixing of the magnetized ejecta with the shocked ambient gas via growth of R-T fingers can give rise to a strong amplification of the magnetic fields behind the forward shock. Full simulations are required to quantify the conditions under which the instability is effective.
The nonlinear distortions of the contact and the shock fronts should generate turbulence in the shocked fluids on both sides of the contact discontinuity. At early stages this may strongly affect particle acceleration and the emission processes. It is tempting to speculate that the lack of observed optical flashes, that are anticipated in the “standard” model, and the fact that the early afterglow emission observed in many sources is inconsistent with the prediction of the blast wave model may be attributed to the instability discussed here. In any case, it is clear that a careful analysis that takes account of this process is required to better understand the observational characteristics of the emission during the early afterglow phase.
The stability analysis of the Blandford-McKee solution performed by Gruzinov (2000) suggests that it may be a very slow attractor. Linear perturbations of the forward shock in the B-M phase decay very slowly. Whether this behavior continues also in the nonlinear regime is unclear yet. If it does then it is anticipated that the growth of R-T fingers and, perhaps, nonlinear oscillations of the forward shock itself that are induced by the convective instability may be a source of vorticity during a long portion of the evolution of the blast wave. As demonstrated recently by Zhang et al. (2009) the induced turbulence can amplify weak magnetic fields. Their simulation seem to converge at a saturation level of $\epsilon_B\sim5\times10^{-3}$, weakly dependent on the initial magnetic field strength. This process may provide an explanation for the origin of the strong magnetic fields inferred behind the collisionless shock in the afterglow phase.
Unfortunately, the linear analysis outlined above is restricted to a limited set of conditions under which the unperturbed self-similar solution of Nakamura & Shigeyama (2006) is applicable. Full 3D MHD simulations are required to study this process in other situations, and to follow the evolution of the convective instability in the nonlinear state. As illustrated above, high resolution simulations that can resolve angular scales $\Delta \theta<<1/\Gamma$ are required, posing a great numerical challenge. We believe that our findings strongly motivate such efforts.
I thank A. Ditkowski for a technical help in the development of the code, and M. Alloy, A. MacFadyen, E. Nakar and E. Waxman for enlightening discussions. This work was supported by an ISF grant for the Israeli Center for High Energy Astrophysics, and by the NORDITA program on Physics of relativistic flows.
[99]{} Chevalier R. A., Blondin, J. M. & Emmering R. 1992, ApJ, 392, 118 Goodman, J. & MacFadyen, A. 2007, arXiv:0706.1818 Gruzinov, A. 2000, arXiv:astro-ph/0012364 Gull, S. F. 1973, MNRAS, 161, 47 Jun, B-I. & Norman, M. L. 1996, ApJ, 465, 800 Milosavljevic, M., Nakar, E. & Zhang, F. 2007, arXiv:0708.1588 Nakamura, K. & Shigeyama, T. 2006, ApJ, 645, 43 Xiaohu, W., Loeb, A. & Waxman, E. 2002, ApJ, 568, 830 Zhang, W., MacFadyen, A. & Wang, P. 2009, ApJ, 692, 40
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---
abstract: 'Given a set of strings, the shortest common superstring problem is to find the shortest possible string that contains all the input strings. The problem is NP-hard, but a lot of work has gone into designing approximation algorithms for solving the problem. We present the first time and space efficient implementation of the classic greedy heuristic which merges strings in decreasing order of overlap length. Our implementation works in $O(n \log \sigma)$ time and bits of space, where $n$ is the total length of the input strings in characters, and $\sigma$ is the size of the alphabet. After index construction, a practical implementation of our algorithm uses roughly $5 n \log \sigma$ bits of space and reasonable time for a real dataset that consists of DNA fragments.'
author:
- Jarno Alanko
- Tuukka Norri
bibliography:
- 'references.bib'
title: Greedy Shortest Common Superstring Approximation in Compact Space
---
Introduction
============
Given a set of strings, the shortest common superstring is the shortest string which contains each of the input strings as a substsring. The problem is NP-hard [@gallant1980finding], but efficient approximation algorithms exist. Perhaps the most practical of the approximation algorithms is the greedy algorithm first analyzed by Tarhio, Ukkonen [@tarhio1988greedy] and Turner [@turner1989approximation]. The algorithm greedily joins together the pairs of strings with the longest prefix-suffix overlap, until only one string remains. In case there are equally long overlaps, the algorithm can make an arbitrary selection among those. The remaining string is an approximation of the shortest common superstring. The algorithm has been proven to give a superstring with length at most $3 \frac{1}{2}$ times the optimal length [@kaplan2005greedy]. It was originally conjectured by Ukkonen and Tarhio [@tarhio1988greedy] that the greedy algorithm never outputs a superstring that is more than twice as long as the optimal, and the conjecture is still open.
Let $m$ be the number of strings, $n$ be the sum of the lengths of all the strings, and $\sigma$ the size of the alphabet. In 1990 Ukkonen showed how to implement the greedy algorithm in $O(n)$ time and $O(n \log n)$ bits of space using the Aho-Corasick automaton [@ukkonen1990linear]. Since then, research on the problem has focused on finding algorithms with better provable approximation ratios (see e.g. [@mucha2013lyndon] for a summary). Currently, algorithm with the best proven approximation ratio in peer reviewed literature is the one by Mucha with an approximation ratio of $2 \frac{11}{23}$ [@mucha2013lyndon], and there is a preprint claiming an algorithm with a ratio of $2 \frac{11}{30}$ [@paluch2014better]. However, we are not aware of any published algorithm that solves the problem in better than $O(n \log n)$ bits of space. Improving the factor $\log n$ to $\log \sigma$ is important in practice. Many of the largest data sets available come from DNA strings which have an alphabet of size only 4, while $n$ can be over $10^9$. We present an algorithm that implements the greedy heuristic in $O(n \log \sigma)$ time and bits of space. It is based on the FM-index enhanced with a succinct representation of the topology of the suffix tree. The core of the algorithm is the iteration of prefix-suffix overlaps of input strings in decreasing order of length using a technique described in [@makinen2015genome] and [@simpson2010efficient], combined with Ukkonen’s bookkeeping [@ukkonen1990linear] to keep track of the paths formed in the overlap graph of the input strings. The main technical novelty of this work is the implementation of Ukkonen’s bookkeeping in $O(n \log \sigma)$ space. We also have a working implementation of the algorithm based on the SDSL-library [@gog2014theory]. For practical reasons the implementation differs slightly from the algorithm presented in this paper, but the time and space usage should be similar.
Preliminaries
=============
Let there be $m$ strings $s_1, \ldots, s_m$ drawn from the alphabet $\Sigma$ of size $\sigma$ such that the sum of the lengths of the strings is $\sum_{i=1}^m |s_i| = n$. We build a single string by concatenating the $m$ strings, placing a separator character \$ $\not\in \Sigma$ between each string. We define that the separator is lexicographically smaller than all characters in $\Sigma$. This gives us the string $S = s_1 \$ s_2 \$ \cdots s_m \$$ of length $n + m$. Observe that the set of suffixes that are prefixed by some substring $\alpha$ of $S$ are adjacent in the lexicographic ordering of the suffixes. We call this interval in the sorted list of suffixes the *lexicographic range* of string $\alpha$. All occurrences of a substring $\alpha$ can be uniquely represented as a triple $(a_\alpha, b_\alpha, d_\alpha)$, where $[a_\alpha, b_\alpha]$ is the lexicographic range of $\alpha$, and $d_\alpha$ is the length of $\alpha$. A string $\alpha$ is *right maximal* in $S$ if and only if there exist two or more distinct characters $y,z \in \Sigma \cup \{\$\}$ such that the strings $\alpha y$ and $\alpha z$ are substrings of $S$. Our algorithm needs support for two operations on substrings: left extensions and suffix links. A left extension of string $\alpha$ with character $x$ is the map $(a_\alpha, b_\alpha, d_\alpha) \mapsto (a_{x\alpha}, b_{x\alpha},
d_{x\alpha})$. A suffix link for the right-maximal string $x \alpha$ is the map $(a_{x\alpha}, b_{x\alpha}, d_{x\alpha})
\mapsto (a_\alpha, b_\alpha, d_\alpha)$.
Overview of the Algorithm
=========================
We use Ukkonen’s 1990 algorithm [@ukkonen1990linear] as a basis for our algorithm. Conceptually, we have a complete directed graph where vertices are the input strings, and the weight of the edge from string $s_i$ to string $s_j$ is the length of the longest suffix of $s_i$ which is also a prefix of $s_j$. If there is no such overlap, the weight of the edge is zero. The algorithm finds a Hamiltonian path over the graph, and merges the strings in the order given by the path to form the superstring. We define the merge of strings $s_i = \alpha \beta$ and $s_j = \beta \gamma$, where $\beta$ is the longest prefix-suffix overlap of $s_i$ and $s_j$, as the string $\alpha\beta\gamma$. It is known that the string formed by merging the strings in the order given by the maximum weight Hamiltonian path gives a superstring of optimal length [@tarhio1988greedy]. The greedy algorithm tries to heuristically find a Hamiltonian path with a large total length.
Starting from a graph $G$ where the vertices are the input strings and there are no edges, the algorithm iterates all prefix-suffix overlaps of pairs of strings in decreasing order of length. For each pair $(s_i,s_j)$ we add an edge from $s_i$ to $s_j$ iff the in-degree of $s_j$ is zero, the out-degree of $s_i$ is zero, and adding the edge would not create a cycle in $G$. We also consider overlaps of length zero, so every possible edge is considered and it is easy to see that in the end the added edges form a Hamiltonian path over $G$.
Algorithm {#sec:algorithm}
=========
Observe that if an input string is a proper substring of another input string, then any valid superstring that contains the longer string also contains the shorter string, so we can always discard the shorter string. Similarly if there are strings that occur multiple times, it suffices to keep only one copy of each. This preprocessing can be easily done in $O(n \log \sigma)$ time and space for example by backward searching all the input strings using the FM-index.
After the preprocessing, we sort the input strings into lexicographic order, concatenate them placing dollar symbols in between the strings, and build an index that supports suffix links and left extensions. The sorting can be done with merge sort such that string comparisons are done $O(\log(n))$ bits at a time using machine word level parallelism, as allowed by the RAM model. This works in $O(n \log \sigma)$ time and space if the sorting is implemented so that it does not move the strings around, but instead manipulates only pointers to the strings. For notational convenience, from here on $s_i$ refers to the string with lexicographic rank $i$ among the input strings.
We iterate in decreasing order of length all the suffixes of the input strings $s_i$ that occur at least twice in $S$ and for each check whether the suffix is also a prefix of some other string $s_j$, and if so, we add an edge from $s_i$ to $s_j$ if possible. To enumerate the prefix-suffix overlaps, we use the key ideas from the algorithm for reporting all prefix-suffix overlaps to build an overlap graph described in [@makinen2015genome] and [@simpson2010efficient], adapted to get the overlaps in decreasing order of length.
We maintain an iterator for each of the input strings. An iterator for the string $s_i$ is a quadruple $(i,\ell,r,d)$, where $[\ell,r]$ is the lexicographic range of the current suffix $\alpha$ of $s_i$ and $d$ is the length of $\alpha$, i.e. the depth of the iterator. Suffixes of the input strings which are not right maximal in the concatenation $S = s_1\$\ldots s_m\$$ can never be a prefix of any of the input strings. The reason is that if $\alpha$ is not right-maximal, then $\alpha$ is always followed by the separator $\$$. This means that if $\alpha$ is also a prefix of some other string $s_j$, then $s_j = \alpha$, because the only prefix of $s_j$ that is followed by a $\$$ is the whole string $s_j$. But then $s_j$ is a substring of $s_i$, which can not happen because all such strings were removed in the preprocessing stage. Thus, we can safely disregard any suffix $\alpha$ of $s_i$ that is not right maximal in $S$. Furthermore, if a suffix $\alpha$ of $s_i$ is not right maximal, then none of the suffixes $\beta\alpha$ are right-maximal either, so we can disregard those, too.
We initialize the iterator for each string $s_i$ by backward searching $s_i$ using the FM-index for as long as the current suffix of $s_i$ is right-maximal. Next we sort these quadruples in the decreasing order of depth into an array $\texttt{iterators}$. When this is done, we start iterating from the iterator with the largest depth, i.e. the first element of $\texttt{iterators}$. Suppose the current iterator corresponds to string $i$, and the current suffix of string $s_i$ is $\alpha$. At each step of the iteration we check whether $\alpha$ is also a prefix of some string by executing a left extension with the separator character $\$$. If the lexicographic range $[\ell',r']$ of $\$\alpha$ is non-empty, we know that the suffixes of $S$ in the range $[\ell',r']$ start with a dollar and are followed by a string that has $\alpha$ as a prefix. We conclude that the input string with lexicographic rank $i$ among the input strings has a suffix of length $d$ that matches a prefix of the strings with lexicographic ranks $\ell' , \ldots , r'$ among the input strings. This is true because the lexicographic order of the suffixes of $S$ that start with dollars coincides with the lexicographic ranks of the strings following the dollars in the concatenation, because the strings are concatenated in lexicographic order.
Thus, according to the greedy heuristic, we should try to merge $s_i$ with a string from the set $s_{\ell'},\ldots,s_{r'}$, which corresponds to adding an edge from $s_i$ to some string from $s_{\ell'},\ldots,s_{r'}$ in the graph $G$. We describe how we maintain the graph $G$ in a moment. After updating the graph, we update the current iterator by decreasing $d$ by one and taking a suffix link of the lexicographic range $[\ell, r]$. The iterator with the next largest $d$ can be found in constant time because the array $\texttt{iterators}$ is initially sorted in descending order of depth. We can maintain a pointer to the iterator with the largest $d$. If at some step $\texttt{iterators}[k]$ has the largest depth, then in the next step either $\texttt{iterators}[k+1]$ or $\texttt{iterators}[1]$ has the largest depth. The pseudocode for the main iteration loop is shown in Algorithm \[alg:main\_iteration\].
$k \gets 1$
Now we describe how we maintain the graph $G$. The range $[\ell', r']$ now represents the lexicographical ranks of the input strings that are prefixed by $\alpha$ among all input strings. Each string $s_j$ in this range is a candidate to merge to string $s_i$, but some bookkeeping is needed to keep track of available strings. We use essentially the same method as Tarhio and Ukkonen [@tarhio1988greedy]. We have bit vectors $\texttt{leftavailable}[1..m]$ and $\texttt{rightavailable}[1..m]$ such that $\texttt{leftavailable}[k] = 1$ if and only if string $s_k$ is available to use as the left side of a merge, and $\texttt{rightavailable}[k] = 1$ if and only if string $s_k$ is available as the right side of a merge. Equivalently, $\texttt{leftavailable}[k] = 1$ iff the out-degree of $s_k$ is zero and $\texttt{rightavailable}[k] = 1$ if the in-degree of $s_k$ is zero. Also, to prevent the formation of a cycle, we need arrays $\texttt{leftend}[1..m]$, where $\texttt{leftend}[k]$ gives the leftmost string of the chain of merged strings to the left of $s_k$, and $\texttt{rightend}[1..m]$, where $\texttt{rightend}[k]$ gives the rightmost string of the chain of merged strings to the right of $s_k$. We initialize $\texttt{leftavailable}[k] =
\texttt{rightavailable}[k] = 1$ and $\texttt{leftend}[k] = \texttt{rightend}[k] = k$ for all $k = 1, \ldots, m$.
When we get the interval $[\ell', r']$ such that $\texttt{leftavailable}[j] = 1$, we try to find an index $j \in [\ell_{\$ \alpha}, r_{\$ \alpha}]$ such that $\texttt{rightavailable}[i] = 1$ and $\texttt{leftend}[j] \neq i$. Luckily we only need to examine at most two indices $j$ and $j'$ such that $\texttt{rightavailable}[j] = 1$ and $\texttt{rightavailable}[j'] = 1$ because if $\texttt{leftend}[j] = i$, then $\texttt{leftend}[j'] \neq i$, and vice versa. This procedure is named $\texttt{trymerge}([l', r'], i)$ in Algorithm \[alg:main\_iteration\].
The problem is now to find up to two ones in the bit vector $\texttt{rightavailable}$ in the interval of indices $[\ell_{\$ \alpha}, r_{\$ \alpha}]$. To do this efficiently, we maintain for each index $k$ in $\texttt{rightavailable}$ the index of the first one in $\texttt{rightavailable}[k+1..m]$, denoted with $\texttt{next\_one}(k)$. If there are two ones in the interval $[\ell_{\$ \alpha}, r_{\$ \alpha}]$, then they can be found at $\texttt{next\_one}(\ell_{\$ \alpha}-1)$ and $\texttt{next\_one}(\texttt{next\_one}(\ell_{\$ \alpha}-1))$. The question now becomes, how do we maintain this information efficiently? In general, this is the problem of indexing a bit vector for dynamic successor queries, for which there does not exist a constant time solution using $O(n \log \sigma)$ space in the literature. However, in our case the vector $\texttt{rightavailable}$ starts out filled with ones, and once a one is changed to a zero, it will not change back for the duration of the algorithm, which allows us to have a simpler and more efficient data structure.
Initially, $\texttt{next\_one}(k) = k+1$ for all $k < m$. The last index does not have a successor, but it can easily be handled as a special case. For clarity and brevity we describe the rest of the process as if the special case did not exist. When we update $\texttt{rightavailable}(k) := \texttt0$, then we need to also update $\texttt{next\_one}[k'] := \texttt{next\_one}(k)$ for all $k' < k$ such that $\texttt{rightavailable}[k'+1..k]$ contains only zeros. To do this efficiently, we store the value of $\texttt{next\_one}$ only once for each sequence of consecutive zeros in $\texttt{rightavailable}$, which allows us to update the whole range at once. To keep track of the sequences of consecutive zeros, we can use a union-find data structure. A union-find data structure maintains a partitioning of a set of elements into disjoint groups. It supports the operations $\texttt{find}(x)$, which returns the representative of the group containing $x$, and $\texttt{union}(x,y)$, which takes two representatives and merges the groups containing them.
We initialize the union-find structure such that there is an element for every index in $\texttt{rightavailable}$, and we also initialize an array $\texttt{next}[1..m]$ such that $\texttt{next}[k] := k+1$ for all $k = 1, \ldots m$. When a value at index $k$ is changed to a zero, we compute $q := \texttt{next}[\texttt{find}(k)]$. Then we will do $\texttt{union}(\texttt{find}(k), \texttt{find}(k-1))$ and if $\texttt{rightavailable}[k+1] = 0$, we will do $\texttt{union}(\texttt{find}(k), \texttt{find}(k+1))$. Finally, we update $\texttt{next}[\texttt{find}(k)] = q$. We can answer queries for $\texttt{next\_one}(k)$ with $\texttt{next}[\texttt{find}(k)]$.
Whenever we find a pair of indices $i$ and $j$ such that $\texttt{leftavailable}[i] = 1$, $\texttt{rightavailable}[j] = 1$ and $\texttt{leftend}[j] \neq i$, we add an edge from $s_i$ to $s_j$ by recording string $j$ as the successor of string $i$ using arrays $\texttt{successor}[1..m]$ and $\texttt{overlaplength}[1..m]$. We set $\texttt{successor}[j] = i$ and $\texttt{overlaplength}[j] = d_i$, where $d_i$ is the length of the overlap of $s_i$ and $s_j$, and do the updates: $$\begin{aligned}
\texttt{leftavailable}[i] &:= 0 \\
\texttt{rightavailable}[j] &:= 0 \\
\texttt{leftend}[\texttt{rightend}[j]] &:= \texttt{leftend}[i] \\
\texttt{rightend}[\texttt{leftend}[i]] &:= \texttt{rightend}[j] \\\end{aligned}$$ Note that the arrays $\texttt{leftend}$ and $\texttt{rightend}$ are only up to date for the end points of the paths, but this is fine for the algorithm. Finally we update the $\texttt{next}$ array with the union-find structure using the process described earlier. We stop iterating when we have done $m-1$ merges. At the end, we have a Hamiltonian path over $G$, and we form a superstring by merging the strings in the order specified by the path.
Time and Space Analysis {#sec:timespace}
=======================
The following space analysis is in terms of number of bits used. We assume that the strings are binary encoded such that each character takes $\lceil \log_2 \sigma \rceil$ bits. A crucial observation is that we can afford to store a constant number of $O(\log n)$ bit machine words for each distinct input string.
\[lemma:mlogn\] Let there be $m$ **distinct** non-empty strings with combined length $n$ measured in $\textbf{bits}$ from an alphabet of size $\sigma > 1$. Then $m \log n \in O(n \log \sigma)$.
Proof. Suppose $m \leq \sqrt n $. Then the Lemma is clearly true, because: $$m \log n \leq \sqrt n \log n \in O(n \log \sigma)$$ We now consider the remaining case $m \geq \sqrt n$, or equivalently $\log n \leq 2 \log m$. This means $m \log n \leq 2 m \log m$, so it suffices to show $m \log m \in O(n \log \sigma)$.
First, note that at least half of the strings have length at least $\log (m) - 1$ bits. This is trivially true when $\log (m) -1 \leq 1$. When $\log (m) - 1 \geq 2$, the number of distinct binary strings of length at most $\log(m)-2$ bits is $$\sum_{i = 1}^{\lfloor \log(m)-2 \rfloor} 2^i
\leq 2^{\log (m) - 1} = \frac{1}{2} m$$ Therefore indeed at least half of the strings have length of at least $\log m - 1$ bits. The total length of the strings is then at least $\frac{1}{2}m(\log m - 1)$ bits. Since the binary representation of all strings combined takes $n \lceil \log_2 \sigma \rceil$ bits, we have $n \lceil \log_2 \sigma \rceil \geq \frac{1}{2}m(\log m - 1)$, which implies $m \log m \leq 2 n \lceil \log_2 \sigma \rceil + 1 \in O(n \log \sigma). \, \qed$
Next, we describe how to implement the suffix links and left extensions. We will need to build the following data structures for the concatenation of all input strings separated by a separator character:
- The Burrows-Wheeler transform, represented as a wavelet tree with support for rank and select queries.
- The $C$-array, which has length equal to the number of characters in the concatenation, such that $C[i]$ is the number of occurrences of characters with lexicographic rank strictly less than $i$.
- The balanced parenthesis representation of the suffix tree topology with support for queries for leftmost leaf, rightmost leaf and lowest common ancestor.
Note that in the concatenation of the strings, the alphabet size is increased by one because of the added separator character, and the total length of the data in characters is increased by $m$. However this does not affect the asymptotic size of the data, because $$(n+m) \log (\sigma + 1) \leq 2n (\log \sigma + 1)
\in \Theta (n \log \sigma)$$ The three data structures can be built and represented in $O(n \log \sigma)$ time and space [@belazzougui2014linear]. Using these data structures we can implement the left extension for lexicographic interval $[\ell, r]$ with the character $c$ by: $$([\ell, r], c) \mapsto [C[c]
+ \texttt{rank}_{BWT}(\ell,c), C[c] + \texttt{rank}_{BWT}(r,c)]$$ We can implement the suffix link for the right maximal string $c \alpha$ with the lexicographic interval $[\ell, r]$ by first computing $$v = \texttt{lca}(\texttt{select}_{BWT} (c, \ell - C[c]),
\texttt{select}_{BWT} (c, r - C[c]))$$ and then $$[\ell, r] \mapsto [\texttt{leftmostleaf}(v), \texttt{rightmostleaf}(v)]$$ This suffix link operation works as required for right-maximal strings by removing the first character of the string, but the behaviour on non-right-maximal strings is slightly different. The lexicographic range of a non-right-maximal string is the same as the lexicographic range of the shortest right-maximal string that has it as a prefix. In other words, for a non-right-maximal string $c \alpha$, the operation maps the interval $[\ell_{c \alpha}, r_{c \alpha}]$ to the lexicographic interval of the string $\alpha \beta$, where $\beta$ is the shortest right-extension that makes $c \alpha \beta$ right-maximal. This behaviour allows us to check the right-maximality of a substring $c \alpha$ given the lexicographic ranges $[\ell_\alpha, r_\alpha]$ and $[\ell_{c \alpha}, r_{c \alpha}]$ in the iterator initialization phase of the algorithm as follows:
The substring $c \alpha$ is right maximal if and only if the suffix link of $[\ell_{c \alpha}, r_{c \alpha}]$ is $[\ell_\alpha, r_\alpha]$.
As discussed above, the suffix link of $[\ell_{c \alpha}, r_{c \alpha}]$ maps to the lexicographic interval of the string $\alpha \beta$ where $\beta$ is the shortest right-extension that makes $c \alpha \beta$ right-maximal. Suppose first that $c \alpha$ is right-maximal. Then $[\ell_{\alpha\beta}, r_{\alpha\beta}] = [\ell_\alpha, r_\alpha]$, because $\beta$ is an empty string. Suppose on the contrary that $c \alpha$ is not right-maximal. Then $[\ell_{\alpha\beta}, r_{\alpha\beta}] \neq [\ell_\alpha, r_\alpha]$, because $\alpha\beta$ and $\alpha$ are distinct right-maximal strings. $\square$
Now we are ready to prove the time and space complexity of the whole algorithm.
The algorithm in Section \[sec:algorithm\] can be implemented in $O(n \log \sigma)$ time and $O(n \log \sigma)$ bits of space.
The preprocessing to remove contained and duplicate strings can be done in $O(n \log \sigma)$ time and space for example by building an FM-index, and backward searching all input strings.
The algorithm executes $O(n)$ left extensions and suffix links. The time to take a suffix link is dominated by the time do the select query, which is $O(\log \sigma)$, and the time to do a left extension is dominated by the time to do a rank-query which is also $O(\log \sigma)$. For each left extension the algorithm does, it has to access and modify the union-find structure. Normally this would take amortized time related to the inverse function of the Ackermann function [@thomas2009introduction], but in our case the amortized complexity of the union-find operations can be made linear using the construction of Gabow and Tarjan [@gabow1985linear], because we know that only elements corresponding to consecutive positions in the array $\texttt{rightavailable}$ will be joined together. Therefore, the time to do all left extensions, suffix links and updates to the union-find data structure is $O(n \log \sigma)$.
Let us now turn to consider the space complexity. For each input string, we have the quadruple $(i,\ell,r,d)$ of positive integers with value at most $n$. The quadruples take space $3 m \log m + m \log n$. The union-find structure of Gabow and Tarjan can be implemented in $O(m \log m)$ bits of space [@gabow1985linear]. The bit vectors $\texttt{leftavailable}$ and $\texttt{rightavailable}$ take exactly 2$m$ bits, and the arrays $\texttt{successor}$, $\texttt{leftend}$, $\texttt{rightend}$ and $\texttt{next}$ take $m \log m$ bits each. The array $\texttt{overlaplength}$ takes $m \log n$ bits of space. Summing up, in addition to the data structures for the left extensions and contractions, we have only $O(m \log n)$ bits of space, which is $O(n \log \sigma)$ by Lemma \[lemma:mlogn\]. $\qed$
Implementation
==============
The algorithm was implemented with the SDSL library [@gog2014theory]. A compressed suffix tree that represents nodes as lexicographic intervals [@ohlebusch2010cst++] was used to implement the suffix links and left extensions. Only the required parts of the suffix tree were built: the FM-index, balanced parentheses support and a bit vector that indicates the leftmost child node of each node. These data structures differ slightly from the description in Section \[sec:timespace\], because they were chosen for convenience as they were readily available in the SDSL library, and they should give very similar performance compared to those used in the aforementioned Section. In particular, the leftmost child vector was needed to support suffix links, but we could manage without it by using the operations on the balanced parenthesis support described in Section \[sec:timespace\]. Our implementation is available at the URL $\texttt{https://github.com/tsnorri/compact-superstring}$
The input strings are first sorted with quicksort. This introduces a $\log n$ factor to the time complexity, but it is fast in practice. The implementation then runs in two passes. First, exact duplicate strings are removed and the stripped compact suffix tree is built from the remaining strings. The main algorithm is implemented in the second part. The previously built stripped suffix tree is loaded into memory and is used to find the longest right-maximal suffix of each string and to iterate the prefix-suffix overlaps. Simultaneously, strings that are substrings of other strings are marked for exclusion from building the superstring.
For testing, we took a metagenomic DNA sample from a human gut microbial gene catalogue project [@qin2010human], and sampled DNA fragments to create five datasets with $2^{26 + i}$ characters respectively for $i = 0,\ldots,4$. The alphabet of the sample was $\{A,C,G,T,N\}$. Time and space usage for all generated datasets for both the index construction phase and the superstring construction phase are plotted in Figure \[fig:time\_space\]. The machine used run Ubuntu Linux version 16.04.2 and has 1.5 TB of RAM and four Intel Xeon CPU E7-4830 v3 processors (48 total cores, 2.10 GHz each). A breakdown of the memory needed for the largest dataset for the different structures comprising the index is shown in Figure \[fig:memory\_pie\].
While we don’t have an implementation of Ukkonen’s greedy superstring algorithm, have a conservative estimate for how much space it would take. The algorithm needs at least the goto- and failure links for the Aho-Corasick automaton, which take at least $2 n \log n$ bits total. The main algorithm uses linked lists named $L$ and $P$, which take at least $2 n \log n$ bits total. Therefore the space usage is at the very least $4 n \log n$. This estimate is plotted in Figure \[fig:time\_space\].
Figure \[fig:space\_as\_function\_of\_time\] shows the space usage of our algorithm in the largest test dataset as a function of time reported by the SDSL library. The peak memory usage of the whole algorithm occurs during index construction, and more specifically during the construction of a compressed suffix array. The SDSL library used this data structure to build the BWT and the balanced parenthesis representation, which makes the space usage unnecessarily high. This could be improved by using more efficient algorithms to build the BWT and the balanced parenthesis representation of the suffix tree topology [@belazzougui2014linear]. These could be plugged in to bring down the index construction memory. At the moment index constructions takes roughly 19 times the size of the input in bits. The peak memory of the part of the algorithm which constructs the superstring is only approximately 5 times the size of the input in bits.
Discussion
==========
We have shown a practical way to implement the greedy shortest common superstring algorithm in $O(n \log \sigma)$ time and bits of space. After index construction, the algorithm consists of two relatively independent parts: reporting prefix-suffix overlaps in decreasing order of lengths, and maintaining the overlap graph to prevent merging a string to one direction more than once and the formation of cycles. The part which reports the overlaps could also be done in other ways, such as using compressed suffix trees or arrays, or a succinct representation of the Aho-Corasick automaton. The only difficult part is to avoid having to hold $\Omega(n)$ integers in memory at any given time. We believe it is possible to engineer algorithms using these data structures to achieve $O(n \log \sigma)$ space as well.
Regrettably, we could not find any linear time implementations of Ukkonen’s greedy shortest common superstring algorithm for comparison. There is an interesting implementation by Liu and Sýkora [@liu2005sequential], but it is too slow for our purposes because it involves computing all pairwise overlap lengths of the input strings to make better choices in resolving ties in the greedy choices. While their experiments indicate that this improves the quality of the approximation, the time complexity is quadratic in the number of input strings. Zaritsky and Sipper [@zaritsky2004preservation] also have an implementation of the greedy algorithm, but it’s not publicly available, and the focus of the paper is on approximation quality, not performance. As future work, it would be interesting to make a careful implementation of Ukkonen’s greedy algorithm, and compare it to ours experimentally.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank anonymous reviewers for improving the presentation of the paper.
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title: '**Square Kilometre Array station configuration using two-stage beamforming**'
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Introduction
============
The Square Kilometre Array (SKA) is being developed as a radio telescope with a combination of unprecedented sensitivity, resolution and field of view (FoV) over a 70 MHz to $>$10 GHz frequency range [@dewdney2009]. The key scientific objectives of this instrument include mapping the distribution of neutral hydrogen gas in the Universe, studying the origin of magnetic fields and measuring the dynamics of pulsars in the Galaxy. To realize these objectives, a very large collecting area is needed (typically one square kilometre) using new antenna technology that can be mass produced with minimum cost.
The SKA design consists of three frequency bands, the lowest of which, termed [@memo125], has been selected to be built in Australia and New Zealand[^1]. will use aperture phased arrays of antennas operating in the 70–450 MHz range and extending across a distance of several hundred kilometres. The wideband antennas convert cosmic radio waves from physical processes in the universe to RF signals, which are digitised, cross-correlated and averaged. Post-correlation aperture synthesis is used to reconstruct the brightness distribution, to achieve high-dynamic range radio pictures of the sky [@HalSch08; @memo125].
Cross-correlating the several million antenna elements signals to create these radio pictures would require an extremely large signal processing system. Instead, the signal processing load is reduced by grouping geographically-adjacent antenna elements into digitally-beamformed stations; this limits the number of signals to be transported and cross-correlated. Additional cost efficiencies can be gained by grouping individual antenna elements into beamformed tiles prior to the station beamformer [@FauAle10]. Figure \[SKA-sch\] shows a schematic of this two-stage beamforming approach for . The cost efficiencies arise because the tile beamformer limits the number of independent signals (analogue or digital) early in the signal path. A first-order approximation of this effect is described in Appendix \[app\_a\].
The actual reduction in station hardware cost due to two-stage beamforming depends on the technologies used. When analogue first-stage and digital second-stage beamforming is implemented in place of two-stage digital beamforming, the cost reduction is larger. This is because the analogue first-stage beamforming reduces, by an order of magnitude, the digitisation, intra-station signal transmission and station beamformer costs. For example, an analysis of SKA-low station hardware costs shows a factor between 1.5 and 5 reduction in station hardware cost compared to all-digital beamforming [@ColHal12], across various architectural implementations. With cost estimates in @FauVaa11-Deployment of €1.8–2.8M per station, this is a potentially significant cost reduction.
Present day telescopes have various approaches to tile or station beamforming. The high band antenna (HBA) of the Low Frequency Array (LOFAR) telescope [@de2009lofar] uses a two-stage beamforming station, but both stages are performed on regular gridded antenna elements, making it an example of a regular tile in a regular station. The low band antenna (LBA) of the same telescope uses a single stage of beamforming on antenna elements in an irregular station layout. The Long Wavelength Array Station 1 (LWA1) telescope [@EllTay12] is an irregular array with a single-stage of beamforming, and is the first station of an envisioned larger correlation array. Instead of a station approach, the Murchison Widefield Array (MWA) telescope [@lonsdale2009murchison; @TinGoe12] feeds a many-input (large-N) correlator with beamformed 16-element regular tiles.
None of these telescopes use beamformed tiles within an irregular station layout. However, stations composed of an irregular array of elements are under active consideration for the SKA [@BijLer11]. Because these stations will have several orders of magnitude more elements per station than these other telescopes, two-stage beamforming is an attractive option to reduce cost. Although antenna elements arranged into beamformed tiles can change key performance characteristics of the station, the effects of two-stage beamforming on the beam pattern and effective area of an irregular station have not been considered previously in the SKA context. This paper examines the implications of placing beamformed tiles within an irregular station grid layout, and compares it with a single-stage irregular station. The aim in doing so is to assess the magnitude of performance degradation which accompanies the potentially lower-cost two-stage layouts. Even though the intra-station layouts considered in this paper are not optimised for performance, they illustrate the distinguishing features of each layout and provide a reference point for future studies of SKA station layout and system design.
The paper is arranged as follows: Section \[sec:background\] sets out the intra-station geometry and the analysis method used. Results are presented in Section \[sec:results\] and discussed in Section \[sec:discussion\].
Theoretical Background {#sec:background}
======================
Aperture arrays (AAs) must be exquisitely well calibrated to meet the high imaging dynamic range requirements for the SKA [@dewdney2009]; one facet of calibration is correcting for a non-ideal synthesized beams, formed from the correlation of signals from AA stations. To do this, the instrumental response needs to be accurately characterized via an astronomical calibration procedure [@wijnholds2010calibration].
The difficulty in calibrating the station beam depends on the station size and the nature of its sidelobes [@WijBre11], and is influenced by the element configuration; a regular array produces strong, predictable sidelobes while an irregular array produces weaker, more diffuse sidelobes. An irregular array smears out in spatial extent the sidelobes that are present at frequencies where a regular array becomes sparse. These sidelobes, or grating lobes in sparse regular arrays, cause strong frequency and direction dependent variations in the main lobe gain [@CapWij06].
For this analysis, the effects of mutual coupling are ignored as they tend to average out for large irregular arrays (or stations) [@gonzalez2011non]. There will be non-trivial mutual coupling effects on a tile level but from a station perspective, these effects will not fundamentally change features such as sidelobe levels and effective area, which we analyse for the two-stage beamforming approach. Future work will incorporate the second order effects of mutual coupling and simulated (or measured) element patterns.
single-stage reg-irreg irreg-irreg
-------------------------------------------------------- -------------- -------------- --------------
3dB beamwidth of element 70$^{\circ}$ 70$^{\circ}$ 70$^{\circ}$
Maximum element footprint 1.2 m 1.2 m 1.2 m
Minimum separation between elements (centre to centre) 1.3 m 1.3 m 1.3 m
Number of elements per tile NA 16 16
Tile breadth (centre to centre) NA 3.9 m 6.3 m
Minimum separation between tiles (centre to centre) NA 5.5 m 6.8 m
Number of tiles in a station NA 699 701
Station diameter (centre to centre) 189 m 189 m 248 m
Total number of elements in station 11 200 11 184 11 216
Number of dual polarised signals paths 11 200 699 701
\
Three representative configurations for station layout are considered in this paper; (i) single-stage irregular array (single-irregular), (ii) regular gridded tile in an irregular array (regular-irregular) and (iii) irregular gridded tile in an irregular array (irregular-irregular). The irregular-irregular configuration uses the same randomised tile layout for all tiles in the station. Both two-stage stations use a single tile layout to enable easily replicated beamforming hardware with a fixed number of inputs. This is likely to reduce the beamformer hardware cost. We note that future work, based in part on the present study, could involve “non-tile” intra-station configurations, wherein signals from free-form antenna groups are beamformed.
Specifications for the representative configurations are given in Table I and the layouts for the two-stage stations are shown in Figure \[fig:gridded\_tile\] and Figure \[fig:gridded\_station\]. For comparison, the current description specifies 11 200 elements within a 180 m diameter station [@Dewbij10]. We note that emerging thinking about the configuration tends to now favour a somewhat larger number of smaller stations (e.g. [@mellema2012reionization]).
The number of elements, station diameter and element layout within the station are key parameters in determining station effective area and field of view. We choose to maintain a fixed number of elements in order to characterise the station effective area in the sparse regime of operation where the station sensitivity depends directly on the number of elements. The number of elements per station () is held approximately constant and a 1.3 m minimum inter-element spacing ($\lambda$/2 at 115 MHz) is enforced. For the two-stage layouts, each tile consists of 16 elements. Due to the randomisation, the irregular tile is necessarily larger than the regular tile; hence a larger station diameter is required for the irregular-irregular layout. An alternative approach is to control station diameter, and vary the number of elements per station. While this approach ensures a similar FoV between station layouts, station costs are not directly comparable because the number of signal chains vary. Furthermore, such an approach for the irregular-irregular layout would result in fewer elements, hence the station high-frequency sensitivity would suffer. Holding constant ensures that both two-stage beamforming stations have the same number of digitised signal chains.
The simulations are carried out in two stages, using Xarray[^2], a tool developed for computing radiation patterns together with effective area and other aperture array parameters. We generate the tile beam pattern using $\cos{\theta}^{3.47}$ as a single element pattern which approximates one of the moderately directive candidate antenna radiation pattern [@Eloy_2012_SKALA]. The station beam pattern uses this tile beam as the sub-array pattern for the two-stage beamforming. For single-stage beamforming we simply use the single element pattern in generating the station beam. The beams are calculated with an antenna radiation efficiency of 90$\%$, which is the minimum requirement for the station [@Dewbij10].
Results {#sec:results}
=======
Conceptually, the SKA-low stations replace the function of dishes in a radio telescope array; each station beam is an input to the correlator. As for dishes, two important metrics describing telescope performance are the station beam sidelobe levels and station effective area ().
Unlike dishes, the effective area of a station is strongly dependent on frequency. This occurs because the aperture is not fully-sampled (or not a dense, highly coupled array). Effective area also changes as a function of zenith angle due to the element pattern and geometric effects. Figure \[fig:aeff\] shows of the three station layouts as a function of frequency, for a zenith angle of 0.
The sidelobe levels are evident in the station beam pattern. The broadside station beam pattern, at 70 MHz and 300 MHz, for all three cases is shown in the u-v plane in Figure \[fig:station-beam\] and Figure \[fig:station-beam:300\]. The same beam at the v = 0 plane for the two frequencies is plotted in Figure \[fig:VO:station-beam\] and Figure \[fig:VO:station-beam:300\]. The u-v coordinates (equivalent to *l, m* coordinates) are given by u = $\sin{\theta}\cos{\phi}$ and v = $\sin{\theta}\sin{\phi}$ where $\theta$ and $\phi$ are zenith angle and azimuth angle respectively.
Statistics on inter-element spacing can provide further insight into the performance of stations of equal diameter (the single-stage irregular and regular–irregular layouts). The minimum separation between elements in the regular–irregular layout is defined by the regular tile layout: 1.3m for every element. The single-stage irregular station has a range of minimum inter-element spacings, defined by physical constraints and randomisation of the elements within these constraints. The 1.3m minimum spacing between elements is imposed by the element’s footprint, while the 189m diameter station boundary limits how far apart the elements can be spaced. Figure \[fig:1stageStats\] shows a frequency count of the minimum distance from each element to all other elements. The mean of these minimum inter-element spacings is 1.38m and the maximum is 2.21m. For comparison, the minimum inter-element spacing of 11200 elements within a 189m diameter single-stage regular gridded station (not shown) is approximately 1.6m.
Discussion {#sec:discussion}
==========
Station effective area
----------------------
The station effective area is an important metric for because, for a constant number of stations, telescope sensitivity is linearly proportional to . A related factor is whether the array is ‘dense’ or ‘sparse’ at the frequency in question. Both and the transition from dense and sparse varies with frequency and the inter-element spacing.
There is no single definition for when an array is dense or sparse. The broad definition used in this paper is that an array is dense when the aperture is fully sampled with $\lambda$/2 or closer element packing, and inter-element mutual coupling is significant. Effective area is then approximately equal to the physical (geometric) area of the station: $$\AEmaths\approx\frac{\pi}{4}\eta\Dstmaths^{2},\label{eq:AeffDense}$$ where $\eta$ is the antenna element radiation efficiency. When the array is sparse, the effective area of each isolated element contributes to the array effective area, such that $$\AEmaths=\NEmaths\frac{\lambda^{2}}{4\pi}\eta\mathcal{D},$$where $\mathcal{D}$ is the directivity of an isolated antenna element [@Bal05]. Wideband aperture arrays have a dense to sparse transition region, which occurs over an inter-element spacing of 0.5-1.5$\lambda$ for dipole-type antennas [@BraCap06], and typically greater than 2$\lambda$ for more directive antennas [@Rog08-DAM70].
As Figure \[fig:aeff\] shows, for all layouts is similar at the higher frequencies, where the elements are in the sparse regime and $\propto\lambda^{2}$. The dense-sparse transition is evident at lower frequencies. Here, effective area is no longer proportional to $\lambda^2$ as mutual coupling effects become active.
The general trend in Figure \[fig:aeff\] is that the regular–irregular station exhibits the strongest departure from $\lambda^2$ at lower frequencies, and the two-stage irregular station the weakest. This is to be expected, as the minimum inter-element spacing of the regular–irregular layout is 1.3 m for every element. Meanwhile, the single-stage irregular station has some elements of larger minimum spacing, but most are still less than 1.4 m (see Figure \[fig:1stageStats\]). For the larger diameter two-stage irregular station, the irregular tile is larger, hence more of the elements remain sparse at lower frequencies.
The regular–irregular and single-stage irregular stations show a significant divergence in effective area at lower frequencies, despite having the same 189 m diameter. The greatest percentage difference in effective area occurs at $\approx$150 MHz, where of the regular–irregular station is 74% of the single-stage irregular station, equating to a similar loss in telescope sensitivity. An inter-element spacing greater than 1.3 m for the regular tiles is a potential solution, but the tile geometry puts limitations on such an increase because it restricts the randomisation of the station layout. A conclusive analysis of the potential loss of effective area due to regular tiles requires a full mutual coupling analysis and optimisation of the layouts, which is beyond the scope of this paper.
Sidelobes
---------
As mentioned, over the full frequency range, the stations (for all layouts) operate as both dense and sparse array. The sidelobe characteristics, for the stations, will change as the array moves from dense to sparse region of operation. To observe these different characteristics, we analyse the sidelobes at two frequencies which are representative of the two regimes.
### Dense region (70 MHz)
The primary sidelobe (sidelobe closest to main lobe) is independent of the intra-station layout. Figure \[fig:station-beam\] and Figure \[fig:VO:station-beam\] show a primary sidelobe level of approximately -18 dB for all three stations; this level is consistent with that of a uniformly distributed circular aperture [@Mai95].
At the lower frequencies (eg. 70 MHz) the element separation is less than half wavelength. Thus, although the element (or tile) placement is random, the station aperture is ‘filled’. This ‘filled’ aperture generates a station beam which contains an annulus of primary and secondary sidelobes. These sidelobes will not be smeared or suppressed by techniques such as station configuration rotation, which has been proposed for sparse arrays [@CapWij06]. All three layouts exhibit these annuli. However, compared to the single-stage beamforming station, the sidelobes for the two-stage beamforming stations are spread over a wider area.
### Sparse region (300 MHz)
At higher frequencies (eg. 300 MHz), for all stations shown in Figures. \[fig:station-beam:300\]–\[fig:VO:station-beam:300\], the sidelobes are suppressed due to the random placement of elements. However, the sidelobes for the single-stage beamforming station are concentrated towards the horizon, away from the main lobe.
At the higher frequency of 300 MHz, the first-stage beamforming of tiles with relatively few elemental antenna inputs influences how the sidelobes are distributed as a function of scan angle and frequency. As Figure \[fig:station-beam:300\] shows, the irregular tiles smear out the strong, predictable sidelobes of the regular tiles. However, the general structure of the tile beam pattern is still visible in the station beam, especially for the regular tile.
The level of the secondary station sidelobes visible in Figure \[fig:VO:station-beam:300\] is affected by the first-stage beamforming. The maximum secondary sidelobe level increases from -35 dB for the single-stage station to -27 dB for the two-stage regular-irregular station. In comparison, the classical 1/N result for distant sidelobes of a random distribution of 11200 isotropic antenna elements is approximately -40 dB. The irregular station does not reach this level because the randomisation is restricted to a relatively small range of element spacings as shown in Figure \[fig:1stageStats\], resulting in the annulus of secondary sidelobes visible in Figure \[fig:station-beam:300\].
The use of predictable sidelobes in calibration is under investigation [@WijBre11; @BraCap06]; if it is useful, then the regular tile beam structure may assist the calibration while maintaining a low sidelobe level. If more randomised sidelobe structure is beneficial, or even acceptable, significant gain in effective area are possible (see Figure \[fig:aeff\]).
Station diameter and cost
-------------------------
Although results will depend on the exact station layout, these examples illustrate the impact of the two-stage beamforming and inter-element spacing on effective area. These trends are important, as the current SKA system description proposes stations of 180m diameter and 11200 elements [@Dewbij10]. The tight constraints on the minimum spacing due to the antenna footprint, and on the maximum spacing from the station diameter, limits the scope of layout optimization.
Given these constraints, then why not increase the diameter? An obvious feature of Figure \[fig:aeff\] is that the irregular–irregular station achieves a larger effective area at lower frequencies, for the same number of elements. Its larger diameter arises from the larger tile area required to achieve some degree of randomisation in the irregular tile, where the larger tiles simply do not fit in an irregular pattern within the 189 m diameter station.
The reasons not to place the same number of elements within a larger station diameter are two-fold: cost and calibratability. At higher frequencies, a larger station diameter reduces the station beam FoV with no commensurate increase in effective area. Therefore, fewer calibration sources are visible within the FoV, making the station calibration more difficult [@WijBre11].
The cost perspective is explored in @ColHal12. The larger physical area leads to increased costs related to on-site infrastructure. There are also increased signal processing costs to counteract the smaller station beam FoV. The number of these beams required to cover a given area of sky (the processed FoV) scales as $\Dstmaths^2$. The digital station beamformer processing costs are approximately proportional to the number of inputs and the number of beams formed (see Appendix \[app:beamforming-computational-cost\]). The cost of data transmission from the station to the correlator and the correlator processing cost itself are also approximately linearly proportional to the number of station beams [@ColHal12]. As an example, a 248 m diameter station requires 1.7 times more beams to achieve the same processed FoV as the 189 m diameter stations, resulting in roughly the same increase in signal processing costs.
Future work
-----------
Further to the illustrative examples in this paper, more complete optimisations will result in station layouts with slightly improved performance. The representative layouts in this paper maintain a constant number of elements per station, resulting in a constant when the station is sparse at the higher frequencies. An alternative approach is change the number of elements or station diameter, to ensure that all layouts instead maintain a relatively constant at the lower frequencies. This will have cost implications, and the preferred optimisation approach depends on the SKA station requirements.
Regardless of the optimisation approach, there is scope for further characterisation of station performance, incorporating mutual coupling between the elements and the overall telescope response. For example, rotating the intra-station configuration between stations of the telescope helps to reduce sidelobes in the correlated beam [@CapWij06]. Similarly, rotating tiles within the station would smear out the station sidelobes arising from the regular tile layout. Increased randomization of elements or tiles in the station layout can reduce sidelobes, but requires more station area, resulting in a smaller station FoV. Down-weighting the edges of the aperture distribution through a spatial taper of the elements or tiles would reduce sidelobes [@willey1962space], again at the cost of increased station area.
Conclusions
===========
We have quantified the first-order effect of two-stage beamforming on the station beam and effective area () of the . We used both regular and irregular tiles within an irregular station layout and compared these to a single-stage irregular station. At higher frequencies the station is same irrespective of the inner station layout. At lower frequencies, regular tiles can significantly reduce the station compared to the other two configurations. Across all frequencies, the primary sidelobes are not affected by the station configuration, being largely a function of the station diameter. However, the secondary sidelobes are dependent on the input beam (tile or single element). While randomised two-stage configurations have the potential for considerable cost savings, the increase in far sidelobe levels by 10 dB or so will need to be assessed in the context of the SKA-low weighted science case. For example, low sidelobe levels are required to characterise the ionosphere at low frequencies [@WijBre11] and enable precision imaging in e.g. the Epoch of Reionization domain. Conversely, time-domain studies at higher frequencies do not require precise secondary sidelobe control [@StaHes11]. In combination with science priority setting, further station configuration studies which include, e.g., mutual coupling will be useful to refine studies of two-stage and other hierarchical beamforming architectures.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Dr. Adrian Sutinjo for the discussions surrounding this topic, and the reviewer for helpful suggestions in finalising our manuscript.
The International Centre for Radio Astronomy Research is a Joint Venture between Curtin University and the University of Western Australia, funded by the State Government of Western Australia and the Joint Venture partners. A. Jiwani is a recipient of a Curtin Strategic International Research Scholarship. T. M. Colegate is a recipient of an Australian Postgraduate Award and a Curtin Research Scholarship.
[^1]: <http://www.skatelescope.org/news/dual-site-agreed-square-kilometre-array-telescope>
[^2]: <http://sites.google.com/site/xarraytool/>
|
---
author:
- 'A. Poudel'
- 'P. Heinämäki'
- 'P. Nurmi'
- 'P. Teerikorpi'
- 'E. Tempel'
- 'H. Lietzen'
- 'M. Einasto'
bibliography:
- 'references.bib'
date: 'Received ; accepted '
title: 'Multifrequency studies of galaxies and groups: I. Environmental effect on galaxy stellar mass and morphology.[^1]'
---
[To understand the role of the environment in galaxy formation, evolution, and present-day properties, it is essential to study the multifrequency behavior of different galaxy populations under various environmental conditions. ]{} [We study the stellar mass functions of different galaxy populations in groups as a function of their large-scale environments using multifrequency observations. ]{} [We cross-matched the SDSS DR10 group catalog with GAMA Data Release 2 and Wide-field Survey Explorer (WISE) data to construct a catalog of 1651 groups and 11436 galaxies containing photometric information in 15 different wavebands ranging from ultraviolet (0.152 $\mu$m) to mid-infrared (22 $\mu$m). We performed the spectral energy distribution (SED) fitting of galaxies using the MAGPHYS code and estimate the rest-frame luminosities and stellar masses. We used the $1/V_\mathrm{max}$ method to estimate the galaxy stellar mass and luminosity functions, and the luminosity density field of galaxies to define the large-scale environment of galaxies.]{} [The stellar mass functions of both central and satellite galaxies in groups are different in low- and high-density, large-scale environments. Satellite galaxies in high-density environments have a steeper low-mass end slope compared to low-density environments, independent of the galaxy morphology. Central galaxies in low-density environments have a steeper low-mass end slope, but the difference disappears for fixed galaxy morphology. The characteristic stellar mass of satellite galaxies is higher in high-density environments and the difference exists only for galaxies with elliptical morphologies.]{} [Galaxy formation in groups is more efficient in high-density, large-scale environments. Groups in high-density environments have higher abundances of satellite galaxies, irrespective of the satellite galaxy morphology. The elliptical satellite galaxies are generally more massive in high-density environments. The stellar masses of spiral satellite galaxies show no dependence on the large-scale environment.]{}
Introduction
============
Galaxies and their dark matter halos have grown hierarchically, forming larger systems, such as groups, clusters, filaments, and superclusters of galaxies, separated by enormous voids. In observations, this large-scale distribution of dark and baryonic matter manifests itself as a complex web of galaxies. This cosmic web includes a wide range of cosmic scales from the mega-parsec (Mpc) galaxy group scale to the supercluster and filament scales, several tens to hundreds of Mpc [@1978IAUS...79..241J; @1978ApJ...222..784G; @1979ApJ...230..648C; @1982Natur.300..407Z; @2014MNRAS.438.3465T]. As a result, galaxies reside in a variety of environments. A natural question is whether the interplay of various physical processes during the hierarchical growth of structures has also affected the properties of galaxies and which scale of environment plays a dominant role in galaxy evolution.
The mass of a galaxy is likely the main driver of its evolution . Until recently, theoretical models of the halo-galaxy relationship have usually assumed that galaxy populations in dark matter halos depend solely on the halo mass and do not depend on its large-scale environment and assembly history [@2010MNRAS.409..936P and references therein]. Recent results, however, suggest that the dependence on halo mass alone cannot explain all galaxy properties. In theoretical studies, older dark matter halos are found to cluster more strongly than more recently formed halos of the same mass [@2005MNRAS.363L..66G]. This effect, which is known as assembly bias, suggests that the properties of a galaxy also depend on its formation history and the environment, where it is embedded.
Observational studies have also hinted at a connection between galaxy and group properties and the large-scale environments wherein they reside: Equal mass groups in superclusters have higher correlation function amplitudes [@2012MNRAS.426..708Y]; late-type brightest group galaxies have higher luminosities and stellar masses, redder colors, lower star formation activity, and longer star formation timescales when embedded in superstructures [@2015MNRAS.448.1483L]; the fraction of red galaxies is higher in high-density environments (superclusters) than in low-density environments ; at a fixed local density, galaxy colors change toward void walls [@2008MNRAS.390L...9C]; galaxies within groups with the same richness and mass are redder/older in high-density regions . These results support environmental effects but are not conclusive about the dependence between large-scale environment and galaxy properties.
The stellar mass is a basic galaxy parameter that correlates strongly with almost all other galaxy properties [@2004MNRAS.353..713K] and also with the mass of its dark matter halo [@2010ApJ...710..903M]. Thus the stellar mass function (SMF) of galaxies is an observational quantity often used to constrain the models of galaxy formation and evolution. Recently, @2015MNRAS.451.3249A reported that the SMF of galaxies also seems to vary in different large-scale structures. There is, however, no common consensus about its global shape and also its behavior in different large-scale environments is still an open question. This is largely associated with the nontrivial definition of large-scale environments and uncertainties in stellar mass estimations, which depend on the models and wavelength coverage used. The multifrequency study of galaxies from ultraviolet (UV) to infrared (IR) wavelengths allows us to constrain dust in galaxies and construct a more detailed and accurate picture of physical properties, including stellar masses, of a wide range of galaxy types. Recently, several large-scale sky surveys have been carried out at various wavelengths, e.g., in the far-ultraviolet (FUV) and near-ultraviolet (NUV) by the Galaxy Evolution Explorer (GALEX), in the optical by the Sloan Digital Sky Survey (SDSS), in the near-infrared (NIR) by the UKIRT Infrared Deep Sky Survey (UKIDSS), and in NIR and mid-infrared (MIR) by NASA’s Wide-field Infrared Survey Explorer (WISE). By combining the data from these different surveys, one can construct broadband spectral energy distribution (SED) of galaxies with appropriate galaxy models. The best state-of-the-art model available for this purpose is the model developed by @2008MNRAS.388.1595D. It is based on the stellar population synthesis model of @2003MNRAS.344.1000B and the two-component dust model of @2000ApJ...539..718C. By fitting the observed SEDs with models, we can estimate rest-frame luminosities and other physical properties related to stellar and dust contents in galaxies. This method was successfully applied to nearby star-forming galaxies from the Spitzer Infrared Nearby Galaxy Survey (SINGS), 3258 low-redshift galaxies from the SDSS Data Release 6 [@2010MNRAS.403.1894D], 250 $\mu$m selected galaxies from Herschel-ATLAS Survey [@2012MNRAS.427..703S], and the Andromeda galaxy . It was also tested with simulated galaxies from hydrodynamical simulations [@2015MNRAS.446.1512H].
Physical mechanisms like strangulation [@1980ApJ...237..692L], ram-pressure stripping [@1972ApJ...176....1G], and harassment [@1996Natur.379..613M], which are believed to transform galaxy properties on group and cluster scales, mainly operate on satellite galaxies. On much larger scales, the number of satellite galaxies in groups are found to vary depending on whether they are present within or outside filaments [@2015ApJ...800..112G]. Any interplay between the inner group and large-scale environments may influence the efficiency of typical physical mechanisms occurring in groups and result in differences observed in properties of satellite galaxies in different large-scale environments. The interdependence of environment and central and satellite galaxies on cosmological scales still remains a less explored topic in galaxy evolution. We take a novel step toward understanding the large-scale environmental effect on central and satellite galaxy evolution using multifrequency observations and galaxy SED modeling. In this paper, we study the differences of the galaxy SMFs for different combinations of the large-scale structure, central and satellite galaxies, and galaxy morphology. For this purpose, we first expand the optical SDSS DR10 galaxy and group catalog to IR and UV wavelengths with data from different large-scale sky surveys. Using a publicly available MAGPHYS SED fitting code, we determine galaxy rest-frame luminosities and stellar masses in groups inferred from galaxy data at wavelength range 0.15 $\mu$m to 12.33 $\mu$m. Then, we divide the galaxy $r$-band luminosity density field into low- and high-density, large-scale environments and study SMFs of central and satellite galaxies in these two distinct environments. Our data analysis also gives the luminosity functions of galaxies for 15 wavebands from UV to IR. Plots of these are shown in appendix A.
Data
====
Ultraviolet, optical, and near-infrared data
--------------------------------------------
Photometric data from UV to NIR wavelengths come from the GAMA Data Release 2 [@2015MNRAS.452.2087L]. The UV photometric data come from the GalexCoGPhot catalog, which provides GALEX $NUV$ (0.152 $\mu$m) and $FUV$ (0.231 $\mu$m) curve-of-growth photometry at optical positions for all GAMA DR2 objects. Optical and NIR data come from SersicCatAll catalog, which provides photometry as a result of fitting a single-Sersic (1-component only) profile to every GAMA DR2 object in each of the bands $u$ (0.3562 $\mu$m), $g$ (0.4719 $\mu$m), $r$ (0.6185 $\mu$m), $i$ (0.75 $\mu$m), $z$ (0.8961 $\mu$m), $Y$ (1.0319 $\mu$m), $J$ (1.2511 $\mu$m), $H$ (1.6383 $\mu$m), and $K$ (2.2085 $\mu$m). This catalog is constructed using SIGMA v0.9-0 (Structural Investigation of Galaxies via Model Analysis) tool on SDSS and UKIDSS LAS imaging data [@2012MNRAS.421.1007K]. All magnitudes are expressed in AB mags.
Mid-infared data
----------------
The MIR data are taken from the latest data release from the WISE survey [@2010AJ....140.1868W], which has mapped the whole sky with an angular resolution of 6.1$\arcsec$, 6.4$\arcsec$, 6.5$\arcsec$, and 12.0$\arcsec$ in four wavebands, $W1$, $W2$, $W3$, and $W4$ with effective wavelengths 3.4, 4.6, 12, and 22 $\mu$m, respectively. In the unconfused regions on the ecliptic, WISE has achieved 5$\sigma$ point source sensitivities better than 0.08, 0.11, 1 and 6 mJy in the four bands and a sensitivity better than 100 times than that of IRAS in the 12 $\mu$m band.
Optical group catalog
---------------------
The optical group catalog used in this work is constructed from SDSS DR10 galaxies using a friends-of-friends algorithm and a linking length that varies with redshift as explained in detail in . The catalog consists of both isolated (not a member of any group) and grouped galaxies in all of the 588193 galaxies and 82458 groups. The effective area of coverage of the catalog is around 7221 square degrees and extends up to redshift 0.2. The catalog is flux limited with magnitude limit 17.77 mag in the SDSS $r$ band. The redshifts of galaxies are taken from the SDSS spectroscopic sample, the Two-degree Field Galaxy Redshift Survey [2DFRS; @2001MNRAS.328.1039C], the Two Micron All Sky Survey [2MASS; @2003AJ....125..525J; @2006AJ....131.1163S] Redshift Survey [2MRS; @2012ApJS..199...26H], and the Third Reference Catalogue of Bright Galaxies [RC3; @1994AJ....108.2128C]. The morphological classification of galaxies is based on the fraction of luminosity contributed by the de Vaucouleurs profile ($f_{deV}$), the exponential profile axis ratio ($q_{exp}$), $u-r$ and $g-r$ colors (see for details).
Construction of the far-UV to mid-IR catalog
--------------------------------------------
The galaxies in the optical group catalog were first identified in GAMA, which resulted in 11056 galaxies and 1651 groups. In these SDSS groups, 380 galaxies do not have GAMA counterparts and we use SDSS DR10 photometric information for these galaxies. Combining these galaxies, the matched catalog resulted in 11436 galaxies and 1651 groups. GAMA DR2 already provides cross identifications between UV, optical, and NIR data. To combine MIR data with other wavebands, we searched for all the WISE objects lying within an angular distance of 3$\arcsec$ from optical positions of galaxies in the matched catalog. This angular distance was chosen after many cross-matching trials so as to minimize the number of multiple associations and maximize the number of unique associations. Around 1.32$\%$ and 97.56$\%$ of the galaxies were found to have multiple and unique matches with WISE objects, respectively. All WISE objects that are flagged as contaminated by artifacts in one or more WISE bands were deleted. This is about 3$\%$ of all matched WISE objects. Multiple associations were removed in the galaxy SED fitting process by selecting the match that fits better. We restricted our analysis in the redshift range 0.01–0.2. The final galaxy catalog consisted of 1635 groups and 11330 galaxies (about 2$\%$ of the SDSS DR10 galaxy catalog) containing photometric information in GALEX, SDSS, UKIDSS, and WISE wavelengths with the group membership criteria remaining the same as in the optical group catalog. The number of detected galaxies in different wavelengths are shown in column 8 of Table \[sample selection\]. The effective area of coverage of the catalog is about 144 square degrees. It includes all the three equatorial survey regions of GAMA I called G09, G12, and G15.
Quantifying galaxy environments
-------------------------------
The final step in our data preparation involved quantification of large-scale environments in the region of our catalog. Segregating different large-scale environments is not a trivial task, since filaments and superclusters are mostly unvirialized structures and their density is only slightly larger than the cosmic mean density . As is well-known, most of the galaxies are small and have a low luminosity. Hence, a simple number density analysis of galaxies, which counts all the galaxies with the same weight, cannot provide a precise description of the cosmic web. The luminosity density field of galaxies is a powerful and widely used technique to define the large-scale density distribution. In this method, the galaxy luminosities, corrected by observational biases and selection effects, are smoothed with an appropriate kernel, the width of which determines the characteristic spatial scale. This approach leads to the total three-dimensional luminosity density field of the survey, where different density levels represent different characteristic structures: clusters, filaments, voids, and superclusters. Following , we use the normalized optical galaxy luminosity density field with a smoothing scale of 8 $h^{-1}$ Mpc for characterizing the large-scale environment of our galaxy and group sample. We use the optical galaxy density value of 1.5 in units of cosmic mean luminosity density to divide our galaxy sample into low- and high-density, large-scale environments containing 31$\%$ and 69$\%$ of all galaxies in the sample, respectively. This value was also used in to classify galaxies into voids and other large-scale environments.
Methods
=======
Fitting spectral energy distributions
-------------------------------------
The UV to IR part of the SED contains a lot of information about stars and dust in galaxies. However, to extract such information, models are necessary to link physical properties of a galaxy with its observed SED. The best state-of-the-art model available that is consistent with observations and simulations, is that by @2008MNRAS.388.1595D. This model is based on the principle that the light emitted from stellar populations in a galaxy is partly absorbed by dust and re-emitted at longer wavelengths. It computes the spectral evolution of galaxies using the population synthesis model of @2003MNRAS.344.1000B , whereas dust attenuation is computed using the two-component model of @2000ApJ...539..718C. We use a publicly available MAGPHYS SED fitting tool to fit the observed galaxy SEDs to a library of model SEDs extending from UV to IR wavelengths with known physical parameters. The fitting produces the best-fit, rest-frame magnitudes and various other physical parameters related to galaxies and also provides the range of their likelihood values using a Bayesian approach.
There are a few concerns about the model used in this work: the effects of viewing angle, AGN contamination, and dust grain composition. The recent work by [@2015MNRAS.446.1512H], in which they performed MAGPHYS SED fitting on simulated galaxies from hydrodynamical simulation, has shown that MAGPHYS recovers most of the physical parameters well regardless of the viewing angle. They also found that even when the contribution of AGNs to the UV–mm luminosity is around 25 percent, the fits are acceptable and the parameters recovered are accurate. The largest discrepancy appears when assuming a different composition of dust. Both Milky Way type dust and Large Magellanic Cloud type dust produce acceptable fits, but the discrepancy increases for Small Magellanic Cloud type dust composition. Our sample consists of bright spiral galaxies and hence the Milky Way type dust is a reasonable assumption.
Carefully calibrated photometric data for different wavelengths and from different surveys are needed for reliable estimates of SEDs. We use profile-fitted fluxes that give a measure of the total stellar light from a galaxy at all wavelengths as input to the code. In WISE bands, for the sources with signal-to-noise (S/N) ratios of less than two, the given magnitudes are 2 $\sigma$ brightness upper limits in magnitude units. Such sources account for 1.1$\%$, 1.1$\%$, 28.1$\%,$ and 68.0$\%$ of all galaxies in the matched catalog in $W1$, $W2$, $W3,$ and $W4$ bands, respectively. In WISE, photometric measurements of both stars and galaxies are based on profile fits using the point spread function as the source model. Although this method minimizes the effects of source blending, this may lead to underestimation of true fluxes for extended sources. @2015ApJS..219....8C found that the flux underestimation depends on the effective radius ($R_e$) of the source and may be estimated as $$\triangle m = 0.10 + 0.46 \log(R_e) + 0.47 \log(R_e)^2 + 0.08 \log(R_e)^3
.$$ The correction is applied to galaxies with effective radii of greater than 0.5 arcseconds. Following @2015ApJS..219....8C, we use the effective radii obtained after dividing SDSS $r$-band effective radii by a factor of 1.5 to convert galaxy size from optical to infrared. These corrections are added to flux uncertainties in the respective WISE wavebands.
Reducing the color bias in the sample
-------------------------------------
Our sample is $r$-band selected ($m_r$ < 17.77 mag), so that the sample galaxies in the $r$ band do not represent a volume limited (VL) sample, but are affected by a certain amount of Malmquist-type bias. In the other wavelength bands, not originally used for selection, the biases generally differ [cf. the “Gould effect”; @1993ApJ...412L..55G]. Therefore, the form of the average SED of the observed sample differs from the average SED of a representative VL sample. This becomes visible as color biases in the observed sample, as shown in Fig. \[Color bias\]. As a result of this bias, galaxies with specific color are not detected in the sample as seen from the gap at the lower right part of the color magnitude plots. $FUV$, $NUV,$ and $W3$ bands are very strongly color biased. The sample of galaxies fainter than the bright limit (vertical solid line) have negligible color biases but this limit is not used as it reduces the sample size significantly for any statistical analysis. In order to reduce color bias in the sample, we adopt the method used by @2012MNRAS.427.3244D and calculate the mean color of the bands with respect to the $r$ band; we make a magnitude cut as shown in Fig. \[Color bias\] to remove all galaxies fainter than the faint limits (vertical dashed line) in each of the wavebands separately. Although this method does not completely remove the color bias, it is a good compromise between the bias and sample size. This procedure is applied after estimating rest-frame magnitudes from the SED fitting as was explained in the previous section. The number of galaxies in different wavebands after such color cuts is tabulated in column 9 of Table \[sample selection\].
Fit results
===========
Figure \[magnitude difference\] shows the difference between the observed and model fitted magnitudes at various wavebands used. The median difference is close to the zero line in all wavebands except $W4$ band. In the $W4$ band, the median difference is about +1.5 mag. The upper limit in source fluxes in WISE bands may contribute to the large offset seen in the $W4$ band. The real signal in these bands may be much weaker than these upper limits, which result in offset toward positive values. Also, $W4$ filter transmission curve has been revised, but MAGPHYS models are not yet updated in the $W4$ band. Similar offset in $W4$ band is also seen in @2016MNRAS.455.3911D.
The fitting also provides chi square ($\chi^2$) goodness of fit values. The $\chi^2$ values are reasonably small with few outliers. To test whether neglecting AGN heating in the model results in bad fits, we divided galaxies in the catalog into star-forming galaxies and AGNs based on classifications provided in GAMA, which uses the @2001ApJ...556..121K classification scheme. No significant dependence of the chi square ($\chi^2$) goodness of fit values on galaxy types was found and the distribution of $\chi^2$ values (Fig. \[chi square values\]) are very similar for both the star-forming galaxies and AGNs. Based on the analysis of the model SEDs of MAGPHYS, @2012MNRAS.427..703S derived a relation (equation B1) between the number of bands used during SED fitting and the number of degrees of freedom. Our 15 band photometry corresponds to 9 degrees of freedom, which means that above a chi square value 20 there is a probability of less than 1 percent that the observations are consistent with the models. We set this limit when making any statistical analysis with the sample. Within this limit, MAGPHYS was found to reproduce the galaxy parameters of simulated galaxies with good accuracy [@2015MNRAS.446.1512H]. As the WISE catalog contains both stars and galaxies, such a $\chi^2$ limit also reduces the stellar contamination present in the sample when matching galaxies with WISE sources.
![Difference between observed and MAGPHYS fitted magnitudes at different wavelengths. The white solid line shows the median values. []{data-label="magnitude difference"}](fit_diff.pdf){width="\hsize"}
![Chi square ($\chi^2$) goodness of fit values obtained from SED fitting. Different line types indicate GAMA classifications. []{data-label="chi square values"}](chi_square.pdf){width="\hsize"}
-------- ------------ ----------- -------------- ------------ ------------- ----------------- ------- ---------- ------------------- --
Filter Wavelength M$_\odot$ Bright limit Mean color Faint limit M$_X$ limit N(z) N(color) N($\chi^2$+color)
($X$) ($\mu$m) (AB mag) (AB mag) ($X-r$) (AB mag) (AB mag)
$FUV$ 0.152 16.02 19.0 5.26 23.0 $-11.5$,$-19.5$ 7678 4931 3985
$NUV$ 0.231 10.18 19.0 4.17 22.0 $-13.4$,$-20.0$ 7948 5355 4260
$u$ 0.356 6.38 19.0 2.22 20.0 $-13.4$,$-20.5$ 10581 9008 7362
$g$ 0.472 5.15 18.0 0.77 18.5 $-14.4$,$-22.0$ 11328 10091 8317
$r$ 0.618 4.71 18.0 0.00 17.8 $-14.4$,$-23.0$ 11330 11043 9228
$i$ 0.750 4.56 17.2 -0.34 17.5 $-14.9$,$-23.0$ 11316 10981 9179
$z$ 0.896 4.54 17.0 -0.61 17.2 $-15.4$,$-23.5$ 11115 10465 8737
$Y$ 1.032 4.52 16.5 -0.78 17.0 $-15.9$,$-24.0$ 10591 9739 8089
$J$ 1.251 4.57 16.5 -0.92 16.9 $-15.9$,$-24.0$ 10569 9601 7985
$H$ 1.638 4.71 16.0 -1.05 16.9 $-16.4$,$-24.0$ 10751 9950 8289
$K$ 2.208 5.19 16.0 -1.01 16.9 $-16.4$,$-24.0$ 10717 9655 8055
$W1$ 3.379 5.94 16.5 -0.38 17.5 $-15.9$,$-23.5$ 11230 10209 8572
$W2$ 4.629 6.61 17.0 0.06 17.8 $-15.4$,$-23.0$ 11230 9675 8098
$W3$ 12.33 8.40 15.0 0.53 18.0 $-17.4$,$-24.5$ 11230 8014 6706
$W4$ 22.00 9.10 15.0 3.92 18.0 $-17.4$,$-24.5$ 11230 10472 8728
-------- ------------ ----------- -------------- ------------ ------------- ----------------- ------- ---------- ------------------- --
Physical parameters from SED fitting
------------------------------------
![Histograms of differences between our stellar mass estimates (indexed as MAGPHYS) with SDSS (solid black) and GAMA (dashed blue) values. Masses are expressed in their log values. []{data-label="stellar mass"}](stellar_mass.pdf){width="\hsize"}
Stellar masses
--------------
The fitting gives many physical parameters characterizing galaxies such as age, metallicity, stellar mass, and star formation rate. It also provides the likelihood distribution of stellar masses and we take median values as the best estimates. The stellar masses obtained agree very well with SDSS and GAMA [@2011MNRAS.418.1587T] values in general (Fig. \[stellar mass\]). The median and standard deviation of the differences between our and SDSS values are found to be $-0.098$ and $0.886,$ respectively. For GAMA, these values are found to be $-0.029$ and $0.477,$ respectively. For estimating stellar mass in galaxies, SDSS and GAMA use SED fitting to optical bands. The observed similarities show that using only optical band information can also provide good estimates for stellar masses in galaxies.
Rest-frame absolute magnitudes
------------------------------
The SED fitting also provides estimates of the expected rest-frame fluxes in different bands by simultaneously fitting the observed SEDs with models. It also provides flux estimates of galaxies in bands where detection has not been possible because of observational limitations. However, we only use rest-frame fluxes in bands which have detections and restrict our analysis to galaxies with $\chi^2 < 20$. Such a restriction increases the reliability of our results. We convert the rest-frame (k-corrected) fluxes in different bands obtained from SED fitting into absolute magnitudes in AB units for our analysis. The absolute magnitudes are not dust de-attenuated.
Statistical properties of the sample
====================================
In order to check the reliability of the sample and its rest-frame luminosities obtained from SED fitting, we construct the galaxy stellar mass and luminosity functions in different bands. We compare our galaxy SMF with the results from @2008MNRAS.388..945B and @2014MNRAS.444.1647K. We also compare the luminosity functions in different bands with SDSS [@2003ApJ...592..819B] and GAMA [@2012MNRAS.427.3244D] results. We use the Schechter fit parameters in the Table. 4 in @2012MNRAS.427.3244D for luminosity function comparisons.
Galaxy luminosity function
--------------------------
The galaxy LFs at all bands are estimated using the $1/V_\mathrm{max}$ statistical method [@1976ApJ...203..297S] along with some modifications made by @2012MNRAS.427.3244D. This method minimizes the effect of the color bias in the sample by adjusting the $1/V_\mathrm{max}$ volume according to each object’s color. However, it is less effective at the low luminosity end, where a galaxy with a specific color may not be detected in our sample. At a waveband $X$ ($X$ = $FUV$, $NUV$, $u$, $g$, $i$, $z$, $Y$, $J$, $H$, $K$, $W1$, $W2$, $W3$ or $W4$), the maximum volume available for each galaxy in the sample is estimated using the magnitude limit as either $17.77 - (mag_r - mag_X)$ or the faint limit in that band, depending on which is brighter. The luminosity distribution is then obtained by $$\mathrm{\phi (\mathit{M})} = \frac{c}{\eta} \mathrm{\sum_i \frac{I_{\mathit{(M,M+dM)}} (\mathit{M_i})}{\mathit{V_\mathrm{max,i}}}},$$ where $c$ is the ratio of the number of galaxies before $\chi^2$ cut to the number of galaxies after $\chi^2$ cut. $V_\mathrm{max,i}$ is the maximum comoving volume over which the $i^{th}$ galaxy can be observed, I$_A(x)$ is the indicator function that selects the galaxies belonging to a particular absolute magnitude bin (taken here as 0.5 AB mag) and the sum runs over all galaxies in the sample. $\eta$ is the cosmic variance correction factor [$\eta$ = 0.85, @2011MNRAS.413..971D]. For each waveband, the absolute magnitude limits fainter than which the method fails to correct for the color biases are calculated using the bright limits indicated in Table \[sample selection\]. The faint absolute magnitude limit values at different wavebands are shown in column 7 of Table \[sample selection\]. The absolute magnitude bins that contain fewer than ten galaxies toward the brightest part of the magnitude distribution do not contribute significantly to the overall luminosity density. The bright absolute magnitude limit values at different wavebands are shown in column 7 of Table \[sample selection\].
Figure \[lumf\] shows the obtained luminosity functions as well as a comparison with the GAMA and @2003ApJ...592..819B results for redshift 0.1. Within the selection boundaries, the optical galaxy LFs agree well with @2003ApJ...592..819B results. Compared to our estimates, the GAMA LFs are shifted toward the brighter end at optical and NIR bands except the K band. At FUV and NUV wavelengths, the GAMA luminosity functions have higher amplitudes than our results. These differences are mainly due to different magnitude limits, redshift ranges, photometry, and sampling. The GAMA sample is about 1.6 magnitude deeper than @2003ApJ...592..819B and our samples. The GAMA results use r-band matched aperture photometry, @2003ApJ...592..819B use Petrosian magnitudes, whereas we use Sérsic profile fitted magnitudes to construct the LFs at optical and NIR wavelengths. Sérsic magnitudes are brighter than Petrosian and aperture magnitudes for galaxies with higher Sérsic indices. Sérsic LFs were found to be shifted toward the bright relative to the LFs based on Petrosian and aperture magnitudes in the r band by @2011MNRAS.412..765H. This shows that different magnitude types used affect the derived LFs noticably.
Galaxy stellar mass function
----------------------------
The galaxies in the sample are divided into logarithmic stellar mass bins to estimate the galaxy SMF. For each bin, the galaxy SMF is given by $$\mathrm{\phi(\log \mathit{M_{st}}) d(\log \mathit{M_{st}})} = \frac {c}{\triangle \log \it M_{st}} \mathrm{\sum_\mathit{i} \frac{I_{\it(M_{st},M_{st}+dM_{st})} (\log \it M_{{st},i})}{\mathit{V_{max,i}}}}
.$$\
Here, $c$ is the ratio of the number of galaxies before $\chi^2$ cut to the number of galaxies after $\chi^2$ cut, and $V_{max,i}$ is the maximum comoving volume over which the $i^{th}$ galaxy can be observed and is obtained in the same way as explained in section 5.1. I$_A(x)$ is the indicator function that selects the galaxies belonging to a particular stellar mass bin (taken as $10^{0.5}$ M$_\odot$) and the sum runs over all galaxies in the sample. Recent studies have shown that the galaxy SMF has a distinctive bump around $10^{10.6}$ M$_{\odot}$ and can be best approximated by a double Schechter form with a combined knee [see @2008MNRAS.388..945B; @2010ApJ...721..193P; @2012MNRAS.421..621B; @2014MNRAS.444.1647K]. It can be expressed as $$\begin{gathered}
\mathrm{ \phi (\log \mathit{M_{st}}) d(\log\mathit{M_{st}}) = \ln(10) exp(-10^{\log\mathit{(M_{st}/M_{st}^\star)}}) } \\
\mathrm{[\phi_1^\star 10^{\mathit{\log(M_{st}/M_{st}^\star)(\alpha_1 + 1)}} + \phi_2^\star 10^{\log\mathit{(M_{st}/M_{st}^\star)(\alpha_2 + 1)}}] d(\log\mathit{M_{st})}}, \end{gathered}$$ where $M_{st}^\star$ is the characteristic mass corresponding to the position of the distinctive “knee” in the mass function. The terms $\alpha_1$ and $\alpha_2$ are the slope parameters and $\phi_1^{\star}$ and $\phi_2^{\star}$ are normalization constants. The double Schechter function accurately models the bump observed in the galaxy SMF around $M_{st}^\star$, with one Schechter function dominating at stellar masses greater than $M_{st}^\star$ and the second one dominating at lower masses. The resulting galaxy SMF (red circular points) and the double Schechter fit including errors (red shaded region) are shown in the stellar mass range $10^{8.0}$–$10^{12.0}$ in Fig. \[stellar mass function: All galaxies\]. The curve fitting of the double Schechter function is performed using nonlinear least squares fitting technique with five free parameters. The fit limits and errors in number density of galaxies in each stellar mass bin are estimated using the bootstrap resampling technique. The fit parameters with errors are shown in Table \[double schechter stellar mass function\]. The best-fit paramaters highly deviate from the results of @2008MNRAS.388..945B (B08 hereafter) and @2014MNRAS.444.1647K (K14 hereafter). Considering the errors in fit parameters, $M_{st}^\star$ has higher values than B08 and K14. Both, $\phi_1^{\star}$ and $\alpha_1$ have lower values than B08 but lie within the errors when compared to K14 estimates. The remaining two parameters ($\phi_2^{\star}$ and $\alpha_2$) agree within the errors with both B08 and K14 values.
![Stellar mass function of galaxies obtained from SED fitting and comparison with @2008MNRAS.388..945B (black dotted line) and @2014MNRAS.444.1647K (blue dashed line) results. The red solid line represents the best double Schechter function fit. The light red lines, which form a shaded region, represent the individual double Schechter function fits to the galaxy stellar mass functions obtained from all bootstrap samples. The errors in density estimates at each stellar mass bins are calculated using bootstrap resampling technique. []{data-label="stellar mass function: All galaxies"}](smf_fit.pdf){width="\hsize"}
Unlike luminosity, which is a directly observable quantity, the stellar mass is commonly estimated from the multifrequency data by fitting the observed SEDs with a library of model SEDs with known physical properties. A number of different models exist regarding the initial mass function (IMF), star formation history (SFH), stellar populations, and dust that are used to build the model SEDs. Many studies have tried to quantify the effect of these assumptions on stellar mass estimates . These studies commonly agree that stellar masses can be reliably constrained but uncertainties up to 0.5 dex may exist depending on different models used. @2009ApJ...701.1765M quantified how the uncertainties associated with the assumptions made in the SED fitting method may affect the galaxy SMF. They find that the shape of the galaxy SMF at the low-mass end may change depending on the assumed metallicity and dust model. SDSS and GAMA used only optical wavelengths to estimate stellar masses from the SED fitting. Here, we attempt to include UV and MIR wavelengths. We do not find significant differences in our estimates although our values have small systematic offsets compared to SDSS and GAMA estimates. These differences can be attributed to a number of different factors such as inclusion of the UV and IR wavelengths, different assumptions in the SED fitting, and different photometry. These differences also affect the shape of the global galaxy SMF as seen in our results.
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Galaxy stellar mass functions in different environments
========================================================
All galaxies {#ssec:num1}
------------
To study the large-scale environmental dependence on galaxy stellar mass, we divide our galaxy sample into low-density (Den8 < 1.5) and high-density (Den8 > 1.5) environments according to their spatial location in the luminosity density field. We also construct the SMFs of galaxies in these two distinct environments. High-density environments have a higher number density of galaxies than low-density environments. To take this effect into account and to facilitate direct comparisons, we draw equal numbers of galaxies randomly from each of these samples when constructing the SMFs. For each sample, we calculate the mean densities and error in densities in different stellar mass bins by combining all the SMFs obtained from 1000 different bootstrap subsamples and fit a single Schechter function to these values to get the best fit. For each sample, we also estimate all the single Schechter function fits to galaxy SMFs obtained from all the bootstrap subsamples. Figure \[stellar mass functions: High and low density environments\](a) visualizes the results and Table \[Single schechter stellar mass functions: All galaxies\] shows the best-fit Schechter parameters with errors for galaxies in low- and high-density environments. Compared to low-density environments, the best Schechter function fit is shifted toward higher stellar masses for galaxies in high-density environments. In Fig. \[Error ellipses\](a), we show the 1$\sigma$ (dark colored region) and 2$\sigma$ (light colored region) error contours of the corresponding Schechter fit parameters, characteristic stellar mass ($M_{st}^{\star}$) and low-mass slope ($\alpha$) at 95$\%$ confidence level. The SMFs of galaxies are different in high- and low-density environments. At 2$\sigma$, high-density environments have higher $M_{st}^{\star}$ than low-density environments, but $\alpha$ does not show any clear difference. For comparison, we also show the results (points with errorbars) from @2015MNRAS.451.3249A in three different environments, namely voids, tendrils, and filaments.These environments are defined using the filament finder algorithm based on a minimal spanning tree method [@2014MNRAS.438..177A]. As the definition of environment in our method is completely different, our results can only be compared qualitatively with those in @2015MNRAS.451.3249A. Qualitatively, our results agree with @2015MNRAS.451.3249A; $M_{st}^{\star}$ increases toward high-density environments.
All centrals and satellites
---------------------------
We further divide each of our high- and low-density galaxy samples into centrals and satellites to study the environmental dependence on the stellar content of central and satellite galaxies in groups in different large-scale environments. Central galaxies are those with the highest stellar mass in groups and the remaining members are classified as satellites. Our central galaxy samples also include isolated galaxies. We then construct the SMFs of centrals and satellites in both high- and low-density, large-scale environments (Fig. \[stellar mass functions: High and low density environments\]b). We use the same method as explained in section 6.1 to construct the galaxy SMFs. The corresponding single Schechter function fit parameters with errors are shown in Table \[Single schechter stellar mass functions: All galaxies\]. We also draw the 1$\sigma$ (dark colored region) and 2$\sigma$ (light colored region) error ellipses of $M_{st}^{\star}$ and $\alpha$ at 95$\%$ confidence level (Fig. \[Error ellipses\]b). For central galaxies, the error ellipses are well separated from each other, suggesting that the SMFs of central galaxies are different in high- and low-density environments. High-density environments have higher $\alpha$ value than low-density environments, but the $M_{st}^{\star}$ value shows no clear difference between different environments. Similar to central galaxies, the SMFs of satellite galaxies are different in high- and low-density environments. High-density environments have steeper $\alpha$ value than low-density environments. The $M_{st}^{\star}$ value is clearly higher in high-density environments at 1$\sigma$ only. The SMFs of central and satellite galaxies may have variations depending on the choice of the central galaxy definition and the group catalogs used (see appendix C for details).
Centrals and satellites with fixed morphology
---------------------------------------------
We divide central and satellite galaxy samples in both high- and low-density environments into spiral and elliptical populations to study the environmental dependence on the stellar content of central and satellite galaxies with fixed morphology in different large-scale environments. We further construct the SMFs of central and satellite galaxies with different morphologies across various large-scale environments using the method explained in section 6.1.
### Spiral morphology
Figure \[stellar mass functions: High and low density environments\](c) shows the SMFs of central and satellite galaxies with spiral morphologies in low- and high-density, large-scale environments as indicated. The single Schechter fit parameters are shown in Table \[Single schechter stellar mass functions: Spiral galaxies\]. The 2$\sigma$ error ellipses of $M_{st}^{\star}$ and $\alpha$ for central galaxies with spiral morphologies are well separated (Fig. \[Error ellipses\]c). However, there is no clear difference in $M_{st}^{\star}$ and $\alpha$ values in high- and low-density environments within the error region considered.
The 2$\sigma$ error ellipses of $M_{st}^{\star}$ and $\alpha$ for satellite galaxies with spiral morphologies are overlapping, but the 1$\sigma$ error ellipses are well separated (Fig. \[Error ellipses\]c). At 1$\sigma$, the $\alpha$ value is steeper in high-density environments than that in low-density environments. However, the $M_{st}^{\star}$ values for satellite galaxies are similar in both environments.
### Elliptical morphology
Figure \[stellar mass functions: High and low density environments\](d) shows the SMFs of central and satellite galaxies with elliptical morphologies in low- and high-density, large-scale environments as indicated. The single Schechter fit parameters are shown in Table \[Single schechter stellar mass functions: Elliptical galaxies\]. Both 1$\sigma$ and 2$\sigma$ error ellipses of $M_{st}^{\star}$ and $\alpha$ for central galaxies with elliptical morphologies in different environments are overlapping (Fig. \[Error ellipses\]d). There is no clear difference in $M_{st}^{\star}$ and $\alpha$ values in high- and low-density environments within the error region considered.
The 2$\sigma$ error ellipses of $M_{st}^{\star}$ and $\alpha$ for satellite galaxies with elliptical morphologies are overlapping, but the 1$\sigma$ error ellipses are well separated (Fig. \[Error ellipses\]d). At 1$\sigma$, the $\alpha$ value is steeper in high-density environments than that in low-density environments. At 1$\sigma$, the $M_{st}^{\star}$ value for satellite galaxies is higher in high-density environments compared to that in low-density environments.
Discussion
==========
Galaxy stellar mass and halo relation
-------------------------------------
According to the theory of galaxy formation, stars can form from cooling gas inside a virialized and gravitationally bound dark matter halo. The gas cooling and star formation rates of the galaxy depend mainly on the host halo mass [@1978MNRAS.183..341W]. In this scenario, the stellar mass of a galaxy can be expected to correlate strongly with the mass of its halo in which the galaxy formed. Several observational and simulation studies have tried to quantify the relation between the stellar mass of a galaxy and its halo mass. The stellar mass of a galaxy has been found to correlate with its halo mass with finite scatter. It increases rapidly with the halo mass when the halo mass is small, but slowly for large halo masses with a break in the relation around halo mass of 10$^{12}$ M$_{\odot}$ and galaxy stellar mass of 5 $\times$ 10$^{10}$ M$_{\odot}$ [@2010ApJ...710..903M]. The relation is very shallow at the high-mass end and the scatter in the relation is also higher. We note that our sample consists mainly of small groups and, thus, the correlation between the galaxy stellar mass and halo mass in our sample is expected to have less scatter. Some studies have also pointed out that the galaxy stellar and halo mass relation may differ between various morphologies and late-type galaxies reside in less massive halos than early types [@2015ApJ...799..130R and references therein]. Using hydrodynamical simulations, @2015ApJ...812..104T recently found a higher stellar to halo mass ratio in halos within the mass range 10$^{11}$–10$^{12.9}$ M$_{\odot}$ in large-scale denser environments. The dark matter halo mass function varies across different large-scale environments [@2007MNRAS.375..489H]. The low-mass end has the same slope in clusters, voids, filaments, and sheets, but the position of the high-mass cutoff depends on environment. The cutoff mass is the lowest in voids and gradually increases toward sheets, filaments, and clusters. In the halo mass range M < 5 $\times$ 10$^{12.9}h^{-1}$ M$_{\odot}$ , halos of a given mass tend to be older in high-density, large-scale environments and independent of environment at larger masses [@2007MNRAS.375..489H]. The present day halos in high-density environments are hence more massive and evolved than those in low-density environments as they have undergone many merger and accretion events during their evolution. The shift of the characteristic stellar mass toward higher stellar masses observed in our analysis of the SMFs of all galaxies in high-density, large-scale environments compared to the low-density environments may simply be the result of the mass and age segregation of halos in these environments.
Stellar mass growth of central and satellite galaxies
-----------------------------------------------------
![Mean richness of groups as a function of group dynamical mass. The red points and lines represent the relations in high-density scale environments, while blue points and lines represent low-density, large-scale environments. The square, circular, and asterisk points represent the results from all galaxy, spiral galaxy, and elliptical galaxy samples, respectively.[]{data-label="mean richness vs group mass"}](mean_richness.pdf){width="\hsize"}
When a larger halo accretes a smaller halo with its central galaxy, the accreted halo turns into a subhalo and its galaxy becomes a satellite of the accreting halo. Stellar mass growth of galaxies then occurs mainly by star formation, accretion of smaller satellite galaxies, or major mergers. In low-mass halos, the stellar mass growth occurs mainly by star formation, while mergers play the dominant role in high-mass halos [@2012ApJ...746..145Z]. The galaxy merger rate in the semianalytical models is found to depend strongly on environment, where higher density regions have larger merger rates [@2012ApJ...754...26J]. According to our results, the characteristic stellar mass of the central galaxy SMF shows no clear differences in high- and low-density environments but the low-mass end slope is steeper in low-density environments. Such trends may be attributed to higher abundances of low mass and younger halos with low stellar mass central galaxies in low-density environments. In contrast to central galaxies, the satellite galaxies in high-density environments have higher characteristic stellar mass and the shift is only observed for satellite galaxies of elliptical morphologies and not for spiral satellites. Such differences may occur because galaxies in high-density environments reside in relatively more massive and evolved halos than low-density environments. The low-mass end in the SMFs of satellite galaxies have steeper slopes in high-density environments irrespective of their morphologies. This suggests that groups in high-density environments have higher abundances of satellite galaxies than those in low-density environments. We investigate further whether this is true at a fixed halo mass. We use the dynamical mass of a group as a proxy of its halo mass and study it as a function of group richness. It is clear from Fig. \[mean richness vs group mass\] that groups in high-density environments are richer in galaxies, i.e., show higher abundances of satellite galaxies in all dynamical mass bins than low-density environments. Similar trends are also seen in the richness of galaxies with spiral and elliptical morphologies. Recently, @2015ApJ...800..112G have also shown that central galaxies in filaments have more satellites than elsewhere.
All these results support the scenario where groups with similar halos in high-density environments have high levels of substructures [@2007ApJ...666L...5E] and higher stellar mass content in those substructures. This suggests that galaxy formation is more efficient in subhalos located in high-density environments. A plausible scenario is that the high-density environments have more filaments, which supply cold gas to halos resulting in increased galaxy formation efficiency.
Conclusion
==========
We have used GAMA DR2 data, WISE data, SDSS DR10 group catalog, and the MAGPHYS code to construct a multifrequency catalog of groups of galaxies with wavelengths ranging from UV to MIR. We studied the galaxy SMFs of different galaxy populations in groups in different large-scale environments. The main results are summarized below:
1. The global galaxy SMF obtained from multifrequency (0.152–22$\mu$m) galaxy SED fitting differs from results in the literature. It has a slightly higher characteristic stellar mass than results from @2008MNRAS.388..945B and @2014MNRAS.444.1647K.
2. The SMFs of central galaxies in groups differ in high- and low-density environents. The low-mass end slope is steeper in high-density environments compared to low-density environments, but the characteristic stellar mass are similar in both environments.
3. The SMFs of satellite galaxies in groups vary in different large-scale environments. High-density environments have a steeper low-mass slope and a higher characteristic stellar mass than that in low-density environments.
4. The SMFs of central galaxies with spiral morphologies are different in high- and low-density, large-scale environments. However, the differences in the Schechter fit parameters are not clear. The galaxy SMFs of central galaxies of elliptical morphologies are similar in high- and low-density, large-scale environments.
5. The SMFs of satellite galaxies with spiral morphologies are different in high- and low-density, large-scale environments. In high-density environments the low-mass end slope is higher than that in low-density environments, but the characteristic stellar mass is similar in both environments. The SMFs of satellite galaxies with elliptical morphologies are different in different large-scale environments. High-density environments have a higher low-mass end slope and a higher characteristic stellar mass than that in low-density, large-scale environments.
Our first paper in the series of planned environmental studies of groups using multifrequency analysis of galaxies shows that the large-scale environment plays a significant role in shaping the SMFs of different galaxy populations in groups. Groups in high-density environments have higher abundances of satellite galaxies, irrespective of the satellite galaxies morphology. For ellipticals, this trend is more obvious. Elliptical satellite galaxies are in general more massive in high-density environments. Stellar masses of spiral satellite galaxies do not show any dependence on large-scale environment. We also note that the SMFs of satellite and central galaxies may differ with group selection and central galaxy definition. The small sample size has restricted us to using a simple division of groups into void and nonvoid regions. However, the luminosity density method also allows us to separate high-density regions into finer structures such as filaments and superclusters. Using a much larger sample and wider wavelength coverage from the whole GAMA panchromatic data release [@2016MNRAS.455.3911D], we plan to study stellar masses, luminosities, and star formation properties of groups located in these structures. This will allow us to constrain physical mechanisms by which large-scale environments may transform galaxy properties.
GAMA is a joint European-Australian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalog is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KIDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT, and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/.
This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. AP acknowledges financial support from the University of Turku Graduate School (UTUGS). HL acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (MINECO) under the 2011 Severo Ochoa Program MINECO SEV- 2011-0187. ET and ME acknowledge the support from the ESF grants IUT26-2 and IUT40-2.
Luminosity function plots
=========================
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Stellar mass function parameters
================================
------------ ------------------------- ------------------------- ------------------------ ------------------------- ------------------------
References $\log {M_{st}^\star}$ $\alpha_1$ $\phi_1^{\star}$ $\alpha_2$ $\phi_2^{\star}$
(M$_\odot$) ($10^{-3}$X$^{-3}$) ($10^{-3}$X$^{-3}$)
This study $10.73_{-0.01}^{+0.02}$ $-0.77_{-0.08}^{+0.13}$ $9.73_{-2.01}^{+1.06}$ $-1.43_{-0.09}^{+0.15}$ $1.66_{-0.92}^{+1.87}$
1 $10.64\pm{0.01}$ $-0.46\pm{0.05}$ $4.26\pm{0.09}$ $-1.58\pm{0.02}$ $0.58\pm{0.07}$
2 $10.64\pm{0.07}$ $-0.43\pm{0.35}$ $4.18\pm{1.52}$ $-1.50\pm{0.22}$ $0.74\pm{1.13}$
------------ ------------------------- ------------------------- ------------------------ ------------------------- ------------------------
: Double Schechter stellar mass function fit parameters of all galaxies and comparisons with other results. []{data-label="double schechter stellar mass function"}
-------------- --------------- ------------------------- -------------------------
Sample Subsample $\log {M_{st}^\star}$ $\alpha$
(M$_\odot$)
All galaxies den8 > 1.5 $10.85_{-0.07}^{-0.02}$ $-1.16_{-0.00}^{+0.08}$
den8 < 1.5 $10.75_{-0.08}^{-0.03}$ $-1.12_{-0.02}^{+0.10}$
Centrals den8 > 1.5 $10.81_{-0.08}^{+0.04}$ $-0.25_{-0.12}^{+0.23}$
den8 < 1.5 $10.75_{-0.05}^{+0.04}$ $-0.76_{-0.08}^{+0.15}$
Satellites den8 > 1.5 $10.67_{-0.07}^{+0.06}$ $-1.64_{-0.07}^{+0.10}$
den8 < 1.5 $10.52_{-0.05}^{+0.04}$ $-1.17_{-0.08}^{+0.11}$
-------------- --------------- ------------------------- -------------------------
: Single Schechter stellar mass function parameters of galaxies in different samples.[]{data-label="Single schechter stellar mass functions: All galaxies"}
------------ --------------- ------------------------- -------------------------
Sample Subsample $\log {M_{st}^\star}$ $\alpha$
(M$_\odot$)
Centrals den8 > 1.5 $10.85_{-0.09}^{+0.04}$ $-0.93_{-0.07}^{+0.19}$
den8 < 1.5 $10.74_{-0.12}^{+0.07}$ $-1.06_{-0.10}^{+0.14}$
Satellites den8 > 1.5 $10.62_{-0.18}^{+0.19}$ $-1.77_{-0.10}^{+0.14}$
den8 < 1.5 $10.58_{-0.18}^{+0.24}$ $-1.48_{-0.13}^{+0.12}$
------------ --------------- ------------------------- -------------------------
: Single Schechter stellar mass function parameters of galaxies with spiral morphologies in different samples.[]{data-label="Single schechter stellar mass functions: Spiral galaxies"}
------------ --------------- ------------------------- -------------------------
Sample Subsample $\log {M_{st}^\star}$ $\alpha$
(M$_\odot$)
Centrals den8 > 1.5 $10.81_{-0.18}^{+0.07}$ $0.06_{-0.30}^{+0.84}$
den8 < 1.5 $10.68_{-0.15}^{+0.12}$ $-0.04_{-0.33}^{+0.65}$
Satellites den8 > 1.5 $10.77_{-0.21}^{+0.12}$ $-1.47_{-0.17}^{+0.42}$
den8 < 1.5 $10.46_{-0.14}^{+0.07}$ $-0.63_{-0.25}^{+0.52}$
------------ --------------- ------------------------- -------------------------
: Single Schechter stellar mass function parameters of galaxies with elliptical morphologies in different samples.[]{data-label="Single schechter stellar mass functions: Elliptical galaxies"}
Central and satellite galaxy stellar mass functions
===================================================
In this section, we perform a simple exercise to check how different definitions of central galaxies and use of different group catalogs may affect the SMFs of central and satellite galaxies. For this, we compare our results with that obtained from the GAMA group catalog [@2011MNRAS.416.2640R]. First, we find the galaxies in the whole GAMA catalog with the highest stellar mass in their groups and define these as central galaxies and the remaining members as satellite galaxies. Then, we cross-match these galaxies with our multifrequency catalog, in which the central galaxies also have the same definition. Then, we extract two samples, one containing galaxies that are defined as centrals in the GAMA group catalog and other consisting of galaxies that are defined as centrals in our group catalog. Next, we make 1000 different random samplings from each of these samples and estimate the corresponding SMFs. In both cases, we use the stellar mass estimates from our multifrequency catalog. The red and green ellipses shown in Fig. \[stellar mass function comparisons: GAMA and SDSS\] represent the error ellipses of Schechter fit parameters, $M_{st}^\star$ and $\alpha$, for central galaxies in our and GAMA catalogs, respectively. The error ellipses are well separated from each other. However, the low-mass end slopes are similar but $M_{st}^\star$ seem to have lower values in GAMA centrals. Similarly, we also obtain the error ellipses of the Schechter fit parameters, $M_{st}^\star$ and $\alpha$ for satellite galaxies in GAMA and our catalog separately. The 2$\sigma$ error ellipses are overlapping but the 1$\sigma$ error ellipses are well separated. The low-mass end slopes are similar, but the GAMA satellites clearly have higher $M_{st}^\star$.
![Error ellipses at 95$\%$ confidence level for stellar mass function fit parameters (${M_{st}^\star}$ and $\alpha$) of central and satellite galaxies for different catalogs and central galaxy definitions. MMGs represent the most massive galaxies in groups and BCGs represent the brightest galaxies in the groups. Red and blue (green and black) ellipses represent the error ellipses for the MMGs and not MMGs in our multifrequency (GAMA) group catalog. The golden and brown ellipses represent the error ellipses for the BCGs and not BCGs in our multifrequency group catalog, respectively.[]{data-label="stellar mass function comparisons: GAMA and SDSS"}](gama_sdss_group.pdf){width="12cm"}
In order to see if the definition of central galaxy based on luminosity affects the SMFs of the satellite and central galaxies, we selected the brightest galaxy in our sample as the central galaxy and remaining members as satellites. The golden and brown ellipses in Fig. \[stellar mass function comparisons: GAMA and SDSS\] show the corresponding error ellipses of the Schechter fit parameters, $M_{st}^\star$ and $\alpha$ for central and satellite galaxies, respectively. The error ellipse for the brightest central galaxies is almost the same as that for the most massive central galaxies. The error ellipse for the satellite galaxies of the brightest central galaxies overlaps with that for the satellite galaxies of the most massive central galaxies. However, $M_{st}^\star$ and $\alpha$ have similar values. This shows that the SMFs of central and satellite galaxies may differ based on the definition of the central galaxy and the group catalog used.
[^1]: The multifrequency catalog is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via <http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/>
|
---
address: ' Department of mathematics, Faculty of Science, Kobe University, Kobe, 657, Japan'
author:
- Kōta Yoshioka
title: A note on moduli of vector bundles on rational surfaces
---
Introduction
============
Let $(X,H)$ be a pair of a smooth rational surface $X$ and an ample divisor $H$ on $X$. Assume that $(K_X,H)<0$. Let $\overline{M}_H(r,c_1,\chi)$ be the moduli space of semi-stable sheaves $E$ of ${\operatorname{rk}}(E)=r$, $c_1(E)=c_1$ and $\chi(E)=\chi$. To consider relations between moduli spaces of different invariants is an interesting problem. If $(c_1,H)=0$ and $\chi \leq 0$, then Maruyama [@Ma:4], [@Ma:3] studied such relations and constructed a contraction map $\phi:\overline{M}_H(r,c_1,\chi) \to \overline{M}_H(r-\chi,c_1,0)$. Moreover he showed that the image is the Uhlenbeck compactification of the moduli space of $\mu$-stable vector bundles. In particular, he gave an algebraic structure on Uhlenbeck compactification which was topologically constructed before. After Maruyama’s result, Li [@Li:1] constructed the birational contraction for general cases, by using a canonical determinant line bundle, and gave an algebraic structure on Uhlenbeck compactification. Although Maruyama’s method works only for special cases, his construction is interesting of its own. Let us briefly recall his construction. Let $E$ be a semi-stable sheaf of ${\operatorname{rk}}(E)=r$, $c_1(E)=c_1$ and $\chi(E)=\chi$. Then $H^i(X,E)=0$ for $i=0,2$. We consider a universal extension $$\label{eq:uni-ext}
0 \to E \to F \to H^1(X,E) \otimes {\cal O}_X \to 0.$$ Maruyama showed that $F$ is a semi-stable sheaf of ${\operatorname{rk}}(F)=r-\chi$, $c_1(F)=c_1$ and $\chi(F)=0$. Then we have a map $\phi:\overline{M}_H(r,c_1,\chi) \to
\overline{M}_H(r-\chi,c_1,0)$. He showed that $\phi$ is an immersion on the open subscheme consistings of $\mu$-stable vector bundles and the image of $\phi$ is the Uhlenbeck compactification. For the proof, the rigidity of ${\cal O}_X$ is essential. In this note, we replace ${\cal O}_X$ by other rigid and stable vector bundles $E_0$ and show that similar results hold, if $E_0$-twisted degree $\deg_{E_0}(E):=(c_1(E_0^{\vee} \otimes E),H)=0$. If $H$ is a general polarization, then we also show that ${\operatorname{im}}\phi$ is normal (Theorem \[thm:normal\]).
We are also motivated by our study of sheaves on K3 surfaces. For K3 and abelian surfaces, integral functor called Fourier-Mukai functor gives an equivalence of derived categories of coherent sheaves, and under suitable conditions, we get a birational correspondence of moduli spaces (cf. [@Y:5], [@Y:7], [@Y:9]). For rational surfaces, we can rarely expect such an equivalence (cf. [@Br:2]). For example, an analogue of Mukai’s reflection [@Mu:4] (which is given by ) may lose some information. Indeed we get our contraction map $\phi:\overline{M}_H(r,c_1,\chi) \to
\overline{M}_H(r-\chi,c_1,0)$.
In section \[sect:deg=1\], we also consider the relation of different moduli spaces in the case where $\deg_{E_0}E=1$. Then we find some relations on (virtual) Hodge numbers (or Betti numbers) of moduli spaces. If $X={\Bbb P}^2$, by using known results on Hodge numbers ([@E-S:1], [@Y:1]), we calculate Hodge numbers of some low dimensional moduli spaces. We also determine the boundary of ample cones in some cases.
Preliminaries {#sect:pre}
=============
Twisted stability
-----------------
Let $X$ be a smooth projective surface. Let $K(X)$ be the Grothendieck group of $X$. For $x \in K(X)$, we set $$\gamma(x):=({\operatorname{rk}}x,c_1(x),\chi(x)) \in {\Bbb Z} \oplus {\operatorname{NS}}(X) \oplus
{\Bbb Z}.$$ Then $\gamma:K(X) \to {\Bbb Z} \oplus {\operatorname{NS}}(X) \oplus
{\Bbb Z}$ is a surjective homomorphism and $\ker \gamma$ is generated by ${\cal O}_X(D)-{\cal O}_X$ and ${\Bbb C}_P-{\Bbb C}_Q$, where $D \in {\operatorname{Pic}}^0(X)$ and $P,Q \in X$. For $\gamma=(r,c_1,\chi) \in {\Bbb Z} \oplus {\operatorname{NS}}(X) \oplus
{\Bbb Z}$, we set ${\operatorname{rk}}\gamma=r$, $c_1(\gamma)=c_1$ and $\chi(\gamma)=\chi$. $K(X)$ is equipped with a bilinear form $\chi(\;\;,\;\;)$: $$\begin{matrix}
K(X) \times K(X) &\to &{\Bbb Z}\\
(x,y) & \mapsto &\chi(x,y)
\end{matrix}$$ It is easy to see that
$\chi(x,y)=\chi(y,x)+(K_X,c_1(y^{\vee} \otimes x))$, $x,y \in K(X)$.
$\chi(\;\;,\;\;)$ induces a bilinear form on ${\Bbb Z} \oplus {\operatorname{NS}}(X) \oplus
{\Bbb Z}$. We also denote it by $\chi(\;\;,\;\;)$: $\chi(\gamma(x),\gamma(y))=\chi(x,y)$.
Let ${\cal M}_H(\gamma)^{\mu \text{-}ss}$ (resp. ${\cal M}_H(\gamma)^{\mu \text{-}s}$) be the moduli stack of $\mu$-semi-stable sheaves (resp. $\mu$-stable sheaves) $E$ such that $\gamma(E)=\gamma \in {\Bbb Z} \oplus {\operatorname{NS}}(X) \oplus
{\Bbb Z}$.
For $G \in K(X) \otimes{\Bbb Q}$ of ${\operatorname{rk}}G>0$, we define $G$-twisted rank, degree, and Euler characteristic of $x \in K(X) \otimes {\Bbb Q}$ by $$\begin{split}
{\operatorname{rk}}_{G}(x)&:={\operatorname{rk}}(G^{\vee} \otimes x)\\
\deg_{G}(x)&:=(c_1(G^{\vee} \otimes x),H)\\
\chi_{G}(x)&:=\chi(G^{\vee} \otimes x).
\end{split}$$ For $t \in {\Bbb Q}_{>0}$, we get $$\frac{\deg_G(x)}{{\operatorname{rk}}_G(x)}=\frac{\deg_{tG}(x)}{{\operatorname{rk}}_{tG}(x)},\;
\frac{\chi_G(x)}{{\operatorname{rk}}_G(x)}=\frac{\chi_{tG}(x)}{{\operatorname{rk}}_{tG}(x)}.$$
We shall define $G$-twisted stability.
Let $E$ be a torsion free sheaf on $X$. $E$ is $G$-twisted semi-stable (resp. stable) with respect to $H$, if $$\frac{\chi_G(F(nH)}{{\operatorname{rk}}_G(F)} \leq \frac{\chi_{G}(E(nH))}{{\operatorname{rk}}_{G}(E)},
n \gg 0$$ for $0 \subsetneq F \subsetneq E$ (resp. the inequality is strict).
It is easy to see that the following relations hold: $$\text{$\mu$-stable} \Rightarrow \text{$G$-twisted stable}
\Rightarrow \text{$G$-twisted semi-stable}
\Rightarrow \text{$\mu$-semi-stable}.$$ For a ${\Bbb Q}$-divisor $\alpha$, we define $\alpha$-twisted stability as ${\cal O}_X(\alpha)$-twisted stability. This is nothing but the twisted stability introduced by Matsuki and Wentworth [@M-W:1]. It is easy to see that $G$-twisted stability is determined by $\alpha=\det(G)/{\operatorname{rk}}G$. Hence $G$-twisted stability is the same as the Matsuki-Wentworth stability.
For $ \gamma \in {\Bbb Z} \oplus {\operatorname{NS}}(X) \oplus
{\Bbb Z}$, let ${\cal M}_{H}^{G}(\gamma)^{ss}$ be the moduli stack of $G$-twisted semi-stable sheaves $E$ of $\gamma(E)=\gamma$ and ${\cal M}_{H}^{G}(\gamma)^{s}$ the open substack consisting of $G$-twisted stable sheaves. For usual stability, i.e, $G={\cal O}_X$, we denote ${\cal M}_{H}^{{\cal O}_X}(\gamma)^{ss}$ by ${\cal M}_{H}(\gamma)^{ss}$.
There is a coarse moduli scheme $\overline{M}_H^G(\gamma)$ of $S$-equivalence classes of $G$-twisted semi-stable sheaves $E$ of $\gamma(E)=\gamma$.
Construction of contraction map {#sect:contraction}
===============================
From now on, we assume that $(X,H)$ is a pair of a rational surface $X$ and an ample divisor $H$ on $X$. Then $\gamma:K(X) \to {\Bbb Z} \oplus {\operatorname{NS}}(X) \oplus
{\Bbb Z}$ is an isomorphism. Assume that $(K_X,H)<0$. Let $E_0$ be a exceptional vector bundle which is stable with respect $H$. Let $e_0 \in K(X)$ be the class of $E_0$ in $K(X)$. We set $\gamma_0:=\gamma(E_0)$ and $\omega:=\gamma({\Bbb C}_P)$, $P \in X$. We define homomorphism $L_{e_0},R_{e_0}:K(X) \to K(X)$ by $$\begin{split}
L_{e_0}(x)&:=x-\chi(x,e_0)e_0, x \in K(X),\\
R_{e_0}(x)&:=x-\chi(e_0,x)e_0, x \in K(X).
\end{split}$$ Then the following relation holds.
$\chi(x,R_{e_0}(y))=\chi(L_{e_0}(x),y)$ for $x,y \in K(X)$.
Existence of $\mu$-stable vector bundle
---------------------------------------
In this subsection, we shall give a sufficient condition for ${\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}s}$ to be non-empty.
${\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}ss}$ is smooth of $\dim {\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}ss}=2ra {\operatorname{rk}}E_0-r^2$.
For $E \in {\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}ss}$, ${\operatorname{Ext}}^2(E,E) \cong {\operatorname{Hom}}(E,E(K_X))^{\vee}=0$. Hence ${\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}ss}$ is smooth and $\dim {\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}ss}=\dim {\operatorname{Ext}}^1(E,E)-
\dim {\operatorname{Hom}}(E,E)=-\chi(E,E)=2ra {\operatorname{rk}}E_0-r^2$.
\[lem:necessary\] If ${\cal M}_H^{E_0}(r\gamma_0-a \omega)^s \ne \emptyset$, then $r=1$ and $a=0$, or $a {\operatorname{rk}}E_0-r \geq 0$.
Let $E$ be an element of ${\cal M}_H^{E_0}(r\gamma_0-a \omega)^s$. Since $E$ is simple and ${\operatorname{Ext}}^2(E,E)=0$, $1 \geq \chi(E,E)=r^2-2ra {\operatorname{rk}}E_0$. Hence $a \geq \frac{1}{2 {\operatorname{rk}}E_0}(r-\frac{1}{r}) \geq 0$. If $\chi(E_0,E)=r-a {\operatorname{rk}}E_0 > 0$, then there is a non-zero homomorphism $E_0 \to E$. Then $$\frac{1}{{\operatorname{rk}}E_0}=\frac{\chi(E_0,E_0)}{{\operatorname{rk}}E_0} \leq
\frac{\chi(E_0,E)}{r {\operatorname{rk}}E_0}=\frac{r-a {\operatorname{rk}}E_0}{r {\operatorname{rk}}E_0}.$$ Therefore $a=0$ and $r=1$.
\[lem:ev\] Let $E$ be a $\mu$-semi-stable sheaf of $\deg_{E_0}(E)=0$. Then $ev:{\operatorname{Hom}}(E_0,E) \otimes E_0 \to E$ is injective and ${\operatorname{coker}}(ev)$ is $\mu$-semi-stable.
We set $G:=\ker(ev)$. Assume that $G \ne 0$. Let $G_0$ be a $\mu$-stable locally free subsheaf of $G$ such that $\deg_{E_0} G_0=0$. Then we get a non-zero homomorphism $\phi:G_0 \to E_0$. Since $G_0$ is locally free, $\phi$ must be an isomorphism. Hence ${\operatorname{Hom}}(E_0,G_0) \ne 0$. On the other hand, $ev$ induces an isomorphism ${\operatorname{Hom}}(E_0,{\operatorname{Hom}}(E_0,E) \otimes E_0) \to {\operatorname{Hom}}(E_0,E)$. Hence ${\operatorname{Hom}}(E_0,G)=0$, which is a contradiction. Therefore $G=0$. We next show that $I:={\operatorname{coker}}(ev)$ is $\mu$-semi-stable. Assume that $I$ has a torsion submodule $T$. Then $J:=\ker(E \to I/T)$ is a submodule of $E$ containing ${\operatorname{im}}(ev)$. By the $\mu$-semi-stability of $E$, $0 \leq \deg_{E_0}(J)=\deg_{E_0}(T)$. Hence $T$ is of dimension 0. Since ${\operatorname{im}}(ev)$ is locally free, $J={\operatorname{im}}(ev)$. Thus $I$ is torsion free. Then it is easy to see that ${\operatorname{coker}}(ev)$ is $\mu$-semi-stable.
If ${\cal M}_H^{E_0}(r\gamma_0-a \omega)^{\mu \text{-}ss}\ne \emptyset$, then $a \geq 0$.
If $a <0$, then $\dim {\operatorname{Hom}}(E_0,E) >r$ for $E \in {\cal M}_H^{E_0}(r\gamma_0-a \omega)^{\mu \text{-}ss}$. By Lemma \[lem:ev\], we get a contradiction.
\[prop:exist\] ${\cal M}_H^{E_0}(r\gamma_0-a \omega)^{\mu \text{-}s} \ne \emptyset$, if $r-a {\operatorname{rk}}E_0 \leq 0$. Moreover, there is a $\mu$-stable locally free sheaf $E$ of $\gamma(E)=r\gamma_0-a \omega$.
Let $W$ be a closed substack of ${\cal M}_H^{E_0}(r\gamma_0-a \omega)^{\mu \text{-}ss}$ such that $E$ belongs to $W$ if and only if there is a quotient $E \to G$ such that $(c_1(G)/{\operatorname{rk}}G,H)=(c_1(\gamma_0)/{\operatorname{rk}}\gamma_0,H)$ but $c_1(G)/{\operatorname{rk}}G \ne c_1(\gamma_0)/{\operatorname{rk}}\gamma_0$. Let $f:E_0^{\oplus r} \to \oplus_{i=1}^a{\Bbb C}_{x_i}$, $x_i \in X$ be a surjective homomorphism. Then $E:=\ker f$ is $\mu$-semi-stable and does not belong to $W$. Hence ${\cal M}_H^{E_0}(r\gamma_0-a \omega)^{\mu \text{-}ss} \setminus W$ is a non-empty open substack of ${\cal M}_H^{E_0}(r\gamma_0-a \omega)^{\mu \text{-}ss}$. For pairs of integers $(r_1,a_1)$ and $(r_2,a_2)$ such that $r_1,r_2>0$, $a_1,a_2 \geq 0$ and $(r_1+r_2,a_1+a_2)=(r,a)$, let $N(r_1,a_1;r_2,a_2)$ be the substack of ${\cal M}_H^{E_0}(r\gamma_0-a \omega)$ consisting of $E$ which fits in an exact sequence: $$0 \to E_1 \to E \to E_2 \to 0$$ where $E_1$ is a $\mu$-stable sheaf of $\gamma(E_1)=r_1 \gamma_0-a_1 \omega$ and $E_2$ is a $\mu$-semi-stable sheaf of $\gamma(E_2)=r_2 \gamma_0-a_2 \omega$. By [@D-L:1 sect. 1] or [@Y:8 Lem. 5.2], $$\begin{split}
{\operatorname{codim}}N(r_1,a_1;r_2,a_2)& \geq -\chi(E_1,E_2)\\
&=(a_1r_2+a_2r_1){\operatorname{rk}}E_0-r_1r_2.
\end{split}$$ By Lemma \[lem:necessary\], $(a_1+a_2) {\operatorname{rk}}E_0-(r_1+r_2) \geq 0$. Hence if $a_1=0$ or $a_2=0$, then we get $(a_1r_2+a_2r_1){\operatorname{rk}}E_0-r_1r_2
\geq 0$. If $a_1,a_2>0$, then by using Lemma \[lem:necessary\] again, we see that $(a_1r_2+a_2r_1){\operatorname{rk}}E_0-r_1r_2\geq
a_2r_1 {\operatorname{rk}}E_0>0$. Therefore $N(r_1,a_1;r_2,a_2)$ is a proper substack of ${\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}ss}
\setminus W$, which implies that ${\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}s} \ne \emptyset$. By [@Y:1 Thm. 0.4], the locus of non-locally free sheaves is of codimension $r{\operatorname{rk}}E_0-1>0$ (use ). Hence ${\cal M}_H(r\gamma_0-a \omega)^{\mu \text{-}s}$ contains a locally free sheaf.
Universal extension and the contraction map
-------------------------------------------
We define a coherent sheaf ${\cal E}$ on $X \times X $ by the following exact sequence $$\label{eq:family}
0 \to {\cal E} \to p_1^*(E_0^{\vee}) \otimes p_2^*(E_0)
\overset{ev}{\to}
{\cal O}_{\Delta} \to 0.$$ Then ${\cal E}$ is $p_2$-flat and ${\cal E}_x:={\cal E}_{|\{x \} \times X}$ is a $E_0$-twisted stable sheaf of $\gamma({\cal E}_x)={\operatorname{rk}}(E_0)\gamma(E_0)-\omega$. In particular $\chi(E_0,{\cal E}_x)=0$.
\[lem:WIT1\] For a $\mu$-semi-stable sheaf $E$ of $\deg_{E_0}(E)=0$, $$p_{2*}({\cal E} \otimes p_1^*(E_0))=
R^2p_{2*}({\cal E} \otimes p_1^*(E_0))=0.$$
For $E \in {\cal M}_H(\gamma)^{\mu \text{-}ss}$, Lemma \[lem:ev\] implies that $ev:{\operatorname{Hom}}(E_0,E) \otimes E_0 \to E$ is injective. Hence $p_{2*}({\cal E} \otimes p_1^*(E))=0$. Since $(K_X,H)<0$, $\deg_{E_0}(E(-K_X))>\deg_{E_0}(E)=0$. Hence ${\operatorname{Ext}}^2(E_0,E)={\operatorname{Hom}}(E(-K_X),E_0)^{\vee}=0$. Then $R^2p_{2*}({\cal E} \otimes p_1^*(E)) \cong {\operatorname{Ext}}^2(E_0,E) \otimes E_0=0$.
The following is our main theorem of this section.
\[thm:contract\] Let $e \in K(X)$ be a class such that ${\operatorname{rk}}e>0$ and $\deg_{E_0}(e)=0$. Then we have a morphism $\phi_{\gamma(e)}:\overline{M}_H(\gamma(e)) \to
\overline{M}_H^{E_0}(\gamma(\hat{e}))$ sending $E$ to the $S$-equivalence class of $R^1p_{2*}({\cal E} \otimes p_1^*(E))$ and the restriction of $\phi_{\gamma(e)}$ to $M_H(\gamma)^{\mu \text{-}s,loc}$ is an immersion, where $\hat{e}=R_{e_0}(e)$ and $M_H(\gamma(e))^{\mu \text{-}s,loc}$ is the open subscheme consisting of $\mu$-stable vector bundles. If $\oplus_i E_i$ is the $S$-equivalence class of $E$ with respect to $\mu$-stability, then $\phi_{\gamma(e)}(E)$ is uniquely determined by $\oplus_i E_i^{\vee \vee}$ and the location of pinch points of $\oplus_i E_i$.
In order to prove this theorem, we prepare some lemmas.
$${\bf R} p_{2*}({\cal E} \otimes p_1^*(E_0)))=0.$$
By , we have an exact sequence $${\operatorname{Hom}}(E_0,E_0) \otimes E_0 \overset{ev}{\to} E_0 \to
R^1 p_{2*}({\cal E} \otimes p_1^*(E_0))) \to {\operatorname{Ext}}^1(E_0,E_0) \otimes E_0.$$ Since $ev$ is isomorphic and ${\operatorname{Ext}}^1(E_0,E_0)=0$, we get that $R^1 p_{2*}({\cal E} \otimes p_1^*(E_0))=0$. Therefore we get our claim.
\[lem:hom=0\] For a $\mu$-semi-stable sheaf $E$ of $\deg_{E_0}(E)=0$, $${\operatorname{Hom}}(E_0,R^1 p_{2*}({\cal E} \otimes p_1^*(E)))=0.$$
By Leray spectral sequence and projection formula, $${\operatorname{Hom}}(E_0,R^1 p_{2*}({\cal E} \otimes p_1^*(E)))=
H^1(X \times X,{\cal E} \otimes p_1^*(E) \otimes p_2^*(E_0^{\vee})).$$ Since ${\bf R} p_{1*}({\cal E} \otimes p_2^*(E_0^{\vee}))=0$, ${\bf R} p_{1*}({\cal E}\otimes p_1^*(E) \otimes p_2^*(E_0^{\vee}))
={\bf R} p_{1*}({\cal E} \otimes p_2^*(E_0^{\vee}))
\overset{\bf L}{\otimes} E=0$.
For simplicity, we set $\widehat{E}:=R^1 p_{2*}({\cal E} \otimes p_1^*(E))$.
\[prop:contract\] For a $\mu$-semi-stable sheaf $E$ of $\deg_{E_0}(E)=0$, $\widehat{E}$ is a $E_0$-twisted semi-stable sheaf of $\chi(E_0,\widehat{E})=0$.
By , $\widehat{E}$ fits in an exact sequence $$0 \to
{\operatorname{Hom}}(E_0,E) \otimes E_0 \overset{ev}{\to}
E \to \widehat{E} \to {\operatorname{Ext}}^1(E_0,E) \otimes E_0 \to 0$$ By Lemma \[lem:ev\], $ \widehat{E}$ is $\mu$-semi-stable. It is easy to see that $\chi(E_0,\widehat{E})=0$. Assume that $\widehat{E}$ is not semi-stable and let $G$ be a destabilizing subsheaf. Then $\chi(E_0,G)/{\operatorname{rk}}G>0$. By our assumption on $H$, ${\operatorname{Ext}}^2(E_0,G)=0$. Hence ${\operatorname{Hom}}(E_0,G) \ne 0$, which contradicts to Lemma \[lem:hom=0\].
If $E$ is $E_0$-twisted semi-stable such that $\chi(E_0,E) \leq 0$, then $\widehat{E}$ fits in an exact sequence $$\label{eq:univ.ext}
0 \to
E \to \widehat{E} \to {\operatorname{Ext}}^1(E_0,E) \otimes E_0 \to 0.$$ By Lemma \[lem:hom=0\], is a universal extension.
Let $E$ be a $\mu$-stable vector bundle of $\deg_{E_0}(E)=0$. Then $\widehat{E}$ is $E_0$-twisted stable.
We may assume that $E \ne E_0$. Then $\widehat{E}$ fits in a universal extension $$0 \to E \to \widehat{E} \to E_0^{\oplus h} \to 0$$ where $h=\dim {\operatorname{Ext}}^1(E_0,E)$. Assume that $\widehat{E}$ is not $E_0$-twisted stable. Then there is a $E_0$-twisted stable subsheaf $G_1$ of $\widehat{E}$ such that $G_2:=\widehat{E}/G_1$ is $E_0$-twisted semi-stable. If $E$ is contained in $G_1$, then we get a homomorphism $E_0^{\oplus h} \to G_2$. Since $\chi(E_0, G_2)/{\operatorname{rk}}G_2=0<\chi(E_0,E_0^{\oplus h})/h {\operatorname{rk}}E_0$, we get a contradiction. Hence $E$ is not contained in $G_1$. Since $E$ is $\mu$-stable, we get $E \cap G_1=0$. Hence $G_1 \to E_0^{\oplus h}$ is injective. Let $G'$ be a $\mu$-stable locally free subsheaf of $G_1$. Then we see that $G' \cong E_0$, which implies that $G_1$ is not $E_0$-twisted stable. Therefore $\widehat{E}$ is $E_0$-twisted stable.
[*Proof of Theorem \[thm:contract\]:*]{} Let $\{{\cal F}_s \}_{s \in S}$ be a flat family of $\mu$-semi-stable sheaves of $\deg_{E_0}({\cal F}_s)=0$. Then Lemma \[lem:WIT1\] and Proposition \[prop:contract\] imply that $\{\widehat{{\cal F}_s} \}_{s \in S}$ is also a flat family of $E_0$-twisted semi-stable sheaves (cf. [@Mu:5 Thm. 1.6]). Hence we get a morphism $\phi_{\gamma(e)}:
\overline{M}_H(\gamma(e)) \to \overline{M}_H(\gamma(\hat{e}))$. Let $E$ be a $\mu$-stable vector bundle of $\deg_{E_0}(E)=0$ and $\varphi:E \to T$ be a quotient such that $T$ is of dimension 0. Then for $F:=\ker \varphi$, we get an exact sequence $$0 \to p_{2*}({\cal E} \otimes p_1^*(T)) \to \widehat{F} \to
\widehat{E} \to 0.$$ Let $0 \subset T_1 \subset T_2 \subset \dots \subset
T_n=T$ be a filtration such that $T_i/T_{i-1} \cong {\Bbb C}_{x_i}$, $x_i \in X$ (i.e, Jordan-Hölder filtration with respect to Simpson’s stability). Then $G:=p_{2*}({\cal E} \otimes p_1^*(T))$ has a filtration $0 \subset G_1 \subset G_2 \subset \dots \subset
G_n=G$ such that $G_i/G_{i-1} \cong {\cal E}_{x_i}$. Since $\widehat{E}$ is stable, the $S$-equivalence class of $\widehat{F}$ is $\widehat{E} \oplus
\oplus_{i=1}^n {\cal E}_{x_i}$.
For a $\mu$-semi-stable sheaf $E$ of $\deg_{E_0}(E)=0$, let $\oplus_{i=1}^n E_i$ be an $S$-equivalence class of $E$ with respect to $\mu$-stability. Let $\oplus_j {\Bbb C}_{x_{{i,j}}}$ be the $S$-equivalence class of $E_i^{\vee \vee}/E_i$ as a purely 0-dimensional sheaf. Then the $S$-equivalence class of $\widehat{E}$ with respect to $E_0$-twisted stability is $\oplus_{i=1}^n(\widehat{E_i^{\vee \vee}} \oplus \oplus_{j}
{\cal E}_{x_{{i,j}}})$. By Proposition \[prop:classification\] and Remark \[rem:classification\] below, $\widehat{E_i^{\vee \vee}}$ is uniquely determined by $E_i^{\vee \vee}$. Hence the $S$-equivalence class of $\widehat{E}$ is uniquely determined by $E_i^{\vee \vee}$ and $x_{i,j}$.
\[prop:classification\] Let $F$ be an $E_0$-twisted stable sheaf such that $\deg_{E_0}(F)=0$ and $\chi(E_0,E)=0$. Then
1. $F={\cal E}_x$, $x \in X$, or
2. $F$ fits in an exact sequence $$\label{eq:classification}
0 \to E \to F \to E_0^{\oplus n} \to 0,$$ where $E$ is a $\mu$-stable locally free sheaf.
If $F$ is $\mu$-stable, then we see that $F^{\vee \vee} \cong E_0$, and hence ${\operatorname{rk}}E_0=1$ and $F \cong {\cal E}_x$, $x \in X$. Assume that there is an exact sequence $$0 \to G_1 \to F \to G_2 \to 0,$$ where $G_1$ is a $\mu$-stable sheaf of $\deg_{E_0}(G_1)=0$ and $G_2$ is a $\mu$-semi-stable sheaf of $\deg_{E_0}(G_2)=0$. Then we get an exact sequence $$0 \to \widehat{G_1} \to \widehat{F} \to \widehat{G_2} \to 0.$$ Since $F$ is $E_0$-twisted stable, $\widehat{F}=F$. In particular $\widehat{F}$ is $E_0$-twisted stable. By the stability of $G_1$, $\chi(E_0,G_1)<0$, which implies that $\widehat{G_1} \ne 0$. Therefore $\widehat{G_1} \cong \widehat{F}$ and $\widehat{G_2}=0$. By using , we see that ${\operatorname{Hom}}(E_0,G_2) \otimes E_0 \to G_2$ is an isomorphism. We note that $\widehat{G_1}$ fits in an exact sequence $$0 \to p_{2*}({\cal E} \otimes p_1^*(G_1^{\vee \vee}/G_1)) \to
\widehat{G_1} \to \widehat{G_1^{\vee \vee}} \to 0.$$ By the stability of $F$, (i) $G_1^{\vee \vee}/G_1=0$, or (ii) $G_1^{\vee \vee}/G_1={\Bbb C}_x$, $x \in X$ and $\widehat{G_1^{\vee \vee}}=0$. Therefore $G_1$ is locally free, or $F={\cal E}_x$.
\[rem:classification\] If $F$ fits in the exact sequence , then $E=\ker(F \to {\operatorname{Hom}}(F,E_0)^{\vee} \otimes E_0)$. Thus $E$ is uniquely determined by $F$.
Assume that $(X,H)=({\Bbb P}^2,{\cal O}_{{\Bbb P}^2}(1))$ and $E_0=\Omega_X(1)$. Then we have a contraction $$\overline{M}_H(2,-H,-n) \to \coprod_{0 \leq k \leq n}
M_H(2,-H,-k)^{\mu \text{-}s,loc} \times S^{n-k} X$$ sending $E$ to $(E^{\vee \vee},gr(E^{\vee \vee}/E))$, where $gr(E^{\vee \vee}/E)$ is the $S$-equivalence class of $E^{\vee \vee}/E$.
For a $\mu$-semi-stable sheaf $E$ of $\deg_{E_0}(E)=0$, ${\cal H}(E):={\operatorname{Ext}}^1_{p_1}(p_2^*(E),{\cal E})$ is a semi-stable sheaf such that $\deg_{E_0^{\vee}}({\cal H}(E))=0$ and $\chi(E_0^{\vee},{\cal H}(E))=0$. Indeed, it is easy to see that ${\cal H}(E)$ is a $\mu$-semi-stable sheaf such that $\deg_{E_0^{\vee}} {\cal H}(E)=0$ and $\chi(E_0^{\vee},{\cal H}(E))=0$. Since ${\operatorname{Hom}}(E_0^{\vee},{\cal H}(E))=
{\operatorname{Ext}}^1(p_2^*(E),{\cal E} \otimes p_1^*(E_0))=0$, ${\cal H}(E)$ is semi-stable. Hence we have a morphism $\psi_{\gamma}:
\overline{M}_H^{E_0}(\gamma(e)) \to
\overline{M}_H^{E_0^{\vee}}(\gamma(\hat{e}^{\vee}))$. It is easy to see that $\psi_{\delta}$ is an isomorphism and we get a commutative diagram. $$\begin{matrix}
&& \overline{M}_H^{E_0}(\gamma(e))&&&&&
\overline{M}_{H}^{E_0^{\vee}}(\gamma(e^{\vee}))\cr
&\llap{$\phi_{\gamma(e)}$}
\swarrow&&\searrow\rlap{$\psi_{\gamma(e)}$}&&
\llap{$\phi_{\gamma(e^{\vee})}$}\swarrow &\cr
\overline{M}_H^{E_0}(\gamma(\hat{e}))&&
\overset{\psi_{\gamma(\hat{e})}}{\to}
&&\overline{M}_H^{E_0^{\vee}}(\gamma(\hat{e}^{\vee}))&&\cr
\end{matrix}$$
The image of the contraction {#sect:image}
============================
Brill-Noether locus
-------------------
We set $\widehat{\gamma}:=m \gamma_0-c \omega$. Assume that $H$ is general with respect to $\widehat{\gamma}$, that is, $H$ does not lie on walls with respect to $\widehat{\gamma}$ (cf. [@M-W:1], [@Y:2],[@Y:8]). Hence ${\cal M}_H^{E_0}(\widehat{\gamma})^{ss}={\cal M}_H(\widehat{\gamma})^{ss}$. We define Brill-Noether locus by $${\cal M}_H(\widehat{\gamma},n):=
\{F \in {\cal M}_H(\widehat{\gamma})^{ss}|\dim {\operatorname{Hom}}(F,E_0) \geq n \}$$ and the open substack ${\cal M}_H(\widehat{\gamma},n)_0={\cal M}_H(\widehat{\gamma},n) \setminus{\cal M}_H(\widehat{\gamma},n+1)$. By using determinantal ideal, ${\cal M}_H(\widehat{\gamma},n)$ has a substack structure. Indeed, let $Q(\widehat{\gamma})$ be a standard open covering of ${\cal M}_H(\widehat{\gamma})^{\mu \text{-}ss}$, that is, $Q(\widehat{\gamma})$ is an open subscheme of a quot-scheme ${\operatorname{Quot}}_{{\cal O}_X(-k)^{\oplus N}/X/{\Bbb C}}$, $k \gg 0$, $N=\chi(\widehat{\gamma}(k))$ whose points consist of quotients ${\cal O}_X(-k)^{\oplus N} \to F$ such that
1. $F \in {\cal M}_H(\widehat{\gamma})^{\mu \text{-}ss}$,
2. $H^0(X,{\cal O}_X^{\oplus N}) \to H^0(X,F(k))$ is an isomorphism and $H^i(X,F(k))=0$ for $i>0$.
We may assume that $$\label{eq:k>>0}
H^i(X,E_0(k))=0, i>0.$$ Let ${\cal O}_{Q(\widehat{\gamma}) \times X}(-k)^{\oplus N} \to {\cal Q}$ be the universal quotient and ${\cal K}$ the universal subsheaf. We set $$\begin{split}
V:&=
{\operatorname{Hom}}_{p_{Q(\widehat{\gamma})}}({\cal O}_{Q(\widehat{\gamma}) \times X}(-k)^{\oplus N},
{\cal O}_{Q(\widehat{\gamma})} \otimes E_0),\\
W:&={\operatorname{Hom}}_{p_{Q(\widehat{\gamma})}}({\cal K},{\cal O}_{Q(\widehat{\gamma})} \otimes E_0).
\end{split}$$ Since ${\operatorname{Ext}}^2({\cal Q}_q,E_0)=0$ for all $i>0$ and $q \in Q(\widehat{\gamma})$, implies that ${\operatorname{Ext}}^i({\cal K}_q,E_0)=0$ for all $q \in Q(\widehat{\gamma})$. Hence $V$ and $W$ are locally free sheaves on $Q(\widehat{\gamma})$ and we have an exact sequence $$0 \to {\operatorname{Hom}}({\cal Q}_q,E_0) \to
V_q \to W_q \to {\operatorname{Ext}}^1({\cal Q}_q,E_0) \to 0, q \in Q(\widehat{\gamma}).$$ Therefore we shall define the stack structure on ${\cal M}_H(\widehat{\gamma},n)$ as the zero locus of $\wedge^r V \to \wedge^r W$.
Let ${\cal M}_H(\widehat{\gamma},n\gamma_0)$ be the moduli stack of isomorphism classes of $F \to E_0^{\oplus n}$ such that $F \in {\cal M}_H(\widehat{\gamma})^{\mu \text{-}ss}$ and ${\operatorname{Hom}}(E_0^{\oplus n},E_0) \to {\operatorname{Hom}}(F,E_0)$ is injective. We have a natural projection ${\cal M}_H(\widehat{\gamma},n\gamma_0) \to {\cal M}_H(\widehat{\gamma},n)$. Let ${\cal M}_H(\widehat{\gamma},n\gamma_0)_0$ be the open substack of ${\cal M}_H(\widehat{\gamma},n\gamma_0)$ such that ${\operatorname{Hom}}(E_0^{\oplus n},E_0) \to {\operatorname{Hom}}(F,E_0)$ is isomorphic. By [@ACGH:1 Chap.II sect. 2,3], ${\cal M}_H(\widehat{\gamma},n\gamma_0)_0$ is isomorphic to ${\cal M}_H(\widehat{\gamma},n)_0$.
We shall show that ${\cal M}_H(\widehat{\gamma},n)$ is Cohen-Macaulay and normal. By [@ACGH:1 Chap.II Prop.(4.1)], if ${\cal M}_H(\widehat{\gamma},n)$ has an expected codimension, that is, ${\operatorname{codim}}{\cal M}_H(\widehat{\gamma},n) =n^2$, then ${\cal M}_H(\widehat{\gamma},n)$ is Cohen-Macaulay. We shall estimate the dimension of substack ${\cal M}_H(\widehat{\gamma};n,p,a)$ of ${\cal M}_H(\widehat{\gamma})^{\mu \text{-}ss}$ consisting of $F \in {\cal M}_H(\widehat{\gamma})^{\mu \text{-}ss}$ such that $\dim F^{\vee \vee}/F=p$ and $F^{\vee \vee}$ fits in an exact sequence $$0 \to E \to F^{\vee \vee} \to G \to 0$$ where $E$ is a $\mu$-semi-stable sheaf of $\gamma(E)=
r\gamma_0-b\omega$, $G^{\vee \vee} \cong E_0^{\oplus n}$ and $\gamma(G)=n\gamma_0-a\omega$.
\[lem:estimate\] ${\operatorname{codim}}{\cal M}_H(\widehat{\gamma};n,p,a) \geq n^2+(r {\operatorname{rk}}E_0-1)(a+p)$.
For a locally free sheaf $L$, [@Y:1 Thm. 0.4] implies that $$\label{eq:quot}
\dim {\operatorname{Quot}}_{L/X/{\Bbb C}}^a=({\operatorname{rk}}L+1)a.$$ Let $N$ be the substack of ${\cal M}_H((r+n)\gamma_0-(a+b)\omega)^{\mu \text{-}ss}$ consisting of $F$ which fits in an exact sequence $$0 \to E \to L \to G \to 0$$ where $E$ is a $\mu$-semi-stable sheaf of $\gamma(E)=
r\gamma_0-b\omega$, $G^{\vee \vee} \cong E_0^{\oplus n}$ and $\gamma(G)=n\gamma_0-a\omega$. By [@Y:8 Lem. 5.2], we see that $$\begin{split}
\dim N& \leq \dim {\cal M}_H(r \gamma_0-b \omega)^{\mu \text{-}ss}+
\dim ([{\operatorname{Quot}}_{E_0^{\oplus n}/X/{\Bbb C}}^a/{\operatorname{Aut}}(E_0^{\oplus n})])-
\chi(G,E)\\
&=
(2rb {\operatorname{rk}}E_0-r^2)+((n {\operatorname{rk}}E_0+1)a-n^2)+
((ra+nb){\operatorname{rk}}E_0-rn))\\
&=(r+n)((a+b){\operatorname{rk}}E_0-(r+n))+n(r+n)+a+br {\operatorname{rk}}E_0-n^2.
\end{split}$$ Hence by using and the assumption $(a+b+p){\operatorname{rk}}E_0=r+n$, we see that $$\begin{split}
\dim {\cal M}_H(\widehat{\gamma};n,p,a)&=\dim N+((r+n){\operatorname{rk}}E_0+1)p\\
& \leq n(r+n)+a+p+br {\operatorname{rk}}E_0-n^2.
\end{split}$$ Therefore we get $$\begin{split}
{\operatorname{codim}}{\cal M}_H(\widehat{\gamma};n,p,a) &\geq (r+n)(2(a+b+p) {\operatorname{rk}}E_0-(r+n))
-(n(r+n)+a+p+br {\operatorname{rk}}E_0-n^2)\\
&=(r+n)^2-n(r+n)-(a+p+br {\operatorname{rk}}E_0-n^2)\\
&=n^2+(r {\operatorname{rk}}E_0-1)(a+p).
\end{split}$$
If $r:=m-n \geq 1$, then ${\cal M}_H(\widehat{\gamma};n)$ is Cohen-Macaulay.
Assume that $r {\operatorname{rk}}E_0 \geq 2$. Since ${\operatorname{codim}}_{{\cal M}_H(\widehat{\gamma};n)} {\cal M}_H(\widehat{\gamma};n+1)
\geq 2n+1$, we shall show that ${\cal M}_H(\widehat{\gamma};n)_0 \cong {\cal M}_H(\widehat{\gamma},n\gamma_0)_0$ is regular in codimension 1. For an element $ F \to E_0^{\oplus n}$ of ${\cal M}_H(\widehat{\gamma},n\gamma_0)_0$, the obstruction for smoothness belongs to ${\operatorname{Ext}}^2(F,F \to E_0^{\oplus n})$.
\[lem:obstruction\] If $F \to E_0^{\oplus n}$ is surjective or $F$ is locally free, then ${\operatorname{Ext}}^2(F,F \to E_0^{\oplus n})=0$.
We have an exact sequence $${\operatorname{Ext}}^2(F,E) \to
{\operatorname{Ext}}^2(F,F \to E_0^{\oplus n}) \to
{\operatorname{Ext}}^2(F,G \to E_0^{\oplus n}),$$ where $G:={\operatorname{im}}(F \to E_0^{\oplus n})$. Then ${\operatorname{Ext}}^2(F,E)={\operatorname{Hom}}(E,F(K_X))^{\vee}=0$. Since ${\operatorname{Ext}}^2(F,G \to E_0^{\oplus n})=
{\operatorname{Ext}}^1(F,E_0^{\oplus n}/G)$, we get our claim.
If $a+p \geq 2$, then ${\operatorname{codim}}{\cal M}_H(\widehat{\gamma};n,p,a) \geq 2$. If $a+p \leq 1$, then Lemma \[lem:obstruction\] implies that ${\cal M}_H(\widehat{\gamma},n)$ is smooth on ${\cal M}_H(\widehat{\gamma};n,p,a)$. Hence ${\cal M}_H(\widehat{\gamma},n)$ is regular in codimension 1. By Serre’s criterion, ${\cal M}_H(\widehat{\gamma};n)$ is normal.
\[prop:image\] Assume that $r {\operatorname{rk}}E_0 \geq 2$. Then ${\cal M}_H(\widehat{\gamma};n)$, $n:=m-r$ is normal and general member $F$ fits in an exact sequence $$0 \to E \to F \to E_0^{\oplus n} \to 0,$$ where $E \in {\cal M}_H(\widehat{\gamma}-n\gamma_0)^{\mu \text{-}s, loc}$ and ${\operatorname{Hom}}(E,E_0)=0$.
The following is a partial answer to [@Ma:3 Question 6.5].
\[thm:normal\] Assume that $r {\operatorname{rk}}E_0 \geq 2$. For $n:=m-r$, we set $$\overline{M}_H(\widehat{\gamma};n):=
\{F \in \overline{M}_H(\widehat{\gamma})|\dim {\operatorname{Hom}}(F,E_0) \geq n \}.$$ Then $\overline{M}_H(\widehat{\gamma};n)$ is normal, $\overline{M}_H(\widehat{\gamma};n)=\phi_{\gamma}(\overline{M}_H(\gamma))$ and we have an identification $$\overline{M}_H(\widehat{\gamma};n)
=\coprod_{r_i,a_i,n_i,l}
\prod_i S^{n_i} M_H(r_i \gamma_0-a_i \omega)^{\mu \text{-}s, loc}
\times S^l X$$ where $r_i,a_i,n_i,l$ satisfy that $a_i {\operatorname{rk}}E_0 \geq r_i$, $(r_i,a_i) \ne (r_j,a_j)$ for $i \ne j$, $l+\sum_i n_i a_i=a$ and $\sum_i n_i r_i \leq r=m-n$ Therefore $\phi_{\gamma}(\overline{M}_H(\gamma))$ is normal.
By Proposition \[prop:image\], $\overline{M}_H(\widehat{\gamma};n)$ is normal. Moreover $\phi_{\gamma}(M_H(\gamma)^{\mu \text{-}s, loc})$ is a dense subset of $\overline{M}_H(\widehat{\gamma};n)$. Hence $\overline{M}_H(\widehat{\gamma};n)=
\phi_{\gamma}(\overline{M}_H(\gamma))$. Let $F$ be a poly-stable sheaf of $\gamma(F)=\widehat{\gamma}$, i.e, $F$ is a direct sum of $E_0$-twisted stable sheaves. By Proposition \[prop:classification\], there are $\mu$-stable locally free sheaves $E_i$, $1 \leq i \leq k$ of $\gamma(E_i)=r_i \gamma_0-a_i \omega$ and points $x_j \in X$, $1 \leq j \leq
l$ such that $F=\oplus_{i=1}^k\widehat{E_i} \oplus \oplus_{j=1}^l {\cal E}_{x_j}$. Since $\dim {\operatorname{Hom}}(\widehat{E_i},E_0)=a_i {\operatorname{rk}}E_0-r_i$ and $\dim {\operatorname{Hom}}( {\cal E}_{x_j},E_0)={\operatorname{rk}}E_0$, we see that $$\begin{split}
\dim {\operatorname{Hom}}(F,E_0)&=\sum_i(a_i {\operatorname{rk}}E_0-r_i)+l {\operatorname{rk}}E_0\\
&=a {\operatorname{rk}}E_0-\sum_i r_i=m-\sum_i r_i.
\end{split}$$ Hence $F$ belongs to $\overline{M}_H(\widehat{\gamma};n)$ if and only if $\sum_i r_i \leq m-n=r$. Then the last claim follows from this.
The case where $\deg_{E_0}(E)=1$ {#sect:deg=1}
================================
Twisted coherent systems and correspondences
--------------------------------------------
In this section, we shall treat the case where the twisted degree is $1$. This case was highly motivated by Ellingsrud and Strømme’s paper [@E-S:1]. Assume that ${\operatorname{rk}}e_0 (-K_X,H)>1$. Let $e$ be a class in $K(X)$ such that ${\operatorname{rk}}e>0$ and $\deg_{e_0}(e)=1$. We set $\gamma:=\gamma(e)$ and $\gamma_0:=\gamma(e_0)$. For a stable sheaf $E$ of $\gamma(E)=\gamma$, ${\operatorname{Hom}}(E,E_0)=0$. Since $\deg_{E_0}(E(K_X))=\deg_{E_0}(E)+{\operatorname{rk}}E {\operatorname{rk}}E_0 (K_X,H)<0$, we get ${\operatorname{Ext}}^2(E,E_0)={\operatorname{Hom}}(E_0,E(K_X))^{\vee}=0$. Hence $-\chi(e,e_0) \geq 0$.
\[prop:fine\] $M_H(\gamma)$ is compact and there is a universal family on $M_H(\gamma) \times X$.
Since $\deg_{e_0}(e)={\operatorname{rk}}e_0 (c_1(e),H)-
{\operatorname{rk}}e (c_1(e_0),H)=1$, ${\operatorname{rk}}e$ and $(c_1(e),H)$ are relatively prime. Hence there is a universal family.
In order to construct a correspondence, we consider $E_0$-twisted coherent systems. Let ${\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ be the moduli space of $E_0$-twisted coherent systems: $${\operatorname{Syst}}(E_0^{\oplus n},\gamma):=\{(E,V)|E \in M_H(\gamma), V \subset
{\operatorname{Hom}}(E_0,E), \dim V=n \}.$$ ${\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ is a projective scheme over $M_H(\gamma)$ (cf. [@L:1]).
We set $$M_H(\gamma)_i:=\{E \in M_H(\gamma)|
\dim {\operatorname{Hom}}(E_0,E)=i \}.$$ If $i \geq n$, then ${\operatorname{Syst}}(E_0^{\oplus n},\gamma)
\times_{M_H(\gamma)} M_H(\gamma)_i \to M_H(\gamma)_i$ is $Gr(i,n)$-bundle.
[@Y:5 Lem. 2.1]\[lem:basic\] For $E \in M_H(\gamma)$ and $V \subset {\operatorname{Hom}}(E_0,E)$,
1. $ev:V \otimes E_0 \to E$ is injective and ${\operatorname{coker}}(ev)$ is stable.
2. $ev:V \otimes E_0 \to E$ is surjective in codimension 1 and $\ker(ev)$ is stable.
\[lem:D\] If $ev:V \otimes E_0 \to E$ is surjective in codimension 1, then
1. $D(E):={\cal E}xt^1(V \otimes E_0 \to E,{\cal O}_X)$ is a stable sheaf of $\deg_{E_0^{\vee}}D(E)=1$.
2. ${\operatorname{Ext}}^1(E_0,E)=0$.
In particular $\chi(\gamma_0,\gamma) \geq n$.
We have an exact sequence $${\cal E}xt^1({\operatorname{im}}(ev) \to E,{\cal O}_X) \to
{\cal E}xt^1(V \otimes E_0 \to E,{\cal O}_X) \to
{\cal H}om(\ker(ev),{\cal O}_X) \to
{\cal E}xt^2({\operatorname{im}}(ev) \to E,{\cal O}_X).$$ By Lemma \[lem:basic\], $\ker (ev)$ is stable and ${\operatorname{coker}}(ev)$ is of $0$-dimensional. Then ${\cal E}xt^1({\operatorname{im}}(ev) \to E,{\cal O}_X) \cong
{\cal E}xt^1({\operatorname{coker}}(ev),{\cal O}_X)=0$ and ${\cal E}xt^2({\operatorname{im}}(ev) \to E,{\cal O}_X) \cong
{\cal E}xt^2({\operatorname{coker}}(ev),{\cal O}_X)$ is of $0$-dimensional. Hence $D(X)$ is stable.
We next show that ${\operatorname{Ext}}^1(E_0,E)=0$. Since $\ker(ev)$ is stable, we get $${\operatorname{Ext}}^2(E_0,\ker(ev))={\operatorname{Hom}}(\ker(ev),E_0(K_X))^{\vee}=0.$$ Combining the fact ${\operatorname{Ext}}^1(E_0,E_0)=0$, we see that ${\operatorname{Ext}}^1(E_0,{\operatorname{im}}(ev))=0$. Since ${\operatorname{Ext}}^1(E_0,{\operatorname{coker}}(ev))=0$, we get ${\operatorname{Ext}}^1(E_0,E)=0$.
${\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ is smooth and $\dim {\operatorname{Syst}}(E_0^{\oplus n},\gamma)=
\dim M_H(\gamma)-n(n-\chi(\gamma_0,\gamma))$.
Let $(E,V) \in {\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ be a $E_0$-twisted coherent system. Since $V \subset {\operatorname{Hom}}(E_0,E)$, we have a homomorphism $${\operatorname{Hom}}(V \otimes E_0,V \otimes E_0) \to
{\operatorname{Hom}}(V \otimes E_0,E) \to {\operatorname{Ext}}^1(V \otimes E_0 \to E,E).$$ Then the cokernel is the Zariski tangent space of ${\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ and the obstruction space is ${\operatorname{Ext}}^2(V \otimes E_0 \to E,E)$. If ${\operatorname{rk}}(\gamma-n\gamma_0) \geq 0$, then ${\operatorname{Ext}}^2(V \otimes E_0
\overset{ev}{\to} E,E) \cong {\operatorname{Ext}}^2({\operatorname{coker}}(ev),E)=0$. If ${\operatorname{rk}}(\gamma-n\gamma_0) < 0$, then by using Lemma \[lem:D\] and an exact sequence $${\operatorname{Ext}}^1(V \otimes E_0,E) \to {\operatorname{Ext}}^2(V \otimes E_0 \to E,E) \to {\operatorname{Ext}}^2(E,E),$$ we see that ${\operatorname{Ext}}^2(V \otimes E_0 \to E,E)=0$. Hence ${\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ is smooth. Then we see that $$\begin{split}
\dim {\operatorname{Syst}}(E_0^{\oplus n},\gamma)&=
\dim {\operatorname{Ext}}^1(V \otimes E_0 \to E,E)-\dim PGL(V)\\
&=-\chi(E,E)+n\chi(E_0,E)-n^2\\
&=\dim M_H(\gamma)-n(n-\chi(\gamma_0,\gamma)).
\end{split}$$
\[prop:Gr\] We set $m:=-\chi(\gamma,\gamma_0)$.
1. If ${\operatorname{rk}}\gamma \geq n {\operatorname{rk}}\gamma_0$, then ${\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ is a $Gr(m+n,n)$-bundle over $M_H(\gamma-n\gamma_0)$.
2. If ${\operatorname{rk}}\gamma < n {\operatorname{rk}}\gamma_0$, then ${\operatorname{Syst}}(E_0^{\oplus n},\gamma) \cong
{\operatorname{Syst}}((E_0^{\vee})^{\oplus n},n\gamma_0^{\vee}-\gamma^{\vee})$. In particular ${\operatorname{Syst}}(E_0^{\oplus n},\gamma)$ is a $Gr(m+n,n)$-bundle over $M_H(n\gamma_0^{\vee}-\gamma^{\vee})$.
We first assume that ${\operatorname{rk}}\gamma \geq n {\operatorname{rk}}\gamma_0$. For $(E,V) \in {\operatorname{Syst}}(E_0^{\oplus n},\gamma)$, Lemma \[lem:basic\] implies that $ev:V \otimes E_0 \to E$ is injective and ${\operatorname{coker}}(ev)$ is stable. Thus we have a morphism $\pi_n:{\operatorname{Syst}}(E_0^{\oplus n},\gamma) \to
M_H(\gamma-n \gamma_0)$. Conversely for $G \in M_H(\gamma-n\gamma_0)$ and an $n$-dimensional subspace $U$ of ${\operatorname{Ext}}^1(G,E_0)$, we have an extension $$0 \to U^{\vee} \otimes E_0 \to E \to G \to 0$$ whose extension corresponds to the inclusion $U \hookrightarrow {\operatorname{Ext}}^1(G,E_0)$. Then $E$ is stable. Since $\dim {\operatorname{Ext}}^1(G,E_0)=-\chi(G,E_0)=-\chi(\gamma-n\gamma_0,\gamma_0)$ and there is a universal family, we see that $\pi_n$ is a (Zariski locally trivial) $Gr(m+n,n)$-bundle. Therefore we get our claim.
We next treat the second case. For $(E,V) \in {\operatorname{Syst}}(E_0^{\oplus n},\gamma)$, $D(E):={\cal E}xt^1(V \otimes E_0 \to E,{\cal O}_X)$ fits in an exact sequence $$0 \to E^{\vee} \to (V \otimes E_0)^{\vee} \to
D(E) \to {\cal E}xt^1(E,{\cal O}_X) \to 0.$$ Hence $(V \otimes E_0)^{\vee} \to D(E)$ defines a point of ${\operatorname{Syst}}((E_0^{\vee})^{\oplus n},n\gamma_0^{\vee}-\gamma^{\vee})$. Thus we get a morphism $\psi:{\operatorname{Syst}}(E_0^{\oplus n},\gamma) \to
{\operatorname{Syst}}((E_0^{\vee})^{\oplus n},n\gamma_0^{\vee}-\gamma^{\vee})$. Conversely for $(F,U) \in
{\operatorname{Syst}}((E_0^{\vee})^{\oplus n},n\gamma_0^{\vee}-\gamma^{\vee})$, we get a homomorphism $U^{\vee} \otimes E_0 \to
{\cal E}xt^1(U \otimes E_0^{\vee} \to F,{\cal O}_X)$. It gives the inverse of $\psi$ (for more details, see [@K-Y:1 Prop. 5.128]).
1. If ${\operatorname{rk}}(\gamma-\chi(\gamma_0,\gamma)\gamma_0) \geq 0$, then $\overline{M_H(\gamma)_i} =\emptyset$ for ${\operatorname{rk}}(\gamma-i\gamma_0)<0$.
2. If ${\operatorname{rk}}(\gamma-\chi(\gamma_0,\gamma)\gamma_0)<0$, then $M_H(\gamma)_{\chi(\gamma_0,\gamma)}=M_H(\gamma)$.
If $\dim(E_0,E)=i$ with ${\operatorname{rk}}(\gamma-i\gamma_0)<0$, then Lemma \[lem:D\] implies that $\chi(\gamma_0,\gamma) \geq i$. Hence ${\operatorname{rk}}(\gamma-\chi(\gamma_0,\gamma)\gamma_0)<0$. By Lemma \[lem:D\], ${\operatorname{Ext}}^1(E_0,E)=0$ for all $E \in M_H(\gamma)$. Hence $M_H(\gamma)_{\chi(\gamma_0,\gamma)}=M_H(\gamma)$.
By using Proposition \[prop:Gr\], we get the following theorem.
We set $\zeta:=\gamma(L_{e_0}(e))=\gamma-\chi(\gamma,\gamma_0) \gamma_0$ and $s:=-(K_X,c_1(e_0^{\vee} \otimes e))$. Assume that $n:=-\chi(\gamma,\gamma_0)>0$. Then $M_H(\gamma) \cong {\operatorname{Syst}}(E_0^{\oplus n},\zeta)$ and we get a morphism $\lambda_{\gamma_0,\gamma}:M_H(\gamma)
\to M_H(\zeta)$ by sending $E$ to a universal extension $$0 \to E_0 \otimes {\operatorname{Ext}}^1(E,E_0)^{\vee} \to
\lambda_{\gamma_0,\gamma}(E) \to E \to 0.$$ Hence we have a stratification $$M_H(\gamma)= \coprod_{i \geq s}
\lambda_{\gamma_0,\gamma}^{-1}(M_H(\zeta)_i)$$ such that $\lambda_{\gamma_0,\gamma}^{-1}(M_H(\zeta)_i) \to M_H(\zeta)_i$ is a $Gr(i,n)$-bundle. In particular, $M_H(\gamma)_0 \to M_H(\zeta)_{n}$ is an isomorphism for $n \geq s$.
If $0>\chi(e_0,e)=-k\geq -s$, then $M_H(\gamma(e))\to M_H(\gamma(L_{e_0}(e)))$ is birationally $Gr(s,k)$-bundle. In particular, if $\chi(e_0,e)=-s$, then $M_H(\gamma(e)) \to M_H(\gamma(L_{e_0}(e)))$ is a birational map.
Assume that $(X,H)=({\Bbb P}^1 \times {\Bbb P}^1,
{\cal O}_{{\Bbb P}^1 \times {\Bbb P}^1}(1,n))$, $n>0$. We set $L:={\cal O}_{{\Bbb P}^1 \times {\Bbb P}^1}(-1,n+1)$. Then $(L,H)=1$, $s=(L,-K_X)=2n$ and $\chi(L)=0$. Hence $M_H(1+r,L,r) \cong Gr(2n,r)$.
Virtual Hodge polynomial
------------------------
We set $a:=-\chi(\gamma,\gamma_0)$. Assume that ${\operatorname{rk}}(\gamma-\chi(\gamma_0,\gamma)\gamma_0) \geq 0$. We shall consider vitrual Hodge polynomial of $M_H(\gamma+k \gamma_0)_i$. For an algebraic set $Z$, $$e(Z):=\sum_{p,q}(-1)^{p+q}h^{p,q}(Z)x^p y^q$$ is the virtual Hodge polynomial of $Z$ (cf. [@D-K:1]). We set $t:=xy$. Then $$\begin{split}
e(M_H(\gamma+k \gamma_0)_j)&=
e(Gr(a+j-k,j))e(M_H(\gamma+(k-j) \gamma_0)_0)\\
&=\frac{[a+j-k]!}{[a-k]![j]!}e(M_H(\gamma+(k-j) \gamma_0)_0),
\end{split}$$ where $$[n]:=\frac{t^n-1}{t-1}, \quad [n]!:=[n][n-1]\cdots[1].$$ By summing up all $e(M_H(\gamma+k \gamma_0)_k)$, we get $$\sum_k [a-k]! e(M_H(\gamma+k \gamma_0))y^k
=\left(\sum_j \frac{1}{[j]!}y^j \right)
\left(\sum_l [a-l]! e(M_H(\gamma+l \gamma_0)_0)y^l \right).$$ Since $$\left(\sum_j \frac{1}{[j]!}y^j \right)^{-1}=
\sum_j \frac{(-1)^jt^{j(j-1)/2}}{[j]!}y^j,$$ we get that
If ${\operatorname{rk}}(\gamma-\chi(\gamma_0,\gamma)\gamma_0) \geq 0$, then $$e(M_H(\gamma+l \gamma_0)_0)=\sum_{j \geq 0} (-1)^{j} t^{j(j-1)/2}
\frac{[a+j-l]!}{[a-l]![j]!}e(M_H(\gamma+(l-j) \gamma_0)).$$ In particular $$e(M_H(\gamma+k \gamma_0)_i)=\sum_{j \geq 0} (-1)^{j} t^{j(j-1)/2}
\frac{[a-k+i+j]!}{[a-k]![i]![j]!}e(M_H(\gamma+(k-i-j) \gamma_0)).$$ Since $M_H(\gamma+l \gamma_0)_0 = \emptyset$ for $a-s<l \leq a$, we also get the following relations: $$\sum_{j \geq 0} (-1)^{j} t^{j(j-1)/2}
\frac{[a+j-l]!}{[a-l]![j]!}e(M_H(\gamma+(l-j) \gamma_0))=0$$ for $a-s<l \leq a$.
Examples on ${\Bbb P}^2$
------------------------
From now on, we assume that $X$ is ${\Bbb P}^2$. Then $s=-(K_X,{\cal O}_X(1))=3$. Hence we get the following relations: $$\label{eq:relation2}
\begin{split}
&\sum_{j \geq 0} (-1)^{j} t^{j(j-1)/2}
e(M_H(\gamma+(a-j) \gamma_0))=0,\\
&\sum_{j \geq 0} (-1)^{j} t^{j(j-1)/2}
[j+1]e(M_H(\gamma+(a-1-j) \gamma_0))=0,\\
&\sum_{j \geq 0} (-1)^{j} t^{j(j-1)/2}
\frac{[j+2][j+1]}{[2]!}e(M_H(\gamma+(a-2-j) \gamma_0))=0.
\end{split}$$
By a simple calculation, we get
\[prop:relation\] $$\label{eq:relation}
\begin{split}
e(M_H(\gamma+(a-2) \gamma_0)) & =\sum_{j \geq 0} (-1)^{j} t^{(j+1)j/2}
\frac{[j+3][j+2]}{[2]!}e(M_H(\gamma+(a-3-j) \gamma_0)),\\
e(M_H(\gamma+(a-1) \gamma_0))& =\sum_{j \geq 0} (-1)^{j}
t^{(j+1)j/2}[j+3][j+1]e(M_H(\gamma+(a-3-j) \gamma_0)),\\
e(M_H(\gamma+a \gamma_0))&=\sum_{j \geq 0} (-1)^{j} t^{(j+1)j/2}
\frac{[j+2][j+1]}{[2]!}e(M_H(\gamma+(a-3-j) \gamma_0)).
\end{split}$$
Assume that $E_0:={\cal O}_X$. We set $\gamma:=\gamma({\cal O}_X(1))$. Then $$\begin{split}
M_H(\gamma-a \omega-\gamma_0)&=\{{\cal O}_l(1-a) |
\text{ $l$ is a line on ${\Bbb P}^2$} \}\\
& \cong {\Bbb P}^2.
\end{split}$$ Hence $M_H(\gamma-a \omega)_1$, $a \geq 2$ is a ${\Bbb P}^a$-bundle over ${\Bbb P}^2$. By the morphism $M_H(\gamma-a \omega) \to M_H(\gamma-a\omega+a\gamma_0)$, the fibers of $M_H(\gamma-a \omega)_1 \to {\Bbb P}^2$ are contracted.
If $a=2$, then $M_H(\gamma-2\omega+2\gamma_0) \cong M_H(\gamma^2-\gamma_0) \cong
{\Bbb P^2}$. That is, $E \in M_H(\gamma-2\omega+2\gamma_0)$ fits in a universal extension $$0 \to {\cal O}_X^{\oplus 3} \to E \to {\cal O}_l(-1) \to 0.$$ Moreover we see that $M_H(\gamma-2\omega+i\gamma_0)$, $i=0,1$ are ${\Bbb P}^2$-bundle over $M_H(\gamma-2\omega+2\gamma_0) \cong {\Bbb P}^2$.
If $a=3$, then $M_H(\gamma-3\omega) \to M_H(\gamma-3\omega+3\gamma_0)$ is the blow-up of $M_H(\gamma-3\omega+3\gamma_0)_4 \cong M_H(\gamma-3\omega-\gamma_0)$. This was obtained by Drezet [@D:3 IV].
By [@E-S:1] and [@Y:1], we know $e(M_H(r,H,\chi))$ for $r=1,2$. By using Proposition \[prop:relation\], we get the following:
$$\begin{split}
e(M_H(1,H,0))&=1+2t+5t^2+6t^3+5t^4+2t^5+t^6,\\
e(M_H(2,H,1))&=1+2t+6t^2+9t^3+12t^4+9t^5+6t^6+2t^7+t^8,\\
e(M_H(3,H,2))&=1+2t+5t^2+8t^3+10t^4+8t^5+5t^6+2t^7+t^8,\\
e(M_H(4,H,3))&=1+t+3t^2+3t^3+3t^4+t^5+t^6.
\end{split}$$
$$\begin{split}
e(M_H(1,H,-1))&=1+2t+6t^2+10t^3+13t^4+10t^5+6t^6+2t^7+t^8,\\
e(M_H(2,H,0))&=1+2t+6t^2+13t^3+24t^4+35t^5+41t^6+
35t^7+24t^8+13t^9+6t^{10}+2t^{11}+t^{12},\\
e(M_H(3,H,1))&=1+2t+6t^2+12t^3+24t^4+38t^5+54t^6+
59t^7+54t^8+38t^9+24t^{10}+12t^{11}+6t^{12}+2t^{13}+t^{14},\\
e(M_H(4,H,2))&=1+2t+5t^2+10t^3+18t^4+28t^5+38t^6+
42t^7+38t^8+28t^9+18t^{10}+10t^{11}+5t^{12}+2t^{13}+t^{14},\\
e(M_H(5,H,3))&=1+t+3t^2+5t^3+8t^4+10t^5+12t^6+
10t^7+8t^8+5t^9+3t^{10}+t^{11}+t^{12}.
\end{split}$$
If $E_0:=\Omega_X(1)$, then $\deg_{E_0}({\cal O}_X)=1$. We set $\gamma=\gamma({\cal O}_X)$. Then
- $M_H(\gamma-a \omega) \to M_H(\gamma-a \omega+2a \gamma_0)$ is a closed immersion for $a \geq 2$.
- If $a=2$, then $M_H(\gamma-2\omega+\gamma_0) \to
M_H(\gamma-2\omega+4 \gamma_0)$ is the blow-up along $M_H(\gamma-2\omega)$.
Here we remark that Drezet showed that $M_H(\gamma-2\omega+4\gamma_0)=M_H(9,-4H,-1)
\cong Gr(6,2)$ (see [@D:1 Appendice]). Since $e(M_H(1,0,-1))=1+2t+3t^3+2t^3+t^4$ and $e(M_H(3,-H,-1))=e(M_H(3,H,2))$, Proposition \[prop:relation\] implies that $$\begin{split}
e(M_H(3,-H,-1))&=1+2t+5t^2+8t^3+10t^4+8t^5+5t^6+2t^7+t^8,\\
e(M_H(5,-2H,-1))&=1+2t+5t^2+8t^3+13t^4+14t^5+13t^6+8t^7+5t^8+2t^9+t^{10},\\
e(M_H(7,-3H,-1))&=1+2t+4t^2+6t^3+9t^4+10t^5+9t^6+6t^7+4t^8+2t^9+t^{10},\\
e(M_H(9,-4H,-1))&=1+t+2t^2+2t^3+3t^4+2t^5+2t^6+t^7+t^8
(=e(Gr(6,2))).
\end{split}$$
### Line bundles on $M_H(\gamma)$
Let $p_{M_H(\gamma(e))}:M_H(\gamma(e)) \times X \to
M_H(\gamma(e))$ and $q:M_H(\gamma(e)) \times X \to X$ be projections, and let ${\cal E}$ be a universal family on $M_H(\gamma(e)) \times X$. We define a homomorphism $\theta_e:e^{\perp} \to {\operatorname{Pic}}(M_H(\gamma(e)))$ by $$\theta_e(x):=\det p_{M_H(\gamma(e))!}({\cal E}^{\vee} \otimes q^*(x)),$$ where $e^{\perp}:=\{x \in K(X)|\chi(e,x)=0 \}$. The following is a special case of Drezet’s results.
[@D:2] Assume that $\dim M_H(\gamma(e))=1-\chi(e,e)>0$. Then $\theta_e$ is surjective and
1. $\theta_e$ is an isomorphism, if $\chi(e,e)<0$,
2. $\ker \theta_e={\Bbb Z}e_0$, if $\chi(e,e_0)=0$.
We set $\tilde{e}:=L_{e_0}(e)$. By a simple calculation, we see that the following diagram is commutative: $$\begin{CD}
e^{\perp} @<{R_{e_0}}<< \tilde{e}^{\perp}/e_0\\
@V{\theta_e}VV @VV{\theta_{\tilde{e}}}V \\
{\operatorname{Pic}}(M_H(\gamma(e))) @<<{\lambda_{\gamma(e_0),\gamma(e)}^*}<
{\operatorname{Pic}}(M_H(\gamma(\tilde{e})))
\end{CD}$$ We set $\alpha_e:=-({\operatorname{rk}}e) {\cal O}_H+\chi(e,{\cal O}_H){\Bbb C}_P$. Then it gives a map to the Uhlenbeck compactification [@Li:1]. $\beta_e:=R_{e_0}(\alpha_{\tilde{e}})$ gives the map $\lambda_{\gamma(e_0),\gamma(e)}:
M_H(\gamma(e)) \to M_H(\gamma(\tilde{e}))$.
- If $E_0={\cal O}_X$, ${\operatorname{rk}}e>0$ and $\chi(e,e_0)<0$, then the nef. cone of $M_H(\gamma(e))$ is generated by $\alpha_e$ and $\beta_e$.
This is a generalization of [@S:1].
For $\gamma:=(3,H,5-a)$, we set $\gamma_0:=(1,0,1)$, $\gamma_1:=\gamma(\Omega_X(1))=(2,-H,0)$, $\delta:=\gamma+a\gamma_0$ and $\eta:=\gamma^{\vee}+(2a-3)\gamma_1$. Then we get the following diagram:
$$\begin{matrix}
&& {M}_H(\gamma)&&\leftarrow \cdots \rightarrow &&
{M}_{H}(\gamma^{\vee})&&&\cr
&\llap{$\lambda_{\gamma_0,\gamma}$}
\swarrow&&\searrow\rlap{}&&
\llap{}\swarrow
&&\searrow\rlap{$\lambda_{\gamma_1,\gamma^{\vee}}$}\cr
M_H(\delta) &&
&& N_H(\gamma)&&&&
{M}_H(\eta)\cr
\end{matrix}$$
where $N_H(\gamma)$ is the Uhlenbeck compactification of $M_H(\gamma)^{\mu\text{-}s,loc}$. $M_H(\gamma^{\vee})$ contains ${\Bbb P}^{2a-3}$-bundle over $M_H(1,0,2-a)$ and $\lambda_{\gamma_0,\gamma^{\vee}}$ contracts the fibers. ${\lambda_{\gamma_0,\gamma}}_{|M_H(\gamma)_i}$ is a $Gr(a-2+i,a-2)$-bundle over $M_H(\delta)_{a-2+i} \cong M_H(\gamma-i\gamma_0)_0$. Then it is easy to see that $M_H(3,H,5-a) \not \cong M_H(3,-H,2-a)$.
[*Acknowledgement.*]{} A starting point of this note is [@E-S:1]. I would like to thank M. Maruyama for giving me a preprint version of [@E-S:1].
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|
---
abstract: 'We report a simultaneous 44 and 95 GHz class I methanol maser survey toward 144 sources from the 95 GHz class I methanol maser catalog. The observations were made with the three telescopes of the Korean very long baseline interferometry network operating in single-dish mode. The detection rates are 89% at 44 GHz and 77% at 95 GHz. There are 106 new discoveries at 44 GHz. Comparing the previous 95 GHz detections with new observations of the same transitions made using the Purple Mountain Observatory 13.7 m radio telescope shows no clear evidence of variability on a timescale of six years. Emission from the 44 and 95 GHz transitions shows strong correlations in peak velocity, peak flux density, and integrated flux density, indicating that they are likely cospatial. We found that the peak flux density ratio $S_{\rm pk,95}$/$S_{\rm pk,44}$ decreases as the 44 GHz peak flux density increases. We found that some class I methanol masers in our sample could be associated with infrared dark clouds, while others are associated with H [ii]{} regions, indicating that some sources occur at an early stage of high-mass star formation, while others are located toward more evolved sources.'
author:
- Wenjin Yang
- Ye Xu
- Yoon Kyung Choi
- 'Simon P. Ellingsen'
- 'Andrej M. Sobolev'
- Xi Chen
- Jingjing Li
- Dengrong Lu
title: '44 GHz methanol masers: Observations toward 95 GHz methanol masers'
---
Introduction {#sec:intro}
============
Methanol masers are commonly found in star formation regions (SFRs). The methanol molecule has numerous transitions within the centimeter- and millimeter-wavelength ranges and hence provides a powerful tool for the study of star-forming regions. According to the empirical scheme of [@1991ASPC...16..119M], methanol masers are divided into two categories. Class II methanol masers, such as the 6.7 GHz (5$_{1}$–6$_{0}$A$^{+}$) and 12.2 GHz (2$_{0}$–3$_{-1}$E) transitions [@1987Natur.326...49B; @1991ApJ...380L..75M], are found in close proximity to ultracompact H [II]{} regions, infrared sources, and OH masers [@1998MNRAS.301..640W], and are known to be pumped by radiative processes [@1992MNRAS.259..203C; @2005MNRAS.360..533C]. In contrast, class I methanol masers are thought to be tracers of shocked regions related to outflows and expanding H[ii]{} regions [e.g., @2013ApJ...763....2G; @2014MNRAS.439.2584V] and are produced by collisional pumping. On the basis of the latest class I methanol maser models , such masers can be divided into several families: the (J+1)$_{0}$–J$_{1}$-A type, the (J+1)$_{-1}$–J$_{0}$-E type, the J$_2$–J$_1$-E series at 25 GHz, and the J$_{-2}$–(J$-$1)$_{-1}$-E series at 9 GHz. Within the Milky Way, methanol maser emission from the 44 GHz 7$_{0}$–6$_{1}$A$^{+}$ [e.g., @1985ApJ...288L..11M; @1992AZh....69.1002K; @1994MNRAS.268..464S] is the most widespread and strongest transition, with the 36 GHz 4$_{-1}$–3$_{0}$ E [e.g., @1989ApJ...339..949H; @1996AA...314..615L], 84 GHz 5$_{-1}$–4$_{0}$ E [e.g., @2001ARep...45...26K; @2019MNRAS.484.5072B], and 95 GHz 8$_{0}$–7$_{1}$A$^{+}$ [e.g., @2011ApJS..196....9C; @2013ApJS..206....9C] transitions also being relatively common.
The first detection of 44 GHz methanol masers was reported by [@1985ApJ...288L..11M]. Subsequent single-dish surveys searching for 44 GHz methanol masers have targeted galactic SFRs [@1990ApJ...354..556H], water masers [@1990AA...240..116B], cold Infrared Astronomical Satellite (IRAS) sources [@1992AZh....69.1002K], H [ii]{} regions [@1994MNRAS.268..464S], 6.7 GHz methanol masers [@2010AA...517A..56F], intermediate-mass young stellar objects [YSOs; @2011ApJS..196...21B], and 2.122 $\mu$m emission [@2012AJ....144..151L]. These searches have laid the crucial groundwork for establishing the nature of the 44 GHz methanol maser transition. Recently, a series of interferometric observations targeting outflows [@2016ApJS..222...18G] and high-mass protostellar objects [HMPOs; @2017ApJS..233....4R], have revealed the relationship between the class I methanol masers and other star-forming phenomena at high angular resolution, which has allowed more detailed investigations of the evolutionary stage and the maser excitation conditions. In addition, variability of class I masers has only been the subject of limited studies, and to date, no detectable variability has been reported [e.g., @2012ApJ...759..136C].
The 8$_{0}$–7$_{1}$A$^{+}$ transition of methanol at 95 GHz is a member of the same transition family as the 44 GHz maser transition. Systematic searches for 95 GHz methanol masers toward Spitzer Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE) extended green objects [EGOs; @2011ApJS..196....9C; @2013ApJS..206....9C], molecular outflow sources [@2013ApJ...763....2G], cross-matching between the GLIMPSE point sources and the Bolocam Galactic Plane Survey (BGPS) sources [@2012ApJS..200....5C], and BGPS sources [@2017ApJS..231...20Y] have significantly increased the number of known 95 GHz methanol masers. [@2017ApJS..231...20Y] compiled a catalog summarizing the past two decades of searches for 95 GHz methanol masers. This catalog contains 481 maser sources and a further 37 candidates, and it currently represents the largest and most complete catalog of 95 GHz methanol masers.
In this paper, we report a search for 44 GHz methanol masers targeting known 95 GHz masers. This search was made using the three 21 m antennas from the Korean very long baseline interferometry network (KVN) in single-dish telescope mode. In Section \[sec:obs\] we describe the source selection and observations, then we give the results of the detections, compare them with the previous observations in Section \[sec:results\], and follow this by discussing the flux density ratio between the two maser transitions and their occurrence at different stages of the star formation process (Section \[sec:discuss\]) and a summary in Section \[sec:sum\].
Source Selection and Observations {#sec:obs}
=================================
Source selection {#sec:source}
----------------
The 95 GHz methanol maser sources in which we searched for emission from the 44 GHz methanol transition were selected using the following criteria: (1) The sources are listed in the catalog of 95 GHz methanol masers and maser candidates [@2017ApJS..231...20Y]. The @2017ApJS..231...20Y catalog of 95 GHz methanol masers contains 481 sources and 37 maser candidates, identified in searches toward BGPS sources [version 1.0.1; @2010ApJS..188..123R] and other searches undertaken over the past two decades [@1994AAS..103..129K; @1995AZh....72...22V; @2000MNRAS.317..315V; @2005MNRAS.359.1498E; @2006ARep...50..289K; @2010AA...517A..56F; @2011ApJS..196....9C; @2012ApJS..200....5C; @2013ApJS..206....9C; @2013ApJ...763....2G]. Sources in which only broad spectral components [$>$ 2.5 ; @2017ApJS..231...20Y] are observed at 95 GHz were considered maser candidates. (2) Sources with declination greater than $-$25$^\circ$ (which are observable with the KVN 21m antennas). (3) Sources with previously reported 44 GHz maser detections from single-dish observations were excluded [@1990ApJ...354..556H; @1990AA...240..116B; @1992AZh....69.1002K; @1994MNRAS.268..464S; @2006ARep...50..289K; @2008AJ....135.1718P; @2010AA...517A..56F; @2011ApJS..196...21B; @2012AJ....144..151L; @2015ApJS..221....6K]. (4) Sources with previous KVN observations, separated by less than 30$\arcsec$ from a @2017ApJS..231...20Y 95 GHz source from targeting of class I masers [private communication; @1994ApJS...91..659K; @1998AA...336..339M; @2005AA...432..737P; @2006ApJ...641..389R; @2007ApJ...656..255P; @2008AJ....136.2391C; @2010MNRAS.404.1029C; @2010MNRAS.409..913G] are also excluded. Based on these criteria, a total of 144 95 GHz methanol maser sources were identified (see Table \[tab:144\_obs\]). Of these, 123 ($\sim$ 85%) are sources selected from [@2012ApJS..200....5C] and [@2017ApJS..231...20Y] that target the position of peak emission of associated BGPS sources [version 1.0.1; @2010ApJS..188..123R] whose beam-averaged H$_2$ column density is greater than 10$^{22.1}$ cm$^{-2}$, the others are targeting star-forming regions or outflow/EGO sources with a 95 GHz methanol maser detection. Ten of the 144 target sources are 95 GHz methanol maser candidates, and these are identified with a double dagger in Table \[tab:144\_obs\].
KVN observations {#sec:kvnobs}
----------------
The observations of the 44.06943 GHz (7$_{0}$–6$_{1}$A$^{+}$) and 95.16946 GHz (8$_{0}$–7$_{1}$A$^{+}$) class I methanol maser transitions took place in the period 2016 October to December using the KVN 21m radio antennas in single-dish telescope mode. There are three KVN stations: KVN Yonsei telescope (Seoul), KVN Ulsan telescope (Ulsan), and KVN Tamna telescope (Jeju island). Each of the KVN antennas are equipped with a multifrequency receiving systems able to simultaneously operate in the 43 and 86 GHz bands, which enables observing 44 and 95 GHz methanol transitions at the same time. The beam size and antenna efficiency of each antenna are listed in Table \[tab:tele\_para\]. Telescope pointing observations were performed at least once an hour, resulting in a pointing accuracy better than $\sim$6$\arcsec$. Digital spectrometers with 4096 spectral channels were used as back-ends, each with a bandwidth of 64 MHz, which provides velocity resolutions of 0.11 and 0.05 at 44 GHz and 95 GHz, respectively.
A low-order polynomial baseline was fit to the line-free channels in the spectra. Hanning smoothing was applied once (twice) to improve the signal-to-noise ratio for the 44 GHz (95 GHz) methanol line spectra. This results in a velocity-channel width of 0.21 and 0.20 for the 44 and 95 GHz transitions, respectively. The system temperature varied between 150 K and 300 K depending on the weather conditions and the elevation of the telescope. The on-source integration times were between 5 and 15 minutes, which achieved a typical 1$\sigma$ noise level of between 0.4 and 1 Jy for both of the methanol transitions at the $\sim$0.2 resolution obtained after smoothing. The data were reduced and analyzed using the GILDAS/CLASS package [@2005sf2a.conf..721P]. To characterize the spectra, we undertook Gaussian fitting of each peak in the spectrum for each source.
Twenty-four sources which were observed in bad weather and were not reobserved. These sources are indicated by a dagger in Table \[tab:144\_obs\]. These sources may have unreliable flux densities for any detected emission, and they have been excluded from our analysis of intensity ratios in Section \[sec:95\_compare\], \[sec:peak\_ratio\], \[sec:iras\], \[sec:sec\_irdc\], and \[sec\_hii\].
PMO observations
----------------
The current KVN search did not detect 95 GHz methanol emission toward a small number of targets for which detections of this transition have been reported previously (see Section \[sec:kvn\_results\] for details). The majority of 95 GHz methanol masers/candidates as KVN targets were detected using the Purple Mountain Observatory (PMO) 13.7m radio telescope with a typical on-source integration time of 10–20 minutes [@2017ApJS..231...20Y], and have a peak flux density greater than 4 Jy. To eliminate potential effects due to different beam sizes and effectively determine whether these 95 GHz methanol masers show intrinsic intensity variability, we reobserved 13 sources using the PMO 13.7m radio telescope. All the 13 sources (see Table \[tab:f\_comparison\]) were observed using the PMO 13.7m in previous studies with a 95 GHz peak flux density stronger than 4 Jy. Nine of them were not detected in the current KVN 21m observations, the other four sources were randomly selected among the KVN detections.
Observations toward thirteen 95 GHz class I methanol maser sources were made with the PMO 13.7m radio telescope in Delingha, China on 2018 December 4 and 6. A 3$\times$3 multibeam sideband-separating superconducting spectroScopic array receiver system was used to observe the 95 GHz methanol transition. This receiver operates over a frequency range of 85–115 GHz, and the beam in the middle of the first row of the receiver was pointed at the target position in most cases. The spectra were recorded using a fast Fourier transform spectrometer with 16,384 spectral channels across a bandwidth of 1 GHz, yielding a frequency resolution of 61 kHz and an effective velocity resolution of 0.19 km s$^{-1}$ for the 95 GHz methanol transition. The system temperature for the 95 GHz methanol maser observations was in the range 135–200 K, depending on the weather conditions and telescope elevation. Most sources were observed in a position-switching mode with an off-position offset by 15$\arcmin$ in R.A. For some sources, a different reference position was chosen to ensure that the reference spectrum was free from emission. The pointing rms was better than 5$\arcsec$. The standard chopper-wheel calibration technique [@1976ApJS...30..247U] was applied to measure the antenna temperature, $T^{*}_{\rm A}$, corrected for atmospheric absorption. The beam size of the telescope is approximately 55$\arcsec$ at 95 GHz, with a main-beam efficiency $\eta_{\rm mb}$ of 62%. The antenna efficiency is 48%, corresponding to a factor of 39.0 Jy K$^{-1}$ to convert antenna temperature into flux density. An on-source integration time of 15 minutes for each source achieved rms noise levels of approximately 1.4 Jy for the 95 GHz methanol transition.
The data were reduced and analyzed using the GILDAS/CLASS package [@2005sf2a.conf..721P]. The data reduction procedure was similar to that of the KVN observations. We only analyzed the data for the one beam used to track the target position. A low-order polynomial baseline subtraction and Hanning smoothing (the velocity resolution is $\sim$ 0.39 after smoothing, reaching a typical rms noise levels of 0.9 Jy) were performed on the averaged spectrum for each target source.
Results {#sec:results}
=======
KVN Detection {#sec:kvn_results}
--------------
We only consider a source/transition to be a detection where the peak intensity is greater than the 3$\sigma$ noise level in the spectrum. A total of 144 sites were searched for 44 and 95 GHz methanol masers simultaneously, with 128 (89%) and 111 (77%) detections in the 44 and 95 GHz transitions, respectively. Table \[tab:144\_obs\] summarizes the position of each of the target sources in equatorial coordinates (J2000), whether emission was detected for each transition, whether 44 GHz emission was a new detection, and the noise level for the 44 and 95 GHz spectra, respectively. Tables \[tab:44\_para\] and \[tab:95\_para\] contain the list of all detected sources along with the fitted Gaussian parameters of their spectral features. The spectra of the 144 observed sources are shown in Figure \[fig:spectra\].
After cross-matching the latest and largest class I methanol maser catalog [@2019AJ....158..233L], a total of 106 detections at 44 GHz are the first reported observations of maser emission in the target sources. The exceptions are 17 sources detected in previous KVN observations [@2016ApJS..227...17K; @2018ApJS..236...31K; @2019ApJS..244....2K] and 7 sources with previous interferometric observations (indicated in Table \[tab:144\_obs\]). One source, G033.390+00.008, which is detected to have weak 44 GHz emission (peak flux density 2 Jy) by [@2018ApJS..236...31K], did not meet the criteria to be considered a detection in the current observations.
The line width of individual 44 GHz emission components obtained from Gaussian fitting ranges from 0.21 to 5.45 with a mean of 1.04 and a median of 0.74 . Approximately 80% (103/128) of the 44 GHz methanol spectra have one or more narrow components (line width $\leqslant$ 1 ). The line width of individual 95 GHz emission components ranges from 0.23 to 9.33 with a mean of 1.07 and a median of 0.82 . Approximately 82% (92/111) of the 95 GHz methanol detections have at least one component with a line width $\leqslant$ 1 .
The better velocity resolution of the current observations shows that nine of the sources at 95 GHz considered maser candidates by @2017ApJS..231...20Y [$\sim$ 0.39 after Hanning smoothing] because they only contained broad spectral components (line width of $>$ 2.5 ) do also exhibit narrower spectral components. With the exception of BGPS2190 (which is better fit with a single broad Gaussian profile), these sources have been fitted with multiple Gaussian components. From the current observations there are three sources (BGPS4048, BGPS5057, BGPS7208) at 44 GHz and six sources (BGPS2152, BGPS2190, BGPS4048, BGPS6502, NGC 7538IRAS11, BGPS7208) at 95 GHz that are best fit with only a single broad spectral component. BGPS7208 (also known as IRAS 23139+5939), has been observed with Very Large Array (VLA), which confirms that although the spectral profile is broad, it is still due to maser emission [@2017ApJS..233....4R]. Single-dish observations alone cannot identify whether these broad emission sources are due to maser or quasi-thermal emission, and future interferometric observations are required to determine this. For the purposes of our analysis, we have assumed that the detected 44 and 95 GHz methanol emission in these single broad profile sources is due to maser emission. In the unlikely event that all of these broader lines are found to be pure thermal emission, the number (fewer than 10) is too small to significantly influence our statistics.
In total, 107 sources have both 44 and 95 GHz emission, 21 sources have only 44 GHz emission, and 4 sources have only 95 GHz emission. Previous searches have only detected 95 GHz maser emission toward sources which also show 44 GHz masers [@2010AA...517A..56F; @2018ApJS..236...31K]. For the four sources where we have a 95 GHz methanol detection without corresponding 44 GHz emission (G183.72$-$3.66, G008.458$-$00.224, BGPS2190, BGPS7318), the 95 GHz emission is weak and only reaches the 3$\sigma$ requirement. One of the sources, G183.72$-$3.66 (also known as GGD4), was observed at 44 GHz in 2007 March/April using the VLA in the D configuration [@2016ApJS..222...18G], but no emission was detected. Additional observations of these four sources are required to confirm the 95 GHz detection, and higher resolution observations will be required to determine whether it is maser emission or thermal emission.
A total of 33 sources were not detected in the 95 GHz methanol emission in the current KVN search. There are several potential reasons for nondetections: (1) The majority of nondetected sources were previously observed using the PMO 13.7 m radio telescope, which has a beam size at 95 GHz of $\sim$ 55$\arcsec$. This is similar to the beam size of the KVN 21m antennas at 44 GHz, but significantly larger than that at 95 GHz, so masing regions detected in previous searches could be located outside the KVN beam at 95 GHz, but within the beam at 44 GHz. There are 21 sources with 44 GHz maser detection but without 95 GHz detection. (2) For weak masers, the limited integration time means that there is emission with a signal-to-noise ratio lower than three. This emission cannot be unambiguously distinguished from the noise in the spectrum. (3) The source may have exhibited variability in the 95 GHz class I methanol maser emission. (4) Some of the sources where a previous detection has been claimed based on data with poor signal-to-noise ratio may have been incorrect, in which case we do not expect to detect either 95 GHz or 44 GHz emission in these sources.
Comparison with previous detections
-----------------------------------
### 95 GHz methanol masers: single-dish observations {#sec:95_compare}
Excluding the 24 sources with unreliable flux densities due to observations in bad weather conditions (see Section \[sec:kvnobs\]), there are 99 detected 95 GHz methanol masers, and of these, 95 were previously detected in observations with the PMO 13.7m, while the other 4 were previously detected in observations with the Mopra 22m telescope. These 99 sources in KVN observations were smoothed to a similar velocity resolution as the previous detections, i.e., $\sim$ 0.2 for Mopra, and $\sim$ 0.4 for PMO. Figure \[fig:ff\] shows the peak flux density ratio of the 95 GHz methanol masers detected in the current KVN observations compared to previous observations with either PMO or Mopra. More than 85% of the sources have a peak flux density ratio between the KVN observations and previous PMO observations ($S_{\rm pk,KVN}$/$S_{\rm pk,PMO}$) that lies in the range 0.4–1.1. In addition to the flux calibration and pointing accuracy, which affect the ratio, different beam sizes during the observations will lead systematic bias in the ratio. The beam size of the PMO 13.7m is larger by a factor of two than that of the KVN 21m at 3 mm wavelength. This means that any maser emission that is offset from the pointing center of the observations will be relatively closer to the edge of the KVN beam than PMO beam. This will lower the value of $S_{\rm pk,KVN}$/$S_{\rm pk,PMO}$.
The beam size and velocity resolution of the Mopra 22m are $\sim$ 36$\arcsec$ and 0.22 , respectively, which are comparable to those of the KVN 21m. The Mopra observations (2009 August) were made approximately 7 yr prior to the KVN observations (2016 October–December). For the four sources observed with Mopra, the peak flux density ratio is close to one, suggesting no significant variability in the maser emission over that period.
In total, 13 sources that were previously detected in PMO observations were reobserved using the PMO 13.7m. Nine of them were not detected in the current KVN observations, and four were chosen randomly in the ratio of $S_{\rm pk,KVN}$/$S_{\rm pk,PMO}$. Table \[tab:f\_comparison\] compares the new PMO observations with the KVN results and previous PMO observations. Some sources were observed more than once in order to increase the on-source integration time to improve the signal-to-noise ratio. Because there is no evidence of variability in the class I methanol masers, we averaged all the observed spectra. Figure \[fig:95reobs\] shows a comparison of the spectra for each source, with red and black representing the current and previous PMO spectra, respectively. Except for BGPS1917 and BGPS2011, whose peak flux densities are consistent between the two epochs of PMO observations, the peak flux densities of other sources are all slightly lower in the current data than before.
Of the nine sources that are not detected with the KVN at 95 GHz, five were detected in the current PMO 13.7m observations. This is consistent with the KVN nondetection arising because the class I methanol maser emission in these sources is offset from the pointing center and lies at the edge of or beyond the narrower KVN beam. For a further two sources (BGPS7252 and BGPS7351), the PMO peak flux density is significantly stronger than that of KVN data, and again, this is likely because the maser emission in these sources is offset from the pointing center and toward the edge of the KVN beam. Two sources (BGPS1917 and BGPS4252) that show a peak flux density in KVN observations that is twice as strong as the PMO detection. When we smooth the KVN spectra to the same velocity resolution as the PMO, the intensity is almost the same, suggesting that these sources have narrow spectral components.
No methanol emission was detected in the current PMO or KVN observations for four sources (BGPS2784, BGPS3319, BGPS4063, and G031.013+00.781). For BGPS3319 there was no detected 44 GHz methanol emission either, suggesting that this may not be a class I methanol maser source and that the previously reported detection may not be real, but a misidentification due to a poor signal-to-noise ratio. For the other three sources, we detect weak 44 GHz methanol emission, and when we take the typical intensity ratio between 44 and 95 GHz methanol two transitions into account, the 95 GHz emission might be too weak to be detected. While we cannot rule out that the uncertainty in the flux density calibration is partly responsible for differences in the measured flux densities between different epochs, an offset between the location of the detected maser emission and the pointing center of the observations appears to be the primary reason for the nondetection of some previously detected 95 GHz methanol masers in the current KVN observations.
To summarize, when we compare the previous 95 GHz detections with new observations of the same transitions made using PMO 13.7 m radio telescope, there is no clear evidence of variability on a timescale of six years (e.g., BGPS3322, BGPS7252, and BGPS7322 in Table \[tab:f\_comparison\]).
### 44 GHz methanol masers: interferometric observations {#sec:44_compare}
For nine sources in our sample, 44 GHz interferometric observations from the VLA are available. These sources are indicated with an asterisk after the name in Table \[tab:144\_obs\].
*G183.72$-$3.66 (GGD4).* GGD4 is a low-mass SFR with a known outflow. This source was observed during 2007 March to April using the VLA in D configuration, but there was no detection of 44 GHz methanol emission [channel rms $\sim$ 52 mJy beam$^{-1}$ @2016ApJS..222...18G]. In the current observations, we do not detect any emission above the noise at 44 GHz. [@2013ApJ...763....2G] reported the detection of a weak 95 GHz class I methanol maser toward this source, and the current observations also show methanol emission at 95 GHz. Higher angular resolution observations are needed to determine whether the detected 95 GHz methanol emission is a maser or quasi-thermal.
*BGPS7501 (S255N).* @2004ApJS..155..149K imaged the 44 GHz methanol masers in S255N on 2000 September 25 using the VLA in the D configuration. The current KVN observations have a similar velocity resolution as those of @2004ApJS..155..149K, with two major maser features. The strongest maser component has a velocity of 11 and an intensity of about 250 Jy in the current KVN observations, compared to approximately 150 Jy in the VLA observations. The secondary maser feature at 10 has a peak flux density of about 60 Jy in both the current and previous observations. An interferometric observation resolves out emission structures larger than a specific angular scale that is determined by the shortest baseline and the observing frequency. [@2017MNRAS.471.3915J] analyzed the relative strength of compact maser emission (interferometric measurement) to diffuse emission (simultaneous single-dish measurement), of a sample of 44 GHz class I methanol masers and found that class I methanol emission can be confined to compact structures, extended structures or a combination of the two. We cannot rule out the possibility that the single-dish KVN spectrum includes additional thermal or diffuse contributions that are resolved out in the interferometric observations, so that only additional high-resolution observations can determine whether this spectral component has varied or not. In addition to the main spectral features, the VLA observations show a series of weak methanol emission components in the velocity range 7–10 , which are also seen in the KVN spectrum. The VLA image shows that the two strongest components are offset from the pointing center of the KVN observations by $\sim$ 11$\arcsec$. If the 95 GHz methanol masers are cospatial with the 44 GHz transition, the peak flux density of the 95 GHz methanol masers might be underestimated. The S255 molecular complex lies between two H [ii]{} regions, S255 and S257, and is at a distance of 1.59 kpc [@2010AA...511A...2R]. It consists of three massive star forming regions, S255N, S255IR and S255S. The class I methanol maser emission is located near S255N, whose evolutionary stage is between that of S255IR (more evolved) and S255S (less eveolved). Outflow activity in this region has been studied through a number of different tracers, including CO, shocked H$_2$ and SiO emission [@1997ApJ...488..749M; @2007AJ....134..346C; @2011AA...527A..32W].
*BGPS3016 (IRAS 18308$-$0841).* [@2017ApJS..233....4R] reported five 44 GHz methanol maser components in their VLA observations from 2008 September 7. The three components that are stronger than 6 Jy can be clearly seen in the KVN spectrum and are located approximately 11$\arcsec$ from the pointing center of the KVN beam. The peak flux density of the 75.8 , 76.4 , and 76.9 components measured with the VLA are about 6 Jy, 15 Jy, and 6 Jy, respectively, similar to the current KVN observations, which show 7 Jy, 15 Jy, and 7 Jy, respectively. Despite the different spatial resolution and flux density calibration uncertainties, we can see that with a similar velocity resolution ($\sim$ 0.16 ), there is no evidence for variability of the 44 GHz methanol masers in this source over a period of 8 yr (2008–2016).
*BGPS3307 (IRAS 18337$-$0743).* [@2017ApJS..233....4R] reported two maser components observed with the VLA on 2008 September 8. The high-resolution observations show that the class I methanol masers are coincident with the northeastern edge of a H [ii]{} region traced by bright 8 $\mu$m emission [see figure 1 in @2017ApJS..233....4R]. The H [ii]{} region, G024.400$-$0.190, has radio recombination line (RRL) emission that peaks at a velocity of 54.7 [@2015ApJ...810...42A], consistent with the methanol maser velocity range (58–62 ). The H [ii]{} region is surrounded by a Spitzer dark cloud [SDC; @2009AA...505..405P], G024.405$-$0.187, which is seen in Herschel Hi-GAL data [@2016AA...590A..72P]. The location of the class I methanol masers suggests that they may be located at the interface between an expanding H [ii]{} region and an infrared dark cloud (IRDC). On the other hand, [@2002ApJ...566..931S] were unable to confirm CO line outflows/wings in this source due to confusion with Galactic plane emission. Higher resolution observations are necessary to identify the possibility of the presence of outflow(s) and further confirm the origin of 44 GHz maser emission. The two maser components are separated from the KVN pointing center by only 1$\arcsec$, suggesting that the measured flux densities for both the 44 and 95 GHz transitions are reliable. The peak flux density ratio of the two transitions for both maser components is 0.7, suggesting similar environmental parameters, such as temperature and density. When we use the kinematic distance estimate of 3.7 kpc obtained from the model of @2014ApJ...783..130R, the linear separation between the two components is about 0.02 pc.
*G058.471$+$00.433 (IRAS 19368$+$2239).* The KVN observations of this source were made in bad weather, leading to potentially unreliable flux density estimates. [@2016ApJS..222...18G] detected 12 maser components in their VLA imaging of this source. The primary and secondary components from the VLA observations are at velocities of 37.9 and 36.1 , and the separation of these two components from the KVN pointing center are 19$\arcsec$ and 21$\arcsec$, respectively. These offsets place two of the components beyond the FWHM of the KVN beam at 95 GHz, but strong emission at almost the same velocities was still detected.
*G65.78$-$2.61 (IRAS 20050$+$2720).* [@2016ApJS..222...18G] detected a single maser component toward this source with a peak flux density of 1.16 Jy. The angular separation of the 44 GHz methanol maser component compared to the KVN beam center is 5$\arcsec$. Neither 44 nor 95 GHz class I methanol transitions were detected in the current KVN observations (RMS noise level $\sim$0.5 Jy). Given the low peak flux density of the VLA detection, this is likely because of insufficient sensitivity in the KVN observations.
*BGPS6712 (IRAS 20286$+$4105).* [@2016ApJS..222...18G] detected two 44 GHz methanol maser components in this source. The primary component at a velocity of $-$4.0 has an angular separation of $\sim$ 14$\arcsec$ from the KVN beam center. The peak flux density measured in the KVN observations (6 Jy) is comparable to that observed by the VLA. If the 95 GHz methanol maser component at this velocity is cospatial with the 44 GHz counterpart, it lies close to the edge of the KVN beam and the measured flux density will be unreliable. Comparison with previous 95 GHz methanol observations [@2017ApJS..231...20Y] shows the peak flux density measured in the KVN spectrum is approximately half of that in previous observations.
*BGPS7208 (IRAS 23139$+$5939).* [@2017ApJS..233....4R] report a single weak 44 GHz maser component with peak flux density of 0.91 Jy in VLA observations made on 2008 September 12. The KVN single-dish spectrum shows a peak flux density at 44 GHz above 1 Jy, suggesting that perhaps the maser emission is mixed with thermal/diffuse emission. The maser component is offset from the KVN beam center by approximately 14$\arcsec$. Of interest, is that the 95 GHz methanol emission is stronger than the 44 GHz counterpart in this source.
*G111.24$-$1.24 (IRAS 23151$+$5912).* [@2017ApJS..233....4R] detected two maser components in their VLA observations from 2008 September 12. For the $-$52.7 component, which is located 5$\arcsec$ from the KVN beam center, the peak flux density detected by both the VLA and KVN is $\sim$4 Jy. Similarly, for the $-$54.7 component, offset from the KVN beam center by 1$\arcsec$, the peak flux density measured in both single-dish and interferometric observations is 3.5 Jy. So once again, there is no evidence of variability in this source. The maser spots are close to the KVN beam center, and hence the measured flux densities of both the 95 and 44 GHz methanol masers will be reliable. When we assume that the 95 GHz methanol masers are cospatial with 44 GHz transition, the peak flux density ratio between the two transitions measured from the current observations can be considered reliable. The $S_{\rm pk,95}/S_{\rm pk,44}$ for the $-$52.7 component is about 0.5, while for the $-$54.7 component, it is about 0.9. The difference in the flux density ratio between the two components may reflect a difference in the physical conditions of the environment hosting the two maser components. The angular separation of the two components is 6.$\arcsec$6, corresponding to a linear separation of about 0.1 pc for a parallax distance of 3.33 kpc [@2014ApJ...790...99C].
Emission with single peaks and multiple peaks
---------------------------------------------
Figure \[fig:spectra\] shows a wide variety of spectral profiles for the class I methanol masers. Some show a single, narrow, and strong maser feature, such as BGPS1509, BGPS1584, and BGPS2147, while others (e.g., BGPS3018, BGPS3026, and BGPS3212) contain multiple peaks. Some sources have broad spectral components, which could be quasi-thermal emission, indicating that the maser emission may in some cases be mixed the thermal emission. To reliably count the number of maser components, we only considered sources without broad emission components (line width $>$ 2.5 ), resulting in 111 and 96 methanol maser sources in our analysis for the 44 and 95 GHz transitions, respectively. For the 44 GHz methanol transition, the ratio of the number of sources with a singlep eak to those with multiple peaks is 0.66 (44/67). For the 95 GHz methanol transition, this ratio is 0.57 (35/61). It is not clear why some sources have a single peak of maser emission while others have multiple peaks, but possible reasons include (1) precession of the jet/outflow driven by the central object. (2) multiple outflows, and (3) the distribution or structure of the ambient molecular clouds with which the outflow interacts.
Discussion {#sec:discuss}
==========
The relationship between the two class I maser transitions
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### V$_{\rm pk}$ of methanol maser features {#sec:veldiff}
From Figure \[fig:spectra\], it is clear that the 44 and 95 GHz methanol transitions typically cover the same velocity range with very similar radial velocities for individual maser components. This has been noted in several previous investigations that compared the emission from these two transitions [e.g., @2000MNRAS.317..315V; @2018ApJS..236...31K].
In our sample, a total of 107 sources have detections for both the 44 and 95 GHz methanol transitions. We have smoothed the spectra from the two transitions to have the same velocity resolution ($\sim$ 0.2 ), and the peak velocity of each maser features were determined by Gaussian fitting. For sources that have the same number of maser features in each transition (e.g., BGPS3307), we matched all of the components individually to analyze the difference in peak velocity. For sources that have a different number of maser components for the 44 and 95 GHz transitions (e.g., BGPS1917), we compared the velocity of the strongest spectral feature.
Figure \[fig:Vpeak\] shows a comparison of the velocity of a total of 161 methanol peak velocities for the 44 and 95 GHz transitions. A very strong correlation exists between these two velocities with $V_{\rm pk,95}=(0.9998\pm0.0010)V_{\rm pk,44}-(0.0470\pm0.0614)$, with the Pearson coefficient $r =0.9999$. For 68% (109/161) of the maser components, the peak velocity of the two transitions agrees within 0.2 , suggesting that the emission from the 44 and 95 GHz transitions arises in the same environment and is likely cospatial. The peak velocity difference between the 44 GHz masers and their 95 GHz counterparts, $V_{\rm pk,44}-V_{\rm pk,95}$, ranges from $-$3.15 to 1.37 , with an average of 0.05 and a median of 0.04 . BGPS2718 shows the largest velocity deviation between the two transitions, with the strongest emission from the 44 GHz transition at $\sim$54 , but the peak of the 95 GHz transition is at $\sim$57 . There is a component at 54 in the 95 GHz spectrum, but it is not the strongest emission from this transition. The large difference in the radial velocity of the strongest emission between two transitions in this source suggests that there may be significant environmental differences between the two components. The blue dots in Figure \[fig:Vpeak\] highlight the 28 sources that have a single maser component for both transitions, which are not subject to the uncertainties caused by fitting multiple Gaussian components. In these sources, the peak emission velocities of two transitions has a tighter correlation, and the deviation ranges from $-$0.23 to 0.2 , with a mean and median for both of $\sim$0.02 .
### Peak flux density {#sec:peak_ratio}
Excluding the sources with unreliable flux densities, the flux density of each component of the 111 44 GHz maser sources was derived from Gaussian fitting of the spectra, ranging from 0.6 to 335 Jy (mean of 12.1 Jy, median 4.7 Jy). For the 95 GHz masers (99 sources), the flux density range is from 0.6 to 165.4 Jy (mean of 7.4 and median of 3.0 Jy). BGPS4252 and BGPS7501 are the 44 and 95 GHz methanol masers with the highest detected flux density in our sample, respectively. There are 95 sources with reliable flux densities in both transitions, and of these, 14 sources ($\sim$ 15%) have a higher 95 GHz peak flux density than for the 44 GHz methanol maser counterpart.
Based on the discussion of the velocity difference in Section \[sec:veldiff\] and the velocity resolution of the spectra for the two transitions, we can confidently match the emission from the two transitions where there are components with a velocity difference within 0.2 . Similar to the analysis we have undertaken of the peak velocity, for sources that have the same number of maser features in each transition, we matched all of the components individually to analyze the peak flux density ratio. For sources that have a different number of maser components for the 44 and 95 GHz transitions, we calculated the ratio of the strongest emission in each transition. In total, 119 pairs were investigated. Figure \[fig:flux1\] compares their peak flux density and the corresponding integrated flux density. The peak flux density ratio, $S_{\rm pk,95}$/$S_{\rm pk,44}$, ranges from 0.1 to 2.8. G123.07$-$6.31 shows the highest ratio, while BGPS4258 has the lowest ratio.
The blue dots in Figure \[fig:flux1\] highlight the 24 sources with a single maser component in both transitions that are not affected by Gaussian fitting, thus their flux density ratios are more reliable. There are four sources with a single maser component ($\sim$ 17%) where the 95 GHz peak flux density surpasses the 44 GHz counterpart. The ratios, $S_{\rm pk,95}$/$S_{\rm pk,44}$, range from 0.3 to 1.4, with BGPS6547 having the highest value and G8.72$-$0.36 the lowest. The integrated flux density ratio, $S_{\rm 95}$/$S_{\rm 44}$, is significantly affected by Gaussian fitting, and so we decided to only investigate the range obtained from the sources with a single maser component. The integrated flux density ratio ranges from 0.36 to 2.5, and this encompasses the ratio for the majority components from the multiple-peak sources.
Figure \[fig:flux1\] shows strong correlations between the intensity of matched components from the two transitions. Linear fits yield the following results:\
$$S_{\rm pk,95}=(0.49\pm0.01)S_{\rm pk,44}+(1.88\pm0.66), r=0.96 \\$$
$$log(S_{\rm pk,95})=(0.79\pm0.04) log(S_{\rm pk,44})+(0.03\pm0.04), r=0.88\\$$
$$S_{\rm 95}=(0.58\pm0.01)S_{\rm 44}+(1.55\pm0.59), r=0.93 \\$$
$$log(S_{\rm 95})=(0.80\pm0.05) log(S_{\rm 44})+(0.04\pm0.05), r=0.83\\$$
Our linear fit for the peak emission is consistent with that obtained by @2018ApJS..236...31K [which gives $S_{\rm pk,95}$/$S_{\rm pk,44}$ of 0.56 $\pm$ 0.08], and slightly lower than the ratio of 0.71$\pm$0.08 found by [@2015ApJS..221....6K]. When we consider only the 24 sources with a single maser component, the correlation between the intensity of the two transitions becomes stronger, $S_{\rm pk,95}$=(0.43$\pm$0.03)$S_{\rm pk,44}$+(3.79$\pm$2.04), with a Pearson coefficient of 0.96. A recent investigation using interferometric data from the Australia Telescope Compact Array (ATCA) with an angular resolution of better than 6$\arcsec$ for both transitions found $S_{\rm pk,95}$/$S_{\rm pk,44}$ to be 0.35$\pm$0.10 [@2018MNRAS.477..507M]. This result agrees with earlier estimates of 0.31 and 0.32 [@2000MNRAS.317..315V; @2015MNRAS.448.2344J]. However, for the three studies that have obtained ratios of about 0.31–0.35, these ratios were not obtained under the same conditions, such as the same velocity resolution, observing mode, and epoch. The three studies all used data from different epochs. Moreover, [@2015MNRAS.448.2344J] compared their 44 GHz maser fluxes from ATCA observations with 95 GHz fluxes from previous single-dish studies, and [@2018MNRAS.477..507M] compared the two data sets at very different velocity resolutions. The current KVN observations observed the two transitions simultaneously with the same pointing, but for the majority of sources, we do not have interferometric observations, and so the position of the maser components within the beam is unknown. We cannot rule out the possibility that some masers are close to the edge of the KVN beam at 95 GHz, but this will cause the detected flux density to be lower than the real value and hence lead to a peak flux density ratio of 95 and 44 GHz transitions that underestimates the real situation.
Class I maser pumping models find that the line ratio between maser transitions is sensitive to the physical conditions of the environment. The different line ratios between the two transitions found in previous studies may be due to differences in the sample selection criteria. In addition, interferometric observations [@2017MNRAS.471.3915J] have found class I methanol emission can consist of only compact components, extended components, or a combination of the two. We cannot rule out that the single-dish spectra are a blend of maser and quasi-thermal emission in some sources, which may cause line ratio differences between interferometric and single-dish observations. The class I methanol masers reported in [@2015ApJS..221....6K] and [@2018MNRAS.477..507M] all have an associated 6.7 GHz methanol maser, but the former were observed by the KVN in single-dish mode, while the latter were observed using ATCA, and the line ratio between the 95 and 44 GHz methanol transitions measured from the two studies is quite different.
It should be noted that $S_{\rm pk,95}$/$S_{\rm pk,44}$ noticeably decreases as the flux density of the 44 GHz transition increases. Figure \[fig:ratio\] shows $S_{\rm pk,95}$/$S_{\rm pk,44}$ against the peak flux density of 44 GHz methanol emission. The best linear fit is $S_{\rm pk,95}$/$S_{\rm pk,44}$=($-$0.45 $\pm$ 0.08)$log(S_{\rm pk,44})$ + (1.22 $\pm$ 0.08), with a Pearson coefficient of $-$0.47. As the 44 GHz peak flux density increases, the $S_{\rm pk,95}$/$S_{\rm pk,44}$ decreases and flattens to an approximately constant value lower than one. From both Fig. \[fig:flux1\] (left panel) and Fig. \[fig:ratio\], it is clear that when the peak flux density of the 44 GHz methanol maser is lower, the ratio of $S_{\rm pk,95}$/$S_{\rm pk,44}$ has more scatter. In addition, for the 10 components (from 119 methanol transition pairs) where the peak flux density of the 44 GHz transition is greater than $\sim$ 40 Jy, there is no case where the 95 GHz methanol emission exceeds that of the 44 GHz counterpart. A possible explanation for this behavior is that the sensitivity of the 44 and 95 GHz transitions to the physical parameters is different, e.g., the methanol specific column density , which is a factor in the maser optical depth of the maser and therefore determines the maser brightness. Pumping models for class I methanol masers (A.M. Sobolev 2020, private communication, and S.Yu. Parfenov 2020, private communication) show that the 95 GHz transition becomes relatively weaker than the 44 GHz transition with the increase in maser flux. This dependence is not linear and is more pronounced when the masers have lower fluxes.
Masers and Central Objects {#sec:iras}
--------------------------
Because the angular resolution of the current single-dish observations is limited, we cannot study the infrared emission where the maser arises in depth, but only the overall infrared environment of the central YSOs that host the maser.
The Infrared Astronomical Satellite (IRAS[^1]) performed an unbiased sky survey at 12, 25, 60, and 100 $\mu$m, with an angular resolution ranging from 30$\arcsec$ to 2$\arcmin$. The data from this survey were used to create the IRAS Point Source Catalog (version 2.1) which we have used to analyze the environment of the maser host. Of the 128 detected 44 GHz methanol masers, 59 have an IRAS counterpart within 1$\arcmin$, and Table \[tab:iras\] gives the angular separation and the name of the *IRAS* counterpart of each source, as well as the *IRAS* band fluxes, distance, and bolometric luminosity. We used $F_{x}$ to denote the flux density in the $x$ $\mu$m IRAS band in Jy, and $Q_{x}$ to denote the corresponding flux quality, with values of 1, 2, and 3 representing an upper limit, moderate quality, and high quality, respectively. We also calculated the bolometric luminosity of the central object based on the IRAS fluxes [@2007AJ....133.1528C]. Distances for each of the target sources were tabulated by [@2017ApJS..231...20Y], primarily using kinematic distances calculated from the [@2014ApJ...783..130R] model.
The majority of the IRAS counterparts (93%; 55/59) has a bolometric luminosity greater than 10$^3L_\odot$, indicating that they are probably associated with high-mass star forming regions. About 83% (49/59) of the IRAS counterparts have $F_{12}<F_{25}<F_{60}<F_{100}$, which is characteristic of YSOs. Figure \[fig:iras\] shows IRAS color-color diagrams in which the youngest sources have the steepest spectra and are located toward the top right corner. As a source evolves, the infrared colors become bluer and sources shift toward the lower left corner. The green lines in the left panel depict the criteria of @1989ApJ...340..265W [hereafter WC89] which identify sources likely to be asscociated with an ultra-compact H [ii]{} (UCH [ii]{}) region, ($log(F_{25}/F_{12})\geqslant0.57$ and $log(F_{60}/F_{12})\geqslant1.3$). The left-hand panel in Fig. \[fig:iras\] shows that a significant fraction (61% ; 36/59) of sources meet the WC89 criteria region. This is a lower fraction than is observed for 6.7 GHz class II methanol masers [87%; @2003ChJAA...3...49X]. This seems to be consistent with the maser-based evolutionary sequence for high-mass star formation, which suggests that while both class I methanol masers and 6.7 GHz class II methanol masers can occur during the UCH [ii]{} phase, the class I methanol masers are thought to disappear earlier than those of class II and so have less overlap with the UCH [ii]{} phase [@2010MNRAS.401.2219B]. However, there are a number of potential biases that may impact our analysis. First, the WC89 criteria are for UCH [ii]{} candidates, but many of them have been found to be HMPOs rather than UCH[ii]{} regions [e.g., @1996AA...308..573M; @2002ApJ...566..931S]. Secondly, the beam sizes of 6.7 GHz maser observations (e.g., 3$\arcmin$–6$\arcmin$) were usually much larger than those (0.$\arcmin$5–1$\arcmin$) of 44 or 95 GHz maser observations, and hence there were more likely to be two or more massive YSOs in different evolutionary stages within the beam of 6.7 GHz maser observations. The green line in the right panel of Fig. \[fig:iras\] represents an improved H [ii]{} region selection criterion [@2018MNRAS.476.3981Y], and we find that 48 sources satisfy this. Note that in applying the [@2018MNRAS.476.3981Y] criterion, we did not exclude sources with $Q_{60}=1$ or $Q_{100}=1$ as is suggested because under those circumstances the number of sources remaining is too small for a statistical analysis.
After the methanol masers with an unreliable flux density are removed, a total of 42 sources that have both 44 and 95 GHz class I methanol maser emission and an IRAS source counterpart. Fig.\[fig:LL\] shows a plot of the isotropic maser luminosity as a function of the bolometric luminosity for the 44 GHz (left) and 95 GHz (right) methanol masers in our sample. The isotropic luminosities of the 44 and 95 GHz methanol masers ($L_{44}$ and $L_{95}$) can be calculated from the integrated flux density $\int S_v dv$ and the distance to the source [see, e.g., @2011ApJS..196...21B]. The black line in each panel marks the best linear fit to the data. The best fit for the 44 GHz methanol masers is $log (L_{\rm 44}) =(0.77\pm0.11)\times log (L_{\rm bol})-(8.43\pm0.45)$ with a Pearson coefficient of 0.75. The best fit for the 95 GHz methanol masers is $log (L_{\rm 95}) =(0.60\pm0.10) \times log (L_{\rm bol})-(7.49\pm0.43)$ with a Pearson coefficient of 0.68. Both $L_{44}$ and $L_{95}$ have a similar correlation with $L_{\rm bol}$, but the $L_{44}$ shows a slightly better correlation with $L_{\rm bol}$ than $L_{95}$.
For 44 GHz methanol masers, the relation between $L_{44}$ and $L_{\rm bol}$ has been studied toward low-mass YSOs [$L_{\rm bol}<10^2 L_{\odot}$; @2013ARep...57..120K], intermediate-mass YSOs [$L_{\rm bol}\sim 10^2-10^3 L_{\odot}$; @2011ApJS..196...21B], HMPO candidates ($L_{\rm bol}>10^3 L_{\odot}$) from Red Midcourse Space Experiment Sources [@2018ApJS..236...31K], and UCH [ii]{} regions [@2019ApJS..244....2K]. Combining the data from these previous studies with our sample, the best fit for all sources is $log (L_{\rm 44}) =(0.56\pm0.05)\times log (L_{\rm bol})-(7.66\pm0.22)$ with a Pearson coefficient of 0.63. Fig. \[fig:LL2\] shows the isotropic maser luminosity as a function of the bolometric luminosity for all objects. The bolometric luminosity is a good measure of the central object mass. The correlation between maser luminosity and bolometric luminosity may arise when sources with higher mass sources drive more powerful outflows, or at least outflows that result in a larger volume of gas that produces class I methanol masers, where those outflows interact with the surrounding molecular environment [e.g., @2011ApJS..196...21B; @2019ApJS..244....2K].
44 GHz methanol masers occur at different evolutionary stages
-------------------------------------------------------------
It is widely accepted that the earliest stages of high-mass star formation often take place while the source is embedded within an IRDC [@2006ApJ...641..389R]. Investigation of molecular gas within IRDCs compared to HMPOs, shows that as the sources evolve, the temperatures increase and the densities and masses rise. The next evolutionary phase is associated with UCH [ii]{} regions, which are produced by young massive stars as they reach the main sequence [e.g., @2018ARAA..56...41M]. Thus molecular clumps associated with IRDCs, HMPOs, and UCH [ii]{}s represent an approximate evolutionary sequence of massive star formation.
[@2006ApJ...638..241E] found that the infrared color of sources that are only associated with class I methanol masers are redder than the color of sources associated with both class I and class II methanol masers, indicating that the class I methanol masers trace an earlier stage than class II methanol masers. On the other hand, a few previous studies suggested that the detection rate of 44 GHz class I methanol masers increases as the central objects evolve [e.g., @2011ApJS..196...21B; @2019ApJS..244....2K]. These studies indicate that 44 GHz class I methanol masers can be associated with both early and late evolutionary stages of high-mass star formation, but the masers are more frequently detected in later evolutionary stages. Theoretical prediction and interferometric observations [e.g., @2014MNRAS.439.2584V] also found that class I methanol maser could be pumped by expanding H [ii]{} regions, demonstrating that some class I methanol masers are associated with later stages of high-mass star formation. We would like to determine whether the current sample contains class I masers likely associated with both younger and older SFRs.
We investigated the infrared emission detected by the *all-sky Wide-field Infrared Survey Explorer* [*WISE*[^2]; @2010AJ....140.1868W] toward each of our target sources. The *WISE* images can be used to determine which class I methanol maser sources may be associated with an infrared dark background or mid-infrared morphologies suggesting H[ii]{} regions.
WISE used a 40 cm telescope to image the entire sky in four mid-IR bands at 3.4, 4.6, 12 and 22 $\mu$m. From 3.4 to 22 $\mu$m, WISE achieved sensitivities for point sources of 0.08, 0.11, 1, and 6 mJy and angular resolution of 6$\arcsec$.1, 6$\arcsec$.4, 6$\arcsec$.5 and 12$\arcsec$.0 in the four bands, respectively. Mid-IR emission can be used to trace star-forming activity through polycyclic aromatic hydro-carbon emission, seen in the WISE 12$\mu$m band, and through the correlation with thermal emission from warm dust, predominantly see in the WISE 22 $\mu$m band.
We inspected the 12$\arcmin$ $\times$ 12$\arcmin$ three-color images for each of our target maser sources (see Figure \[fig:infra\] as an example). From our sample, we found that some class I methanol masers could be associated with IRDCs and some could be associated with H [ii]{} regions, suggesting that there are both younger and older class I methanol maser sources during star formation.
### Infrared dark clouds {#sec:sec_irdc}
The IRDCs are seen as a dark silhouette against the bright background emission in images at wavelengths between 7 and 25 $\mu$m [e.g., @2006ApJ...639..227S]. Investigations of molecular line and dust continuum emission shows that they are regions of dense, cold gas and have high column densities, consistent with the expectations for the initial conditions for high-mass star formation [e.g., @2006AA...450..569P; @2006ApJ...641..389R].
[@2009AA...505..405P] compiled an unbiased sample of candidate IRDCs (11303 in total) in the 10$^\circ < \mid l \mid <65 ^\circ$, $\mid b \mid < 1^\circ$ region of the Galactic plane using Spitzer 8 $\mu$m extinction. Subsequently, 76($\pm$19)% of these cataloged SDCs were confirmed through association with a peak in Herschel column density maps constructed from 160 $\mu$m and 250 $\mu$m data from Herschel Galactic plane survey Hi-GAL [@2016AA...590A..72P].
In the WISE three-color images, 98% (125/128) have at least one WISE point source within 30$\arcsec$ (about half of the beam size at 44 GHz) of the pointing center. Two of the exceptions are G208.97$-$19.37 (Orion-KL) and BGPS2152 (M17), where the three-color images are saturated and only one source (BGPS2054) has no close WISE point source in the AllWISE catalog. For the majority of sources with a 95 GHz class I methanol maser detection, we can surmise that the emission is located within the 15$\arcsec$ of the targeted center (because the sensitivity of the observations drops very rapidly at larger separations). When Orion-KL and M17 are excluded, 25% (31/126) of the targeted locations have no WISE point source within 15$\arcsec$ (about half of the beam size in 95 GHz) of the pointing center. We examined the WISE three-color images of these 31 masers and found that some of them show an absence of bright 22 $\mu$m emission, perhaps indicating the presence of an IRDC.
With the exception of IRDC18223-3 (where the observed target is a known IRDC), for the 31 sources without nearby ($\sim$15$\arcsec$) WISE counterparts, we cross-matched with the Herschel-confirmed SDC catalog, finding that 5 of them may be associated with IRDCs (See Table \[tab:irdc\]). The ratios, $S_{\rm pk,95}$/$S_{\rm pk,44}$, for the masers that may be associated with IRDCs are no greater than one, but for IRDC18223-3, the peak emission of the 95 GHz methanol maser is stronger than that of the 44 GHz counterpart of $S_{\rm pk,95}$/$S_{\rm pk,44}$ $\sim$ 1.2.
### masers in the vicinity of H [ii]{} regions {#sec_hii}
[@2014ApJS..212....1A] compiled a catalog of more than 8000 Galactic H[ii]{} regions and candidates by visually and automatically searching for their characteristic mid-IR morphology using WISE data. Approximately 1500 of the mid-infrared selected sample have been detected in RRL emission and are thus confirmed to be H [ii]{} regions. Furthermore, as a part of the Green Bank Telescope H [ii]{} Region Discovery Survey, new RRL observations are being undertaken [@2015ApJ...810...42A; @2018ApJS..234...33A], expanding the catalog of confirmed H [ii]{} regions.
We cross-matched the class I methanol masers targeted in the current observations with the WISE catalog of Galactic H [ii]{} regions v2.0 from <http://astro.phys.wvu.edu/wise/>[see also @2018ApJS..234...33A]. The WISE name, coordinates, the approximate circular radius encompassing the WISE mid-IR emission and corresponding RRL velocity information were extracted from the catalog. We consider the class I methanol maser to be possibly associated with a H [ii]{} region when it meets the following criteria: (1) The angular separation between maser and H [ii]{} region is smaller than the radius of the corresponding H [ii]{} region. (2) To avoid including the effects from other complex environments, the angular separation between maser and H [ii]{} region is no greater than 60$\arcsec$ (approximately the beam size at 44 GHz). It should be noted that this criterion may result in not identifying cases where the maser is located at the edge of an evolved and extended H [ii]{} region. (3) The H [ii]{} region has detected RRL emission, and the absolute difference between the velocity of the RRL and class I methanol maser is less than $\sim$ 10 [@2014ApJS..212....1A].
The criteria identified a total of 22 class I methanol masers (see Table \[tab:HII\]) that may be associated with H [ii]{} regions, all of which also have BGPS 1.1 mm emission. There are multiple WISE H [ii]{} region counterparts within 30$\arcsec$ (about half of the beam size in 44 GHz) of two masers (BGPS1116 and BGPS3155), suggesting that they are part of a larger H [ii]{} region complex. We listed all the possible WISE H [ii]{} regions, because when the accurate position of maser is unknown, we only interested in which masers may be associated with H [ii]{} regions rather than which specific H [ii]{} region it is associated with. Table \[tab:HII\] lists the names of the methanol masers, the possible associated WISE H [ii]{} regions, the angular separation between the BGPS source and H[ii]{} region, the angular sizes of the H [ii]{} regions, the velocity range of class I methanol masers and the velocity of the RRLs from H [ii]{} regions, as well as the ratio of the peak flux density for the two class I methanol transitions. For the source with an unreliable flux density or a 95 GHz nondetection, the ratio of the peak flux density between the two transitions is not listed. The ratios, $S_{\rm pk,95}$/$S_{\rm pk,44}$, for the masers that may be associated with H [ii]{} regions are all lower than one.
It should be noted that a large number of observations have shown that CO outflows are common toward both HMPOs and UCH [ii]{}s [e.g., @1996ApJ...472..225S; @2001ApJ...552L.167Z; @2002ApJ...566..931S], and numerous YSOs are observed at the peripheries of compact or extended H [ii]{} regions . Thus it is difficult to differentiate whether class I methanol masers are produced by the expansion of H[ii]{} regions or the outflows of associated YSOs, based on low-resolution radio and IR data. High-resolution observations are necessary to clarify the issue.
Four sources (BGPS1954, BGPS3026, BGPS3307, and BGPS4933) which may be associated with a H [ii]{} region and also a nearby IRDC (within $\sim$ 1 $\arcmin$). The VLA observation of BGPS3307 (see Sec.\[sec:44\_compare\]) shows that the maser components are located at the interface between a H [ii]{} region and an IRDC. For the other three sources, the exact position of the maser emission is unknown, and hence we cannot determine whether they are associated with an IRDC.
Summary {#sec:sum}
=======
We have used the three KVN antennas in single-dish mode to simultaneously observed the 44 and 95 GHz class I methanol masers toward 144 sources with previous 95 GHz detections. The main results are listed below.\
1. 128 44 GHz and 111 95 GHz methanol were detected, corresponding to detection rates of 89% and 77%, respectively. This is the first reported detection of 44 GHz class I methanol masers for 106 sites.\
2. Through comparison with previous observations with PMO 13.7m new observations, no clear evidence was found for variability in 95 GHz methanol masers.\
3. The 44 and 95 GHz methanol masers show a strong correlation in peak velocity, peak flux density, and integrated flux density. The peak flux density ratio $S_{\rm pk,95}$/$S_{\rm pk,44}$ ranges from 0.1 to 2.8, and the best fit for $S_{\rm pk,95}$/$S_{\rm pk,44}$ in our sample is $S_{\rm pk,95}$=(0.49$\pm$0.01)$S_{\rm pk,44}$+(1.88$\pm$0.66). We found that the peak flux density ratio $S_{\rm pk,95}$/$S_{\rm pk,44}$ decreases as the 44 GHz peak flux density increases. No 95 GHz methanol maser are stronger than the 44 GHz counterpart when the peak flux density of the 44 GHz maser is stronger than 40 Jy, and in only $\sim$ 15% of the sources in our sample dose the 95 GHz peak flux density surpass that of the 44 GHz counterpart.\
4. Class I methanol masers occur at early stage and more evolved stage of high-mass star formation. There are some class I methanol masers in our sample that could be associated with IRDCs and others with H[ii]{} regions.\
We acknowledge the anonymous referee for helpful comments that have improved this paper. We are grateful to the staff of the KVN observatory and PMO observatory for their assistance during the observations. We also thank Dr. Yan Gong and Dr. Qingzeng Yan for helpful discussions and suggestions. This work was sponsored by the MOST under grand No. 2017YFA0402701, the National Natural Science Foundation of China (grant numbers: 11933011, 11873019 and 11673066), and the Key Laboratory for Radio Astronomy, CAS. A.M.S was supported by the Ministry of Science and Education, FEUZ-2020-0030. W.Y. thanks her parents for feeding her tons of foods during the COVID-19 outbreak period.
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![The distribution of the peak flux density ratio for the 95 GHz methanol masers detected in the current KVN observations compared to previous PMO (purple) or Mopra (red) observations.\[fig:ff\]\
](ff)
![Comparison of the velocity of the 44 and 95 GHz methanol peak emission. The two dashed gray lines show a deviation of 3 on both sides. The blue dots represent the 28 sources with only one maser component in both transitions. \[fig:Vpeak\]\
](Vpk)
![The peak flux density ratio of $S_{\rm pk,95}$/$S_{\rm pk,44}$ against with the peak flux density of the 44 GHz methanol maser. As shown before, the red dots represent the matched methanol emission component in sources with multiple peaks, and the blue dots represent the 24 sources with a single methanol emission peak. Error bars are shown in the peak flux density figure. The dash-doted orange line depicts where $S_{\rm pk,95}$/$S_{\rm pk,44}$ equals 1 when the peak flux density of 44 GHz is stronger than 40 Jy (dashed green line). The solid black line denotes the best linear fit. \[fig:ratio\]\
](ratio_vs_flux)
![The IRAS color-color diagrams for 59 sources. The solid green lines in the left-hand panel show the criteria of WC89 [@1989ApJ...340..265W], and the UCH[ii]{} regions lie in the upper right corner. In the right-hand panel, the green line shows an improved H[ii]{} region selection criterion [@2018MNRAS.476.3981Y], and the H [ii]{} regions lie above the green line.\[fig:iras\]\
](iras2 "fig:") ![The IRAS color-color diagrams for 59 sources. The solid green lines in the left-hand panel show the criteria of WC89 [@1989ApJ...340..265W], and the UCH[ii]{} regions lie in the upper right corner. In the right-hand panel, the green line shows an improved H[ii]{} region selection criterion [@2018MNRAS.476.3981Y], and the H [ii]{} regions lie above the green line.\[fig:iras\]\
](iras6 "fig:")
![The luminosity of class I methanol masers (left: 44 GHz; right: 95 GHz) versus the luminosity of IRAS counterparts for 42 sources. The black line depicts the best linear least-squares fit. \[fig:LL\]\
](L44iras "fig:") ![The luminosity of class I methanol masers (left: 44 GHz; right: 95 GHz) versus the luminosity of IRAS counterparts for 42 sources. The black line depicts the best linear least-squares fit. \[fig:LL\]\
](L95iras "fig:")
![ Isotropic maser luminosity as a function of the bolometric luminosity of the central star for 44 GHz methanol masers. Crosses, pluses, open circles, pentagrams, and solid circles represent low-mass YSOs [@2013ARep...57..120K], intermediate-mass YSOs [@2011ApJS..196...21B], high-mass protostellar object candidates from RMS sources [@2018ApJS..236...31K], UCH[ii]{} regions [@2019ApJS..244....2K] and our sample. The dashed and solid lines are the best linear least-squares fit for our sample and for all objects. \[fig:LL2\]\
](L44all)
[llrccccc]{} G123.07$-$6.31$^\ddagger$ & 00 52 23.9 & 56 33 45 & Y & Y & Y & 0.6 & 0.6\
BGPS7351 & 02 25 30.8 & 62 06 18 & Y & Y & N & 0.4 & 0.4\
G208.97$-$19.37 & 05 35 15.5 & $-$05 20 41 & Y & Y & Y & 0.6 & 0.7\
G208.77$-$19.24 & 05 35 21.8 & $-$05 07 37 & Y & Y & Y & 0.3 & 0.5\
G173.58$+$2.44 & 05 39 27.5 & 35 40 43 & Y & Y & Y & 0.3 & 0.5\
[lccccc]{} KVN Yonsei & 43 & 63 & 0.48 & 0.64 & 15.5\
& 86 & 32 & 0.41 & 0.53 & 18.7\
KVN Ulsan & 43 & 63 & 0.47 & 0.62 & 16.0\
& 86 & 32 & 0.44 & 0.57 & 17.4\
KVN Tamna & 43 & 63 & 0.49 & 0.64 & 15.5\
& 86 & 32 & 0.44 & 0.57 & 17.4\
[lcccccccc]{} BGPS1917 &9 &0.9 &201605,201405 &9 & 0.7 &23 &18 &a\
BGPS2011 &8 &0.8 &201406 &8 & 1.0 &13 &N &b\
BGPS2720 &5 &0.9 &201406 &7 & 0.9 &12 &N &b\
BGPS2784 &N &1.0 &201605,201306,201206 &5 & 0.8 &4 &N &c\
BGPS3319 &N &0.9 &201206 &6 & 1.3 &N &N &d\
BGPS3322 &5 &0.9 &201206 &8 & 1.5 &11 &N &b\
BGPS4063 &N &1.0 &201406,201307 &5 & 1.1 &2 &N &c\
G031.013+00.078 &N &1.0 &201103 &4 & 0.8 &6 &N &c\
BGPS4252 &60 &1.3 &201206 &75 & 1.6 &280 &130 &a\
BGPS7125 &5 &0.8 &201505,201205 &6 & 0.7 &26 &N &b\
BGPS7252 &9 &0.8 &201205 &11 & 0.6 &9 &2 &b\
BGPS7322 &4 &0.8 &201205 &5 & 0.9 &6 &N &b\
BGPS7351 &4 &0.8 &201605,201205 &5 & 0.8 &2 &2 &b\
[lcrrcrc]{} G123.07$-$6.31 & a & $-$33.20 (0.21) & 2.15 (0.21) & 0.8 (0.6) & 1.9 (0.2) & 8.8\
G123.07$-$6.31 & b & $-$30.34 (0.21) & 2.88 (0.21) & 2.2 (0.6) & 6.9 (0.2) &\
BGPS7351 & a & $-$45.41 (0.24) & 1.40 (0.49) & 0.8 (0.4) & 0.9 (0.3) & 5.7\
BGPS7351 & b & $-$42.25 (0.14) & 2.92 (0.37) & 1.9 (0.4) & 4.8 (0.5) &\
G208.97$-$19.37 & a & 7.26 (0.05) & 0.30 (0.24) & 1.8 (0.6) & 0.6 (0.3) & 6.8\
G208.97$-$19.37 & b & 8.62 (0.11) & 1.10 (0.38) & 3.1 (0.6) & 3.7 (1.0) &\
G208.97$-$19.37 & c & 10.00 (0.13) & 0.85 (0.42) & 2.8 (0.6) & 2.5 (1.0) &\
G208.77$-$19.24 & & 11.22 (0.05) & 0.27 (0.07) & 1.5 (0.3) & 0.4 (0.1) & 0.4\
G173.58+2.44 & & $-$17.03 (0.02) & 0.63 (0.06) & 4.1 (0.3) & 2.8 (0.2) & 2.8\
[lcrrcrc]{} G123.07$-$6.31 & a & $-$32.94 (0.20) & 1.33 (0.20) & 2.8 (0.6) & 3.9 (0.4) & 23.0\
G123.07$-$6.31 & b & $-$31.58 (0.20) & 0.92 (0.20) & 4.8 (0.6) & 4.7 (0.4) &\
G123.07$-$6.31 & c & $-$30.52 (0.20) & 1.12 (0.20) & 6.1 (0.6) & 7.3 (0.4) &\
G123.07$-$6.31 & d & $-$29.30 (0.20) & 1.20 (0.20) & 3.9 (0.6) & 5.0 (0.4) &\
G123.07$-$6.31 & e & $-$27.89 (0.20) & 1.32 (0.20) & 1.5 (0.6) & 2.1 (0.4) &\
BGPS7351 & a & $-$45.56 (0.16) & 0.73 (0.25) & 0.8 (0.4) & 0.7 (0.2) & 4.2\
BGPS7351 & b & $-$43.79 (0.09) & 1.21 (0.24) & 1.6 (0.4) & 2.1 (0.3) &\
BGPS7351 & c & $-$42.49 (0.08) & 0.54 (0.15) & 1.2 (0.4) & 0.7 (0.2) &\
BGPS7351 & d & $-$41.60 (0.04) & 0.41 (0.12) & 1.7 (0.4) & 0.7 (0.2) &\
G208.97$-$19.37 & a & 8.09 (0.20) & 0.95 (0.36) & 1.4 (0.7) & 1.4 (0.4) & 7.5\
G208.97$-$19.37 & b & 8.93 (0.07) & 0.33 (0.24) & 1.7 (0.7) & 0.6 (0.4) &\
G208.97$-$19.37 & c & 9.95 (0.05) & 1.20 (0.16) & 4.3 (0.7) & 5.5 (0.6) &\
G208.77$-$19.24 & a & 11.25 (0.06) & 0.55 (0.31) & 1.8 (0.5) & 1.0 (0.4) & 1.4\
G208.77$-$19.24 & b & 11.95 (0.07) & 0.30 (0.31) & 1.3 (0.5) & 0.4 (0.3) &\
G173.58+2.44 & & $-$17.23 (0.08) & 0.82 (0.19) & 1.8 (0.3) & 1.5 (0.3) & 1.5\
[lcccccccccccc]{} G123.07$-$6.31 &1 &00494+5617 &1.803 &13.36 &329.6 &1166.0 &1 &3 &3 &3 &2.82 &9.0\
G208.77$-$19.24 &39 &05329$-$0508 &0.3883 &24.8 &297.1 &43.82 &1 &3 &1 &1 &0.42 &0.1\
G173.58$+$2.44 &1 &05361+3539 &1.182 &6.715 &29.15 &1310.0 &3 &3 &3 &1 &1.7 &2.7\
G207.27$-$1.81 &0 &06319+0415 &78.44 &375.2 &958.8 &995.2 &3 &3 &3 &3 &1.3 &5.5\
BGPS1062 &33 &17542$-$2447 &4.165 &4.401 &13.51 &344.3 &1 &1 &3 &1 &2.1 &1.2\
BGPS1116 &28 &17545$-$2357 &11.48 &106.6 &793.9 &1667.0 &3 &3 &3 &3 &1.9 &8.2\
BGPS1138 &43 &17571$-$2401 &24.43 &67.23 &12790.0 &26780.0 &1 &3 &1 &1 &2.3 &159.8\
[ccc]{} BGPS1584 & SDC011.081$-$0.532 &0.4\
BGPS2054 & SDC014.493$-$0.143 &\
BGPS2718 & SDC020.731$-$0.055 &0.6\
BGPS3018 & SDC023.210$-$0.371 &1\
BGPS4557 & SDC030.811$-$0.110 &0.6\
[cclcccc]{} BGPS7501 & 7 & G192.584$-$00.043 & 60 & 6–12 & 7.5 & 0.6\
BGPS1116 & 24 & G005.637$+$00.232 & 19 & 6–10 & 6.3 &\
& 28 & G005.633$+$00.238 & 59 & 6–10 & 6.3 &\
BGPS1954 & 27 & G013.880$+$00.285 & 144 & 49–53 & 51 &\
BGPS2011 & 48 & G014.207$-$00.193 & 608 & 38–43 & 36.1 &\
BGPS2275 & 18 & G016.360$-$00.211 & 33 & 47–51 & 46.4 &\
BGPS2784 & 2 & G021.386$-$00.255 & 57 & 90–92 & 91.2 &\
BGPS3026 & 13 & G023.264$+$00.077 & 60 & 76–82 & 78.2 & 0.4\
BGPS3155 & 19 & G023.708$+$00.174 & 171 & 112–115 & 103.8 & 0.6\
& 22 & G023.713$+$00.175 & 44 & 112–115 & 103.8 & 0.6\
& 24 & G023.705$+$00.165 & 46 & 112–115 & 103.8 & 0.6\
BGPS3183 & 21 & G023.872$-$00.119 & 60 & 71–79 & 73.8 & 0.5\
BGPS3337 & 20 & G024.498$-$00.039 & 60 & 107–112 & 108.1 & 0.7\
BGPS3474 & 27 & G025.220$+$00.289 & 42 & 45–47 & 42.4 & 0.9\
G024.920$+$00.085 & 21 & G024.923$+$00.079 & 42 & 40–50 & 42.4 &\
BGPS3307 & 13 & G024.397$-$00.191 & 47 & 58–62 & 54.7 & 0.8\
BGPS3774 & 21 & G027.279$+$00.143 & 33 & 30–34 & 36.3 & 0.7\
BGPS4014 & 7 & G028.651$+$00.026 & 33 & 100–105 & 102.4 & 0.6\
BGPS4048 & 36 & G028.801$+$00.170 & 60 & 102–108 & 107.6 & 0.5\
BGPS4933 & 13 & G032.152$+$00.131 & 60 & 91–96 & 95 & 0.8\
BGPS5539 & 15 & G035.051$-$00.520 & 52 & 48–53 & 48 & 0.6\
BGPS5821 & 44 & G037.200$-$00.430 & 60 & 34–38 & 38 & 0.6\
BGPS5874 & 9 & G037.820$+$00.414 & 52 & 12–22 & 22.3 & 0.7\
BGPS6418 & 11 & G053.188$+$00.209 & 53 & $-$1–2 & 5.3 &\
BGPS6547 & 11 & G076.155$-$00.286 & 60 & $-$32–$-$30 & $-$28.2 &\
[^1]: <http://irsa.ipac.caltech.edu/Missions/iras.html>
[^2]: <http://irsa.ipac.caltech.edu/Missions/wise.html>
|
---
abstract: 'Magnetic Resonance Imaging (MRI) is a noninvasive imaging technique that provides exquisite soft-tissue contrast without using ionizing radiation. The clinical application of MRI may be limited by long data acquisition times; therefore, MR image reconstruction from highly undersampled k-space data has been an active area of research. Many works exploit rank deficiency in a Hankel data matrix to recover unobserved k-space samples; the resulting problem is non-convex, so the choice of numerical algorithm can significantly affect performance, computation, and memory. We present a simple, scalable approach called Convolutional Framework (CF). We demonstrate the feasibility and versatility of CF using measured data from 2D, 3D, and dynamic applications.'
address: |
\*Department of Electrical and Computer Engineering, \*\*Department of Biomedical Engineering,\
The Ohio State University
bibliography:
- 'root.bib'
title: Convolutional Framework for Accelerated Magnetic Resonance Imaging
---
parallel imaging, low rank, calibrationless, multi-level block Hankel.
Introduction {#sec:intro}
============
MRI reconstruction from undersampled data implicitly relies on priors to provide regularizing assumptions. Priors in MRI reconstruction may be considered in two classes: coil properties and image structure. The spatial smoothness of coil sensitivity maps may be modeled as filters with finite k-space support, leading to a shift-invariant prediction property for multi-coil k-space data. GRAPPA [@griswold2002generalized] uses a fully-sampled auto-calibration signal (ACS) region to solve for a linear prediction kernel and applies the kernel to recover missing k-space samples. SPIRiT [@lustig2010spirit] is a generalization of GRAPPA that enforces the linear predictability across the entire k-space. PRUNO[@zhang2011parallel] generalizes SPIRiT further by using multiple kernels satisfying the linear prediction property. The shift invariant linear prediction property may be expressed as a nullspace of a convolution operator, which is a multi-level Hankel-structured matrix. And, the finite k-space support of coil sensitivities translates to low-rank for the structured Hankel data matrix. Calibration-free methods, such as SAKE [@shin2014calibrationless], employ low-rank structured matrix completion to recover missing k-space samples. To improve computational efficiency, alternatives to the Cadzow’s algorithm (used in SAKE) have been proposed for single-coil [@ongie2017fast] and multi-coil [@haldar2016p] MRI. Sparsity in the image domain also leads to existence of approximate annihilating filters in the k-space and yields the shift-invariant linear prediction property [@haldar2019linear]. Any linear filtering that sparsifies the image also yields annihilating filters in the resulting weighted k-space. In this vein, ALOHA uses a low-rank matrix completion method to recover missing samples in the weighted k-space [@jin2016general].
We present a viewpoint and computational approach, called Convolutional Framework (CF), that unifies many reconstruction techniques and provides an algorithmic approach for structured matrix completion. Memory efficiency permits high-dimensional imaging cases not demonstrated previously. Numerical evaluations are presented for three imaging applications: 2D, 2D cine, and 3D. Discussion and summary conclude the manuscript.
Methods
=======
We first define notation for organizing k-space data into structured arrays. A tensor denotes a multidimensional array. Let $*$, $\circledast$, and $\boxast$ denote linear convolution, circular convolution, and valid convolution, respectively. Valid convolution maps to output points that do not depend on any boundary conditions (i.e., no padding of the input). For example, for $n$-point vector $\bm{a}$ and $m$-point vector $\bm{b}$, $m \geq n$, the lengths of $\bm{a}*\bm{b}$ and $\bm{a} \boxast \bm{b}$ are $m+n-1$ and $m-n+1$, respectively. Let $\bm{s}(\mathbb{A})$ be the row vector that lists the sizes of tensor $\mathbb{A}$. Let $\mathscr{H}_{\bm{s}(\mathbb{A})}\{ \mathbb{B} \}$ denote multi-level block Hankelization of the tensor $\mathbb{B}$ such that right multiplication of the matrix $\mathscr{H}_{\bm{s}(\mathbb{A})}\{ \mathbb{B} \}$ with vectorized $\mathbb{A}$ $(\bm{vec} \{\mathbb{A}\})$ is the vectorization of the valid convolution result between $\mathbb{B}$ and $\mathbb{A}$, i.e., $$\begin{aligned}
\mathscr{H}_{\bm{s}(\mathbb{A})} \{\mathbb{B}\} \bm{vec} \{\mathbb{A}\} = \bm{vec} \{\mathbb{B} \boxast \mathbb{A} \}.\end{aligned}$$ Finally, $\circ$ denotes Hadamard multiplication between tensors of the same size, and $(\cdot)^H$ denotes conjugate transpose.
For generality, we denote the k-space by $\mathbb{D}$ with five dimensions: frequency encoding, first phase encoding, second phase encoding, coil, and time; dimensions are ordered and indexed by $k_x, k_y, k_z, l, t$. We choose the kernel size $\bm{s} = [f_x, f_y, f_z, N_c, f_t]$, where $N_C$ is the number of coils.
Based on the shift-invariant linear predictability assumption, CF leverages linear prediction in all dimensions, i.e., annihilating filters exist for jointly processing all dimensions. An iterative algorithm to apply the property has two simple steps: (i) Estimate all annihilating filters from the current estimate of k-space; and (ii) Enforce annihilation for the whole k-space and update the unobserved k-space. The CF processing is summarized in Algorithm 1.
\
$\mathbb{D}_o$: Observed k-space with zero filling\
$\mathbb{M}$: Sampling mask\
$\bm{s}$: Kernel size\
$r$: Rank\
$\text{tol}$: Tolerance \
$\hat{\mathbb{D}}^{(n)}$: Recovered k-space\
Initialization: $n = 0, \delta = \infty, \hat{\mathbb{D}}^{(n)} = \mathbb{D}_o$ $n = n+1$ $[\bm{\Lambda}^2, \bm{V}] = \text{EIG}(\mathscr{H}_{\bm{s}}^H (\hat{\mathbb{D}}^{(n)}) \mathscr{H}_{\bm{s}}(\hat{\mathbb{D}}^{(n)}))$ $\bm{V} = [\bm{V}_\parallel~|~\bm{V}_\perp]$ based on $r$ Split, reshape and flip $\bm{V}_\perp$ into kernels $\mathbb{F}_1, \mathbb{F}_2, \cdots$ $\hat{\mathbb{D}}_u^{(n)} = \text{argmin}_{\mathbb{X}} \sum_i \| (\mathbb{D}_o + \mathbb{X})\boxast \mathbb{F}_i \|_F^2$ s.t. $\mathbb{X} \circ \mathbb{M} = 0$ $\hat{\mathbb{D}}^{(n)} = \mathbb{D}_o + \hat{\mathbb{D}}_u^{(n)}$ $\delta = \|\hat{\mathbb{D}}^{(n)} - \hat{\mathbb{D}}^{(n-1)} \|_F / \|\hat{\mathbb{D}}^{(n-1)} \|_F$
\
The eigendecomposition in Step 3 extracts a null space ($\bm{V}_\perp$) of the product of two Hankel operators; this avoids a singular value decomposition of the explicit – and very large – convolutional matrix $\mathscr{H}_{\bm{s}}^H$. The operator product in Step 3 may be calculated with limited memory and computation using convolution with small kernels, in lieu of explicit matrices. Step 6 is a large scale least squares problem; to limit memory requirements, we avoid direct computation and instead rely on the implicit convolution operator in a gradient descent (GD) method with exact linear search (ELS). The memory requirement for GD + ELS is approximately only the original data size, in contrast to explicit construction of Hankel matrices. Step 6 minimizes null space energy while simultaneously preserving Hankel structure and data consistency. In contrast, SAKE enforces the Hankel structure, rank deficiency, and data consistency as three separate projections. Also, spatial, coil, and time dimensions are jointly incorporated in CF. If there is an ACS region, we may directly estimate the null space $\bm{V}_\perp$ from $\mathbb{D}_{\text{ACS}}$, then enforce it to hold for the whole k-space. This variant, which for 2D static MRI shares the assumptions with PRUNO, avoids the iterative step to estimate $\bm{V}_\perp$. Also, the type of convolution (linear, circular, or valid) may differ across k-space dimensions; for example, we may adopt circular convolution for the time dimension and valid convolution for the others.
Experiments and Results
=======================
We implement CF and compare to several existing techniques for 2D, 3D, and 2D cine (“2D+t”) imaging. For a fair comparison, we set the kernel size to be $5 k_x \times 5 k_y \times N_C$ for CF, SAKE, and ALOHA. Since P-LORAKS [@haldar2016p] uses a disk-shaped kernel, we choose a radius $3$ for P-LORAKS (“C” version) which leads to a similar but slightly larger kernel including $29> 5\times 5 = 25$ k-space points per coil. The reconstruction SNR generally increases with the kernel size but so does the computation burden. The stopping tolerance, relative change in the whole k-space, is $10^{-3}$ for all methods. A maximum of $200$ iterations is used, except for SAKE, in which case we also continue beyond $200$ iterations until the SNR of SAKE matches that of CF. This reconstruction is referred to as SAKE\*. We fine-tuned rank selection for all methods with respect to the first dataset, then applied that choice of rank to all other datasets. For other parameters, e.g., regularization parameter $\lambda$ for P-LORAKS, we employ published default values. For the recruitment and consent of human subjects used in this study, the ethical approval was given by an Internal Review Board (2005H0124) at The Ohio State University.
For 2D, we retrospectively downsampled three 3T brain datasets using three different acceleration rates, R $=4,6,$ and $8$ and four different sampling patterns: (i) 2D random; (ii) 2D random + $7k_x \times 7 k_y$ ACS region; (iii) 2D random + $17 k_x \times 17 k_y$ ACS region; (iv) 1D random + $5$ ACS readout lines. The data were compressed to eight virtual coils. Quantitative results and representative frames are in Table \[Tab:1\] and Figure \[Fig:1\], respectively. Not surprisingly, the performances of CF and SAKE\* are similar because they essentially solve the same 2D problem, but CF converges in fewer iterations and has a significantly smaller memory footprint.
For 3D, we truncated a 3T knee dataset (downloaded from mridata.org) to $160k_x \times 80k_y \times 64k_z$, then retrospectively downsampled in $k_y$ and $k_z$ using 2D random sampling, with $15 k_y \times 15k_z$ ACS and fully sampled $k_x$. The data were compressed to four virtual coils for faster processing. A representative slice is shown in Figure \[Fig:2\]. Since 3D CF reconstruction utilizes similarity and redundancy in three dimensions, with smaller degrees of freedom, it is able to outperform 2D CF reconstruction, which was separately applied to individual 2D slices. Due to prohibitive memory requirements, it was not feasible to extend SAKE’s implementation to 3D.
For 2D+t, we retrospectively downsampled three 3T cardiac cine datasets at four different acceleration rates, R $=4,6,8,$ and $10$ using a variable density sampling pattern [@ahmad2015variable]. The data were compressed to four virtual coils for faster processing. We fine-tuned parameters for all methods with respect to the first dataset, then applied these parameters to other two datasets. We averaged the reconstruction SNR for the other two different datasets. Quantitative results and representative frames are in Table \[Tab:2\] and Figure \[Fig:3\]. We compare to SENSE-based techniques, as SAKE and LORAKS do not provide extension to these cases. The CF is consistently better than L+S [@otazo2015low] and TV [@lustig2007sparse] by $1.1$ to $2.3$dB in terms of k-space SNR. With square-root sum of squared coils (SSoS), the margin is even larger.
-------------------------------------------------------------------------
R=4 R=6 R=8
---- -------------------------------------------------------- ----- -----
CF **[22.4dB]{} & **[19.4dB]{} & **[15.8dB]{}\
& 147.3s & 306.6s & 382.7s\
SAKE & 20.5dB & 16.3dB & 13.7dB\
& 301.2s & 320.7s & 293.7s\
SAKE$^*$ & **[22.4dB]{} & **[19.4dB]{} & **[15.8dB]{}\
& 593.5s & 772.2s & 722.8s\
P-LORAKS & 22.3dB & 18.2dB & 14.4dB\
& **[123.6s]{} & **[158.6s]{} & **[156.2s]{}\
******************
-------------------------------------------------------------------------
: 2D k-space reconstruction SNR (dB) and time (s) comparison. Median time and average k-space SNR are computed across two datasets and four different sampling patterns. SAKE$^*$ denotes continuing SAKE past $200$ iterations until reaching CF SNR. []{data-label="Tab:1"}
----------------------------------------------------------------------------
R=4 R=6 R=8 R=10
---- ---------------------------------------------------- ----- ----- ------
CF **[29.5]{} & **[28.2]{} & **[27.3]{} & **[26.3]{}\
L+S & 27.1 & 26.4 & 25.5 & 25.1\
TV & 27.2 & 26.1 & 25.1 & 24.1\
********
----------------------------------------------------------------------------
: 2D+t k-space reconstruction SNR (dB) averaged over two datasets. CF offers more than one dB advantage across all acceleration rates.[]{data-label="Tab:2"}
Discussion and Conclusion
=========================
Many parallel imaging approaches interpolate missing k-space points by solving a rank-deficient matrix completion. Choice of solution method can affect performance and memory requirements for this non-convex problem. CF provides an effective, memory-efficient solution and can subsume modeling choices found in existing GRAPPA-inspired methods. The memory requirement for CF processing (EIG + GD + ELS) is approximately the storage of the fully sampled k-space tensor $\mathbb{D}$. For example, for a 3D static double-precision complex k-space tensor with size $256 k_x \times 256 k_y \times 256 k_z \times 8 coils$ and a kernal size $10k_x \times 10k_y \times 10k_z \times 8coils$, a SAKE computation requires $2$ TB of memory, while CF only needs $2$ GB.
Computation speed of CF can be further enhanced by using the following properties and processing steps. *Automatic filter size adaptation*: Many annihilating relationships can be efficiently and approximately captured by a small kernel. Thus, the CF computation can be sped up by adaptively progressing from small kernels to larger ones as iterations evolve. *Automatic center to full k-space reconstruction*: Because the annihilation relationship holds for the entire k-space, we can apply CF for a small region of k-space and extract the null space, then enforce the estimated null space for the whole k-space. *Highly parallizable*: All convolutions inside each iterative step are independent and thus highly parallizable. Thus, we can fully utilize multi-cluster, multi-core CPU, or GPU architectures to accelerate the processing. *Johnson–Lindenstrauss Lemma (JLL)*: The dimensionality of the null space and thus the size of the least squares problem can be further reduced by applying JLL in each iteration. In summary, simple conceptual framework, broad applicability, unifying nature, memory efficient computation, and a potential to further improve the computation speed make CF an attractive framework for MRI reconstruction.
|
[**Role of different model ingredients in the exotic cluster-decay of $^{56}$Ni$^*$**]{}\
[^1]\
[*Govt. Sr. Sec. School, Summer Hill, Shimla -171005, India*]{}
We present cluster decay studies of $^{56}$Ni$^*$ formed in heavy-ion collisions using different Fermi density and nuclear radius parameters proposed by various authors. Our study reveals that different technical parameters do not alter the transfer structure of fractional yields significantly. The cluster decay half-lives of different clusters lies within $\pm$10% for different Fermi density parameters and nuclear radius, therefore, justify the current set of parameters used in the literature for the calculations of cluster decay.
Introduction
============
In earlier days, nucleus was considered to have a uniform density and sharp radius. With the passage of time, the density distribution was found to be more complicated. Several different forms (direct or indirect) exist in literature that can explain these complicated nuclear density distributions. The first method is the direct parametrization which involves the choice of a suitable functional form where parameters are varied to fit the experimental data. The two parameter Fermi density distribution is an example of such a parametrization. The second method is of indirect parametrization of density distribution proceeds via nuclear models. The nuclear models like shell model contains certain parameters which are determined by other physical considerations and it is then used to calculate the nuclear density distribution without further adjustments. The experimental data can be described accurately with two-parameter Fermi density distribution at relatively low momentum. Among all the density distributions two-parameters Fermi density has been quite successful in the low, medium and heavy mass regions. The systematic study of charge distributions have been carried out in Refs.[@Angeli; @Weso; @Fried]. We shall use this density distribution here.
Since the nuclear systems obey quantum laws, therefore, their surfaces are not well defined. The nuclear density remains constant up to certain distance but fall more rapidly close to the surface region where the nucleons are free to move about. The nuclear densities provide important information about the structure of nuclear matter at low energies and other important information regarding the equation of state at intermediate energies [@sk; @rk].
Various methods have been developed for exploring the nuclear structure and radius. The electron scattering/ electrically charged particles of high energy are employed as probe to explore the proton distribution of the nuclei (i.e charge radii), whereas neutral nuclear probes such as neutrons will give the effect of nuclear forces over the nuclear surface (i.e. interaction radii). The charge radii are often used to extract the information about nuclear radii. The electron scattering experiments shows that the charge distribution within a nucleus either follow Fermi trapezoidal shape or modified Gaussian distribution. These studies have shown that nuclear charge density does not decrease abruptly but has a finite diffuseness.
A model that uses density distribution such as two parameter Fermi density (as shown in Fig. 1) has to rely on the information about nuclear radius (or half density radii $R_0$), central density $\rho_0$, and surface diffuseness ($a$). Interestingly, several different experimental as well as theoretical values of these parameters are available in literature [@8Ngo; @8Puri92; @8SM; @8EW; @8HS; @Elton]. In addition, several different names such as central radii, equivalent sharp radii, root mean square radii etc. have also been used in the literature to define different functional forms. The role of different radii was examined in exotic cluster decay half-lives [@8Rkg] and interestingly two different forms of radii were found to predict five order of magnitude different half-lives within the same theoretical model. Similarly, the use of different values of surface diffuseness also varies from author to author. The effect of these model ingredients on the fusion process at low incident energy have been studied in Ref. [@8RashmiT] and there was found that the effect of different radii is more than marginal and therefore this parameter should be used with a more fundamental basis. Unfortunately, no systematic study is still available in the cluster decay process. In this paper, we plan to study the role of Fermi density parameters in the cluster decay of $^{56}$Ni$^*$ when formed in heavy-ion collisions. This study is still missing in the literature.
Heavy-ion reactions provide a very good tool to probe the nucleus theoretically. This includes low energy fusion process [@id], intermediate energy phenomena [@qmd] as well as cluster-decay and/or formation of super heavy nuclei [@gupta; @kp]. In the last one decade, several theoretical models have been employed in the literature to estimate the half-life times of various exotic cluster decays of radioactive nuclei. These outcome have also been compared with experimental data. Among all the models employed preformed cluster model (PCM) [@rkg88; @mal89; @kum97] is widely used to study the exotic cluster decay. In this model the clusters/ fragments are assumed to be pre-born well before the penetration of the barrier. This is in contrast to the unified fission models (UFM) [@7Poen; @7Buck; @7Sand], where only barrier penetration probabilities are taken into account. In either of these approach, one needs complete knowledge of nuclear radii and densities in the potential.
Cluster decay of $^{56}$Ni is studied when formed as an excited compound system in heavy-ion reactions. Since $^{56}$Ni has negative $Q$-value (or $Q_{out}$) and is stable against both fission and cluster decay processes. However, if is is produced in heavy-ion reactions depending on the incident energy and angular momentum involved, the excited compound system could either fission, decay via cluster emissions or results in resonance phenomenon. The $^{56}$Ni has a negative $Q_{out}$ having different values for various exit channels and hence would decay only if it were produced with sufficient compound nucleus excitation energy $E^{\ast}_{CN}~(=E_{cm} + Q_{in})$, to compensate for negative $Q_{out}$, the deformation energy of the fragments $E_d$, their total kinetic energy ($TKE$) and the total excitation energy ($TXE$), in the exit channel as: $$E^{\ast}_{CN} = \mid Q_{out} \mid + E_{d} + TKE + TXE.
\label{eq:1}$$ (see Fig. 2, where $E_d$ is neglected because the fragments are considered to be spherical). Here $Q_{in}$ adds to the entrance channel kinetic energy $E_{cm}$ of the incoming nuclei in their ground states.
Section 2 gives some details of the Skyrme energy density model and preformed cluster model and its simplification to unified fission model. Our calculations for the decay half-life times of $^{56}$Ni compound system and a discussion of the results are presented in Section 3. Finally, the results are summarized in Section 4.
Model
=====
Skyrme Energy Density Model
---------------------------
In the Skyrme Energy Density Model (SEDM) [@8Puri92], the nuclear potential is calculated as a difference of energy expectation value $E$ of the colliding nuclei at a finite distance $R$ and at complete isolation (i.e. at $\infty$) [@8Puri92; @VB72]. $$V_{N} (R) = E(R)- E(\infty), \label{eq:2}$$
where $E = \int H (\vec{r})\vec{dr}$, with $H(\vec{r})$ as the Skyrme Hamiltonian density which reads as: $$\begin{aligned}
H(\rho,\tau,\vec{J}) & = & \frac{\hbar^2}{2m} \tau +\frac{1}{2}t_0
[(1+
\frac{1}{2}x_0)\rho^2-(x_0+\frac{1}{2})(\rho_n^2+\rho_p^2)]\nonumber\\
& &+\frac{1}{4}( t_1+t_2)\rho\tau +\frac{1}{8}(t_2-t_1)(\rho_n
\tau_n+\rho_p \tau_p) \nonumber \\ & & +\frac{1}{16} (t_2-3t_1)
\rho \nabla^2 \rho + \frac{1}{4}t_3 \rho_n \rho_p \rho \nonumber
\\ & &+\frac{1}{32} (3t_1+t_2) (\rho_n \nabla^2 \rho_n+\rho_p
\nabla^2 \rho_p) \nonumber \\ && -\frac {1}{2} W_0(\rho
\vec{\nabla}\cdot\vec{J} +\rho_n \vec{\nabla}\cdot\vec{J}_n+
\rho_p \vec{\nabla} \cdot\vec{J}_p). \label{eq:3}\end{aligned}$$ Here $\vec{J} = \vec{J}_{n} + \vec{J}_{p}$ is the spin density which was generalized by Puri et al. [@8Puri92], for spin-unsaturated nuclei and $\tau=\tau_{n} + \tau_{p}$ is the kinetic energy density calculated using Thomas Fermi approximation [@gupta85; @2Von], which reduces the dependence of energy density $H(\rho ,\tau ,\vec{J})$ to be a function of nucleon density $\rho$ and spin density $\vec{J}$ only. Here strength of surface correction factor is taken to be zero (i.e. $\lambda=0$). The remaining term is the nucleon density $\rho=\rho_{n} + \rho_{p}$ is taken to be well known two-parameter Fermi density. The Coulomb effects are neglected in the above energy density functional, but will be added explicitly. In Eq. (\[eq:3\]), six parameters $t_0$, $t_1$, $t_2$, $t_3$, $x_0$, and $W_0$ are fitted by different authors to obtain the best description of the various ground state properties for a large number of nuclei. These different parameterizations have been labeled as S, SI, SII, SIII etc. and known as Skyrme forces for light and medium colliding nuclei. Other Skyrme forces are able to reproduce the data for heavy systems better. The Skyrme force used for the present study is SIII with parameters as: $t_0=-1128.75$ MeVfm$^3$, $t_1=395.00$ MeVfm$^5$, $t_2=-95.00$ MeVfm$^5$, $t_3=14000.00$ MeVfm$^6$, $x_0=0.45$, and $W_0=120.00$ MeVfm$^5$. It has been shown in previous studies that SIII force reproduces the fusion barrier much better than other sets of Skyrme forces for light and medium nuclei. Other Skyrme forces such as SKa, SKm, however, are found to be better for heavier masses.
From Eq. (\[eq:3\]), one observes that the Hamiltonian density $H(\rho ,\tau ,\vec{J})$ can be divided into two parts: (i) the spin-independent part $V_{P}(R)$, and (ii) spin-dependent $V_{J}(R)$ [@8Puri92] as: $$\begin{aligned}
V_{N}(R) &=&\int \left\{H(\rho)- \left[H_{1}(\rho_{1}) +
H_{2}(\rho_{2}) \right] \right\}d\vec{r}
\nonumber \\
&+&\int \left\{H(\rho, \vec{J})- \left[H_{1}(\rho_{1},
\vec{J}_{1})
+ H_{2}(\rho_{2}, \vec{J}_{2}) \right] \right\}d\vec{r}\nonumber \\
&=& V_{P}(R)+V_{J}(R) \label{eq:4}\end{aligned}$$ We apply the standard Fermi mass density distribution for nucleonic density: $$\rho (R)=\frac{\rho_{0} }{ 1+ \exp\left\{\frac{R - R_{0} }{a}
\right\} },~~~~~~~~~~~~ - \infty \leq R \leq \infty \label{eq:5}$$ Here $\rho_{0}$, $R_{0}$ and [*“a”*]{} are respectively, the average central density, half-density radius and the surface diffuseness parameter. The $R_{0}$ gives the distance where density drops to the half of its maximum value and the surface thickness $s~(=4.4a)$ has been defined as the distance over which the density drops from 90% to 10% of its maximum value is the average central density $\rho_{0}$. The systematic two parameter Fermi density distribution is shown in Fig. 1.
Another quantity, which is equally important is the r.m.s. radius $\langle r^{2}\rangle _{m}$ defined as: $$\begin{aligned}
\left<r^{2} \right>_{m} &= & \int r^{2}\rho \left(\vec{r}
\right)d\vec{r}
= 4\pi \int\limits_{0}^{\infty } \rho \left(\vec{r} \right)r^{4}d^{3}r.
\label{eq:6}\end{aligned}$$ One can find the half density radius by varying surface diffuseness [*“a”*]{} and keeping r.m.s. radius $\langle
r^{2}\rangle _{m}$ constant or from normalization condition: $$R_{0}= \frac{1}{3}\left[5 \left<r^{2} \right>_{m} -7\pi ^{2}a^{2}
\right], \label{eq:7}$$ The average central density $ \rho_{0}$ given by [@8Stancu] $$\rho_{0} =\frac{3A}{4 \pi R^{3}_{0}}
\left[1+\frac{\pi^{2}a^{2}}{R^{2}_{0}} \right]^{-1}. \label{eq:8}$$ Using Eq. (\[eq:5\]), one can find the density of neutron and proton individually as: $$\rho _{n}= \frac{N}{A}\rho ,~~~~~~~~~~\rho _{p}= \frac{Z}{A}\rho.
\label{eq:9}$$ For the details of the model, reader is referred to Ref. [@8Puri92].
In order to see the effect of different Fermi density parameters on the cluster decay half-lives, we choose the following different Fermi density parameters proposed by various authors.
1. **H. de Vries *et al.* [@Elton]:** Here, we use the interpolated experimental data [@8Puri91] of Elton and H. de Vries for half density radius $R_{0}$ and surface thickness $a$. Using $R_{0}$ and $a$, central density $\rho_{0}$ can be computed using Eq. (\[eq:7\]). This set of parameters is labeled as DV.
2. **Ngô-Ngô [@8Ngo]:** In the version of Ngô-Ngô, a simple analytical expression is used for nuclear densities instead of Hartree-Fock densities. These densities are taken to be of Fermi type and written as: $$\rho_{n, p} (R)= \frac{\rho_{n, p}(0)}{1+\exp [(R-C_{n,
p})/0.55]}~, \label{eq:10}$$ $\rho_{n, p}(0)$ are then given by: $$\rho_n(0)=\frac{3}{4\pi}\frac{N}{A}\frac{1}{r^3_{0_{n}}},~~~~~~~\rho_p(0)=\frac{3}{4\pi}\frac{Z}{A}\frac{1}{r^3_{0_{p}}}~.
\label{eq:11}$$ where $C$ represents the central radius of the distribution. $$C= R\left[1-\frac{1}{R^{2}}\right],
\label{eq:12}$$ and $$R= \frac{NR_{n} +ZR_{p}}{A}. \label{eq:13}$$ The sharp radii for proton and neutron are given by, $$R_{p}= r_{0_{p}}A^{1/3},~~~~~~~~~~~~~R_{n}= r_{0_{n}}A^{1/3},
\label{eq:14}$$ with $$r_{0_{p}}= 1.128~fm,~ r_{0_{n}}= 1.1375 + 1.875\times 10^{-4} A.
\label{eq:15}$$ This set of parameters is labeled as Ngo.
3. **S.A. Moszkwski [@8SM]:** The Fermi density parameters due to Moszkwski has central density $\rho_0 = 0.16$ nucl./fm$^3$, the surface diffuseness parameters $a$ is equal to 0.50 fm and radius $R_0 = 1.15A^{1/3}$. This set of parameters is labeled as SM.
4. **E. Wesolowski [@8EW]:** The expressions for Fermi density parameters taken by E. Wesolowski reads as: The central density $$\rho_0 = \left[\frac{4}{3} \pi R^3_0 \left\{1 + \left(\pi
a/R_0\right)^2 \right\} \right]^{-1}. \label{eq:16}$$
The surface diffuseness parameters $a$ = 0.39 fm and half density radius, $$R_0 = R^{\prime}\left[1 - \left(\frac{b}{R^{\prime}}\right)^2 +
\frac{1}{3}\left( \frac{b}{R^{\prime}}\right)^6 +
\cdot\cdot\cdot\cdot\cdot\right];\label{eq:17}$$ with $$R^{\prime} = \left[1.2 - \frac{0.96}{A^{1/3}}\left(
\frac{N-Z}{A}\right)
\right]A^{1/3},~\mbox{and}~b=\frac{\pi}{\sqrt3}a. \label{eq:18}$$ This set of parameters is labeled as EW.
5. **H. Schechter *et al.* [@8HS]:** The value of Fermi density parameters taken by H. Schechter *et al.* can be summarized as: central density $\rho_0 = 0.212/(1 + 2.66A^{-2/3})$, the surface diffuseness parameters $a$ is equal to 0.54 fm and radius $R_0 = 1.04A^{1/3}$ in single folding model for one of the nucleus. This set of parameters is labeled as HS.
In the spirit of proximity force theorem, the spin independent potential $V_{P}(R)$ of the two spherical nuclei, with radii C$_1$ and C$_2$ and whose centers are separated by a distance $R=s+C_1+C_2$ is given by $$V_{P}(R) = 2\pi \overline{R} \phi (s), \label{eq:19}$$ where $$\phi (s) =\int \left\{H(\rho)- \left[H_{1}(\rho_{1}) +
H_{2}(\rho_{2}) \right] \right\}dZ, \label{eq:20}$$ and $$\overline{R} =\frac{C_1 C_2}{C_1+C_2}, \label{eq:21}$$ with Süssmann central radius $C$ given in terms of equivalent spherical radius $R$ as $$C =R-\frac{b}{R}. \label{eq:22}$$ Here the surface diffuseness $b=1$ fm and nuclear radius $R$ taken as given by various authors in the literature [@8Ngo; @3Blocki77; @AW95; @3MS00; @Royer; @5Bass73; @5CW].
In the original proximity potential [@3Blocki77], the equivalent sharp radii used are $$R= 1.28A^{1/3}- 0.76 + 0.8A^{-1/3}~~~{\rm fm}.\label{eq:23}$$ This radius is labeled as R$_{Prox77}$.
In the present work, we also used the nuclear radius due to Aage Winther, labeled as R$_{AW}$ and read as [@AW95]: $$R= 1.20A^{1/3}- 0.09~~~{\rm fm}. \label{eq:24}$$ The newer version of proximity potential uses a different form of nuclear radius [@3MS00] $$R= 1.240A^{1/3}\left[1+1.646A^{-1}- 0.191A_{s} \right]~~~{\rm
fm}.\label{eq:25}$$ This radius is labeled as R$_{Prox00}$.
Recently, a newer form of above Eq. (\[eq:25\]) with slightly different constants is reported [@Royer] $$R= 1.2332A^{1/3}+2.8961A^{-2/3}- 0.18688A^{1/3}A_{s} ~~~{\rm
fm},\label{eq:26}$$ and is labeled as R$_{Royer}$.
For Ngô and Ngô [@8Ngo] nuclear radius, we use Eqs. (\[eq:13\])-(\[eq:15\]) and is labeled as R$_{Ngo}$.
The potential based on the classical analysis of experimental fusion excitation functions, used the nuclear radius (labeled as R$_{Bass}$) [@5Bass73] as: $$R= 1.16A^{1/3} - 1.39A^{-1/3}. \label{eq:27}$$ The empirical potential due to Christensen-Winther (CW) uses the same radius form (Eq. (\[eq:27\])) having different constants (labeled as R$_{CW}$) [@5CW]. $$R= 1.233A^{1/3} - 0.978A^{-1/3}.
\label{eq:28}$$
The Preformed Cluster Model
---------------------------
For the cluster decay calculations, we use the Preformed Cluster Model [@rkg88; @mal89; @kum97]. It is based on the well known quantum mechanical fragmentation theory [@7Puri; @rkg75; @7SNG; @7Maruhn74], developed for the fission and heavy-ion reactions and used later on for predicting the exotic cluster decay [@7Sandu80; @Rose; @rkg94] also. In this theory, we have two dynamical collective coordinates of mass and charge asymmetry $\eta=(A_1-A_2)/(A_1+A_2)$ and $\eta_Z=(Z_1-Z_2)/(Z_1+Z_2)$. The decay half-life $T_{1/2}$ and decay constant $\lambda$, in decoupled $\eta$- and $R$-motions is $$\lambda =\frac{\ln 2}{T_{1/2}}= P_{0}\nu _{0}P,
\label{eq:29}$$ where the preformation probability $P_0$ refers to the motion in $\eta$ and the penetrability $P$ to $R$-motion. The $\nu_0$ is the assault frequency with which the cluster hits the barrier. Thus, in contrast to the unified fission models [@7Poen; @7Buck; @7Sand], the two fragments in PCM are considered to be pre-born at a relative separation co-ordinate $R$ before the penetration of the potential barrier with probability $P_0$. The preformation probability $P_0$ is given by $$P_{0}(A_{i})=
\mid \psi(\eta,A_i) \mid^{2}\sqrt{B_{\eta \eta }(\eta
)}\left(\frac{4}{A_i}\right),\,\,\,\,\,\,\,\,\,(i=1~{\rm or}~2),
\label{eq:30}$$ with $\psi^{\nu}(\eta),~~ \nu=0,1,2,3,.....$, as the solutions of stationary Schrödinger equation in $\eta$ at fixed $R$, $$\left[-\frac{\hbar ^{2}}{2\sqrt{B_{\eta\eta }}}\frac{\partial
}{\partial \eta }\frac{1}{\sqrt{B_{\eta \eta }}}\frac{\partial
}{\partial \eta }+V_R(\eta) \right]\psi^{\nu}(\eta)=E^{\nu}
\psi^{\nu}(\eta), \label{eq:31}$$ solved at $R=R_a=R_{min}$ at the minimum configuration i.e. $R_a =
R_{min}$ (corresponding to $V_{min}$) with potential at this $R_a$-value as $V(R_a = R_{min})= \overline{V}_{min}$ (displayed in Fig. 2).
The temperature effects are also included here in this model through a Boltzmann-like function as $$\mid \psi(\eta)\mid^{2}=\sum_{\nu=0}^{\infty }\mid \psi^{\nu}
(\eta) \mid^{2}\exp \left(-\frac{E_{\eta}}{T} \right),
\label{eq:32}$$ where the nuclear temperature $T$ (in MeV) is related approximately to the excitation energy $E^{\ast}_{CN}$, as: $$E^{\ast}_{CN}=\frac{1}{9}A{T}^2-T, \qquad\qquad {(\rm in~ MeV)}.
\label{eq:33}$$ The fragmentation potential (or collective potential energy) $V_R(\eta)$, in Eq. (\[eq:31\]) is calculated within Strutinsky re-normalization procedure, as $$V_R(\eta)=-\sum^2_{i=1} \left[V_{LDM}(A_i,Z_i)+\delta U_i
\exp\left(-\frac{T^2}{T_0^2}\right)\right] +\frac{Z_1 \cdot
Z_2e^2}{R} + V_N(R),
\label{eq:34}$$ where the liquid drop energies ($V_{LDM} = B - \delta U$) with $B$ as theoretical binding energy of Möller et al. [@mol95] and the shell correction $\delta U$ calculated in the asymmetric two center shell model. The additional attraction due to nuclear interaction potential $V_N(R)$ is calculated within SEDM potential using different Fermi density parameters and nuclear radii as discussed earlier. The shell corrections are considered to vanish exponentially for $E^{\ast}_{CN} \ge 60$ MeV, giving $T = 1.5$ MeV. The mass parameter $B_{\eta\eta}$ representing the kinetic energy part of the Hamiltonian in Eq. (\[eq:31\]) are smooth classical hydrodynamical masses of Kröger and Scheid [@kro80].
The WKB action integral was solved for the penetrability $P$ [@rkg94]. For each $\eta$-value, the potential $V(R)$ is calculated by using SEDM for $R \ge R_{d}$, with $R_{d}=R_{min} +
\Delta R$ and for $R \le R_{d}$, it is parameterized simply as a polynomial of degree two in $R$: $$V(R)=\left \{
\begin{array}{ll}
\mid Q_{out} \mid +{a_1}(R-R_0)+{a_2}(R-R_0)^2 & \mbox{for \quad $R_0\leq R\leq R_{d} $}, \\
V_N(R) + Z_1 \cdot Z_2 e^2/R & \mbox{for $ \quad R\geq R_{d}$},
\end{array}
\right. \label{eq:35}$$ where $R_0$ is the parent nucleus radius and $\Delta R$ is chosen for smooth matching between the real potential and the parameterized potential (with second-order polynomial in $R$). A typical scattering potential, calculated by using Eq. (\[eq:35\]) is shown in Fig. 2, with tunneling paths and the characteristic quantities also marked. Here, we choose the first (inner) turning point $R_a$ at the minimum configuration i.e. $R_a
= R_{min}$ (corresponding to $V_{min}$) with potential at this $R_a$-value as $V(R_a = R_{min})= \overline{V}_{min}$ and the outer turning point $R_b$ to give the $Q_{eff}$-value of the reaction $V(R_b) = Q_{eff}$. This means that the penetrability $P$ with the de-excitation probability, $W_i=\exp (-bE_i)$ taken as unity, can be written as $P=P_iP_b,$ where $P_i$ and $P_b$ are calculated by using WKB approximation, as: $$P_i=\exp \left[- \frac{2}{\hbar} \int\limits_{R_a}^{R_i}\{2\mu
[V(R)-V(R_i)]\}^{1/2}dR \right],
\label{eq:36}$$ and $$P_b=\exp\left[- \frac{2}{\hbar} \int\limits_{R_i}^{R_b}\{2\mu
[V(R)-Q_{eff}]\}^{1/2}dR \right],
\label{eq:37}$$ here $R_a$ and $R_b$ are, respectively, the first and second turning points. This means that the tunneling begins at $R =
R_a~(=R_{min})$ and terminates at $R = R_b$, with $V(R_b) =
Q_{eff}$. The integrals of Eqs. (\[eq:36\]) and (\[eq:37\]) are solved analytically by parameterizing the above calculated potential $V(R)$.
The assault frequency $\nu_0$ in Eq. (\[eq:29\]) is given simply as $$\nu_0 = \frac{v}{R_0} = \frac{(2E_2/\mu)^{1/2}}{R_0},
\label{eq:38}$$ where $E_2 = \frac{A_1}{A} Q_{eff}$ is the kinetic energy of the emitted cluster, with $Q_{eff}$ shared between the two fragments and $\mu =m(\frac{A_1 A_2}{A_1+ A_2})$ is the reduced mass.
The PCM can be simplified to UFM, if preformation probability $P_0=1$ and the penetration path is straight to $Q_{eff}$-value.
\[result\]Results and Discussions
=================================
In the following, we see the effect of different Fermi density parameters and nuclear radii on the cluster-decay process using the Skyrme energy density formalism within PCM and UFM.
First of all, to see the effect of different Fermi density parameters on the cluster decay half-lives, we choose the different Fermi density parameters proposed by various authors as discussed earlier.
Fig. 2 shows the characteristic scattering potential for the cluster decay of $^{56}$Ni$^{\ast}$ into $^{16}$O + $^{40}$Ca channel as an illustrative example. In the exit channel for the compound nucleus to decay, the compound nucleus excitation energy $E_{CN}^{\ast}$ goes in compensating the negative $Q_{out}$, the total excitation energy $TXE$ and total kinetic energy $TKE$ of the two outgoing fragments as the effective Q-value (i.e. $TKE=Q_{eff}$ in the cluster decay process). In addition, we plot the penetration paths for PCM and UFM using Skyrme force SIII (without surface correction factor, $\lambda =0$) with DV Fermi density parameters. For PCM, we begin the penetration path at $R_a
= R_{min}$ with potential at this $R_a$-value as $V(R_a =
R_{min})= \overline{V}_{min}$ and ends at $R = R_b$, corresponding to $V(R=R_b) = Q_{eff}$, whereas for UFM, we begin at $R_a$ and end at $R_b$ both corresponding to $V(R_a) =V(R_b)=Q_{eff}$. We have chosen only the case of variable $Q_{eff}$ (as taken in Ref. [@nkdid]), for different cluster decay products to satisfy the arbitrarily chosen relation $Q_{eff}=0.4(28 - \mid
Q_{out} \mid)$ MeV, as it is more realistic [@MKS00]. The scattering potential with SM Fermi density parameters is also plotted for comparison.
Fig. 3(a) and (b) shows the fragmentation potential $V(\eta)$ and fractional yield at $R = R_{min}$ with $V(R_{min})=
\overline{V}_{min}$. The fractional yields are calculated within PCM at $T$ = 3.0 MeV using various Fermi density parameters for $^{56}$Ni$^{\ast}$. From figure, we observe that different parameters have minimal role in the fractional mass distribution yield. The fine structure is not at all disturbed for different sets of Fermi density parameters.
We have also calculated the half-life times (or decay constants) of $^{56}$Ni$^{\ast}$ within PCM and UFM for clusters $\ge
^{16}$O. For $^{16}$O, the cluster decay constant varies by an order of magnitude ten. The variation is much more with SM parameters. In the case of UFM, variation is almost constant.
In Fig. 4, we display the cluster decay half-lives $\log T_{1/2}$ for various Fermi density parameters using PCM. There is smooth variation in half-life times with all the density parameters except for SM parameter. The trends in the variation of cluster half-life times (or decay constants) are similar in both PCM and UFM, but in case of UFM decay constants are more by an order of ten. In SM the decay constants are larger by an order of 14.
In order to quantify the results, we have also calculated the percentage variation in $\log T_{1/2}$ as: $$\left[\log T_{1/2} \right] \% = \frac{(\log T_{1/2})^i- (\log
T_{1/2})^{DV}}{(\log T_{1/2})^{DV}}\times 100, \label{eq:39}$$ where $i$ stands for the half-life times calculated using different Fermi density parameters. The variation in the cluster decay half-lives is studied with respect to DV parameters. In Fig. 5(a) and (b), we display the percentage variation in the half-life times within both the PCM and UFM models as a function of cluster mass $A_2$ using Eq. (\[eq:39\]). For the PCM these variation lies within $\pm$5% excluding SM parameters, whereas including SM parameters it lies within $\pm$13%. In the case of UFM half-lives lies within $\pm$1.5% for all density parameters except of SM. For SM parameters variations lie within $\pm$9%.
Finally, it would be of interest to see how different forms of nuclear radii as discussed earlier would affect the cluter decay half-lives.
In Fig. 6, we display the characteristic scattering potential for the cluster decay of $^{56}$Ni$^{\ast}$ into $^{28}$Si + $^{28}$Si channel for R$_{Bass}$ and R$_{Royer}$ forms of nuclear radius. In the exit channel for the compound nucleus to decay, the compound nucleus excitation energy $E_{CN}^{\ast}$ goes in compensating the negative $Q_{out}$, the total excitation energy $TXE$ and total kinetic energy $TKE$ of the two outgoing fragments as the effective Q-value. We plot the penetration path for PCM using Skyrme force SIII (without surface correction factor, $\lambda
=0$) with nuclear radius R$_{Bass}$. Here again, we begin the penetration path at $R_a = R_{min}$ with potential at this $R_a$-value as $V(R_a = R_{min})= \overline{V}_{min}$ and ends at $R = R_b$, corresponding to $V(R=R_b) = Q_{eff}$ for PCM. The $Q_{eff}$ are same as discussed earlier.
Fig. 7(a) and (b) show the fragmentation potentials $V(\eta)$ and fractional yields at $R = R_{min}$ with $V(R_{min})=
\overline{V}_{min}$. The fractional yields are calculated within PCM at $T$ = 3.0 MeV for $^{56}$Ni$^{\ast}$ using various forms of nuclear radii. From figure, we observe that different radii gives approximately similar behavior, however small changes in the fractional mass distribution yields are observed. The fine structure is not at all disturbed for different radius values.
We have also calculated the half-life times (or decay constants) of $^{56}$Ni$^{\ast}$ within PCM for clusters $\ge^{16}$O. The cluster decay constant for nuclear radius due to Bass varies by an order of magnitude $10^{2}$, where as order of magnitude is same for other radii. In Fig. 8, we display the cluster decay half-lives $\log T_{1/2}$ for various nuclear radii taken by different authors as explained earlier using PCM. One can observe small variations in half-life times.
In order to quantify the results, we have also calculated the percentage variation in $\log T_{1/2}$ as: $$\left[\log T_{1/2} \right] \% = \frac{(\log T_{1/2})^i- (\log
T_{1/2})^{R_{Royer}}}{(\log T_{1/2})^{R_{Royer}}}\times 100,
\label{eq:40}$$ where $i$ stands for the half-life times calculated using different forms of nuclear radii. The variation in the cluster decay half-lives is studied with respect to radius formula given by Royer R$_{Royer}$. In Fig. 9, we display the percentage variation in the half-life times for PCM as a function of cluster mass $A_2$ using Eq. (\[eq:40\]). These variation lies within $\pm$7% excluding Bass radius where it lies within $\pm$10%.
\[summary\]Summary
==================
We here reported the role of various model ingredients as well as radii in the cluster decay constant calculations. Our studies revealed that the effect of different density and nuclear radius parameters on the cluster decay half-life times is about $10\%$. Our study justify the use of current set of parameters for radius as the effect of different prescriptions is very small.\
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![The systematics diagram for two parameter Fermi density.[]{data-label="fig2"}](fig1.eps "fig:"){width="14cm"} -4cm
![The scattering potential $V(R)$ (MeV) for cluster decay of $^{56}$Ni$^{\ast}$ into $^{16}$O + $^{40}$Ca channel for different Fermi density parameters. The distribution of compound nucleus excitation energy E$_{CN}^{*}$ at both the initial ($R=R_{0}$) and asymptotic ($R \to \infty$) stages and $Q$-values are shown. The decay path for both PCM and UFM models is also displayed.[]{data-label="fig2"}](fig2.eps "fig:"){width="14cm"} -4cm
![(a) The fragmentation potential $V(\eta)$ and (b) calculated fission mass distribution yield with different density parameters at $T$ = 3.0 MeV.[]{data-label="fig2"}](fig3.eps "fig:"){width="14cm"} -4cm
![The variation of $\log T_{1/2}$ (sec) using different density parameters for PCM.[]{data-label="fig2"}](fig4.eps "fig:"){width="14cm"} -4cm
![Percentage variation of $\log T_{1/2}$ for different different Fermi density parameters w.r.t. DV parameters.[]{data-label="fig2"}](fig5.eps "fig:"){width="14cm"} -4cm
![Same as fig. 2, but for different radii. The decay path displayed only for PCM.[]{data-label="fig2"}](fig6.eps "fig:"){width="14cm"} -4cm
![Same as fig. 3, but for different radii.[]{data-label="fig2"}](fig7.eps "fig:"){width="14cm"} -4cm
![Same as fig. 4, but for different radii.[]{data-label="fig2"}](fig8.eps "fig:"){width="14cm"} -4cm
![Percentage variation of $\log T_{1/2}$ for different forms of radii with PCM only.[]{data-label="fig2"}](fig9.eps "fig:"){width="14cm"} -4cm
[^1]: Email: [email protected]
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abstract: 'In this paper, we show that the Lie superalgebra $\mathfrak{spo}(2l+2|n)$ is into the intersection of Lie superalgebra of contact vector fields $\mathcal{K}(2l+1|n)$ and the Lie superalgebra of projective vector fields $\mathfrak{pgl}(2l+2|n)$. We use mainly the embedding used by P. Mathonet and F. Radoux in “ *Projectively equivariant quantizations over superspace $\mathbb{R}^{p|q}$. Lett. Math. Phys, 98: 311-331, 2011*”. Explicitly, we use the embedding of a Lie superalgebra constituted of matrices belonging to ${\mathfrak{gl}}(2l+2|n)$ into $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$. We generalize thus in superdimension $2l+1-n$, the matrix realization described in [@MelNibRad13] on $S^{1|2}$. We mention that the intersection ${\mathfrak{spo}}(2l+2|n)={\mathfrak{pgl}}(2l+2|n)\cap{\mathcal{K}}(2l+1|n)$ that we prove here, in super case, has been prooved on ${\mathbb{R}}^{2l+2}$ in even case in [@CoOv12].'
address: 'University of Burundi, Institute of Applied Pedagogy, Department of mathematics, B.P 2523, Bujumbura-Burundi'
author:
- Aboubacar Nibirantiza
title: 'On the matrix realization of the Lie superalgebra of contact projective vector fields $\mathfrak{spo}(2l+2|n)$'
---
Introduction
============
The present paper is based on the concepts of supergeometry. It begins with a brief introduction to the notions that we need in the all sections, i.e: superfunctions, vector fields, differential $1$-superforms, etc on the superspace ${\mathbb{R}}^{m|n}$, where $m$ and $n$ are integers. We describe the supergeometry of ${\mathbb{R}}^{m|n}$ by its supercommutative superalgebra of superfunctions $C^{\infty}({\mathbb{R}}^{m|n})$.\
Using the standard contact structure on ${\mathbb{R}}^{2l+1|n}$, where $l$ is also an integer, we compute the formula of the contact vector fields on ${\mathbb{R}}^{2l+1|n}$. This formula is a generalization of those formulas known in classical geometry, as in [@CoOv12] and in supergeometry in low dimensions, as in [@GarMelOvs07; @Mel09; @MelNibRad13]. We also compute, in the super case, the formula of the Lagrange bracket of the superfunctions $f$ and $g$.\
As in [@MelNibRad13], we consider an superskewsymmetric form $\omega$ defined on the superspace ${\mathbb{R}}^{2l+2|n}$ and we realize thus a Lie superalgebra ${\mathfrak{spo}}(2l+2|n)$ constituted by the matrices $A$ of ${\mathfrak{gl}}(2l+2|n)$ which preserve the form $\omega$. We use the method used by P. Mathonet and F. Radoux in [@MatRad11]. This construction allows us to embed the Lie superalgebra ${\mathfrak{spo}}(2l+2|n)\subset{\mathfrak{pgl}}(2l+2|n)$ into the Lie superalgebra $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$ of vector fields on ${\mathbb{R}}^{2l+1|n}$.\
Thanks to the formula of contact vector fields $X_f$ obtained, for a certain superfuncfion given $f\in C^\infty({\mathbb{R}}^{2l+1|n})$ of degree to most equal two in $z,x_i,y_i$ and $\theta_i$ variables, and to the formulas of projective vector fields of ${\mathfrak{spo}}(2l+2|n)\subset{\mathfrak{pgl}}(2l+2|n)$ obtained, we realize that the Lie superalgebra ${\mathfrak{spo}}(2l+2|n)$ is constituted by the contact projective vector fields, i.e: the Lie superalgebra ${\mathfrak{spo}}(2l+2|n)$ is into the intersection of the Lie superalgebra of projective vector fields ${\mathfrak{pgl}}(2l+2|n)$ and the Lie superalgebra of contact vector fields ${\mathcal{K}}(2l+1|n)$. To justify the terminology of contact projective vector fields for the elements of ${\mathfrak{spo}}(2l+2|n)$, we refer to [@CoOv12].
Superfunctions on ${\mathbb{R}}^{2l+1|n}$
=========================================
We define the geometry of the superspace ${\mathbb{R}}^{2l+1|n}$, where $l\in{\mathbb{N}},n\in{\mathbb{N}}^*$, by describing its associative supercommutative superalgebra of superfunctions on ${\mathbb{R}}^{2l+1|n}$ which we denote by $$C^\infty({\mathbb{R}}^{2l+1|n}):=C^\infty({\mathbb{R}}^{2l+1})\otimes \Lambda{\mathbb{R}}^n$$ and which is constituted by the elements $$\begin{aligned}
f(x,\theta)&=\sum_{0\leqslant|I|\leqslant n}{f_I(x)\theta_I}\\
&=f_0(x)+f_1(x)\theta_1+...+f_n(x)\theta_n+f_{12}(x)\theta_1\theta_2+...
+f_{1...n}(x)\theta_1...\theta_n\end{aligned}$$ where $|I|$ is the length of $I$, $x=(x_i),\quad i=1,\cdots, 2l+1$ is a coordinates system on ${\mathbb{R}}^{2l+1}$ and where $\theta=(\theta_i),\quad i=1,\cdots, n$ is odd Grassmann coordinates on $\Lambda{\mathbb{R}}^n$, i.e. $\theta_i^2=0,\quad \theta_i\theta_j=-\theta_j\theta_i$. We define the parity function $\tilde{.}$ by setting $\tilde{x}=0$ and $\tilde{\theta}=1$.
Vector fields on ${\mathbb{R}}^{2l+1|n}$
========================================
A vector fields on ${\mathbb{R}}^{2l+1|n}$ is a superderivation of the associative supercommutative superalgebra $C^\infty({\mathbb{R}}^{2l+1|n})$. In coordinates, it can be expressed as $$X=\sum_{i=1}^{2l+1}X^i\partial_{x_i}+\sum_{j=1}^nY^j\partial_{\theta_j},$$ where $X^i$ and $Y^j$ are the elements of $C^\infty({\mathbb{R}}^{2l+1|n})$, $\partial_{x_i}=\frac{\partial}{\partial x_i}$ and $\partial_{\theta_j}=\frac{\partial}{\partial \theta_j}$ for all $i=1,2,\cdots, 2l+1$ and $j=1,2,\cdots,n$.\
It can also be expressed as $$X=\sum_{i=1}^{p+q}X^i\partial_{z_i},$$ where $z_i=x_i$ for all $i\in\{1,\ldots,2l+1\}$ and $z_i=\theta_{i-(2l+1)}$ for all $i\in\{2l+2,\ldots, 2l+1+n\}$. The parity function $\tilde{.}$ on vector field $X$ is defined as $$\widetilde{\partial_{x_i}}=0\quad \mbox{and}\quad \widetilde{\partial_{\theta_i}}=1.$$ The superspace of all vector fields on ${\mathbb{R}}^{2l+1|n}$ is a Lie superalgebra, which we shall denote by $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$, by defining the following Lie bracket $$[X,Y]=XY-(-1)^{\tilde{X}\tilde{Y}}YX,$$ for all vector fields $X,Y$.
Differential $1$-superforms on ${\mathbb{R}}^{2l+1|n}$
======================================================
We define the superspace $\Omega^1({\mathbb{R}}^{2l+1|n})$ of differential $1$-superforms on ${\mathbb{R}}^{2l+1|n}$ as a superspace which is constituted by the elements $$\alpha=\sum_{i=1}^{2l+1}{f_i(x_i,\theta_i)dx^i}+\sum_{i=1}^n{g_i(x_i,\theta_i)d\theta^i},$$ where $f_i$ and $g_i$ are elements of $C^\infty({\mathbb{R}}^{2l+1|n})$ and $\widetilde{dx^i}=0,\,\widetilde{d\theta^i}=1$ and where we set $\mathcal{B}'=(dx^i,d\theta^i)$ of $\Omega^1({\mathbb{R}}^{2l+1|n})$ the dual basis of a basis $\mathcal{B}=(\partial_{x_i},\partial_{\theta_i})$ of $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$ such that $$\langle\partial_{x_j},dx^i\rangle=\delta^i_j,\quad \langle\partial_{x_j},d\theta^i\rangle=0 \quad\mbox{and}\quad \langle\partial_{\theta_j},d\theta_i\rangle=-\delta^i_j.$$
These elements $f_i$ and $g_i$ can also be declared at right and in this case we must use the even sign rule known in supergeometry.
When we consider a vector field $X$, we can also define the evaluation of differential $1$-superform on $X$, or the interior product of a differential $1$-superform $\alpha$ by $X$ as follow: $$\alpha(X)=(-1)^{\tilde{X}\tilde{\alpha}}\langle X,\alpha\rangle,\quad\mbox{and}\quad i(X)\alpha=\langle X,\alpha\rangle.$$ Explicitly, if $X=\sum_{i=1}^{2l+1+n}X^i\partial_{z_i}$ and $\alpha=\sum_{j=1}^{2l+1+n}\alpha_jdz_j$, we have via the sign rule, $$\label{coucou1}
\langle X,\alpha\rangle=\langle\sum_{i=1}^{2l+n+1}X^i\partial_{z_i},\sum_{j=1}^{2l+n+1}\alpha_jdz_j\rangle=
\sum_{i,j=1}^{2l+n+1}X^i\alpha_j(-1)^{\tilde{i}\tilde{\alpha_j}}
\langle\partial_{z_i},dz_j\rangle=\sum_{i=1}^{2l+n+1}(-1)^{\tilde{i}(\tilde{\alpha_i}+\tilde{i})}X^i\alpha_i.$$ We can generalize the definition of differential superforms and we have also a version of de de Rham differential which is adapted in the framework of supergeometry. Thus it allows us to define the Lie derivative of differential superforms. These operators have the analogue properties known in classical geometry.
Standard contact structure on ${\mathbb{R}}^{2l+1|n}$
=====================================================
We consider here the standard contact structure on ${\mathbb{R}}^{2l+1|n}$. We can find in [@Gr13] the notions of the contact structure on any supermanifold of dimension $m|n$.
The standard contact structure on ${\mathbb{R}}^{2l+1|n}$ is defined by the kernel of the differential $1$-superforms $\alpha$ on ${\mathbb{R}}^{2l+1|n}$ which, in the system of Darboux coordinates $(z,x_i,y_i,\theta_j),\quad i=1,\cdots,l$ and $j=1,\cdots,n$ it can be written as $$\label{STANDARDCONTACT}
\alpha=dz+\sum_{i=1}^l{(x_idy_i-y_idx_i)}+\sum_{i=1}^n{\theta_id\theta_i}.$$ This differential $1$-superform $\alpha$ is called contact form on ${\mathbb{R}}^{2l+1|n}$ and we denote by $\mathrm{Tan}({\mathbb{R}}^{2l+1|n})$ the space constituted of the elements of the kernel of $\alpha$.
If we denote $q^A=(z,q^r)$ the generalized coordinate where $$\label{gencoordin}
q^A=\left\lbrace
\begin{array}{lcl}
z\quad\mbox{if}\quad A=0\\
x_A \quad\mbox{if}\quad 1\leqslant A\leqslant l,\\
y_{A-l} \quad\mbox{if} \quad l+1\leqslant A\leqslant 2l\\
\theta_{A-2l} \quad\mbox{if}\quad 2l+1\leqslant A\leqslant 2l+n
\end{array}\right.$$ we can write $\alpha$ in the following way $$\alpha=dz+\omega_{rs}q^rdq^s, \quad (\omega_{rs})=\left(
\begin{array}{cc|c}
0& {\mathrm{id}}_l&0\\
-{\mathrm{id}}_l&0&0\\
\hline
0&0&{\mathrm{id}}_n
\end{array}
\right).$$
We denote by $\omega^{sk}$ the matrix so that $(\omega_{rs})(\omega^{sk})=(\delta^k_r)$. We have thus $$(\omega^{rs})=\left(
\begin{array}{cc|c}
0&- {\mathrm{id}}_l&0\\
{\mathrm{id}}_l&0&0\\
\hline
0&0&{\mathrm{id}}_n
\end{array}
\right).$$ and $(\omega^{rs})=-(-1)^{\tilde{r}\tilde{s}}(\omega^{sr}).$
We call the field of Reeb on ${\mathbb{R}}^{2l+1|n}$, the vector field $T_0\in\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$ which, in the system of Darboux coordinates, one write $T_0=\partial_z$.
We can show that the field of Reeb is the unique vector field on ${\mathbb{R}}^{2l+1|n}$ so that $i(T_0)\alpha=1$ and $i(T_0)d\alpha=0$.\
In the system of Darboux coordinates, the elements $T_r$ of $\mathrm{Tan}({\mathbb{R}}^{2l+1|n})$ can be written as follow $$\label{tangentdistr}
T_r=\left\lbrace
\begin{array}{lcl}
A_r &:=& \partial_{x_r}+y_r\partial_z\quad\mbox{if}\quad 1\leqslant r\leqslant l\\
-B_{r-l} &:=& \partial_{y_{r-l}}-x_{r-l}\partial_z\quad\mbox{if}\quad l+1\leqslant r\leqslant 2l\\
\overline{D}_{r-2l}&:=& \partial_{\theta_{r-2l}}-\theta_{r-2l}\partial_z \quad\mbox{if}\quad 2l+1\leqslant r\leqslant 2l+n
\end{array}\right.$$
If we denote by $T_r$ the vector field $T_r=\partial_{q^r}-\langle\partial_{q^r},\alpha\rangle\partial_z,$ and since $\widetilde{\alpha}=0$, we have $$\alpha(T_r)=\langle T_r,\alpha\rangle=
\langle \partial_{q^r},\alpha\rangle-
\langle\langle\partial_{q^r},\alpha\rangle\partial_z,\alpha\rangle=\langle \partial_{q^r},\alpha\rangle-\langle \partial_{q^r},\alpha\rangle=0.$$ We can also show that any vector field $X$ of $\mathrm{Tan}({\mathbb{R}}^{2l+1|n})$ can be written as a linear combination of the vector fields $T_r$. It is useful to compute the vector fields $T_r$ according to the matrix $\omega$. One has, via \[coucou1\], $$T_r=\partial_{q^r}-\alpha_r\partial_z=\partial_{q^r}-\omega_{kr}q^k\partial_z.$$ It is sufficient to vary $r$ in the interval $[1, 2l+n]$ to conclude.
The following formulas are immediate. $$\label{TANGDIST}
T_r(q^k)=\delta_r^k,\quad T_r(z)=-\omega_{kr}q^k,\quad [T_r,T_j]=-2\omega_{rj}\partial_z,\quad T_r(z^2)=-2z\omega_{kr}q^k.$$
Contact vector fields on ${\mathbb{R}}^{2l+1|n}$ {#coucou2}
================================================
We call a contact vector field on ${\mathbb{R}}^{2l+1|n}$ a vector field $X$ that preserves the contact structure, i.e. a vector field $X$ verifying the following condition: $[X,T]\in \mathrm{Tan}({\mathbb{R}}^{2l+1|n}) $ for all $T\in \mathrm{Tan}({\mathbb{R}}^{2l+1|n})$.
The following proposition is known in the classical geometry [@CoOv12] and in supergeometry in small dimensions, i.e: $(1|1)$ and $(1|2)$ in [@MelNibRad13; @Mel09; @GarMelOvs07]. We give here its analogue in supergeometry and generalize it in dimension $(m|n)$. It is our main first result.
A vector field $X$ on ${\mathbb{R}}^{2l+1|n}$ is called contact vector field if and only if it exists a superfunction $f$ that $X=X_f$ where $X_f$ is given by the following formula $$\label{CONTACTFORME}
X_f=f\partial_z-\frac{1}{2}(-1)^{\tilde{f}\tilde{T_r}}\omega^{rs}T_r(f)T_s,$$ We denote by ${\mathcal{K}}(2l+1|n)$ the space of the all contact vector fields on ${\mathbb{R}}^{2l+1|n}$.
Seen the definition of $T_r$, we can say that any vector field on ${\mathbb{R}}^{2l+1|n}$ can be written as $X=f\partial_z+\sum_{i=1}^{2l+n}g_iT_i$. The vector field $X$ is thus called contact vector field if and only if $$[f\partial_z+\sum_{i=1}^{2l+n}{g_iT_i},T_j]\in\,<T_1,\cdots,T_{2l+n}>,\quad\forall j\in
\{1,\cdots,2l+n\}.$$ This formula can be also written as $$\begin{aligned}
[f\partial_z+\sum_{i=1}^{2l+n}{g_iT_i},T_j]&=&-(-1)^{\tilde{f}\tilde{T}_j}[T_j,f\partial_z]-(-1)^{(\tilde{g}_i+\tilde{T}_i)\tilde{T}_j}\sum_{i=1}^{2l+n}{[T_j,g_iT_i]}\\
&=&-(-1)^{\tilde{f}\tilde{T}_j}T_j(f)\partial_z+(-1)^{\tilde{T}_i\tilde{T}_j}2\sum_{i=1}^{2l+n}{g_i\omega_{ji}\partial_z}-(-1)^{(\tilde{g}_i+\tilde{T}_i)\tilde{T}_j}\sum_{i=1}^{2l+n}{T_j(g_i)T_i}.\end{aligned}$$ This vector field $X$ is in the kernel of $\alpha$ if and only if $$-(-1)^{\tilde{f}\tilde{T}_j}T_j(f)-2\sum_{i=1}^{2l+n}g_i\omega_{ij}=0,$$ for all $j\in \{1,\cdots, 2l+n\}$. This equation shows that all proposed vector fields $X_f$ are contact vector fields. In the other hand, this equation implies also that $$-(-1)^{\tilde{f}\tilde{T}_j}T_j(f)\omega^{jk}-2\sum_{i=1}^{2l+n}{g_i\omega_{ij}\omega^{jk}}=0, \quad\forall j\in\{1,\cdots,2l+n\},$$ or, when we sum on j, we have $$-(-1)^{\tilde{f}\tilde{T}_j}\omega^{jk}T_j(f)=2\sum_{i=1}^{2l+n}{g_i\omega_{ij}\omega^{jk}}=2\sum_{i=1}^{2l+n}{g_i\delta_i^k}.$$ We obtain directly that $$g_k=-\frac{1}{2}(-1)^{\tilde{f}\tilde{T}_j}\omega^{jk}T_j(f)$$ and this allows us to conclude.
The following proposition gives, in the super case, the formula of the Lagrange bracket of the superfunctions $f$ and $g$. It is the generalization of the formula given in [@Mel09; @MelNibRad13; @GarMelOvs07].
The set ${\mathcal{K}}(2l+1|n)$ is a Lie sub superalgebra of $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$. More explicitly, if $X_f$ and $X_g$ are the elements of ${\mathcal{K}}(2l+1|n)$, one writes $$\label{LAGRANGE}
[X_f,X_g]=X_{\{f,g\}}$$ where the superfunction $\{f,g\}$ is given by $$\label{LagrBracket}
{\{f,g\}}:=fg'-f'g-\frac{1}{2}(-1)^{\tilde{T}_r\tilde{f}}\omega^{rs}T_r(f)T_s(g),$$ and where $h'=\partial_z(h)$.
The Lie bracket $[X_f,X_g]$ of the two contact vector fields $X_f$ and $X_g$ is also a contact vector field. Indeed, the Lie bracket $$[X_f,X_g]=[f\partial_z-\frac{1}{2}(-1)^{\tilde{T}_r\tilde{f}}\omega^{rs}T_r(f)T_s,g\partial_z-\frac{1}{2}(-1)^{\tilde{T}_k\tilde{g}}\omega^{kl}T_k(g)T_l]$$ is written as $$\begin{gathered}
[f\partial_z,g\partial_z]-\frac{1}{2}(-1)^{\tilde{T}_k\tilde{g}}\omega^{kl}[f\partial_z,T_k(g)T_l]
-\frac{1}{2}(-1)^{\tilde{T}_r\tilde{f}}\omega^{rs}[T_r(g)T_s,g\partial_z]\\
+\frac{1}{4}\omega^{rs}\omega^{kl}[T_r(f)T_s,T_k(g)T_l].
\end{gathered}$$ The sum of the first three Lie brackets equals to $$\begin{gathered}
(fg'-f'g)\partial_z+\frac{1}{2}(-1)^{\tilde{g}(\tilde{T}_k+\tilde{f})}\omega^{kl}T_k(g)T_l(f)\partial_z-\frac{1}{2}(-1)^{\tilde{f}\tilde{T}_r}\omega^{rs}T_r(f)T_s(g)\partial_z\\
-\frac{1}{2}(-1)^{\tilde{g}\tilde{T}_k}\omega^{kl}fT_k(g')T_s+\frac{1}{2}(-1)^{\tilde{f}\tilde{T}_r+\tilde{f}\tilde{g}}\omega^{rs}gT_r(f')T_s
\end{gathered}$$ and the fourth Lie bracket equals to $$\frac{1}{4}(-1)^{\tilde{f}\tilde{T}_r+\tilde{g}\tilde{T}_k}\omega^{rs}\omega^{kl}\left(
T_r(f)T_sT_k(g)T_l-(-1)^{\tilde{f}\tilde{g}}T_k(g)T_lT_r(f)T_s
\right) -\frac{1}{2}(-1)^{(\tilde{T}_r+\tilde{f})\tilde{g}}\omega^{kr}T_k(g)T_r(f)\partial_z.$$
Since the Lie bracket of two contact vector fields is also a contact vector field and since $X_{\{f,g\}}$ is written, via the formula , by $$(\{f,g\})\partial_z-\frac{1}{2}(-1)^{(\tilde{f}+\tilde{g})\tilde{T}_r}\omega^{rs}T_r(\{f,g\})T_s,$$ then we can see that the sum of the coefficients of $\partial_z$ gives the formula of Lagrange bracket. One has thus $$\{f,g\}= fg'-f'g-(-1)^{\tilde{f}\tilde{T}_r}\frac{1}{2}\omega^{rs}T_r(f)T_s(g).$$
Via the Lagrange formula , the Lie bracket of contact vector fields, which defines a Lie superalgebra structure on ${\mathcal{K}}(2l+1|n)$, induces a Lie superalgebra structure on the superspace $C^\infty({\mathbb{R}}^{2l+1|n})$ by the bilinear law given by .\
The following remark is very important:
The Lagrange bracket of superfunctions $f$ and $g$ of degree to most equal two is always a superfunction of degree to most equal two.
This remark allows us to define the Lie superalgebra constituted by the contact vector fields $X_f$ which the associated superfunctions $f$ are of degrees to most equal two in $z,x_i,y_i$ and $\theta_i$ variables. We denote this Lie superalgebra temporarily by $\mathfrak{g}\subset{\mathcal{K}}(2l+1|n)$.
Matrix realization of ${\mathfrak{spo}}(2l+2|n)$
================================================
In this section, we embed a Lie sub-superalgebra ${\mathfrak{spo}}(2l+2|n)$ of ${\mathfrak{gl}}(2l+2|n)$ in the Lie superalgebra $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$. We use the method used in [@MatRad11] and we show that the Lie superalgebra obtained is exactly isomorphic to $\mathfrak{g}$.\
We consider a matrix $G$ defined by $G=\begin{pmatrix}
J&0\\0&{\mathrm{id}}_n
\end{pmatrix} $ such that $J=\begin{pmatrix}
0&-{\mathrm{id}}_{l+1}\\{\mathrm{id}}_{l+1}&0
\end{pmatrix}$. We define on ${\mathbb{R}}^{2l+2|n}$ the following superskewsymmetric form $\omega$ associated to the matrix $G$ as
$$\omega:{\mathbb{R}}^{2l+2|n}\times{\mathbb{R}}^{2l+2|n}\to{\mathbb{R}}:(U,V)\to V^tGU,$$
where $A^t$ is the usual transpose of the matrix $A$.
We define a Lie superalgebra ${\mathfrak{spo}}(2l+2|n)$ constituted by the matrices $A$ of ${\mathfrak{gl}}(2l+2|n)$ which preserve the form $\omega$, i.e such that $$\label{formeOmega}
\omega(AU,V)+(-1)^{\tilde{A}\tilde{U}}\omega(U,AV)=0,\quad \forall U,V\in{\mathbb{R}}^{2l+2|n}.$$
Our second main result is the following:
The Lie superalegbra ${\mathfrak{spo}}(2l+2|n)$ is the space of the matrices $A=\begin{pmatrix}
A_1&A_2\\A_3&A_4
\end{pmatrix}$ that the blocks $A_1,A_2,A_3$ and $A_4$ satisfy the following conditions
1. $A_1^tJ+JA_1=0,i.e: A_1\in \mathfrak{sp}(2l+2)$\
2. $A_4^t+A_4=0, i.e: A_4\in \mathfrak{o}(n)$\
3. $A_3=-A_2^tJ$.
We consider the following matrices $$A=\begin{pmatrix} A_1& A_2\\A_3&A_4\end{pmatrix}\in\mathfrak{gl}(2l+2|n);\quad\mbox{where}\quad A_1\in\mathfrak{gl}(2l+2), A_2\in{\mathbb{R}}^n_{2l+2}, A_3\in{\mathbb{R}}^{2l+2}_n,A_4\in{\mathbb{R}}^n_n.$$ For all vector fields $U=\begin{pmatrix}U_1\\U_2\end{pmatrix},V=\begin{pmatrix}V_1\\V_2
\end{pmatrix}$ of ${\mathbb{R}}^{2l+2|n}$, we compute the matrices $A$ of ${\mathfrak{gl}}(2l+2|n)$ which satisfy . First, we can see that the first term $\omega(AU,V)$ of equals to $$V_1^tJA_1U_1+V_2^tA_3U_1+V_1^tJA_2U_2+V_2^tA_4U_2$$ and the second term $(-1)^{\tilde{A}\tilde{U}}\omega(U,AV)$ equals to $$V_1^tA_1^tJU_1+V_2^tA_2^tJU_1-V_1^tA_3^tU_2+V_2^tA_4^tU_2.$$ It is also easy to see that the formula equals to $$\begin{gathered}
\label{formeOmega2}
V_1^tJA_1U_1+V_2^tA_3U_1+V_1^tJA_2U_2+V_2^tA_4U_2+V_1^tA_1^tJU_1\\
+V_2^tA_2^tJU_1-V_1^tA_3^tU_2+V_2^tA_4^tU_2=0, \quad\forall U,V\in{\mathbb{R}}^{2l+2|n}.\end{gathered}$$ In particular, if $U_2=0$, $V_2=0$, then the equation equals to $$V_1^t(JA_1+A_1^tJ)U_1=0, i.e:JA_1+A_1^tJ=0.$$ This last condition means that the blocks $A_1$ are symplectic matrices.\
If we set $U_1=0$ and $V_1=0$, then the equation becomes $$V_2^t(A_4^t+A_4)U_2=0,i.e: A_4^t+A_4=0.$$ This condition means that the block $A_4$ is an orthogonal matrix.\
Finally, if $U_2=0$ and $V_1=0$, then the equation equals to $$V_2^t(A_3+A_2^tJ)U_1=0,i.e: A_3+A_2^tJ=0.$$
In the following, we describe a basis of ${\mathfrak{spo}}(2l+2|n)$. If we denote by $a_{i,j}$ the number $a\in{\mathbb{R}}$ situated on the $i^{th}$ line and on the $j^{th}$ column, we can see that this basis is constituted by the three following types of matrices:\
The first type of matrices of the basis of ${\mathfrak{spo}}(2l+2|n)$ is associated to the symplectic algebra $\mathfrak{sp}(2l+2)$ and is given by following family of matrices:
$$\begin{gathered}
\label{BaseMatrix}
\left(\begin{array}{cc|ccc|c}
{}&{}&{}&{}&{}&{}\\
1_{i,j}&{}&{}&0&{}&0\\
{}&{}&{}&{}&{}&{}\\
\hline
{}&{}&{}&{}&0&{}\\
{}&{}&{}&{}&{}&{}\\
0&{}&{}&-1_{(l+1+j),(l+1+i)}&{}&0\\
{}&{}&{}&{}&{}&{}\\
{}&{}&0&{}&{}&{}\\
\hline
0&{}&{}&0&{}& 0_{n,n}\\
\end{array}\right);
\left(\begin{array}{cc|cc|c}
{}&{}&0&1_{i,(l+1+j)}&{}\\
0&{}&{}&{}&0\\
{}&{}&1_{j,(l+1+i)}&{}&{}\\
{}&{}&{}&0&{}\\
\hline
{}&{}&{}&{}&{}\\
0&{}&0&{}&0\\
{}&{}&{}&{}&{}\\
\hline
{}&{}&{}&{}&{}\\
0&{}&0&{}&0_{n,n}\\
{}&{}&{}&{}&{}\\
\end{array}\right), \\ \left(\begin{array}{cc|cc|c}
{}&{}&{}&{}&{}\\
0&{}&{}&0&0\\
{}&{}&{}&{}&{}\\
\hline
0&1_{(l+1+i),j}&{}&{}&{}\\
{}&{}&{}&{}&{}\\
1_{(l+1+j),i}&{}&{}&0&0\\
{}&0&{}&{}&{}\\
\hline
{}&{}&{}&{}&{}\\
0&{}&0&{}&0_{n,n}\\
{}&{}&{}&{}&{}\\
\end{array}\right)
\quad 1\leq i,j\leq l.\end{gathered}$$
The second type of matrices is given by the following family of matrices: $$\begin{gathered}
\label{BaseMatrix1}
\left(\begin{array}{cc|cc}
{}&{}&{}&{}\\
0&{}&{}&1_{(i,(2l+2+j))}\\
{}&{}&{}&{}\\
\hline
{}&{}&{}&{}\\
-1_{((2l+2+j),i-(l+1))}&{}&{}&0\\
{}&{}&{}&{}\\
\end{array}\right)\quad \mbox{if}\quad l+1\leq i\leq 2l+2,\quad 1\leq j\leq n\\
\quad\mbox{and}\quad
\left(\begin{array}{cc|cc}
{}&{}&{}&{}\\
0&{}&{}&1_{(i,(2l+2+j))}\\
{}&{}&{}&{}\\
\hline
{}&{}&{}&{}\\
1_{((2l+2+j),(l+1+i))}&{}&{}&0\\
{}&{}&{}&{}\\
\end{array}\right)\quad\mbox{if}\quad 1\leq i\leq l+1,\quad 1\leq j\leq n.\end{gathered}$$ And the third type is associated to the orthogonal algebra $\mathfrak{o}(n)$ and is given by: $$\label{BaseMatrix2}
\left(\begin{array}{cc|ccc}
{}&{}&{}&{}&{}\\
0&{}&{}&0&{}\\
{}&{}&{}&{}&{}\\
\hline
{}&{}&0&{}&{}\\
{}&{}&{}&{}&1_{((2l+2+i),(2l+2+j))}\\
0&{}&{}&{}&{}\\
{}&{}&-1_{((2l+2+j),(2l+2+i))}&{}&{}\\
{}&{}&{}&{}&0
\end{array}\right).$$ Our third main result is given by the following theorem:
The Lie superalgebra $\mathfrak{g}$ made in evidence at the end of the section \[coucou2\] and whose superfunctions $f$ are degrees to most equal two is isomorphic to the Lie superalgebra ${\mathfrak{spo}}(2l+2|n)$.
Because of $Id\notin {\mathfrak{spo}}(2l+2|n)$ we can define the injective homomorphism $$\label{iota3}
\iota:\mathfrak{spo}(2l+2|n)\to\mathfrak{pgl}(2l+2|n):A\mapsto [A].$$ Now, the Lie superalgebra ${\mathfrak{pgl}}(2l+2|n)$ can be embedded into the Lie superalgebra of vector fields on ${\mathbb{R}}^{2l+1|n}$ thanks to the projective embedding define in [@MatRad11] in the following way: $$\label{plongProj}
\left[\left(\begin{array}{ll}0&\xi\\v&B\end{array}\right)\right]\mapsto-\sum_{i=1}^{2l+1+n}v^i
\partial_{t^i}-\sum_{i,j=1}^{2l+1+n}(-1)^{\tilde{j}(\tilde{i}+\tilde{j})}B_j^i t^j\partial_{t^i}+\sum_{i,j=1}^{2l+1+n}(-1)^{\tilde{j}}\xi_j t^j t^i\partial_{t^i},$$ where $v\in {\mathbb{R}}^{2l+1|n}$, $\xi\in ({\mathbb{R}}^{2l+1|n})^*$, $B\in\mathfrak{gl}(2l+1|n)$ and the coordinates $t^{1},t^{2},\cdots,t^{2l+1+n}$ corresponds respectively to $x_1,\cdots, x_l, z,y_1,\cdots,y_l,\theta_{1},\cdots,\theta_{n}.$\
Composing $\iota$ with the projective embedding, we can embed ${\mathfrak{spo}}(2l+2|n)$ into $\mathrm{Vect}({\mathbb{R}}^{2l+1|n})$. If we compute this embedding on the generators of ${\mathfrak{spo}}(2l+2|n)$ written above, we obtain via , the contact projective vector fields $X_f$ for a certain given $f\in C^\infty({\mathbb{R}}^{2l+1|n})$.\
In the following, we study explicitly the three types of matrices described above.\
For the first matrix of , i.e: when $i,j\in[1,l]$, we obtain the following contact projective vector fields.
1. If $i=j=1$, we have $$x_i\partial_{x_i}+y_i\partial_{y_i}+\theta_i\partial_{\theta_i}+2z\partial_z\quad, i.e: \quad 2X_z$$\
2. If $i=1$ and $j\neq 1$, then we have $$x_{j-1}(x_i\partial_{x_i}+y_i\partial_{y_i}+z\partial_z+\theta_i\partial_{\theta_i})+
z\partial_{y_{j-1}},\quad i.e: \quad 2X_{x_{j-1}z}$$\
3. If $i\neq 1$ and $j=1$, then we obtain $$-\partial_{x_{i-1}}+y_{i-1}\partial_z,\quad i.e: \quad
2X_{y_{i-1}}$$
4. If $i\neq 1$ and $j\neq 1$, then $$y_{i-1}\partial_{y_{j-1}}-x_{j-1}\partial_{x_{i-1}}, \quad i.e:\quad
2X_{x_{i-1}y_{i-1}}.$$
For the second matrix of , we obtain the following contact projective vector fields
1. If $i=j=1$, one has $$z(z\partial_z+x_i\partial_{x_i}+y_i\partial_{y_i}+\theta_j\partial_{\theta_j}),
\quad i.e: \quad X_{z^2}$$
2. If $i=j$ and $j\neq 1$, we have $$-y_{i-1}\partial_{x_{i-1}}, \quad i.e: \quad
X_{y^2_{i-1}},$$
3. If $i\neq j$ and $i=1$, one has $$y_{j-1}(x_i\partial_{x_i}+y_i\partial_{y_i}+z\partial_z+\theta_j\partial_{\theta_j})-z\partial_{x_{j-1}}, \quad i.e:\quad
2X_{y_{j-1}z}$$
4. If $i\neq j$ and $i\neq 1$, one has $$-(y_{j-1}\partial_{x_{i-1}}+y_{i-1}\partial_{x_{j-1}}), \quad i.e:\quad
2X_{y_{i-1}y_{j-1}}.$$
For the third matrix of , we obtain the following contact projective vector fields
1. If $i=j=1$, we obtain
$$-\partial_z, \quad i.e: \quad-X_1$$
2. If $i=j$ and $j\neq 1$, we have $$-x_{i-1}\partial_{y_{i-1}}, \quad i.e: \quad -X_{x^2_{i-1}}$$
3. If $i\neq j$ and $i=1$, we write $$-(\partial_{y_{j-1}}+x_{j-1}\partial_z), \quad i.e: \quad -2X_{x_{j-1}}$$
4. If $i\neq j$ and $i\neq 1$, then we have $$-(x_{j-1}\partial_{y_{i-1}}+x_{i-1}\partial_{y_{j-1}}), \quad i.e:\quad -2X_{x_{j-1}x_{i-1}}.$$
Now, we study explicitly any matrix of , i.e: when $l+1\leqslant i\leqslant 2l+2$ and $1\leqslant j \leqslant n$.\
For the first matrix of , we obtain the following contact projective vector fields
1. If $i=l+2$, then we have $$\theta_j\partial_z+\partial_{\theta_j}, \quad i.e:\quad 2X_{\theta_j}$$
2. and if $i\neq l+2$, we obtain $$\theta_j\partial_{y_{i-l-2}}+x_{i-l-2}\partial_{\theta_j}, \quad i.e:\quad 2X_{x_{i-l-2}\theta_j}.$$
For the second matrix of , we have the following contact projective vector fields
1. If $i=1$, one has $$-\theta_j(x_i\partial_{x_i}+y_i\partial_{y_i}+z\partial_z+
\theta_j\partial_{\theta_j})-z\partial_{\theta_j}, \quad i.e:\quad -2X_{z\theta_j}$$
2. and if $i\neq 1$, then we obtain $$\theta_j\partial_{x_{i-1}}-y_{i-1}\partial_{\theta_j}, \quad i.e:\quad -2X_{y_{i-1}\theta_j}.$$
Finally, for the matrix , i.e: when $2l+2\leqslant i,j\leqslant 2l+2+n$, we obtain the contact projective vector field $$\theta_{i-1}\partial_{\theta_{j-1}}-\theta_{j-1}\partial_{\theta_{i-1}}, i<j , \quad i.e:\quad 2X_{\theta_{i-1}\theta_{j-1}}.$$ The Lie superalgebra $\mathfrak{g}$ is isomorphic to ${\mathfrak{spo}}(2l+2|n)$ thanks to the identification of the generators of these two Lie superalgebras.
The Lie superalgebra ${\mathfrak{spo}}(2l+2|n)$ is into the intersection of the Lie superalgebra of contact vector fields ${\mathcal{K}}(2l+1|n)$ and the Lie superalgebra of projective vector fields ${\mathfrak{pgl}}(2l+2|n)$. Thus, as in [@CoOv12], the elements $X_f$ of ${\mathfrak{spo}}(2l+2|n)$ are called contact projective vector fields.
[10]{} D. A. Leites. . Russian Math.Surveys 35:1 (1980),1-64.
F. A. Berezin. , volume 9 of [*Mathematical Physics and Applied Mathematics*]{}. D. Reidel Publishing Co., Dordrecht, 1987. Edited and with a foreword by A. A. Kirillov, With an appendix by V. I. Ogievetsky, Translated from the Russian by J. Niederle and R. Koteck[ý]{}, Translation edited by D. Leites.
V. G. Kac. Lie superalgebras. , 26(1):8–96, 1977.
C. Conley and V. Ovsienko. Linear Differential Operators on Contact manifolds. ,24p, 2012.
D. Leites, E. Poletaeva and V. Serganova. On Einstein Equations on Manifolds. and Supermanifolds. , 9(4):394-425, 2002, math.DG/0306209
P. Mathonet and F. Radoux. Projectively equivariant quantizations over the Superspace ${\mathbb{R}}^{p|q}$. , 98:311–331, 2011.
N. Mellouli, and A. Nibirantiza, and F. Radoux. . , arxiv:math.DG/1302.3727v2 A. Nibirantiza. . http://hdl.handle.net/2268/169770, 2014.
H. Gargoubi, N. Mellouli, and V. Ovsienko. , 79(1):51–65, 2007.
J.Grabowski. ,68(2013),27-58. N. Mellouli. Second-order conformally equivariant quantization in dimension $1|2$. , 5(111), 2009.
|
---
abstract: 'Every person speaks or writes their own flavor of their native language, influenced by a number of factors: the content they tend to talk about, their gender, their social status, or their geographical origin. When attempting to perform Machine Translation (MT), these variations have a significant effect on how the system should perform translation, but this is not captured well by standard one-size-fits-all models. In this paper, we propose a simple and parameter-efficient adaptation technique that only requires adapting the bias of the output softmax to each particular user of the MT system, either directly or through a factored approximation. Experiments on TED talks in three languages demonstrate improvements in translation accuracy, and better reflection of speaker traits in the target text.'
author:
- |
Paul Michel\
Language Technologies Institute\
Carnegie Mellon University\
[[email protected]]{}\
Graham Neubig\
Language Technologies Institute\
Carnegie Mellon University\
[[email protected]]{}\
bibliography:
- 'acl.bib'
title: Extreme Adaptation for Personalized Neural Machine Translation
---
Introduction
============
The production of language varies depending on the speaker or author, be it to reflect personal traits ([*e.g.* ]{}job, gender, role, dialect) or the topics that tend to be discussed ([*e.g.* ]{}technology, law, religion). Current Neural Machine Translation (NMT) systems do not incorporate any explicit information about the speaker, and this forces the model to learn these traits implicitly. This is a difficult and indirect way to capture inter-personal variations, and in some cases it is impossible without external context (Table 1, ).
Source Translation
-------- ------------------------------------
[\[Man\]:]{} Je suis à la maison
[\[Woman\]:]{} Je suis à la maison
[\[Doctor\]:]{} Je
[\[Police\]:]{} Je
: \[tab:perso\_example\]Examples where speaker information influences English-French translation.
Recent work has incorporated side information about the author such as personality [@mirkin-EtAl:2015:EMNLP], gender [@rabinovich-EtAl:2017:EACLlong] or politeness [@sennrich-haddow-birch:2016:N16-1], but these methods can only handle phenomena where there are explicit labels for the traits. Our work investigates how we can efficiently model speaker-related variations to improve NMT models.
In particular, we are interested in improving our NMT system given few training examples for any particular speaker. We propose to approach this task as a domain adaptation problem with an extremely large number of domains and little data for each domain, a setting where we may expect traditional approaches to domain adaptation that adjust all model parameters to be sub-optimal (§\[sec:def\]). Our proposed solution involves modeling the speaker-specific variations as an additional bias vector in the softmax layer, where we either learn this bias directly, or through a factored model that treats each user as a mixture of a few prototypical bias vectors (§\[sec:model\]).
We construct a new dataset of Speaker Annotated TED talks (SATED, §\[sec:dataset\]) to validate our approach. Adaptation experiments (§\[sec:exp\_known\]) show that explicitly incorporating speaker information into the model improves translation quality and accuracy with respect to speaker traits.[^1]
Problem Formulation and Baselines {#sec:def}
=================================
In the rest of this paper, we refer to the person producing the source sentence (speaker, author, etc…) generically as the *speaker*. We denote as $\mathcal{S}$ the set of all speakers.
The usual objective of NMT is to find parameters $\theta$ of the conditional distribution $p(y\mid x;\theta)$ to maximize the empirical likelihood.
We argue that personal variations in language warrant decomposing the empirical distribution into $\vert \mathcal{S}\vert$ speaker specific domains $\mathcal{D}_s$ and learning a different set of parameters $\theta_s$ for each. This setting exhibits specific traits that set it apart from common domain adaptation settings:
1. The number of speakers is very large. Our particular setting deals with $\vert \mathcal{S}\vert\approx 1800$ but our approaches should be able to accommodate orders of magnitude more speakers.
2. There is very little data (even monolingual, let alone bilingual or parallel) for each speaker, compared to millions of sentences usually used in NMT.
3. As a consequence of 1, we can assume that many speakers share similar characteristics such as gender, social status, and as such may have similar associated domains.[^2]
Baseline NMT model {#sec:base}
------------------
All of our experiments are based on a standard neural sequence to sequence model. We use one layer LSTMs as the encoder and decoder and the *concat* attention mechanism described in @luong2015stanford. We share the parameters in the embedding and softmax matrix of the decoder as proposed in @press-wolf:2017:EACLshort. All the layers have dimension 512 except for the attention layer (dimension 256). To make our baseline competitive, we apply several regularization techniques such as dropout [@srivastava2014dropout] in the output layer and within the LSTM (using the variant presented in [@gal2016theoretically]). We also drop words in the target sentence with probability 0.1 according to and implement label smoothing as proposed in with coefficient $0.1$. Appendix \[sec:model\_details\] provides a more thorough description of the baseline model.
Baseline adaptation strategy {#sec:baseline_adaptation}
----------------------------
As mentioned in §\[sec:def\], our goal is to learn a separate conditional distribution $p(y\mid x, s)$ and parametrization $\theta_s$ to improve translation for speaker $s$. The usual way of adapting from general domain parameters $\theta$ to $\theta_s$ is to retrain the full model on the domain specific data [@luong2015stanford]. Naively applying this approach in the context of personalizing a model for each speaker however has two main drawbacks:
#### Parameter cost
Maintaining a set of model parameters for each speaker is expensive. For example, the model in §\[sec:base\] has $\approx$47M parameters when the vocabulary size is 40k, as is the case in our experiments in §\[sec:exp\_known\]. Assuming each parameter is stored as a 32bit float, every speaker-specific model costs $\approx$188MB. In a production environment with thousands to billions of speakers, this is impractical.
#### Overfitting
Training each speaker model with very little data is a challenge, necessitating careful and heavy regularization [@micelibarone-EtAl:2017:EMNLP2017] and an early stopping procedure.
Domain Token {#sec:domain_token}
------------
A more efficient domain adaptation technique is the *domain token* idea used in @sennrich-haddow-birch:2016:N16-1 [@chu-dabre-kurohashi:2017:Short]: introduce an additional token marking the domain in the source and/or the target sentence. In experiments, we add a token indicating the speaker at the start of the target sentence for each speaker. We refer to this method as the `spk_token` method in the following.
Note that in this case there is now only an embedding vector (of dimension 512 in our experiments) for each speaker. However, the resulting domain embedding are non-trivial to interpret (i.e. it is not clear what they tell us about the domain or speaker itself).
Speaker-specific Vocabulary Bias {#sec:model}
================================
In NMT models, the final choice of which word to use in the next step $t$ of translation is generally performed by the following softmax equation $$\label{eq:bias_dist}
p_t=\text{softmax}(E_To_t + b_T)$$ where $o_t$ is predicted in a context-sensitive manner by the NMT system and $E_T$ and $b_T$ are the weight matrix and bias vector parameters respectively. Importantly, $b_T$ governs the overall likelihood that the NMT model will choose particular vocabulary. In this section, we describe our proposed methods for making this bias term speaker-specific, which provides an efficient way to allow for speaker-specific vocabulary choice.[^3]
Full speaker bias {#sec:full_bias}
-----------------
We first propose to learn speaker-specific parameters for the bias term in the output softmax only. This means changing Eq. \[eq:bias\_dist\] to $$\label{eq:full_bias_dist}
p_t=\text{softmax}(E_To_t + b_T + b_s)$$ for speaker $s$. This only requires learning and storing a vector equal to the size of the vocabulary, which is a mere 0.09% of the parameters in the full model in our experiments. In effect, this greatly reducing the parameter cost and concerns of overfitting cited in §\[sec:baseline\_adaptation\]. This model is also easy to interpret as each coordinate of the bias vector corresponds to a log-probability on the target vocabulary. We refer to this variant as `full_bias`.
![\[fig:models\_diagrams\]Graphical representation of our different adaptation models for the softmax layer. From top to bottom is the base softmax, the `full_bias` softmax and the `fact_bias` softmax](models_diagrams.png){width="\columnwidth"}
Factored speaker bias {#sec:factored_bias}
---------------------
The biases for a set of speakers $\mathcal{S}$ on a vocabulary $\mathcal{V}$ can be represented as a matrix: $$B\in\mathbb R^{\vert\mathcal{S}\vert\times\vert\mathcal{V}\vert}$$
where each row of $B$ is one speaker bias $b_s$. In this formulation, the $\vert\mathcal{S}\vert$ rows are still linearly independent, meaning that $B$ is high rank. In practical terms, this means that we cannot share information among users about how their vocabulary selection co-varies, which is likely sub-ideal given that speakers share common characteristics.
Thus, we propose another parametrization of the speaker bias, `fact_bias`, where the $B$ matrix is factored according to: $$\begin{split}
B=&S\tilde{B}\\\ S\in&\mathbb R^{\vert\mathcal{S}\vert\times r},\\ \tilde{B}\in&\mathbb R^{r\times\vert\mathcal{V}\vert}
\end{split}$$ where $S$ is a matrix of speaker vectors of low dimension $r$ and $\tilde B$ is a matrix of $r$ speaker independent biases. Here, the bias for each speaker is a mixture of $r$ “centroid” biases $\tilde B$ with $r$ speaker “weights”. This reduces the total number of parameters allocated to speaker adaptation from $\vert\mathcal{S}\vert\vert\mathcal{V}\vert$ to $r(\vert\mathcal{S}\vert+\vert\mathcal{V}\vert)$. In our experiments, this corresponds to using between $99.38$ and $99.45\%$ fewer parameters than the `full_bias` model depending on the language pair, with $r$ parameters per speaker. In this work, we will use $r=10$.
We provide a graphical summary of our proposed approaches in figure \[fig:models\_diagrams\].
Speaker Annotated TED Talks Dataset {#sec:dataset}
===================================
In order to evaluate the effectiveness of our proposed methods, we construct a new dataset, Speaker Annotated TED (SATED) based on TED talks,[^4] with three language pairs, English-French (`en-fr`), English-German (`en-de`) and English-Spanish (`en-es`) and speaker annotation. The dataset consists of transcripts directly collected from <https://www.ted.com/talks>, and contains roughly 271K sentences in each language distributed among 2324 talks. We pre-process the data by removing sentences that don’t have any translation or are longer than 60 words, lowercasing, and tokenizing (using the Moses tokenizer [@Koehn:2007:MOS:1557769.1557821]).
`en-fr` `en-es` `en-de`
---------------- --------- --------- --------- --
\#talks 1,887 1,922 1,670
\#train 177,743 182,582 156,134
\#dev 3,774 3,844 3,340
\#test 3,774 3,844 3,340
avg. sent/talk 94,2 95.0 93,5
std dev 57,6 57.8 60,3
: \[tab:data\_stats\]Dataset statistics
Some talks are partially or not translated in some of the languages (in particular there are fewer translations in German than in French or Spanish), we therefore remove any talk with less than 10 translated sentences in each language pair.
The data is then partitioned into training, validation and test sets. We split the corpus such that the test and validation split each contain 2 sentence pairs from each talk, thus ensuring that all talks are present in every split. Each sentence pair is annotated with the name of the talk and the speaker. Table \[tab:data\_stats\] lists statistics on the three language pairs.
This data is made available under the Creative Commons license, Attribution-Non Commercial-No Derivatives (or the CC BY-NC-ND 4.0 International, <https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode>), all credit for the content goes to the TED organization and the respective authors of the talks. The data itself can be found at <http://www.cs.cmu.edu/~pmichel1/sated/>.
Experiments {#sec:exp_known}
===========
We run a set of experiments to validate the ability of our proposed approach to model speaker-induced variations in translation.
Experimental setup {#sec:exp_known_setup}
------------------
We test three models `base` (a baseline ignoring speaker labels), `full_bias` and `fact_bias`. During training, we limit our vocabulary to the 40,000 most frequent words. Additionally, we discard any word appearing less than 2 times. Any word that doesn’t satisfy those conditions is replaced with an `UNK` token.[^5]
All our models are implemented with the DyNet [@neubig2017dynet] framework, and unless specified we use the default settings therein. We refer to appendix \[sec:training\] for a detailed explanation of the training process. We translate the test set using beam search with beam size 5.
Does explicitly modeling speaker-related variation improve translation quality?
-------------------------------------------------------------------------------
Table \[tab:known\_results\] shows final test scores for each model with statistical significance measured with paired bootstrap resampling [@koehn:2004:EMNLP]. As shown in the table, both proposed methods give significant improvements in BLEU score, with the biggest gains in English to French ($+0.99$) and smaller gains in German and Spanish ($+0.74$ and $+0.40$ respectively). Reducing the number of parameters with `fact_bias` gives slightly better (`en-fr`) or worse (`en-de`) BLEU score, but in those cases the results are still significantly better than the baseline.
en-fr en-es en-de
------------- ----------- ----------- -----------
`base` 38.05 39.89 26.46
`spk_token` **38.85** 40.04 26.52
`full_bias` **38.54** **40.30** **27.20**
`fact_bias` **39.01** 39.88 **26.94**
: \[tab:known\_results\]Test BLEU. Scores significantly ($p<0.05$) better than the baseline are written in bold
However, BLEU is not a perfect evaluation metric. In particular, we are interested in evaluating how much of the personal traits of each speaker our models capture. To gain more insight into this aspect of the MT results, we devise a simple experiment. For every language pair, we train a classifier (continuous bag-of-n-grams; details in Appendix \[sec:classifier\]) to predict the author of each sentence on the target language part of the training set. We then evaluate the classifier on the ground truth and the outputs from our 3 models (`base`, `full_bias` and `fact_bias`).
The results are reported in Figure \[fig:cbong\_speaker\]. As can be seen from the figure, it is easier to predict the author of a sentence from the output of speaker-specific models than from the baseline. This demonstrates that explicitly incorporating information about the author of a sentence allows for better transfer of personal traits during translations, although the difference from the ground truth demonstrates that this problem is still far from solved. Appendix \[sec:qual\_examples\] shows qualitative examples of our model improving over the baseline.
![\[fig:cbong\_speaker\]Speaker classification accuracy of our continuous bag-of-n-grams model.](cbong_speaker_bylang_bw_safe.pdf){width="\columnwidth"}
Further experiments on the Europarl corpus
------------------------------------------
One of the quirks of the TED talks is that the speaker annotation correlates with the topic of their talk to a high degree. Although the topics that a speaker talks about can be considered as a manifestation of speaker traits, we also perform a control experiment on a different dataset to verify that our model is indeed learning more than just topical information. Specifically, we train our models on a speaker annotated version of the Europarl corpus [@rabinovich-EtAl:2017:EACLlong], on the `en-de` language pair[^6].
We use roughly the same training procedure as the one described in §\[sec:exp\_known\_setup\], with a random train/dev/test split since none is provided in the original dataset. Note that in this case, the number of speakers is much lower (747) whereas the total size of the dataset is bigger ($\approx$300k).
We report the results in table \[tab:europarl\_results\]. Although the difference is less salient than in the case of SATED, our factored bias model still performs significantly better than the baseline ($+0.83$ BLEU). This suggests that even outside the context of TED talks, our proposed method is capable of improvements over a speaker-agnostic model.
Related work
============
Domain adaptation techniques for MT often rely on data selection [@moore-lewis:2010:Short; @li-EtAl:2010:PAPERS3; @chen-EtAl:2017:NMT; @wang-EtAl:2017:EMNLP20174], tuning [@luong2015stanford; @micelibarone-EtAl:2017:EMNLP2017], or adding domain tags to NMT input [@chu-dabre-kurohashi:2017:Short]. There are also methods that fine-tune parameters of the model on each sentence in the test set [@li2016one], and methods that adapt based on human post-edits [@turchi2017continuous], although these follow our baseline adaptation strategy of tuning all parameters. There are also partial update methods for transfer learning, albeit for the very different task of transfer between language pairs [@zoph-EtAl:2016:EMNLP2016].
Pioneering work by introduced ways to incorporate information about speaker role, rank, gender, and dialog domain for rule based MT systems. In the context of data-driven systems, previous work has treated specific traits such as politeness or gender as a “domain” in domain adaptation models and applied adaptation techniques such as adding a “politeness tag” to moderate politeness [@sennrich-haddow-birch:2016:N16-1], or doing data selection to create gender-specific corpora for training [@rabinovich-EtAl:2017:EACLlong]. The aforementioned methods differ from ours in that they require explicit signal (gender, politeness…) for which labeling (manual or automatic) is needed, and also handle a limited number of “domains” ($\approx2$), where our method only requires annotation of the speaker, and must scale to a much larger number of “domains” ($\approx1,800$).
en-de
------------- -----------
`base` 26.04
`spk_token` 26.49
`full_bias` 26.44
`fact_bias` **26.87**
: \[tab:europarl\_results\]Test BLEU on the Europarl corpus. Scores significantly ($p<0.05$) better than the baseline are written in bold
Conclusion
==========
In this paper, we have explained and motivated the challenge of modeling the speaker explicitly in NMT systems, then proposed two models to do so in a parameter-efficient way. We cast this problem as an extreme form of domain adaptation and showed that, even when adapting a small proportion of parameters (the softmax bias, $<0.1\%$ of all parameters), allowed the model to better reflect personal linguistic variations through translation.
We further showed that the number of parameters specific to any person could be reduced to as low as 10 while still retaining better scores than a baseline for some language pairs, making it viable in a real world application with potentially millions of different users.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors give their thanks the anonymous reviewers for their useful feedback which helped make this paper what it is, as well as the members of Neulab who helped proof read this paper and provided constructive criticism. This work was supported by a Google Faculty Research Award 2016 on Machine Translation.
Detailed model description {#sec:model_details}
==========================
#### Word embeddings
We embed source and target words in a low dimensional space with embedding matrices $E_S\in\mathbb{R}^{\vert V_S\vert\times d_{\text{emb}}}$, $E_T\in\mathbb{R}^{\vert V_T\vert\times d_{\text{emb}}}$. Each word vector is initialized at random from $\mathcal{N}(0,\frac{1}{\sqrt{d_{\text{emb}}}})$. We use $d_{\text{emb}} = 512$.
#### Encoder
Our encoder is a one layer bidirectional LSTM with dimension $d_h=512$. For a source sentence $e = e_1, \dots, e_{|e|}$ the concatenated output of the encoder is thus of shape $\vert e\vert\times 2d_h$.
#### Attention
We use a multilayer perceptron attention mechanism: given a query $h_t$ at step $t$ of decoding and encodings $x_1,\ldots,x_{\vert e\vert}$, the context vector $c_t$ is computed according to: $$\begin{split}
\alpha_{it} &=V_a^T\tanh(W_{a}e_i+W_{ah}h_t+b_a)\\
c_t&=\sum_i\alpha_{it}x_i,\\
\end{split}$$ where $V_a, W_a, W_{ah}, b_a$ are learned parameters. We choose $d_a=256$ as the dimension of the intermediate layer.
#### Decoder
The decoder is a single layer LSTM of dimension $d_h=512$. At each timestep $t$, it takes as input the previous word embedding $w_{t-1}$ and the previous context $c_{t-1}$. Its output $h_t$ is used to compute the next context vector $c_t$ and the distribution over the next possible target words $w_t$: $$\label{eq:output_dist}
\begin{split}
o_t &=W_{oh}h_t + W_{oc}c_t + W_{ow} E_{Tw_{t-1}} + b_o\\
p_t&=\text{softmax}(E_To_t + b_T),\\
\end{split}$$ where $W_{o*}, b_o, b_T$ are learned parameters, $E_T$ is the target word embedding matrix and $ E_{Tw_{t-1}}$ is the embedding of the previous target word.
#### Learning paradigm
We employ several techniques to improve training. First, we are using the same parameters for the target word embeddings and the weights of the softmax matrix [@press-wolf:2017:EACLshort]. This reduces the number of total parameters and in practice this gave slightly better BLEU scores.
We apply dropout [@srivastava2014dropout] between the output layer and the softmax layer, as well as within the LSTM (using the variant presented in ). We also drop words in the target sentence with probability 0.1 according to . Intuitively, this forces the decoder to use the conditional information.
In addition to this, we implement label smoothing as proposed in with a smoothing coefficient 0.1. We noticed improvements of up to 1 BLEU point with this additional regularization term.
Training process {#sec:training}
================
We first train each model using the Adam optimizer [@kingma2014adam] with learning rate 0.001 (we clip the gradient norm to 1). The data is split into batches of size 32 where every source sentence has the same length. We evaluate the validation perplexity after each epoch. Whenever the perplexity doesn’t improve, we restart the optimizer with a smaller learning rate from the previous best model [@denkowski2017stronger]. Training is stopped when the perplexity doesn’t go down for 3 epochs. We then perform a tuning step: we restart training with the same hyper-parameters except for using simple stochastic gradient descent and gradient clipping at a norm of 0.1, which improved the validation BLEU by 0.3-0.9 points.
User classifier {#sec:classifier}
===============
In our analysis, we use a classifier to estimate which user wrote each output, which we describe more in this section.
The model uses a continuous bag of $n$-grams where $v_{\text{n-gram}}$ is a parameter vector for a paticular n-gram and the probability of speaker $s$ for sentence $f$ is given by: $$\begin{split}
p(s\mid f)&\propto w_s^Th_f+b_s\\
h_f&=\frac{1}{\#\{\text{n-gram}\in f\}}(\sum_{\text{n-gram}\in f}v_{\text{n-gram}})\\
\end{split}$$ The size of hidden vectors is 128. We limit n-grams to unigrams and bigrams. We estimate the parameters with Adam and a batch size of 32 for 50 epochs.
Qualitative examples {#sec:qual_examples}
====================
Table \[tab:known\_qual\] shows examples where our `full_bias`/`fact_bias` model helped translation by favoring certain words as opposed to the baseline in `en-fr`.
Talk Andrew McAfee : What will future jobs look like?
------------- --------------------------------------------------------------------------------------
Source but the middle class is clearly under huge threat right now .
Reference mais la classe moyenne fait aujourd’ hui face à une grande menace .
`base` mais la classe moyenne est clairement une menace énorme en ce moment .
`full_bias` mais la classe moyenne est clairement maintenant une grande menace .
`fact_bias` mais la classe moyenne est clairement grande menace en ce moment .
Talk Olafur Eliasson : Playing with space and light
Source the show was , in a sense , about that .
Reference le spectacle était , dans un sens , à propos de cela .
`base` le spectacle était , un sens , à propos de .
`full_bias` le spectacle était , un sens , à propos de .
`fact_bias` le spectacle était , un sens , .
Lona Szabo de Carvalho : 4 lessons I learned from taking a stand
against drugs and gun violence
Source we need to make illegal drugs legal .
Reference nous avons besoin de rendre les drogues illégales , légales .
`base` nous devons faire des illégaux .
`full_bias` nous devons faire des illégales .
`fact_bias` nous devons produire des illégales .
Talk Wade Davis: On the worldwide web of belief and ritual
Source a people for whom blood on ice is not a sign of death , but an affirmation of life .
un peuple pour qui du sang sur la glace n’ est pas un signe de mort
mais une affirmation de la vie .
pour qui sang sur la glace n’ est pas un signe de mort ,
mais une affirmation de vie .
pour qui sang sur la glace n’ est pas un signe de mort ,
mais une affirmation de vie .
pour qui sang sur la glace n’ est pas un signe de mort ,
mais une affirmation de vie .
[^1]: Data/code publicly available at <http://www.cs.cmu.edu/~pmichel1/sated/> and <https://github.com/neulab/extreme-adaptation-for-personalized-translation> respectively.
[^2]: Note that the speakers are still unique, and many might use very specific words ([*e.g.* ]{}the name of their company or of a specific medical procedure that they are an expert on).
[^3]: Notably, while this limits the model to only handling word choice and does not explicitly allow it to model syntactic variations, favoring certain words over others can indirectly favor certain phenomena ([*e.g.* ]{}favoring passive speech by increasing the probability of auxiliaries).
[^4]: <https://www.ted.com>
[^5]: Recent NMT systems also commonly use sub-word units [@sennrich-haddow-birch:2016:P16-12]. This may influence on the result, either negatively (less direct control over high-frequency words) or positively (more capacity to adapt to high-frequency words). We leave a careful examination of these effects for future work.
[^6]: available here: <https://www.kaggle.com/ellarabi/europarl-annotated-for-speaker-gender-and-age/version/1>
|
---
author:
- 'W. Wölfli[^1], W. Baltensperger[^2], R. Nufer[^3]'
title: An additional planet as a model for the Pleistocene Ice Age
---
Properties of the Pleistocene Ice Age
=====================================
Earth’s most recent Ice Age Epoch is characterised by unique features, which still require an explanation [@Elkibbi]. The Pleistocene Glaciation began approximately 2 Myr ago, after a gradual decrease of the global temperature in the Upper Plicoene from about 3 to 2 Myr BP. During the Pleistocene the general drop in temperature was interrupted by fluctuations, which augmented in proportion to the global cooling. Cold periods (Stadials) and warm periods (Interstadials) followed each other with a period of about 100 kyr during the last 1 Myr. Sometimes the temperature of the Interstadials even exceeded the average value for the Holocene [@Tiedemann] \[Fig. 1\]. The last Stadial (100 to 11.5 kyr BP) was interrupted about 20 times at irregular intervals by sudden temperature increases lasting from a few hundred to a few thousand years (Dansgaard-Oeschger events) [@Greenland] \[Fig. 2\]. During the Last Glacial Maximum, 20 kyr ago, the continental ice sheets reached the region around the present New York and covered Northern Germany, while Eastern Siberia and part of Alaska were ice-free and inhabited by large herbivorous mammals such as mammoths. Some of these have been excavated in a frozen state, which shows that in these areas the temperature dropped suddenly at the end of the Ice Age Epoch. During the Last Glacial Maximum, the continental ice sheets were centred in a geographically displaced position with relation to the present pole positions [@Petit-Maire]. In the Northern Hemisphere this position was in Greenland, about 18 degrees away from the present North Pole \[Fig. 3\]. According to Fig. 2, the Last Glacial Maximum was suddenly terminated 11.5 kyr ago. At about the same time a catastrophic geological event occurred, the relics of which are recorded in peculiar sediments found all over the Earth [@Allan].
{width="16cm"}
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An additional planet as the basis of the model
==============================================
Usually, the Ice Age Epoch is considered to be the reaction of a highly unstable climate system to the slow insolation variations proposed by Milankovitch [@Milankovitch]. However, the present climate on Earth with its distribution and behaviour follows the basis of the Milankovitch model, suggesting that for the Holocene the climate does not require major non-linearities for its explanation. In contrast, throughout the Ice Age Epoch, the climate was strongly variable, as is evident from Fig. 1. It is therefore reasonable to assume that for a limited time the main driving force of the climate was not the Milankovitch effect but an additional external agent to which Earth’s climate responded linearly. In particular, the asymmetry of the ice distribution in the Northern Hemisphere as well as the presence of Mammoths in arctic Siberia during the Last Glacial Maximum suggest that the geographic position of the North Pole was located somewhere in Central Greenland. If this was the case, then, at the End of the Pleistocene, it had to move to its present position. Such a movement of the geographic position of the Earth’s rotation axis, which in stellar space retains a practically fixed direction, can be induced by a transitory deformation of the Earth. This requires an extremely close passage near the Earth of a mass having at least the size of Mars. We therefore postulate that during the Ice Age Epoch and in the Upper Pliocene such an additional planet existed, henceforth called Z. Since at present Z does not exist any more, the Sun is the most likely agent to have promoted its disappearance. We therefore assume that Z moved in a highly eccentric orbit with a perihelion distance of only about 4 million km, so that during each passage near the Sun, Z was heated by both tidal forces and solar radiation. Thus planet Z was liquid and radiant.
{width="16cm"}
{width="16cm"}
Origin and fate of Z
====================
Since Z is not a member of the present planetary system the crucial question has to be answered how in such a short time interval it could appear and subsequently disappear. Regarding the origin of Z there are several possibilities: Z may have entered into the planetary system from outside, i.e. from the Kuiper belt or the Oort cloud. Alternatively, it may have its origin in the Asteroid belt or as a moon of Jupiter. It then must have lost energy and angular momentum through resonances with other planets [@Murray]. This requires a time of the order of a million years only. Fig. 1 suggests that Z reached an orbit with a small perihelion distance 3 million years ago, thereby creating a gas cloud resulting in the Earth’s Pleistocene. Regarding the termination of Z, it is indispensable to assume that it was fragmented during the final close encounter with the Earth. This process consumed orbital energy, so that the perihelion distances of the fragments were likely to be reduced compared to the perihelion distance of Z. Most importantly, the smaller escape velocity of the fragments increased the evaporation rate. Typically, the molecular binding energy became more important than the escape energy, so that both molecules and clusters could evaporate. These were then blown away by radiation pressure. In this way, the fragments of Z could become dissolved within the Holocene. It is not unlikely that some of the fragments also dropped into the Sun.
{width="7.6cm"}
Frequency of approaches to Earth
================================
In order to obtain a measure of how often Z approached the Earth, the equations of motion of the planetary system including Z were solved for various orbital parameters of Z. Tidal work and any effects from evaporation were disregarded. Earth and Moon were considered separately, and the planets beyond Saturn were omitted [@Nufer]. In the main calculation, the parameters assumed for Z at time J2000.0 were: semi-major axis 0.978 AU, numerical eccentricity 0.973, inclination 0$^\circ$, longitude and argument of the perihelion both 0$^\circ$. The result for a period of 750 kyr is shown in Figs. 4 and 5. The orbit of Z was found to be stable over the time range considered. The semi- major axis varied within 0.95 AU and 1.1 AU without showing a general trend. Similarly, the eccentricity remained between 0.958 and 0.977. The inclination, i.e. the angle between the orbit of Z and the invariant plane, showed an irregular variation between 0$^\circ$ and 13.5$^\circ$, with a period of approximately 9 kyr. In Fig. 4, each approach to Earth to less than 3 million km is marked by a vertical line ending at the closest distance. The Figure shows that the encounters are irregularly clustered, somewhat resembling the pattern of temperature fluctuations of Figs. 1 and 2. In Fig. 5 calculated approaches between Z and Earth to less than twice the Earth-Moon distance are shown. Within 100 kyr there are several passages closer than Moon’s distance. These must have created enormous earthquakes, and during the Ice Age Epoch must have caused ruptures of the continental ice shelves. We tentatively identify these with the Heinrich events [@Heinrich]. The diagram in Fig. 5 contains no approach closer than 30 000 km, i.e., approaches which might induce a polar shift; however, additional calculations suggest that this may occur once in a few Myr.
Mechanics of a polar shift
==========================
The asymmetry of the glaciation in the Northern Hemisphere \[Fig. 3\] is the most conspicuous feature of the Ice Age Epoch. Because of this, a displaced pole position and a fast polar migration have been postulated already at the end of the 19$^{\hbox{\scriptsize{th}}}$ century [@Hapgood]. This migration is a damped precession of the rotation axis on the globe, while the axis remains fixed in stellar space. This was discussed and judged as impossible. Indeed, with an Earth in either the solid or the liquid state, both of which were at that time considered, a deformation would relax too fast to allow an appreciable geographic shift of the poles. However, assuming a plastic Earth with a relaxation time of at least a few hundred days makes possible a shift of the required magnitude of ca. 18$^\circ$ [@Gold]. The rotation of the unperturbed Earth is stabilised by its increased radius at the equator compared to that at the poles. If the Earth had an additional deformation in an oblique direction, it would perform a motion in which the position of the rotation axis on the globe would migrate. The deformation could result from a close passage of a planet-sized object, which would stretch the Earth by tidal forces. A 1 per mil deformation is required for the shift. We consider an initial stretching deformation of the Earth such that, at an angle of 30$^\circ$ with the initial rotation axis, the radius is increased by 6.5 km. In the ensuing process, the Earth relaxes into a new equilibrium shape with a displaced equatorial belt. Angular momentum is strictly conserved, and at all times the rotation axis practically points to the same star. The spiral in Fig. 3 shows the geographic path of the rotation axis. It is the solution of Euler’s equations and a relaxation equation for the inertial tensor. A turn in the spiral takes about 400 days. For details of the calculation, see Appendix and [@Woelfli]. A stretching force is obtained from the tidal action due to a mass near the Earth. The required deformation corresponds to the equilibrium shape of Earth when a Mars-sized mass is at 24000 km distance. The actual deformation problem, with a mass passing near the rotating Earth, is vastly more complex. We expect that the peak tidal force has to be about an order of magnitude larger; this brings the closest distance between the centres of Earth and Z into the range of 12000 to 15000 km. As a result, Z enters the Roche limit of Earth. It is then likely to be torn into two or more parts. Since Z is lighter than the Earth, tidal effects are stronger on Z than on Earth: Z is torn to pieces rather than the Earth. Note that the requirements restrict the mass of Z at the end of the Pleistocene to a range of at least that of Mars and clearly smaller than Earth’s. The polar shift event must have been accompanied on Earth by continental floods, earthquakes and volcanic eruptions, i.e. a world-wide catastrophe, which actually left many stratigraphic evidences and which can also be assumed to be reported in traditions in many countries all over the globe [@Allan].
The gas cloud during the Ice Age Epoch
======================================
Since during the Pleistocene Z is assumed to have been at least Mars-sized, evaporation was limited by the gravitational escape energy. Only single atoms escaped from the hot surface of Z. If an atom has an optical transition from the ground state within the main solar spectrum, then radiation pressure expels it from the planetary system. However, some atoms and many ions can be excited with ultraviolet light only. Apart from the rare gas atoms, these include atoms of Oxygen and Carbon. In these cases the repulsion due to solar radiation is much weaker than the gravitational attraction to the Sun, so that atoms of these elements can remain in bound orbits. These orbits shrink due to the Poynting-Robertson drag. The continued evaporation creates an interplanetary cloud. Its material consists of single atoms and ions and is thus quite distinct from the present zodiacal dust [@Gustafson]. The dynamics of the atomic cloud involves a variety of processes and is complex. We can only tentatively guess its behaviour. Collisions between atoms with planetary velocities are inelastic. They reduce the relative velocity between the colliding particles, so that their outgoing orbits become more similar. This increases the particle densities and thereby the frequency of collisions. This suggests that the range of inclinations of the orbits in the cloud can shrink with a time scale determined by the mean free time for particle collisions. Also, since collisions are inelastic, the semi-axes diminish. Particles with different ratios of repulsion by solar radiation to gravitational attraction intrinsically belong to different orbits and may become separated. If molecules form, these are expelled by radiation pressure. Atoms and molecules may become ionised. The scattering of solar radiation from any material along the line between Sun and Earth lowers the global temperature. Clearly, this screening depends on the density of the cloud and on the relative motions of cloud and Earth. Therefore, the extremely strong variations in temperature characteristic of the Pleistocene may be due to changes in the screening of the Sun. The isotopic and stratigraphic data for the last Myr of the Ice Age Epoch show a 100 kyr period [@Petit]. Now, Earth’s inclination, i.e. the angle between Earth’s orbit and the invariant plane, is governed by a 100 kyr cycle. The maxima of the inclination in fact coincide with the Interstadials, except for the last maximum, where no Interstadial has been observed. This indicates that the orbits of the atoms of the cloud often had inclinations which were smaller than the maxima of Earth’s inclination. Possibly at the end of the Pleistocene the width of the cloud had become too large. A similar solution to the problem of the origin of the 100 kyr cycle has previously been suggested by Muller et al [@Muller95; @Muller97].
The termination of the Ice Age Epoch
====================================
The climatic fluctuations which occurred towards the end of the Pleistocene require further study. We just note here that the last rapid increase of the temperature recorded in the polar ice data occurred at 11500$\pm$65 yr BP \[Fig. 2\]. All radiocarbon dates made on residue material originated during the global catastrophe point to the same age [@Allan; @Martin]). However, these radiocarbon ages are not corrected for the variation of the production rate. The new dendro and U/Th calibration curves indicate that these ages have to be increased by about 1500 yr [@Stuiver; @Bard]). Thus, it appears that the Younger Dryas, which begins at 12700$\pm$100 yr BP [@Greenland] is younger than the polar shift event. At the beginning of the Holocene the temperature increased in two steps. The first fast step was followed by a much slower rise, which reached its maximum about 9000 yr ago \[Fig. 2\]. Since then the temperature remained remarkably constant until today. The possibility should not be a priori discarded that minor, still unexplained climatic features such as the cold events at 8.2 kyr cal BP [@Hu] and 4.166 kyr cal BP [@Beck] as well as the so-called Little Ice Age, 300 years ago, are due to remaining traces of gaseous material.
Facts which become plausible or which can be explained by our model
===================================================================
From the displaced pole positions
---------------------------------
- [The [**asymmetry**]{} of the glaciation in North America and east Siberia; it was the main motivation for considering geographically displaced poles. At present there are no climate models which determine the optimum position of the poles and the amount of screening of the solar radiation compatible with the observed glaciation.]{}
- [The existence of [**mammoths**]{} in arctic East Siberia indicates that there was sufficient sunlight for the growth of the plants on which they lived.]{}
- [[**Archaeological**]{} objects having ages around 40000 years BP were found close to the Arctic Circle in Siberia [@Pavlov]. At the time the place had lower latitude.]{}
- [[**Lake Baikal**]{} was never frozen during the Pleistocene [@Kashiwaya].]{}
- [The [**Tibetan Plateau**]{} was about 15$^\circ$ closer to the equator and not ice-domed during the last 170 kyr [@Schaefer].]{}
- [The [**Atacama Desert**]{} and the [**altiplano Bolivia**]{} were humid [@Baker]. They were then situated closer to the equator than today.]{}
- [The [**Sahara desert**]{} was covered with grass and bushes during the Pleistocene. While for the Western Sahara this may be explained by its higher latitude in the Pleistocene, other reasons such as globally lower temperature should play a role in the Eastern Sahara.]{}
From the interplanetary gas cloud
---------------------------------
- [The [**beginning**]{} of the general temperature decrease, 3 Myr ago can be understood as the time at which the perihelion distance of Z became sufficiently small for the gas cloud to develop.]{}
- [The [**coldest**]{} Ice Age was at the [**end**]{} of the Pleistocene, since due to tidal work the perihelion distance of Z decreased. With the passing of time the cloud became denser.]{}
- [[**The colder**]{} the mean temperature, [**the larger**]{} were the [**variations**]{} of the temperature. Dense clouds throw strong shadows.]{}
- [Some Interstadials had [**higher**]{} temperatures than at present. Not only was the Earth exposed to the regular sunlight, but it also received radiation scattered from clouds or backscattered from material outside Earth’s orbit.]{}
- [The form of the [**Daansgard-Oeschger temperature variations**]{}, typically a gradual decrease followed by a rapid increase, may be connected with the presently still unknown dynamics of the cloud. The observed shape of the variations may help to understand the cloud’s behavior.]{}
- [During the last million years the Interstadials occur approximately every [**100 kyr**]{}, which coincides with the cycle of Earth’s inclination.]{}
From the orbit of Z
-------------------
- [The approaches shown in Fig. 5 produce a similar irregular pattern as the temperature variations of Fig. 2. The events marked in Fig. 5 have distances less than 3 Mio. km, which is the radius to which a gas cloud may expand by thermal motion during the passage from the perihelion of Z to Earth’s distance. This suggests that a screening of the solar radiation by gases may lead to a [**frequency of temperature excursions**]{} as observed in the sequence of the Dansgaard-Oeschger events.]{}
- [Gigantic earthquakes, as listed in Fig. 5, accompanied fly-byes at less than Moon’s distance. Their frequency, about 6 per 100 kyr, corresponds to the [**frequency of Heinrich events**]{}, in which large glaciers broke away from the continent and floated into the Atlantic Ocean.]{}
- [Fig. 5 contains no fly-byes to less than 30000 km, which might induce a polar shift. The [**rarity of polar shifts**]{}, say one event in a few Myr, is compatible with Fig. 5.]{}
From the polar shift catastrophe
--------------------------------
- [The model explains how a [**polar shift**]{} within the time of relaxation of a global deformation, i.e. a few years, can occur.]{}
- [The [**frozen mammoths**]{} in the permafrost of arctic Siberia are a direct consequence of the geographic motion of the polar axis.]{}
- [The catastrophe produced a global [**extinction of**]{} many [**species**]{} of large animals [@Allan; @Martin].]{}
- [ [**Frozen muck**]{} containing broken trees and bones testify for the violence of the event [@Allan].]{}
- [The fragmentation of Z lead to its relatively rapid disappearance. Thus the Ice Age Epoch had an [**end**]{}.]{}
- [Once the continental ice shelf was molten, the [**Holocene**]{} was [**constantly warm**]{} and distinct from the Interstadials, which were interrupted by temperature variations.]{}
- [Human [**civilization reappeared**]{} about 9000 years ago. Notably these populations, which stem from survivors, had an elaborately structured language.]{}
- [Many [**traditions**]{} are related to Z or its fragments, such as the Chinese dragon, a flying animal that spits fire and has a long, indefinite tail [@Velikovsky].]{}
Conclusion
==========
At present, the climate system of the Earth is observed to react to external forcing in a plausible way. If we assume this to be generally true in the geological past the observed asymmetry of the glaciation during the Late Glacial Maximum requires a shifted pole position and a fast migration of the poles at the end of the last glaciation. A pole shift of the order of 18$^\circ$ requires a close encounter with a massive object, which we have here called Z. Its mass was at least Mars sized, but clearly smaller than Earth’s, so that Z could be torn into fragments during the encounter. Since only the Sun can dispose of Z, we have to assume the perihelion of Z so small that Z was intensely heated. Clearly, its aphelion has to be larger than the radius of the Earth’s orbit. The choice of the aphelion determines the frequency of encounters. It had several million passes near the sun and a few close encounters with the Earth as well as with the other inner Planets. The heating of the surface of Z lead to the accumulation of an interplanetary atomic gas cloud having a complex dynamics. The material between the Sun and the Earth reduced the insolation and thereby the global temperature in a time-dependent way. In particular, the increasingly cold and variable climate during the last 3 Myr, until 11.5 kyr ago, is plausibly explained by the slow decrease of the orbital energy of Z and its angular momentum. The perihelion decreases, and this enhances the density of the gas cloud. In the Holocene, after the removal of all fragments of Z, its threat for life on Earth finally ended. Note that this model has only few free parameters. On the other hand, it creates new problems that deserve a more complete treatment in future studies. It may be worthwhile to clarify the relation of our model to a claim held by I. Velikovsky [@Velikovsky] that a close-by passage of Venus and later of Mars had produced a polar shift on Earth. Einstein [@Einstein] in his third letter to Velikovsky resumed his recommendations by the expression “Catastrophes yes, Venus no”. Our model is compatible with this directive of Einstein.
Appendix {#appendix .unnumbered}
========
The tidal force field $F(z)$ (for large distances $R$ compared to the radius $R_E$ of the Earth) is parallel to the $z$-direction, which points to the perturbing mass $M_Z$ . It has the value $$F(z) = 2 M_Z G\; \frac z{R^3},$$ where $G = 6.673\cdot 10^{-11}$ m$^3$ kg$^{-1}$ s$^{-2}$ is the gravitational constant. Earth’s induced deformation is described by an increment to the radius $H(\gamma )$, where $\gamma$ is the angle with the direction $z$ (at latitude 30$^\circ$) $$H(\gamma ) = H_0 \left[\cos(2\gamma )+\frac 13\right] .$$ The energy for such a deformation is minimised by the amplitude $$H_0 = \frac{R_E^2 M_Z G}{2 R^3 g}$$ with $g = 9.8$ m/s$^2$, the gravitational acceleration at the surface of the Earth. For $R = $24000 km, $H_0 = 6.45$ km. In a dynamic theory $R$ will be smaller. The diagonalized inertial tensor of the equilibrium Earth $\Xi_0$ has matrix elements \[1.0033,1,1\] in units $8.01\cdot 10^{37}$ kg m$^2$. For the deformed Earth the initial inertial tensor $\Xi(0)$ has diagonal elements \[1.0018, 0.9995, 1\] and off-diagonal elements $\Xi_{12}(0) = \Xi_{21}(0) = 0.000\thinspace 9$. Deformations are assumed to relax as $$\frac{d \Xi}{dt} = -\frac{\Xi(t) - \Xi_0[\vec\omega (t)]}\tau ,\qquad \tau = 1000\; \hbox{d}.$$ The geographic wandering of the rotation vector $\vec\omega$ in co-ordinates fixed to the Earth is described by the Euler equation for free motion: $$\frac{d \;\Xi \vec\omega}{dt} = [\Xi\vec\omega,\vec\omega].$$ Eq. (4) and (5) are solved numerically.
Acknowledgement {#acknowledgement .unnumbered}
===============
We are indebted to H.-U. Nissen for comments on the manuscript.
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[^1]: Institute for Particle Physics, ETHZ Hönggerberg, CH-8093 Zürich, Switzerland (Prof. emerit.); e-mail: [email protected]
[^2]: Centro Brasileiro de Pesquisas Fí sicas, Rua Dr. Xavier Sigaud,150, 22290 Rio de Janeiro, Brazil;e-mail: [email protected]
[^3]: Im Römergarten 1, CH-4106 Therwil, Switzerland; e-mail: [email protected]
|
---
abstract: |
{width=".2\linewidth"}
We present results for the $I=2$ $\pi\pi$ scattering length using $N_f=2+1+1$ twisted mass lattice QCD for three values of the lattice spacing and a range of pion mass values. Due to the use of Laplacian Heaviside smearing our statistical errors are reduced compared to previous lattice studies. A detailed investigation of systematic effects such as discretisation effects, volume effects, and pollution of excited and thermal states is performed. After extrapolation to the physical point using chiral perturbation theory at NLO we obtain $M_\pi a_0=-0.0442(2)_\mathrm{stat}(^{+4}_{-0})_\mathrm{sys}$.
author:
- |
C. Helmes, C. Jost, B. Knippschild, L. Liu, C. Urbach,\
M. Ueding and M. Werner\
\
- |
C. Liu\
\
\
- 'J. Liu and Z. Wang\'
bibliography:
- 'bibliography.bib'
title: |
Hadron-Hadron Interactions from $N_f=2+1+1$ Lattice QCD:\
isospin-2 $\pi\pi$ scattering length
---
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank the members of ETMC for the most enjoyable collaboration. The computer time for this project was made available to us by the John von Neumann-Institute for Computing (NIC) on the JUDGE and Juqueen systems in J[ü]{}lich. We thank U.-G. Mei[ß]{}ner for granting us access on JUDGE. We thank A. Rusetsky for very useful discussions. We thank K. Ottnad for providing us with the data for $M_\pi/f_\pi$ and S. Simula for the estimates of the finite size corrections to $M_\pi$ and $f_\pi$. We thank C. Michael for helpful comments on the manuscript. This project was funded by the DFG as a project in the Sino-German CRC110. C. Liu, J. Liu and J. Wang are supported in part by the National Science Foundation of China (NSFC) under the project o.11335001. The open source software packages tmLQCD [@Jansen:2009xp], Lemon [@Deuzeman:2011wz] and R [@R:2005] have been used.
|
---
abstract: 'An attempt to formulate the optical model of particle-hole-type excitations (including giant resonances) is undertaken. The model is based on the Bethe–Goldstone equation for the particle-hole Green function. This equation involves a specific energy-dependent particle-hole interaction that is due to virtual excitation of many-quasiparticle configurations and responsible for the spreading effect. After energy averaging, this interaction involves an imaginary part. The analogy between the single-quasiparticle and particle-hole optical models is outlined.'
author:
- 'M.H. Urin'
title: 'Particle-Hole Optical Model: Fantasy or Reality?'
---
Damping of giant resonances (GRs) is a long-standing problem for theoretical studies. There are three main modes of GR relaxation: (i) particle-hole (p–h) strength distribution (Landau damping), which is a result of the shell structure of nuclei; (ii) coupling of (p–h)-type states with the single-particle (s.p.) continuum, which leads to direct nucleon decay of GRs; and, (iii) coupling of (p–h)-type states with many-quasiparticle configurations, which leads to the spreading effect. An interplay of these relaxation modes takes place in the GR phenomenon.
A description of the giant-resonance strength function with exact allowance for the Landau damping and s.p. continuum can be obtained within the continuum-RPA (cRPA), provided the nuclear mean field and p–h interaction are fixed [@ref1]. As for the spreading effect, it is attempted to be described together with other GR relaxation modes within microscopic and semimicroscopic approaches. The coupling of the (p–h)-type states, which are the doorway states (DWS) for the spreading effect, with a limited number of 2p–2h configurations is explicitly taken into account within the microscopic approaches (see, e.g., Refs. [@ref2; @ref3]). Some questions to the basic points of these approaches could be brought up: (i) “Thermalization” of the DWS, which form a given GR, i.e. the DWS coupling with many-quasiparticle states (MQPS) (the latter are complicated superpositions of 2p–2h, 3p–3h, …configurations), is not taken into account. As a result, each DWS may interact with others via 2p–2h configurations. Due to complexity of MQPS one can reasonably expect that after energy averaging the interaction of different DWS via MQPS would be close to zero (the statistical assumption). (ii) The use of a limited basis of 2p–2h configurations does not allow to describe correctly the GR energy shift due to the spreading effect. The full basis of these configurations should be formally used for this purpose. (iii) With the single exception of Ref. [@ref4], there are no studies of GR direct-decay properties within the microscopic approaches.
Within the so-called semimicroscopic approach, the spreading effect is phenomenologically taken into account directly in the cRPA equations in terms of the imaginary part of an effective s.p. optical-model potential [@ref5; @ref6]. Within this approach, the afore-mentioned statistical assumption is supposed to be valid and used in formulation of the approach. The GR energy shift due to the spreading effect is evaluated by means of the proper dispersive relationship, and therefore, the full basis of MQPS is formally taken into account [@ref7]. The approach is applied to description of direct-decay properties of various GRs (the references are given in Ref. [@ref6]). In accordance with the “pole” approximation used for description of the spreading effect within the semimicroscopic approach, the latter is valid only in the vicinity of the GR energy. However, for an analysis of some phenomena it is necessary to describe the low- and/or high-energy tails of various GRs. For instance, the asymmetry (relative to 90$^\circ$) of the ($\gamma \mathrm n$)–reaction differential cross section at the energy of the isovector giant quadrupole resonance is determined, in particular, by the high-energy tail of the isovector giant dipole resonance [@ref8]. Another example is the isospin-selfconsistent description of the IAR damping. In particular, the IAR total width is determined by the low-energy tail of the charge-exchange giant monopole resonance [@ref9; @ref9_5; @ref10].
In the present work, we attempt to formulate a model for phenomenological description of the spreading effect on p–h strength functions at arbitrary (but high enough) excitation energies. The formulation of this semimicroscopic model (simply called as the p–h optical model) is analogous to that of the single-quasiparticle optical model [@ref11; @ref12]. The dispersive version of this model [@ref11] is widely used for description of various properties of single-quasiparticle excitations at relatively high energies (see, e.g., Ref. [@ref13]). This model can be also used for description of direct particle decay of subbarrier s.p. states [@ref14].
The starting point in formulation of the single-quasiparticle optical model is the Fourier-component of the Fermi-system single-quasiparticle Green function, $G(x,x';\varepsilon)$, taken in the coordinate representation (see, e.g., Refs. [@ref11; @ref12]). By analogy with that, we start formulation of the p–h optical model from the Fourier-component of the Fermi-system (generally, nonlocal) p–h Green function, $\mathcal A(x,x';x_1,x'_1;\omega)$, also taken in the coordinate representation. Being a kind of the Fermi-system two-particle Green function (definitions see, e.g., in Ref. [@ref15]), $\mathcal A$ satisfies the following spectral expansion:
$$\mathcal{A}(x,x';x_1,x'_1;\omega) = \sum_s \left(
\frac{\rho_s^\ast(x',x)\rho_s(x_1,x'_1)}{\omega-\omega_s+i0}
-\frac{\rho_s^\ast(x'_1,x_1)\rho_s(x,x')}{\omega+\omega_s-i0}
\right). \label{eq1}$$
Here, $\omega_s = E_s-E_0$ is the excitation energy of an exact state $|s\rangle$ of the system and $\rho_s(x,x') = \langle s|
\widehat\Psi^+(x) \widehat\Psi(x')|0\rangle$ is the transition matrix density ($\widehat\Psi^+(x)$ is the operator of particle creation at the point $x$). In accordance with the expansion of Eq. (\[eq1\]) the p–h Green function determines the strength function $S_V(\omega)$ corresponding to an external (generally, nonlocal) single-quasiparticle field $\widehat V = \int
\widehat\Psi^+(x) V(x,x') \widehat\Psi(x') dxdx' \equiv \left[
\widehat\Psi^+ V \widehat\Psi \right]$: $$S_V(\omega) = -\frac{1}{\pi} \mathrm{Im} \left[ V^+ \mathcal
A(\omega) V \right], \label{eq2}$$ where the brackets $\left[ \dots \right]$ mean the proper integrations.
The free s.p. and p–h Green functions, $G_0(x,x';\varepsilon)$ and $\mathcal A_0(x,x';x_1,x'_1;\omega)$, respectively, are determined by the mean field (via the single-quasiparticle wave functions) and the occupation numbers (only nuclei without nucleon pairing are considered). Being determined by Eq. (\[eq1\]), the free transition matrix densities $\rho_s^{(0)}(x,x')$ are orthogonal: $\left[ \rho_s^{(0)\ast}\rho_{s'}^{(0)} \right]= \delta_{ss'}$. As for the transition densities $\rho_s^{(0)}(x = x')$, which appear in the spectral expansion for the free local p–h Green function $A_0(x,x_1;\omega) = \mathcal A_0(x=x', x_1=x_1';\omega)$, this statement is wrong. The RPA p–h Green function, $\mathcal
A_{RPA}(x,x';x_1,x'_1;\omega)$, is determined also by a p–h (local) interaction $\mathcal F(x,x';x_1,x'_1) =
F(x,x_1)\delta(x-x')\delta(x_1-x'_1)$, which is responsible for long-range correlations leading to formation of GRs. In particular, the Landau–Migdal forces $F(x,x_1) \rightarrow F(x)\delta(x-x_1)$ are used in realizations of the semimicroscopic approach of Refs. [@ref5; @ref6]. The RPA p–h Green function satisfies the expansion, which is similar to that of Eq. (\[eq1\]). In such a case, the RPA states $|d\rangle$ are the DWS for the spreading effect. The local RPA p–h Green function $A_{RPA}(x,x_1;\omega)$ determined by the p–h interaction $F(x,x_1)$ is used for cRPA-based description of the GR strength function corresponding to a local external field $\widehat V = \int \widehat\Psi^+(x) V(x)
\widehat\Psi(x) dx$ [@ref1].
The s.p. and p–h Green functions satisfy, respectively, the Dyson and Bethe–Goldstone integral equations: $$G(\varepsilon) = G_0(\varepsilon) + \left[ G_0(\varepsilon)
\Sigma(\varepsilon) G(\varepsilon) \right] \label{eq3}$$ and $$\mathcal A(\omega) = \mathcal A_{RPA}(\omega) + \left[ \mathcal
A_{RPA}(\omega) \mathcal P(\omega) \mathcal A(\omega) \right],
\label{eq4}$$ where $$\mathcal A_{RPA}(\omega)=\mathcal A_0(\omega)+\left[\mathcal
A_0(\omega)\mathcal F\mathcal A_{RPA}(\omega)\right]. \label{eq5}$$ The self-energy operator $\Sigma(x,x';\varepsilon)$ and the specific p-h interaction (polarization operator) $\mathcal
P(x,x';x_1,x'_1;\omega)$ describe the coupling, correspondingly, of single-quasiparticle and (p–h)-type states with proper MQPS. Analytical properties of $G$ and $\Sigma$ are nearly the same. A similar statement can be made for $\mathcal A$ and $\mathcal P$. The quantities $\Sigma(\varepsilon)$ and $\mathcal P(\omega)$ both exhibit a sharp energy dependence due to a high density of poles corresponding to virtual excitation of MQPS. Concluding consideration of the basic relationships given above in a rather schematic form, we present the alternative equation for the p–h Green function: $$\mathcal A(\omega) = \mathcal A_0(\omega) + \left[ \mathcal
A_0(\omega) \left(\mathcal F+\mathcal P(\omega)\right) \mathcal
A(\omega) \right], \label{eq6}$$ which follows from Eqs. (\[eq4\]), (\[eq5\]).
Since the density of MQPS, $\rho_m$, is large and described by statistical formulae, only the quantities $\bar
\Sigma(x,x';\varepsilon)$ and $\bar {\mathcal
P}(x,x';x_1,x'_1;\omega)$ averaged over an interval $J \gg
\rho_m^{-1}$ can be reasonably parameterized. As applied to $\bar
\Sigma(\varepsilon) = \Sigma\left(\varepsilon +
iJSgn(\varepsilon-\mu)\right)$, it is done, e.g., in Refs. [@ref11; @ref12]: $$\bar \Sigma(x,x';\varepsilon) = Sgn(\varepsilon-\mu)
\left\{-iw(x;\varepsilon)+p(x;\varepsilon)\right\}\delta(x-x').
\label{eq7}$$ Here, $\mu$ is the chemical potential and $w(x;\varepsilon)$ is the imaginary part of a (local) optical-model potential. Assuming that the radial dependencies of $p$ and $w$ are the same, i.e. $w(x;\varepsilon) \rightarrow w(r)w(\varepsilon)$ and $p(x;\varepsilon) \rightarrow w(r)p(\varepsilon)$, the intensity of the real addition to the mean field, $p(\varepsilon)$, has been expressed in terms of $w(\varepsilon)$ via the corresponding dispersive relationship [@ref11]. It is noteworthy, that the optical-model addition to the mean field can be taken as the local one, i.e. $\bar \Sigma(x,x';\varepsilon) \sim \delta(x-x')$, in view of a large momentum transfer (of order of the Fermi momentum) at the “decay” of single-quasiparticle states into MQPS. The energy averaged single-quasiparticle Green function $\bar
G(x,x';\varepsilon)$ satisfies the Eq. (\[eq3\]), which involves in such a case the quantity $\bar \Sigma$ of Eq. (\[eq7\]). Actually, $\bar G$ is the Green function of the Schrödinger equation, which involves the addition to the mean field considerated above.
The energy-averaged polarization operator can be parameterized similarly to Eq. (\[eq7\]):
$$\bar{\mathcal P}(x,x';x_1,x'_1;\omega) = \left\{-i\mathcal
W(x,x';\omega)+P(x,x';\omega)\right\}\delta(x-x_1)\delta(x'-x'_1).
\label{eq8}$$
Assuming that the coordinate dependencies of the quantities $P$ and $\mathcal W$ are the same, i.e. $\mathcal W(x,x';\omega) \rightarrow
\mathcal W(x,x')\mathcal W(\omega)$ and $P(x,x';\omega) \rightarrow
\mathcal W(x,x')P(\omega)$, we can express $P(\omega)$ in terms of $\mathcal W(\omega)$ via the corresponding dispersive relationship. The example of such a relationship is given in Ref. [@ref7]. In accordance with Eqs. (\[eq4\]), (\[eq6\]), (\[eq8\]) the energy-averaged p–h Green function satisfies the equivalent equations: $$\bar{\mathcal A}(\omega) = \mathcal A_{RPA}(\omega) + \left[
\mathcal A_{RPA}(\omega) \bar{\mathcal P}(\omega) \bar{\mathcal
A}(\omega) \right], \label{eq9}$$ $$\bar{\mathcal A}(\omega) = \mathcal A_0(\omega) + \left[ \mathcal
A_0(\omega) \left(\mathcal F+\bar{\mathcal P}(\omega)\right)
\bar{\mathcal A}(\omega) \right]. \label{eq10}$$ Formally, Eqs. (\[eq8\])–(\[eq10\]) are the basic equations of the p–h optical model. In particular, the energy-averaged strength function is determined by Eq. (\[eq2\]) with the substitution $\mathcal A(\omega) \rightarrow \bar{\mathcal A} (\omega)$.
To realize the model in practice, a reasonable parametrization of $\mathrm{Im} \bar{\mathcal P}$ should be done with taking the statistical assumption into account. For this purpose, we consider the quantity $\mathcal A_{RPA}$ within the discrete–RPA (dRPA) in the “pole” approximation. In accordance with Eq. (\[eq1\]), we have $$\mathcal A_{RPA}(x,x';x_1,x'_1;\omega) \rightarrow \sum_d
\frac{\rho_d^\ast(x',x)\rho_d(x_1,x'_1)}{\omega-\omega_d+i0}.
\label{eq11}$$ The statistical assumption $\left[ \rho_d^\ast \bar{\mathcal P}
\rho_{d'} \right] \sim \delta_{dd'}$ is fulfilled, provided that: (i) the intensity $\mathcal W(x,x';\omega)$ is nearly constant within the nuclear volume, i.e. $\mathcal W(x,x';\omega) \rightarrow
\mathcal W(\omega)$; and, (ii) the dRPA transition matrix densities are orthogonal, i.e. $\left[ \rho_d^\ast \rho_{d'} \right] =
\delta_{dd'}$. Under these assumptions, the solution of Eq. (\[eq9\]) can be easily obtained in the pole approximation: $\bar{\mathcal A}(\omega) = \mathcal A_{RPA}(\omega +i\mathcal
W(\omega) - P(\omega))$. As a result, the energy-averaged strength function is the superimposition of the DWS resonances: $$\bar S_V(\omega) = -\frac{1}{\pi}\mathrm{Im}\sum_d
\frac{\left|\left[V
\rho_d\right]\right|^2}{\omega-\omega_d+i\mathcal
W(\omega)-P(\omega)}. \label{eq12}$$ The quantity $2\mathcal W$ can be considered as the mean DWS spreading width $\langle \Gamma_d^\downarrow \rangle $, which might be larger than the mean energy interval between neighboring DWS resonances.
A few points are noteworthy in conclusion of the above-given description of the p–h optical model. Within the model the spreading effect on formation of (p–h)-type excitations is described phenomenologically in terms of the specific (p–h) interaction $\bar{\mathcal P}$. Because the interference between the spreading of particles and holes is taken into account by this interaction, the latter cannot be expressed via the single-quasiparticle self-energy operator $\bar\Sigma$. Formally, the p–h optical model is valid at arbitrary (but high enough) excitation energy. The low limit is determined by the possibility of using the statistical formulae to describe the MQPS density. Within the semimicroscopic approach, the substitution like $\omega
\rightarrow \omega+i\mathcal W(\omega) - P(\omega)$ is used in the cRPA equations to take the spreading effect phenomenologically into account in the “pole” approximation together with the statistical assumption [@ref6; @ref7]. Thus, the parametrization of $\mathcal
W(\omega)$ can be taken in the form widely used for the intensity of the imaginary part of the effective s.p. optical-model potential in implementations of the semimicroscopic approach. Within the s.p. optical model the statistical assumption for “decay” of different single-quasiparticle states with the same angular momentum and parity into MQPS seems to be valid. At high excitation energies $|\varepsilon-\mu|$, when the empirical value of $w(\varepsilon)$ is comparable with the energy interval between the afore-mentioned single-quasiparticle states, the empirical radial dependence $w(r)$ becomes nearly constant within the nuclear volume (see, e.g., Refs. \[14\]).
The p–h optical model can be simply realized in terms of the energy-averaged local p–h Green function $\bar A(x,x_1;\omega)$ to describe the strength function of a “single-level” GR, because in such a case there is no need for the statistical assumption. Being more simple, the equations like (\[eq4\])–(\[eq6\]), (\[eq8\]), (\[eq10\]) are actually the straight-forward extension of the corresponding cRPA equations. In practice, within the cRPA it is more convenient to use the equation for the effective field $\bar V(x,\omega)$, which corresponds to a local external field $V(x)$ and is determined in accordance with the relationship: $\left[ V\bar A(\omega) \right] = \left[ \bar V(\omega) A_0(\omega)
\right]$. The effective field determines the strength function: $$\bar S_V(\omega) = -\frac{1}{\pi}\mathrm{Im}\left[ V_0 A_0(\omega)
\bar V(\omega) \right], \label{eq13}$$ and satisfies the equation: $$\bar V(\omega) = V + \left[ \left(F+\bar \Pi(\omega)\right)
A_0(\omega) \bar V(\omega) \right]. \label{eq14}$$ The energy-averaged local polarization operator can be parameterized similarly to Eq. (\[eq8\]): $$\bar \Pi(x,x_1;\omega) =
C\left\{-iW(x;\omega)+P(x;\omega)\right\}\delta(x-x_1). \label{eq15}$$ Here, $C=300$MeVfm$^3$ is the value often used in parametrization of the Landau–Migdal forces; $W$ and $P$ are the dimensionless quantities, which can be parameterized as follows: $W(x;\omega) \rightarrow W(r) W(\omega)$ and $P(x;\omega)
\rightarrow W(r) P(\omega)$, where $P(\omega)$ is determined by $W(\omega)$ via the corresponding dispersive relationship [@ref7].
Due to strong coupling with s.p. continuum, the high-energy GRs (they are mostly the overtones of corresponding low-energy GRs) can be roughly considered as the “one-level” ones. Being the IAR overtone, the charge-exchange (in the $\beta^{-}$-channel) giant monopole resonance (GMR$^{(-)}$) is related to these GRs. Within the isospin-selfconsistent description of the IAR damping [@ref9_5; @ref10], the low-energy “tail” of the GMR$^{(-)}$ in the energy dependence of the “Coulomb” strength function $\bar
S_C^{(-)}(\omega)$ determines the IAR total width $\Gamma_A$ via the nonlinear equation: $$\Gamma_A = 2\pi S_A^{-1}\bar S_C^{(-)}(\omega = \omega_A).
\label{eq16}$$ Here, $S_A \simeq \left( N-Z \right)$ is the IAR Fermi strength, $\omega_A$ is the IAR energy, and the “Coulomb” strength function corresponds to the external field $V(x) \rightarrow V_C^{(-)} =
\left( U_C(r) - \omega_A +\frac{i}{2} \Gamma_A \right) \tau^{(-)}$, where $U_C(r)$ is the mean Coulomb field. Strength function $\bar
S_C^{(-)}(\omega)$ exhibits a wide resonance corresponding to the GMR$^{(-)}$. In Fig. \[fig1\], we present the strength function calculated for the $^{208}$Pb parent nucleus within: (i) the cRPA (in such a case the strength function $S^{(-)}_C(\omega=\omega_A)$ determines the IAR total escape width found without taking the isospin-forbidden spreading effect into account [@ref16]); (ii) the semimicroscopic approach [@ref10] and, (iii) the p–h optical model by Eqs. (\[eq13\])–(\[eq15\]). All the model parameters, parameterization of the imaginary part of the effective single-quasiparticle optical-model potential $I(r;\omega)$ [@ref10] and parameterization of $W(r;\omega)$ in Eq. (\[eq15\]) are taken the same in both approaches. The intensities of $I(r;\omega)$ and $W(r;\omega)$ are chosen to reproduce in calculations the observable total width of the GMR$^{(-)}$ in $^{208}$Bi ($\simeq 15$ MeV). Both approaches lead to the similar results, which are not exactly the same for the low-energy “tail” of the GMR$^{(-)}$ at $\omega \simeq \omega_A$. Irregularities in the energy dependence of $\bar S_C^{(-)}(\omega)$ calculated within the p–h optical model are explained by the fact that the GMR$^{(-)}$ can be roughly considered as the “one-level” one.
![\[fig1\]The “Coulomb” strength function calculated for the $^{208}$Pb parent nucleus within the cRPA (thin line), the semimicroscopic approach (dash–dotted line), and p–h the optical model (full line). The arrow indicates the IAR energy.](1.eps)
In the present work, the optical model of particle-hole-type excitations has been formulated in terms of the energy-averaged nonlocal particle-hole Green function. The equation for this Green function involves a specific energy-dependent particle-hole interaction, which is due to virtual excitation of many-quasiparticle configurations. The intensity of the imaginary part of this interaction should be taken nearly constant within nuclear volume to satisfy the statistical assumption on the independent spreading of different particle-hole-type states which form a given giant resonance. The strength function of the “single-level” giant resonance can be described in terms of the energy-averaged local particle-hole Green function.
Along with numerical realizations, the particle-hole optical model can be extended to describe direct particle decays of giant resonances. These points are under consideration.
In conclusion, one can say that in formulation of the particle-hole optical model we are on a way from fantasy to reality.
The author thanks M.L. Gorelik for the calculation leading to the results presented in Fig. \[fig1\], and I.V. Safonov for his kind help in preparing the manuscript.
This work is partially supported by RFBR under grant no. 09-02-00926-a.
[99]{} S. Shlomo, G. Bertsch, Nucl. Phys. A 243, 507 (1975). G.F. Bertsch, P.F. Bortignon, R.A. Broglia, Rev. Mod. Phys. 55, 287 (1983). S. Kamerdziev, J. Speth, G. Tertychny, Phys. Rep. 393, 1 (2004). G. Coló, N. Van Giai, P.F. Bortignon, R.A. Broglia, Phys. Rev. C 50, 1496 (1994). M.L. Gorelik, I.V. Safonov, M.H. Urin, Phys. Rev. C 69, 054322 (2004). M.H. Urin, Nucl. Phys. A 811, 107 (2008). B.A. Tulupov, M.G. Urin, Phys. At. Nucl. 72, 737 (2009). M.L. Gorelik, B.A. Tulupov, M.G. Urin, Phys. At. Nucl. 69, 598 (2006). N. Auerbach, Phys. Rep. 98, 273 (1983) I.V. Safonov, M.G. Urin, Bull. Rus. Acad. Sci. Phys. 67, 44 (2003). M.L. Gorelik, V.S. Rykovanov, M.G. Urin, Bull. Rus. Acad. Sci. Phys. 73, 1551 (2009); Phys. At. Nucl. 73 (2010) (in press). C. Mahaux, S. Sartor, Adv. Nucl. Phys. 20, 1 (1991). M.G. Urin, “Relaxation of nuclear excitations”, Energoatomizdat, Moscow, 1991 (in Russian); S.E. Muraviev, M.G. Urin, Particles and Nuclei, 22, 882 (1991). E.A. Romanovskii et al., Phys. At. Nucl. 63, 399 (2000); O.V. Bespalova et al., Phys. At. Nucl. 69, 796 (2006). G.A. Chekomazov, M.H. Urin, Phys. Lett. B 349, 400 (1995); Phys. At. Nucl. 61, 375 (1998). A.B. Migdal, “Theory of finite Fermi-sistems and applications to atomic nuclei”, Interscience, New York, 1967. M.L. Gorelik and M.H. Urin, Phys. Rev. C 63, 064312 (2001).
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abstract: 'Observations of the Ly$\alpha$ forest at $z \sim 3$ reveal an average metallicity ${\rm Z} \sim 10^{-2} Z_{\odot}$. The high-redshift supernovae that polluted the IGM also accelerated relativistic electrons. Since the energy density of the CMB $\propto (1+z)^{4}$, at high redshift these electrons cool via inverse Compton scattering. Thus, the first star clusters emit X-rays. Unlike stellar UV ionizing photons, these X-rays can escape easily from their host galaxies. This has a number of important physical consequences: (i) Due to their large mean free path, these X-rays can quickly establish a universal ionizing background and partially reionize the universe in a gradual, homogeneous fashion. If X-rays formed the dominant ionizing background, the universe would have more closely resembled a single-phase medium, rather than a two-phase medium. (ii) X-rays can reheat the universe to higher temperatures than possible with UV radiation. (iii) X-rays counter the tendency of UV radiation to photo-dissociate ${\rm H_{2}}$, an important coolant in the early universe, by promoting gas phase ${\rm H_{2}}$ formation. The X-ray production efficiency is calibrated to local observations of starburst galaxies, which imply that $\sim 10 \%$ of the supernova energy is converted to X-rays. While direct detection of sources in X-ray emission is difficult, the presence of relativistic electrons at high redshift and thus a minimal level of X-ray emission may be inferred by synchrotron emission observations with the Square Kilometer Array. These sources may constitute a significant fraction of the unresolved hard X-ray background, and can account for both the shape and amplitude of the gamma-ray background. This paper discusses the existence and observability of high-redshift X-ray sources, while a companion paper models the detailed reionization physics and chemistry.'
author:
- |
S. Peng Oh\
Princeton University Observatory, Princeton, NJ 08544; [email protected]
title: |
Reionization by Hard Photons: I.\
X-rays from the First Star Clusters
---
Introduction
============
While the theoretical literature on the epoch of reionization is large and increasing rapidly, our empirical knowledge of this period in the history of the universe is scant and may be succintly summarized: (i) The universe is likely to have been reionized in the period $5.8 < z_{r} < 35$, where the lower bound arises from the lack of Gunn-Peterson absorption in the spectra of high redshift quasars (Fan et al 2000), and the upper bound comes from the observation of small scale power in Cosmic Microwave Background anistropies (Griffiths et al 1999). In CDM cosmologies, non-linear objects above the cosmological Jeans mass $10^{4-5} M_{\odot}$ first collapse during this period. (ii) The presence of metals in the Ly$\alpha$ forest implies that significant star formation took place at high redshift (Songailia & Cowie 1996, Songailia 1997). (iii) COBE constraints on the Compton y-distortion of the CMB (Wright et al, 1994, Fixsen et al 1996) implies that the IGM was not heated to high temperatures. This means it is unlikely that the hydrogen and helium in the IGM were collisionally reionized.
Thus, the current state of observations is consistent with scenarios in which the universe was reionized by an early generation of stars or quasars. While we do not know whether stars or quasars were the dominant source of ionizing photons, the observational and theoretical case for quasars is somewhat more uncertain. Extrapolation of empirical quasar luminosity functions to high redshift do not yield enough ionizing photons to maintain the observed lack of Gunn-Peterson absorption at $z \sim 5$ (Madau, Haardt & Rees 1999). The requisite steepening of the faint end slope at high redshift necessary to boost ionizing photon production is constrained by the lack of red, point-like sources in the Hubble Deep Field (Haiman, Madau & Loeb 1999); the authors find that AGN formation must have been suppressed in halos with $v_{c} < 50-75 \,
{\rm km \, s^{-1}}$. We do not have a sufficiently firm understanding of the formation and fueling of supermassive black holes to assert on theoretical grounds that AGNs must have been present at high redshift. On the other hand, a minimal level of high-redshift star formation is guaranteed by the observed metal pollution of the IGM.
Theoretical scenarios in which stars or quasars figure predominantly have been calculated in detail. Our ignorance of the efficiency of gas fragmentation, and star/black hole formation as a function of halo mass, make the prediction of observable differences between these two scenarios very uncertain. Indeed, if one normalizes assumed emissivities to a fixed reionization epoch, differences between the two scenarios boil down to: (i) stars result in supernovae, which inject dust, metals, and entropy into the host galaxy and surrounding IGM, which affects subsequent chemistry and cooling, (ii) quasars have a significantly harder spectrum than stars. In particular, they produce X-rays.
In this paper, I emphasize a hitherto neglected fact: high redshift supernova also produce X-rays, both by thermal emission from the hot supernova remnant, and inverse Compton scattering of soft photons by relativistic electrons accelerated by the supernova. Considerable X-ray emission is already observed in starburst galaxies at low redshift (e.g., Rephaeli et al 1995), and the efficiency of most proposed X-ray production mechanisms should [*increase*]{} with redshift (e.g., explosions take place in a denser medium at high redshift, hardening expected thermal emission; inverse Compton scattering becomes more efficient since the CMB provides a ready supply of soft photons $U_{CMB} \propto (1+z)^{4}$). Thus, the SED of high-redshift star forming regions is considerably harder than has been previously assumed. This blurs the distinction between stellar/quasar reionization scenarios, and has a number of important physical consequences:
**Escape fraction**
The escape fraction of UV ionizing photons in the local universe is small, $\sim 3-6 \%$ (Leitherer et al 1995, Bland-Hawthorne & Maloney 1999, Dove, Shull & Ferrara 2000) and is expected to decrease with redshift (Wood & Loeb 1999, Ricotti & Shull 1999). On the other hand, X-rays can escape freely from the host galaxy. Thus, processing by the host ISM may imply that the universe was reionized by a significantly harder spectrum than previously assumed.
**Reionization topology**
Photons from stellar spectra have a short mean free path and thus a sharply defined ionization front. This fact gives rise to the conventional picture of expanding HII bubbles embedded in the neutral IGM. The spectra of quasars is significantly harder and exert an influence over a larger distance (which is why it is much more difficult to perform numerical simulations of reionization by quasars; see Gnedin 1999). Nonetheless, for quasars, $\nu L_{\nu} \propto \nu^{-0.8}$ (Zheng et al 1997, although note that the observations were only in the radio-quiet AGN subsample at energies up to 2.6 Ry) and most of the energy for ionization lies just above the Lyman edge. Thus, there is still a sharply defined HII region and a thin ionization front where the ionization fraction drops sharply. By contrast, for the inverse Compton case $\nu L_{\nu}
\sim {\rm const}$, there is equal power per logarithmic interval, and thus there is no preferred energy scale. In particular, there is no preferred scale for the mean free path of ionizing photons. When the universe is largely neutral, it is optically thick even to hard photons and all photons with energies $E < E_{thick}=1.5
{\left( \frac{1+z}{10} \right)}^{0.5} x_{HI}^{1/3} \, {\rm keV}$ (where $x_{HI}$ is the mean neutral fraction) are absorbed across a Hubble volume. While the (more numerous) soft photons can only travel a short distance before ionizing neutral HI and HeI, the (less numerous but more energetic) hard photons will be able to travel further and ionize an equivalent number of photons by secondary ionizations. Thus, even if sources are distributed very inhomogeneously, reionization will be a fairly homogeneous event, with a largely uniform ionizing background and fluctuations in ionization fraction determined mainly by gas clumping. Instead of an two-phase medium in whose HII filling fraction increases with time, the early IGM may have been a single phase medium whose ionization fraction increases with time.
**Increased reheating**
A hard spectrum can reheat the IGM to considerably higher temperatures than soft stellar spectra, both through photoionization heating and Compton heating. A soft spectrum loses thermal contact with the IGM once HI and HeI are completely ionized (at the mean IGM density, HI has a recombination time longer than the Hubble time for $z<10$), and the gas cools adiabatically due to the expansion of the universe (Hui & Gnedin 1997). By contrast, a hard spectrum can continually transfer large amounts of energy from the radiation field to the IGM by ionizing HeII, which recombines rapidly (Miralda-Escude & Rees 1994). This feedback mechanism is important in increasing the Jeans mass, a proposed mechanism for preventing excessive cooling and star formation at high redshift (e.g., Prunet & Blanchard 1999). The higher IGM temperatures may also explain why observed Ly$\alpha$ forest line widths are commonly in excess of that predicted by numerical simulations (Theuns et al 1999, Ricotti et al 2000).
**Early universe chemistry**
${\rm H_{2}}$ is an extremely important coolant in the metal-free early universe. While the neutral IGM is optically thick to UV ionizing photons, it is optically thin to photons longward of the Lyman limit (except at wavelengths corresponding to higher order hydrogen Lyman resonance lines, as well as $H_{2}$ resonance lines). In particular, photons in the 11.2-13.6 eV range quickly establish a soft UV background which photodissociates ${\rm H}_{2}$ via the Solomon process, shutting down subsequent star formation (Haiman, Rees & Loeb 1997, Cicardi, Ferrara & Abel 1998), unless the $H_{2}$ opacity is sufficient to reduce the photo-dissociation rate (Ricotti, Gnedin & Shull 2000). If X-rays are present in the early universe, they can counter this $H_{2}$ destruction. The IGM is also optically thin to X-rays, which can penetrate dense clouds of gas and promote gas phase ${\rm H_{2}}$ formation ${\rm H + e^- \rightarrow H^- + \gamma}$ and ${\rm H^- + H
\rightarrow H_2 + e^-}$ by increasing the abundance of free electrons. Haiman, Abel & Rees (1999) show that if quasars were the dominant ionizing sources in the early universe, gas cooling and thus star formation can continue unabated. In Paper II, I show that in fact even if only stars were present, a self-consistent treatment of the stellar SED incorporating X-rays produced by supernovae favours ${\rm H_{2}}$ formation over destruction in dense regions.
In this paper, I study the emission mechanisms and observational signatures of X-ray bright star clusters at high redshift. In Paper II (Oh 2000a), I address the changes in reionization topology, reheating and early universe chemistry mentioned above due to these X-rays.
In all numerical estimates, I assume a background cosmology given by the ’concordance’ values of Ostriker & Steinhardt (1995): $(\Omega_{m},\Omega_{\Lambda},\Omega_{b},h,\sigma_{8
h^{-1}},n)=(0.35,0.65,0.04,0.65,0.87,0.96)$. This corresponds to $\Omega_{b}
h^{2} =0.017$, compared with $\Omega_{b} h^{2}=0.020 \pm 0.002 \, (95
\% {\rm c.l.})$ (Burles, Nollett & Turner 2000), and $\Omega_{b}
h^{2}=0.0205 \pm 0.0018$ (O’Meara et al 2000) from Big Bang Nucleosynthesis, and the significantly higher values $0.022 < \Omega_{b}
h^{2} < 0.040 (95 \% {\rm c.l.})$ (Tegmark & Zaldarriaga 2000) preferred by recent CMB anisotropy data, such as Boomerang and Maxima.
Emission mechanisms
===================
Star formation at $z > 3$
-------------------------
What fraction of present day stars formed at high redshift? Estimates of the comoving star formation rate as a function of redshift (Madau et al, 1996) should be regarded as lower bounds, particularly at high redshift, due to the unknown effects of dust extinction, and star formation in faint systems below the survey detection threshold. Indeed, Lyman break survey results for $3.6< z <4.5$ (Steidel et al 1999) suggest that after correction for dust extinction, the comoving star formation rate for $z>1$ is constant, rather than falling sharply as previously believed. Furthermore, in recent years compelling evidence has emerged that the majority of stars in ellipticals and bulges formed at high redshift, $z > 3$. This comes from the tightness of correlations between various global properties of ellipticals which indicate a very small age dispersion and thus a high redshift of formation, unless their formation was synchronized to an implausible degree. The evidence includes the tightness of the fundamental plane and color magnitude relations for ellipticals, and the modest shift in zero-points for these relations with redshift (Renzini 1998 and references therein). Since spheroids contain $\sim 30\%$ of all stars in the local universe (King & Ellis 1985, Schechter & Dressler 1987), this would imply that $\sim 30\%$ of all stars have formed at $z>3$. Since $\sim 20 \%$ of baryons have been processed into stars by the present day (Fukugita, Hogan & Peebles 1998), this implies that $\sim 0.2 \times 0.3 \sim 6 \%$ of baryons have been processed into stars by $z \sim 3$. Assuming widespread and uniform enrichment and 1 ${\rm M_{\odot}}$ of metals per 100 ${\rm M_{\odot}}$ of stars formed, this translates into an IGM metallicity of $6 \times 10^{-4} \sim 3
\times 10^{-2} Z_{\odot}$. At $z \sim 3$, the metallicity of damped Ly$\alpha$ systems appears to be $\sim 0.05 Z_{\odot}$ (Pettini et al 1997), in reasonable agreement. The observed metallicity of the Ly$\alpha$ forest at $z \sim 3$, to which most models of reionization have been normalized, is between $10^{-2} Z_{\odot}$ and $10^{-3} Z_{\odot}$ (Songailia & Cowie 1996, Songailia 1997), which would imply that only $0.2-2 \%$ of present day stars formed at $z > 3$. However, its metallicity may be more representative of low density regions, rather than the mean cosmological metallicity (Cen & Ostriker 1999). Note that normalization of high redshift star formation to Lyman $\alpha$ forest metallicities assume efficient metal ejection (which underestimates star formation if a significant fraction of metals are retained) and mixing (which overestimates star formation if Ly$\alpha$ lines are preferentially observed in overdense regions which are sites of star formation). In this context, it is worth mentioning claims that Ly$\alpha$ lines with $10^{13.5} {\rm cm^{-2}} <
{\rm N_{HI}} < 10^{14.5} \, {\rm cm^{-2}}$ reveal lower metallicities by a factor of 10 than clouds with ${\rm N_{HI}} > 10^{14.5} \, {\rm cm^{-2}}$ (Lu et al 1999). Ellison et al (2000) find no break in the power law column density distribution for C IV down to log N(C IV) =11.7, and Schaye et al (2000) detect O VI down to $\tau_{HI} \sim 10^{-1}$ in underdense gas, so it appears that metal pollution was fairly widespread. I regard ${\rm Z} \sim 10^{-3} -2.5 \times 10^{-2} Z_{\odot}$ at $z \sim 3$ as a fairly firm bracket on the range of possibilities. For inverse Compton radiation, this corresponds to an energy release per IGM baryon of $\epsilon^{SN} \sim 10
\left( \frac{Z}{10^{-2} Z_{\odot}} \right) \left( \frac{\epsilon}{0.1}
\right)$eV (where $\epsilon$ is the efficiency of conversion of supernova energy to X-rays), which is comparable to the energy release in stellar UV radiation for the low escape fractions expected, $\epsilon^{stellar} \sim 10
\left( \frac{Z}{10^{-2} Z_{\odot}} \right) \left( \frac{f_{esc}}{0.01}
\right)$eV, where $f_{esc}$ is the escape fraction of ionizing photons from the source.
X-ray emission in local starbursts
----------------------------------
Most models of reionization use population synthesis codes to estimate the spectral energy distribution of starbursts. However, there are many processes associated with star formation that generate UV and X-rays, beside stellar radiation: massive X-ray binaries, thermal emission from supernova remnants and hot gas in galactic halos and winds, inverse Compton scattering of soft photons by relativistic electrons produced in supernovae. Indeed, X-ray emission appears to be ubiquitous among starbursts (e.g., Rephaeli, Gruber, & Persic 1995), and starburst galaxies may account for a significant portion of the XRB (Bookbinder et al 1980, Rephaeli et al 1991, Moran, Lehnert & Helfand 1999). The X-ray emission from these processes, which hardens the spectrum of starbursts and changes both the topology and chemistry of reionization, has to date been neglected in studies of the $z > 5 $ universe.
The X-ray luminosity of starbursts correlates well with other star formation indicators; for example, David et al (1992) find a roughly linear relation between $L_{FIR}$ and $L_{X}$. As a very rough empirical calibration, the starburst galaxies M82 & NGC 3256 observed with ROSAT and ASCA (Moran & Lehnert 1997, Moran, Lehnert & Helfand, 1999) follow the relations (after correction for absorption): $L_{X, 0.2-10 \, {\rm keV}}=8
\times 10^{-4} L_{IR}$, and $L_{X, 5 \, {\rm keV}} = 1.2 \times 10^{4} L_{R, 5 \, {\rm GHz}}$ (note that 5 keV flux density is relatively unaffected by photoelectric absorption or soft thermal emission). The starburst model of Leitherer and Heckman (1995) yields $L_{bol} \sim L_{{\rm FIR}} \sim 1.5
\times 10^{10} ({\rm SFR}/ 1 \, {\rm M_{\odot} yr^{-1}}) L_{\odot}$; as a cross-check, the empirical relation for radio emission is (Condon 1992) $L_{R} = 1.4 \times 10^{28} (\nu/{\rm GHz})^{-\alpha} ({\rm SFR}/{\rm M_{\odot}
yr^{-1}}) \, {\rm erg \, s^{-1} \, Hz^{-1}}$, where $\alpha \sim 0.8$. Together I obtain: $$L_{X}= 5 \times 10^{40} \left( \frac{SFR}{1 \, {\rm M_{\odot} yr^{-1}}}
\right) \ \ {\rm erg \, s^{-1}}
\label{Xray_lum}$$ One should not regard this as more than a rough order of magnitude estimate; a large scatter is expected in this relation. By way of comparison, Rephaeli et al (1995) obtain from the mean of 51 starbursts observed with Einstein and HEAO, $L_{X,2-30 \, {\rm keV}} \sim 8
\times 10^{-3} L_{IR}$, an order of magnitude greater; and David et al (1992) obtain from a sample of 71 normal and starburst galaxies a ratio lower by about an order of magnitude, largely due to the inclusion of normal galaxies (this is consistent with an inverse Compton origin for X-rays, since normal galaxies have much lower radiation field energy densities and would not be expected to show significant inverse Compton emission).
The typical observed X-ray spectrum is a power-law, $L_{\nu}
\propto \nu^{-0.8}$, consistent with a non-thermal origin. An obscured AGN is not likely to be the source of these X-rays, as several observations suggest that the X-ray emission is powered primarily by massive stars. In NGC 3256, observations by ISO fail to detect high excitation emission lines (Rigopoulou et al 1996). In M82, the optical spectrum is HII-like (Kennicutt 1992), discrete nuclear radio sources are spatially resolved (Muxlow et al 1994), its nuclear X-ray emission is extended (Bregman et al 1995), and the expected broad H$\alpha$ emission is not detected (Moran & Lehnert 1997); the HEX continuum and Fe-K line emission of NGC 253 as observed by BeppoSAX is extended (Cappi et al 1999). Moran & Lehnart (1997) and Moran, Lehnart, & Helfand (1999) have modelled the X-ray emission of M82 and NGC 3256, and find inverse-Compton emission to be the most likely mechanism, rather than an obscured AGN or massive X-ray binaries. The ratio between the observed radio and X-ray fluxes (note that $L_{X}/L_{syn} \propto
U_{IR}/U_{B}$, where $U_{IR}$ is the energy density of the infra-red radiation field, and $U_{B}$ is the energy density of the magnetic field) is consistent with an inverse-Compton origin for the X-rays. Furthermore, the X-ray and radio emission have the same spectral slope, as is expected if both types of emission are non-thermal, arising from the same population of electrons. The X-ray luminosity is energetically consistent with a star formation origin. Assuming a Salpeter IMF and that each supernova explosion yields $10^{51}$ erg in kinetic energy yields an energy injection rate into the ISM: $$\dot{E}_{SN} \sim 3 \times 10^{40} {{\rm \left( \frac{SFR}{1 M_{\odot} \, yr^{-1}}
\right)}}{\left( \frac{\epsilon}{0.1} \right)}\, {\rm erg \,
s^{-1}}
\label{IC_lum}$$ where $\epsilon$ is the fraction of energy injected into relativistic electrons; consistency with the observed value, (\[Xray\_lum\]), implies that $\epsilon \sim 10\%$. Hereafter, I shall use the empirical equation (\[Xray\_lum\]) as a fiducial conversion between X-ray luminosity and star formation rate.
Is an acceleration efficiency of $\epsilon \sim 10\%$ reasonable? The acceleration mechanism for relativistic electrons is poorly understood. While first-order Fermi acceleration in shocks is widely accepted as the acceleration mechanism for cosmic rays (Blandford & Eichler 1987, Jones & Ellison 1991), the electron acceleration is thought to be more problematic, due to the smaller electron gyroradius (which leads to greater difficulties in bouncing an electron back and forth across a shock of finite thickness), and the difficulty of initially boosting the electron to relativistic speeds, where Fermi acceleration can operate (Levinson 1994). From measurements of cosmic rays energy density it is inferred that $\sim 10\%$ of the supernova kinetic energy, or $\sim
10^{50}$erg per explosion, is liberated as cosmic rays (Volk, Klein, & Wielebinski 1989), but the division between electrons and protons at the source is not known. Since the measured ratio of cosmic ray protons to electrons is $\sim 75$ (e.g., Gaisser 1990), it might well be that relativistic electrons only constitute $\epsilon \sim 10^{-3}$ of the supernova energy budget. Thus, the reader should be cautioned that $\epsilon \sim 0.1$, which corresponds roughly equal energy division between protons and electrons (e/p=1), may be an overly optimistic estimate of the energy injection into relativistic electrons. Theoretical models of shock acceleration, in which e/p is a free parameter, often set e/p$\sim 1-5
\%$ for consistency with cosmic-ray experiments (Ellison & Reynolds 1991, Ellison et al 2000). However, this is somewhat model-dependent: in models where electrons are injected directly from the thermal pool, $\sim 5 \%$ of the energy in the shock must go to non-thermal electrons in order to match gamma-ray observations (Bykov et al 2000). Furthermore, note that the observed cosmic-ray e/p ratio could equally well be the result of different transport processes and energy loss mechanisms for electrons and protons. In particular, cosmic ray electrons are subject to loss processes which operate on much longer timescales for cosmic ray protons (inverse Compton, synchrotron losses, etc); the cosmic ray flux at earth for electrons could arise from a much smaller effective volume than that for protons. At the source, the energy division between protons and electrons could range between 1 and 100. Perhaps the most reliable means of inferring the proton/electron energy division is by direct observations of supernova remnants. In modelling the observed production of gamma-rays in the supernova remnants IC 443 and $\gamma$ Cygni observed by the EGRET instrument on the Compton Gamma Ray Observatory, Gaisser, Protheroe & Stanev (1998) find that a proton to electron ratio of 3–5 gives the best fit to the observed spectra, implying $\epsilon \sim 0.02-0.03$. Similarly, in modelling the $\gamma$-ray flux from 2EG J1857+0118 associated with supernova remnant W44, de Jager & Mastichiadis (1997) find $\epsilon
\sim 0.09$. They speculate that electron injection by the pulsar may be responsible for the increased electron energy content. Given the large uncertainties, henceforth I shall simply use the empirical relation (\[Xray\_lum\]).
Thus, barring non-standard IMFs, type II detonation energies or alternate sources of relativistic electrons, the empirical relation (\[Xray\_lum\]) implies an acceleration efficiency $\epsilon \sim 0.1$ which is plausible but certainly lies at the upper limit of theoretical expectations. Another possibility is that the X-ray emission cannot be wholly attributed to inverse Compton emission alone (this assumption rests on the arguments of Moran & Lehnart (1997) and Moran, Lehnart, & Helfand (1999) with regards to the slopes and relative intensities of the observed non-thermal radio and X-ray emission). The X-ray emission may instead be due to X-ray binaries, thermal emission from supernova remnants or starburst driven superwinds (e.g., see Natarajan & Almaini 2000). It should be noted that only soft X-rays are relevant for reionization, since the universe is optically thin to photons with energies $E > E_{thick}=1.5 {\left( \frac{1+z}{10} \right)}^{0.5} x_{HI}^{1/3} \, {\rm keV}$ (where $x_{HI}$ is the mean neutral fraction of the IGM). Since equation (\[Xray\_lum\]) is calibrated with the soft bands observed by ROSAT, this implies that even if $\epsilon \ll 0.1$ and the observed X-rays are not predominantly due to inverse Compton emission, the importance of X-rays for reionization (in particular, for changing the topology, for increased reheating, and increased $H_{2}$ production) may still hold. However, observational predictions which focus specifically on the inverse Compton mechanism (e.g., the gamma-ray background (section (\[xray\_obs\])), and detecting synchrotron emission with the SKA (section (\[radio\_obs\])) will no longer be valid. Since the acceleration efficiency is the most uncertain parameter in this paper, wherever relevant I insert the scaling factor ${\left( \frac{\epsilon}{0.1} \right)}$ into numerical estimates.
How does the X-ray luminosity compare with stellar UV ionizing radiation? Assuming a Salpeter IMF with solar metallicity, the Bruzual & Charlot (1999) population synthesis code yields an energy output of $L_{ion}= 3.2 \times 10^{42} ({\rm SFR/1
{\rm M_{\odot} \, yr^{-1}})\, {\rm erg \, s^{-1}}} $ in ionizing photons, which translates into ${\rm \dot{N}_{ion}} = 10^{53} ({\rm SFR}/1
{\rm M_{\odot} \, yr^{-1}}) \, {\rm photons \, s^{-1}}$. However, note that most of these ionizing photons are absorbed locally with the ISM of the star cluster; the escape fraction of ionizing photons into the IGM is expected to be small. Leitherer et al (1995) have observed four starburst galaxies with the Hopkins Ultraviolet Telescope (HUT). Their analyis suggests an escape fraction of only $3\%$, based on a comparison between the observed Lyman continuum flux and theoretical spectral energy distributions. For our own Galaxy, Dove, Shull & Ferrara (2000) find an escape fraction for ionizing photons of $6\%$ and $3\%$ (for coeval and Gaussian star formation histories respectively) from OB associations in the Milky Way disk. Bland-Hawthorn & Maloney (1999) find an escape fraction of $6\%$ is necessary for consistency with the observed H$\alpha$ emission from the Magellanic stream and high velocity clouds. On the other hand, a recent composite spectrum of 29 Lyman break galaxies (LBGs) with redshifts $\langle z \rangle = 3.40 \pm 0.09$ shows significant detection of Lyman continuum flux (Steidel, Pettini & Adelberger 2000); for typical stellar synthesis models, the observed flux ratio L(1500)/L(900)=$4.6 \pm 1.0$ implies little or no photoelectric absorption. The fraction of 900 ${\rm \AA}$ photons which escape, $f_{esc} \sim 15 -20 \%$, is modulated almost entirely by dust absorption. Nonetheless, the authors themselves stress this result should be treated as preliminary; the result could be due to a large number of uncertainties or selection effects, among them the fact that these galaxies were selected from the bluest quartile of LBGs. On the theoretical side, radiative transfer calculations by Woods & Loeb (1999) find that the escape fraction at $z \sim 10$ is $< 1\%$ for stars; calculations by Ricotti & Shull (1999) find that the escape fraction decreases strongly with increasing redshift and halo mass; for a $10^{9} \, M_{\odot}$ halo at $z=9$, the escape fraction is $\sim 10^{-3}$ (note that in the Ricotti & Shull (1999) models, the escape fraction rises towards low masses, and can be considerable for the halos with $M < 10^{7} M_{\odot}$. Since such halos have $T_{vir} < 10^{4}$K, their contribution to reionization depends on whether $H_{2}$ formation and cooling can take place despite photodissociative processes (Haiman, Rees & Loeb 1997, Cicardi, Ferrara & Abel 1998, Ricotti, Gnedin & Shull 2000)). For low escape fractions, the energy release in UV photons is roughly comparable to that in inverse-Compton X-rays : $$L_{UV} = 3 \times 10^{40} \left( \frac{f_{esc}}{0.01} \right) \left
( \frac{\rm SFR}{1 {\rm M_{\odot} yr^{-1}}} \right) \, {\rm erg \, s^{-1}}
\label{lum_stars}$$ Note that to first order the ratio of stellar UV to inverse Compton X-rays is not sensitive to uncertainties in the IMF, as the same massive stars with $M > 20
M_{\odot}$ that dominate the Lyman continuum of a stellar population also explode as supernovae (however, see section (\[metal\_free\]) for some caveats).
Will X-ray emission still be efficient at high redshift? The following changes are expected to take place at high redshift: (i) The ISM is initially free of dust and metals, although rapid enrichment could occur on fairly short timescales ($t \sim
10^{6}-10^{7}$yr). (ii) The average ISM density is significantly higher, $n_{halo} \sim n_{0} 18 \pi^{2} (1+z)^{3} = 0.02
(\frac{1+z}{10})^{3} {\rm cm^{-3}}$ in a halo and $n_{disc} \sim \lambda^{-3} n_{halo} \sim 160
(\frac{1+z}{10})^{3} {\rm cm^{-3}} (\lambda/0.05)^{-3}$ in a disk (where $\lambda$ is the spin parameter). A supernova remnant at high $z$ expands into a denser ISM: since it spends a shorter time in the Taylor-Sedov phase, most of its energy is radiated at a smaller radius, where the effective temperature is higher. This implies a harder spectrum for thermal emission. In addition, the density and temperature of gas in star forming regions is determined by the properties of ${\rm H}_{2}$ cooling (which saturates at $n \sim 10^{4} \,
{\rm cm^{-3}}$ and $T \sim 300$K), rather than metal cooling as in the local universe. (iii) Potential wells are significantly shallower, so pressurised regions (hot gas, strong magnetic fields) cannot be efficiently confined. (iv) The CMB energy density $U_{CMB}
\propto (1+z)^{4}$, so inverse Compton radiation becomes particularly efficient at high redshift. Because of (iii) and (iv), relativistic electrons cool predominantly by inverse Compton scattering rather than synchrotron emission. Since inverse Compton emission is likely to be the most promising mechanism for X-ray emission, I shall consider it at length.
Inverse Compton emission
------------------------
The energy loss rate for a relativistic electron with Lorentz factor $\gamma$ is given by: $$\dot{{\rm E}_{IC}}= \frac{4}{3} \sigma_{T} c \gamma^{2} U_{rad} = 1.12
\times 10^{-16} {\left( \frac{1+z}{10} \right)}^{4} \left( \frac {\gamma}{10^{3}}
\right)^{2} \ {\rm erg \, s^{-1}}
\label{IC_rate}$$ for $U_{rad}= U_{\rm CMB}$. The loss rate by synchrotron radiation is given by substiting $U_{B}$ for $U_{rad}$, $\dot{{\rm E}_{synch}}= \frac{4}{3} \sigma_{T}
c \gamma^{2} U_{B}$. In the local universe, galaxies with relatively quiescent star formation emit most of their electron energy in synchrotron radiation. Starbursts in the local universe can radiate efficiently in IC, as the energy density in the local radiation field is sufficiently high (typically, $U_{r} \sim 10^{-8} \, {\rm erg \, cm^{-3}}$ as opposed to $U_{r} \sim 10^{-12} \, {\rm erg \, cm^{-3}}$ in our Galaxy). Seed photons are provided by IR emission from dust grains. By contrast, at high redshift, [*all*]{} star forming regions will emit in inverse Compton radiation, as the CMB provides a universal soft photon bath of high energy density, $U_{CMB} = 4 \times 10^{-9} (\frac{1+z}{10})^{4}\, {\rm erg \,
cm^{-3}}$. An electron with Lorentz factor $\gamma$ will boost a CMB photon of frequency $\nu_{o}$ to a frequency $\nu= \gamma^{2}
\nu_{o}$. Thus, the rest frame frequency of a CMB photon (at the peak of the blackbody spectrum) which undergoes inverse Compton scattering is: $${\rm E}_{IC}=600 \left( \frac{\gamma}{300} \right)^{2} \left( \frac{1+z}
{10} \right) \, {\rm eV}$$ Note that the observed frequency is independent of source redshift, since the higher initial frequency and redshifting effects cancel out.
Below, I examine in detail the mechanisms by which relativistic electrons lose energy, to see if indeed inverse Compton radiation will predominate.
### Energy loss processes for relativistic electrons
Once relativistic electrons are produced, they can cool via a variety of mechanisms. Let us examine them in turn (for more details see Pacholczyk 1970, Daly 1992).
[**Synchrotron radiation**]{} The relative emission rate in inverse Compton and synchrotron emission is given simply by the relative energy densities in the radiation and magnetic fields, $\dot{E}_{IC}/\dot{E}_{syn}= U_{rad}/U_{m}$. This yields: $$\frac{\dot{E}_{IC}}{\dot{E}_{syn}}= 1.1 \times 10^{3} \left( \frac{B}
{10 \mu G} \right)^{-2} \left( \frac{1+z}{10} \right)^{4}$$ Note that in energy loss terms the CMB may be characterized as having an effective magnetic field strength $B_{CMB,eff}=3.24 \times 10^{2}
\left( \frac{1+z} {10} \right)^{2} \ {\rm \mu G}$. Could magnetic fields in proto-galaxies possibly reach these high values? A reasonal assumption is that $P_{B} \sim P_{rel} < P_{gas}$, where $P_{B}$ is the magnetic field pressure, $P_{rel}$ is the pressure in relativistic particles, and $P_{gas}$ is the thermal gas pressure. Local observations of synchrotron and inverse Compton emission from radio galaxies are consistent with equipartition $P_{B} \sim P_{rel}$ (Kaneda et al 1995). I have assumed that the energy injection into relativistic particles is $\sim 10 \%$ of the total kinetic energy of a supernova, so $P_{rel} < P_{gas}$ should be a strict upper bound. Thus, $P_{B} < P_{gas}$ gives the upper bound: $$B < 6 \left( \frac{n}{1 {\rm cm^{-3}}} \right)^{1/2} \left( \frac{T}{10^{4}
K} \right) \, \mu G$$ where $n$ is the baryon number density (note that gas with temperatures $> 10^{4-5}$K will escape from the shallow potential wells of the first proto-galaxies). If the magnetic field exceeds the above value, the over-pressurised lobe will expand on the dynamical time scale until the magnetic pressure drops. Thus, for $z>5$, it seems likely that synchrotron energy losses will be unimportant.
[**Ionization & Cherenkov losses**]{} Interactions with the non-relativistic gas will result in energy losses via ionization and Cherenkov emission of plasma waves at a rate independent of the Lorentz factor, $\dot{E}_{ion} \approx 9 \times 10^{-19} n \, {\rm
erg \, s^{-1}}$, which implies that the relative energy loss rate is: $$\frac{\dot{E}_{IC}}{\dot{E}_{ion}} = 120 \left( \frac{n}{1 {\rm cm^{-3}}}
\right)^{-1} \left( \frac{1+z}{10} \right)^{4} \left( \frac
{\gamma}{10^{3}} \right)^{2}$$ Thus, for $$\gamma > \gamma_{break} \approx 100 \left( \frac{n}{1 {\rm cm}^{-3}}
\right)^{1/2} (\frac{1+z}{10})^{-2}
\label{gamma_break}$$ inverse Compton losses are more important than ionization losses. The cutoff Lorenz factor at the lower end $\gamma_{co}$, is of interest since the lower end accounts for most of the electrons, both in terms of number and energy: $N_{electrons} (> \gamma)
\propto \gamma^{-2 \alpha} = \gamma_{co}^{-1.6} \ $; $E_{electrons} (> \gamma)
\propto \gamma^{-2 \alpha+1} = \gamma_{co}^{-0.6} \ (\alpha=0.8) $. Note that the electrons that cool via ionization losses do not drop out completely, but merely form a flattened distribution with $\gamma_{f}
\approx \gamma_{i} -350 \, n ({\rm t/10^{7} yr})$. These electrons with lower Lorentz factors could scatter CMB photons to optical and UV frequencies.
[**Free-free radiation**]{} Free-free radiation results in an energy loss rate $\dot{E}_{free-free} = 6 \times 10^{-22} n \gamma \,
{\rm erg \, s^{-1}}$. Thus, free-free radiation only dominates over ionization and Cherenkov radation for $\gamma > 1500$. However, in this regime inverse Compton losses dominate, since $$\frac{\dot{E}_{IC}}{\dot{E}_{ff}}= 190 \left( \frac{n}{1 cm^{-3}}
\right)^{-1} \left( \frac{1+z}{10} \right)^{4} \left( \frac
{\gamma}{10^{3}} \right)$$ Thus, free-free emission is never important in cooling relativistic electrons at high redshift.
In summary, the inverse Compton emission is the dominant energy loss mechanism between a lower and upper energy cutoff. From equation (\[gamma\_break\]), the lower frequency break is determined by the competition between the inverse Compton loss rate and ionization and atomic cooling losses, below $$E_{lower} = \gamma_{break}^{2} h \nu_{{\rm CMB}}= 70 {\left( {\rm \frac{{\rm n}}{1 \, {\rm cm^{-3}}}}
\right)}\left(
\frac{1+z}{10} \right)^{-3} \, {\rm eV},$$ where I assume the seed photon $\nu_{{\rm CMB}}=1.6 \times 10^{12} {\left( \frac{1+z}{10} \right)}$GHz lies at the peak of the CMB blackbody spectrum. The competition between inverse Compton cooling losses and the rate of energy injection by Fermi acceleration determines the upper energy cutoff. The timescale for losses by inverse Compton radiation is: $$t_{life}= \frac{E}{\dot{E}} = 7.9 \times 10^{5} \left( \frac{1+z}{10}
\right)^{-4} \left( \frac{\bar{\gamma}}{300} \right)^{-1} \ {\rm
years}
\label{lifetime}$$ Equating the Fermi acceleration timescale $t_{acc} \sim r_{L}c/v_{sh}^{2}= 1.3
\times 10^{-3} \left( \frac{\gamma}{300} \right) \left( \frac{B}{10
\mu G} \right)^{-1} \left( \frac{v_{sh}}{2000 {\rm km s^{-1}}}
\right)^{-2}$yr (where $r_{L}$ is the Larmour gyroradius, and $v_{sh}$ is the typical shock velocity) to $t_{life}$ (equation (\[lifetime\]), I obtain for the maximum Lorentz factor $\gamma_{max}= 7.4 \times 10^{6} {{\rm \left( \frac{B}{10 \mu G} \right) }}^{1/2} \left(
\frac{v_{sh}}{2000 \, {\rm km \, s^{-1}}} \right) {\left( \frac{1+z}{10} \right)}^{-2}$ which corresponds to an upper energy cutoff: $$E_{upper} \sim 360 {{\rm \left( \frac{B}{10 \mu G} \right) }}\left(
\frac{v_{sh}}{2000 {\rm km \, s^{-1}}} \right)^{2} {\left( \frac{1+z}{10} \right)}^{-3} \, {\rm
GeV}.
\label{Emax_eqn}$$ In section (\[IC\_spectrum\_section\]), I derive the form of the spectrum. For the power law spectrum obtained, $L_{\nu} \propto
\nu^{-1}$, the specific luminosity depends only logarithmically on the energy cutoffs: $L_{\nu} = \frac{L_{tot}}{{\rm log
(\nu_{upper}/\nu_{lower}})} \nu^{-1}$.
### Inverse Compton spectrum {#IC_spectrum_section}
The Fermi shock acceleration mechanism for cosmic rays involves a steady growth of particle energy as a particle scatters back and forth across the shock front. This naturally produces a power law electron energy spectrum with energy spectrum $dn/d\gamma \propto \gamma^{-p}$, where $p=(\chi+2)/(\chi-1)$ and $\chi$ is the compression ratio for the shock (e.g., Jones & Ellison 1991). The final emission spectrum $L_{\nu} \propto
\nu^{-\alpha}$ depends on the electron energy spectrum; the synchrotron or inverse Compton spectral index is $\alpha=(p-1)/2$. Thus, the spectral index is determined by the shock structure rather than the details of the scattering process. For a strong adiabatic shock $\chi =4$ and $dn/d\gamma \propto \gamma^{-2}$; for an isothermal shock, $\chi \gg 1$ and $dn/d\gamma \propto
\gamma^{-1}$. The assumption of adiabaticity is most appropriate for the Taylor-Sedov phase, when most of the electrons are accelerated. Galactic SNR show a mean radio spectral index $\alpha= 0.5 \pm 0.15$ (Droge et al 1987), which agrees with $dn/d\gamma \propto \gamma^{-2}$. I shall use this as the canonical electron injection spectrum in this paper. The steepening of the diffuse synchrotron emission to $\alpha \sim 0.8$ is likely to be due to energy losses as the electrons age. I examine the emission spectrum in detail below. I assume that there is one supernova for every 100 $M_{\odot}$ of stars formed, that each supernova liberates $\sim 10^{51}$erg in kinetic energy, and $\sim 1 M_{\odot}$ of metals.
The equation for the evolution of the electron population is given by (Ginzburg & Syrovatskii 1964, Sarazin 1999): $$\frac{ \partial N(\gamma)}{\partial t} = \frac{\partial}{\partial
\gamma} [b(\gamma)N(\gamma)] + Q(\gamma)
\label{electron_pop_eqn}$$ where $N(\gamma)$ is the number of electrons in the range $\gamma$ to $\gamma + d\gamma$, the rate of production of new relavistic electrons is given by $Q(\gamma)$, and the rate of energy loss of an individual particle is given by $b(\gamma) \equiv d\gamma/dt$. From equation (\[lifetime\]), we see that the electron lifetime is shorter than both the Hubble time for the redshifts under consideration $z < 30$ and typical timescales for starbursts ($\sim 10^{7}$ years). We can thus assume that the electron population quickly reaches a steady state where the energy losses due to inverse Compton scattering (for high $\gamma$) and Coulomb collisions (for low $\gamma$) balance the injection rate due to supernova explosions, and set the time derivative to zero. Since injected electrons survive for less than a Hubble time, I ignore evolution in loss rate due to the evolution of the CMB energy density. Assuming that each supernova injects a population of electrons with $Q(\gamma) = Q_{o} \gamma^{-p}$, where $p \sim 2$, then the steady state solution to equation (\[electron\_pop\_eqn\]) is given by: $$\begin{aligned}
N(\gamma) &=& 4 \times 10^{61} {\left( \frac{1+z}{10} \right)}^{-4} {{\rm \left( \frac{SFR}{1 M_{\odot} \, yr^{-1}}
\right)}}\gamma^{-(p+1)} \ \ \ ; \gamma >
\gamma_{break}
\label{electron_pop_soln} \\
N(\gamma) &=& 4 \times 10^{57} {\left( {\rm \frac{{\rm n}}{1 \, {\rm cm^{-3}}}}
\right)}^{-1} {{\rm \left( \frac{SFR}{1 M_{\odot} \, yr^{-1}}
\right)}}\gamma^{-(p-1)} \ \ \ ; \gamma<
\gamma_{break}
\nonumber \end{aligned}$$ where $\gamma_{break}$ (equation (\[gamma\_break\])) is the transition energy between the regimes where ionization and inverse Compton losses dominate. Thus, the electron population flattens by one power at low energies and steepens by one power at high energies, as compared with the distribution function for the injected population. Since $\alpha=-(p-1)/2$, the emitted spectrum flattens by a half power and steepens by a half power at low and high energies respectively. In particular, for $p=2$, as is appropriate for adiabatic shocks, $L_{\nu} \propto {\rm const}$ at low energies and $L_{\nu} \propto \nu^{-1}$ at high energies.
The spectrum of inverse Compton radiation is given by (Sarazin 1999): $$L_{\nu} = 12 \pi \sigma_{T} \int_{1}^{\infty} N(\gamma) d \gamma
\int_{0}^{1} J(\frac{\nu}{4 \gamma^{2} x}) F(x) dx
\label{IC_spectrum}$$ where $N(\gamma)$ is the electron energy spectrum, $J(\nu)=
B_{\nu}(T_{CMB,o}(1+z))$ is the CMB blackbody spectrum, and $F(x)=1+x+2x {\rm ln}(x) -2 x^{2}$. This differs only by a normalization correction from assuming that all photons reside at the peak of the blackbody spectrum, $\nu = 1.6 \times 10^{11} (1+z)$Hz. This is because the seed photon spectrum is narrow compared to the electron energy distribution. In figure (\[spectrum\]), I show example spectra computed with equations (\[electron\_pop\_soln\]) and (\[IC\_spectrum\]). Note that the efficiency of Coulomb cooling depends on the assumed electron density; I assume that $n_{e} \sim
\delta n_{o} (1+z)^{3}$, where I assume the overdensity to be either $\delta \sim 200$ (to mimic the overdensity at the virial radius) or $\delta \sim 10^{4}$ (to mimic the overdensity for collapsed gas in dense star forming regions). At $z\sim 10$, this assumption corresponds to $n_{e} \sim 4 \times 10^{-2} \, {\rm cm^{-3}}$ and $n_{e} \sim 2 \ {\rm cm^{-3}}$ respectively. Also plotted for reference is the spectrum of the same starburst with a Salpeter IMF, assuming an escape fraction of $\sim 1 \%$, and the spectrum of a mini-quasar with spectrum $L_{\nu} \propto
\nu^{-1.8}$, normalized to have the same energy release above the Lyman limit as the starburst. Note that the inverse Compton case has the hardest spectrum of all. Since the spectrum is so hard, photoelectric absorption by the host galaxy does not significantly attenuate the ionizing flux. Most UV photons produced will not escape the host galaxy, whereas X-ray photons with $E > 270 \left( \frac {N_{HI}}{10^{21} cm^{-2}} \right) \,
{\rm eV} $ where $N_{HI}$ is the column density in the host galaxy, will escape unimpeded. Since $\nu L_{\nu} \sim {\rm const}$, most of the energy in ionizing photons escape. Note, incidentally, that the escape fraction for UV photons produced by inverse Compton may be significantly higher than photons of the same frequencies produced by stars, as relativistic electrons can disperse from the star forming region, where gas densities are highest and most photon captures take place.
Zero-metallicity star formation {#metal_free}
-------------------------------
The stellar IMF at high redshift under conditions of low or zero metallicity is unknown. Up to now, I have assumed a Salpeter IMF, in which one supernova expodes for every $\sim 100 \, M_{\odot}$ of stars formed, and each supernova deposits on average $\sim 10^{50} {\left( \frac{\epsilon}{0.1} \right)}$erg in relativistic electrons and $\sim 1 \,
M_{\odot}$ of metals. Since the estimated amount of star formation at high redshift is calibrated to the observed IGM metallicity at $ z
\sim 3$, it is worth asking whether the energy injected into X-rays per solar mass of metals produced, $\epsilon_{Z} \sim 10^{50} \, {\rm erg \,
M_{\odot}^{-1}}$, could change significantly at high redshift. I have argued that variations in the IMF will not effect significant changes in $\epsilon_{Z}$ or the ratio of energy emitted in stellar UV ionizing photons to inverse-Compton X-rays, since the same massive OB stars which produce UV ionizing photons also explode as supernova, producing both relativistic electrons and metals.
However, zero-metallicity star formation may result in very different stellar populations from that seen at low redshift. The lack of efficient cooling mechanisms could result in extremely top heavy stellar IMFs (Larson 1998, Larson 1999) and in particular the production of “Very Massive Objects” (VMOs) in the range $10^{2}-10^{5} \, M_{\odot}$ (Carr, Bond & Arnett 1984). Such stars have high effective temperatures and produce a much harder spectrum than ordinary stars (Tumlinson & Shull, 2000, Bromm et al, 2000). Moreover, the distribution of stellar endpoints is very different from a normal IMF (Heger, Woosley & Waters 2000). Stars with masses between 10 and 35 $M_{\odot}$ explode as type II supernovae. While it is not known whether metal free stars with masses between 35 and 100 $M_{\odot}$ will explode–they might collapse to form black holes–stars with masses $\sim 100-250 M_{\odot}$ are disrupted by the pair production instability, once again producing an energetic supernova event and dispersing metals. Stars more massive than $250 M_{\odot}$ should collapse completely to black holes, without ejecting any metals (unless they eject their envelopes during hydrogen shell burning, in which case it is possible for them to explode). The latter provides an obvious mechanism for seeding supermassive black holes to form AGN. A number of studies of zero-metallicity star formation suggest that the Jeans/Bonner-Ebert mass and hence the lower mass cutoff for star formation is very high, $M_{*} > 10^{2}-10^{3} \, M_{\odot}$ (Abel, Bryan & Norman 1999, Padoan, Nordlund & Jones 1997).
There are three main observations to make: (i) VMOs with initial stellar masses $100 M_{\odot} < M < 250
M_{\odot}$ which are disrupted by the pair-production stability show $\epsilon_{Z}(PP) \approx
\epsilon_{Z}({\rm type \ II})$, which imply that even if these objects are abundant in the early universe, our estimates of the level of inverse Compton X-ray emission do not change strongly. In particular, $E_{\rm explosion}= 6.3 \times 10^{52} \left( \frac{M}{10^{2} M_{\odot}}
\right)^{2.8} \frac{{\rm min}[0.13,(1-\phi_{L})^{2.8}]}{0.13} \, {\rm
erg}$ while the yield in elements heavier than helium is $Z_{\rm ej} =
{\rm min}\left[ (1- \phi_{L}),0.5 \right]$, where $\phi_{L}$ is the fraction of the initial mass lost during hydrogen burning (Carr, Bond & Arnett, 1984). Thus, $\epsilon_{Z}(PP)= 1.3 \times 10^{50} \left( \frac{M}{10^{2} M_{\odot}}
\right)^{1.8} \frac{{\rm min}[0.25,(1-\phi_{L})^{1.8}]}{0.25} \, {\rm
erg \ M_{\odot}^{-1}} \approx \epsilon_{Z}({\rm type \ II})$. (ii) Stars which directly collapse to form black holes represent an additional, unaccounted source of ionizing photons, since they are not included in the metal pollution budget (of course, their most important contribution could lie in seeding AGN formation). (iii) Zero metallicity is a singularity: true zero-metallicity stellar populations differ greatly from low metallicity ones. Even the introduction of trace amounts of metals $Z \sim 10^{-4} Z_{\odot}$ introduces important changes in stellar structure and evolution (Heger, Woosley & Waters 2000). Furthermore, trace amounts of metals drastically reduces the abundance of VMOs by allowing efficient cooling past the 300K barrier imposed by ${\rm H}_{2}$ cooling, reducing the Jeans mass and thus the minimum mass for star formation. Prompt initial enrichment is quite plausible: the lifetime of massive stars before they explode to pollute the IGM with metals is $\sim 10^{6}$ years, which is a short fraction of the Hubble time $t_{H} \sim 8 \times 10^{8} (\frac{1+z}{10})^{-3/2}$yrs even at high redshift. Although the degree of metal mixing is uncertain (e.g. Gnedin & Ostriker 1997, Nath & Trentham 1997, Ferrara, Pettini & Shchekinov 2000), note that first star forming regions are very highly biased, and subsequent generations of the halo hierarchy collapse in proto-clusters and filaments very close to the first star clusters. Thus, stars of finite metallicity could quickly predominate, even if the mean metallicity of the universe is close to primordial (Cen & Ostriker 1999). It is therefore possible and even likely that true zero metallicity star formation was confined to a very small fraction of the stars formed at high redshift, and thus negligible in terms of the energy budget for reionization.
Observational signatures
========================
In this section, to estimate number counts I use the Press-Schechter based high redshift star formation models of Haiman & Loeb (1997), which are normalised to the observed metallicity of the z=3 IGM, ${\rm Z = 10^{-3}-10^{-2}
Z_{\odot}}$. In these models, in every halo capable of atomic cooling (i.e., with a virial temperature $T_{vir} > 10^{4}$K), a fixed fraction of the gas $f_{star}=1.7, 17 \%$ (for ${\rm Z}(z=3)= 10^{-3}, 10^{-2} {\rm Z_{\odot}}$ respectively) fragments in a starburst lasting $t_{o} \sim
10^{7}$yrs. The correspondence between halo mass and star formation rate is thus ${\rm SFR} \approx 2 \left( {\rm \frac{M_{halo}} {10^{9}
\, M_{\odot}} } \right) \left( \frac{f_{star}}{0.17} \right) \
{\rm M_{\odot} \, yr^{-1}}$, and each halo is only visible for some fraction of the time $\frac{t_{o}}{t_{H}(z)}$.
HeII recombination lines
------------------------
One possible signature of the hard spectrum produced by inverse Compton X-rays would be HeII recombination lines from the host galaxy. Indeed, such lines may well be detectable from the first luminous objects with sufficiently hard spectra, such as mini-quasars or metal-free stars (Oh, Haiman & Rees 2000). The different sources may perhaps be distinguished on the basis of line widths and line ratios (Tumlinson, Giroux & Shull, 2000). However, inverse Compton emission produces too few HeII ionizing photons for such recombination lines to be detectable with NGST at high redshift. For a Salpeter IMF, $\dot{N}_{ion,HI}
= 10^{53} {{\rm \left( \frac{SFR}{1 M_{\odot} \, yr^{-1}}
\right)}}\ {\rm photons \, s^{-1}}$ from stellar UV radiation. On the other hand, since secondary ionizations of HeII are negligible (Shull & van Steenberg 1985), the production rate of HeII ionizing photons from non-thermal emission is $\dot{N}_{ion, HeII}= \int_{4 \nu_{L}}^{\nu_{thin}} \frac{L_{\nu}}{h \nu} \approx
2.5 \times 10^{49} \left( \frac{SFR}{ 1 \, M_{\odot} \, yr^{-1}} \right) \ {\rm photon
\, s^{-1}}$, where $\nu_{thin}$ is the frequency at which the halo becomes optically thin. Thus, $Q \equiv \dot{N}^{\rm HeII}_{\rm ion}/\dot{N}^{\rm
HI}_{\rm ion} \approx 2.5 \times 10^{-4}$, as compared with $Q \approx
0.05$ for stellar emission from metal-free stellar population, and $Q
\approx 4^{-\alpha} = 0.08-0.25$ for a QSO, where $L_{\nu} \propto
\nu^{-\alpha}$ and $\alpha = 1-1.8$, implying that the relative flux in HeII and H$\alpha,\beta$ recombination lines is much smaller for inverse Compton emission than metal-free stars or AGN. The luminosity in a helium recombination line $i$ may be estimated as $L_{\rm i} = Q f_{\rm i} L_{\rm
H\alpha}$, where $L_{\rm H\alpha}$ is the Balmer $\alpha$ line luminosity, $f_{\rm i}\equiv \frac{j_{i}}{j_{H \alpha}} \left
( \frac{\alpha_{B}(HI) n_{HII}}{\alpha_{B}(HeII) n_{HeIII}} \right)$, $\epsilon$ is the fraction of SN energy which emerges as IC emission, and $j_{i},j_{H \alpha}$ may be obtained from Seaton (1978). The observed flux is $J_{i}= \frac{L_{\rm i}}{4 \pi d_{\rm L}^{2}}
\frac{1}{\delta \nu} = 0.04 \left( \frac{q_{\rm i}}{0.5} \right) \left(\frac{1+z}{10} \right)^{-1} \left(\frac{\epsilon}{0.1} \right) \left(
\frac{R} {1000} \right) \left( \frac{{\rm SFR}}{1 {\rm M_{\odot} yr^{-1}}}
\right) \, {\rm nJy} $ where ${\rm R}$ is the spectral resolution, and $q_{\rm i} \equiv
f_{\rm i} \nu_{\rm H\alpha}/ \nu_{\rm i}$, where $f_{\lambda
4686}=0.74, f_{\lambda 1640}=4.7, f_{\lambda 3203}=0.30$. This is undetectable by NGST, which in $10^{5}$s integration time requires for a 10 $\sigma$ detection a flux $F(10\sigma) \sim 30$nJy at these frequencies (observed wavelengths $1 < \lambda < 5.5 \mu$m) and spectral resolution $R \sim 1000$ (see Oh, Haiman & Rees 2000 for details).
It is worth mentioning that a hard source may produce relatively little recombination line flux and yet play an important role in reionizing the universe. The recombination line flux from a source is $\propto
(1-f_{esc})$, whereas the ionizing flux escaping into the IGM is $\propto
f_{esc}$ (where $f_{esc}$ is significantly higher for hard sources: all photons with $E > E_{halo, thin} = 270
\left(\frac{N_{HI}}{10^{21} {\rm cm}^{-2}} \right)$eV can escape freely from the host). Secondary ionizations are unimportant within a host halo: much of the gas in fully ionized, in which case the energetic photoelectron deposits its energy as heat. In any case, the halo is optically thin to the most energetic photons $E > E_{halo,thin}$ which are important for secondary ionizations. Thus, in recombination line flux what matters is the total [*number*]{} of ionizing photons produced, whereas for the reionization of the IGM what matters is the total output [*energy*]{} (since in a largely neutral medium with $x_{e} < 0.1$ the total number of ionizations for a photon of energy $E_{photon}$, including secondary ionizations, is $N_{ion} \sim E_{photon}/37 {\rm eV}$ (Shull & van Steenberg 1985). Note, however, that as the medium becomes more ionised an increasing fraction of the energy is deposited as heat, and an additional source of soft photons is needed for reionization to proceed (Oh 2000a)). Thus, for instance, for $f_{esc} \sim 1 \%$, inverse Compton radiation makes a negligible contribution to the H$\alpha$ luminosity, $\frac{L_{H \alpha}^{IC}}{L_{H \alpha}^{stellar}} \approx
\frac{\dot{N}_{ion}^{IC}}{\dot{N}_{ion}^{stellar}} = 2.5 \times
10^{-4}$, but the ionizing IC and stellar radiation escaping from the host are energetically comparable (from equations (\[Xray\_lum\]) and (\[lum\_stars\]), $L_{X}^{IC} \approx
L_{UV}^{stellar}$), and thus they produce roughly equal number of ionizations in the IGM.
X-rays and gamma-rays {#xray_obs}
---------------------
Unfortunately, direct detection of inverse Compton X-rays from high-redshift star clusters is unlikely. From equation (\[Xray\_lum\]), the observed flux in X-rays from a star forming region is: $$f=\frac{L}{4 \pi d_{L}^{2}}= 5 \times 10^{-20}
{\left( \frac{\epsilon}{0.1} \right)}\left
( \frac{\rm SFR}{1 {\rm M_{\odot} yr^{-1}}} \right) \left( \frac{1+z}{10}
\right)^{-2} \, {\rm erg \, s^{-1} \, cm^{-2}}
\label{point_flux}$$ By contrast, the Chandra X-ray Observatory (CXO) sensitivity (see http://chandra.harvard.edu/) for a 5$\sigma$ detection of a point source in an integration time of $10^{5}$s in the 0.2–10 keV range is $F_{X} = 4 \times 10^{-15} \, {\rm erg \, s^{-1} \,
cm^{-2}}$. The next generation X-ray telescope, Constellation-X (http://constellation.gsfc.nasa.gov/) is optimised for X-ray spectroscopy and will not have significantly greater point source sensitivity: for a similar bandpass and integration time it will be able to detect objects out to a flux limit of $F_{X} = 2 \times
10^{-15} \, {\rm erg \, s^{-1} \, cm^{-2}}$. Thus, in a week CXO and Constellation-X will at best be able to detect starbursts (with seed photons provided by the IR radiation field from dusty star forming regions) with star formation rates SFR $\sim 100 \,
{\rm M_{\odot} yr^{-1}}$ (corresponding to $L_{X}\sim 4.8 \times 10^{42}
\, {\rm erg \, s^{-1}}$) out to redshift $z \sim 1$ (note that the Lyman-break galaxies detected at $z\sim 3$ (Steidel et al 1996) typically have have inferred star formation rates ${\rm SFR} \sim 100 {\rm M_{\odot} \, yr^{-1}}$), while the very brightest starbursts, with star formation rates of SFR$\sim 1000
\, {\rm M_{\odot} yr^{-1}}$, might be detectable out to $z
\sim 3$. As discussed in section \[radio\_obs\], simultaneous detection of synchrotron radiation with the Square Kilometer Array will then constrain magnetic field strengths in these objects.
Detection in gamma ray emission is also unlikely. The upcoming Gamma-ray Large Area Space Telescopy (GLAST) (http://glastproject.gsfc.nasa.gov/) will have a flux sensitivity of $\sim 2 \times 10^{-9} {\rm photons \, cm^{-2} \, s^{-1}}$ in the 20MeV–300GeV range, whereas star-forming regions at high redshift will have fluxes of at most $F_{\gamma} \sim 10^{-15} {\left( \frac{\epsilon}{0.1} \right)}{{\rm \left( \frac{SFR}{1 M_{\odot} \, yr^{-1}}
\right)}}{\left( \frac{1+z}{10} \right)}^{-2} {\rm photons \,
cm^{-2} s^{-1}}$.
What fraction of the unresolved X-ray and gamma-ray background could be due to star formation at high redshift $z>3$? Observations of absorption in CIV and other metals in Ly$\alpha$ forest absorption lines with $10^{14.7} {\rm cm^{-2}} < N_{HI} < 10^{16} {\rm cm^{-2}}$ indicate that $Z\sim
10^{-2} Z_{\odot} \sim 2 \times 10^{-4}$. Each supernova produces about $\sim 1 M_{\odot}$ of metals (Woosley & Weaver 1995). Assuming $(\Omega_{b},h)=(0.04,0.65)$, this implies one supernova every $\sim
1000 \, {\rm kpc^{3}}$ (comoving). If each supernova injects $\sim
10^{50}$ ergs in hard X-rays, the comoving X-ray energy density was $U_{X} \sim \frac{c}{4 \pi} \frac{U_{X}}{(1+
\bar{z})} \sim 2 \times 10^{-6} {\rm eV \,
cm^{-3}}$. If the mean source redshift was $\bar{z} \sim 5$, the distant sources produced a diffuse flux of $J \sim 1
{\rm keV s^{-1}
cm^{-2} sr^{-1}}$. One can make a more detailed estimate using the equation of cosmological transfer (Peebles 1993): $$J(\nu_{o},z_{o})= \frac{1}{4 \pi} \int_{z_{o}}^{\infty} dz
\frac{dl}{dz} \frac{(1+z_{o})^{3}}{(1+z)^{3}} \epsilon(\nu,z) e^{-\tau_{{\rm
eff}}(\nu_{o},z_{o},z)}
\label{rad_transfer}$$ where $z_{o}$ is the observer redshift, $\nu=\nu_{o}(1+z)/(1+z_{o})$, and $\epsilon(\nu,z)$ is the comoving X-ray emissivity. Using the star formation model of Haiman & Loeb (1997), I obtain the estimate: $$\nu J_{\nu}= 0.8 {\left( \frac{\epsilon}{0.1} \right)}\left( \frac{Z(z=3)}{10^{-2} Z_{\odot}}
\right) {\rm keV \, s^{-1}
cm^{-2} sr^{-1}} \ \ {\rm for} \ z_{source} \ge 3$$ In Fig (\[background\_fig\]), I display this predicted level of emission against the observed X-ray and gamma-ray background in the 1 keV –100 GeV range, from analytic fits to the ASCA and HEAO A2,A4 data in the 3-60 keV range (Boldt 1987), the HEAO 1 A-4 data in the 80-400 keV range (Kinzer et al 1997), the COMPTEL data in the 800 keV – 30MeV range (Kappadath et al 1996). Data points from EGRET in the 30 MeV – 100 GeV range (Sreekumar et al 1998) are also shown. The level of the unresolved background of course depends on the sensitivity of the instrument; recently CXO resolved $\sim 80 \%$ of the hard X-ray background in the 2-10 keV range into point sources (Mushotzky et al 2000). This agrees well with the predictions of XRB synthesis models (e.g., Madau, Ghisellini & Fabian 1994), which use AGN unification schemes to reproduce the observed spectral shape of the XRB. A prediction of the IC scenario presented here is that a non-trivial fraction of the X-ray/gamma-ray background will not be resolved into point sources with upcoming missions, due to the extreme faintness of high redshift sources.
It is particularly intriguing that both the amplitude and spectral shape of the gamma-ray background as observed by EGRET is well-matched by the predicted level of gamma-ray emission in this model. This raises the exciting possibility that the majority of the observed gamma-ray background comes from inverse Compton emission at high redshifts. At present, the origin of the gamma-ray background is still unknown. The most favoured scenario for some time was that it is due to unresolved gamma-ray blazars (Bignami et al 1979, Kazanas & Protheroe 1983, Stecker & Salamon 1996): the observed blazar $\gamma$-ray spectrum has an average spectral index compatible with the observed GRB (Chiang & Mukherjee 1998). However, extrapolation of the observed EGERT blazar luminosity function implies that unresolved blazars can account for at most $\sim 25\%$ of the diffuse $\gamma$-ray background (Chiang & Mukherjee 1998). The unresolved blazar model will most likely be decisively tested by GLAST (Stecker & Salamon 1999), which will be two orders of magnitude more sensitive than EGRET. A host of other models include pulsars expelled into the halo by asymmetric supernova explosions (Dixon et al 1998, Hartmann 1995), primordial black hole evaporation (Page & Hawking 1976), supermassive black holes at very high redshift (Gnedin & Ostriker 1992), annihilation of weakly interactive big bang remnants (Silk & Srednicki 1984, Rudaz & Stecker 1991) and finally, inverse Compton radiation from cosmic ray electrons in our own Galaxy (Strong & Moskalenko 1998, Dar & De Rujula 2000), and from collapsing clusters (Loeb & Waxman 2000). However, to date the possibility of inverse Compton emission from high redshift supernovae has not been discussed.
A scenario in which the majority of the gamma-ray background comes from inverse Compton emission at high redshift make a number of firm predictions: (i) As previously mentioned, the majority of the GRB will remain unresolved by GLAST, due to the extreme faintness of the contributing sources. (ii) After removal of the Galactic contribution, which is correlated with the structure of our Galaxy and our position within it (Dixon et al 1998, Dar et al 1999), a highly isotropic component of the GRB will still be present. (iii) The GRB should be extremely smooth, and exhibit significant fluctuations only at extremely small angular scales. The fluctuations should be dominated by the Poisson rather than the clustering contribution (see Oh 1999). Using $\langle
S^{n} \rangle = \int_{0}^{S_{c}} \frac{dN}{dS} S^{n}$, where $S_{c}$ is the cut-off flux for point source removal, and the star formation model of Haiman & Loeb (1997), I find that for $z > 3$ sources (which are too faint to be removed as point sources), $\frac{\langle I^{2} \rangle^{1/2}} {\langle I
\rangle}= 3.4 \times 10^{-2} \left( \frac{\theta}{5^{\prime}} \right)^{-1}$, where $I$ is surface brightness (the angular resolution of GLAST is expected to be of order $1-5$ arcmin). (iv) The gamma-ray background at $E > 100$GeV should be attenuated, due to pair production opacity against IR/UV photons (e.g., Salamon & Stecker 1998, Oh 2000b). High energy photons initiate an electromagnetic cascade which transfers energy from high energy photons to the lower energy portion of the spectrum, where the universe is optically thin (Coppi & Aharonian 1997). Note that the EGRET spectrum was directly determined with data only up to 10 GeV; beyond 10 GeV larger uncertainties exist due to backsplash in the NaI calorimeter, and Monte-Carlo simulations were used to determine the differential flux in the 10-30,30-50 and 50-120 GeV range (Sreekumar et al 1998). Thus, the EGRET data points with $E > 10$GeV are less reliable. It would be intriguing to see if GLAST (sensitive out to 300 GeV) indeed shows an absorption edge to the gamma-ray background at higher energies. It would also be interesting to look for absorption of the gamma-ray background in a line of sight passing through a massive cluster (indicating that the gamma-rays come from higher redshifts than the cluster); the pair-production optical depth through a cluster is of order $\tau \sim 2 n_{\gamma} \sigma_{T}
r_{vir} \sim 0.4$ (assuming $n_{\gamma} \sim 0.1 {\rm cm^{-3}}$ and $r_{vir} \sim 1$Mpc).
CMB constraints
---------------
Will the upscattering of CMB photons by relativistic electrons at high redshift produce an observable signal in the CMB, or violate any present observational constraints? When the electrons are relativistic, CMB photons are inverse Compton scattered to such high energies (completely out of the detector bandpass) the process may simply be thought of as absorption, $\delta I/I \sim \tau_{e}^{rel}$, where $\tau_{e}^{rel}$ is the optical depth of relativistic electrons, and $I$ is the number flux of CMB photons. The flux decrement due to the absorption of CMB photons may be estimated as: $$\Delta S_{\nu} \approx J_{CMB} \int d\Omega
\int dl n_{e}^{rel} \sigma_{T} =
J_{CMB} \sigma_{T}
\frac{N_{e}^{rel}}{d_{A}^{2}}
\label{SZ_flux}$$ where $J_{CMB}$ is the blackbody surface brightness of the CMB, and $N_{e}^{rel}$ is the total number of relativistic electrons in the system. The steady state number of relativistic electrons is given by $N_{e}^{rel} = \int d \gamma N(\gamma)$ where $N(\gamma)$ is given by equation (\[electron\_pop\_soln\]). Since the CMB flux peaks at $\nu \sim 10^{11}$Hz, the absolute magnitude of the flux decrement is maximized by going to similar frequencies. At 20 GHz, the highest frequency detectable by SKA, the SKA has an rms sensitivity of $\sim 6$nJy for a $10^{5}$s integration. On the other hand, the flux decrement is $\Delta S_{\nu} (20 \, {\rm GHz}) \sim 3.5 \times 10^{-5} (\frac{1+z}{10})^{2}
(N_{e}^{rel}/3 \times 10^{58} {\rm electrons})$ nJy, which is unobservably small.
One might hope to detect the mean signal of all the CMB photons upscattered by relativistic electrons. In particular, since the number of CMB photons is no longer conserved, the absorption might be detectable as a chemical potential distortion of the CMB. Let us estimate the number of CMB photons destroyed. Each supernova upscatters at most $N_{scattered} \sim \epsilon E_{SN}/\langle E_{X} \rangle \sim 10^{60}$ CMB photons, where I have set the average photon energy $\langle E_{X} \rangle \sim 100$eV (note that since the number of photons $N_{\nu}
\propto \nu^{-1}$, most of the upscattered photons are of low energy). For a metallicity of $Z \sim 10^{-2} Z_{\odot}$ at $z \sim
3$, one supernova has gone off every comoving $V_{SN} \sim 1000 \, {\rm kpc^{3}}$, and thus the comoving number density of upscattered CMB photons is $\delta n
\sim N_{scattered}/V_{SN} \sim 4 \times
10^{-8} \, {\rm cm^{3}}$. Since $n_{\gamma} \sim 400 \, {\rm cm^{-3}}$, we have $\delta n/n \sim 10^{-10}$, which results in an undetectably small chemical potential distortion. Thus, the upscattering of CMB photons at high redshift does not violate any distortion constraints on the CMB.
If the IGM is reionized inhomogeneously, as in canonical models, then secondary CMB anistropies will be created by CMB photons Thompson scattering off moving ionized patches (Agahanim et al 1996, Grusinov & Hu 1998, Knox et al 1998). The power spectrum is generally white noise, with $\Delta T/T \sim 10^{-6}-10^{-7}$, peaking at arc-minute to sub arc-minute scales. However, if the IGM is reionized fairly homogeneously by X-rays then over a line of sight the positive and negative contributions of the velocity field will cancel out. In this case, only the second-order Ostriker-Vishniac effect due to coupling between density and velocity fields will be present. A null detection of the inhomogeneous reionization anisotropy could place upper limits on the patchiness of reionization, although this will be a difficult measurement as the inhomogeneous reionization and Ostriker-Vishniac signals are likely to be of comparable strength (Haiman & Knox 1999).
Radio observations {#radio_obs}
------------------
The supernovae that exploded generate magnetic turbulence and magnetic fields, which allow Fermi acceleration to take place (Jones & Ellison 1991, Blandford & Eichler 1987); observations of local radio galaxies in non-thermal radio and X-ray emission yield field strengths consistent with equipartition between relativistic particles and the magnetic field (Kaneda et al 1995). If such magnetic fields are present in the first star clusters, the relativistic electron population is a source of synchrotron radio emission as well as inverse-Compton emission. Below I find that for a given relativistic electron population, the Square Kilometer Array (SKA) will be much more sensitive to non-thermal radio emission than CXO or Constellation-X will be to inverse Compton X-ray emission. Radio observations will thus allow one to establish the presence of relativistic electrons in objects too faint to observe directly in X-ray emission. Since one knows the CMB energy density exactly as a function of redshift, given reasonable assumptions for the magnetic field the observed radio emission allows one to immediately estimate the amount of inverse Compton X-ray emission which must be taking place, $${\rm L_{X}}= \frac{\rm U_{CMB}(z)}{\rm U_{B}}{\rm L}_{\rm synch}
\label{X_radio}$$
The transition Lorentz factor $\gamma_{break}$ at which electron energy losses are dominated by inverse Compton rather than ionization losses is given by equation (\[gamma\_break\]). Above an observed radio frequency $\nu_{break}= \nu_{L} \gamma_{break}^{2}/(1+z) = 2.8 \times 10^{4} {{\rm \left( \frac{B}{10 \mu G} \right) }}{\left( {\rm \frac{{\rm n}}{1 \, {\rm cm^{-3}}}}
\right)}{\left( \frac{1+z}{10} \right)}^{-5}$ Hz (where $\nu_{L} \equiv \frac{eB}{2 \pi m_{e} c}$ is the electron gyrofrequency), the steady state electron population is determined by balance between supernova injection and inverse Compton losses, and is given by equation (\[electron\_pop\_soln\]). From standard formulae for synchrotron emissivity (Rybicki & Lightman 1979) $\epsilon_{\nu}^{synch}=
\sigma_{T} \gamma^{2} \beta^{2} U_{mag} c \, {\rm n(E) dE/d\nu}$ (where n(E) is the number density of emitting electrons in $dE$) this yields a synchrotron luminosity: $$L_{\nu}^{sync}= 2.7 \times 10^{28} {{\rm \left( \frac{B}{10 \mu G} \right) }}^{1+\alpha} { \left( \frac{\nu}{1 \, {\rm GHz}} \right)}^{-\alpha} {{\rm \left( \frac{SFR}{1 M_{\odot} \, yr^{-1}}
\right)}}{\left( \frac{\epsilon}{0.1} \right)}f(z,U_{\gamma}) \ {\rm erg \, s^{-1} Hz^{-1}}$$ where $f(z,U_{\gamma})= { {\rm min} \left[ {\left( \frac{1+z}{10} \right)}^{-4},
\left( \frac{U_{\gamma}}{ 4.2 \times 10^{-9} \, {\rm erg \, cm^{-3}}}
\right)^{-1} \right] }$ (the latter term in brackets is used if the stellar radiation field has a higher energy density than the CMB). By contrast, the thermal free-free emission is given by (Oh 1999): $$L_{\nu}^{ff}=1.2 \times 10^{27} \left( \frac{{\rm SFR}}{1 {\rm
M_{\odot} yr^{-1}}} \right) \ {\rm erg \, s^{-1} \, Hz^{-1}}$$ Thus, assuming $\alpha=1$, at observed frequencies $\nu < \nu_{trans} = 2.3 {\left( \frac{1+z}{10} \right)}^{-1}
{{\rm \left( \frac{B}{10 \mu G} \right) }}^{2} {\left( \frac{\epsilon}{0.1} \right)}f(z,U_{\gamma}) \, {\rm GHz}$ synchrotron emission dominates over free-free emission. This is well within the 0.1–20 GHz capability of the SKA. However, this is only true if the power law for electron population extends to high energies. If it is truncated at some maximum Lorentz factor $\gamma_{max}$, then synchrotron emission is only observable for $\nu < \nu_{max}= \nu_{L} \gamma^{2}_{max} = 0.28 {\left( \frac{1+z}{10} \right)}^{-1} {{\rm \left( \frac{B}{10 \mu G} \right) }}\left( \frac{\gamma_{max}}{10^{4}} \right)^{2} \, {\rm GHz} $. If $\nu_{max}< \nu_{trans}$, this will be manifested by an abrupt drop in radio flux at $\nu_{max}$, beyond which the emission takes the flat spectrum free-free emission form (this does not take place for expected values of $\gamma_{max}= 7.4 \times 10^{6} {{\rm \left( \frac{B}{10 \mu G} \right) }}^{1/2} \left(
\frac{v_{sh}}{2000 \, {\rm km \, s^{-1}}} \right) {\left( \frac{1+z}{10} \right)}^{-2}$; see equation (\[Emax\_eqn\])). Observation of such a drop will yield valuable constraints on $B,\gamma_{max}$ in the first objects; the value of $\gamma_{max}$ in turn constrains the upper energy cutoff in the inverse Compton X-ray/gamma-ray spectrum, which could be important in determining whether high-redshift starbursts make a significant contribution to the gamma-ray background.
Can radio emission from star clusters at high redshift be detected by the proposed Square Kilometer Array? Non-thermal emission can be distinguished from free-free emission with multi-frequency observations to identify frequency regimes where the spectral slope is steep.The SKA detector noise may be estimated as: $$S_{\rm instrum}= \frac{2 kT_{\rm sys}}{A_{\rm eff} \sqrt{2 t \Delta \nu}}= 25
\left(\frac{\Delta \nu}{160 \, {\rm MHz}} \right)^{-1/2} \left( \frac{t}
{10^{5} \, s} \right)^{-1/2} {\rm nJy}
\label{noise}$$ where I have used $A_{\rm eff}/T_{\rm sys}=2 \times 10^{8} {\rm
cm^{2}/K}$ for the SKA (Braun et al 1998), and I assume a bandwidth $\Delta \nu \approx 0.5 \nu$. The flux density due to non-thermal emission from a high-redshift star cluster, assuming $\alpha=1$, is: $$S_{\nu_{o}}= \frac{L_{\nu}(\nu_{o}(1+z)}{4 \pi d_{L}^{2}} (1+z)= 8.5 {{\rm \left( \frac{B}{10 \mu G} \right) }}^{1+
\alpha} \left( \frac{\nu}{320 MHz} \right)^{-\alpha} {{\rm \left( \frac{SFR}{1 M_{\odot} \, yr^{-1}}
\right)}}{\left( \frac{\epsilon}{0.1} \right)}{\left( \frac{1+z}{10} \right)}^{-2} f(z,U_{\gamma}) {\rm nJy}$$ Thus, a source with SFR$ \sim 10 {\rm M_{\odot} yr^{-1}}$ at $z \sim 9$ can be detected as a 10$\sigma$ detection in 10 days. To estimate the number of sources detectable by SKA, I use the Press-Schechter based high redshift star formation models of Haiman & Loeb (1997), and define an efficiency factor $f_{radio} = {{\rm \left( \frac{B}{10 \mu G} \right) }}^{1+\alpha} {\left( \frac{\epsilon}{0.1} \right)}\left
( \frac{f_{star}}{0.17} \right)$, where $f_{star}$ is the fraction of halo gas which fragments to form stars. In Figure (\[synchrotron\_fig\]), I display the number of sources above a given redshift which may be detected in non-thermal emission in the $1^{\circ}$ SKA field of view, assuming $f_{radio} = 1, 0.01$. Also shown is the number of sources which can be detected in free-free emission at 4 GHz. One should be able to detect a large number of sources at high redshift, $z>5$. Thus, radio observations of non-thermal emission can serve as a useful proxy for X-ray observations in allowing one to estimate $L_{X}$, and thus the overal level of X-ray emission at high redshift.
Multi-wavelength observations
-----------------------------
While a detailed study of high-redshift multi-wavelength campaigns is beyond the scope of this paper, below I describe some possible follow-up observations if synchrotron emission is detected at high redshift.
**Redshift estimates**
Thus far the most efficient way to select high-redshift radio galaxies has proven to be the observation of steep radio spectra $\alpha < -1.3$ (Chambers et al 1990, van Breugel et al 1999). This is at least partially due to the fact that extremely bright radio galaxies have SEDs which steepen with frequency; the k-correction then implies that sources at increasing redshift have steeper observed spectra. Steepening with redshift due to inverse Compton losses, as well as selection effects (i.e., brighter sources have stronger magnetic fields, and thus more rapid synchrotron losses) could also play a role (Krolik & Chen 1991). This technique may fail for fainter sources at high redshift since for $\nu > {\rm
min}(\nu_{trans},\nu_{max})$, the radio flux will be dominated by free-free emission and the spectra will appear flat. For instance, for $ B < 3 {\left( \frac{1+z}{10} \right)}^{2.5} {\left( \frac{\epsilon}{0.1} \right)}^{0.5} \, \mu$G, we have $\nu_{trans} < 160$MHz and SKA will only detect free-free emission within its frequency coverage. An efficient way to select high-redshift objects prior to reionization would be to perform broad-band deep field imaging with NGST and select Lyman-break dropouts as has been done at $z \sim 3$ (Steidel et al 1996); before the epoch of reionization one must select ’Gunn-Peterson dropouts’, i.e. galaxies with no flux shortward of rest-frame HI Ly$\alpha$. Yet another method of selecting high-redshift objects would be to perform a joint survey in the submm with the Atacama Large Millimeter Array (ALMA); at low redshift the submm dust emission scales almost linearly with other star formation indicators, such as radio and UV emission. However, for $z \approx 0.5-10$ and a given dust emission SED the K-correction almost balances the cosmological dimming of a source, implying a flux density almost independent of redshift (e.g., Blain et al 2000). This has spawned suggestions to obtain approximate redshifts by the flux density ratio between submm and radio wavebands (Carilli & Yun 1999, 2000), although uncertainties include an AGN contribution to the radio flux and the dust temperature in the galaxy, with a degeneracy between hotter galaxies at high redshift and cooler ones nearby (Blain 1999).
**Relative importance of X-ray and UV stellar emission for reionization**
As previously noted, from a measurement of the radio synchrotron flux we can use equation (\[X\_radio\]) to estimate the level of inverse Compton X-ray emission which must be taking place. Redshifts and thus ${\rm U_{CMB}(z)}$ can be determined with Balmer line spectroscopy with NGST, while ${\rm U_{B}}$ can be estimated by assuming a magnetic field strength required to minimise the total energy density of the system (this is close to the value for energy equipartition between relativistic particles and magnetic fields). Observations of local radio galaxies in non-thermal radio and X-ray emission yield field strengths consistent with the minimum energy value (Kaneda et al 1995). NGST can constrain the rest-frame UV emission, and a joint measurement of the rest-frame UV flux longward of Ly$\alpha$ and the Balmer line flux (where ${\rm L_{H\alpha}} \propto (1- f_{esc})
\dot{N}_{ion}$) can constrain $f_{esc}$, the escape fraction of ionizing photons, where one roughly expects $J_{H\alpha} \sim 40 (1-
f_{esc}) J_{IR} (R/1000)$. Thus, joint SKA/NGST observations could place limits on whether inverse Compton X-rays or stellar UV photons were energetically dominant and thus more important in reionizing the universe.
**Measuring magnetic fields in bright sources**
The very brightest sources out to $z \sim 3$ will also be visible in X-ray emission with CXO; the fact that the same population of relativistic electrons is responsible for non-thermal X-ray and radio emission can be confirmed by comparing respective spectral slopes. In this case, one can estimate the strength of magnetic fields, ${\rm U_{B}} \approx \frac{L_{X}}{L_{sync}} {\rm
U_{CMB}(z)}$. Since dust obscuration is unimportant at both hard X-ray and radio wavelengths, the measurement should be fairly robust.
**Distinguishing between synchrotron emission from AGN and supernovae**
To confirm that such emission arises from star-forming regions rather than an AGN, one might look for signs of diffuse emission (note that the angular resolution of the SKA is $\sim 0.1''$, while the angular scale of the virial radius of typical objects will be $\theta_{vir} \sim 0.5'' (M/10^{9} M_{\odot})^{1/3}$). AGNs can be selected on the basis of color, as has been successfully carried out at lower redshifts (Fan 1999), using broad band NIR and MIR imaging with NGST. Finally, the line widths of the H$\alpha$, H$\beta$ lines as observed with NGST may be considerably broader for an AGN ($\sigma
\sim 1000 \, {\rm km \, s^{-1}}$), due to line broadening by the accretion disc.
Conclusions
===========
X-ray emission from a early star forming regions is predicted to be large and energetically comparable to UV emission. Non-thermal inverse Compton emission, which provides a good fit to local observations and should become increasingly important at high redshift, due to the evolution of the CMB energy density, is predicted to be the dominant source of X-rays. It introduces a whole host of physical consequences: the topology of reionization changes, becoming more homogeneous with much fuzzier delineation between ionized and neutral regions; reheating temperatures increase, with implications for feedback on structure formation and the observed width of Ly$\alpha$ forest lines; the abundance of free electrons in dense regions increases, promoting gas phase $H_{2}$ formation, cooling, and star formation. These effects will be considered in a companion paper (Oh 2000a). While direct detection of individual sources with CXO or Constellation-X appears difficult, we can hope to confirm the presence of relativistic electrons in high redshift objects by detecting non-thermal radio emission with the Square Kilometer Array. Given the CMB energy density at that epoch, this yields a minimal level of inverse Compton X-ray emission. Combined with NGST observations of rest frame UV emission, this will determine if stellar radiation or inverse Compton X-rays were the dominant factor in reheating and reionizing the universe. In addition, in this scenario a non-trivial fraction of the hard X-ray and gamma-ray background comes from inverse Compton emission at high redshift. In particular, it is possible to reproduce both the shape and amplitude of the gamma-ray background observed by EGRET, and predict that the majority of the smooth, isotropic gamma-ray background will remain unresolved by GLAST, which should display attenuation above $\sim 100$GeV, from the pair production opacity due to ambient UV/IR radiation fields.
Many of the conclusions in this paper depend upon a scenario in which the escape fraction of UV ionizing photons is small, $f_{esc} < 10 \%$. This assumption is well supported by observations in the local universe, as well as theoretical radiative transfer calculations of high redshift star forming regions, which predict very low escape fractions, $f_{esc} \sim 0.01$ (Wood & Loeb 1999, Ricotti & Shull 1999, although note that the latter authors predict a substantial escape fraction at low masses $10^{7}
M_{\odot}$). However, note that the latter ignore gas clumping and the multi-phase structure of the ISM. If for some reason the escape fraction is unexpectedly high (e.g. the supernovae blow holes in the ISM through which UV photons can escape), then the X-ray component is energetically subdominant and stellar UV radiation dominates the reionization of the universe. Even in this regime, the X-ray component still plays a role in heating the gas above 15 000 K, He II reionization, and promoting gas phase ${\rm H_{2}}$ formation, as these tasks cannot be accomplished by soft photons. The escape fraction of UV ionising photons in high redshift objects may eventually be deduced by comparing the IR and $H\alpha$ fluxes observed by NGST (Oh 1999).
The most uncertain aspect of this paper is the assumed level of X-ray luminosity, $L_{X} \sim 0.1 \dot{E}_{SN}$. I have calibrated the conversion rate via local X-ray and gamma-ray observations of starburst galaxies and individual supernova remnants within our galaxy, and argued that efficiency of all proposed X-ray production mechanisms either remains constant or increases with redshift. If the X-ray emission in local starbursts is primarily due to inverse Compton scattering of soft IR photons (Moran & Lehnart 1997, Moran, Lehnart & Helfand 1999), the empirical relation between star formation rate and X-ray luminosity (equation (\[Xray\_lum\])) implies an electron acceleration efficiency of $\epsilon \sim 10 \%$, which lies at the upper limit of theoretical expectations. If instead the empirical relation (\[Xray\_lum\]) is approximately correct but the X-rays arise from a variety of emission mechanisms, the importance of X-rays for reionization still holds, but specific observational tests which rely on the inverse-Compton mechanism, such as the gamma-ray background observations(section (\[xray\_obs\])) and observations of radio synchrotron emission with the SKA (section (\[radio\_obs\])) will fail.
Acknowledgements
================
I am very grateful to my advisor David Spergel for his encouragement and advice. I also thank Roger Blandford, Andrea Ferrara, Zoltan Haiman and Ed Turner for helpful conversations, and Bruce Draine and Michael Strauss for detailed and helpful comments on an earlier manuscript, and the anonymous referee for helpful comments. I thank the Institute of Theoretical Physics, Santa Barbara for its hospitality during the completion of this work. This work is supported by the NASA ATP grant NAG5-7154, and by the National Science Foundation, grant number PHY94-07194.
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|
---
abstract: 'We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell\neq p$.'
author:
- Fabien Trihan and Christian Wuthrich
bibliography:
- 'parity.bib'
title: Parity conjectures for elliptic curves over global fields of positive characteristic
---
Introduction
============
Let $K$ be a global field and let $E$ be an elliptic curve defined over $K$. The conjecture of Birch and Swinnerton-Dyer asserts that the rank of the Mordell-Weil group $E(K)$ is equal to the order of vanishing of the Hasse-Weil $L$-function $L(E/K,s)$ as $s=1$. A weaker question is to know whether these two integers have at least the same parity. This seems more approachable because the parity of the order of vanishing on the analytic side can by expressed in more algebraic terms through local root numbers – at least when the $L$-function is known to have an analytic continuation. Let $w(E/K)\in \{\pm 1\}$ be the global root number of $E$ over $K$ which is equal to the product of local root numbers $\prod_v w(E/K_v)$ as $v$ runs over all places in $K$. The local terms $w(E/K_v)$ were defined by Deligne without reference to the $L$-function, see [@rohrlich_wd] for the definition. So we can formulate the following conjecture.
We have $(-1)^{\operatorname{rank}E(K)} = w(E/K)$.
This conjecture is unproven except for specific cases. We will focus on the following easier question. Let $\Sha(E/K)$ be the Tate-Shafarevich group defined as the kernel of the localisation maps $H^1(K,E) \to \prod_{v} H^1(K_v,E)$ in Galois cohomology. For a prime $\ell$, the $\ell$-primary Selmer group $\operatorname{Sel}_{\ell^{\infty}}(E/K)$ fits into an exact sequence $$\label{mwselsha_eq}
0 \to E(K)\otimes {}^{{\mathbb{Q}}_{\ell}}\!/\!{}_{{\mathbb{Z}}_{\ell}} \to \operatorname{Sel}_{\ell^{\infty}}(E/K) \to \Sha(E/K)[\ell^{\infty}] \to 0.$$ If the characteristic of $K$ is prime to $\ell$, we may define it as the preimage of $\Sha(E/K)[\ell^{\infty}]$ under the map $H^1(K,E[\ell^\infty])\to H^1(K,E)[\ell^{\infty}]$. If the characteristic is equal to $\ell$, then one should use flat instead of Galois cohomology, see section \[selmer\_sec\] for definitions. The theorem of Mordell-Weil shows that the dual of $\operatorname{Sel}_{\ell^{\infty}}(E/K)$ is a finitely generated ${\mathbb{Z}}_{\ell}$-module, whose rank we will denote by $r_{\ell}$. In particular, is a short exact sequence of finite cotype ${\mathbb{Z}}_l$-modules for any prime $\ell$. Since it is conjectured that $\operatorname{rank}E(K) = r_{\ell}$, we can make the following conjecture, which seems more approachable as it links two algebraically defined terms.
Let $\ell$ be a prime. We have $(-1)^{r_{\ell}} = w(E/K)$.
These conjectures have attracted much attention in recent years and the $\ell$-parity conjecture is now known in many cases, in particular when the ground field is $K={\mathbb{Q}}$ by work of the Dokchitser brothers [@dok_isogeny; @dok_nonab; @dok_reg; @dok_modsquares; @dok_09], Kim [@kim], Mazur and Rubin [@mazur_rubin], Nekovář [@nekovar2; @nekovar3; @nekovar4], Coates, Fukaya, Kato, and Sujatha [@cfks] and others.
In this article, we restrict our attention to the case of positive characteristic. So, we suppose from now on that $K$ is a global field of characteristic $p>3$ with constant field ${\mathbb{F}}_q$. The main result of this article is the following theorem.
\[pparity\_thm\] The $p$-parity conjecture holds for any elliptic curve $E$ over a global field $K$ of characteristic $p>3$.
The proof consists of two steps: first a local calculation linking the local root number to local data on the Frobenius isogeny on $E$, carried out in section \[local\_sec\]; followed by the use of global duality in section \[duality\_sec\]. Luckily, we do not have to treat all individual cases of bad reduction for the local considerations, since we are able to use a theorem of Ulmer [@ulmer_gnv] to reduce to the semistable case. This is done in section \[red\_ss\_sec\].
The proof follows closely the arguments in [@dok_isogeny] and Fisher’s appendix of [@dok_nonab]. We repeat it here in details, both for completeness and to make the reader aware of a few subtleties; for instance, it is to note that the Frobenius isogeny $F$ and its dual $V$ do not play an interchangeable role.
The hardest part concerns the global duality. The relevant dualities that we need for our conclusion have never appeared in the literature and we are forced to prove them. We think that it is worthwhile to include in section \[pparity\_sec\] a general formula for the parity of the corank of the $p$-primary Selmer group and a local formula for the root number in section \[rootno\_sec\].
Originally, global dualities have appeared in Cassels’ work [@cassels] on the invariance under isogenies of the conjecture of Birch and Swinnerton-Dyer. It should be noted that one could use our duality statements to prove this in the case of characteristic $p>0$, but there is no need for this. In fact it is known by [@kato_trihan] that the conjecture of Birch and Swinnerton-Dyer is equivalent to the finiteness of the Tate-Shafarevich group – and it is clear that the latter question is invariant under isogeny.
The second main result of this paper is a proof of the $\ell$-parity conjecture when $\ell\neq p$ in some cases. Write $\mu_\ell$ for the $\ell$-th roots of unity.
\[ellparity\_thm\] Let $E/K$ be an elliptic curve and let $\ell> 2$ be a prime different from $p$. Furthermore assume that
1. $a=[K(\mu_{\ell}):K]$ is even, and
2. the analytic rank of $E$ does not grow by more than $1$ in the constant quadratic extension $K\cdot {\mathbb{F}}_{q^2}/K$.
Then the $\ell$-parity conjecture holds for $E/K$.
The proof will be presented in section \[ell\_sec\]. Its main ingredients are the non-vanishing results of Ulmer in [@ulmer_gnv] and the techniques for proving the parity conjectures from representation theoretic considerations as explained in [@dok_modsquares; @dok_sd].
Although the conditions will be fulfilled for many curves, the methods in this paper fail to give a complete proof of the $\ell$-parity conjecture. See the remarks at the beginning of section \[ell\_sec\] and the more detailed section \[fail\_sec\] for an explanation of why we are not able to extend the proof any further.
Notations
---------
The constant field of the global field $K$ of characteristic $p>3$ is the finite field ${\mathbb{F}}_q$ for some power $q$ of $p$. Let $C$ be a smooth, geometrically connected, projective curve over ${\mathbb{F}}_q$ with function field $K$. Let $E/K$ be an elliptic curve, which we will assume to be non-isotrivial (i.e. the $j$-invariant of $E$ is transcendental over ${\mathbb{F}}_q$). We fix a Weierstrass equation $$\label{w_eq}
E \colon\quad y^2\, =\, x^3\, +\, A\,x\, +\, B$$ with $A$ and $B$ in $K$ and the corresponding invariant differential $\omega = \tfrac{dx}{2y}$. By $F\colon E \to E'$ we denote the Frobenius isogeny of degree $p$ whose dual $V\colon E'\to E$ is the Verschiebung.
If $f\colon A\to B$ is a homomorphism of abelian groups, we write $$z(f) = \frac{\# \operatorname{coker}(f)}{\# \ker(f)}$$ provided that the kernel and the cokernel of $f$ are finite. For any abelian group (or group scheme) $A$ and integer $m$, we denote by $A[m]$ the $m$-torsion part of it; and, for any prime $\ell$, the $\ell$-primary part will be denoted by $A[\ell^{\infty}]$.
The Pontryagin dual of an abelian group $A$ is written $A^{\vee}$. If the Pontryagin dual of $A$ is a finitely generated ${\mathbb{Z}}_{\ell}$-module for some prime $\ell$, then we write $\operatorname{div}(A)$ for its maximal divisible subgroup and let $A_{\operatorname{div}}$ denote the quotient of $A$ by $\operatorname{div}(A)$.
Reduction to the semistable case {#red_ss_sec}
================================
Before, we start we should mention that the conjecture of Birch and Swinnerton-Dyer is known for isotrivial curve $E$ by the work of Milne [@milne_isotrivial]. So for the rest of the paper we will assume that $E$ is not isotrivial as otherwise the parity conjectures are known. In particular, it follows from this assumption that $E/K$ is ordinary. The parity conjecture is also known in the following cases:
\[an\_rk0&1\] Let $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell$ be a prime ($\ell=p$ is allowed). The analytic rank of $A/K$ is always greater or equal to the $\ell$-corank of the Selmer group. If the analytic rank of $A/K$ is zero, then the conjecture of Birch and Swinnerton-Dyer holds. If the analytic rank is $1$ then it coincides with the ${\mathbb{Z}}_\ell$-corank of the $\ell$-primary Selmer group.
Note that if we restricted ourselves to elliptic curves and to the case $\ell \neq p$, then this result could already be deduced from the work of Artin and Tate [@tate].
By 3.2 in [@kato_trihan], the Hasse-Weil $L$-function of $A/K$ can be expressed as an alternating product of characteristic polynomials of some operators $\phi^i_\ell$ acting on a finite dimensional ${\mathbb{Q}}_\ell$-vector space $H^i_{{\mathbb{Q}}_\ell}$, with $i=0,1,2$. Then by 3.5.1 in [@kato_trihan], the order at $s=1$ of the Hasse-Weil $L$-function can be interpreted as the multiplicity of the eigenvalue 1 for the operator $\phi^1_\ell$ on $H^1_{{\mathbb{Q}}_\ell}$. Following the notations of 3.5 in [@kato_trihan], let $I_{3,\ell}$ denote the part of $H^1_{{\mathbb{Q}}_\ell}$ on which the operator $\operatorname{id}-\phi^1_\ell$ acts nilpotently and let $I_{2,\ell}$ denote the kernel of $\operatorname{id}-\phi^1_\ell$, such that we have the inclusions: $$I_{2,\ell}\subset I_{3,\ell}\subset H^1_{{\mathbb{Q}}_\ell}.$$ Since by 3.5.1 in [@kato_trihan], the operator $\operatorname{id}-\phi^i_\ell$ is an isomorphism for $i=0,2$, it follows that the analytic rank of $A/K$ is equal to the dimension of $I_{3,\ell}$. On the other hand, it follows from 3.5.5 and 3.5.6 in [@kato_trihan], that the $\ell$-corank of the Selmer group of $A/K$ is the dimension of $I_{2,\ell}$ so that we deduce that the analytic rank of $A/K$ is always greater or equal to the $\ell$-corank of the Selmer group of $A/K$. If the analytic rank of $A/K$ is trivial, so is the dimension of $I_{3,\ell}$. It implies that the dimension of $I_{2,\ell}$ is zero and by 3.5.6 in [@kato_trihan], we conclude that the Mordell-Weil group is also of rank zero. We then conclude the proof of the assertion thanks to the main result 1.8 of [@kato_trihan]. If the analytic rank of $A/K$ is one, then $\phi^1_\ell$ acts like the identity on $I_{3,\ell}$ and therefore $I_{2,\ell}=I_{3,\ell}$ and the second assertion immediately follows.
The following proposition will be used at several places to reduce the conjecture to easier situations.
\[red\_prop\] Let $E/K$ be a non-isotrivial curve and $L/K$ a separable extension. Let $\ell$ be a prime. Assume one of the following three conditions:
1. \[odd\_cond\] $\ell\neq p$ and the extension $L/K$ is a Galois extension of odd degree.
2. \[zero\_cond\] The analytic rank of $E$ does not grow in $L/K$.
3. \[one\_cond\] $\ell\neq p$ and the analytic rank of $E$ does not grow by more than $1$ in $L/K$.
Then the $\ell$-parity conjecture for $E/K$ holds if and only if the $\ell$-parity conjecture for $E/L$ is known.
If condition (\[odd\_cond\]) holds then the conclusion follows directly from Theorem 1.3 in [@dok_sd]. Note already here that the complete paper [@dok_sd] and its proofs hold in our situation as long as $\ell\neq p$.
Suppose now as in condition (\[zero\_cond\]) that the analytic rank does not grow in $L/K$. Denote by $A/K$ the Weil restriction of $E$ under $L/K$ and by $B/K$ the quotient of $A$ by the natural image of $E$ in it. Since $$L(E/L,s) = L(A/K,s) = L(E/K,s)\cdot L(B/K,s)$$ we see that that the analytic rank of $B/K$ is zero and therefore by Proposition \[an\_rk0&1\], the full Birch and Swinnerton-Dyer conjecture holds. In particular, the Mordell-Weil rank of $B/K$ is zero and its Selmer group is a finite group. Moreover, we have an exact sequence $$\label{sels_ses}
\operatorname{Sel}_{\ell^{\infty}}(E/K) \to \operatorname{Sel}_{\ell^{\infty}}(A/K) \to \operatorname{Sel}_{\ell^{\infty}}(B/K),$$ and the kernel of the first map lies in $B(K)[\ell^{\infty}]$, which is a finite group. Hence we conclude that $r_{\ell}$ is equal to the corank of $\operatorname{Sel}_{\ell^{\infty}}(A/K)$ and, by Proposition 3.1 in [@mazur_rubin], this is the same as the corank of $\operatorname{Sel}_{\ell^{\infty}}(E/L)$. So we are able to deduce the $\ell$-parity for $E/K$ from the $\ell$-parity for $E/L$.
Finally, suppose that $\ell\neq p$ and that the analytic rank grows exactly by $1$; so we are under condition (\[one\_cond\]). Then we know by Proposition \[an\_rk0&1\] that the rank of $\operatorname{Sel}_{\ell^{\infty}}(B/K)$ is less or equal to $1$. We wish to exclude the possibility that it is $0$, so assume by now that $\operatorname{Sel}_{\ell^{\infty}}(B/K)$ is finite. But this means that $\Sha(B/K)[\ell^{\infty}]$ is finite and hence the full conjecture of Birch and Swinnerton-Dyer holds by [@kato_trihan] again. So we reach a contradiction since we would have $0 = \operatorname{rank}B(K) = \operatorname{ord}_{s=1} L(B/K,s) = 1$. Hence we have shown that the corank of $\operatorname{Sel}_{\ell^{\infty}}(B/K)$ is $1$.
Note that the left-hand map in still has finite kernel. We will show now that right-hand map has finite cokernel, too. Let $\Sigma$ be the finite set of places in $K$ of bad reduction for $E$. Write $G_{\Sigma}(K)$ for the Galois group of the maximal separable extension of $K$ which is unramified outside $\Sigma$. Note that from the definition of the Selmer group, we find the following diagram with exact rows and columns $$\xymatrix@C-8pt{
0\ar[d] & 0\ar[d] & \\
\operatorname{Sel}_{\ell^{\infty}}(A/K)\ar[d]\ar[r] &
\operatorname{Sel}_{\ell^{\infty}}(B/K) \ar[d]& \\
H^1\bigl(G_{\Sigma}(L),A[\ell^{\infty}]\bigr)\ar[d]\ar[r] &
H^1\bigl(G_{\Sigma}(K),B[\ell^{\infty}]\bigr)\ar[d]\ar[r] &
H^2\bigl(G_{\Sigma}(K),E[\ell^{\infty}]\bigr)\ar[r]^{r} &
H^2\bigl(G_{\Sigma}(K),A[\ell^{\infty}]\bigr) \\
\bigoplus_{v\in \Sigma} H^1(K_v,A)[\ell^{\infty}] \ar[r] &
\bigoplus_{v \in \Sigma} H^1(K_v,B)[\ell^{\infty}]
}$$ We know that the bottom groups are finite as they are dual to $\varprojlim A(K_v)/\ell^k$ and $\varprojlim B(K_v)/\ell^k$ respectively. Hence we see from the snake lemma that we only have to prove that the kernel of $r$ is finite. Shapiro’s lemma shows that $H^2(G_{\Sigma}(K), A[\ell^\infty])$ is isomorphic to $H^2(G_{\Sigma}(L),E[\ell^\infty])$ and hence the map $r$ is simply the restriction map. As its kernel will only get larger when increasing $L$, we may assume that $L/K$ is Galois. Then the kernel of the restriction is contained in the part of $H^2(G_{\Sigma}(K),E[\ell^{\infty}])$ that is killed by $[L:K]$. Hence it is finite, because $H^2(G_{\Sigma}(K),E[\ell^{\infty}])$ is a discrete abelian group with finite ${\mathbb{Z}}_{\ell}$-corank.
Therefore, we conclude again that the corank of the $\ell$-primary Selmer group increased by exactly $1$ in $L/K$.
\[toss\_cor\] If the $\ell$-parity conjecture holds for all semistable elliptic curves, then it holds for all elliptic curves.
Theorem 11.1 in [@ulmer_gnv] proves that there is a separable extension $L/K$ such that the reductions of $E$ becomes semistable and the analytic rank does not grow in $L/K$.
The same argument also reduces the full parity conjecture to the semistable case.
Local computations {#local_sec}
==================
The following computations are purely local and we change the notations for this section. Let $K$ be a local field of characteristic $p>3$ with residue field ${\mathbb{F}}_q$. The ring of integers is written $\mathcal{O}$, the maximal ideal $\mathfrak{m}$ and the normalised valuation by $v$. The elliptic curve $E/K$ is given by the equation . By changing the equation, if necessary, we may suppose for this section that $A$ and $B$ are in $\mathcal{O}$.
Define $L$ to be the minimal extension of $K$ such that $E'(L)[p] ={{}^{{\mathbb{Z}}}\!/\!{}_{p {\mathbb{Z}}}}$, or equivalently that $E[F]$ is isomorphic to $\mu[p]$ as a group scheme over $L$. There is a representation $$\rho\colon \operatorname{Gal}(L/K)\to \operatorname{Aut}(E'(L)[p]) \cong ({{}^{{\mathbb{Z}}}\!/\!{}_{p {\mathbb{Z}}}})^{\times}$$ which shows that $[L:K]$ divides $p-1$. Define $\bigl(\frac{-1}{L/K}\bigr)\in\{\pm 1\}$ to be the image of $-1$ under the composition of the reciprocity map and $\rho$ $$K^{\times} \to \operatorname{Gal}(L/K)\to ({{}^{{\mathbb{Z}}}\!/\!{}_{p {\mathbb{Z}}}})^{\times}.$$ So $\bigl(\frac{-1}{L/K}\bigr) = +1$ if and only if $-1$ is a norm from $L^{\times}$ to $K^{\times}$.
We will also consider ${z_{{\scriptscriptstyle}V}}= z\bigl(V\colon E'(K) \to E(K)\bigr)$, which is a certain power of $p$. Put $$\sigma = \sigma(E/K) = \begin{cases} +1 \quad&\text{ if $z_V$ is a square and} \\ -1 &\text{ otherwise.} \end{cases}$$ It is important to note that we cannot define $z\bigl(F\colon E(K) \to E'(K)\bigr)$ since its cokernel will never be finite.
Finally, as in the introduction, we let $w = w(E/K)$ be the local root number of $E$ over $K$, as defined by Deligne and well explained in [@rohrlich_wd]. The aim of this section is to show the following theorem.
\[local\_thm\] Let $K$ be a local field of characteristic $p>3$. For any non-isotrivial elliptic curve $E/K$ whose reduction is not additive and potentially good, we have $w(E/K) = \bigl(\frac{-1}{L/K}\bigr) \cdot \sigma(E/K)$.
We will prove this theorem by treating each type of reduction separately. In the last section of this paper, we will prove this local theorem without the assumption on the reduction using global methods. See Conjecture 5.3 in [@dok_09] for the analogue in characteristic zero. In particular, the following computations show that the analogy should take places above $p$ in characteristic zero to supersingular places in characteristic $p$.
Recall the definition of the Hasse invariant $\alpha = A(E,\omega)$ associated to the given integral equation . Write $\mathcal{F}$ for the formal group of $E$ over $\mathcal{O}$, and similarly $\mathcal{F}'$ for the formal group for the isogenous curve $E'$ given by the integral equation $$E'\colon \quad y'^2 \, = \, x'^3\, + \,A^p\, x'\,+\,B^p.$$
Choose $t=-\frac{x'}{y'}$ as the parameter of the formal group $\mathcal{F}'$. Then the formal isogeny $V_1$ of the Verschiebung $V$ is of the form $$\xymatrix@R=3mm{
V_1 \colon \mathcal{F}'(\mathfrak{m})\ar@{-o>}[r] & \mathcal{F}(\mathfrak{m}) \\
t \ar@{|-o>}[r] & \alpha \cdot G(t) + H(t^p)
}$$ for some $G(t) = t+\cdots \in \mathcal{O}[\![t]\!]$ and $H(t) = u\cdot t +\cdots \in \mathcal{O}[\![t]\!]$ with $u$ in $\mathcal{O}^{\times}$. See section 12.4 in [@katz_mazur] for other descriptions of the Hasse invariant $\alpha$.
We begin now the proof of Theorem \[local\_thm\]. For the computation of the local root number $w$, we can simply refer to Proposition 19 in [@rohrlich_wd], where we find that $w=-1$ if the reduction is split multiplicative and $w=+1$ if it is good or non-split multiplicative.
Good reduction
--------------
\[good\_prop\] Suppose $E/K$ has good reduction. Then $w=+1$. The quantities $\sigma$ and $\bigl(\frac{-1}{L/K}\bigr)$ are $+1$ if and only if $q^{v(\alpha)}$ is a square. In particular, if the reduction is ordinary then $\sigma=\bigl(\frac{-1}{L/K}\bigr)=+1$.
We may suppose that the equation is minimal, i.e. that it has good reduction. We then have the diagram $$\xymatrix@C=10mm{
0 \ar@{-o>}[r] & \mathcal{F}'(\mathfrak{m}) \ar@{-o>}[r] \ar[d]_{V_1} & E'(K) \ar@{-o>}[r] \ar[d]_{V} & \tilde{E'}({\mathbb{F}}_q) \ar@{-o>}[r]\ar[d] & 0 \\
0 \ar@{-o>}[r] & \mathcal{F}(\mathfrak{m}) \ar@{-o>}[r] & E(K) \ar@{-o>}[r] & \tilde{E}({\mathbb{F}}_q) \ar@{-o>}[r] & 0
}$$ where $\tilde{E}$ denotes the reduction of $E$. The isogenous curves $\tilde{E}$ and $\tilde{E'}$ over ${\mathbb{F}}_q$ have the same number of points, so the kernel and cokernel of this map have the same size. Hence ${z_{{\scriptscriptstyle}V}}= z(V_1)$.
For any $N\geqslant 1$, $$\frac{\mathcal{F}(\mathfrak{m}^N)}{\mathcal{F}(\mathfrak{m}^{N+1})} \cong \frac{\mathfrak{m}^N}{\mathfrak{m}^{N+1}} \cong \frac{\mathcal{F}'(\mathfrak{m}^N)}{\mathcal{F}'(\mathfrak{m}^{N+1})}$$ and so the same argument shows that ${z_{{\scriptscriptstyle}V}}= z(V_1) = z\bigl(V_N\colon \mathcal{F}'(\mathfrak{m}^{N}) \to \mathcal{F}(\mathfrak{m}^{N}) \bigr)$.
We claim that if $N>\upsilon(\alpha)$ then $V_N$ maps $\mathcal{F}'(\mathfrak{m}^N)$ bijectively onto $\mathcal{F}(\mathfrak{m}^{N+\upsilon(\alpha)})$. If $t$ has valuation at least $N$, then the valuation of $\alpha\,t$ is smaller than the valuation of $u\cdot t^p$. Therefore $\upsilon(V_N(t)) = \upsilon(\alpha) + \upsilon(t)$. This shows that $V_N$ maps $\mathcal{F}'(\mathfrak{m}^N)$ injectively to $\mathcal{F}(\mathfrak{m}^{N+\upsilon(\alpha)})$. In particular, the kernel $\ker(V_N)$ is trivial.
Let $s$ have valuation $\upsilon(s) \geqslant N+\upsilon(\alpha)$. Put $t_0 = s/\alpha$. Then $t_0$ is close to a zero of $g(t) = V_N(t) - s$. Namely $g(t_0) = \alpha \,a\, t_0^2 +\cdots +u\, t_0^p+\cdots$ has valuation at least $2\,\upsilon(s)-\upsilon(\alpha) \geqslant 2\,N+\upsilon(\alpha) > 2\,\upsilon(\alpha)$, if we write $G(t) = t+a\,t^2+\cdots$ for some $a\in\mathcal{O}$. Since $g'(t_0) = \alpha + 2\,\alpha\, a \, t_0 + \cdots$ has valuation $\upsilon(\alpha)$, Hensel’s lemma shows that there is a $t$ close to $t_0$ such that $g(t) = 0$, i.e. such that $V_N(t) = s$.
We conclude that the cokernel of $V_N$ is equal to the index of $\mathcal{F}(\mathfrak{m}^{N+\upsilon(\alpha)})$ in $\mathcal{F}(\mathfrak{m}^N)$. Hence ${z_{{\scriptscriptstyle}V}}= z(V_N) = q^{\upsilon(\alpha)}$. In particular ${z_{{\scriptscriptstyle}V}}=1$ if the reduction is ordinary, i.e. when $\alpha$ is a unit in $\mathcal{O}$.
Let ${e_{{\scriptscriptstyle}L/K}}$ be the ramification index of $L/K$. If the reduction is good ordinary, then the inertia group acts trivially on $T_p E'$, which is a ${\mathbb{Z}}_p$-module of rank 1. Hence $L/K$ is unramified and we have immediately that $\bigl(\frac{-1}{L/K}\bigr)=+1$.
The parity of ${v(\alpha)}$ is equal to the parity of $\frac{p-1}{{e_{{\scriptscriptstyle}L/K}}}$.
If $E$ has good ordinary reduction, then ${v(\alpha)}=0$, $e_{L/K}=1$ and $p-1$ is even so that the assertion is true. If $E$ has good supersingular reduction, then since $E'(L)[p]$ contains a non-trivial point $P=(x'_P,y'_P)$, but the reduction does not contain a point of order $p$, there exist a $t_P = -x'_P/y'_P$ in the maximal ideal $\mathfrak{m}_L$ of $L$ such that $V_1(t_P) = 0$. From $V_1 (t_P) = \alpha\,t_P +\cdots + u\, t_P^p +\cdots$, we see that the valuation of $\alpha t_P$ and $ut_P^p$ must cancel out. Hence $\upsilon_L(\alpha) = {e_{{\scriptscriptstyle}L/K}}\cdot\upsilon(\alpha) = (p-1)\cdot \upsilon_L(t_P)$, where $\upsilon_L$ denotes the normalised valuation in $L$. So if the valuation of $t_P$ is odd, we have proved the assertion.
Assume that $\upsilon_L(t_P)$ is even. Then $\upsilon(\alpha)$ is also even and we have to show that $\frac{p-1}{{e_{{\scriptscriptstyle}L/K}}}$ is even. The extension $L/K(x'_P)$ is generated by $t_P$ whose square belongs to $K(x'_P)$; so this extension is either unramified quadratic or trivial. If $L=K(x'_P)$, then $\operatorname{Gal}(L/K)$ acts on the set of $\{x'_P\vert O\neq P\in E'(L)[p]\}$ and hence $[L:K]$ divides $\frac{p-1}{2}$, so $\frac{p-1}{[L:K]}$ is even. Otherwise, if $L$ is an unramified quadratic extension of $K(x'_P)$, then ${e_{{\scriptscriptstyle}L/K}}= e_{K(x'_p)/K}$ and $p-1$ is divisible by $[L:K] = 2 {e_{{\scriptscriptstyle}L/K}}f_{K(x'_p)/K}$. So $\frac{p-1}{{e_{{\scriptscriptstyle}L/K}}}$ is even.
Now we can conclude the proof of Proposition \[good\_prop\]. Lemma 12 in [@dok_isogeny], whose proof is valid even if the characteristic of $K$ is not zero, says that $\bigl(\frac{-1}{L/K}\bigr)=+1$ if and only if $q$ is a square or if $\frac{p-1}{e_{L/K}}$ is even. The previous lemma suffices now to conclude.
In the good supersingular case, $L/K$ may or may not be totally ramified. We illustrate this with two examples.
We take $p=5$, $w>0$ any integer, and the curve $E$ given by the minimal Weierstrass equation $$y^2\ = \ x^3 \ + \ T^w \cdot x \ + \ 1$$ over $\mathcal{O}={\mathbb{F}}_5[\![T]\!]$. The Hasse invariant is $\alpha = 2\cdot T^w$. The reduction is good, but supersingular. The division polynomial $f_V$ associated to the isogeny $V$ can be computed to be equal to $$f_V(x) = 2\, T^w\,x^2+4\,T^{2w}\,x + (4+3\,T^{3w}+T^{6w})\,.$$
First we take the case $w=2\,m$ is even. Then we can make the substitution $X = T^m \cdot x$ to get $$f_V(x) = 2\, X^2+4\,T^{3m}\,X + (4+3\,T^{6m}+T^{12m})\,.$$ We see that $K(x_P)$ is a quadratic unramified extension of $K$. The quantity $\frac{p-1}{{e_{{\scriptscriptstyle}L/K}}}$ will certainly be even.
Now, take $w=2\,m-1$ to be odd with $m>1$. This time the substitution $X=T^m\cdot x$ gets us to $$T\cdot f_V(x) = 2\, X^2+4\,T^{3m-2}\,X + T\cdot (4+3\,T^{6m-3}+T^{12m-6})\,.$$ Therefore $K(x_P)/K$ will be a ramified extension of degree $2$. The valuation of $x_P$ over $K(x_P)$ is odd, so we have to make a further extension $L/K(x_P)$, again ramified of degree $2$, to have a $p$-torsion point in $E'(L)$. So ${e_{{\scriptscriptstyle}L/K}}=4$ and $\frac{p-1}{{e_{{\scriptscriptstyle}L/K}}}$ is odd.
Split multiplicative
--------------------
\[split\_prop\] Suppose $E/K$ has split multiplicative reduction. Then $w(E/K) = -1$, $\bigl(\frac{-1}{L/K}\bigr) = +1$, and $\sigma(E/K) = -1$.
Let ${q_{{\scriptscriptstyle}E}}\in K^{\times}$ be the parameter of the Tate curve which is isomorphic to $E$ over $K$. Then the isogenous curve $E'$ has parameter ${{q_{{\scriptscriptstyle}E}}}^p$ and the Frobenius map $$\xymatrix{V\colon \frac{ K^{\times} }{ ({{q_{{\scriptscriptstyle}E}}}^p)^{{\mathbb{Z}}} } \ar@{-o>}[r] & \frac{ K^{\times} }{ ({q_{{\scriptscriptstyle}E}})^{{\mathbb{Z}}} } }$$ is induced by the identity on $K^{\times}$. Hence $V$ has a kernel with $p$ elements and is surjective, so $z_V = \frac{1}{p}$ and $\sigma = -1$.
Since $E'$ has already a $p$-torsion point over $K$, we have $L=K$ and $\bigl(\frac{-1}{L/K}\bigr)=+1$.
Non-split multiplicative
------------------------
\[nonsplit\_prop\] Suppose $E/K$ has non-split multiplicative reduction. Then $$w(E/K) =\bigl(\frac{-1}{L/K}\bigr)=\sigma(E/K) = +1.$$
There is a quadratic extension $K'$ over which $E$ has split multiplicative reduction. So either $L=K$ or $L=K'$. Let $E^{\dagger}$ be the quadratic twist of $E$ over $K'$. Up to $2$-torsion groups, we have $E(L)= E(K)\oplus E^{\dagger}(K)$. Since $E^{\dagger}$ has split multiplicative reduction over $K$ there is a $p$-torsion point in $E^{\dagger}(K)$. So $L=K'$.
From the previous section, we know that ${z_{{\scriptscriptstyle}V}}$ for $E/L$ and $E^{\dagger}/K$ both are equal to $\frac{1}{p}$. So by the above formula for $E(L)$ up to $2$-torsion, we get that ${z_{{\scriptscriptstyle}V}}$ for $E/K$ is $1$. So $\sigma = +1$. Since $L/K$ is unramified, $ \bigl(\frac{-1}{L/K}\bigr) = +1$.
Additive potentially multiplicative
-----------------------------------
Suppose $E/K$ has additive, potentially multiplicative reduction. Let $\chi\colon K^{\times} \to \{\pm 1\}$ be the character associated to the quadratic ramified extension over which $E$ has split multiplicative reduction. Then $w(E/K) =\bigl(\frac{-1}{L/K}\bigr)=\chi(-1)$ and $\sigma(E/K) = +1$.
The root number is computed by Rohrlich [@rohrlich_wd 19.ii]. The proof that $\sigma = +1$ is the same as in the non-split multiplicative case. The formula $ \bigl(\frac{-1}{L/K}\bigr) = \chi(-1)$ is clear, too.
Selmer groups {#selmer_sec}
=============
We return to the global situation and we wish to define properly the Selmer groups involved in $p$-descent in characteristic $p$ using flat cohomology.
From now on, $K$ is again a global field with field of constants ${\mathbb{F}}_q$ and $E/K$ is a semistable, non-isotrivial elliptic curve. We denote by $\mathcal{E}$ the Néron model of $E/K$ over $C$ and $\mathcal{E}^0$ its connected component containing the identity. Let $U$ be a dense open subset of $C$ such that $\mathcal{E}$ has good reduction on $U$. The group schemes $\mathcal{E}$ and $\mathcal{E}^0$ coincide over $U$ and we define for any $v\not\in U$ the group of connected components $\Phi_v$ in the fibre above $v$. So we have the following short exact sequence: $$0 \to \mathcal{E}^0 \to \mathcal{E} \to \bigoplus_{v\not \in U} \Phi_v \to 0.$$
Following 2.2 in [@kato_trihan], the discrete $p^{\infty}$-Selmer group of $E/K$ is defined as $$\operatorname{Sel}_{p^\infty}(E/K) := \ker \Bigl[ {H_{\operatorname{fl}}}^1 (K,E[p^\infty])\rightarrow
\prod_{v} {H_{\operatorname{fl}}}^1 (K_v,E) \Bigl]$$ where $E[p^{\infty}]$ is the $p$-divisible group associated to $E$ and ${H_{\operatorname{fl}}}$ stands for flat cohomology. It is known that $\operatorname{Sel}_{p^{\infty}}(E/K)$ fits into the following exact sequence: $$\label{ses1_seq}
0 \to E(K)\otimes {}^{{\mathbb{Q}}_p}\! /\!{}_{{\mathbb{Z}}_p} \to \operatorname{Sel}_{p^\infty}(E/K) \to \Sha(E/K)[p^\infty] \to 0 .$$ This follows from the fact that the Tate-Shafarevich group can also be computed using flat cohomology as the kernel of ${H_{\operatorname{fl}}}^1 (K,E)\to \prod_v {H_{\operatorname{fl}}}^1 (K_v,E)$ since for the elliptic curve $E$ over $K$ or $K_v$, the étale and flat cohomology groups coincide (see Theorem 3.9 in [@Milne80]). Note also that the dual of $\operatorname{Sel}_{p^{\infty}}(E/K)$ is a finitely generated ${\mathbb{Z}}_p$-module by the theorem of Mordell-Weil and the finiteness of $\Sha(E/K)[p]$ (see e.g. [@ulmer_pdescent]).
Let $\phi\colon E\to E'$ be an isogeny of elliptic curves. The map $\phi$ induces a short exact sequence of sheaves in the flat topology $$\label{ses2_seq}
\xymatrix@1{
0 \ar[r] & E[\phi] \ar[r] & E \ar[r]^{\phi} & E' \ar[r] & 0.
}$$ The Selmer group $\operatorname{Sel}_\phi(E/K)$ is defined to be the set of elements in ${H_{\operatorname{fl}}}^1(K,E[\phi])$ whose restrictions to ${H_{\operatorname{fl}}}^1(K_v,E[\phi])$ lie in the image of the connecting homomorphism $E(K_v)\to {H_{\operatorname{fl}}}^1(K_v,E[\phi])$ for all $v$. If $U$ is any open subset of $C$ where $E$ has good reduction, we can also describe $\operatorname{Sel}_\phi(E/K)$ as the kernel of the composed map $$\xymatrix@1{
{H_{\operatorname{fl}}}^1(U,\mathcal{E}[\phi]) \ar[r] &
\prod_{v\not\in U} {H_{\operatorname{fl}}}^1(K_v,E[\phi]) \ar[r] &
\prod_{v\not\in U} {H_{\operatorname{fl}}}^1(K_v,E)[\phi].
}$$ Passing to cohomology, the short exact sequence induces the short exact sequence of finite groups $$\label{selsha_seq}
\xymatrix@1{
0 \ar[r] & E'(K)/\phi(E(K)) \ar[r] & \operatorname{Sel}_\phi (E/K) \ar[r] & \Sha(E/K)[\phi] \ar[r] & 0,
}$$ where $\Sha(E/K)[\phi]$ is the kernel of the induced map ${\phi_{\smallSha}}\colon \Sha(E/K) \to \Sha(E'/K)$.
Global Euler characteristics {#duality_sec}
============================
We prove in the next three sections a few results on global dualities for the $p$-primary part of the Tate-Shafarevich group in characteristic $p$ using flat cohomology. The main reference will be [@milne], but we wish to point the reader to related results in [@GT] and [@Go].
Note that most results in these three sections do not need any condition on the reduction. Also, except where mentioned, the characteristic $p$ can be any prime.
We give a short review of the Oort-Tate classification of finite flat group schemes of order $p$ (see [@oort_tate] for details). Let $X$ be a scheme of characteristic $p>0$. The data of a finite flat group scheme $N$ of order $p$ over $X$ is equivalent to the data of an invertible sheaf $\mathcal{L}$, a section $a\in H^0(C,\mathcal{L}^{\otimes(p-1)})$ and a section $b\in H^0(C,\mathcal{L}^{\otimes(1-p)})$ such that $a\otimes b=0$. We use the notation $N_{\mathcal{L},a,b}$. If $N$ is of height one, then $a=0$. The Cartier dual of $N_{\mathcal{L},a,b}$ is $N_{\mathcal{L}^{-1},b,a}$.
For a scheme $S$ of characteristic $p>0$ and a finite flat group scheme $N/S$ we define the Euler characteristic of $N/S$ as $$\chi(S,N) := \prod_i\Bigl(\#{H_{\operatorname{fl}}}^i(S,N)\Bigr)^{(-1)^i}$$ whenever the groups ${H_{\operatorname{fl}}}^i(S,N)$ are finite.
\[globalchi\_lem\] Assume that the prime $p$ is odd. Let $N$ be a finite flat group scheme of order $p$ over $C$. Assume that the Cartier dual $N^D$ of $N$ is of height 1. Then the groups ${H_{\operatorname{fl}}}^i(C,N)$ are finite and $\chi(C,N)$ is a square in ${\mathbb{Q}}^{\times}$.
The cohomology is finite by Lemma III.8.9 in [@milne] since $N$ is finite flat. If $N^D$ has height $1$ then $N$ corresponds to a group scheme $N_{\mathcal{L},a,b}$ with $b=0$. Now we follow the explanation after problem III.8.10 in [@milne]. Since $N$ is the dual of a group scheme of height 1, we have that there is a sequence $$\xymatrix@R-4ex@C+1ex{0\ar[r]& N \ar[r] & \mathcal{L}\ar[r] & \mathcal{L}^{\otimes p}\ar[r] &0,\\
& & z \ar @{|->} [r] & z^{\otimes p}-a\otimes z & }$$ which is exact by the definition of $N_{\mathcal{L},a,b}$. See also Example III.5.4 in [@milne]. Hence we have that $\chi(C,N) = q^{\chi(\mathcal{L})-\chi(\mathcal{L}^{\otimes p})}$. Using Riemann-Roch, we get $$\begin{aligned}
\chi(\mathcal{L}) &= \deg(\mathcal{L}) + 1 - g\\
\chi(\mathcal{L}^{\otimes p}) &= p\cdot \deg(\mathcal{L}) + 1 - g\end{aligned}$$ and therefore we find the formula $$\chi(C,N) = q^{(p-1)\deg \mathcal{L}}.$$ So the lemma follows from the fact that $p$ is odd.
For any place $v$ in $K$, we denote by $\vert\cdot\vert_v$ the normalised absolute value of the completion $K_v$. In particular the absolute value of a uniformiser is $q_v^{-1}$ where $q_v$ denotes the number of elements in the residue field.
\[localchi\_lem\] Let $N=N_{\mathcal{L},a,b}$ be a finite flat group scheme of order $p>2$ over the ring $O_v$ of integers in $K_v$. Assume that $N_{K_v}$ is étale. Then the Euler characteristic of $N$ is well-defined and we have $$\chi(O_v,N)\equiv\vert a\vert_v^{-1}$$ modulo squares in ${\mathbb{Q}}^{\times}$.
The invertible sheaf $\mathcal{L}$ is $c^{-1}\cdot O_v$ for some $c\in K_v^{\times}$. Then by III.0.9.(c) in [@milne], we have $N_{\mathcal{L},a,b}\cong N_{O_v,ac^{p-1},bc^{1-p}}$. Using Remark III.7.6 and the Example after Theorem III.1.19 on page 244 of [@milne], we have $\chi(O_v,N)=\vert a\cdot c^{p-1}\vert_v^{-1}\equiv \vert a\vert_v^{-1}$ modulo squares in ${\mathbb{Q}}^{\times}$.
For a scheme $S$ of characteristic $p>0$ and a scheme $X/S$, we denote by $X'$ the fibre product $X\times_S S$ where the map $S\to S$ in this product is the absolute Frobenius of $S$. By the universal property of the fibre product, we have a map $F\colon X\to X'$ called the relative Frobenius. If moreover $X/S$ is a flat group scheme, then there exists a map $V\colon X'\to X$ called the Verschiebung such that $V\circ F$ and $F\circ V$ induce $[p]$, the multiplication by $p$ (see [@SGA3], VII). In particular, $F\colon E\to E'$ is a $p$-isogeny of elliptic curves which extends to the Néron models of $E$ and $E'$ by its universal property. Since the Néron model of $E'$ is $\mathcal{E}'$, this map is just the relative Frobenius $F\colon \mathcal{E}\to \mathcal{E}'$.
Over the field $K$, or more generally over any open subset $U$ in $C$ where $E$ has good reduction, Proposition 2.1 of [@ulmer_pdescent] shows that $E[F]=N_{{\underline{\omega}}^{-1},0,\alpha}$ and $E[V]=N_{{\underline{\omega}},\alpha,0}$, where $\alpha$ is the Hasse invariant of $E$ and where ${\underline{\omega}}$ is the invertible sheaf $\pi_{*} \Omega^1_{E/K}$ with $\pi\colon E \to \operatorname{Spec}(K)$ being the structure morphism.
\[global\_chi\_prop\] Let $E/K$ be a non-isotrivial elliptic curve. There exists a dense open subset $U$ of $C$ such that $E$ has everywhere good ordinary reduction and $\chi(U,\mathcal{E}[F])$ is a well-defined square in ${\mathbb{Q}}^{\times}$.
By the Oort-Tate classification, $E[F]/K$ is isomorphic to $N_{{\underline{\omega}}^{-1},0,\alpha}$. By Proposition B.4 in [@milne] and its proof, it extends to a finite flat group scheme ${\mathcal N}/C$ of order $p$ of the form $N_{\mathcal{O}_C(W),0,\alpha}$ for some Weil divisor $W\leqslant 0$ such that $(\alpha)\geqslant W$. Let $U_1$ be a dense open subset of $C$ over which $\mathcal{E}$ has good reduction. As in the proof of Theorem III.8.2 in [@milne] on page 291, we replace $U_1$ by a smaller open set $U_2$, over which $\mathcal{N}\vert_{U_2}\simeq\mathcal{E}[F]\vert_{U_2}$. Finally, we set $U$ equal to the open subset of $U_2$ where we have removed all places $v$ for which $E/K$ has good supersingular reduction.
Write $\mathcal{N}^{D}$ for the Cartier dual of $\mathcal{N}$. By Proposition III.0.4.(c) and Remark III.0.6.(b) in [@milne], we have a long exact sequence $$\xymatrix@1{\cdots \ar[r]& {H_{\operatorname{fl},c}}^i\bigl(U,\mathcal{N}^{D} \bigr)\ar[r] &
{H_{\operatorname{fl}}}^i\bigl(C, \mathcal{N}^D\bigr)\ar[r] &
\prod_{v\not\in U} {H_{\operatorname{fl}}}^i\bigl(O_v, \mathcal{N}^D\bigr)\ar[r]&
\cdots.}$$ Global duality (Theorem III.8.2 in [@milne]) shows that $${H_{\operatorname{fl},c}}^i\bigl(U,\mathcal{N}^{D} \bigr) = {H_{\operatorname{fl},c}}^i\bigl(U, \mathcal{E}'[V]\bigr) \quad\text{ is dual to}\quad
{H_{\operatorname{fl}}}^i\bigl(U, \mathcal{E}[F]\bigr).$$ By the multiplicative property of the Euler characteristic, we get $$\chi(U,\mathcal{E}[F])=\frac{\chi(C,\mathcal{N}^D)}{\prod_{v\not\in U}\chi(O_v,\mathcal{N}^D)}.$$ Since $\mathcal{N}^{D} = N_{\mathcal{O}_C(-W), \alpha, 0}$ is finite flat of order $p$ over $C$, Lemma \[globalchi\_lem\] shows that $\chi(C,\mathcal{N}^D)$ is a square. Furthermore, Lemma \[localchi\_lem\] yields $$\chi(U,\mathcal{E}[F]) \equiv \prod_{v\not\in U}\chi(O_v,\mathcal{N}^D)^{-1} \equiv \prod_{v\not\in U}\vert\alpha\vert_v \pmod{\square}.$$ Since the places of $U$ are places of good ordinary reduction where $\vert\alpha\vert_v$ is a square by Proposition \[good\_prop\], we have, using the product formula, $$\chi(U,\mathcal{E}[F]) \equiv \prod_{v\not\in U}\vert\alpha\vert_v^{-1}\equiv\prod_{v}\vert\alpha\vert_v^{-1}=1\pmod{\square}.
\qedhere$$
The Cassels-Tate pairing {#cassels-tate}
========================
Recall that there exist a pairing (proof of Theorem II.5.6 in [@milne]) called the Cassels-Tate pairing $$\langle\!\langle\cdot,\cdot\rangle\!\rangle \colon \Sha(E/K)\times \Sha(E/K)\to {}^{{\mathbb{Q}}}\!/\!{}_{{\mathbb{Z}}}.$$ As claimed in Proposition III.9.5 in [@milne] its left and right kernels are the divisible part $\operatorname{div}(\Sha(E/K))$ of the Tate-Shafarevich group. We are calling the attention of the reader to the fact that the initial proof in [@milne] is wrong as noticed by D. Harari and T. Szamuely in [@HS]. The first correct published proofs that the Cassels-Tate pairing of [@milne], Theorem II.5.6(a), annihilates only maximal divisible subgroups appear in [@HS] (for prime-to-$p$ primary components) and in [@Go] (for $p$-primary components) when the 1-motive considered in these references is taken to be ($0\to E$). This pairing is alternating and hence the order of $\Sha(E/K)_{\operatorname{div}}$ is a square. This last fact is not always true if we consider general abelian varieties.
\[adjoint\_lem\] Let $\phi\colon E\to E'$ be an isogeny of elliptic curves and $\hat \phi$ the dual isogeny. Then the induced map ${\phi_{\smallSha}}\colon \Sha(E/K)\to\Sha(E'/K)$ and ${\hat{\phi}_{\smallSha}}\colon\Sha(E'/K)\to\Sha(E/K)$ are adjoints with respect to the Cassels-Tate pairings, i.e. $$\langle\!\langle{\phi_{\smallSha}}(\eta),\xi\rangle\!\rangle_{E'}=\langle\!\langle \eta,\hat{\phi_{\smallSha}}(\xi)\rangle\!\rangle_{E}$$ for every $\eta\in \Sha(E/K)$ and $\xi\in \Sha(E'/K)$.
The proof is analogous to the proof in the number field case (see Remark I.6.10 in [@milne] or §2 of [@cassels]) and is deduced from the functoriality of the local pairings in flat cohomology.
\[orthongal\_prop\] The orthogonal complement of $\Sha(E'/K)[V]$ in $\Sha(E'/K)[p^{\infty}]$ under the Cassels-Tate pairing $$\Sha(E'/K)[p^\infty]\times \Sha(E'/K)[p^\infty]\to {}^{{\mathbb{Q}}}\!/\!{}_{{\mathbb{Z}}}$$ is the image of $F\colon \Sha(E/K)[p^{\infty}]\to \Sha(E'/K)[p^{\infty}]$.
Note that the proposition follows immediately from the previous lemma if the pairing is perfect. Else, by the previous Lemma \[adjoint\_lem\], it is immediate that $F\bigl(\Sha(E/K)[p^{\infty}]\bigr)$ is contained in the orthogonal of $\Sha(E'/K)[V]$. Let $\xi$ be an element in $\Sha(E'/K)[p^{\infty}]$ orthogonal to the kernel of $V$. Let $D'$ denote the maximal divisible subgroup of $\Sha(E'/K)[p^\infty]$ and $D$ the maximal divisible subgroup of $\Sha(E/K)[p^\infty]$. Then there is a perfect paring on the quotients $\Sha(E'/K)[p^{\infty}]/D'$ and $\Sha(E/K)[p^{\infty}]/D$. Since $V$ and $F$ map divisible elements to divisible elements, they induce maps between these quotients. $$\xymatrix{ 0 \ar[r] & D' \ar[r] \ar@<1ex>[d]^{V}& \Sha(E'/K)[p^{\infty}] \ar[r]\ar@<1ex>[d]^{V} & \Sha(E'/K)[p^{\infty}]/D' \ar[r]\ar@<1ex>[d]^{V} & 0 \\
0 \ar[r] & D \ar[r] \ar@<1ex>[u]^{F}& \Sha(E/K)[p^{\infty}] \ar[r]\ar@<1ex>[u]^{F} & \Sha(E/K)[p^{\infty}]/D \ar[r]\ar@<1ex>[u]^{F} & 0 }$$ The element $\xi + D'$ in the quotient $\Sha(E'/K)[p^{\infty}]/D'$ is orthogonal to the kernel of $V$. Since the pairing is perfect there, we have an element $\eta$ in $\Sha(E/K)[p^{\infty}]$ such that $F$ maps $\eta+D$ to $\xi+D'$ in the quotients. Hence $F(\eta) = \xi + \delta$ for some $\delta \in D'$. But since the map $F\circ V = [p]$ is surjective on $D'$, the map $F$ maps $D$ onto $D'$. Hence $\delta$ is in the image of $F$ and so is $\xi$.
The short exact sequence of finite flat group schemes $$\label{ses5_seq}
0 \to E[F]\to E[p] \to E'[V] \to 0,$$ induces, when passing to flat cohomology, the top row of the following exact commutative diagram. $$\xymatrix@R=4mm{
\dots\ar[r] & E'(K)[V]\ar[r] & {H_{\operatorname{fl}}}^1(K,E[F]) \ar[r]\ar[d] & {H_{\operatorname{fl}}}^1(K,E[p])\ar[r]^F \ar[d] & {H_{\operatorname{fl}}}^1(K,E'[V]) \ar[d] \\
& 0 \ar[r] & \prod_v {H_{\operatorname{fl}}}^1(K_v,E)[F] \ar[r] &\prod_v {H_{\operatorname{fl}}}^1(K_v,E)[p] \ar[r]^F &\prod_v {H_{\operatorname{fl}}}^1(K_v,E')[V]
}$$ From the above diagram, we obtain an exact sequence $$\label{ses6_seq}
\xymatrix@R=3mm{
0 \ar[r] & E(K)[F] \ar[r] & E(K)[p] \ar[r]^{F} & E'(K)[V] \ar[r] & &\\
\ar[r] & \operatorname{Sel}_F(E/K)\ar[r] & \operatorname{Sel}_p(E/K) \ar[r]^{F} & \operatorname{Sel}_V(E'/K) \ar[r] & T \ar[r] & 0,
}$$ where $T$ is the cokernel of the map induced by $F$ on the Selmer groups. Parallel to this, we have a long exact (kernel-cokerel) sequence $$\label{kcok_seq}
\xymatrix@R=3mm{
0 \ar[r] & E(K)[F] \ar[r] & E(K)[p] \ar[r]^{F} & E'(K)[V] \ar[r] &\\
\ar[r] & E'(K)/F(E(K)) \ar[r]^V & E(K)/p E(K) \ar[r] & E(K)/V(E'(K)) \ar[r] & 0,
}$$ We may quotient the exact sequence by the exact sequence , using Kummer maps in the short exact sequence . We get an alternative description of $T$ by an exact sequence. $$\label{ses7_seq}
\xymatrix@1{
0 \ar[r] & \Sha(E/K)[F]\ar[r] & \Sha(E/K)[p] \ar[r]^{F} & \Sha(E'/K)[V] \ar[r] & T\ar[r] & 0.
}$$
\[T\_square\_cor\] Let $E/K$ be an elliptic curve. The order of $T$ is a square. In other words, $$\#\Sha(E/K)[F]\cdot \#\Sha(E'/K)[V] \equiv \Sha(E/K)[p] \pmod{\square}.$$
By restriction the Cassels-Tate pairing induces a pairing on $\Sha(E'/K)[V]$ with values in ${{}^{{\mathbb{Z}}}\!/\!{}_{p {\mathbb{Z}}}}$. By the previous proposition the right and left kernels of this pairing are equal to the intersection of $F\bigl(\Sha(E/K)[p^{\infty}]\bigr)$ and $\Sha(E'/K)[V]$, which is equal to $F\bigl(\Sha(E/K)[p]\bigr)$. So the pairing induces a non-degenerate alternating pairing on $T$; hence the order of $T$ is a square.
\[rp\_lem\] We have $$p^{r_p} \equiv \frac{\#E(K)[F] \cdot \#\operatorname{Sel}_V(E'/K)}{\#E'(K)[V] \cdot \#\operatorname{Sel}_F(E/K)} \pmod{\square}.$$
Of course, we have $\# E(K)[F] = 1$, but we include it here so as to make the formula resemble the symmetric formula in the classical case, like in Fisher’s appendix to [@dok_nonab].
By the short exact sequence , $r_p=r+ \operatorname{corank}_{{\mathbb{Z}}_p}\Sha(E/K)[p^\infty]$, where $r=\operatorname{rank}_{{\mathbb{Z}}}(E(K))$ and $r_p$ is the ${\mathbb{Z}}_p$-rank of the dual of $\operatorname{Sel}_{p^\infty}(E/K)$. Now, since $\Sha(E/K)[p^\infty]$ is cofinitely generated as ${\mathbb{Z}}_p$-module, we have $$\dim_{{\mathbb{F}}_p} \Sha(E/K)[p] = \operatorname{corank}_{{\mathbb{Z}}_p} \bigl(\operatorname{div}\Sha(E/K) [p^\infty]\bigr) + \dim_{{\mathbb{F}}_p} \bigl( \Sha(E/K)_{\operatorname{div}} [p] \bigr)$$ As noticed at the beginning of section \[cassels-tate\], $\#\Sha(E/K)_{\operatorname{div}}$ (and therefore $\#\Sha(E/K)_{\operatorname{div}}[p]$) is a square. We deduce that $$r_p \equiv r+\dim_{{\mathbb{F}}_p} \Sha(E/K)[p] \pmod{2}.$$ On the other hand, the short exact sequence applied to $[p]$ implies that $$\dim_{{\mathbb{F}}_p} \operatorname{Sel}_p(E/K) = r + \dim_{{\mathbb{F}}_p} E(K)[p] + \dim_{{\mathbb{F}}_p}\Sha(E/K)[p],$$ since $E(K)/pE(K) \simeq E(K)[p]\oplus ({{}^{{\mathbb{Z}}}\!/\!{}_{p {\mathbb{Z}}}})^r$. So we get the formula $$r_p\equiv \dim_{{\mathbb{F}}_p}E(K)[p] +\dim_{{\mathbb{F}}_p}\operatorname{Sel}_p(E/K) \pmod{2}.$$ The assertion results then from the exact sequence and Corollary \[T\_square\_cor\].
Global duality
==============
\[global\_duality\_prop\] Let $E/K$ be a non-isotrivial elliptic curve and let $U$ be an open subset of $C$ over which $E$ has good reduction. Then we have $$\frac{\#E(K)[F] \cdot \#\operatorname{Sel}_V(E'/K)}{\#E'(K)[V] \cdot \#\operatorname{Sel}_F(E/K)} = \frac{1}{\chi(U,\mathcal{E}[F])}\cdot \prod_{v\not \in U} z(V_{E'(K_v)}).$$
We insist once more that the roles of $F$ and $V$ here are not interchangeable, e.g. the terms $z(F_{E(K_v)})$ in the product would not be finite.
The long exact sequence for flat cohomology deduced from the definition of ${H_{\operatorname{fl},c}}^i$ in Proposition III.0.4.(a) in [@milne] reads $$\xymatrix@1{\cdots \ar[r]& {H_{\operatorname{fl},c}}^i(U,\cdot)\ar[r]&{H_{\operatorname{fl}}}^i(U,\cdot)\ar[r]& \bigoplus_{v\not\in U} {H_{\operatorname{fl}}}^i(K_v,\cdot)\ar[r]& {H_{\operatorname{fl},c}}^{i+1}(U,\cdot)\ar[r]&\cdots}$$ The global duality in Theorem III.8.2 of [@milne] implies that the group ${H_{\operatorname{fl},c}}^i(U, \mathcal{E}[F])$ is dual to ${H_{\operatorname{fl}}}^{3-i}(U,\mathcal{E}'[V])$ since $\mathcal{E}[F]$ is finite and flat over $U$. We find the following long exact sequence $$\xymatrix@R-1ex{ & {H_{\operatorname{fl}}}^1\bigl(U,\mathcal{E}[F]\bigr) \ar[r]& \bigoplus_{v \not \in U} {H_{\operatorname{fl}}}^1\bigl(K_v, E[F]\bigr) \ar[r]& {H_{\operatorname{fl}}}^1\bigl(U,\mathcal{E}'[V]\bigr)^{\vee} \ar[r]& \\
\ar[r] & {H_{\operatorname{fl}}}^2\bigl(U,\mathcal{E}[F]\bigr) \ar[r]& \bigoplus_{v \not \in U} {H_{\operatorname{fl}}}^2\bigl(K_v, E[F]\bigr) \ar[r]& {H_{\operatorname{fl}}}^0\bigl(U,\mathcal{E}'[V]\bigr)^{\vee} \ar[r]& 0
}$$ Local duality as in Theorem III.6.10 in [@milne] shows that ${H_{\operatorname{fl}}}^2(K_v,E[F])$ is dual to $E'(K_v)[V]$. Our aim is to replace the local term ${H_{\operatorname{fl}}}^1(K_v,E[F])$ by the cokernel of the map from $E'(K_v)/F(E(K_v))$. By local duality (Theorem III.7.8 in [@milne] and the functoriality of biextensions), this term is dual to ${H_{\operatorname{fl}}}^1(K_v,E')[V]$. So we will quotient the term ${H_{\operatorname{fl}}}^1\bigl(U,\mathcal{E}'[V]\bigr)^{\vee}$ by the image of the map on the right hand side in the following commutative diagram. $$\label{useless_eq}
\xymatrix@R=9mm{
& \bigoplus_{v \not\in U} {}^{E'(K_v)}\!/\!{}_{F(E(K_v))} \ar[r]^{\cong} \ar[d] & \bigoplus_{v \not \in U} \bigl({H_{\operatorname{fl}}}^1(K_v,E')[V]\bigr)^{\vee} \ar[d] & \\
{H_{\operatorname{fl}}}^1(U,\mathcal{E}[F]) \ar[r] & \bigoplus_{v \not \in U} {H_{\operatorname{fl}}}^1\bigl(K_v, E[F]\bigr) \ar[r]& {H_{\operatorname{fl}}}^1\bigl(U,\mathcal{E}'[V]\bigr)^{\vee} \ar[r] & \dots
}$$ Because of the exact Kummer sequence $$\xymatrix@1{ 0\ar[r] & {}^{E'(K_v)}\!/\!{}_{F(E(K_v))} \ar[r] & {H_{\operatorname{fl}}}^1(K_v,E[F]) \ar[r] & {H_{\operatorname{fl}}}^1(K_v,E)[F] \ar[r] & 0. }$$ the cokernel of the map on the left in is $\bigoplus_{v\not\in U}{H_{\operatorname{fl}}}^1(K_v,E)[F]$, which, again by local duality, is dual to $\bigoplus_{v \not \in U} E(K_v)/V(E'(K_v))$. By definition the cokernel of the map on the right in is the dual of the Selmer group $\operatorname{Sel}_V(E'/K)$.
Putting all these results together, we obtain the long exact sequence $$\xymatrix@R-2ex{ 0\ar[r] & \operatorname{Sel}_F(E/K) \ar[r] & {H_{\operatorname{fl}}}^1\bigl(U,\mathcal{E}[F]\bigr) \ar[r]& \bigoplus_{v \not \in U} \Bigl({}^{E(K_v)}\!/\!{}_{V(E'(K_v))}\Bigr)^{\vee} \ar[r] & \\
\ar[r] & \operatorname{Sel}_V(E'/K)^{\vee} \ar[r] & {H_{\operatorname{fl}}}^2\bigl(U,\mathcal{E}[F]\bigr) \ar[r]& \bigoplus_{v \not \in U} \Bigl(E'(K_v)[V]\Bigr)^{\vee} \ar[r] & \\
\ar[r] & \Bigl(E(K)[V]\Bigr)^{\vee} \ar[r]& 0
}$$ Since all other terms in the sequence are finite, the groups ${H_{\operatorname{fl}}}^i(U,\mathcal{E}[F])$ are finite, too. The alternating product of its orders gives the result.
If $E/K_v$ is a non-isotrivial, semistable elliptic curve then one can show that the group scheme $\mathcal{E}[F]$ is finite and flat. So the result of Proposition \[global\_duality\_prop\] can be extended to any open subset $U$ such that $E$ has semistable reduction over all places in $U$. In particular $U$ can be taken to be equal to $C$, if $E/K$ is semistable.
The proof of the $p$-parity {#pparity_sec}
===========================
We now pass to the proof of Theorem \[pparity\_thm\]. We return now to our running assumptions. $K$ has characteristic $p>3$ and $E/K$ is not isotrivial. We present first the main results coming from global duality and the local computations and then we just have to put them together. But both these statements are interesting in their own right.
\[parity\_corank\_thm\] Let $E/K$ be a non-isotrivial elliptic curve. We have $$p^{r_p} \equiv \prod_v z\Bigl( V_{E'(K_v)}\colon E'(K_v)\to E(K_v)\Bigr) \pmod{\square}$$ where the product runs over all places $v$ in $K$.
Proposition \[global\_chi\_prop\] provides us with an open subset $U$ in $C$ such that $E$ has good ordinary reduction at all places in $U$. It follows from Lemma \[rp\_lem\], Proposition \[global\_duality\_prop\], and Proposition \[global\_chi\_prop\] that $$\begin{aligned}
p^{r_p} &\equiv \frac{\#E(K)[F] \cdot \#\operatorname{Sel}_V(E'/K)}{\#E'(K)[V] \cdot \#\operatorname{Sel}_F(E/K)} \pmod{\square}\\
&= \frac{1}{\chi(U,\mathcal{E}[F])}\cdot \prod_{v\not \in U} z\Bigl( V_{E'(K_v)}\colon E'(K_v)\to E(K_v)\Bigr) \\
& \equiv \prod_{v\not \in U} z\Bigl( V_{E'(K_v)}\colon E'(K_v)\to E(K_v)\Bigr) \pmod{\square}
\end{aligned}$$ Finally from Proposition \[good\_prop\], we know that $z(V_{E'(K_v)})$ is a square for all places $v \in U$ as $E$ has good ordinary reduction there.
Next, we collect from section \[local\_sec\] the following result.
\[rootno\_prop\] Let $E/K$ be a semistable elliptic curve. Then the root number is $w(E/K) = (-1)^s$ where $s$ is the number of split multiplicative primes for $E/K$. Furthermore $s$ has the same parity as the $p$-adic valuation of $$\prod_{v} \frac{ c_v(E/K) }{c_v(E'/K)}$$ where $c_v$ are Tamagawa numbers.
From Theorem \[local\_thm\] we deduce that $$w(E/K) = \prod_v w(E/K_v) = \prod_v \sigma(E/K_v) \cdot \Bigl(\frac{-1}{L_w/K_v}\Bigr) = \prod_v \sigma(E/K_v)$$ by the product formula for the norm symbols $\prod_v \bigl(\frac{-1}{L_w/K_v}\bigr)$ with $L$ being the extension of $K$ over which $E[F]=\mu_p$ and $w$ is any place above $v$. Using the Propositions \[good\_prop\], \[split\_prop\], and \[nonsplit\_prop\], we see that $\sigma(E/K_v)$ is $-1$ if and only if $E$ has split multiplicative reduction at $v$.
If the reduction at $v$ is split multiplicative, then we have $c_v(E'/K) = p\cdot c_v(E/K)$ since the parameters in the Tate parametrisation satisfy $q_{E'} = {q_E}^p$. If the reduction is non-split multiplicative then the Tamagawa numbers can only be 1 or 2.
Note that we could have used the known modularity and the Atkin-Lehner operators to prove this statement without the computations in section \[local\_sec\], at least if $E$ has at least one place of split multiplicative reduction.
First, we use Corollary \[toss\_cor\] which allows us to assume that $E/K$ is semistable. Then by the previous Proposition \[rootno\_prop\] we have $w(E/K) = \prod_v \sigma(E/K_v)$ and Theorem \[parity\_corank\_thm\] states that $(-1)^{r_p}= \prod_v \sigma(E/K_v)$.
Local Root Number Formula {#rootno_sec}
=========================
We now prove Theorem \[local\_thm\] without any hypothesis on the reduction. We use, without repeating the definitions, the notations from section \[local\_sec\].
\[local2\_thm\] Let $K$ be a local field of characteristic $p>3$. For any non-isotrivial elliptic curve $E/K$, we have $w(E/K) = \bigl(\frac{-1}{L/K}\bigr) \cdot \sigma(E/K)$.
As mentioned in section \[local\_sec\] this answers positively a conjecture in [@dok_09] for the isogeny $V$. This theorem could certainly be shown by local computations only, but they would tend to be very tedious for additive potentially supersingular reduction. We can avoid this here by using a global argument. This is a similar idea as in the proof of Theorem 5.7 in [@dok_09].
By Theorem \[local\_thm\], we may assume that $E/K$ has additive, potentially good reduction. Let $n\geqslant 12$. We can find a minimal integral equation $y^2 = x^3 +A\,x + B$ for $E/K$. Choose a global field $\mathcal{K}$ of characteristic $p$ with a place $v_0$ such that $\mathcal{K}_{v_0} = K$. Choose another place $v_1 \neq v_0$ in $\mathcal{K}$ and choose a large even integer $N$ such that $(N-1)\cdot \deg(v_1) > 2g - 1 + n\deg(v_0)$, where $g$ is the genus of $\mathcal{K}$. For a divisor $D$ on the projective smooth curve $\mathcal{C}$ corresponding to $\mathcal{K}$, we write $L(D)$ for the Riemann-Roch space $H^0(\mathcal{C},\mathcal{O}_C(D))$. The inequality on $N$ guarantees that the dimensions of the Riemann-Roch spaces in the exact sequence $$\xymatrix@1{ 0 \ar[r]& L\bigl(N (v_1) - n (v_0) \bigr) \ar[r]& L\bigl(N (v_1) \bigr) \ar[r] & \mathcal{O}_{v_0}/\mathfrak{m}_{v_0}^n \ar[r]& 0 }$$ are positive – e.g. equal to $N\deg(v_1) -n\deg(v_0) + 1 -g> g+\deg(v_1)$ for the smaller space. Choose an element $a$ in $L(N (v_1))$ which maps to $A+\mathfrak{m}_{v_0}^n$ on the right. We can even impose that it does not lie in $L((N-1)(v_1))$, since this is a subspace of codimension $\deg(v_1)>0$ in $L(N(v_1))$. Then $a$ has a single pole of order $N$ at $v_1$ and it satisfies $v_0(A-a) \geq n$. Next, we use that $N$ is even and we choose an element $b$ in $\mathcal{K}$ such that $v_1(b) = -\tfrac{3}{2}N$, and $v_0(B-b)\geq n$. We can also impose that $v_1(4 a^3 + 27 b^2) > -3N$. Furthermore we impose that the zeroes of $b$ are distinct from the zeroes of $a$; this excludes at worst $N\deg(v_1)$ subspaces of codimension $1$ in $L\bigl(\tfrac{3N}{2}(v_1)\bigr)$.
Let $\mathcal{E}/\mathcal{K}$ be the elliptic curve given by $y^2=x^3 +a\,x +b$. By the congruences on $a$ and $b$ at $v_0$ and the continuity of Tate’s algorithm, the reduction of $\mathcal{E}$ at $v_0$ is additive, potentially good. At the place $v_1$ the valuation of the $j$-invariant $j(\mathcal{E}) = 2^8\cdot 3^3\cdot a^3 /(4 \,a^3+27\, b^2)$ will be negative by our choices. Hence the reduction is either multiplicative or potentially multiplicative. For any other place $v$ with $v(a)>0$, we have $v(4 \,a^3+27\, b^2) = 0$ and hence the curve has good reduction at $v$, and for any other place $v$ with $v(a) = 0$, either the reduction is good or $v(j(\mathcal{E}))< 0$.
Therefore we have constructed an elliptic curve $\mathcal{E}/\mathcal{K}$ with a single place $v_0$ of additive, potentially good reduction. So for all other places Theorem \[local\_thm\] applies. Let $\mathcal{L}$ be the extension of $\mathcal{K}$ such that $E[F]\cong\mu_p$ over $\mathcal{L}$. Now we use the results of Theorem \[parity\_corank\_thm\] and the proven $p$-parity in Theorem \[pparity\_thm\] to compute $$\begin{aligned}
w(\mathcal{E}/K) &= \frac{w(\mathcal{E}/\mathcal{K})}{\prod_{v\neq v_0} w(\mathcal{E}/\mathcal{K}_{v})}
= \frac{(-1)^{r_p}}{
\prod_{v\neq v_0}\bigl(\frac{-1}{\mathcal{L}_w/\mathcal{K}_v}\bigr)
\sigma(\mathcal{E}/\mathcal{K}_{v})} \\
& = \frac{\bigl(\frac{-1}{\mathcal{L}_{w_0}/K}\bigr)}{
\prod_{\textnormal{all }v} \bigl(\frac{-1}{\mathcal{L}_w/\mathcal{K}_v}\bigr)}
\cdot\frac{\prod_{\textnormal{all }v} \sigma(\mathcal{E}/\mathcal{K}_{v})}{
\prod_{v\neq v_0}\sigma(\mathcal{E}/\mathcal{K}_{v})}
= \Bigl(\frac{-1}{\mathcal{L}_{w_0}/K}\Bigr)\cdot\sigma(\mathcal{E}/K).
\end{aligned}$$ Once again we used the product formula for the norm symbol. Now we argue that the three terms are all continuous in the topology of $K$ as $a$ and $b$ varies: For the local root number this is exactly the statement of Proposition 4.2 in [@helfgott]. The field $\mathcal{L}_{w_0}$ and the order of the kernel $E'(K)[V]$ of Verschiebung are locally constant because they are defined by continuously varying separable polynomials. Finally, the order of the cokernel of $V\colon E'(K)\to E(K)$ is locally constant, because the group of connected components and the reduction and the induced map $V$ on them will not change and on the formal group the cokernel is determined by the valuation of the Hasse invariant (which again is a polynomial in $a$ and $b$) by the argument in the proof of Proposition \[good\_prop\]. Since all three terms take value $\pm 1$, they will eventually, for big enough $n$, be equal to the corresponding values for $E$.
On the $\ell$-parity conjecture {#ell_sec}
===============================
We switch now to investigating the $\ell$-parity conjecture when $\ell\neq p$. As mentioned in the introduction, we have only a partial result in this case. Recall that $p>3$ is a prime and that $K$ is a global field of characteristic $p$ with constant field ${\mathbb{F}}_q$.
For any $n$ and any extension $L$ of $K$, we denote by $L_n$ the field $L\cdot {\mathbb{F}}_{q^n}$. The aim of this section is to show the following partial result (given as Theorem \[ellparity\_thm\] in the introduction).
\[ell\_thm\] Let $E/K$ be an elliptic curve and let $\ell$ be an odd prime different from $p$. Furthermore assume that
1. \[a\_as\] $a=[K(\mu_{\ell}):K]$ is even and that
2. \[b\_as\] the analytic rank of $E$ does not grow by more than $1$ in the extension $K_2/K$.
Then the $\ell$-parity conjecture holds for $E/K$.
Note first that we believe that (\[a\_as\]) holds for roughly two thirds of the $\ell$ as $a$ is also the order of $q$ in the group $({{}^{{\mathbb{Z}}}\!/\!{}_{\ell {\mathbb{Z}}}})^{\times}$. The second condition should hold quite often as it says that the analytic rank of the twist of $E$ by the unramified quadratic character is less or equal to $1$. So for instance if $b$ as in Lemma 11.3.1 in [@ulmer_gnv] is odd, then (\[b\_as\]) holds.
See the next section for a discussion about why we were not able to extend the proof here to any situation without these hypotheses.
Corollary \[toss\_cor\] allows us to assume that $E$ is semistable and we may assume that $E$ is not isotrivial as for isotrivial curves even BSD is known. First we use a non-vanishing result, to produce from the analytic information a useful extension of $K$, which we want to link to the algebraic side later.
We write $\mathfrak{n}$ for the conductor of $E/K$. The degree of $\mathfrak{n}$ is linked to the degree of the polynomial $L(E/K,T)$ in $T=q^{-s}$ by the formula of Grothendieck-Ogg-Shafarevich (as used in formula (5.1) of [@ulmer_analogies]): $$\deg (\mathfrak{n}) = \deg \bigl(L(E/K,T)\bigr) - 2(2g_{K} -2).$$ We can factor the polynomial to $$L(E/K,T) = (1-qT)^r \cdot (1+qT)^{r'}\cdot \prod_{i} (1-\alpha_i T)(1-\bar\alpha_i T)$$ where $\alpha_i$ are non-real, complex numbers of absolute value $q$. By definition $r$ is the analytic rank of $E/K$ and it is easy to see that $r+r'$ is the analytic rank of $E/K_2$ since the analytic rank of $E/K_2$ is the number of inverse zeroes $\alpha$ of $L(E/K,T)$ such that $\alpha^2 = q^2$. So we get $$\label{nran_eq}
\sum_{v \text{ bad}} \deg(v) = \deg(\mathfrak{n})\equiv \deg \bigl(L(E/K,T)\bigr) \equiv r+r' = \operatorname{ord}_{s=1} L(E/K_2,s) \pmod{2}.$$
We are now going to use Theorem 5.2 in [@ulmer_gnv] to construct suitable extensions of $K$. The argument is very similar to the proof of Step 2 in 11.4.2 of [@ulmer_gnv]. The following is a very special case of this very general and powerful theorem.
\[ulmer\_main\_thm\] Let $K$ be a global field of characteristic $p>3$, let $S$ be a finite non-empty set of places in $K$, let $\ell\neq p$ be an odd prime, and let $E/K$ be a semistable elliptic curve. Assume that $a=[K(\mu_{\ell}):K]$ is even and suppose that the sum of the degree of the bad places not belonging to $S$ is even. Then there exists an integer $n$ coprime to $a$ and a element $z\in K_n^{\times}$ such that the extension $K_n(\sqrt[\ell]{z})/K_n$ is totally ramified at all places above $S$ and unramified at all bad places not in $S$ and such that the analytic rank of $E$ does not grow in it.
All notations and results in this proof refer to [@ulmer_gnv]. We use Theorem 5.2.(1) with $F=K$, $\alpha_n=q^n$, $d=\ell$, $S_r = S$ and $\rho$ the symplectically self-dual representation of weight $w=1$ attached to $E$ on the Tate module $V_{\ell}(E)$ as in section 11. We can choose the sets $S_s$ and $S_i$ arbitrarily as long as we make sure that $S$, $S_s$, and $S_i$ are disjoint. The conditions (especially from his section 3.1) are satisfied. Let $o$ be an orbit in $({{}^{{\mathbb{Z}}}\!/\!{}_{\ell {\mathbb{Z}}}})^{\times}$ for the multiplication by $q$. Then $d_{o}=\ell$ and $a_{o} = a$. So we can conclude the existence of $n$ and $z$ such that $L(\rho\otimes\sigma_{o,z},K_n,T)$ does not have $\alpha_n$ as an inverse root in 5.2.(1) unless we are in the exceptional cases (i) to (iv) in 5.1.1.1. Now, (iv) cannot hold because $\rho$ is not orthogonally self-dual and (i) and (ii) are impossible because $d=\ell$ is odd. However, all the condition in (iii) are satisfied apart from maybe the condition 4.2.3.1. (In particular, we know that $-o=o$ because $a$ is even.)
We now have to show that the hypothesis on $S$ imposes that the condition 4.2.3.1 fails. Since $E$ is semistable, the local exponent of the conductor $\operatorname{cond}_v(\rho)$ is $1$. Let $v$ be a bad place in $S$ and $\chi_v$ be a totally ramified character of the decomposition group $D_v$ which has exact order $\ell$. Then the conductor $\operatorname{cond}_v(\rho\otimes\chi_v)=2$ again because $E$ has multiplicative reduction at $v$. So the first condition in 4.2.3.1 saying that this has constant parity as $\chi_v$ varies is always fulfilled. In order to make the condition 4.2.3.1 fail, we must have that $$\sum_{\text{bad }v \in S} \operatorname{cond}_v(\rho\otimes\chi_v)\deg(v) + \sum_{\text{bad } v \not\in S} \operatorname{cond}_v(\rho) \deg(v)$$ is even. That is exactly what the hypothesis in the theorem imposes.
\[ellext\_lem\] To prove Theorem \[ell\_thm\], we may assume that there exists a non-constant Kummer extension $L/K$ of degree $\ell$ in which the analytic rank does not grow and such that
- if the analytic rank of $E/K_2$ is even then no place of bad reduction ramifies in $L/K$, or
- if the analytic rank of $E/K_2$ is odd then exactly one place of bad reduction ramifies. Moreover, in the latter case, the degree of this place is odd.
If the analytic rank is even we choose the finite non-empty set of places $S$ to be disjoint from the set of bad places. If the analytic rank is odd, then the congruence shows that there is at least one bad place $v$ of odd degree. So we choose $S$ to contain this as the only bad place. Then shows that the hypothesis in Theorem \[ulmer\_main\_thm\] with the above choice for $S$ holds. So we have an integer $n$ and an element $z \in K_n^{\times}$. Now we use the first item in Proposition \[red\_prop\] to replace $K$ by its odd Galois extension $K_n$. So $L=K(\sqrt[\ell]{z})$ is the requested extension.
We now come to the algebraic part of the argument. Using the previous two lemmata, we have now a Kummer extension $L/K$ of degree $\ell$ in which the analytic rank does not grow. The Galois closure of $L/K$ is $L_a$ containing $L_2$. We have the following picture of extensions $$\xymatrix@R-1ex@C+1.5ex{
&& L_a \ar@{-}[dddll]_{\langle\sigma\rangle}^{\ell}
\ar@{-}[rd]^{\langle\tau^2\rangle}_{a/2}
\ar@{-}@/^3pc/[rrdd]_{a}^{\langle\tau\rangle}&& \\
&&& L_2 \ar@{.}[dddll]^{\ell} \ar@{-}[rd]_{2} & \\
&&&& L \ar@{.}[dddll]^{\ell} \\
K_a \ar@{-}[rd]_{a/2} &&&&\\
& K_2 \ar@{-}[rd]_{2}&&& \\
&& K &&
}$$ The dotted lines are non-Galois extensions. We have written the degree under each inclusion. The Galois group $G=\operatorname{Gal}(L_a/K)$ is a meta-cyclic group generated by elements $\sigma$ and $\tau$ of order $\ell$ and $a$ respectively, with $L=(L_a)^{\tau}$. We have $$G = \bigl\langle \sigma,\tau\bigl\vert \tau^a=\sigma^{\ell} = 1, \tau\sigma\tau^{-1} = \sigma^q\bigr\rangle.$$
We list the irreducible ${\mathbb{Q}}_{\ell}[G]$-modules. By $\mathbbm{1}$ we denote the trivial representation. Fix a primitive character $\chi\colon\langle\tau\rangle \cong {{}^{{\mathbb{Z}}}\!/\!{}_{a {\mathbb{Z}}}}\cdot\tau\to {\mathbb{Q}}_{\ell}^{\times}$ that we can view as a character of $G$ by setting $\chi(\sigma)=1$. (Note that $a$ divides $\ell-1$, so $\chi$ is indeed realisable over ${\mathbb{Q}}_{\ell}$.) The non-trivial $1$-dimensional representations of $G$ are exactly the $\chi^i$ for $1\leqslant i \leqslant a-1$. There is only one non-trivial irreducible ${\mathbb{Q}}_{\ell}[\langle\sigma\rangle]$-module. It is of degree $\ell-1$. We can represent it as $\rho = {\mathbb{Q}}_{\ell}[\xi]$ where $\xi$ is a primitive $\ell$^th^ root of unity and $\sigma$ acts on $\rho$ by multiplication with $\xi$. (Over $\bar{\mathbb{Q}}_{\ell}$ is would split into the $\ell-1$ non-trivial characters of $\langle\sigma\rangle\cong{{}^{{\mathbb{Z}}}\!/\!{}_{\ell {\mathbb{Z}}}}$.) We make $\rho$ into a $G$-module by defining the ${\mathbb{Q}}_{\ell}$-linear action of $\tau$ by $\tau(\xi^j) = \xi^{qj}$ for all $0\leqslant j\leqslant \ell-2$. It is easy to see that $\rho$ is an irreducible ${\mathbb{Q}}_\ell[G]$-module of degree $\ell-1$ and in fact it is the only higher dimensional irreducible ${\mathbb{Q}}_\ell[G]$-module. (Note that $\rho\otimes\overline{\mathbb{Q}}_{\ell}$ decomposes into $\tfrac{\ell-1}{a}$ irreducibles of degree $a$ corresponding to the orbits of the multiplication by $q$ on $({{}^{{\mathbb{Z}}}\!/\!{}_{\ell {\mathbb{Z}}}})^{\times}$.) We have $${\mathbb{Q}}_{\ell}[G] = \mathbbm{1} \oplus\bigoplus_{i=1}^{a-1} \chi^i \oplus \rho^a.$$ For convenience we will denote $\chi^{a/2}$ by $\varepsilon$. The fixed field of the kernel of $\varepsilon$ is $K_2$.
To announce the next lemma, we need to introduce the corrected product of Tamagawa numbers. Fix the invariant $1$-form $\omega$ on $E/K$ corresponding to the fixed Weierstrass equation. For each place $v$, write $c_v(E/K)$ for the Tamagawa number and define $$C_v(E/K,\omega) = c_v(E/K)\cdot \Biggl\vert \frac{\omega}{\omega^{o}_{v}}\Biggr\vert_v$$ where $\omega^o_v$ is a Néron differential for $E/K_v$. The global product over all places $v$ of $K$ $$C(E/K) = \prod_v C_v(E/K,\omega)$$ is no longer dependent on the choice of $\omega$ by the product formula.
For any irreducible ${\mathbb{Q}}_\ell[G]$-module $\psi$, write $m_{\psi}$ for the multiplicity of the $\psi$-part of the $\ell$-primary Selmer group $\operatorname{Sel}_{\ell^{\infty}}(E/L_2)$.
We have $$m_{\mathbbm{1}} + m_{\varepsilon} + m_{\rho} \equiv \operatorname{ord}_{\ell} \Biggl( \frac{C(E/L_2)}{C(E/K_2)} \Biggr) \pmod{2}.$$
We are interested in the following relation between permutations representations (in the terminology of Dokchitsers’ work, say 2.3 in [@dok_modsquares]) $$\Theta = 2 \cdot G + \langle \tau^2 \rangle - 2 \cdot \langle \tau\rangle - \langle \sigma,\tau^2\rangle$$ corresponding to the equality of $L$-functions $$L(E/K,s)^2 \cdot L(E/L_2,s) = L(E,\mathbbm{1},s)^3 \cdot L(E,\varepsilon,s)\cdot L(E,\rho,s)^2
= L(E/K_2,s) \cdot L(E/L,s)^2.$$ It can be seen that the regulator constants (as defined in 2.11 of [@dok_modsquares]) satisfy $$C_{\Theta}(\mathbbm{1})\equiv C_{\Theta}(\varepsilon)\equiv C_{\Theta}(\rho)\equiv\ell\pmod{\square}$$ in ${\mathbb{Q}}^{\times}$ modulo squares. For $\mathbbm{1}$ and $\varepsilon$ this is straightforward; for $\rho$ we best use Theorem 4.(4) of [@dok_reg] with $D=\langle\tau\rangle$, implying that $$C_{\Theta}(\rho)\cdot C_{\Theta}(\mathbbm{1}) = C_{\Theta}\Bigl({\mathbb{Q}}_\ell[G/\langle\tau\rangle]\Bigr) = 1.$$ So $S_{\Theta} = \{\mathbbm{1},\varepsilon,\rho\}$ in Dokchitsers’ notation in [@dok_sd].
In short everything looks just like if $L_2/K$ were a dihedral extension (which it is not unless $a=2$). For $a=2$ this is computed in Example 1 in [@dok_reg] and Example 4.5 in [@dok_modsquares] and Example 3.5 in [@dok_sd]. For $a=\ell-1$, this is Example 2.20 in [@dok_modsquares] and Example 3.6 in [@dok_sd].
Now, Theorem 1.6 in [@dok_sd] shows that $$m_{\mathbbm{1}} + m_{\varepsilon} + m_{\rho} \equiv \operatorname{ord}_{\ell} \Biggl( \frac{C(E/K)^2\cdot C(E/L_2)}{C(E/L)^2\cdot C(E/K_2)} \Biggr) \pmod{2}$$ which proves the lemma.
\[fracc\_lem\] Suppose that no bad place ramifies in $L/K$, then the $\ell$-adic valuation of the integer $C(E/L_2)/C(E/K_2)$ is even. If there is only one bad place that ramifies in $L/K$ and this place is of odd degree, then the $\ell$-adic valuation of $C(E/L_2)/C(E/K_2)$ is odd.
The more general statement for $a=2$ can be found in Remark 4.18 in [@dok_modsquares].
Let $v$ be a place of $K_2$. Write $y$ for $\omega/\omega^o_v$. Then $$\frac{\prod_{w\mid v} C_w(E/L_2,\omega)}{C_v(E/K_2,\omega)}
= \frac{\prod_{w\mid v} c_w(E/L_2)}{c_v(E/K_2)}\cdot \frac{\prod_{w\mid v} \vert y \vert_w}{\vert y \vert_v}
\equiv \frac{\prod_{w\mid v} c_w(E/L_2)}{c_v(E/K_2)} \pmod{\square}$$ because $\prod_{w\mid v} \vert y \vert_w / \vert y \vert_v = \vert y \vert_v^{\ell-1}$ is a square. If the place $v$ is unramified, then the type of reduction and the Tamagawa number do not change and we have $$\frac{\prod_{w\mid v} c_w(E/L_2)}{c_v(E/K_2)} =
\begin{cases}
c_v(E/K_2)^{\ell-1} \quad\ & \text{ if $v$ decomposes in $L_2/K_2$ and}\\
1 & \text{ if $v$ is inert.}
\end{cases}$$ In either case it is a square. If the reduction is good at $v$ then $c_w(E/L_2) = c_v(E/K_2) = 1$. This proves the first case.
Suppose now $v$ is a place in $K_2$ which lies above a place of odd degree in $K$ and which ramifies in $L_2/K_2$. Then the place is inert in $K_2/K$ and hence the reduction of $E/K_2$ at $v$ is necessarily split multiplicative. Let $q$ be the Tate parameter of $E$ at $v$. Then $c_v(E/K_2) = v(q)$ and $c_w(E/L_2) = w(q) = \ell\cdot v(q)$ for the place $w$ above $v$. So the quotient is $\ell$ which has odd $\ell$-adic valuation. This proves the second statement.
Finally we can finish the proof of Theorem \[ell\_thm\]. By construction, we have $\operatorname{ord}_{s=1} L(E,\rho,s) = 0$. As in the proof of Proposition \[red\_prop\], this implies that $m_{\rho} = 0$; in fact $L(E,\rho,s)$ is $L(B/K,s)$ for the extension $L/K$. So the last two lemmata show that $m_{\mathbbm{1}}+m_{\varepsilon}$, which is the corank of the $\ell$-primary Selmer group $\operatorname{Sel}_{\ell^{\infty}}(E/K_2)$, has the same parity as the analytic rank of $E/K_2$. This proves the $\ell$-parity conjecture for $E/K_2$. Assumption (\[b\_as\]) and Proposition \[red\_prop\] prove that the $\ell$-parity holds over $K$, too.
Failure to extend {#fail_sec}
=================
Although it is not usual to write in a mathematical article about unsuccessful attempts to prove a result, we wish to include in this last section a short explanation of why we were unable to extend the proof in the previous section. We hope this might be the starting point for a complete proof of the $\ell$-parity conjecture. We try to outline here the missing non-vanishing result for $L$-functions, which might be accessible using automorphic methods.
The main ingredient for proving Theorem \[ell\_thm\] was the existence of a Kummer extension of degree $\ell$ in which the analytic rank does not grow. Moreover this extension was linked in a “non-commutative way” to an even abelian extension. The machinery using representation theory set up by Tim and Vladimir Dokchitser is then sufficient to prove the parity.
First, if condition (\[b\_as\]) in Theorem \[ell\_thm\] does not hold but condition (\[a\_as\]) still holds, then there is no hope that a Kummer extension will do. In order to obtain a Galois extension of $K$ from a Kummer extension, we need to make the extension $K_2/K$. But without any control about the growth of the analytic rank in this quadratic extension, we do not know how to prove the $\ell$-parity over $K$. With some extra work, one can conclude that the $\ell$-parity conjecture holds for $E/K_2$. In this case, we would need a non-vanishing result for an extension of $K$ of degree dividing $\ell$ which is not a Kummer extension.
Suppose now that the condition (\[a\_as\]) does not hold. Then we would need to find the “dihedral” extension somewhere else. The Proposition \[ell\_non\_prop\] below formulates this in a positive way.
\[ell\_non\_prop\] Suppose $E/K$ is semi-stable and non-isotrivial. Let $F/K$ be a quadratic extension such that the analytic rank of $E$ grows at most by $1$ in $F/K$ and the analytic rank of $E/F$ is even. Assume
1. $a=[F(\mu_{\ell}):F]$ is odd and
2. there exists an odd $n\geqslant 1$ and a $z\in F_n^{\times}$ with the property that $L=F_n(\sqrt[\ell]{z})$ is an extension of degree $\ell$ of $F_n$ such that $L_a/K_{an}$ is a dihedral extension, no bad place of $E/F_n$ ramifies in $L/F_n$, and the analytic rank of $E$ does not grow in $L/F_n$.
Then the $\ell$-parity conjecture holds for $E/K$.
Remark that there is a large supply of quadratic extensions $F/K$ by Theorem 11.2 in [@ulmer_gnv]. The main problem here seems to find the extension $L_a/F_{an}$. Theorem 5.2 in [@ulmer_gnv] provides us with many extensions that satisfy all the properties except that we can not guarantee that $L_a/K_{an}$ is dihedral. We first had hopes that Ulmer’s proof could be adapted to enforce that $L/K_{n}$ is dihedral. In the notations of [@ulmer_gnv], we may sketch the problem. Let $D$ be a divisor of large degree as in section 6.2. Then the density (as $n$ grows) of elements in the Riemann-Roch space $H^1(\mathcal{C}\times {\mathbb{F}}_{q^n}, \mathcal{O}(D))$ which give rise to a dihedral extension of $F_n$ with respect to the fixed quadratic extension $F/K$ will be tending very fast to 0. So we would need to modify the parameter space $X$ and it is not clear how to find a nice variety parametrising such dihedral extensions.
This is very similar to the proof of Theorem \[ell\_thm\]. By Proposition \[red\_prop\], we may assume that $a=1$ and $n=1$. So $L/K$ is a dihedral extension with group $G$. Let $\rho$ be the irreducible ${\mathbb{Q}}_{\ell}[G]$-module of degree $\ell-1$ and let $\varepsilon$ the character corresponding to the quadratic extension $F/K$. The usual relation of induced representation $\Theta$ for $G$, as in Example 3.5 in [@dok_sd], yields the congruence $$m_{\mathbbm{1}} + m_{\varepsilon} + m_{\rho} \equiv \operatorname{ord}_{\ell}\Bigl(\frac{C(E/L)}{C(E/F)}\Bigr) \pmod{2}.$$ We have $m_{\rho} = 0$ and from Lemma \[fracc\_lem\], we know that the assumption that no bad place ramifies in $L/K$ implies that $\frac{C(E/L)}{C(E/F)}$ has even $\ell$-adic valuation. This implies that $m_{\mathbbm{1}} + m_{\varepsilon}$ is even, i.e. the $\ell$-parity is valid for $E/F$. By Proposition \[red\_prop\] implies that the $\ell$-parity also holds for $E/K$.
Acknowledgements {#acknowledgements .unnumbered}
================
We express our gratitude to Tim and Vladimir Dokchitser, Jean Gillibert, Christian Liedtke, James S. Milne, Takeshi Saito, Ki-Seng Tan, Douglas Ulmer, and the anonymous referee for useful comments on the preliminary versions of this paper.
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---
abstract: 'We consider the difference Schr[ö]{}dinger equation $\psi(z+h)+\psi(z-h)+ v(z)\psi(z)=E\psi(z)$ where $z$ is a complex variable, $E$ is a spectral parameter, and $h$ is a small positive parameter. If the potential $v$ is an analytic function, then, for $h$ sufficiently small, the analytic solutions to this equation have standard semi-classical behavior that can be described by means of an analog of the complex WKB method for differential equations. In the present paper, we assume that $v$ has a simple pole and, in its neighborhood, we study the asymptotics of meromorphic solutions to the difference Schr[ö]{}dinger equation.'
address:
- 'St. Petersburg State University, 7/9 Universitetskaya nab., St.Petersburg, 199034, Russia'
- 'Sorbonne Universit[é]{}, Universit[é]{} Paris Diderot, CNRS, Institut de Math[é]{}matiques de Jussieu - Paris Rive Gauche , F-75005, Paris, France'
author:
- 'Alexander Fedotov and Fr[é]{}d[é]{}ric Klopp'
title: Semiclassical asymptotics of meromorphic solutions of difference equations
---
Introduction
============
We study the difference Schr[ö]{}dinger equation $$\label{main}
\psi(z+h)+\psi(z-h)+v(z)\psi(z)=E\psi,$$ where $z$ is a complex variable, $v$ is a given meromorphic function called [*potential*]{}, $E$ is a [*spectral*]{} parameter, and $h$ is a small positive [*shift*]{} parameter [^1].\
Instead of , one often considers equations of the form $$\label{main:1}
\phi_{k+1}+\phi_{k-1}+v(kh+\theta)\phi_k=E\phi_k,$$ where $k\in{{\mathbb Z}}$ is an integer variable, and $\theta\in{{\mathbb R}}$ is a parameter. There is a simple relation between and : if $\psi$ is a solution to , then the formula $\phi_k=\psi(kh+\theta)$ yields a solution to . We note that, when $h$ is small, the coefficient in front of $\phi_k$ in varies slowly in $k$.\
Formally, $\psi(z+h)=\sum_{l=0}^\infty \frac{h^l}{l!}
\frac{d^l\psi}{dz^l}(z)=e^{h\frac{d}{dz}}\psi(z)$; thus, in $h$ is a small parameter in front of the derivative. So, $h$ is a standard semiclassical parameter.\
The semi-classical asymptotics of solutions to ordinary differential equations, e.g., the Schr[ö]{}dinger equation $$\label{differential-eq}
-h^2\frac{d^2\psi}{dz^2}(z)+v(z)\psi(z)=E\psi(z),$$ are described by means of the well-known WKB method (called so after G. Wentzel, H. Kramers and L. Brillouin). There is a huge literature devoted to this method and its applications. If $v$ in is analytic, one uses the variant often called the complex WKB method (see, e.g., [@Wa:87; @Fe:93]). This powerful and classical asymptotic method is used to study solutions to on the complex plane. Even when studying this equation on the real line, the complex WKB method is used to compute exponentially small quantities (such as the overbarier tunneling coefficient or the exponentially small lengths of spectral gaps in the case of a periodic $v$) or to simplify the asymptotic analysis (e.g. by going to the complex plane to avoid turning points), see, e.g., [@Fe:93]. The case of meromorphic coefficients is a classical topic in the complex WKB theory. The analog of the complex WKB method for difference equations is being developed in [@B-F:94; @F-Shch:18; @F-K:18] and in the present paper, where we turn to meromorphic solutions to \[main\].\
Difference equations on ${{\mathbb R}}$ or on ${{\mathbb C}}$ and on ${{\mathbb Z}}$ with a small $h$ arise in many fields of mathematics and physics. In quantum physics, for example, one encounters such equations when studying in various asymptotic situations an electron in a two-dimensional crystal submitted to a constant magnetic field (see, e.g. [@Wi] and references therein). The electron is described by a magnetic Schr[ö]{}dinger operator with a periodic electric potential. And, for example, in the semi-classical limit, in certain cases its analysis asymptotically reduces to analyzing an $h$-pseudo-differential operator with the symbol $H(x,p)= 2\cos p+2\cos x$ (see [@H-S:88]). Its eigenfunctions satisfy \[main\] with $v(z)=2\cos z$. The parameter $h$ is proportional to the magnetic flux through the periodicity cell, and the case when $h$ is small is a natural one. The reader can find more references and examples in [@GHT:89]. We add only that \[main:1\] for $v(z)=\lambda \cos(2\pi z)$, $\lambda$ being a coupling constant, is the famous almost Mathieu equation (see, e.g., [@A-J:09]), and, for $v(z)=\lambda \cot(\pi z)$, it is the well known Maryland equation (Maryland model) introduced by specialists in solid state physics in [@GFP].\
Difference equations in the complex plane (with analytic or meromorphic coefficients) arise in many other fields of mathematics and physics, in particular, in the study of the diffraction of acoustic waves by wedges (see, e.g., [@B-L-G:2008]) or in the theory of differential quasi-periodic equations (see, e.g., [@F-K:02]). Small shift parameters arise in the problem of diffraction by thin wedges (the shift parameter is proportional to the angle of the wedge (see [@B-L-G:2008])) and for quasi-periodic equations with two periods of small ratio (the shift parameter is proportional to this ratio (see, e.g., [@F-K:02])). The semi-classical analysis of difference equations is also used to study the asymptotics of orthogonal polynomials (see, e.g., [@G-at-al; @Dobro; @Wo:03]).\
Even when studying on ${{\mathbb Z}}$, it is quite natural to pass to the analysis of on ${{\mathbb R}}$ or on ${{\mathbb C}}$ as for this equation one can fruitfully use numerous analytic ideas developed in the theory of differential equations, e.g., tools of the theory of pseudo-differential operators and of the complex WKB method. If the coefficient $v$ is periodic, for equation one can use ideas of the Floquet theory for differential equations with periodic coefficients, which leads to a natural renormalization method (see [@F:13]).\
B. Helffer and J. Sj[ö]{}strand (e.g., in [@H-S:88]) and V. Buslaev and A. Fedotov (see, e.g. [@F:13]) studied the cantorian geometrical structure of the spectrum of the Harper operator in the semiclassical approximation. Therefore, V. Buslaev and A. Fedotov began to develop the complex WKB method for difference equations in [@B-F:94]. We are going to use the results of the present paper to study in the semiclassical approximation the multiscale structure of the (generalized) eigenfunctions of the Maryland operator (by means of the renormalization method described in [@F-S:15]). In the “anti-semiclassical” case, for the almost Mathieu operator such a problem was solved in [@J-W].\
In this paper, for small $h$, we describe uniform asymptotics of meromorphic solutions to \[main\] near a simple pole of $v$. In the case of a differential equation, say, \[differential-eq\] with a meromorphic $v$, the solutions may have singularities (branch points or isolated singular points) only at poles of $v$. In the case of \[main\], the behavior of its solutions is completely different.\
Let $d_x>0$, $d_y>0$, and $S=\{z\in{{\mathbb C}}\,:\,|{{\rm Re}\,}z|<d_x, \,|{{\rm Im}\,}z|<d_y\}$. We assume that $v$ is analytic in $S\setminus\{0\}$ and has a simple pole at zero. Let $\psi$ be a solution to \[main\] that is analytic in $\{z\in S\,:\,{{\rm Re}\,}z<0\}$. implies that $\psi(z)=-\psi(z-2h)-v(z-h)\psi(z-h)$. Therefore, for sufficiently small $h$, $\psi$ can be meromorphically continued into $S$. It may have poles at the points of $h{{\mathbb N}}$. When $h$ becomes small, these points become close one to another. We describe the semi-classical asymptotics of such meromorphic solutions in $S$.\
Below, unless stated otherwise, the estimates of the error terms in the asymptotic formulas are locally uniform for $z$ in the domain that we consider (i.e., uniform on any given compact subset of such a domain).\
Instead of saying that an asymptotic representation is valid for sufficiently small $h$, we write that it is valid as $h\to0$.\
In the sequel, we shall not distinguish between a meromorphic function and its meromorphic continuation to a larger domain.\
We also use the notations ${{\mathbb R}}_\pm=\{z\in{{\mathbb C}}\,:\, {{\rm Im}\,}z=0, \ \pm{{\rm Re}\,}z\ge 0\}$, ${{\mathbb R}}_\pm^*={{\mathbb R}}_\pm\setminus{0}$ and ${{\mathbb C}}_\pm=\{z\in{{\mathbb C}}\,:\, \pm{{\rm Im}\,}z>0\}$.
Main results
============
The complex WKB method in a nutshell {#intro:WKB}
------------------------------------
Formally \[main\] can be written in the form $$\begin{gathered}
\label{differential}
(2\cos\hat p+v(z))\psi(z)=0,\quad \hat p=-ih\frac{d}{dz}\;.\end{gathered}$$ One of the main objects of the complex WKB method is [*the complex momentum*]{} $p$ defined by the formula $$\label{eq:p}
2\cos\,p(z)+v(z)=0.$$ It is an analytic multivalued function. Its branch points are solutions to $v(z)=\pm 2$. The points where $v(z)=\pm 2$ are called [*turning points*]{}. A subset $D$ of the domain of analyticity of $v$ is [*regular*]{} if it contains no turning points.\
Let $D$ be a regular simply connected domain, and $p_0$ be a branch of the complex momentum analytic in $D$. Any other branch of the complex momentum that is analytic in $D$ is of the form $s p_0+2\pi m$ for some $s\in\{\pm 1\}$ and $m\in{{\mathbb Z}}$.\
In terms of the complex momentum, one defines [*canonical*]{} domains. The precise definition of a canonical domain can be found in \[ss:basics-of-WKB\]. Here, we note only that the canonical domains are regular and simply connected and that any regular point is contained in a canonical domain (independent of $h$).\
One of the basic results of the complex WKB method is
\[th:wkb\_main\] Let $K\subset \mathbb C$ be a bounded canonical domain; let $p$ be a branch of the complex momentum analytic in $K$ and pick $z_0\in K$. For sufficiently small $h$, there exists $\psi$, a solution to \[main\] analytic in $K$ and such that, in $K$, one has $$\label{standard_asymptotic}
\psi(z)=\frac1{\sqrt{\sin p(z)}}\, e^{\frac{i}{h}\int_{z_0}^z
p(\zeta)\,d\zeta+o(1)},\qquad h\to 0,$$ where $z\mapsto\sqrt{\sin(p(z))}$ is analytic in $K$.
\[rem:sin\] The function $\sin p$ does not vanish in regular domains; indeed, $\sin p$ only vanishes at the points where $v(z)=-2\cos p(z)\in\{\pm 2\}$, i.e., at the turning points.
In the case of the Harper equation (for unbounded canonical domains), the analog of Theorem \[th:wkb\_main\] was proved in [@B-F:94].\
Let us underline that the branch $p$ of the complex momentum in Theorem \[th:wkb\_main\] need not be the one with respect to which $K$ is canonical.
Asymptotics of a meromorphic solution {#ss:main}
-------------------------------------
Let us turn to the problem discussed in the present paper. Recall that 0 is a simple pole of $v$. Since $v(z)\to\infty$ as $z\to0$, reducing somewhat $d_x$ and $d_y$ if necessary, we can and do assume that the set $S\setminus \{0\}$ is regular and that the imaginary part of the complex momentum does not vanish there.
### The solution we study {#sss:main}
Let $S'=S\setminus{\mathbb R}_+$. In $S'$, fix an analytic branch $p$ of the complex momentum satisfying ${{\rm Im}\,}p(z)<0$.\
Pick a point $z_0$ in $S'\cap {{\mathbb R}}_-$. As this point is regular, there exists a solution $\psi$ to \[main\] that is analytic in a neighborhood of $z_0$ independent of $h$ and that admits the asymptotic representation in this neighborhood.\
Adjusting $d_x$ and $d_y$ if necessary, we can and do assume that there exists $c\in (0, d_x)$ such that $\psi$ is analytic and admits the asymptotic representation in the domain $S_c=\{z\in S\,:\, {{\rm Re}\,}z<-c\}$.\
As ${{\rm Im}\,}p(z)<0$ in $S'$, the expression $z\mapsto \left| e^{\frac{i}{h}\int_{z_0}^z p(\zeta)\,d\zeta}\right|$ (compare with the leading term in ) increases as $z$ in $S'$ moves to the right parallel to ${{\mathbb R}}$.\
If $h$ is sufficiently small, the solution $\psi$ is meromorphic in $S$; its poles belong to $h{{\mathbb N}}$ and they are simple.
### The uniform asymptotics of $\psi$ in $S$ {#sec:asympt-psi-whole}
To describe the asymptotics of $\psi$, we define an auxiliary function. Clearly, the complex momentum $p$ has a logarithmic branch point at zero. In ${{\mathbb C}}\setminus{{\mathbb R}}_+$, we fix the analytic branch of $z\mapsto \ln(-z)$ such that $\ln(-z)|_{z=-1}=0$. In \[sec:case-where-im-1\] we check
\[le:p-ln\] The function $z\mapsto p(z)-i\ln (-z)$ is analytic in $S$. The function $z\mapsto z\sin p(z)$ is analytic and does not vanish in $S$.
For $z\in S'$, we set $$\label{eq:G0}
G_0(z)=\frac{\sqrt{h/2\pi}}{\sqrt{- z\sin p(z)}} \,
e^{\textstyle\ \frac{z}{h}\ln\frac1h+
\frac{i}h\int_0^z(p(\zeta)-i\ln(-\zeta))\,d\zeta}.$$ Here and below, $\sqrt{h/2\pi}$ and $\ln\frac1h$ are positive; $\sqrt{- z\sin p(z)}= \sqrt{-z}\sqrt{\sin p(z)}$; the branch of $z\mapsto\sqrt{\sin p(z)}$ coincides with the one from .\
In view of , $G_0$ is analytic in $S$.\
Our main result is
\[th:main\] In $S$, the solution $\psi$ admits the asymptotic representation $$\label{eq:psi-gamma}
\psi(z)=\Gamma\,\left(1-\frac{z}h\right)\,G_0(z)\,
e^{\textstyle\frac{i}h\int_{z_0}^0p\,dz+o(1)},\quad\text{ as
}h\to 0$$ where $\Gamma$ is the Euler $\Gamma$-function and the integration path stays in $S$.
So, the special function describing the asymptotic behavior of $\psi$ near the poles generated by a simple pole of $v$ is the $\Gamma$-function.
### The asymptotics of $\psi$ outside a neighborhood of ${{\mathbb R}}_+$ {#sec:asympt-psi-outs}
For large values of $|z/h|$, the $\Gamma$-function in can be replaced with its asymptotics. Let us give more details.\
Fix $\epsilon>0$. We recall that, uniformly in the sector $|\arg \zeta|\le \pi -\epsilon$, one has $$\label{eq:stirling}
\Gamma(1+\zeta)=\sqrt{2\pi\zeta}\,e^{\zeta (\ln\zeta -1)+o(1)}, \
|\zeta|\to\infty,$$ where the functions $\zeta\mapsto \sqrt{\zeta}$ and $\zeta\mapsto\ln\zeta$ are analytic in this sector and satisfy the conditions $\sqrt{1}=1$ and $\ln 1=0$.\
Fix $\delta$ positive sufficiently small. Using , one checks that, in $S$ outside the $\delta$-neighborhood of ${{\mathbb R}}_+$, the representation turns into .\
By construction, $\psi$ admits the asymptotic representation in $S_c$. implies that the representation remains valid in $S'$. This reflects the standard semi-classical heuristics saying that an asymptotic representation of a solution remains valid as long as the leading term is increasing; in the present case, the modulus of the exponential in the leading term in increases in $S'$ as long as $z$ moves to the right parallel to ${{\mathbb R}}$.
### The asymptotics of $\psi$ near ${{\mathbb R}}_+$ away from $0$ {#sec:asymptotics-psi-near}
Assume that $z$ is inside the $\delta$-neighborhood of ${{\mathbb R}}_+$ but outside the $\delta$-neighborhood of 0. In this case, to simplify , we first use the relation $$\label{eq:two-gammas}
\Gamma(1-\zeta)=\frac{\pi}{\sin(\pi\zeta)}\,\frac1{\Gamma(\zeta)}$$ and, next, the asymptotic representation . This yields the asymptotic representation $$\label{as:in-out}
\psi(z)=\frac{e^{\textstyle \
\frac{i}{h}\int_{z_0}^zp(\zeta)\,d\zeta+o(1)}}{(1-e^{2\pi iz/h})\;
\sqrt{\sin(p(z))}}\,,\quad h\to0,$$ where $p$, $z\mapsto\int_{0}^zp(\zeta)\,d\zeta$ and $\sqrt{\sin(p)}$ are obtained by analytic continuation from $S'\cap {{\mathbb C}}_+$ into the domain under consideration.
A basis of solutions {#intro:second-sol}
--------------------
The set of solutions to is a two-dimensional module over the ring of $h$-periodic functions (see section \[s:space-of-sol\]). We now explain how to construct a basis of this module.
###
As the first solution, we take $f_+(z)=e^{-\frac{i}{h}\int_{z_0}^0p(z)\,dz}\,\psi(z)$. In $S'$, it admits the asymptotic representation $$\label{eq:f-plus}
f_+(z)\sim \frac1{\sqrt{\sin p(z)}}\,
e^{\textstyle \;\frac{i}{h}\int_{0}^z p(\zeta)\,d\zeta}, \qquad h\to 0,$$ and has simple poles at the points of $h{{\mathbb N}}$.\
We note that $\left|e^{\frac{i}{h}\int_{0}^z p(\zeta)\,d\zeta}\right|$ increases when $z$ moves to the right parallel to ${{\mathbb R}}$.
###
Fix $z_1\in S\cap{{\mathbb R}}_+^*$. Possibly reducing $S$ somewhat, similarly to $\psi$, one constructs a solution $\phi$ that, in $S\setminus {{\mathbb R}}_-$, admits the asymptotic representation $$\label{eq:phi}
\phi(z)\sim\frac1{\sqrt{\sin(p(z))}}\, e^{-\frac{i}{h}\int_{z_1}^z
p(\zeta)\,d\zeta},\qquad h\to 0,$$ and has simple poles at the points of $-h{{\mathbb N}}$. The branches of the complex momenta appearing in and coincide in ${{\mathbb C}}_+$.\
Note that $\left|e^{-\frac{i}{h}\int_{z_1}^z p(\zeta)\,d\zeta}\right|$ increases when $z$ moves to the left parallel to ${{\mathbb R}}$.\
The function $z\mapsto 1-e^{2\pi iz/h}$ being $h$-periodic, we define another solution to \[main\] by the formula $f_-(z):=(1-e^{2\pi iz/h}) e^{\frac{i}{h}\int_{z_1}^0
p(z)\,dz}\,\phi(z)$. The function $f_-$ is analytic in $S$; its zeroes are simple and located at the points of $h{{\mathbb N}}\cup\{0\}$. As we prove in \[sec:second-solution\], in $S'$, the solution $f_-$ has the asymptotics $$\label{eq:f-minus}
f_-(z)\sim\frac1{\sqrt{\sin(p(z))}}\, e^{-\frac{i}{h}\int_{0}^z
p(\zeta)\,d\zeta+o(1)},\qquad h\to 0.$$
###
In section \[sec:second-solution\], we shall see that, for sufficiently small $h$, $f_+$ and $f_-$ form a basis of the space of solutions to \[main\] meromorphic in $S$ (possibly reduced somewhat).
The idea of the proof of and the plan of the paper {#ss:plan}
--------------------------------------------------
To prove Theorem \[th:main\], we consider the function $z\mapsto f(z)=\psi(z)/\Gamma(1-z/h)$. It is analytic in $S$. Using tools of the complex WKB method for difference equations, outside a disk $D$ centered at 0 (and independent of $h$), we compute the asymptotics of $f$ and obtain $f(z)=e^{\frac ih\int_{z_0}^0p\,dz}G_0(z)(1+o(1))$. The factor $G_0$ is analytic and does not vanish in $S$. Therefore, the function $z\mapsto e^{-\frac ih\int_{z_0}^0p\,dz}f(z)/G_0(z)-1$ is analytic in $D$, and, as it is small outside $D$, the maximum principle implies that it is small also inside $D$.\
The plan of the paper is the following. In \[s:preliminaries\] we describe basic facts on \[main\] and the main tools of the complex WKB method for difference equations. In \[s:outside-as\], we derive the asymptotics of the solution $\psi$ in $S$ outside a neighborhood of 0. In \[s:uniform-as\], we finally prove the asymptotic representation . In \[s:basis\], we briefly discuss the solution $\phi$ mentioned in \[intro:second-sol\].
Acknowledgments {#sec:acknowledgments}
---------------
This work was supported by the CNRS and the Russian foundation of basic research under the French-Russian grant 17-51-150008.
Preliminaries {#s:preliminaries}
=============
We first recall basic facts on the space of solutions to \[main\]; next, we recall basic constructions of the complex WKB method for difference equations and prepare an important tool, . We will use it various times to obtain the asymptotics of solutions to .
The space of solutions to \[main\] {#s:space-of-sol}
----------------------------------
The observations that we now discuss are well-known in the theory of difference equations and are easily proved. We follow [@F-K:18].\
Fix $(X_1,X_2, Y)\in {\mathbb R}^3$ so that $X_1+2h<X_2$. We discuss the set $\mathcal{M}$ of solutions to \[main\] on $I:=\{z\in {{\mathbb C}}\,:\,X_1<{{\rm Re}\,}z<X_2,\; {{\rm Im}\,}z=Y\}$.\
Let $\psi_\pm\subset \mathcal{M}$. The expression $$\label{eq:wronskian}
w(\psi_+(z),\psi_-(z))=\psi_+(z+h)\psi_-(z)-\psi_+(z)\psi_-(z+h),
\quad z,\,z+h\in I,$$ is called [*the Wronskian*]{} of $\psi_+$ and $\psi_-$. It is $h$-periodic in $z$.\
If the Wronskian of $\psi_+$ and $\psi_-$ does not vanish, they form a basis in $\mathcal{M}$, i.e, $\psi\in \mathcal{M}$ if and only if $$\label{eq:three-solutions}
\psi(z)=a(z)\psi_+(z)+b(z)\psi_-(z),\quad z\in I,$$ where $a$ and $b$ are $h$-periodic complex valued functions. One has $$\label{eq:periodic-coef}
a(z)=\frac{w(\psi(z),\,\psi_-(z))}{w(\psi_+(z),\,\psi_-(z))}
\quad\text{and}\quad
b(z)=\frac{w(\psi_+(z),\,\psi\,(z))}{w(\psi_+(z),\,\psi_-(z))}.$$ The set $\mathcal{M}$ is a two-dimensional module over the ring of $h$-periodic functions.
Basic constructions of the complex WKB method {#ss:basics-of-WKB}
---------------------------------------------
We begin by defining [*canonical curves*]{} and [*canonical domains*]{}, the main geometric objects of the method.
### Canonical curves {#sec:canonical-curves}
For $z\in{{\mathbb C}}$, we put $x={{\rm Re}\,}z$, $y={{\rm Im}\,}z$. A connected curve $\gamma \subset \mathbb{C}$ is called [*vertical*]{} if it is the graph of a piecewise continuously differentiable function of $y$.\
Define the complex momentum, turning points and regular domains as in \[intro:WKB\].\
Let $\gamma$ be a regular vertical curve parameterized by $z=z(y)$, and $p$ be a branch of the complex momentum that is analytic near $\gamma$. We pick $z_0\in\gamma$. The curve $\gamma$ is called [*canonical*]{} with respect to $p$ if, at the points where $z'(\cdot)$ exists, one has $$\label{def:can}
\frac{d}{dy} {{\rm Im}\,}\int_{z_0}^zp(\zeta)\,d\zeta> 0
\quad\text{ and }\quad
\frac{d}{dy} {{\rm Im}\,}\int_{z_0}^z(p(\zeta)-\pi) \,d\zeta< 0,$$ and at the points of jumps of $z'(\cdot)$, these inequalities are satisfied for both the left and right derivatives.
### Canonical domains {#sec:canonical-domains}
In this paper we discuss only bounded canonical domains.\
Let $K\subset{{\mathbb C}}$ be a bounded simply connected regular domain and let $p$ be a branch of the complex momentum analytic in it. The domain $K$ is said to be [*canonical*]{} with respect to $p$ if, on the boundary of $K$, there are two regular points, say, $z_1$ and $z_2$ such that, for any $z\in K$, there exists a curve $\gamma\subset K$ passing through $z$ and connecting $z_1$ to $z_2$ that is canonical with respect to $p$. In this paper, we use the local canonical domains described by
\[le:1\] For any regular point, there exists a canonical domain that contains this point.
This lemma is an analog of Lemma 5.3 in [@F-K:05]; mutatis mutandis, their proofs are identical.
### Standard asymptotic behavior {#sec:stand-asympt-behav}
Let $U\subset {{\mathbb C}}$ be a regular simply connected domain; pick $z_0\in U$ and assume $z\mapsto p(z)$ and $z\mapsto \sqrt{\sin p(z)}$ are analytic in $U$.\
We say that a solution $\psi$ to \[main\] has the standard (asymptotic) behavior $$\label{st_beh}
\psi(z)\sim \frac1{\sqrt{\sin(p(z))}}\, e^{\frac{i}{h}\int_{z_0}^z
p(\zeta)\,d\zeta}$$ in $U$ if, for $h$ sufficiently small, $\psi$ is analytic and admits the asymptotic representation in $U$.\
says that, for any given bounded canonical domain $K$, for any branch of $z\mapsto p(z)$ analytic in $K$, there exists a solution with the standard asymptotic behavior in $K$. To study its asymptotic behavior outside $K$, we use the construction described in the next subsection.
A continuation principle {#sec:cont-princ}
------------------------
Assume the potential $v$ in \[main\] is analytic in a domain in ${{\mathbb C}}$. Let $z_0$ be a regular point, $V_0$ be a regular simply connected domain containing $z_0$ and $p$ be a branch of the complex momentum analytic in $V_0$.\
Finally, let $\psi$ be a solution to having the standard asymptotic behavior in $V_0$. One has
\[th:rectangle\] Let $z_1\in V_0$. Consider the straight line $L=\{z\in{\mathbb C}\,:\,{\rm Im}\,z= {\rm Im}\, z_1\}$. Pick $z_2\in L$ such that ${\rm Re}\,z_2> {\rm Re}\,z_1$. Assume the segment $I=\{z\in L\,:\,{\rm Re}\,z_1\le {\rm Re}\,z\le {\rm Re}\,z_2\}$ is regular.\
If ${\rm Im}\, p\,(z)<0$ along $I$, then there exists $\delta>0$ such that the $\delta$-neighborhood of $I$ is regular and $\psi$ has the standard behavior in this neighborhood.
Theorem \[th:rectangle\] roughly says that the asymptotic formula stays valid along a horizontal line as long as the leading term grows exponentially. It is akin to Lemma 5.1 in [@F-K:05] that deals with differential equations. The proof of Theorem \[th:rectangle\] given below follows the plan of the proof of Lemma 5.1 in [@F-K:05].\
Let $\tilde \psi$ be a solution to \[main\] with the standard behavior $\tilde \psi(z)\sim \frac{e^{-\frac{i}{h}\int_{z_0}^z
p(\zeta)\,d\zeta}}{\sqrt{\sin(p(z))}}$ in $V_0$. If ${{\rm Im}\,}p<0$ in $V_0$, then the analogue, mutatis mutandis, of Theorem \[th:rectangle\] on the behavior of $\tilde \psi$ to the left of $V_0$ holds.
Clearly, for $\zeta\in I$, there exists an open disk $D$ centered at $\zeta$ such that $D$ is regular and ${{\rm Im}\,}p(z)<0$ in $D$. In view of , if $D$ is sufficiently small, then there exists two solutions $\psi_\pm$ having the standard behavior $\psi_\pm(z)\sim \frac{e^{\pm\frac{i}{h}\int_{z_0}^z
p(\zeta)\,d\zeta}}{\sqrt{\sin(p(z))}}$ in $D$ (here, we first integrate from $z_0$ to $z_1$ in $V_0$, next from $z_1$ to $\zeta$ along $I$ and finally from $\zeta$ to $z$ inside $D$).\
The segment $I$ being compact, we construct finitely many open disks $(D_j)_{0\leq j\leq J}$, each centered in a point of $I$, covering $I$ and such that
1. for $0\leq j\leq J$, the disk $D_j$ is regular and one has ${{\rm Im}\,}p(z)<0$ in $D_j$;
2. $z_1\in D_0$, $z_2\in D_J$, and $\psi$ has the standard behavior in $D_0$;
3. for $0\leq j\leq J$, there exists two solutions $\psi^j_\pm$ having the standard behavior $\psi_\pm^j\sim\frac1{\sqrt{\sin p(z)}} e^{\pm \frac
ih\int_{z_0}^z p d\zeta}$ in the domain $D_j$.
Denote the rightmost point of the boundary of $D_j$ by $w_j$. Possibly, excluding some of the disks $D_j$ from the collection $(D_j)_{0\leq j\leq J}$ and reordering them, we can and do assume that, for $1\leq j\leq J$, $D_{j}\setminus D_{j-1}\not=\emptyset$ and $w_{j}>w_{j-1}$. Indeed, to choose $D_1$, consider the point $w_0$. If $w_0$ is to the right of $z_2$, we can keep only $D_0$ in the collection. Otherwise, in our collection, there is a disk that contains $w_0$. Denote it by $D_1$. We then obtain the set of disks by induction.\
For $r>0$ we define $$S(r)=\{z\in {{\mathbb C}}\,:\, |{{\rm Im}\,}(z-z_1)|< r\}.$$ For $1\le j\le J$, let $r_j=\min\{r>0\,:\, D_j\cap D_{j-1}\subset S(r)\}$. Pick ${\displaystyle}0<\delta <\min_{1\le j\le J} r_j$ sufficiently small so that $I_\delta$, the $\delta$-neighborhood of $I$, be a subset of $\cup_{j=0}^J D_j$.\
Let us prove that $\psi$ has the standard behavior in $I_\delta$.\
First we note that, for $h$ sufficiently small, by means of the formula $\psi(z)=-\psi(z-2h)-v(z)\psi(z-h)$ (i.e., by means of \[main\]), $\psi$ can be analytically continued in $I_\delta$. It clearly satisfies equation in $I_\delta$.\
Let us justify the asymptotic representation in $I_\delta$. For $0\le j\le J$, we define $d_j=D_j\cap I_\delta$. For $j=1,2,\dots J$, we consecutively prove that $\psi$ has the standard behavior in $d_j$ to the right of $d_{j-1}$. Therefore, we let $d^0=d_{j-1}$, $d=d_{j}$, $\psi_\pm=\psi_\pm^{j}$, and then proceed in the following way.\
Using the standard asymptotic behavior of $\psi_+$ and $\psi_-$, one proves the asymptotic formula $$\label{w:f-pm}
w(\psi_+(z),\psi_-(z))=2i+o(1), \quad z, z+h\in D_{j},\qquad h\to 0.$$ As the Wronskians are $h$-periodic, for sufficiently small $h$, formula is valid uniformly in $ I_{\delta}$. This implies in particular that, for sufficiently small $h$, in $I_\delta$, the solution $\psi$ is a linear combination of the solutions $\psi_\pm $ with $h$-periodic coefficients, and one has and .\
The leading terms of the asymptotics of $\psi$ and $\psi_+$ coincide in $d^0\cap d$. Thus, one has $$\label{6.1}
a(z)=1+o(1), \quad z, z+h\in d^0\cap d,\qquad h\to 0.$$ Due to the $h$-periodicity of $a$, for sufficiently small $h$, formula stays valid in the whole $I_\delta$.\
One also has $ b(z)= o\left(e^{\frac{2i}h\int_{z_0}^{z}p(\zeta) d\zeta}\right)$ for $z\in d^0\cap d$. For sufficiently small $h$, the $h$-periodicity of $b$ yields $$\label{eq:1}
b(z)=o\left(e^{\frac{2i}h\int_{z_0}^{\tilde z}p(\zeta)
d\zeta}\right),\quad z\in I_\delta,$$ where $\tilde z\in d^0\cap d$ and $\tilde z=z$ mod $h$.\
Estimates and imply that, in $d$, one has $$ \psi(z)=a(z)\psi_+(z)+b(z)\psi_-(z)=\frac{e^{\frac
ih\int_{z_0}^{z} p(\zeta)d\zeta}}{\sqrt{\sin p(z)}} \left (1+o(1)+
o\left(e^{-\frac{2i}h\int_{\tilde z}^{z}p(\zeta)
d\zeta}\right)\right).$$ Assume that $z\in d$ is located to the right of $d_0$. As ${{\rm Im}\,}p<0$ in $d$, one has ${{\rm Re}\,}\left(i\int_{\tilde z}^{z} p(\zeta)d\zeta\right)>0$. This implies that $\psi$ has the standard behavior in $d$ to the right of $d^0$ and completes the proof of Theorem \[th:rectangle\].
The asymptotics outside a neighborhood of 0 {#s:outside-as}
===========================================
We consider the solution $\psi$ described in \[sss:main\] and derive its asymptotics outside a neighborhood of $0$, the pole of $v$.
The asymptotics outside a neighborhood of ${{\mathbb R}}_+$ {#sec:asympt-outs-neghb}
-----------------------------------------------------------
Recall that
- in the rectangle $S_c$, the solution $\psi$ has the standard behavior ;
- $S'$ is regular;
- in $S'$, the branch $p$ appearing in satisfies the inequality ${{\rm Im}\,}p(z)<0$.
yields the asymptotics of $\psi$ in $S'$ to the right of $S_c$, namely
\[le:psi-in-S-prime\] The solution $\psi$ admits the standard asymptotic behavior in the domain $S'$.
Let us underline that the obstacle to justify the standard behavior of $\psi$ in the whole domain $S$ is the pole of $v$ at 0.
Asymptotics in a neighborhood of ${{\mathbb R}}_+$ outside a neighborhood of 0 {#sec:asympt-neghb-r_+}
------------------------------------------------------------------------------
We note that the function $z\mapsto (1-e^{2\pi i z/h})$ is $h$-periodic. As $\psi$ satisfies \[main\], so does $z\mapsto (1-e^{2\pi i z/h})\,\psi(z)$. Moreover, as $\psi$ has poles only at the points of $h{{\mathbb N}}$ and as these poles are simple, the solution $z\mapsto (1-e^{2\pi i z/h})\,\psi(z)$ is analytic in $S$.\
For $\delta>0$, let $P(\delta)=\{z\in S\,:\, {{\rm Re}\,}z>0, \, |{{\rm Im}\,}z|< \delta\}$. In this subsection, we prove
\[pro:in-out\] Let $\delta>0$ be sufficiently small. In $P(\delta)$, the solution $z\mapsto (1-e^{2\pi i z/h})\,\psi(z)$ has the standard behavior $$\label{eq:psi-factor}
(1-e^{2\pi i z/h})\,\psi(z)\sim n_0\,
\frac{e^{\textstyle\frac ih\int_0^z p_{up}(\zeta)\,d\zeta}}{\sqrt{\sin p_{up}(z)}}\,,
\qquad n_0=e^{\frac{i}{h}\int_{z_0}^0 p(z)\,dz};$$ here, $p_{up}$ and $\sqrt{\sin p_{up}}$ are respectively obtained from $p$ and $\sqrt{\sin p}$ by analytic continuation from $S'\cap{{\mathbb C}}_+$ to $P(\delta)$.
To prove Proposition \[pro:in-out\], it suffices to check that, for any point $z_*\in P(\delta)$, there exists a neighborhood, say, $V_*$ of this point (independent of $h$) where the solution $z\mapsto (1-e^{2\pi i z/h})\psi(z)$ has the standard behavior.\
As their study is simpler, we begin with the points $z_*\not\in {{\mathbb R}}$.
### Points $z_*$ in ${{\mathbb C}}_+$ {#sec:case-where-im}
Let $z_*\in P(\delta)\cap {{\mathbb C}}_+$. Let $V_*\subset P(\delta)\cap{{\mathbb C}}_+$ be an open disk (independent of $h$) centered at $z_*$. In $V_*$ one has $1-e^{2\pi i z/h}=1+o(1)$ as $h\to0$. Furthermore, by , $\psi$ has the standard behavior in $V_*$. Therefore, in $V_*$, one computes $$\label{eq:psi-with-factor}
(1-e^{2\pi i z/h})\,\psi(z)= \frac{e^{\frac{i}{h}\int_{z_0}^z
p(\zeta)\,d\zeta+o(1)}}{\sqrt{\sin p(z) }}=
n_0\,\frac{e^{\frac{i}{h}\int_{0}^z
p_{up}(\zeta)\,d\zeta+o(1)}}{\sqrt{\sin p_{up}(z)}}.$$ This implies the standard behavior in $V_*$.
### Points $z_*$ in ${{\mathbb C}}_-$ {#sec:case-where-im-1}
Pick now $z_*\in P(\delta)\cap {{\mathbb C}}_-$ and let $V_*\subset P(\delta)\cap{{\mathbb C}}_-$ be an open disk (independent of $h$) centered at $z_*$.\
We use
\[le:p\_up-down\] For $z\in P(\delta)\cap {{\mathbb C}}_-$, one has $$\label{eq:p-p-d}
p(z)=p_{up}(z)-2\pi, \quad \sqrt{\sin p(z)}=-\sqrt{\sin p_{up}(z)}.$$
Lemma \[le:p\_up-down\] yields $$\label{eq:123}
\frac{e^{\frac{i}{h}\int_{z_0}^z p(\zeta)\,d\zeta}}{\sqrt{\sin p(z)}} =
-n_0 \,\frac{e^{\frac{i}{h}\int_{0}^z
(p_{up}(z)-2\pi)\,dz}}{\sqrt{\sin p_{up}(z)}},\quad z\in V_*.$$ By , $\psi$ has the standard behavior in $V_*$. Therefore, implies that, in $V_*$, one has $$(1-e^{2\pi i z/h})\psi(z) = n_0\,(1-e^{-2\pi i z/h})
\frac{e^{\frac{i}{h}\int_{0}^zp_{up}(\zeta)\,d\zeta+o(1)}}{\sqrt{\sin p_{up}(z)}}=
n_0\,\frac{e^{\frac{i}{h}\int_{0}^zp_{up}(\zeta)\,d\zeta+o(1)}}{\sqrt{\sin p_{up}(z)}},$$ and $z\mapsto (1-e^{2\pi i z/h})\psi(z)$ has the standard behavior in $V_*$.\
To prove , we shall use
\[le:p-near-zero\] In $S'$, one has $$\label{eq:p-as-0}
p(z)=i\ln(z)+C+g(z)$$ where $\ln$ is a branch of the logarithm analytic in ${{\mathbb C}}\setminus{{\mathbb R}}_+$, $C$ is a constant, and $g$ is a function analytic in $S\cup{0}$ vanishing at $0$.
By definition (see ), $p$ satisfies $ e^{2ip(z)}+v(z)e^{ip(z)}+1=0$. Therefore, $$e^{ip(z)}=-v(z)/2+\sqrt{ (v(z)/2)^2-1},$$ where the branch of the square root is to be determined. Since $v(z)\to\infty$ as $z\to0$, we rewrite this formula in the form $$\label{exp-i-p}
e^{ip(z)}=-v(z)/2\,(1+\sqrt{1-(2/{v(z)})^2}).$$ As ${{\rm Im}\,}p(z)<0$ in $S'$ and as $v(z)\to\infty$ when $z\to0$, \[eq:p\] implies that $e^{ip(z)}\to\infty$ when $z\to 0$. Therefore, in , the determination of the square root is to be chosen so that $\sqrt{1-(2/{v(z)})^2}=1+o(1/(v(z))^2)$ as $z\to 0$. Then, yields the representation $$\label{exp-i-p:1}
e^{ip(z)}=-v(z)+\tilde g(z)$$ where $\tilde g$ is analytic in a neighborhood of 0 vanishing at $0$. By assumption, $v$ has the Laurent expansion $v(z)=v_{-1}/z +v_0+v_1 z+\dots$, $v_{-1}\ne 0$, in a neighborhood of 0. Thus, implies representation in a neighborhood of 0.\
The function $z\mapsto g(z):=p(z)-i\ln(z)-C$ is analytic in $S$ in a neighborhood of ${{\mathbb R}}_-$. To check that it is analytic in the whole domain $S$, we consider two analytic continuations of $z\mapsto p(z)-i\ln z -C$ into a neighborhood of ${{\mathbb R}}_+$ in $S$, one from $S'\cap{{\mathbb C}}_+$ and another from $S'\cap {{\mathbb C}}_-$. As they coincide near 0, they coincide in the whole connected component of $0$ in their domain of analyticity. So, $g$ is analytic in $S$. This completes the proof of .
It now remains to prove Lemma \[le:p\_up-down\]. Therefore, we first prove .
The statements of on the analyticity of the functions $z\mapsto p(z)-i\ln(-z)$ and $z\mapsto z\sin p(z)$ follow directly from . This lemma also implies that the second function does not vanish at 0. Finally, this function does not vanish in $S\setminus \{0\}$ in view of . The proof of is complete.
Let $z\in P(\delta)\cap {{\mathbb C}}_-$. The first formula in follows directly from the representation . By , the function $z\mapsto \sin p(z)$ is analytic and does not vanish in $S\setminus\{0\}$. So, for $z\in P(\delta)\cap {{\mathbb C}}_-$, $\sqrt{\sin p(z)}$ and $\sqrt{\sin p_{up}(z)}$ either coincide or are of opposite signs. In view of , we obtain the second relation in .
Having proved , the analysis for $z\in {{\mathbb C}}_-$ is complete.
### Real points $z_*$: construction of two linearly independent solutions {#sec:case-z_inr.-two}
To treat the case of real points, we define two linearly independent solutions to \[main\] that have standard asymptotic behavior to the right of 0 and express $\psi$ in terms of these solutions.\
Below, in the proof of , we always assume that $z_*$ is a point in ${{\mathbb R}}_+^*\cap S$. Let $x_\pm\in S$ be two points such that $0<x_+<z_*<x_-$.\
We recall that the set $S\setminus\{0\}$ is regular (see the beginning of \[ss:main\]). For $\bullet\in\{+,-\}$, the point $x_\bullet$ is regular. By and , there exists a regular $\epsilon_\bullet$-neighborhood $V_\bullet$ of $x_\bullet$ in $P(\delta)$ such that there exists a solution $\psi_\bullet$ to with the standard asymptotic behavior $\psi_\bullet\sim \frac1{\sqrt{\sin p_{up}(z)}}e^{\bullet\frac {i}h
\int_{x_\bullet}^z p_{up}(\zeta)\,d\zeta}$ in the domain $V_\bullet$.\
Let $\epsilon=\min\{\epsilon_+, \epsilon_-\}$ and $V$ be the $\epsilon$-neighborhood of the interval $(x_+,x_-)$.\
As the imaginary part of the complex momentum can not vanish in $S\setminus \{0\}$, ${{\rm Im}\,}p_{up}$ is negative in $V$. Therefore, by , the solution $\psi_+$ has the standard asymptotic behavior $$\label{eq:3}
\psi_+(z)\sim \frac{e^{\frac{i}{h}\int_{x_+}^z
p_{up}(\zeta)\,d\zeta}}{\sqrt{\sin(p_{up}(z))}}$$ in $V$ to the right of $V_+$.\
Similarly, one shows that $\psi_-$ has the standard asymptotic behavior $$\label{eq:4}
\psi_-(z)\sim \frac{e^{-\frac{i}{h}\int_{x_-}^z
p_{up}(\zeta)\,d\zeta}}{\sqrt{\sin(p_{up}(z))}}$$ in $V$ to the left of $V_-$.\
Then, and yield that, as $h\to 0$, one has $$\label{eq:w_psi_pm}
w(\psi_+(z),\psi_-(z))=2ie^{\frac{i}{h}\int_{x_+}^{x_-}p_{up}(z)\,dz+o(1)}, \qquad z\in V.$$ At cost of reducing $\varepsilon$, this asymptotic is uniform in $V$. We note that the error term is analytic in $z$ together with $\psi_\pm$.\
In view of , for sufficiently small $h$, the solutions $\psi_\pm$ form a basis of the space of solutions to \[main\] defined in $V$; for $z\in V$, we have and . Our next step is to compute the asymptotics of $a$ and $b$ in .
### The coefficient $a$ {#sec:coefficient-a}
We prove
\[le:2\] As $h\to 0$, $$\label{as:a}
a(z)=\frac{n_0\;e^{\frac{i}{h}\int_{0}^{x_+} p_{up}(z)\,dz+o(1)}}
{1-e^{2\pi i z / h}}, \quad z\in V,$$ where the error term is analytic in $z$.
For $z\in V$, we shall compute the asymptotics of the Wronskian $w_-(z)=w(\psi(z),\psi_-(z))$ appearing in the formula for $a$ in .\
As those of $\psi$, the poles of $w_-$ in $S$ are contained in $h{{\mathbb N}}$ and they are simple. We first compute the asymptotics of $w_-$ in $V$ outside the real line. Then, the information on the poles yields a global asymptotic representation for $w_-$ in $V$ and, thus, .\
First, we assume that $z\in V\cap {{\mathbb C}}_+$. Then, $p_{up}$ and $\sqrt{\sin p_{up}} $ coincide respectively with $p$ and $\sqrt{\sin p}$; using the asymptotics of $\psi$ and $\psi_-$ yields $$\label{a:w-up}
w_-(z)=2in_0\; e^{\frac{i}{h}
\int_{0}^{x_-} p_{up}(z)\,dz+o(1)},
\quad z\in V\cap {{\mathbb C}}_+, \quad h\to 0.$$ Now, we assume that $z\in V\cap {{\mathbb C}}_-$. Then, implies that $$\label{eq:5}
\psi(z)=-\frac{n_0\;e^{\frac
ih\int_{0}^z(p_{up}(\zeta)-2\pi)\,d\zeta
+o(1)}}
{\sqrt{\sin p_{up}(z)}}, \qquad h\to0.$$ This representation and yield $$\label{a:w-down}
w_-(z)=-2 i n_0\;e^{-2\pi i z/h}
e^{\frac ih\int_{0}^{x_-}p_{up}(z)\,dz}(1+o(1)),
\quad z\in V\cap {{\mathbb C}}_-, \quad h\to 0.$$ Let $$f\,:\,z\mapsto n_0^{-1}\,(1-e^{2\pi iz/h}) \, e^{-\frac ih
\int_{0}^{x_-}p_{up}(z)\,dz}w_-(z)-2i.$$ Representations and imply that $$\label{f:est}
f(z)=o(1),\quad z\in V\setminus {{\mathbb R}},\quad h \to 0.$$ We recall that $f$ is an $h$-periodic function (as is $w_-$). Therefore, at the cost of reducing $\epsilon$ somewhat, we get the uniform estimate $f(z)=o(1)$ for $|{{\rm Im}\,}z|=\epsilon$ and $h$ sufficiently small. Moreover, the description of the poles of $w_-$ implies that $f$ is analytic in the strip $\{|{{\rm Im}\,}z|\le \epsilon\}$.\
Now, let us consider $f$ as a function of $\zeta=e^{2\pi i z/h}$. It is analytic in the annulus $\{e^{-2\pi\epsilon/h}\le |\zeta|\le e^{2\pi\epsilon/h}\}$ and, on the boundary of this annulus, it admits the uniform estimate $|f(\zeta)|=o(1)$. By the maximum principle, the estimate $f(\zeta)=o(1)$ holds uniformly in the annulus.\
Thus, as a function of $z$, the function $f$ satisfies the uniform estimate $ f(z)=o(1)$ for $|{{\rm Im}\,}z|\le\epsilon$ and, therefore, in the whole domain $V$.\
This estimate, the representation and the definition of $a$ (see ) imply .
### The coefficient $b$ {#sec:coefficient-b}
We estimate $b$ in $V$ for $h$ sufficiently small. To state our result, let $\gamma$ be the connected component of $x_+$ in the set of $z\in P(\delta)$ satisfying $${{\rm Im}\,}\int_{x_+}^{z}p(z)\,dz=0.$$ As $$\frac{\partial}{\partial x}{{\rm Im}\,}\int_{x_+}^{z}p(\zeta)\,d\zeta
={{\rm Im}\,}p(z)\ne 0\quad \text{in } P(\delta),$$ the Implicit Function Theorem guarantees that $\gamma$ is a smooth vertical curve in a neighborhood of $x_+$. Reducing $\epsilon$ if necessary, we can and do assume that $\gamma$ intersects both the lines $\{{{\rm Im}\,}z=\pm \epsilon\}$. We prove
\[le:3\] In $V$ (with $\epsilon$ reduced somewhat if necessary), on $\gamma$ and to the right of $\gamma$, one has $$\label{est:b}
b(z)=o(1)\; \frac{n_0\;e^{\frac{i}h\int_{0}^{x_+} p_{up}(z)\,dz-
\frac{i}h\int_{x_+}^{x_-} p_{up}(z)\,dz}}{1-e^{2\pi i z / h}},
\quad h\to 0,$$ where $o(1)$ is analytic in $z$.
Let us estimate the Wronskian $w_+(z)=w(\psi_+(z),\psi(z))$, the numerator in .\
First, we assume that $z\in V\cap{{\mathbb C}}_+$. Then, $p_{up}$ and $\sqrt{\sin p_{up}} $ coincide respectively with $p$ and $\sqrt{\sin p}$. Thus, the leading terms of the asymptotics of $\psi$ and $\psi_+$ coincide up to a constant factor; this yields $$\label{b:w-up:1}
w_+=o(1)\,n_0\;e^{\frac{i}h\int_{0}^{x_+}p_{up}(z)\,dz+\frac{2i}h
\int_{x_+}^{z}p_{up}(\zeta)\,d\zeta},\qquad h\to 0.$$ We recall that the Wronskians are $h$-periodic (see \[s:space-of-sol\]). Let us assume additionally that $z$ is either between $\gamma$ and $\gamma+h$ or on one of these curves. Pick $\tilde z\in\gamma$ such that ${{\rm Im}\,}\tilde z={{\rm Im}\,}z$. In view of the definition of $\gamma$, as $p_{up}$ is analytic in $P(\delta)$, one has $$\left|e^{\frac{2i}h\int_{x_+}^{z}p_{up}(z)\,dz}\right|=
\left|e^{\frac{2i}h\int_{\tilde
z}^{z}p_{up}(\zeta)\,d\zeta}\right|\le e^C,
\quad h\to0;$$ here, $C$ is a positive constant independent of $h$. This estimate and imply that, for $z\in V\cap {{\mathbb C}}_+$ either between the curves $\gamma$ and $\gamma+h$ or on one of them, one has $$\label{b:w-up:2}
w_+=o(1)\,n_0\;e^{\frac{i}h\int_{0}^{x_+}p_{up}(z)\,dz},\qquad h\to 0.$$ Reducing $\epsilon$ somewhat if necessary, we can and do assume that holds on the line ${{\rm Im}\,}z=\epsilon$ between $\gamma$ and $\gamma+h$ or on one of these curves. Then, thanks to the $h$-periodicity of $w_+$, it holds for all $z$ on the line $\{{{\rm Im}\,}z=\epsilon\}$.\
Now, we assume that $z\in V\cap{{\mathbb C}}_-$. Using the asymptotics and , we compute $$w_+(z)=o(1)\,n_0\;e^{-2\pi i z/h}\;
e^{\frac{i}h\int_{0}^{x_+}p_{up}(z)\,dz+
\frac{2i}h\int_{x_+}^{z}p_{up}(\zeta)\,d\zeta}
\qquad h\to 0.$$ Arguing as when proving , we finally obtain $$\label{b:w-down}
w_+(z)=o(1)\,n_0\;e^{-2\pi i
z/h}\;e^{\frac{i}h\int_{0}^{x_+}p_{up}(z)\,dz},
\quad {{\rm Im}\,}z=-\epsilon,\quad h\to 0.$$ Let $g\,:\,z\mapsto (1-e^{2\pi iz/h}) w_+(z)$. Estimates and imply that, for $|{{\rm Im}\,}z|=\epsilon$, $$\label{w-plus:est}
n_0^{-1}\,e^{-\frac{i}h\int_{0}^{x_+}p_{up}(z)\,dz}g(z)=o(1),\quad
h\to 0.$$ As those of the solution $\psi$ do, the poles of $w_+$ in $P(\delta)$ belong to the set $h{{\mathbb N}}$ and are simple. So, the function $g$ is analytic in the strip $\{|{{\rm Im}\,}z|\leq\epsilon\}$. Moreover, it is $h$-periodic as $w_+$ is. Thus, the maximum principle implies that holds in the whole strip $\{|{{\rm Im}\,}z|\le \epsilon\}$. Estimate , representation and the definition of $b$ (see ) yield . This completes the proof of Lemma \[le:3\].
### Completing the proof of {#sec:compl-proof-crefpr}
Let $V_*\subset V$ be a disk independent of $h$, centered at $z_*$ and located to the right of $\gamma$ (i.e., such that, for any $z\in V_*$, there exists $\tilde z\in \gamma$ such that ${{\rm Re}\,}\tilde z<{{\rm Re}\,}z$ and ${{\rm Im}\,}\tilde z={{\rm Im}\,}z$).\
Below we assume that $z\in V_*$.\
Using and , the asymptotic representations for $\psi_\pm$, and and , the representations for $a$ and $b$, we compute $$\frac{b(z)\,\psi_-(z)}{a(z)\psi_+(z)}=o(1)
e^{-\frac{2i}h\int_{x_+}^{z}p_{up}(\zeta)d\zeta},\quad h\to 0.$$ As before, let $\tilde z\in\gamma$ be such that ${{\rm Im}\,}\tilde z={{\rm Im}\,}z$. Then, we have $$\left|e^{-\frac{2i}h\int_{x_+}^{z}p_{up}(\zeta)d\zeta}\right|=
\left|e^{-\frac{2i}h\int_{\tilde
z}^{z}p_{up}(\zeta)d\zeta}\right|\le 1.$$ Here, we used the definition of $\gamma$ and the fact that ${{\rm Im}\,}p_{up}<0$ in $V$. As a result, we have $\frac{b(z)\,\psi_-(z)}{a(z)\psi_+(z)}=o(1)$. Formula then yields $$\psi(z)=a(z)\,\psi_+(z)\left(1+\frac{b(z)\,\psi_-(z)}
{a(z)\psi_+(z)}\right)=a(z)\,\psi_+(z)(1+o(1)).$$ where $o(1)$ is analytic in $z\in V_*$. This and the asymptotic representations for $\psi_+$ and $a$ yields in $V_*$. This completes the proof of .
Global asymptotics {#s:uniform-as}
==================
The proof of {#sec:proof-crefth:main}
-------------
We follow the plan outlined in \[ss:plan\]. Recall that $G_0$ is defined in . We prove
\[pro:on-the-circle\] Let $\delta>0$ be sufficiently small. In $S$, outside the $\delta$-neighborhood of 0, one has $$\label{eq:on-the-circle}
\psi(z)/\Gamma(1-z/h)= n_0\,G_0(z)\,(1+o(1)),\quad h\to 0.$$
Let us check that follows from Proposition \[pro:on-the-circle\].\
We recall that the poles of $\psi$ belong to $h{{\mathbb Z}}$ and are simple. Furthermore, in view of , $G_0$ is analytic in $S$. Clearly, $G_0$ has no zeros in $S$. These observations imply that the function $$f\,:\,z\ \longrightarrow \ \frac{\psi(z)}{n_0\,G_0(z)\,\Gamma(1-z/h)}-1$$ is analytic in $S$. By , it satisfies the estimate $f(z)=o(1)$ in $S$ outside the $\delta$-neighborhood of 0. Therefore, by the maximum principle, it satisfies this estimate in the whole of $S$. This implies the statement of . To complete the proof of this theorem, it now suffices to check .
The proof of {#sec:proof-crefpr-circle}
-------------
Fix $\varepsilon$ sufficiently small positive. The proof of consists of two parts: first, we prove in the sector $S_\pi=\{z\in S\,:\, |z|\ge \delta,\ |\arg z-\pi|\le \pi-\varepsilon\}$, and, then, in the sector $S_0=\{z\in S\,:\, |z|\ge \delta,\ |\arg z|\le \varepsilon\}$.
### The asymptotic in the sector $S_\pi$ {#sec:asympt-sect-s_pi}
If $z\in S_\pi$ and $h\to 0$, we can use formula for $\Gamma(1-z/h)$ and the standard asymptotic representation for $\psi$, see . This immediately yields in $S_\pi$.
### The asymptotic in the sector $S_0$ {#sec:asympt-sect-s_0}
Let $\varepsilon$ be so small that $S_0$ be a subset of $P(\delta)$ (defined just above ). For $z\in S_0$ and $h$ small, we express $\Gamma(1-z/h)$ in terms of $\Gamma(z/h)$ by formula , then, we use formula for $\Gamma(z/h)$ and the standard asymptotic representation for $(1-e^{2\pi i/z})\psi(z)$ (see ). This yields $$\label{eq:psi-sur-gamma}
\psi(z)/\Gamma(1-z/h)=n_0\,\tilde G_0(z) \,(1+o(1)),\quad h\to 0,\\$$ where $$\label{eq:tildeG0}
\tilde G_0(z)=\frac{i\;\sqrt{h/2\pi}}{\sqrt{z\sin p_{up}(z)}} \,
e^{\;\textstyle \frac{z}{h}\ln\frac1h+
\frac{i}h\int_0^z(p_{up}(\zeta)-i(\ln(\zeta)-i\pi))\,d\zeta},$$ and the functions $z\mapsto \ln z$ and $z\mapsto \sqrt{z}$ are analytic in ${{\mathbb C}}\setminus{{\mathbb R}}_-$ and positive respectively if $z>1$ and $z>0$. We note that by , $\tilde G_0$ is analytic in $S_0$.\
Define the functions $z\mapsto \sqrt{-z}$ and $z\mapsto \ln (-z)$ as in , i.e. so that they be analytic in ${{\mathbb C}}\setminus{{\mathbb R}}_+$ and positive if $z<0$ and $z<-1$ respectively. Then, these functions are related to the functions $z\mapsto \ln z$ and $z\mapsto \sqrt{z}$ from by the formulas $$\sqrt{-z}=-i\sqrt{z},\quad \ln(-z)=\ln z-i\pi,\quad z\in {{\mathbb C}}_+.$$ Furthermore, in $S_0\cap {{\mathbb C}}_+$ the functions $p$ and $p_{up}$ coincide. These two observations imply that one has $\tilde G_0=G_0$ in $S_0\cap C_+$.
As both $G_0$ and $\tilde G_0$ are analytic in $S_0$, they coincide in the whole of $S_0$. This and imply the representation for $z\in S_0$. This completes the analysis in the sector $S_0$ and the proof of .
A basis for the space of solutions {#s:basis}
==================================
We finally discuss a basis of the space of solutions to \[main\] that are meromorphic in $S$. First, we describe the two solutions forming the basis and, second, we compute their Wronskian.
First solution {#sec:first-solution}
--------------
As the first solution, we take $f_+(z)=\psi(z)/n_0$. It has the standard behavior in $S'$ and simple poles at the points of $h{{\mathbb N}}$. We recall that the modulus of the exponential factor in increases when $z$ moves to the right parallel to ${{\mathbb R}}$.
Second solution {#sec:second-solution}
---------------
Let $z_1>0$ be a point in $S$. Mutatis mutandis, in the way we constructed $\psi$, in $S$ (possibly reduced somewhat), we construct a solution $\phi$ in $S\setminus {{\mathbb R}}_-$ that has the standard asymptotic behavior and such that the quasi-momenta $p$ (and the functions $\sqrt{\sin p}$) in and coincide in ${{\mathbb C}}_+$.\
The modulus of the exponential from (\[eq:phi\]) increases when $z$ moves to the left parallel to ${{\mathbb R}}$.\
The solution $\phi$ has simple poles at the points of $-h{{\mathbb N}}$ and, in $S$, it admits the asymptotic representation $$\begin{gathered}
\label{as:phi}
\phi(z)=n_1\Gamma(1+z/h)\,G_1(z)(1+o(1)),\quad
n_1=e^{\frac{i}h\int_{z_1}^0 p(z)\,dz},\quad h\to 0,\\
\label{eq:F0}
G_1(z)=\frac{\sqrt{h/2\pi}}{\sqrt{z\sin p(z)}} \, e^{\;\textstyle
-\frac{z}{h}\ln\frac1h-
\frac{i}h\int_0^z(p(\zeta)-i\ln(\zeta))\,d\zeta}.\end{gathered}$$ Here, the functions $z\mapsto \sqrt{z}$ and $z\mapsto\ln z$ are analytic in ${{\mathbb C}}\setminus {{\mathbb R}}_-$ and positive, respectively, if $z>0$ and $z>1$. The factor $G_1$ is analytic in $S$.\
We define the second solution to be $f_-(z)= 1/n_1\;(1-e^{2\pi i z/h})\,\phi (z)$. The solution $f_-$ is analytic in $S$. It has simple zeros at the points $z\in h{{\mathbb N}}$ and at 0. By means of , one can easily check that, in $S'$, it has the standard behavior .
The Wronskian of the basis solutions {#sec:wronsk-basis-solut}
------------------------------------
Using the asymptotic representations for $f_{\pm}$ in $S'$, one easily computes $$\label{as:wf}
w(f_+(z), f_-(z))=2i+o(1),\quad h\to 0.$$ As the Wronskian is $h$-periodic, this representation is valid in the whole domain $S$. We see that, for sufficiently small $h$, the leading term of the Wronskian does not vanish, and, thus, $f_\pm$ form a basis in the space of solutions to \[main\] in $S$ (possibly reduced somewhat for to be uniform).
[1]{} A. Avila and S. Jitomirskaya. The Ten Martini Problem. [*Annals Math.,*]{} 170:303-342, 2009. V. Babich, M. Lyalinov and V. Grikurov. Oxford, Alpha Science, 2008. V. Buslaev and A. Fedotov. Complex WKB method for Harper equation. , 6(3):495-517, 1995. S. Yu. Dobrokhotov and A. V. Tsvetkova. On Lagrangian manifolds related to asymptotics of Hermite polynomials. , 104(6):810-822, 2018. A. Eckstein. Unitary reduction for the two-dimensional Schr[ö]{}dinger operator with strong magnetic field. , 282(4):504-525, 2009. A. Fedoryuk. Springer-Verlag, Berlin, Heidelberg, 2009. A. Fedotov. Monodromization method in the theory of almost-periodic equations. [*St. Petersburg Math. J.,*]{} 25:303-325, 2014. A. Fedotov and F. Klopp. Anderson transitions for a family of almost periodic [S]{}chr[ö]{}dinger equations in the adiabatic case. , 227:1-92, 2002. A. Fedotov and F. Klopp. On the absolutely continuous spectrum of an one-dimensional quasi-periodic Schr[ö]{}dinger operator in adiabatic limit. , 357:4481-4516, 2005. A. Fedotov and F. Klopp. Quasiclassical asymptotics of solutions to difference equations with meromorphic coefficients. In [*Proceedings of the conference “Days on Diffraction 2017”*]{}, 110-112, IEEE, St. Petersburg, 2017. A. Fedotov and F. Klopp. The complex WKB method for difference equations and Airy functions. HAL, https://hal.archives-ouvertes.fr/hal-01892639 A. Fedotov and F. Sandomirskiy. An exact renormalization formula for the Maryland model. , 334(2):1083-1099, 2015. A. Fedotov and E. Shchetka. A complex WKB method for difference equations in bounded domains. , 224:157-169, 2017. A. Fedotov and E. Shchetka. Complex WKB method for the difference Schr[ö]{}dinger equation with the potential being a trigonometric polynomial. , 29:363-381, 2018. D. Grempel, S. Fishman, R. Prange. Localization in an incommensurate potential: An exactly solvable model. *Phys. Rev. Letters*, 49:833-836, 1982. J. S. Geronimo, O. Bruno and W. Van Assche. WKB and turning point theory for second-order difference equations. , 69:269-301, 1992. J. P. Guillement, B. Helffer and P. Treton. Walk inside Hofstadter’s butterfly. 50:2019-2058, 1989. B.Helffer and J.Sj[ö]{}strand. Analyse semi-classique pour l’[é]{}quation de Harper (avec application [à]{} l’[é]{}quationde Schr[ö]{}dinger avec champ magn[é]{}tique). , 34:1-113, 1988. S. Jitomirskaya, W. Liu. Universal hierarchical structure of quasiperiodic eigenfunctions. , 187(3) : 721-776, 2018. W. Wasow. [*Asymptotic expansions for ordinary differential equations.*]{} Dover Publications, New York, 1987. M. Wilkinson. An exact renormalisation group for Bloch electrons in a magnetic field., 20: 4337-4354, 1987. R.Wong, Z. Wang. Asymptotic expansions for second-order linear difference equations with a turning point. , 94: 147-194, 2003.
[^1]: The results were announced in the conference proceedings [@F-K:18a]
|
---
abstract: |
In this paper we perform a parallel analysis to the model proposed in [majidbeggs]{}. By considering the central co-tetrad (instead of the central metric) we investigate the modifications in the gravitational metrics coming from the noncommutative spacetime of the $\kappa$-Minkowski type in four dimensions. The differential calculus corresponding to a class of Jordanian $%
\kappa$-deformations provides metrics which lead either to cosmological constant or spatial-curvature type solutions of non-vacuum Einstein equations. Among vacuum solutions we find pp-wave type.
author:
- Andrzej Borowiec
- Tajron Jurić
- Stjepan Meljanac
- 'Anna Pacho[ł]{}'
title: Central tetrads and quantum spacetimes
---
Introduction
============
The quantum gravity effects at the Planck scale might modify the structure of spacetime leading to its noncommutativity [@DFR94; @DFR95]. From algebraic point of view in the quantum phase space, besides the non trivial Heisenberg relations between coordinates and momenta, the coordinate relations will be modified and one has to introduce noncommutativity of coordinates themselves [@2; @Zakrzewski]. Such modification of spacetime might have influence on physical solutions, e.g. in gravitational and cosmological effects [@Schupp; @Schenkel; @Mairi; @BTZ]. The (noncommutative) modification of spacetime should be therefore included in the theoretical predictions for (astrophysical) measurements [@link]. Understandably any corrections to classical solutions would be of the order of the Planck scale which makes them difficult to detect with today’s technology. However the theoretical models can suggest new directions to be developed. The finding of any falsifiable prediction to be tested in the real (astrophysical) experiments and observations would be very important in the experimental search for quantum gravity effects and high energy physics. The considerations on deformation of gravitational solutions as well as on cosmological effects coming from noncommutativity are also very timely due to LIGO and PLANCK experiments. In noncommutative spacetimes approach it is assumed that effects of noncommutativity should be visible in quantum gravity and would allow us to model these in an effective description without full knowledge of quantum gravity itself. One of the most known types of noncommutative spacetime is when coordinates satisfy the Lie algebra type commutation relations. Such deformation was inspired by the $\kappa$-deformed Poincaré algebra [@1] as deformed symmetry for the $\kappa$-Minkowski spacetime [@2; @Zakrzewski].
The investigations proposed in this paper focus on $\kappa $ type of noncommutativity, where the $\kappa $-Minkowski commutation relations are as follows: $$\lbrack \hat{x}^{i},\hat{x}^{j}]=0,\quad \lbrack \hat{x}^{0},\hat{x}^{i}]=%
\frac{i}{\kappa }\hat{x}^{i}, \label{kappa1}$$where $\kappa $ is the deformation parameter usually related to some quantum gravity scale or Planck mass [@1; @2], for example certain bounds on this scale were found in [@bgmp10; @hajume].
To study the modifications in the gravitational effects coming from this deformed spacetime one needs to introduce the appropriate differential calculus (compatible with such noncommutativity). There are few approaches in constructing the deformed differential calculi. For example there are the twisted approach [@Aschieri] and the bicovariant differential calculi formulation based on quantum groups framework [@Woronowicz1]. Moreover it has been shown that in the case of time-like $\kappa $-Minkowski spacetime the four-dimensional bicovariant differential calculi compatible with $\kappa $-Poincaré algebra does not exist, but one can construct a five-dimensional one, which is bicovariant [@Sitarz; @Gonera; @Mercati]. On the other hand considering light-like version of $\kappa $ -Minkowski spacetime the differential calculus can be bicovariant and four-dimensional [@hep-th/0307038; @toward]. Differential calculi of classical dimension (number of basis one-forms equal to number of coordinates) compatible with $\kappa
$ -Minkowski algebra (for time-, light- and spacelike deformations) were classified in [@toward]. These differential calculi are bicovariant with respect to other (larger) symmetries than $\kappa $-Poincaré algebra (except for the light-like case). Alternative approaches to differential calculus on $\kappa $-Minkowski space-time were also considered in [@Bu; @KJ; @41; @EPJC; @oeckl].
Our aim in this paper, inspired by the papers [@majidbeggs; @majid2014], is to investigate the noncommutative (quantum) metrics coming from the families of differential calculi introduced in [@41] (also included in [@toward]).
In [@majidbeggs; @majid2014] the authors have investigated the possible noncommutative metrics $g$, which belong to the center of the $\kappa $-Minkowski algebra (\[kappa1\]) for certain differential calculi, see also \[27\]. This condition, necessary in the noncommutative Riemannian geometry to allow contractions, defines set of equations for coefficients in the metric. In the classical limit when $\kappa \rightarrow \infty $ the influence of the noncommutativity remains in the form of the metric and leads to modifications in the known solutions in GR. In this approach authors were able to identify the Einstein tensor built from the central metric with that of the perfect fluid for positive pressure, zero density, and for negative pressure and positive density [@majidbeggs]. In a follow up paper [majid2014]{} they showed that dark energy (cosmological constant) case can be obtained from the algebraic constraint steaming from the central metric approach.
Our aim in this paper is to consider the central tetrad fields $\omega^a$ in the $\kappa$-Minkowski algebra (\[kappa1\]) instead of the central metric. This way the metric $g=\eta_{ab}\omega^a\otimes \omega^b$ has a Lorentzian signature by definition provided that the flat metric $\eta_{ab}$ has the same signature. In the central metric formalism one has to impose the Lorentzian signature condition as an additional constraint.
Once we calculate the tetrads related with certain differential calculus (compatible with the $\kappa$-Minkowski algebra) we can consider the corresponding gravitational metric in the classical limit as induced from noncommutativity. Classical limit is obtained by $\kappa \longrightarrow
\infty $ and then noncommutative objects (coordinates, differentials etc.) will become commutative as follows: $$\begin{split}
& \hat{x}^{k}\longrightarrow x^{k}\quad ,\quad \hat{x}^{0}\longrightarrow t
\\
& \hat{\xi}^{k}\longrightarrow \text{d}x^{k}\quad ,\quad \hat{\xi}%
^{0}\longrightarrow \text{d}t \\
& g=\hat{g}_{\mu \nu }\hat{\xi}^{\mu }\otimes \hat{\xi}^{\nu
}\longrightarrow g_{\mu \nu }\text{d}x^{\mu }\otimes \text{d}x^{\nu }
\end{split}%$$ Of course, we impose that the metric derived from central tetrad in the classical limit has to satisfy Einstein equations.
Then we focus on the non-vacuum Einstein equations in orthonormal tetrad form $G^{a b }=8\pi G T^{a b
} $ with $G^{a b} =R^{a b}-\frac{1}{2} R \eta^{a b}$ and the energy momentum tensor $T^{a b}=(\rho, p, p, p)$ corresponding to the perfect isotropic and barotropic fluid. In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number, the so-called barotropic factor $w$ equal to the ratio of its pressure $p$ to the energy density $\rho $: $w=p/\rho $. For example the most known cases are: $w=-1$ (cosmological constant or dark energy), $w=0$ (dust or dark matter), $w=1/3$ (radiation) and $w=1$ (stiff matter). The value $w=-1/3$ corresponds to spatial curvature and separates two cases: for $w\geqslant -1/3$ the strong energy condition $\rho+3p\geq 0$ is preserved, for $w<-1/3$ it is violated. The last case characterizes accelerating universe while the former decelerating one.
The main result of this paper is that the effects of the noncommutativity are encoded in the constraints coming from the central tetrad formalism which induces a very special and generic classical solutions: universe with a spatial curvature type of barotropic factor $w=-\frac{1}{3}$ and a universe with dark energy (cosmological constant) with barotropic factor $%
w=-1$.
$\protect\kappa$-Minkowski algebra and related quantum differential calculus
============================================================================
$\protect\kappa$-Minkowski algebra
----------------------------------
$\kappa$-Minkowski algebra $\hat{\mathcal{A}}=\mathbb{C}[\hat{x}^{\mu }]/%
\mathcal{I}$ with $\mu =0,..,3$ where $\mathcal{I}$ is a two-sided ideal generated by the commutation relations $$\label{kappa}
[\hat{x}^{i},\hat{x}^{j}]=0, \quad [\hat{x}^{0},\hat{x}^{i}]=\frac{i}{\kappa}%
\hat{x}^{i},$$ where $\kappa$ is the deformation parameter usually related to some quantum gravity scale or Planck mass. Eq. represents the time-like deformations of the usual Minkowski space. We can also look at more general Lie algebraic deformations of Minkowski space $$\label{kappalie}
[\hat{x}^{\mu},\hat{x}^{\nu}]=iC^{\mu\nu}{}_\lambda \hat{x}^{\lambda},$$ where $\hat{x}^{\mu}=(\hat{x}^{0},\hat{x}^{i})$ and structure constants $%
C^{\mu\nu}{}_\lambda$ satisfy $$C^{\mu\alpha}{}_\beta C^{\nu\lambda}{}_\alpha+C^{\nu\alpha}{}_\beta
C^{\lambda\mu}{}_\alpha +C^{\lambda\alpha}{}_\beta C^{\mu\lambda}{}_\alpha=0.$$ $${C^{\mu\nu}}_\lambda =- {C^{\nu\mu}}_\lambda.$$ For $C^{\mu\nu}{}_\lambda=a^{\mu}\delta_{\lambda}^{\nu}-a^{\nu}\delta_{%
\lambda}^{\mu}$ we get $$\label{kappaminkowski}
[\hat x^\mu, \hat x^\nu] = i(a^\mu\hat x^\nu - a^\nu\hat x^\mu).$$ Generally, $a^\mu={\frac{1}{\kappa}} u^\mu$, where a fixed vector $u^\mu \in
\mathbb{R}^4$ belongs to the classical space (Minkowski space). For $u^{\mu}=(1,\vec{0})$ we get back to eq. as a special case. It turns out to be the most general case either since by linear change of generators one can always transform relations into (cf. [@BP2014]). We should be also aware that these relations and hence corresponding noncommutative algebras representing quantum spacetimes are metric independent. Dependence of a metric may come from covariance property under suitable quantum group action. For example, taking the $\kappa-$Poincaré (quantum) group with metric of Lorentzian signature $(-,+,+,+)$ one has to distinguish three cases, where $u^2=-1$ for time-like deformations, $u^2=0$ for light-like deformations and $u^2=1$ for space-like deformations.
Differential calculus of classical dimension
--------------------------------------------
We denote the algebra of differential 1-forms as $d\hat{x}^{\mu }\in \Omega
^{1}( \hat{\mathcal{A}}) $ with $d:\hat{\mathcal{A}}\rightarrow \Omega ^{1}(
\hat{\mathcal{A}}) $. We define the basis 1-forms $d\hat{x}^{\mu }\equiv
\hat{\xi}^{\mu }$ in a usual way, where $d$ is the exterior derivative with the property $d^{2}=0$ and satisfies Leibniz rule.
In [@toward] the construction of the most general algebra of differential one-forms $\hat{\xi}^{\mu }$ compatible with $\kappa $-Minkowski algebra (\[kappa\]) that is closed in differential forms (the differential calculus is of classical dimension) is presented. The commutators between one forms and coordinates are given by $$\lbrack \hat{\xi}^{\mu },\hat{x}^{\nu }]=iK^{\mu }{}^{\nu}_{\alpha }\hat{\xi}%
^{\alpha }, \label{forme}$$where $K^{\mu }{}^{\nu}_{\alpha }\in \mathbb{R}$ and after imposing super-Jacobi identities and compatibility condition[^1] one gets two constraints on $K^{\mu
}{}^{\nu}_{\alpha }$ $$K^{\lambda \mu }{}_{\alpha }K^{\alpha \nu }{}_{\rho }-K^{\lambda \nu
}{}_{\alpha }K^{\alpha \mu }{}_{\rho }=C^{\mu \nu }{}_{\beta }K^{\lambda
\beta }{}_{\rho }. \label{conditionK}$$$$K^{\mu \nu }{}_{\alpha }-K^{\nu \mu }{}_{\alpha }=C^{\mu \nu }{}_{\alpha }.
\label{consist}$$There are only four solutions (three of them are one parameter solutions) to the above equations, which the authors of [@toward] denoted by $\mathcal{%
C}_{1}$, $\mathcal{C}_{2}$, $\mathcal{C}_{3}$ and $\mathcal{C}_{4}$ ($\mathcal{C}_{4}$ is valid only for $a^{2}=0$). In this paper we will be interested in the original $%
\kappa $-Minkowski algebra, that is $a^{2}\neq 0$, and we are left with just three special cases of $\mathcal{C}_{1}$, $\mathcal{C}_{2}$, $\mathcal{C}%
_{3} $. So, if we take $a^{2}\neq 0$ we get three families of algebras which we denote by[^2] $\mathcal{D}_{1}$, $\mathcal{D}_{2}$ and $\mathcal{D}%
_{3} $: $$\begin{aligned}
& \mathcal{D}_{1}:~~~~[\hat{\xi}^{\mu },\hat{x}^{\nu }]=i\frac{s}{a^{2}}%
a^{\mu }a^{\nu }(a\hat{\xi})-ia^{\nu }\hat{\xi}^{\mu } \\
& \mathcal{D}_{2}:~~~~[\hat{\xi}^{\mu },\hat{x}^{\nu }]=i\frac{s}{a^{2}}%
a^{\mu }a^{\nu }(a\hat{\xi})-isa^{\nu }\hat{\xi}^{\mu }+i(1-s)a^{\mu }\hat{%
\xi}^{\nu } \\
& \mathcal{D}_{3}:~~~~[\hat{\xi}^{\mu },\hat{x}^{\nu }]=i\frac{s}{a^{2}}%
a^{\mu }a^{\nu }(a\hat{\xi})-i(1+s)\eta ^{\mu \nu }(a\hat{\xi})-ia^{\nu }%
\hat{\xi}^{\mu }\end{aligned}$$For $a^{\mu }=(\frac{1}{\kappa },\vec{0})$ algebras $\mathcal{D}_{1}$ and $%
\mathcal{D}_{2}$ can be found in [@41]. For $s=1$ we see that algebras $%
\mathcal{D}_{1}^{s=1}$ and $\mathcal{D}_{2}^{s=1}$ coincide. This case was in detail investigated in [@EPJC]. In [@oeckl] (see Corollary 5.1.) the cases $\mathcal{D}_{1}^{s=0}$ and $\mathcal{D}_{2}^{s=0}$ were obtained from a different construction.
In this paper we will focus on the differential algebra $\mathcal{D}_{1}$, since the algebras $\mathcal{D}_{2,3}$ will only lead to degenerate central metric. For $a^{\mu }=(\frac{1}{\kappa },\vec{0})$, $\mathcal{D}_{1}$ family of algebra of differential forms has the following commutation relations with $\kappa $-Minkowski coordinates (\[kappa\]): $$\left[ \hat{\xi}^{0},\hat{x}^{0}\right] =\frac{i}{\kappa }s\hat{\xi}%
^{0};\qquad \left[ \hat{\xi}^{k},\hat{x}^{0}\right] =-\frac{i}{\kappa }\hat{%
\xi}^{k};\qquad \left[ \hat{\xi}^{\mu },\hat{x}^{j}\right] =0
\label{family1}$$where $s\in \mathbb{R}$ is a free parameter.
Let us consider a one-parameter family of Drinfeld Jordanian twist (cf. [BP2009]{}) $$\label{twist}
\mathcal{F}=\exp \left\{ \ln Z\otimes \left( \frac{1}{s-1}%
L_{i}^{i}-L_{0}^{0}\right) \right\}$$where $Z=[1+\frac{s-1}{\kappa }P_{0}]$ and $L_{\beta }^{\alpha }$ are generators of the Lie algebra $\mathfrak{igl}$ of inhomogeneous general linear transformations with the commutation relations $[L_{\nu }^{\mu
},L_{\lambda }^{\rho }]=\delta _{\nu }^{\rho }L_{\lambda }^{\mu }-\delta
_{\lambda }^{\mu }L_{\nu }^{\rho };\quad \lbrack L_{\nu }^{\mu },P_{\lambda
}]=-\delta _{\lambda }^{\mu }P_{\nu }$. These twists provide from one hand $%
\kappa -$Minkowski spacetime algebra (\[kappa\]) and from the other a family of differential calculus (\[family1\]) in such a way that they are bicovariant with respect to the action of $U^{\mathcal{F}}(\mathfrak{igl)}$-Hopf algebra (for more details see [@toward], [@EPJC]).
Central co-tetrad and the corresponding deformed metric
=======================================================
In this section we want to build up the quantum metric tensor from quantum (noncommutative) co-tetrad $\hat{\omega}^{a}:$ $$\label{tetradmetric}
\hat{g}=\eta _{ab}\hat{\omega}^{a}\otimes \hat{\omega}^{b}=\hat{\omega}%
^{a}\otimes \hat{\omega}_{a}$$where the flat classical metric $\eta _{ab}=diag\left( -,+,+,+\right) $ is assumed to bear Lorentzian signature. In the classical limit co-tetrad consists of four linearly independent one-forms $\omega^a=e^a_\mu dx^\mu$ which by construction are orthonormal with respect to the metric ([tetradmetric]{}). The dual object composed of vector fields (named tetrad or vierbein): $e_a=e_a^\mu\partial_\mu$ stands for famous E. Cartan *repére mobile*. The matrices defining the tetrad $e_a^\mu$ and co-tetrad $%
e^a_\mu$ have to be mutually inverse each other. In fact, this approach provides an effective link between flat and curved spacetime formalism and is also useful in noncommutative setting (see e.g. [@Vitale] and references therein). In addition the metric signature is controlled by the signature of a flat metric.
In [@majidbeggs] the authors investigated the differential calculi compatible with $\kappa $-Minkowski algebra of the classical dimension. In our notation this type of differential calculus corresponds to the family $%
\mathcal{D}_{1}^{s=0}$. Later on [@majid2014] they also investigated certain 2-dimensional differential calculi and they extended their investigations to $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ families (which was called $\alpha $ and $\beta $ family there). They showed that the $%
\mathcal{D}_{2}$ case leads to the degenerate central metrics, except for the case: $\mathcal{D}_{1}^{s=1}\equiv \mathcal{D}_{2}^{s=1}$. In our approach also the family $\mathcal{D}_{3}$ lead to degenerate metric as well. Therefore in this section we look for other solutions in the $\mathcal{%
D}_{1}$ family.
We define the central co-tetrad $\hat{\omega}^{a}$ as a collection of four linearly-independent one-forms that commute with all the noncommutative coordinates $\hat{x}^{\mu }$ i.e. $$\label{commtetrad}
\lbrack \hat{\omega}^{a},\hat{x}^{\mu }]=0.$$and $\hat{\omega}^{a}$ can be written as $$\hat{\omega}^{a}=e_{\mu }^{a}\hat{\xi}^{\mu },$$where the components $e_{\mu }^{a}$ are functions of noncommutative coordinates ($\in \hat{\mathcal{A}}$) that are yet to be determined. Moreover, in the classical limit the matrix $e_{\mu }^{a}$ has to be invertible (which provides an additional condition on functions $e_{\mu
}^{a} $).
In the algebra of $\kappa$-Minkowski coordinates the commutator ([commtetrad]{}) is given by: $$\left[ \hat{\omega}\left( \hat{x}^{\mu }\right) ,\hat{x}^{0}\right] =-\frac{i%
}{\kappa }\sum_{i}\hat{x}^{i}\frac{\partial }{\partial \hat{x}^{i}}\hat{%
\omega} \left( \hat{x}^{j},\hat{x}^{0}\right) ;\qquad \left[ \hat{\omega}%
\left( \hat{x}^{\mu }\right) ,\hat{x}^{k}\right] =\hat{x}^{k}\left( \omega
\left( \hat{x}^{i},\hat{x}_{0}+\frac{i}{\kappa }\right) -g\left( \hat{x}%
^{\mu }\right) \right) \label{diff_eq}$$
We start with a single central one-form $\hat{\omega}=e_{\mu }\xi ^{\mu }$ such that $[\hat{\omega},\hat{x}^{\alpha }]=0$.$$\lbrack \hat{\omega},\hat{x}^{0}]=[e_{\mu }\hat{\xi}^{\mu },\hat{x}%
^{0}]=e_{0}\left[ \hat{\xi}^{0},\hat{x}^{0}\right] +e_{k}\left[ \hat{\xi}%
^{k},\hat{x}^{0}\right] +[e_{0},\hat{x}^{0}]\hat{\xi}^{0}+[e_{k},\hat{x}^{0}]%
\hat{\xi}^{k}=0$$$$\lbrack \hat{\omega},\hat{x}^{j}]=[e_{\mu }\hat{\xi}^{\mu },\hat{x}%
^{j}]=e_{0}\left[ \hat{\xi}^{0},\hat{x}^{j}\right] +e_{k}\left[ \hat{\xi}%
^{k},\hat{x}^{j}\right] +[e_{0},\hat{x}^{j}]\hat{\xi}^{0}+[e_{k},\hat{x}^{j}]%
\hat{\xi}^{k}=0$$After using the relations (\[family1\]) of differential calculus we get the following conditions:$$\lbrack e_{0},\hat{x}^{0}]=-\frac{i}{\kappa }se_{0}\quad ;\quad \lbrack
e_{k},\hat{x}^{0}]=\frac{i}{\kappa }e_{k}$$and $$\lbrack e_{0},\hat{x}^{j}]=0\quad ;\quad \lbrack e_{k},\hat{x}^{j}]=0$$Therefore for the algebra $\mathcal{D}_{1}$ the requirement that the tetrad $%
\hat{\omega}$ is a central element in $\hat{\mathcal{A}}$ leads to the following commutation relations $$\begin{split}
& [e_{0},\hat{x}^{0}]=-\frac{i}{\kappa }se_{0}\quad ;\quad \lbrack e_{0},%
\hat{x}^{k}]=0 \\
& [e_{k},\hat{x}^{0}]=\frac{i}{\kappa }e_{k}\quad ;\quad \lbrack e_{k},\hat{x%
}^{i}]=0
\end{split}%$$It turns out that all $e_{\mu }$ are only functions of $\hat{x}^{i}\,\ $(as they commute with $\hat{x}^{j}$) and have to satisfy the following differential equations (cf. \[diff\_eq\]): $$\label{25}
\hat{x}^{k}\frac{\partial }{\partial \hat{x}^{k}}e_{0} =se_{0}$$ $$\label{rt}
\hat{x}^{k}\frac{\partial }{\partial \hat{x}^{k}}e_{j} =-e_{j}$$ The solutions of and are (see also Appendix 1): $$\begin{aligned}
e_{0} &=&\hat{r}^{s}E_{0}\left( \frac{\hat{x}^{k}}{\hat{r}}\right) \\
e_{j} &=&\hat{r}^{-1}E_{j}\left( \frac{\hat{x}^{k}}{\hat{r}}\right)\end{aligned}$$So the central one-form reads as: $$\hat{\omega}=\hat{r}^{s}E_{0}\left( \frac{\hat{x}^{k}}{\hat{r}}\right) \hat{%
\xi}^{0}+\hat{r}^{-1}E_{j}\left( \frac{\hat{x}^{k}}{\hat{r}}\right) \hat{\xi}%
^{j}$$Similarly for the collection of four linearly independent one-forms satisfying $[\hat{\omega}^{a},\hat{x}^{\alpha }]=0$ we get the following solution:$$\label{tetradmetric1}
\hat{\omega}^{a}=\hat{r}^{s}E_{0}^{a}\left( \frac{\hat{x}^{k}}{\hat{r}}%
\right) \hat{\xi}^{0}+\hat{r}^{-1}E_{j}^{a}\left( \frac{\hat{x}^{k}}{\hat{r}}%
\right) \hat{\xi}^{j}$$because $$e_{0}^{a}=\hat{r}^{s}E_{0}^{a}\left( \frac{\hat{x}^{k}}{\hat{r}}\right)
\quad ;\quad e_{j}^{a}=\hat{r}^{-1}E_{j}^{a}\left( \frac{\hat{x}^{k}}{\hat{r}%
}\right)$$In result we have 16 arbitrary functions $E_{\mu }^{a}\left( \frac{\hat{x}%
^{k}}{\hat{r}}\right) $ of variables $\frac{\hat{x}^{k}}{\hat{r}}$ and $\hat{%
r}=\sqrt{\hat{x}^{k}\hat{x}_{k}}$ (note that in spherical coordinates such functions will only depend on the angles).
Now the metric can be built up from the central tetrads (\[tetradmetric\]) as follows: $$\begin{aligned}
\hat{g} &=&\left( \hat{r}^{s}E_{0}^{a}\left( \frac{\hat{x}^{k}}{\hat{r}}%
\right) \hat{\xi}^{0}+\hat{r}^{-1}E_{i}^{a}\left( \frac{\hat{x}^{k}}{\hat{r}}%
\right) \hat{\xi}^{i}\right) \otimes \left( \hat{r}^{s}E_{a0}\left( \frac{%
\hat{x}^{k}}{\hat{r}}\right) \hat{\xi}^{0}+\hat{r}^{-1}E_{aj}\left( \frac{%
\hat{x}^{k}}{\hat{r}}\right) \hat{\xi}^{j}\right) = \\
&=&\hat{r}^{2s}E_{0}^{a}E_{a0}\hat{\xi}^{0}\otimes \hat{\xi}^{0}+\hat{r}%
^{s-1}E_{a0}E_{j}^{a}\hat{\xi}^{0}\otimes \hat{\xi}^{j}+\hat{r}%
^{s-1}E_{i}^{a}E_{a0}\hat{\xi}^{i}\otimes \hat{\xi}^{0}+\hat{r}%
^{-2}E_{i}^{a}E_{aj}\hat{\xi}^{i}\otimes \hat{\xi}^{j}\end{aligned}$$ One can show that this type of the metric belongs to the center of the algebra $\hat{\mathcal{A}}$ of $\kappa $-Minkowski type (\[kappa\]) as well and has vanishing commutators: $$\left[ \hat{g},\hat{x}^{\mu }\right] =0 \label{metric_cent}$$i.e. it falls in the framework introduced in [@majidbeggs].
Classical limit
---------------
In the classical limit $\kappa \longrightarrow \infty $ when the noncommutative objects (coordinates, differentials etc.) become commutative, the functions $E_{\mu }^{a}$ will become the arbitrary functions of $\frac{{x%
}^{k}}{{r}}$ and $r=\sqrt{x^{k}x^{k}}$ commutative coordinates.
In this case the metric for algebra $\mathcal{D}_{1}$ (\[family1\]) in the classical limit reads as: $$g=g_{\mu \nu }dx^{\mu }\otimes dx^{\nu }=\frac{1}{r^{2}}E_{i}^{a}E_{aj}{dx}%
^{i}\otimes {dx}^{j}+r^{\left( s-1\right) }E_{i}^{a}E_{a0}\left( {dx}%
^{i}\otimes {dt}+{dt}\otimes {dx}^{i}\right) +{r}^{2s}E_{0}^{a}E_{a0}{dt}%
\otimes {dt}$$One can see that in the above metric the functions $E_{\mu }^{a}$ do not depend on time therefore such metrics could describe only stationary solutions.
When we introduce the spherical coordinates $$\left( t,r,\theta ,\phi \right):\quad x=r\sin \theta \cos \phi, \quad
y=r\sin \theta \sin \phi, \quad z=r\cos \theta$$ the functions $E_{\mu }^{a}\left(\theta,\phi \right) $ are arbitrary functions of the angles $\left( \theta,\phi\right) $ only. Therefore, in what follows, we shall use spherical coordinate system to write down the metric in the form: [^3]
$g=\left(
\begin{array}{cccc}
r^{2s}E_{0}^{a}\left( \theta,\phi \right) E_{a0}\left( \theta,\phi \right) &
r^{\left( s-1\right) }E_{1}^{a}\left(\theta, \phi \right) E_{a0}\left(
\theta,\phi \right) & r^{s}E_{2}^{a}\left( \theta,\phi \right) E_{a0}\left(
\theta,\phi \right) & r^{s}E_{3}^{a}\left( \theta,\phi \right) E_{a0}\left(
\theta,\phi \right) \\
r^{\left( s-1\right) }E_{1}^{a}\left( \theta,\phi \right) E_{a0}\left(
\theta,\phi \right) & \frac{E_{1}^{a}\left( \theta,\phi \right) E_{a1}\left(
\theta,\phi \right) }{r^{2}} & \frac{E_{1}^{a}\left( \theta,\phi \right)
E_{a2}\left( \theta,\phi \right) }{r} & \frac{E_{1}^{a}\left( \theta,\phi
\right) E_{a3}\left( \theta,\phi \right) }{r} \\
r^{s}E_{2}^{a}\left( \theta,\phi \right) E_{a0}\left( \theta,\phi \right) &
\frac{E_{1}^{a}\left(\theta,\phi \right) E_{a2}\left( \theta,\phi \right) }{r%
} & E_{2}^{a}\left( \theta,\phi \right) E_{a2}\left( \theta,\phi \right) &
E_{2}^{a}\left( \theta,\phi \right) E_{a3}\left( \theta,\phi \right) \\
r^{s}E_{3}^{a}\left( \theta,\phi \right) E_{a0}\left( \theta,\phi \right) &
\frac{E_{1}^{a}\left( \theta,\phi \right) E_{a3}\left(\theta,\phi \right) }{r%
} & E_{2}^{a}\left( \theta,\phi \right) E_{a3}\left( \theta,\phi \right) &
E_{3}^{a}\left( \theta,\phi \right) E_{a3}\left( \theta,\phi \right)%
\end{array}%
\right) $
We are looking for non degenerate solutions so from the condition on non-zero determinant $\det \left( g\right)\neq 0$ we get additional constraints on the functions $E_{\mu }^{a}$.
We can simplify the notation by introducing $$\begin{aligned}
r^{-2}E_{i}^{a}\left( \theta,\phi \right) E_{aj}\left( \theta,\phi \right)
&=&r^{-2}a_{ij}\left( \theta,\phi \right) \\
r^{s-1}E_{0}^{a}\left( \theta,\phi \right) E_{aj}\left( \theta,\phi \right)
&=&r^{s-1}b_{j}\left( \theta,\phi \right) \\
r^{2s}E_{0}^{a}\left( \theta,\phi \right) E_{a0}\left( \theta,\phi \right)
&=&r^{2s}c\left( \theta,\phi \right)\end{aligned}$$
The Einstein equations are written in the form ($G$ being the universal Newtonian constant of gravity): $$\label{ee}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G\,T_{\mu\nu}$$ Given a specified distribution of matter and energy $T_{\mu\nu}$, the equations (\[ee\]) are understood to be equations for the metric tensor $%
g_{\mu\nu}$, as both the Ricci tensor $R_{\mu\nu}$ and scalar curvature $R$ depend on the metric in a complicated nonlinear manner. One can write the Einstein equations in a more compact form by introducing the Einstein tensor $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R$. Since in the classical limit our metric has to satisfy the Einstein equations, this leads us to some coupled differential equations for tetrad functions $E_{\mu }^{a}$ which turn out to be non trivial to solve. In order to find some solutions, we consider a special case with the ’constant coefficients’, i.e. such that the metric depends only on $r$ coordinate and: $$E_{i}^{a}E_{aj}=a_{ij}=const\quad ;\quad E_{a0}E_{j}^{a}=b_{j}=const\quad
;\quad E_{0}^{a}E_{a0}=c=const$$ The metric looks as follows: $$\label{qmetric}
g=\left(
\begin{array}{cccc}
r^{2s}c & r^{\left( s-1\right) }b_{1} & r^{s}b_{2} & r^{s}b_{3} \\
r^{\left( s-1\right) }b_{1} & \frac{a_{11}}{r^{2}} & \frac{a_{12}}{r} &
\frac{a_{13}}{r} \\
r^{s}b_{2} & \frac{a_{12}}{r} & a_{22} & a_{23} \\
r^{s}b_{3} & \frac{a_{13}}{r} & a_{23} & a_{33}%
\end{array}%
\right)$$ and the determinant: is assumed to be non-zero.
In the following we will look for the solutions of non-vacuum $G^{\mu \nu
}=8\pi G\,T^{\mu \nu }$ and vacuum $G^{\mu \nu }=0$ Einstein equations.
### Non vacuum solutions
In the tetrad formalism we can work with Einstein equations in the Lorentzian frame: $G^{ab}=8\pi G\,T^{ab}$. The passage to the coordinate frame is determined by standard formulae: $g_{\mu \nu }=\eta _{ab}e_{\mu }^{a}e_{\nu }^{b}$ with $%
\eta _{ab}=diag\left( -,+,+,+\right) $ and $G_{\nu }^{\mu }=e_{a}^{\mu
}e_{\nu }^{b}G_{b}^{a}$, etc.
On the left hand side of the non-vacuum Einstein equations we assume the energy-momentum of perfect and isotropic fluid, i.e. $T^{ab}=(\rho
+p)u^{a}u^{b}+p\eta ^{ab}=diag(\rho ,p,p,p)$ where $\rho $ is energy density, $p$ is the pressure of the fluid and the vector $u^{a}$ represents its four-velocity. Assuming further that the fluid is co-moving with respect to the tetrad, i.e. $u^{a}=(1,0,0,0)$ we can diagonalize the energy-momentum tensor $T_{b}^{a}=diag(-\rho ,p,p,p)$. Therefore to look for solutions with perfect fluid one should firstly diagonalize the Einstein tensor $%
G_{\mu}^{\nu}.$ [^4] For this purpose we calculate the characteristic polynomial: $\det \left[ G_{\mu}^{\nu}-\lambda \delta _{\mu}^{\nu}\right]
\Leftrightarrow\det \left[ G_{\mu\nu}-\lambda g _{\mu\nu}\right]=0$ which will give us the diagonal form of the $G_{b}^{a}$ with the roots of this equation on the diagonal. Having the multiplicity $1$ of one solution $%
\tilde{\lambda}$ and multiplicity $3$ of another $\lambda $ will allow us to write down the equation with the (diagonal) momentum energy tensor for the perfect fluid as: $$G_{b}^{a}=\left(
\begin{array}{cccc}
\tilde{\lambda} & & & \\
& \lambda & & \\
& & \lambda & \\
& & & \lambda%
\end{array}%
\right) =8\pi G\,\left(
\begin{array}{cccc}
-\rho & & & \\
& p & & \\
& & p & \\
& & & p%
\end{array}%
\right)$$
The equation of state of barotropic fluid is characterized by a dimensionless number $w$ equal to the ratio of its pressure $p$ to the energy density $\rho $: $w=p/\rho =-\lambda /\tilde{\lambda}$. For example the most known from cosmology cases are:
- cosmological constant (dark energy) which corresponds to $w=-1$,
- dust matter (dark or/and ordinary baryonic matter) ($w=0$),
- radiation ($w=1/3$).
It turns out that spatial curvature of FLRW metric can be also described by the barotropic factor $w=-{\frac{1}{3}}$ satisfying strong energy condition $\rho+3p\geq 0$. Therefore after the diagonalization of the Einstein tensor we can see which kind of the equation of state can be derived from the quantum metric ([qmetric]{}). In the case under consideration one obtains only two possible solution of the barotropic type (there is no other solutions).
1. Quantum Universe with spatial curvature type barotropic factor $w=-1/3$;
$\left( G_{b}^{a}\right) _{_{I}}=\left(
\begin{array}{cccc}
\lambda ^{I} & 0 & 0 & 0 \\
0 & \frac{1}{3}\lambda ^{I} & 0 & 0 \\
0 & 0 & \frac{1}{3}\lambda ^{I} & 0 \\
0 & 0 & 0 & \frac{1}{3}\lambda ^{I}%
\end{array}%
\right) =8\pi G\,\left(
\begin{array}{cccc}
-\rho & 0 & 0 & 0 \\
0 & p & 0 & 0 \\
0 & 0 & p & 0 \\
0 & 0 & 0 & p%
\end{array}%
\right) $
$\left( G^{ab}\right) _{_{I}}=\left(
\begin{array}{cccc}
-\lambda ^{I} & 0 & 0 & 0 \\
0 & \frac{1}{3}\lambda ^{I} & 0 & 0 \\
0 & 0 & \frac{1}{3}\lambda ^{I} & 0 \\
0 & 0 & 0 & \frac{1}{3}\lambda ^{I}%
\end{array}%
\right) $
where $\lambda ^{I}=$ $-\frac{{a}_{{22}}s^{2}}{4\left( {a}_{{12}}^{2}-{a}_{{%
11}}{a}_{{22}}\right) }.$This eigenvalue is given for simplified metric with the choice $c=0;b_{1}=0;b_{2}=0$, which does not change the barotropic factor. The full expression for $\lambda ^{I}$ corresponding exactly to ([qmetric]{}) can be found in Appendix 2. From “polynomial constraints” (see Appendix 2) it follows that $a_{23}=\sqrt{a_{22}a_{33}}$ . The determinant has to be non-zero $\det \left( g_{I_{simpl}}\right)
=r^{2s-2}b_{3}^{2}\left( a_{12}^{2}-a_{11}a_{22}\right) \neq 0$ and $\det
\left( g_{I_{simpl}}\right) <0$ iff $a_{12}^{2}<a_{11}a_{22}$. The corresponding metric is then:
$$g_{I_{simpl}}=\left(
\begin{array}{cccc}
0 & 0 & 0 & r^{s}b_{3} \\
0 & \frac{a_{11}}{r^{2}} & \frac{a_{12}}{r} & \frac{a_{13}}{r} \\
0 & \frac{a_{12}}{r} & a_{22} & \sqrt{a_{22}a_{33}} \\
r^{s}b_{3} & \frac{a_{13}}{r} & \sqrt{a_{22}a_{33}} & a_{33}%
\end{array}%
\right)$$
2. Universe with dark energy (cosmological constant) with $w=-1$;
$\left( G_{b}^{a}\right) _{II}=\left(
\begin{array}{cccc}
\lambda ^{II} & 0 & 0 & 0 \\
0 & \lambda ^{II} & 0 & 0 \\
0 & 0 & \lambda ^{II} & 0 \\
0 & 0 & 0 & \lambda ^{II}%
\end{array}%
\right) =8\pi G\,\left(
\begin{array}{cccc}
-\rho & 0 & 0 & 0 \\
0 & p & 0 & 0 \\
0 & 0 & p & 0 \\
0 & 0 & 0 & p%
\end{array}%
\right) $
$\left( G^{ab}\right) _{II}=\left(
\begin{array}{cccc}
-\lambda ^{II} & 0 & 0 & 0 \\
0 & \lambda ^{II} & 0 & 0 \\
0 & 0 & \lambda ^{II} & 0 \\
0 & 0 & 0 & \lambda ^{II}%
\end{array}%
\right) $
where $\lambda ^{II}=\frac{s^{2}}{2{a}_{{11}}}$ (is the simplified version for the choice of the coefficients $a_{12}=0;a_{23}=0;a_{13}=0;$ $%
b_{1}=0=b_{3}$). Note that $s\neq 0.$ The expression for $\lambda _{2}$ depending on all constant coefficients is given in the Appendix 2 as well. The metric itself looks as follows: $$g_{II_{simpl}}=\left(
\begin{array}{cccc}
r^{2s}\frac{{b}_{{2}}^{2}}{2{a}_{22}} & 0 & r^{s}b_{2} & 0 \\
0 & \frac{a_{11}}{r^{2}} & 0 & 0 \\
r^{s}b_{2} & 0 & a_{22} & 0 \\
0 & 0 & 0 & a_{33}%
\end{array}%
\right)$$The determinant of the metric: $\det \left( g_{II_{simpl}}\right) =-\frac{1}{%
2}b_{2}^{2}a_{33}a_{11}\neq 0$ and $\det g_{II_{simpl}}$ can be chosen to be negative for $a_{33}a_{11}>0$.
However, choosing $s=0$ in this case, i.e. $\left(
\begin{array}{cccc}
\frac{{b}_{{2}}^{2}}{2{a}_{22}} & 0 & b_{2} & 0 \\
0 & \frac{a_{11}}{r^{2}} & 0 & 0 \\
b_{2} & 0 & a_{22} & 0 \\
0 & 0 & 0 & a_{33}%
\end{array}%
\right) $ we get an example of vacuum (but flat) solution of Einstein equations, for which $G_{\mu \nu }=0$ (note that $G^{\mu \nu }=0\Rightarrow
R^{\mu \nu }=0$). Below we present yet other interesting and not flat vacuum solutions, both for $s\neq 0$ and $s=0$ cases.
### Vacuum solutions
1. Vacuum solution of the pp-wave type.
One of the examples of the vacuum solution can be provided for the special choice of non-zero, constant coefficients $b_{1},a_{22},a_{33}$ and restoring the dependence on the angles in the coefficient $a_{11}\left( \phi
,\theta \right) $ i.e. choosing the metric as: $$g_{1vac}=\left(
\begin{array}{cccc}
0 & r^{\left( s-1\right) }b_{1} & 0 & 0 \\
r^{\left( s-1\right) }b_{1} & \frac{a_{11}\left( \phi ,\theta \right) }{r^{2}%
} & 0 & 0 \\
0 & 0 & a_{22} & 0 \\
0 & 0 & 0 & a_{33}%
\end{array}%
\right) \label{vac1}$$Then the corresponding Einstein tensor vanishes for any choice of the parameter $s\neq 0$ but under the following additional conditions on the derivatives of function $a_{11}\left( \phi ,\theta \right) $ as follows: $%
\frac{\partial a_{11}}{\partial \phi }\neq 0;\frac{\partial a_{11}}{\partial
\theta }\neq 0;\frac{\partial ^{2}a_{11}}{\partial \phi \partial \theta }%
\neq 0$ and $\frac{\partial ^{2}a_{11}}{\partial \phi ^{2}}=0=\frac{\partial
^{2}a_{11}}{\partial \theta ^{2}}$ \[which amounts to $a_{11}\left( \phi
,\theta \right) =\xi _{1}+\xi _{2}\theta +\xi _{3}\phi +\xi _{4}\theta \phi $, $\xi _{i}$ are constants\]. The Riemann tensor however does not vanish: $%
R_{t\theta \phi r}=\frac{r^{-\left( s+1\right) }}{2b_{1}}\frac{\partial
^{2}a_{11}}{\partial \phi \partial \theta }=R_{t\phi \theta r};R_{\theta
r\phi r}=-\frac{1}{2a_{22}r^{2}}\frac{\partial ^{2}a_{11}}{\partial \phi
\partial \theta };R_{\phi r\theta r}=-\frac{1}{2a_{33}r^{2}}\frac{\partial
^{2}a_{11}}{\partial \phi \partial \theta }$. Therefore (\[vac1\]) constitutes vacuum but non-flat solution.
2. Another example of vacuum solution, also of pp-wave type can be provided (for $s=0$), by: $$g_{2vac}=\left(
\begin{array}{cccc}
c\left( \phi ,\theta \right) & r^{\left( -1\right) }b_{1} & 0 & 0 \\
r^{\left( -1\right) }b_{1} & 0 & 0 & 0 \\
0 & 0 & a_{22} & 0 \\
0 & 0 & 0 & a_{33}%
\end{array}%
\right) \label{vac2}$$The corresponding Einstein tensor vanishes by imposing additional conditions (analogous to the above ones) on the derivatives of function $c\left( \phi
,\theta \right) $ as follows: $\frac{\partial c}{\partial \phi }\neq 0;\frac{%
\partial c}{\partial \theta }\neq 0;\frac{\partial ^{2}c}{\partial \phi
\partial \theta }\neq 0$ and$\frac{\partial ^{2}c}{\partial \phi ^{2}}=0=%
\frac{\partial ^{2}c}{\partial \theta ^{2}}$ \[which amounts to $c\left(
\phi ,\theta \right) =\zeta _{1}+\zeta _{2}\theta +\zeta _{3}\phi +\zeta
_{4}\theta \phi ,\zeta _{i}$ are constants\]. However the Riemann tensor does not vanish ($R_{r\theta \phi t}=\frac{r}{2b_{1}}\frac{\partial ^{2}c}{%
\partial \phi \partial \theta }=R_{r\phi \theta t};R_{\theta t\phi t}=-\frac{%
1}{2a_{22}}\frac{\partial ^{2}c}{\partial \phi \partial \theta };R_{\phi
t\theta t}=-\frac{1}{2a_{33}}\frac{\partial ^{2}c}{\partial \phi \partial
\theta }$). Therefore (\[vac2\]) is as well vacuum non-flat (Ricci-flat) solution.
It is rather known that the so called pp-waves (plane-fronted waves with parallel rays) are exact wave like solutions of Einstein equations which represent gravitational radiation propagating with the speed of light in a direction determined by light-like Killing vector field. They are analogous to source-free photons in Maxwell electrodynamics. They also appear in many places of Theoretical Physics e.g. as string or D-brane background, and supersymmetry, etc..
Note that the non-vacuum and vacuum solutions for $s=0$ are interesting from the quantum symmetry point of view, since in this case the Jordanian twist (\[twist\]) reduces to $$\mathcal{F}=\exp \left\{ \ln \left( 1-\frac{1}{\kappa }P_{0}\right) \otimes
D\right\}$$and corresponds to Poincaré-Weyl symmetry (one generator extension of the Poincaré algebra, with adding dilatation generator $D$) as a minimal quantum group providing symmetry algebra [@BP2009].
Final remarks
=============
We have investigated some of the properties of quantum spaces using the central co-tetrad formalism. We used a sort of a toy model for quantum gravity effects, which should be encoded in the noncommutative $\kappa $-Minkowski algebra and by analyzing a certain bicovariant differential calculus (compatible with such noncommutative structure) we found equations for the components of the noncommutative central co-tetrads, which we solve in general . We show that this formalism gives rise to the same quantum central metric as proposed in [majidbeggs]{}, but here the benefit is that the Lorentzian signature is built in from the very beginning.
Analyzing the classical limit of our quantum metric, and by imposing the validity of Einstein equations in that limit, we found (under further assumptions which simplify the calculations) new vacuum and non-vacuum solutions. Namely, the solutions of Einstein equations for our simplified cases (metric with constant coefficients and only $r$-dependence) contribute to the description of the (quantum) Universe with the cosmological constant and the spacial curvature. Also, a vacuum solution corresponding to pp-wave spacetime is obtained.
The idea is to investigate more complicated and more general solution of . One can see that in the quantum metric , the functions $E_{\mu }^{a}$ do not depend on time therefore such metrics could describe only static solutions for $\mathcal{D}%
_{1}$. Maybe for more general differential calculi $\mathcal{C}_{1}...,%
\mathcal{C}_{4}$ (classified in [@toward]) one can recover some new interesting cosmological and maybe even black hole solutions.
Appendix 1 {#appendix-1 .unnumbered}
==========
We want to find the general solution for the following differential equation $$\hat{x}^{i}\frac{\partial }{\partial \hat{x}^{i}}f=\gamma f \label{master}$$We denote the dilatation operator $\hat{D}=\hat{x}^{i}\frac{\partial }{%
\partial \hat{x}^{i}}$. For any dimension we can define the spherical coordinates and write the radial vector and nabla operators as $$\vec{\hat{r}}=\hat{x}_{k}\vec{e}_{k}=\hat{r}\ \vec{\hat{r}}_{0}\quad \vec{%
\nabla}=\vec{e}_{k}\frac{\partial }{\partial \hat{x}_{k}}=\vec{\hat{r}}_{0}%
\frac{\partial }{\partial \hat{r}}+\text{terms in directions perpendicular to%
}\ \vec{\hat{r}}_{0}$$where $\vec{e}_{k}$ are constant orthonormal vectors of basis in Cartesian coordinate system and $\vec{\hat{r}}_{0}$ is radial unit vector. Now, for the dilatation operator we have $$\hat{D}=\hat{x}^{i}\frac{\partial }{\partial \hat{x}^{i}}=\vec{\hat{r}}\cdot
\vec{\nabla}=\hat{r}\frac{\partial }{\partial \hat{r}}$$so, differential equation can be solved by direct integration which gives the following solution $$f=\text{const.}\ \hat{r}^{\gamma }$$Notice that $\hat{D}\frac{x_{k}}{\hat{r}}=0$, and so any arbitrary function of $\frac{x_{k}}{\hat{r}}$ satisfies $\hat{D}F(\frac{x_{k}}{\hat{r}})=0$ the most general const. is $F(\frac{x_{k}}{%
\hat{r}})$ which gives the most general solution for is
$$f=\hat{r}^{\gamma }F\left( \frac{x_{k}}{\hat{r}}\right)$$
This enables us to solve differential equations.
Appendix 2: Full solution for the metric (\[qmetric\]) {#appendix-2-full-solution-for-the-metric-qmetric .unnumbered}
======================================================
In section III.A.1 we focused on non vacuum solutions with isotropic metric (\[qmetric\]) , i.e. for $a_{ij}=const,b_{i}=const$ and $c=const)$ and only with dependence on $r:$ $$g=\left(
\begin{array}{cccc}
r^{2s}c & r^{\left( s-1\right) }b_{1} & r^{s}b_{2} & r^{s}b_{3} \\
r^{\left( s-1\right) }b_{1} & \frac{a_{11}}{r^{2}} & \frac{a_{12}}{r} &
\frac{a_{13}}{r} \\
r^{s}b_{2} & \frac{a_{12}}{r} & a_{22} & a_{23} \\
r^{s}b_{3} & \frac{a_{13}}{r} & a_{23} & a_{33}%
\end{array}%
\right)$$
After diagonalization of the Einstein tensor $G_{\nu }^{\mu }$ wrt this metric we obtain only two possible solutions of the perfect fluid type (there is no other solutions). The characteristic equation is of the form : $%
\det \left[ G_{\nu }^{\mu }-\lambda \delta _{\nu }^{\mu }\right] =\alpha
\cdot \beta \cdot \gamma =0$ where:
$\alpha =\left( \left( -{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}%
}-{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}+{x\lambda }_{1}\right) ^{2}$$\beta =\left( \left( -3{a}_{{33}}{b}_{{2}}^{2}+6{a}_{{23}}{b}_{{2}}{b}_{{3}%
}-3{a}_{{22}}{b}_{{3}}^{2}-4{a}_{{23}}^{2}c+4{a}_{{22}}{a}_{{33}}c\right)
s^{2}+{x\lambda }_{2}\right) $$\gamma =\left( \left( -{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}%
}-{a}_{{22}}{b}_{{3}}^{2}-4{a}_{{23}}^{2}c+4{a}_{22}{a}_{{33}}c\right)
s^{2}+x\lambda _{3}\right) $
and $x=-4\left( {a}_{{23}}^{2}-{a}_{{22}}{a}_{{33}}\right) b_{{1}%
}^{2}-4\left( {a}_{{12}}^{2}-{a}_{{11}}{a}_{{22}}\right) b_{{3}}^{2}-4\left(
{a}_{{13}}^{2}-{a}_{{11}}{a}_{{33}}\right) b_{{2}}^{2}$
$+8a_{{13}}a_{{23}}b_{{1}}b_{{2}}-8a_{{12}}a_{{33}}b_{{1}}b_{{2}}-8a_{{13}%
}a_{{22}}b_{{1}}b_{{3}}+8a_{{12}}a_{{23}}b_{{1}}b_{{3}}$
$+8a_{{12}}a_{{13}}b_{{2}}b_{{3}}-8a_{{11}}a_{{23}}b_{{2}}b_{{3}}+4\left(
a_{13}^{2}a_{22}-2a_{12}a_{13}a_{23}+a_{11}a_{23}^{2}+a_{12}^{2}a_{33}-a_{11}a_{22}a_{33}\right) c
$
I. To find the solution of multiplicity 3 and 1 (corresponding to subcase 1. in Sec.III.A.1 ) we notice that for: $-4{a}_{{23}}^{2}c+4{a}_{{22}}{a}_{{33}%
}c=0$ we have the following:
$\alpha =\left( \left( -{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}%
}-{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}+{x\lambda }_{1}^{I}\right) ^{2}$
$\beta =\left( \left( -3{a}_{{33}}{b}_{{2}}^{2}+6{a}_{{23}}{b}_{{2}}{b}_{{3}%
}-3{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}+{x\lambda }_{2}^{I}\right) $
$\gamma =\left( \left( -{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}%
}-{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}+x\lambda _{3}^{I}\right) $
Therefore $\lambda _{1}^{I}=\lambda _{3}^{I}$ (the root of multiplicity 3) and the characteristic equation is as follows:
$\left( \left( -{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}}-{a}_{{%
22}}{b}_{{3}}^{2}\right) s^{2}+{x\lambda }_{1}^{I}\right) ^{3}\left( 3\left(
-{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}}-{a}_{{22}}{b}_{{3}%
}^{2}\right) s^{2}+{x\lambda }_{2}^{I}\right) =0\cdot $
It leads to: $\lambda _{1}^{I}=$ $\frac{\left( a{_{{33}}b}_{{2}}^{2}-2{a}_{{%
23}}{b}_{{2}}{b}_{{3}}+{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}}{x}$ as the triple multiplicity root and $\lambda _{2}^{I}=\frac{3\left( a{_{{33}}b}_{{2}%
}^{2}-2{a}_{{23}}{b}_{{2}}{b}_{{3}}+{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}}{x}%
=3\lambda _{1}^{I}$ as a single multiplicity root for $s\neq 0$ .
The additional condition $-4{a}_{{23}}^{2}c+4{a}_{{22}}{a}_{{33}}c=0$ we shall call “polynomial constraint” which relates the coefficients of the metric in the following way: $$a_{23}^{2}=a_{22}a_{33}$$
This results in $\lambda _{2}^{I}=3\lambda _{1}^{I}$ and therefore barotropic factor $w=-1/3$ (Quantum Universe with spatial curvature type barotropic factor):
$$\left( G^{ab}\right) _{_{I}}=\left(
\begin{array}{cccc}
-\lambda ^{I} & 0 & 0 & 0 \\
0 & \frac{1}{3}\lambda ^{I} & 0 & 0 \\
0 & 0 & \frac{1}{3}\lambda ^{I} & 0 \\
0 & 0 & 0 & \frac{1}{3}\lambda ^{I}%
\end{array}%
\right)$$
And the corresponding metric tensor:
$$g_{I}=\left(
\begin{array}{cccc}
r^{2s}c & r^{\left( s-1\right) }b_{1} & r^{s}b_{2} & r^{s}b_{3} \\
r^{\left( s-1\right) }b_{1} & \frac{a_{11}}{r^{2}} & \frac{a_{12}}{r} &
\frac{a_{13}}{r} \\
r^{s}b_{2} & \frac{a_{12}}{r} & a_{22} & \sqrt{a_{22}a_{33}} \\
r^{s}b_{3} & \frac{a_{13}}{r} & \sqrt{a_{22}a_{33}} & a_{33}%
\end{array}%
\right)$$
II\. Another solution is obtained with the polynomial constraint of the form: $-4{a}_{{23}}^{2}c+4{a}_{22}{a}_{{33}}c=2\left( {a}_{{33}}{b}_{{2}}^{2}-2{a}%
_{{23}}{b}_{{2}}{b}_{{3}}+{a}_{{22}}{b}_{{3}}^{2}\right) $
With this we get the following:
$\alpha =\left( \left( -{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}%
}-{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}+{x\lambda }_{1}\right) ^{2}$$\beta =\left( \left( -{a}_{{33}}{b}_{{2}}^{2}+2{a}_{{23}}{b}_{{2}}{b}_{{3}}-%
{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}+{x\lambda }_{2}\right) $$\gamma =\left( \left( {a}_{{33}}{b}_{{2}}^{2}-2{a}_{{23}}{b}_{{2}}{b}_{{3}}+%
{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}+x\lambda _{3}\right) $
which leads to:
- the root of multiplicity 3 is $\lambda _{1}^{II}=\frac{\left( {a}_{{33}}{b}%
_{{2}}^{2}-2{a}_{{23}}{b}_{{2}}{b}_{{3}}+{a}_{{22}}{b}_{{3}}^{2}\right) s^{2}%
}{x}=\lambda _{2}^{II}$
- the root of multiplicity 1 is $\lambda _{3}^{II}=-\frac{\left( {a}_{{33}}{b%
}_{{2}}^{2}-2{a}_{{23}}{b}_{{2}}{b}_{{3}}+{a}_{{22}}{b}_{{3}}^{2}\right)
s^{2}}{x}=-\lambda _{1}^{II}$
and contributes to a barotropic factor $w=-1$ (Universe with dark energy - cosmological constant) with the following Einstein tensor:$$\left( G^{ab}\right) _{II}=\left(
\begin{array}{cccc}
-\lambda ^{II} & 0 & 0 & 0 \\
0 & \lambda ^{II} & 0 & 0 \\
0 & 0 & \lambda ^{II} & 0 \\
0 & 0 & 0 & \lambda ^{II}%
\end{array}%
\right)$$
From the additional condition $-4{a}_{{23}}^{2}c+4{a}_{22}{a}_{{33}%
}c=2\left( {a}_{{33}}{b}_{{2}}^{2}-2{a}_{{23}}{b}_{{2}}{b}_{{3}}+{a}_{{22}}{b%
}_{{3}}^{2}\right) $ we get the relation for the coefficients of the metric in the following way: $$c=-\frac{\left( {a}_{{33}}{b}_{{2}}^{2}-2{a}_{{23}}{b}_{{2}}{b}_{{3}}+{a}_{{%
22}}{b}_{{3}}^{2}\right) }{2\left( {a}_{{23}}^{2}-{a}_{22}{a}_{{33}}\right) }$$
Therefore the metric corresponding to this case is:
$$g_{II}=\left(
\begin{array}{cccc}
-r^{2s}\frac{\left( {a}_{{33}}{b}_{{2}}^{2}-2{a}_{{23}}{b}_{{2}}{b}_{{3}}+{a}%
_{{22}}{b}_{{3}}^{2}\right) }{2\left( {a}_{{23}}^{2}-{a}_{22}{a}_{{33}%
}\right) } & r^{\left( s-1\right) }b_{1} & r^{s}b_{2} & r^{s}b_{3} \\
r^{\left( s-1\right) }b_{1} & \frac{a_{11}}{r^{2}} & \frac{a_{12}}{r} &
\frac{a_{13}}{r} \\
r^{s}b_{2} & \frac{a_{12}}{r} & a_{22} & a_{23} \\
r^{s}b_{3} & \frac{a_{13}}{r} & a_{23} & a_{33}%
\end{array}%
\right)$$
Acknowledgments {#acknowledgments .unnumbered}
===============
The work by T.J. and S.M. has been fully supported by Croatian Science Foundation under the project (IP-2014-09-9582). T.J. would like to thank B. Klajn for the fruitful discussions that led to Appendix 1. A. Much is acknowledged for the discussions with T.J. A.P. acknowledges support from the European Union’s Seventh Framework programme for research and innovation under the Marie Skłodowska-Curie grant agreement No 609402 - 2020 researchers: Train to Move (T2M). Part of this work was also supported by Polish National Science Center project 2014/13/B/ST2/04043. AB is supported by NCN project DEC-2013/09/B/ST2/03455.
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[^1]: for more details see [@toward].
[^2]: Where $\mathcal{D}$ stands for $differential\ \ algebra$ and was already used in [@41].
[^3]: In fact, the coefficient functions $E^a_\mu (\theta, \phi)$ appearing in formulas (35) and below are in general not identical. They can be express each other as linear combinations with trigonometric functions of $(\theta,
\phi)$.
[^4]: The Lorentz indices $a,b.\ldots $ are raised and lowered by means of the flat metric and its inverse.
|
---
abstract: 'We study the random graph ${\ensuremath{G(n,p)}}$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $\{a\to s\}$ and $\{s\to b\}$. We prove that asymptotically the correlation is negative for small $p$, $p<\frac{C_1}n$, where $C_1\approx0.3617$, positive for $\frac{C_1}n<p<\frac2n$ and up to $p=p_2(n)$. Computer aided computations suggest that $p_2(n)=\frac{C_2}n$, with $C_2\approx7.5$. We conjecture that the correlation then stays negative for $p$ up to the previously known zero at $\frac12$; for larger $p$ it is positive.'
address:
- 'Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06, Uppsala, Sweden.'
- 'Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06, Uppsala, Sweden.'
- 'Department of Mathematics, KTH-Royal Institute of Technology, SE-100 44, Stockholm, Sweden.'
author:
- Sven Erick Alm
- Svante Janson
- Svante Linusson
date: '25 March, 2013'
title: 'First critical probability for a problem on random orientations in $G(n,p)$.'
---
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Introduction {#S:Intro}
============
Let $G(n,p)$ be the random graph with $n$ vertices where each edge has probability $p$ of being present independent of the other edges. We further orient each present edge either way independently with probability $\frac{1}{2}$, and denote the resulting random directed graph by [[$\vec G(n,p)$]{}]{}. This version of orienting edges in a graph, random or not, is natural and has been considered previously in e.g. [@AL; @AJL; @G00; @M81].
Let $a,b,s$ be three distinct vertices and define the events $A:=\{a\to
s\}$, that there exists a directed path in ${{\ensuremath{\vec G(n,p)}}}$ from $a$ to $s$, and $B:=\{s\to b\}$. In a previous paper, [@AJL], we showed that, for fixed $p$, the correlation between $A$ and $B$ asymptotically is negative for $p<\frac12$ and positive for $p>\frac12$. Note that we take the covariance in the combined probability space of ${\ensuremath{G(n,p)}}$ and the orientation of edges, which is often referred to as the annealed case, see [@AJL] for details. We say that a probability $p\in(0,1)$ is *critical* (for a given $n$) if the covariance ${{\operatorname{Cov}}}(A,B)=0$. We have thus shown in [@AJL] that there is a critical probability $\frac12+o(1)$ for large $n$. (Moreover, this is the largest critical probability, since the covariance stays positive for all larger $p<1$.) We also conjectured that for large $n$, there are in fact (at least) three critical probabilities when the covariance changed sign. Based on computer aided computations we guessed that the first two critical probabilities would be approximately $\frac{0.36}{n}$ and $\frac{7.5}{n}$. In this note we prove that there is a first critical probability of the conjectured order, where the covariance changes from negative to positive, and thus there must be at least three critical probabilities. Our theorem is as follows.
\[T:firstzero\] With $p=\frac{2c}n$ and sufficiently large $n$, the covariance ${{\operatorname{Cov}}}(A,B)$ is negative for $0<c<c_1$ and positive for $c_1<c<1$, where $c_1\approx0.180827$ is a solution to $(2-c)(1-c)^3=1$. Furthermore, for fixed $c$ with $0\le c<1$, $$\label{main}
{{\operatorname{Cov}}}(A,B)
=\left(1-(2-c)(1-c)^3\right)\cdot \frac{c^3}{(1-c)^5}\cdot \frac1{n^3}
+O\Big(\frac1{n^4}\Big).$$
In fact, the proof shows that holds uniformly in $0\le c\le c'$ for any $c'<1$; moreover, we may (with just a little more care) for such $c$ write the error term as $O(c^4n^{-4})$. This implies that for large $n$, the critical $p\approx2c_1/n$ is indeed the first critical probability, and that the covariance is negative for all smaller $p>0$.
In a random orientation of any given graph $G$, it is a fact first observed by McDiarmid that ${\operatorname{\mathbb P{}}}(a\to s)$ is equal to ${\operatorname{\mathbb P{}}}(a\leftrightarrow s)$ in an edge percolation on the same graph with probability $1/2$ for each edge independently, see [@M81]. Hence the events $A$ (and thus $B$) have the same probability as ${\operatorname{\mathbb P{}}}(a\leftrightarrow s)$ in $G(n,p/2)$. With $p=2c/n$ it is well known that for $c<1$ this probability is $\frac{c}{(1-c)}n{^{-1}}+O(n{^{-2}})$, see e.g. [@JL]. Hence the covariance in is of the order $O({\operatorname{\mathbb P{}}}(A){\operatorname{\mathbb P{}}}(B)/n)$.
The outline of the proof is as follows, see Sections [\[S:pf\] and \[S:Lemmas\]]{} for details.
Let $p:=2c/n$, where $c<1$. Let ${X_A}:=\#\{a\to s\}$ be the number of paths from $a$ to $s$ in [[$\vec G(n,p)$]{}]{} and ${X_B}:=\#\{s\to b\}$. (In the proof below, for technical reasons, we actually only count paths that are not too long.) We first show that, in our range of $p$, the probability that ${X_A}\ge2$ or ${X_B}\ge2$ is small, and that we can ignore these events and approximate ${{\operatorname{Cov}}}(A,B)$ by ${{\operatorname{Cov}}}({{X_A}},{{X_B}})$. The latter covariance is a double sum over pairs of possible paths [$({\alpha},{\beta})$]{}, where ${\alpha}$ goes from $a$ to $s$ and ${\beta}$ goes from $s$ to $b$, and we show that the largest contribution comes from configurations of the following two types:
Type 1
: The two edges incident to $s$, [i.e[=1000]{}]{} the last edge in ${\alpha}$ and the first edge in ${\beta}$, are the same but with opposite orientations; all other edges are distinct. See Figure \[F:Type1\].
![Configurations of Type 1 ($i,j\ge0$, $i+j\ge1$).[]{data-label="F:Type1"}](Figure1){width="5cm"}
Type 2
: ${\alpha}$ and ${\beta}$ contain a common subpath with the same orientation, but all other edges are distinct. See Figure \[F:Type2\].
![Configurations of Type 2 ($i,j\ge0$, $k,l,m\ge1$).[]{data-label="F:Type2"}](Figure2){width="7cm"}
If [$({\alpha},{\beta})$]{} is of Type 1, then ${\alpha}$ and ${\beta}$ cannot both be paths in [[$\vec G(n,p)$]{}]{}, since they contain an edge with opposite orientations. Thus each such pair [$({\alpha},{\beta})$]{} gives a negative contribution to ${{\operatorname{Cov}}}({X_A},{X_B})$. Pairs of Type 2, on the other hand, give a positive contribution. It turns out that both contributions are of the same order ${n^{-3}}$, see Lemmas [\[L:Type1\] and \[L:Type2\]]{}, with constant factors depending on $c$ such that the negative contribution from Type 1 dominates for small $c$, and the positive contribution from Type 2 dominates for larger $c$.
It would be interesting to find a method to compute also the second critical probability, which we in [@AJL] conjectured to be approximately $\frac{7.5}{n}$. (The methods in the present paper apply only for $c<1$.) Even showing that the covariance is negative when $p$ is of the order $\frac{\log n}{n}$ is open. Moreover we conjecture that (for large $n$ at least) there are only three critical probabilities, but that too is open.
Proof of Theorem \[T:firstzero\] {#S:pf}
================================
We give here the main steps in the proof of [Theorem \[T:firstzero\]]{}, leaving details to a sequence of lemmas in Section \[S:Lemmas\]. By a *path* we mean a directed path ${\gamma}=v_0e_1\cdots e_\ell v_\ell$ in the complete graph $K_n$. We use the conventions that a path is self-avoiding, i.e. has no repeated vertex, and that the length $|{\gamma}|$ of a path is the number of edges in the path.
We let ${\Gamma}$ be the set of all such paths and let, for two distinct vertices $v$ and $w$, ${\Gamma}_{vw}$ be the subset of all paths from $v$ to $w$.
If ${\gamma}\in{\Gamma}$, let ${I_{\gamma}}$ be the indicator that ${\gamma}$ is a path in [[$\vec G(n,p)$]{}]{}, [i.e[=1000]{}]{}, that all edges in ${\gamma}$ are present in ${{\ensuremath{\vec G(n,p)}}}$ and have the correct orientation there. Thus $$\label{a1}
{\operatorname{\mathbb E{}}}{I_{\gamma}}={\operatorname{\mathbb P{}}}({I_{\gamma}}=1)={{\left(\frac{p}{2}\right)}}^{|{\gamma}|}
= {{\left(\frac{c}{n}\right)}}^{|{\gamma}|}.$$
Let ${I_A}$ and ${I_B}$ be the indicators of $A$ and $B$. Note that the event $A$ occurs if and only if $\sum_{{\alpha}\in{\Gamma}_{as}}{I_{\alpha}}\ge1$, and similarly for $B$.
It will be convenient to restrict attention to paths that are not too long, so we introduce a cut-off $L{:=}\log^2n$ and let ${{\Gamma}^L_{vw}}$ be the set of paths in ${\Gamma}_{vw}$ of length at most $L$. Let $$\begin{aligned}
{{X_A}}{:=}\sum_{{\alpha}\in{{\Gamma}^L_{as}}} {I_{\alpha}}&&
\text{and}
&&&
{{X_B}}{:=}\sum_{{\beta}\in{{\Gamma}^L_{sb}}} {I_{\beta}},\end{aligned}$$ [i.e[=1000]{}]{}, the numbers of paths in [[$\vec G(n,p)$]{}]{} from $a$ to $s$ and from $s$ to $b$, ignoring paths of length more than $L$.
Write ${X_A}={I_A'}+{X_A'}$ and ${X_B}={I_B'}+{X_B'}$, where ${I_A'}$ and ${I_B'}$ are the indicators for the events ${{X_A}}\ge1$ and ${{X_B}}\ge1$ respectively, so that $$\begin{aligned}
{I_A'}&=\min({{X_A}},1),
\\
{X_A'}&=({{X_A}}-1)_+
=\begin{cases}0&\text{if }{X_A}\le1,\\
{X_A}-1&\text{if }{X_A}>1.\end{cases}\end{aligned}$$
We have ${I_A}\ge{I_A'}$. Let ${J_A}{:=}{I_A}-{I_A'}$ and ${J_B}{:=}{I_B}-{I_B'}$. Thus $$\label{ijsplit}
{{\operatorname{Cov}}}(A,B)={{\operatorname{Cov}}}({I_A},{I_B})={{\operatorname{Cov}}}({I_A'},{I_B'})+{{\operatorname{Cov}}}({I_A'},{J_B})+{{\operatorname{Cov}}}({J_A},{I_B}).$$ We will show in [Lemma \[L:L\]]{} below that the last terms are small: $O(n{^{-99}})$. (The exponent 99 here and below can be replaced by any fixed number.)
Similarly, since ${I_A'}={{X_A}}-{X_A'}$, $$\label{E:split}
{{\operatorname{Cov}}}({I_A'},{I_B'})={{\operatorname{Cov}}}({{X_A}},{{X_B}})-{{\operatorname{Cov}}}({{X_A}},{X_B'})-{{\operatorname{Cov}}}({X_A'},{{X_B}})+{{\operatorname{Cov}}}({X_A'},{X_B'}),$$ where [Lemma \[L:rest\]]{} shows that the last three terms are $O({n^{-4}})$. Hence, it suffices to compute $$\label{a3c}
{{\operatorname{Cov}}}({X_A},{X_B})={{\operatorname{Cov}}}{\Bigl(\sum_{{\alpha}\in{{\Gamma}^L_{as}}}{I_{\alpha}},\sum_{{\beta}\in{{\Gamma}^L_{sb}}}{I_{\beta}}\Bigr)}
=\sum_{{\alpha}\in{{\Gamma}^L_{as}}}\sum_{{\beta}\in{{\Gamma}^L_{sb}}}{\operatorname{Cov}}({I_{\alpha}},{I_{\beta}}).$$ Lemmas \[L:Type1\] and \[L:Type2\] yield the contribution to this sum from pairs [$({\alpha},{\beta})$]{} of Types 1 and 2, and [Lemma \[L:3\]]{} shows that the remaining terms contribute only $O({n^{-4}})$. Using – and the lemmas in [Section \[S:Lemmas\]]{} we thus obtain $$\begin{split}
{{\operatorname{Cov}}}(A,B)&={{\operatorname{Cov}}}({I_A'},{I_B'})+O(n{^{-99}})
={{\operatorname{Cov}}}({{X_A}},{{X_B}})+O({n^{-4}})
\\&
=\left(-\frac{2c^3-c^4}{(1-c)^2}+\frac{c^3}{(1-c)^5}\right)\frac1{n^3}
+O\Big(\frac1{n^4}\Big),
\\&=\frac{c^3}{(1-c)^5}\cdot{\Bigl(1-(2-c)(1-c)^3\Bigr)}\frac1{n^3}
+O\Big(\frac1{n^4}\Big),
\end{split}$$ which is .
The polynomial $1-(2-c)(1-c)^3=-c^4+5c^3-9c^2+7c-1$ is negative for $c=0$ and has two real zeros, for example because its discriminant is $-283 <0$, see e.g. [@W1898]; a numerical calculation yields the roots $c_1\approx0.180827$ and $c_2\approx2.380278$, which completes the proof.
Lemmas {#S:Lemmas}
======
We begin with some general considerations. We assume, as in [Theorem \[T:firstzero\]]{}, that $p=2c/n$ and $0\le c<1$.
Consider a term ${{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})$ in . Suppose that ${\alpha}$ and ${\beta}$ have lengths ${{\ell_{\alpha}}}$ and ${{\ell_{\beta}}}$. Furthermore, suppose that ${\beta}$ contains ${{\delta}}\ge0$ edges not in ${\alpha}$ (ignoring the orientations) and that these form ${\mu}\ge0$ subpaths of ${\beta}$ that intersect ${\alpha}$ only at the endvertices. (We will use the notation ${\beta}\setminus{\alpha}$ for the set of (undirected) edges in ${\beta}$ but not in ${\alpha}$.) The number ${{\ell_{{\alpha}{\beta}}}}$ of edges common to ${\alpha}$ and ${\beta}$ (again ignoring orientations) is thus ${{\ell_{\beta}}}-{{\delta}}$. By , ${\operatorname{\mathbb E{}}}{I_{\alpha}}=(c/n)^{{{\ell_{\alpha}}}}$ and ${\operatorname{\mathbb E{}}}{I_{\beta}}=(c/n)^{{{\ell_{\beta}}}}$.
If ${\alpha}$ and ${\beta}$ have no common edge, then ${I_{\alpha}}$ and ${I_{\beta}}$ are independent and $$\label{cov0}
{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})=0.$$
If all common edges have the same orientation in ${\alpha}$ and ${\beta}$, then $$\label{cov+}
{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})= {\operatorname{\mathbb E{}}}({I_{\alpha}}{I_{\beta}})-{\operatorname{\mathbb E{}}}{I_{\alpha}}{\operatorname{\mathbb E{}}}{I_{\beta}}={{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\delta}}}
-{{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}}.$$
If some common edge has different orientations in ${\alpha}$ and ${\beta}$, then ${\operatorname{\mathbb E{}}}({I_{\alpha}}{I_{\beta}})=0$ and $$\label{cov-}
{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})=-{\operatorname{\mathbb E{}}}{I_{\alpha}}{\operatorname{\mathbb E{}}}{I_{\beta}}=-{{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}}.$$
We denote the falling factorials by $(n)_\ell{:=}n(n-1)\dotsm(n-\ell+1)$. Note that the total number of paths of length $\ell$ in ${\Gamma}_{vw}$ is $(n-2)_{\ell-1}{:=}(n-2)\dotsm(n-\ell)$, since the path is determined by choosing $\ell-1$ internal vertices in order, and all vertices are distinct.
\[L:L\] ${\operatorname{Cov}}({I_A'},{J_B})=O(n{^{-99}})$ and ${\operatorname{Cov}}({J_A},{I_B})=O(n{^{-99}})$.
${J_A}$ is the indicator of the event that there is a path in [[$\vec G(n,p)$]{}]{} from $a$ to $s$, and that every such path has length $>L=\log^2 n$. Thus, $$0\le{J_A}\le\sum_{{\alpha}\in{\Gamma}_{as},\,|{\alpha}|>L}{I_{\alpha}}$$ and thus, using and the fact that there are $(n-2)_{\ell-1}\le
n^{\ell-1}$ paths of length $\ell$ in ${\Gamma}_{as}$, $$0\le{\operatorname{\mathbb E{}}}{J_A}\le
\sum_{{\alpha}\in{\Gamma}_{as},\,|{\alpha}|>L}{{\left(\frac{c}{n}\right)}}^{|{\alpha}|}
\le\sum_{\ell=L}^\infty n^{\ell-1}{{\left(\frac{c}{n}\right)}}^{\ell}
\le\sum_{\ell=L}^\infty c^{\ell}
=O(c^L)=O(n{^{-99}}).$$ Since ${J_A},{I_{\beta}}\in[0,1]$, $$|{{\operatorname{Cov}}}({J_A},{I_B})|\le{\operatorname{\mathbb E{}}}({J_A}{I_B})+{\operatorname{\mathbb E{}}}{J_A}{\operatorname{\mathbb E{}}}{I_B}\le2{\operatorname{\mathbb E{}}}{J_A}=O(n{^{-99}}).$$ Similarly, $|{{\operatorname{Cov}}}({I_A'},{J_B})|=O(n{^{-99}})$.
\[L:Type1\] Pairs of Type 1 contribute $-\frac{1}{n^3}\frac{2c^3-c^4}{(1-c)^2}+O(\frac1{n^4})$ to the covariance ${{\operatorname{Cov}}}({X_A},{X_B})$.
Let the path ${\alpha}$ from $a$ to $s$ consist of $i+1$ edges, where the last edge is the first in the path ${\beta}$ of length $j+1$ from $s$ to $b$, see Figure \[F:Type1\]. The paths must not share any more edges, but could have more common vertices. Here $i,j\ge0$ and $i+j\ge1$ since $a\neq b$. Let $R_{i,j}$ be the number of such pairs of paths, for given $i$ and $j$. If $j\ge1$, the paths are determined by the choice of $i$ distinct vertices for ${\alpha}$ and then $j-1$ distinct vertices for ${\beta}$; if $j=0$, then $i\ge1$ and the paths are determined by the choice of $i-1$ distinct vertices for ${\alpha}$. Order is important so, for $i,j\le L$, with a minor modification if $j=0$, $$(n-2)_{i}\cdot (n-3)_{j-1}\ge R_{i,j}\ge
(n-2)_{i+j-1},$$ Thus $R_{i,j}= n^{i+j-1}{\Bigl(1+O{{\Bigl(\frac{(i+j)^2}{n}\Bigr)}}\Bigr)}$ and summing over all such pairs [$({\alpha},{\beta})$]{} gives by a contribution to ${{\operatorname{Cov}}}({X_A},{X_B})$ of $$\begin{aligned}
-\sum_{i+j\ge 1}&R_{i,j}\Big(\frac cn\Big)^{i+j+2}
=
-\sum_{\substack{i+j\ge 1\\i,j\le L}}
n^{i+j-1}{\Bigl(1+O{{\Bigl(\frac{(i+j)^2}{n}\Bigr)}}\Bigr)}
\Big(\frac cn\Big)^{i+j+2}
\\&
=
-\sum_{\substack{i+j\ge 1\\i,j\le L}} c^{i+j+2}{n^{-3}}
+\sum_{i+j\ge 1}O{\bigl((i+j)^2\bigr)} c^{i+j+2}{n^{-4}}
\\&
=-{n^{-3}}{\Bigl(
2\sum_{j\ge 1}c^{j+2}+\sum_{i,j\ge 1}c^{i+j+2}+O(c^L)\Bigr)}+{O{\bigl(n^{-4}\bigr)}}
\\
&=-{n^{-3}}\left(\frac{2c^3}{1-c}+\frac{c^4}{(1-c)^2}\right)+{O{\bigl(n^{-4}\bigr)}}
=-{n^{-3}}\cdot\frac{2c^3-c^4}{(1-c)^2}+{O{\bigl(n^{-4}\bigr)}}.
\qedhere\end{aligned}$$
\[L:Type2\]Type 2 pairs contribute $\frac1{n^3}\cdot\frac{c^3}{(1-c)^5}+O\big(\frac1{n^4}\big)$ to the covariance ${{\operatorname{Cov}}}({X_A},{X_B})$.
A pair [$({\alpha},{\beta})$]{} of paths of Type 2 must contain a directed cycle containing $s$, from which there are $m\ge1$ edges to a vertex $x$ to which there is a directed path of length $i\ge 0$ from $a$. The cycle continues from $x$ with $k\ge1$ edges to a vertex $y$, which connects to $b$ via $j\ge0$ edges. The cycle is completed by $l\ge1$ edges from $y$ to $s$, see Figure \[F:Type2\]. By , then $${{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})={{\Bigl(\frac{c}{n}\Bigr)}}^{i+j+k+l+m}{\left(1-{{\Bigl(\frac{c}{n}\Bigr)}}^k\right)}.$$
Let $R_{i,j,k,l,m}$ be the number of such pairs [$({\alpha},{\beta})$]{} with given $i,j,k,l,m$. The path ${\alpha}$ is determined by $i+k+l-1$ distinct vertices and given ${\alpha}$, if $j\ge1$, then the path ${\beta}$ is determined by choosing $m+j-2$ vertices; if $j=0$ then $b$ lies on ${\alpha}$, so ${\alpha}$ is determined by choosing $i+k+l-2$ vertices, and then ${\beta}$ is determined by choosing $m-1$ further vertices. Reasoning as in the proof of [Lemma \[L:Type1\]]{} we have $$R_{i,j,k,l,m}=
n^{i+j+k+l+m-3}{\Bigl(1+O{{\Bigl(\frac{(i+j+k+l+m)^2}{n}\Bigr)}}\Bigr)}.$$ Due to our cut-off, we have to have $i+k+l\le L$ and $j+k+m\le L$, but we may for simplicity here allow also paths ${\alpha},{\beta}$ with lengths larger than $L$; the contribution below from pairs with such ${\alpha}$ or ${\beta}$ is $O(c^L)=O(n{^{-99}})$. Summing over all possible configurations gives $$\begin{aligned}
\sum_{i,j\ge0,\,k,l,m\ge1}&R_{i,j,k,l,m}\Big(\frac cn\Big)^{i+j+k+l+m}
\cdot\left(1-\Big(\frac cn\Big)^{k}\right)\\
&=\frac1{n^3}\cdot\sum_{i,j\ge0,\,k,l,m\ge1}c^{i+j+k+l+m}
\cdot\left(1-\Big(\frac cn\Big)^{k}\right)+O\Big(\frac1{n^4}\Big)\\
&=\frac1{n^3}\cdot\frac{c^3}{(1-c)^5}+O\Big(\frac1{n^4}\Big).
\qedhere\end{aligned}$$
\[L:3\] The sum $\sum|{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})|$ over all pairs [$({\alpha},{\beta})$]{} with ${\alpha}\in{{\Gamma}^L_{as}}$, ${\beta}\in{{\Gamma}^L_{sb}}$ and [$({\alpha},{\beta})$]{} not of Type 1 or 2 is ${O{\bigl(n^{-4}\bigr)}}$.
Consider pairs ${\ensuremath{({\alpha},{\beta})}}$ with some given ${{\ell_{\alpha}}},{{\delta}},{\mu}$. The path ${\alpha}$, which has ${{\ell_{\alpha}}}-1$ interior vertices, may be chosen in $\le n^{{{\ell_{\alpha}}}-1}$ ways. The $2{\mu}$ endvertices of the $\mu$ subpaths of ${\beta}\setminus{\alpha}$ are either $b$ or lie on ${\alpha}$, and given ${\alpha}$, these may be chosen (in order) in $\le ({{\ell_{\alpha}}}+2)^{2{\mu}}$ ways. The ${{\delta}}-{\mu}$ internal vertices in the subpaths can be chosen in $\le
n^{{{\delta}}-{\mu}}$ ways. They can be distributed in $\binom{{{\delta}}-1}{{\mu}-1}$ (interpreted as 1 if ${\mu}={{\delta}}=0$) ways over the subpaths. The path ${\beta}$ is determined by these endvertices, the sequence of ${{\delta}}-{\mu}$ interior vertices in the subpaths between these endvertices and which vertices belong to which subpath; hence the total number of choices of ${\beta}$ is $\le\binom{{{\delta}}-1}{{\mu}-1}({{\ell_{\alpha}}}+2)^{2{\mu}}n^{{{\delta}}-{\mu}}$.
For each such pair ${\ensuremath{({\alpha},{\beta})}}$, we have by – $|{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})|\le (c/n)^{{{\ell_{\alpha}}}+{{\delta}}}$. Consequently, the total contribution to $\sum|{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})|$ from the paths with given ${{\ell_{\alpha}}},{{\delta}},{\mu}$ is at most $$\label{c1}
\binom{{{\delta}}-1}{{\mu}-1}({{\ell_{\alpha}}}+2)^{2{\mu}}n^{{{\ell_{\alpha}}}-1+{{\delta}}-{\mu}}{{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\delta}}}
=\binom{{{\delta}}-1}{{\mu}-1}({{\ell_{\alpha}}}+2)^{2{\mu}}c^{{{\ell_{\alpha}}}+{{\delta}}} n^{-{\mu}-1}.$$
We consider several different cases and show that each case yields a contribution ${O{\bigl(n^{-4}\bigr)}}$, noting that we may assume that ${{\ell_{\beta}}}>{{\delta}}$, since otherwise ${\alpha}$ and ${\beta}$ are edge-disjoint, and thus ${{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})=0$ by .
[${\mu}\ge4$]{} Using that $\binom{{{\delta}}-1}{{\mu}-1}\le{{\delta}}^{{\mu}-1}\le L^{{\mu}}$, and summing over ${{\delta}}\ge0$ and ${{\ell_{\alpha}}}\le L$, yields for a fixed ${\mu}$ a contribution $$\label{c2}
\le
(L+2)^{3{\mu}}(1-c)^{-2}n^{-{\mu}-1},
$$ and the sum of these for ${\mu}\ge4$ is $$O{\bigl(L^{12}n^{-5}\bigr)}
=
O{\bigl(n^{-5}\log^{24} n\bigr)}
={O{\bigl(n^{-4}\bigr)}}.
$$
[${\mu}=3$]{} Using that, with ${\mu}=3$, $\binom{{{\delta}}-1}{{\mu}-1}=\binom{{{\delta}}-1}2\le{{\delta}}^2$, and summing over all ${{\ell_{\alpha}}},{{\delta}}\ge0$ yields a contribution of at most $$\sum_{{{\ell_{\alpha}}},{{\delta}}\ge0}{{\delta}}^2({{\ell_{\alpha}}}+2)^6 c^{{{\ell_{\alpha}}}+{{\delta}}} n^{-4}
\le
\sum_{{{\ell_{\alpha}}}\ge0}({{\ell_{\alpha}}}+2)^6 c^{{{\ell_{\alpha}}}}
\sum_{{{\delta}}\ge0}{{\delta}}^2 c^{{{\delta}}} n^{-4}
={O{\bigl(n^{-4}\bigr)}}.
$$ It remains to consider ${\mu}\le2$.
[${\mu}=0$]{} In this case, ${\beta}\subset{\alpha}$, and thus ${{\delta}}=0$ and ${{\ell_{\alpha}}}>{{\ell_{\beta}}}$ (because $a\neq b$). Given ${{\ell_{\alpha}}}$ and ${{\ell_{\beta}}}$, we can choose ${\beta}$ in $\le n^{{{\ell_{\beta}}}-1}$ ways and then ${\alpha}$ in $\le n^{{{\ell_{\alpha}}}-{{\ell_{\beta}}}-1}$ ways; for each choice applies since the edges in ${\beta}$ have opposite orientations in ${\alpha}$, and thus the contribution to $\sum|{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})|$ is at most $$\label{kaw}
n^{{{\ell_{\beta}}}-1+{{\ell_{\alpha}}}-{{\ell_{\beta}}}-1}{{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}}
=
{c}^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}}n^{-{{\ell_{\beta}}}-2}.$$ If ${{\ell_{\beta}}}=1$, then ${\ensuremath{({\alpha},{\beta})}}$ is of Type 1, see [Figure \[F:Type1\]]{} ($j=0$). Since we have excluded such pairs, we may thus assume that ${{\ell_{\beta}}}\ge2$. Summing over ${{\ell_{\alpha}}}>{{\ell_{\beta}}}\ge2$ yields ${O{\bigl(n^{-4}\bigr)}}$.
[${\mu}\in{\ensuremath{\{1,2\}}}$ and ${\alpha}$ and ${\beta}$ have some common edge with opposite orientations]{} \[l3d\] In this case, applies, and $\binom{{{\delta}}-1}{{\mu}-1}\le{{\delta}}\le{{\ell_{\beta}}}$. Thus, if we let ${{\ell_{{\alpha}{\beta}}}}={{\ell_{\beta}}}-{{\delta}}\ge1$ be the number of common edges in ${\alpha}$ and ${\beta}$, then the total contribution to $\sum|{{\operatorname{Cov}}}({I_{\alpha}},{I_{\beta}})|$ for given ${{\ell_{\alpha}}},{{\ell_{\beta}}},{\mu},{{\ell_{{\alpha}{\beta}}}}$ (which determine ${{\delta}}={{\ell_{\beta}}}-{{\ell_{{\alpha}{\beta}}}}$) is at most, in analogy with but using , $$\label{j1}
{{\ell_{\beta}}}({{\ell_{\alpha}}}+2)^{2{\mu}}n^{{{\ell_{\alpha}}}-1+{{\delta}}-{\mu}}{{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}}
=
({{\ell_{\alpha}}}+2)^{2{\mu}}{{\ell_{\beta}}}\,c^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}} n^{-1-{{\ell_{{\alpha}{\beta}}}}-{\mu}}.$$ For fixed ${\mu}$, the sum of over ${{\ell_{\alpha}}},{{\ell_{\beta}}}\ge1$ and ${{\ell_{{\alpha}{\beta}}}}\ge3-{\mu}$ is ${O{\bigl(n^{-4}\bigr)}}$, so we only have to consider $1\le{{\ell_{{\alpha}{\beta}}}}\le2-{\mu}$. In this case we must have ${\mu}=1$ and ${{\ell_{{\alpha}{\beta}}}}=1$ (and $\binom{{{\delta}}-1}{{\mu}-1}=1$); thus ${\alpha}$ and ${\beta}$ have exactly one common edge, which is adjacent to one of the endvertices of ${\beta}$. If the common edge is adjacent to $s$, we have a pair ${\ensuremath{({\alpha},{\beta})}}$ of Type 1, see [Figure \[F:Type1\]]{}; we may thus assume that the common edge is not adjacent to $s$. Then, ${{\ell_{\beta}}}\ge2$ and the common edge is adjacent to $b$, which implies $b\in{\alpha}$. Given ${{\ell_{\alpha}}}$, the number of paths ${\alpha}$ that pass through $b$ is $({{\ell_{\alpha}}}-1)(n-3)_{{{\ell_{\alpha}}}-2}$, since $b$ may be any of the ${{\ell_{\alpha}}}-1$ interior vertices. The choice of ${\alpha}$ fixes the last interior vertex of ${\beta}$ (as the successor of $b$ in ${\alpha}$), and the remaining ${{\ell_{\beta}}}-2$ interior vertices may be chosen in $\le n^{{{\ell_{\beta}}}-2}$ ways. The total contribution from this case is thus at most $$ ({{\ell_{\alpha}}}-1)n^{{{\ell_{\alpha}}}-2+{{\ell_{\beta}}}-2}{{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}}
=
({{\ell_{\alpha}}}-1)c^{{{\ell_{\alpha}}}+{{\ell_{\beta}}}} n^{-4},$$ and summing over ${{\ell_{\alpha}}}$ and ${{\ell_{\beta}}}$ we again obtain ${O{\bigl(n^{-4}\bigr)}}$.
[${\mu}\in{\ensuremath{\{1,2\}}}$ and all common edges in ${\alpha}$ and ${\beta}$ have the same orientation]{} The edge in ${\beta}$ at $s$ does not belong to ${\alpha}$ (since it would have opposite orientation there), so one of the ${\mu}$ subpaths of ${\beta}$ outside ${\alpha}$ begins at $s$. If ${\mu}=1$, or if ${\mu}=2$ and $b\notin{\alpha}$, then ${\ensuremath{({\alpha},{\beta})}}$ is of Type 2, see [Figure \[F:Type2\]]{} ($j=0$ and $j\ge1$, respectively). We may thus assume that ${\mu}=2$ and $b\in{\alpha}$. As in case \[l3d\], given ${{\ell_{\alpha}}}$, we may choose ${\alpha}$ in $({{\ell_{\alpha}}}-1)(n-3)_{{{\ell_{\alpha}}}-2}\le {{\ell_{\alpha}}}n^{{{\ell_{\alpha}}}-2}$ ways. The ${\mu}=2$ subpaths of ${\beta}$ outside ${\alpha}$ have 4 endvertices belonging to ${\alpha}$; one is $s$ and the others may be chosen in $\le {\ell_{\alpha}}^3$ ways. For any such choice, the remaining ${{\delta}}-2$ vertices of ${\beta}$ may be chosen in $\le n^{{{\delta}}-2}$ ways. The total contribution for given ${{\ell_{\alpha}}}$ and ${{\delta}}$ is thus, using , at most $${\ell_{\alpha}}^{4}n^{{{\ell_{\alpha}}}-2+{{\delta}}-2}{{\left(\frac{c}{n}\right)}}^{{{\ell_{\alpha}}}+{{\delta}}}
=
{\ell_{\alpha}}^{4}c^{{{\ell_{\alpha}}}+{{\delta}}} n^{-4},$$ and summing over all ${{\ell_{\alpha}}},{{\delta}}$ we obtain ${O{\bigl(n^{-4}\bigr)}}$.
\[L:rest\] With notation as before, we have ${{\operatorname{Cov}}}({X_A},{X_B'})={{\operatorname{Cov}}}({X_A'},{X_B})=O(n^{-4})$ and ${{\operatorname{Cov}}}({X_A'},{X_B'})=O(n^{-4})$.
We only need to consider paths in $\Gamma^L$, which is assumed throughout the proof. Define $Y_A:=\binom {X_A}2$, the number of pairs of (distinct) paths from $a$ to $s$, and similarly $Y_B:=\binom {{{X_B}}}2$. Then $0\le {X_A'}\le Y_A$ and $0\le{X_B'}\le Y_B$. Let ${Y_A'}:=Y_A-{X_A'}$ and ${Y_B'}:=Y_B-{X_B'}$. Then ${Y_A'}=0$ unless $X_A\ge3$.
Further, let $Z_A:=\binom {X_A}3$, the number of triples of (distinct) paths from $a$ to $s$. Then $0\le {Y_A'}\le Z_A$.
To show that ${{\operatorname{Cov}}}({X_A},{X_B'})={{\operatorname{Cov}}}({X_A'},{X_B})=O(n^{-4})$, we write ${{\operatorname{Cov}}}({X_A'},{X_B})={{\operatorname{Cov}}}({Y_A}-{Y_A'},{X_B})={{\operatorname{Cov}}}({Y_A},{X_B})-{{\operatorname{Cov}}}({Y_A'},{X_B})$. Here, ${{\operatorname{Cov}}}({Y_A'},{X_B})={\operatorname{\mathbb E{}}}({Y_A'}{X_B})-{\operatorname{\mathbb E{}}}({Y_A'})\cdot{\operatorname{\mathbb E{}}}({X_B})$, where ${\operatorname{\mathbb E{}}}({Y_A'}{X_B})\le{\operatorname{\mathbb E{}}}({Z_A}{X_B})$, which we will show is $O(n^{-4})$. Further we will show that ${\operatorname{\mathbb E{}}}({X_A})={\operatorname{\mathbb E{}}}({X_B})=O(n^{-1})$ and that ${\operatorname{\mathbb E{}}}({Y_A'})\le{\operatorname{\mathbb E{}}}({Z_A})=O(n^{-3})$, so that ${{\operatorname{Cov}}}({Y_A'},{X_B})=O(n^{-4})$. Finally we will show that ${{\operatorname{Cov}}}({Y_A},{X_B})=O(n^{-4})$ finishing the proof of the first part of the lemma.
For the second part we write ${{\operatorname{Cov}}}({X_A'},{X_B'})={\operatorname{\mathbb E{}}}({X_A'}{X_B'})-{\operatorname{\mathbb E{}}}({X_A'})\cdot{\operatorname{\mathbb E{}}}({X_B'})$. We prove that ${\operatorname{\mathbb E{}}}({X_A'}{X_B'})\le{\operatorname{\mathbb E{}}}({Y_A}{Y_B})=O(n^{-4})$ and that ${\operatorname{\mathbb E{}}}({X_A'})={\operatorname{\mathbb E{}}}({X_B'})\le{\operatorname{\mathbb E{}}}({Y_A})=O(n^{-2})$, which finishes the proof.\
**(i)** ${\operatorname{\mathbb E{}}}({X_A})=O(n^{-1})$:\
Let $\alpha$ denote an arbitrary path from $a$ to $s$ (in $\Gamma^L$) with length $l\ge1$. Then, $${\operatorname{\mathbb E{}}}({X_A})={\operatorname{\mathbb E{}}}\Big(\sum_{\alpha}I_{\alpha}\Big)=\sum_{\alpha}{\operatorname{\mathbb E{}}}(I_{\alpha})\le\sum_{l=1}^L n^{l-1}\left(\frac cn\right)^{l}\le\frac c{1-c}\cdot n^{-1}=O(n^{-1}).$$
**(ii)** ${\operatorname{\mathbb E{}}}({Y_A})=O(n^{-2})$:\
Let $\alpha_1$ and $\alpha_2$, with lengths $l_1$ and $l_2$ be two distinct paths from $a$ to $s$. Further, let $\delta=|\alpha_2\setminus\alpha_1|$ be the number of edges in $\alpha_2$ not in $\alpha_1$, which form $\mu>0$ subpaths of $\alpha_2$ with no interior vertices in common with $\alpha_1$. The number of choices for $\alpha_2$ is (compare the proof of Lemma \[L:3\]) at most $n^{\delta-\mu}(l_1+1)^{2\mu}\binom{\delta-1}{\mu-1}$, which gives $$\begin{aligned}
{\operatorname{\mathbb E{}}}({Y_A})&=\sum_{\alpha_1\ne\alpha_2}{\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2})\le\sum_{l_1,\delta,\mu}n^{l_1-1}\left(\frac cn\right)^{l_1}n^{\delta-\mu}(l_1+1)^{2\mu}\binom{\delta-1}{\mu-1}\cdot\left(\frac cn\right)^{\delta}\\
&=\sum_{l_1,\delta,\mu}n^{-\mu-1}(l_1+1)^{2\mu}c^{l_1+\delta}\binom{\delta-1}{\mu-1}.\end{aligned}$$ *Case 1:* $\mu\ge2$.\
Here, $(l_1+1)^{2\mu}\le(L+1)^{2\mu}$, $\binom{\delta-1}{\mu-1}\le(\delta-1)^{\mu-1}\le\delta^{\mu}\le L^{\mu}$, so that the terms are at most $n^{-\mu-1}c^{l_1+\delta}(L+1)^{3\mu}$. Summing over $l_1$ and $\delta$ gives at most $\frac{c^2}{(1-c)^2}(L+1)^{3\mu}n^{-\mu-1}$, which summed for $\mu\ge2$ is $O(L^6n^{-3})=O(n^{-3}\log^{12}n)=O(n^{-2})$.\
*Case 2:* $\mu=1$.\
Here, $\binom{\delta-1}{\mu-1}=1$, and $$\sum_{l_1,\delta}{\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2})
\le n^{-2}\sum_{l_1\ge1}(l_1+1)^2c^{l_1}\sum_{\delta\ge1}c^{\delta}
= O(n^{-2}).$$
**(iii)** ${\operatorname{\mathbb E{}}}({Z_A})=O(n^{-3})$:\
We have $${\operatorname{\mathbb E{}}}({Z_A})=\sum_{\alpha_1,\alpha_2,\alpha_3}{\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\alpha_3}),$$ where $\alpha_1$, $\alpha_2$ and $\alpha_3$ denote three distinct paths from $a$ to $s$.
Let $l_1$ denote the length of $\alpha_1$, let $\delta_2=|\alpha_2\setminus\alpha_1|$ be the number of edges in $\alpha_2$ not in $\alpha_1$ forming $\mu_2>0$ subpaths of $\alpha_2$ intersecting $\alpha_1$ only at the endvertices, and let $\delta_3=|\alpha_3\setminus(\alpha_1\cup\alpha_2)|$ be the number of edges in $\alpha_3$ not in $\alpha_1$ or $\alpha_2$ forming $\mu_3\ge0$ subpaths of $\alpha_3$ whose interior vertices are not in $\alpha_1$ or $\alpha_2$. Note that $\mu_3=0$ is possible if $\mu_2\ge2$, as then $\alpha_3$ can be formed by one part from $\alpha_1$ and one part from $\alpha_2$; however, if $\mu_2=1$ then $\mu_3\ge1$. Hence, $\mu_2+\mu_3\ge2$.
If all common edges of the three paths have the same direction, ${\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\alpha_3})=\left(\frac
cn\right)^{l_1+\delta_2+\delta_3}$, otherwise it is 0, so we need only consider paths with the same direction. The number of choices for $\alpha_2$ is, as in (ii), at most $n^{\delta_2-\mu_2}\cdot(l_1+1)^{2\mu_2}\cdot\binom{\delta_2-1}{\mu_2-1}$ and the number of choices for $\alpha_3$ is at most $n^{\delta_3-\mu_3}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\binom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}$, where the last factor is an upper bound for the possible number of choices between segments of $\alpha_1$ and $\alpha_2$. Thus, with summation over $l_1\ge1,\delta_2\ge\mu_2\ge1,\delta_3\ge\mu_3\ge0$, with $\mu_2+\mu_3\ge2$, $$\label{siii}
\begin{split}
\sum &{\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\alpha_3})\le\sum n^{l_1-1}\cdot n^{\delta_2-\mu_2}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot\\
&\cdot n^{\delta_3-\mu_3}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}\cdot\left(\frac cn\right)^{l_1+\delta_2+\delta_3}\\
&=\sum n^{-\mu_2-\mu_3-1}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}\cdot c^{l_1+\delta_2+\delta_3}.
\end{split}$$ *Case 1:* $\mu_2+\mu_3\ge3$.\
Here, $(l_1+1)^{2\mu_2}\le(L+1)^{2\mu_2}$, $\tbinom{\delta_2-1}{\mu_2-1}\le
L^{\mu_2}$, $(l_1+\delta_2-\mu_2+1)^{2\mu_3}\le(2L+1)^{2\mu_3}\le (L+1)^{3\mu_3}$ (assuming as we may $L\ge4$), $\tbinom{\delta_3-1}{\mu_3-1}\le L^{\mu_3}$ and $2^{\mu_2}\le L^{\mu_2}$, so that the sum over $l_1,\delta_2, \delta_3$ is at most $$\label{siiia}
n^{-\mu_2-\mu_3-1}\cdot (L+1)^{4\mu_2+4\mu_3}\cdot\sum c^{l_1+\delta_2+\delta_3}
\le (1-c)^{-3}\cdot n^{-\mu_2-\mu_3-1}\cdot(L+1)^{4(\mu_2+\mu_3)}.$$ Summing over $\mu_2$ and $\mu_3$, with $\mu_2+\mu_3\ge3$ gives $$O(n^{-4}\cdot L^{12})=O(n^{-4}\log^{24}n)=O(n^{-3}).$$ *Case 2:* $\mu_2+\mu_3=2$.\
Here, $(\mu_2,\mu_3)\in\{(2,0),(1,1)\}$, so that $(l_1+1)^{2\mu_2}\le(l_1+1)^4$, $\tbinom{\delta_2-1}{\mu_2-1}\le\delta_2$, $(l_1+\delta_2-\mu_2+1)^{2\mu_3}\le(l_1+\delta_2)^2$, $\tbinom{\delta_3-1}{\mu_3-1}=1$ and $2^{\mu_2}\le4$, so that summing over $l_1,\delta_2, \delta_3$ and $\mu_2+\mu_3=2$ gives at most $$2\cdot4\cdot
n^{-3}\cdot\sum_{l_1,\delta_2,\delta_3}(l_1+1)^4\cdot\delta_2\cdot(l_1+\delta_2)^2\cdot
c^{l_1+\delta_2+\delta_3}
=O(n^{-3}).$$
**(iv)** ${\operatorname{\mathbb E{}}}({Z_A}\cdot{X_B})=O(n^{-4})$:\
${\operatorname{\mathbb E{}}}({Z_A}\cdot{X_B})=\sum{\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\alpha_3}I_{\beta})$, where $
\alpha_1$, $\alpha_2$ and $\alpha_3$ are three distinct paths from $a$ to $s$ and $\beta$ is a path from $s$ to $b$. We need only consider paths where all common edges have the same direction, as ${\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\alpha_3}I_{\beta})=0$ otherwise.
As in (iii) the three $\alpha$ paths are described by $l_1,\delta_2,\mu_2,\delta_3,\mu_3$. Let $\delta_4:=|\beta\setminus(\alpha_1\cup\alpha_2\cup\alpha_3)|$ be the number of edges in $\beta$, not in any of the $\alpha$ paths, and let these form $\mu_4$ subpaths of $\beta$ whose endvertices lie on $\alpha_1,\alpha_2,\alpha_3$ but share no other vertices with those paths. The number of choices for the $\alpha$ paths are the same as in (iii) and given those, and $\delta_4,\mu_4$, the $\beta$ path can be chosen in at most $n^{\delta_4-\mu_4}\cdot(l_1+\delta_2-\mu_2+\delta_3-\mu_3+1)^{2\mu_4}\cdot\tbinom{\delta_4-1}{\mu_4-1}\cdot3^{2(\mu_2+\mu_3)}$ ways, where the last factor is a crude upper bound for the number of ways $\beta$ can choose different sections from the $\alpha$ paths, as there are at most $2(\mu_2+\mu_3)$ vertices where a choice can be made and there are at most 3 possible choices at each of these. Clearly, ${\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\alpha_3}I_{\beta})=(\frac
cn)^{l_1+\delta_2+\delta_3+\delta_4}$ since all common edges have the same direction.
Note that $\mu_4\ge1$ for non-zero terms as otherwise the first edge in $\beta$ from $s$ would be the last edge in one of the $\alpha$ paths, and therefore would have opposite direction. Further, $\mu_2\ge1$, $\mu_3\ge0$, but $\mu_2+\mu_3\ge2$ as $\mu_2=1,\mu_2=0$ would imply that $\alpha_3=\alpha_1$ or $\alpha_3=\alpha_2$.
Summing over $l_1\ge1$, $\mu_2\ge1$, $\delta_2\ge\mu_2$, $\mu_3\ge0$, $\delta_3\ge\mu_3$, $\mu_4\ge1$ and $\delta_4\ge\mu_4$ with $\mu_2+\mu_3\ge2$ gives at most $$\label{siv}
\begin{split}
\sum &n^{l_1-1}\cdot n^{\delta_2-\mu_2}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot n^{\delta_3-\mu_3}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}\cdot\\
&\cdot n^{\delta_4-\mu_4}\cdot(l_1+\delta_2-\mu_2+\delta_3-\mu_3+1)^{2\mu_4}\cdot\tbinom{\delta_4-1}{\mu_4-1}\cdot3^{2(\mu_2+\mu_3)}\cdot\left(\frac cn\right)^{l_1+\delta_2+\delta_3+\delta_4}\\
&=\sum n^{-\mu_2-\mu_3-\mu_4-1}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}\cdot\\
&\cdot (l_1+\delta_2-\mu_2+\delta_3-\mu_3+1)^{2\mu_4}\cdot\tbinom{\delta_4-1}{\mu_4-1}\cdot3^{2(\mu_2+\mu_3)}\cdot c^{l_1+\delta_2+\delta_3+\delta_4}.
\end{split}$$ *Case 1:* $\mu_2+\mu_3+\mu_4\ge4$.\
Here, using the same type of estimates as in (iii) and summing over $l_1,\delta_2,\delta_3,\delta_4$ gives at most $$n^{-\mu_2-\mu_3-\mu_4-1}\cdot(L+1)^{7\mu_2+7\mu_3+4\mu_4}\sum c^{l_1+\delta_2+\delta_3+\delta_4}\le(1-c)^{-4}n^{-\mu_2-\mu_3-\mu_4-1}\cdot(L+1)^{7(\mu_2+\mu_3+\mu_4)},$$ which summed over $\mu_2+\mu_3+\mu_4\ge4$ is $$O(n^{-5}\cdot L^{28})=O(n^{-5}\cdot\log^{56}n)=O(n^{-4}).$$ *Case 2:* $\mu_2+\mu_3+\mu_4=3$.\
Here, $(\mu_2,\mu_3,\mu_4)\in\{(2,0,1),(1,1,1)\}$ so that $(l_1+1)^{2\mu_2}\le(l_1+1)^{4}$, $\tbinom{\delta_2-1}{\mu_2-1}\le\delta_2$, $(l_1+\delta_2-\mu_2+1)^{2\mu_3}\le(l_1+\delta_2)^{2}$, $\tbinom{\delta_3-1}{\mu_3-1}=\tbinom{\delta_4-1}{\mu_4-1}=1$, $2^{\mu_2}\le4$, $(l_1+\delta_2-\mu_2+\delta_3-\mu_3+1)^{2\mu_4}\le(l_1+\delta_2+\delta_3)^{2}$ and $3^{2(\mu_2+\mu_3)}=3^4=81$, so that the sum over $l_1,\delta_2,\delta_3,\delta_4$ is finite and the total contribution is $O(n^{-4})$.
**(v)** ${\operatorname{\mathbb E{}}}({Y_A}\cdot{Y_B})=O(n^{-4})$:\
${\operatorname{\mathbb E{}}}({Y_A}\cdot{Y_B})=\sum{\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\beta_3}I_{\beta_4})$, where $\alpha_1$ and $\alpha_2$ are two distinct paths from $a$ to $s$ and $\beta_3$ and $\beta_4$ are two distinct paths from $s$ to $b$. As above, we need only consider paths where all common edges have the same direction. As before, $\alpha_1$ and $\alpha_2$ are described by $l_1=|\alpha_1|\ge1$, $\delta_2=|\alpha_2\setminus\alpha_1|\ge1$, the number of edges in $\alpha_2$ not in $\alpha_1$, and $\mu_2\ge1$, the number of subpaths they form that intersect $\alpha_1$ in (and only in) the endvertices. Then $\beta_3$ is described by $\delta_3=|\beta_3\setminus(\alpha_1\cup\alpha_2)|$, the number of edges in $\beta_3$ not in $
\alpha_1$ or $\alpha_2$, and $\mu_3$, the number of subpaths they form with no interior vertices in common with $\alpha_1,\alpha_2$. Similarly, $\beta_4$ is described by $\delta_4=|\beta_3\setminus(\alpha_1\cup\alpha_2\cup\beta_3)|\ge0$, the number of edges in $\beta_4$ not in $\alpha_1$, $\alpha_2$ or $\beta_3$ and $\mu_4\ge0$, the number of subpaths they form which intersect $\alpha_1,\alpha_2,\beta_3$ in (and only in) the endvertices. Note that $\mu_3\ge1$ for every non-zero term, as otherwise the first edge in $\beta_3$ from $s$ would be the last edge in one of the $\alpha$ paths, and therefore would have opposite direction.
The number of choices for the $\alpha$ paths are the same as in (ii) and given those, and $\delta_3,\mu_3,\delta_4,\mu_4$, the $\beta$ paths can be chosen in at most $n^{\delta_3-\mu_3}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}\cdot
n^{\delta_4-\mu_4}\cdot(l_1+\delta_2-\mu_2+\delta_3-\mu_3+1)^{2\mu_4}\cdot\tbinom{\delta_4-1}{\mu_4-1}\cdot3^{2(\mu_2+\mu_3})$, where the last factor is an upper bound for the number of ways $\beta_4$ can choose different sections from the $\alpha$ paths and $\beta_3$.
When all common edges have the same direction, ${\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\beta_3}I_{\beta_4})=(\frac
cn)^{l_1+\delta_2+\delta_3+\delta_4}$. Summing over $l_1\ge1$, $\mu_2\ge1$, $\delta_2\ge\mu_2$, $\mu_3\ge1$, $\delta_3\ge\mu_3$, $\mu_4\ge0$ and $\delta_4\ge\mu_4$ gives at most $$\begin{aligned}
\sum &n^{l_1-1}\cdot n^{\delta_2-\mu_2}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot n^{\delta_3-\mu_3}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}\cdot\\
&\cdot n^{\delta_4-\mu_4}\cdot(l_1+\delta_2-\mu_2+\delta_3-\mu_3+1)^{2\mu_4}\cdot\tbinom{\delta_4-1}{\mu_4-1}\cdot3^{2(\mu_2+\mu_3)}\cdot\left(\frac cn\right)^{l_1+\delta_2+\delta_3+\delta_4}\\
&=\sum n^{-\mu_2-\mu_3-\mu_4-1}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot2^{\mu_2}\cdot\\
&\cdot (l_1+\delta_2-\mu_2+\delta_3-\mu_3+1)^{2\mu_4}\cdot\tbinom{\delta_4-1}{\mu_4-1}\cdot3^{2(\mu_2+\mu_3)}\cdot c^{l_1+\delta_2+\delta_3+\delta_4}.\end{aligned}$$ We sum the same terms as in , so the sum over all terms with $\mu_4\ge1$ is $O(n^{-4})$ by the estimates in part [(iv)]{}. Hence it suffices to consider the terms with $\mu_4=0$ and thus ${\delta}_4=0$.\
*Case 1:* $\mu_2+\mu_3\ge4$, $\mu_4=0$.\
Here, each term is $3^{2(\mu_2+\mu_3)}$ times the corresponding term in . Hence, the estimates in (iii) show that, cf. , summing over $l_1,\delta_2,\delta_3$ gives at most $$(1-c)^{-3}n^{-\mu_2-\mu_3-1}\cdot(L+1)^{6(\mu_2+\mu_3)},$$ which summed over $\mu_2+\mu_3\ge4$ is $$O(n^{-5}\cdot L^{24})=O(n^{-5}\cdot\log^{48}n)=O(n^{-4}).$$ *Case 2:* $\mu_2+\mu_3=3$, $\mu_4=0$.\
Here, $\mu_2,\mu_3\le2$ so that $(l_1+1)^{2\mu_2}\le(l_1+1)^{4}$, $\tbinom{\delta_2-1}{\mu_2-1}\le\delta_2$, $(l_1+\delta_2-\mu_2+1)^{2\mu_3}=(l_1+\delta_2)^{4}$, $\tbinom{\delta_3-1}{\mu_3-1}\le{\delta}_3$, $2^{\mu_2}\le4$, and $3^{2(\mu_2+\mu_3)}=3^6=729$, so that the sum over $l_1,\delta_2,\delta_3$ is $O(n^{-\mu_2-\mu_3-1})$ and the contribution is $O(n^{-4})$.\
*Case 3:* $\mu_2+\mu_3=2$, $\mu_4=0$.\
This can only occur if $\mu_2=\mu_3=1$. Thus, $\beta_3$ starts with an edge not in any of the $\alpha$ paths and, as this is its only excursion it must end up at one of the $\alpha$ paths and follow it to $b$ (if $\beta_3$ were to go straight to $b$ without coinciding with any of the $\alpha$ paths then $\beta_4$ would have to do the same, so that $\beta_3=\beta_4$). $\beta_4$ must start as $\beta_3$ until it encounters an $\alpha$ path and must have the possibility to chose a different path to $b$ than $\beta_3$ along the $\alpha$ paths. This means that both $\alpha$ paths must pass through $b$ and that they only differ somewhere between $a$ and $b$. Thus, see Figure \[F:3\], there must be three vertices $x$ (possibly $x=a$), $y$ (possibly $y=x$) and $z$ (possibly $z=b$) between $a$ and $b$, so that both $\alpha$ paths pass in order $a,x,y,z,b,s$, and both $\beta$ paths pass in order $s,x,y,z,b$. Both the two $\alpha$ paths and the two $\beta$ paths follow different subpaths between $y$ and $z$. Let the number of edges between $a$ and $x$ be $i\ge0$, between $x$ and $y$ be $j\ge0$, between $y$ and $z$ be $k\ge1$ and $l\ge1$ for the two possibilities (with $k+l\ge3$), between $z$ and $b$ be $m\ge0$, between $s$ and $x$ be $r\ge1$ and between $b$ and $s$ be $t\ge1$.
![Configurations for *Case 3* of **(v)**: $\mu_2+\mu_3+\mu_4=2$.[]{data-label="F:3"}](Figure3){width="7cm"}
Then, ${\operatorname{\mathbb E{}}}(I_{\alpha_1}I_{\alpha_2}I_{\beta_3}I_{\beta_4})=\left(\frac cn\right)^{i+j+k+l+m+r+t}$ and the number of possibilities is at most $2n^{i+j+k+l+m+r+t-4}$, so that the sum over $i,j,k,l,m,r,t$ is $O(n^{-4})$.
**(vi)** ${{\operatorname{Cov}}}({Y_A},{X_B})=O(n^{-4})$:$$|{{\operatorname{Cov}}}({Y_A},{X_B})|=|\sum_{\alpha_1\ne\alpha_2}\sum_{\beta}{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|\le\sum_{\alpha_1\ne\alpha_2}\sum_{\beta}|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|,$$ where $${{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})={\operatorname{\mathbb E{}}}(I_{\alpha_1}\cdot I_{\alpha_2}\cdot I_{\beta})-{\operatorname{\mathbb E{}}}(I_{\alpha_1}\cdot I_{\alpha_2})\cdot{\operatorname{\mathbb E{}}}(I_{\beta}),$$ which is 0 if $\alpha_1$ and $\alpha_2$ have a common edge with opposite directions, or if $\beta$ has no edge in common with the $\alpha$ paths.\
Let as above $\alpha_1$ have length $l_1$, $\alpha_2$ have $\delta_2$ edges not in $\alpha_1$ forming $\mu_2$ subpaths of $\alpha_2$ intersecting $\alpha_1$ in (and only in) the endvertices. Let also $\beta$ have length $l_{\beta}$ with $\delta_3$ edges not in $\alpha_1$ or $\alpha_2$ forming $\mu_3$ subpaths of $\beta$ intersecting $\alpha_1, \alpha_2$ in (and only in) the endvertices. Then, if all common edges of $\beta$ and $\alpha_1\cup\alpha_2$ have the same direction, $$|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|=
\left|\left(\frac cn\right)^{l_1+\delta_2+\delta_3}-\left(\frac cn\right)^{l_1+\delta_2+l_\beta}\right|\le\left(\frac cn\right)^{l_1+\delta_2+\delta_3},$$ and if $\beta$ has at least one common edge in opposite direction, $$|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|=
\left(\frac cn\right)^{l_1+\delta_2+l_\beta}\le\left(\frac cn\right)^{l_1+\delta_2+\delta_3.}$$ The number of ways of choosing $\alpha_1$, $\alpha_2$ and $\beta$ is at most, as in (iii) above, $$n^{l_1-1}\cdot n^{\delta_2-\mu_2}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot n^{\delta_3-\mu_3}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot4^{2\mu_2}.$$ The last factor is $4^{2\mu_2}$ in this case as $\beta$ can have opposite direction in the common subpaths. If there is a crossing between $\alpha_1$ and $\alpha_2$ there may be 4 choices for $\beta$ and there are at most $2\mu_2$ such vertices. Thus, $$\begin{aligned}
\hskip2em&\hskip-2em\sum_{\alpha_1\ne\alpha_2}\sum_{\beta}|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|\le\sum_{l_1,\mu_2,\delta_2,\mu_2,\delta_3}n^{l_1-1}\cdot n^{\delta_2-\mu_2}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot\\
&\cdot n^{\delta_3-\mu_3}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot4^{2\mu_2}\cdot\left(\frac cn\right)^{l_1+\delta_2+\delta_3}\\
&\le\sum
n^{-\mu_2-\mu_3-1}\cdot(l_1+1)^{2\mu_2}\cdot\tbinom{\delta_2-1}{\mu_2-1}\cdot(l_1+\delta_2-\mu_2+1)^{2\mu_3}\cdot\tbinom{\delta_3-1}{\mu_3-1}\cdot4^{2\mu_2}\cdot
c^{l_1+{\delta}_2+{\delta}_3}.\end{aligned}$$ Here, $l_1\ge1$, $\mu_2\ge1$, $\delta_2\ge\mu_2$, $\mu_3\ge0$ and $\delta_3\ge\mu_3$. Note that the terms in the final sum are the same as in , except that $2^{\mu_2}$ is replaced by $4^{2\mu_2}$.\
*Case 1:* $\mu_2+\mu_3\ge4$.\
Here, using the same estimates as in (iii), see , the sum over $l_1, {\delta}_2,{\delta}_3$ is, for $L\ge16$, at most $$\begin{aligned}
(1-c)^{-3}\cdot n^{-\mu_2-\mu_3-1}\cdot(L+1)^{4(\mu_2+\mu_3)}.\end{aligned}$$ Summing over $\mu_2+\mu_3\ge4$ gives $O(n^{-5}\cdot L^{16})=O(n^{-5}\log^{32}n)=O(n^{-4})$.\
*Case 2:* $\mu_2+\mu_3=3$.\
Here, $(\mu_2,\mu_3)\in\{(3,0),(2,1),(1,2)\}$ and $(l_1+1)^{2\mu_2}\le(l_1+1)^6$, $\tbinom{\delta_2-1}{\mu_2-1}\le\delta_2^2$, $(l_1+\delta_2-\mu_2+1)^{2\mu_3}\le(l_1+\delta_2)^4$, $\tbinom{\delta_3-1}{\mu_3-1}\le\delta_3$ and $4^{2\mu_2}\le4^6=4096$. Summing over $l_1,\delta_2,\mu_2,\delta_3,\mu_3$ gives at most $$3n^{-4}\sum_{l_1,\delta_2,\delta_3}4096\cdot(l_1+1)^6\cdot\delta_2^2\cdot
(l_1+\delta_2)^4\cdot \delta_3\cdot c^{l_1+\delta_2+\delta_3}=O(n^{-4}).$$ *Case 3:* $\mu_2=\mu_3=1$.\
We need only consider the situation when $\beta$ has at least one edge in common with $\alpha_1\cup\alpha_2$, as otherwise the covariance is 0.\
*Subcase 3.1: At least one common edge has opposite direction.*\
$|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|=c^{l_1+\delta_2+l_{\beta}}\cdot n^{-l_1-\delta_2-l_{\beta}}$. Here, $l_{\beta}\ge2$, as $l_{\beta}=1$ would imply that $\mu_3=0$. Further, $l_1+\delta_2\ge3$, as otherwise $\alpha_1=\alpha_2$. Let $l_{\alpha\beta}=|{\beta}\cap({\alpha}_1\cup{\alpha}_2)|=l_{\beta}-\delta_3\ge1$. Then, estimating the number of possible choices of the paths as above, $$\begin{aligned}
\sum_{l_1,\delta_2,\delta_3,l_{\beta}}&|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|\\&\le\sum n^{l_1-1}\cdot n^{\delta_2-1}\cdot(l_1+1)^2\cdot n^{\delta_3-1}\cdot(l_1+\delta_2)^2\cdot2\cdot c^{l_1+\delta_2+l_{\beta}}\cdot n^{-l_1-\delta_2-l_{\beta}}\\
&=2\cdot\sum_{l_1,\delta_2,\delta_3,l_{\alpha\beta}}(l_1+1)^2\cdot(l_1+\delta_2)^2\cdot c^{l_1+\delta_2+\delta_3+l_{\alpha\beta}}\cdot n^{-3-l_{\alpha\beta}}=O(n^{-4}).\end{aligned}$$ *Subcase 3.2: All common edges have the same direction.*\
The first edge of $\beta$, from $s$, must be disjoint with $\alpha_1\cup\alpha_2$. Let $\beta$ start with $i\ge1$ disjoint steps and then join one of the $\alpha$ paths, $\alpha_1$ say, for a further $j\ge1$ steps to $b$. Further, let $\alpha_1$ have $k\ge0$ steps before joining $\beta$ and ending with $l$ steps from $b$ to $s$. As before, $\alpha_2$ is determined by two vertices on $\alpha_1$ and $\delta_2-1$ exterior vertices giving at most $(l_1+1)^2\cdot n^{\delta_2-1}$ possibilities. Further, $\beta$ can join either of the $\alpha$ paths, and may then do an excursion along the other path, giving at most 4 possibilities. Then, as $l_1=k+j+l$, $$\begin{aligned}
\sum&|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|\\
&\le4\cdot\sum_{i\ge1}\sum_{k\ge0}\sum_{j\ge1}\sum_{l\ge1}\sum_{\delta_2\ge1}n^{i-1}\cdot
n^{k+j+l-2}\cdot(l_1+1)^2\cdot n^{\delta_2-1}\cdot\left(\frac
cn\right)^{i+k+j+l+\delta_2}\\
&=4n^{-4}\cdot\sum_{i,k,j,l,{\delta}_2}(k+j+l+1)^2\cdot c^{i+k+j+l+\delta_2}=O(n^{-4}).\end{aligned}$$ *Case 4:* $\mu_3=0$, $\mu_2\in\{1,2\}$.\
$\mu_3=0$ implies that $\beta\subset(\alpha_1\cup\alpha_2)$, so that the first edge in $\beta$ has opposite direction in $\alpha_1\cup\alpha_2$. Furthermore, at least one of the $\alpha$ paths, $\alpha_1$ say, must pass through $b$, so that $l_1\ge2$. $\alpha_2$ can be chosen in at most $(l_1+1)^{2\mu_2}\cdot n^{\delta_2-\mu_2}$ ways and there are at most $2^{\mu_2}$ ways for $\beta$ to choose between the $\alpha$ paths, giving at most $n^{l_1-2}\cdot(l_1+1)^{2\mu_2}\cdot n^{\delta_2-\mu_2}\cdot2^{\mu_2}\le4\cdot(l_1+1)^4\cdot n^{l_1+\delta_2-\mu_2-2}$ ways of choosing $\alpha_1$, $\alpha_2$ and $\beta$. The covariance is $-\left(\frac cn\right)^{l_1+\delta_2+l_{\beta}}$. Summing over $l_1\ge2$, $\mu_2=1,2$, $\delta_2\ge\mu_2$ and $l_{\beta}\ge1$ gives $$\begin{aligned}
\sum|{{\operatorname{Cov}}}(I_{\alpha_1}\cdot I_{\alpha_2},I_{\beta})|&\le4\sum(l_1+1)^4\cdot n^{l_1+\delta_2-\mu_2-2}\cdot\left(\frac cn\right)^{l_1+\delta_2+l_{\beta}}\\
&=4\sum(l_1+1)^4\cdot c^{l_1+\delta_2+l_{\beta}}\cdot n^{-\mu_2-l_{\beta}-2}=O(n^{-4}),\end{aligned}$$ which finishes the proof.
[99]{}
Sven Erick Alm and Svante Linusson, A counter-intuitive correlation in a random tournament, [*Combin. Probab. Comput.*]{} [**20**]{} (2011), no. 1, 1–9.
Sven Erick Alm, Svante Janson and Svante Linusson, Correlations for paths in random orientations of $G(n,p)$ and $G(n,m)$, [*Random Structures Algorithms* ]{} [**39**]{} (2011), no. 4, 486–506.
Geoffrey R. Grimmett, Infinite paths in randomly oriented lattices, [*Random Structures Algorithms*]{} [**18**]{} (2001), no. 3, 257–266.
Svante Janson and Malwina Luczak, Susceptibility in subcritical random graphs. *J. Math. Phys.* [**49**]{}:12 (2008), 125207.
Colin McDiarmid, General percolation and random graphs, *Adv. in Appl. Probab.* [**13**]{} (1981), 40–60.
Heinrich Weber, *Lehrbuch der Algebra*, Zweite Auflage, Erster Band. Friedrich Vieweg und Sohn, Braunschweig (1898).
[^1]: Svante Janson is supported by the Knut and Alice Wallenberg Foundation
[^2]: Svante Linusson is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.
[^3]: This research was initiated when all three authors visited the Institut Mittag-Leffler (Djursholm, Sweden).
|
---
author:
- |
S. Jesgarz [^1], S. Lerma H.[^2], P. O. Hess$^1$ [^3],\
O. Civitarese$^2$ [^4], and M. Reboiro$^2$ [^5],\
\
[*$^{1}$Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,*]{}\
[*Apdo. Postal 70-543, México 04510 D.F.*]{}\
[*$^{2}$ Departamento de Física, Universidad Nacional de La Plata,* ]{}\
[*c.c. 67 1900, La Plata, Argentina.* ]{}\
$\;\;\;\;\;\;\;\;\;\;$\
title: 'A schematic model for QCD, II: finite temperature regime.'
---
[: A schematic model for QCD, developed in a previous paper, is applied to calculate meson properties in the high temperature (up to 0.5 GeV) regime. It is a Lipkin model for quark-antiquark pairs coupled to gluon pairs of spin zero. The partition function is constructed with the obtained meson spectrum and several thermodynamical observables are calculated, like: the energy density, heat capacity, as well as relative production rates of mesons and absolute production rates for pions and kaons. The model predictions show a qualitative agreement with data. Based on these results we advocate the use of the model as a toy model for QCD. ]{}\
[ ]{}
Introduction
============
In Ref. [@paperI] (hereon referred to as (I)) a simple model, representative of QCD, was introduced and applied to the calculation of the spectrum of mesons. It is a Lipkin type model [@lipkin] for the quark sector, coupled to a boson level which is occupied by gluon pairs with spin zero. The four parameters of the model were adjusted in order to reproduce 13 known meson states with spin zero or one. The calculated spectra, for mesons with spin different from the ones used in the fit, were found to be in qualitative agreement with data. As reported in (I), the calculated meson states contain many quarks, antiquarks and gluons. The gluon contributions were found to be of the order of 30$\%$. The model predictions (I) are free of the so-called multiplicity problem, i.e. that a given state can be described in many ways, which is removed due to the action of particle mixing interaction. The model itself resembles the one of Ref. [@schutte] which treats nucleons coupled to pions. Also, it is related to the work of Ref. [@pittel], which describes quarks and uses particle conserving interactions. Generally speaking, the model of (I) belongs to the class of models described in Refs. [@stuart; @betabeta]. The gluon part in (I) is fixed [@gluons99] and does not contain any new parameters. The validity of the basic theoretical assumptions, and the applications to low and high temperature regimes, has been studied for mesons with flavor (0,0) and spin 0 [@simple]. The aim of these studies was to formulate a manageable, schematic, albeit realistic, model to describe qualitatively QCD at low and high energies. Since the model is algebraic, i.e. all matrix elements are analytic, and exactly solvable, it can provide a non-perturbative description based on QCD relevant degrees of freedom, like quarks, antiquarks, and gluons. This, in turn, allows to test other microscopic many body techniques previously applied to the non-perturbative treatment of real QCD [@swan; @bonn]. Although the proposed model (I) is probably too simple to describe real QCD, it contains all basic ingredients of real QCD. These are the correct number of degrees of freedom associated to color, flavor and spin, and the orbital degree of freedom , which is contained in the degeneracy $2 \Omega$ of each of the quark levels.
In this work we investigate the behavior of the model, in the finite temperature regime. By starting from the model predictions of the meson spectrum, we calculate the partition function and different thermodynamical quantities, like the energy density and the heat capacity as a function of temperature. Next, we focus on the calculation of meson production rates. As we shall show, these production rates are in qualitative agreement with the experiments. Also, we calculate absolute production rates for pions and kaons. Finally, we concentrate on the transition from the Quark-Gluon-Plasma (QGP) [@qgp; @qmd] to the hadron gas. The results support the notion that the present model may be taken as a toy model for QCD.
The paper is organized as follows: In section 2 the model is shortly outlined, since the details have been presented in (I). In section 3 we calculate the partition function and give the expressions for the relevant observables. In section 4 the model is applied to the description of the QGP. There, we present and discuss the results corresponding to some branching ratios and absolute production rates. Finally, conclusions are drawn in section 5.
The model
=========
As described in (I), the fermion (quarks and antiquarks) sector of the model consists of two levels at energies $+\omega_f$ and $-\omega_f$, each level with degeneracy $2\Omega$ $=$ $n_cn_fn_s$, where $n_c=3$, $n_f=3$ and $n_s=2$ are the color, flavor and spin degrees of freedom, respectively (see Fig. 1 of (I)). Each level can be occupied by quarks. Antiquarks are described by holes in the lower level. Equivalently, one can use only the positive energy level and fill it with quarks and antiquarks with positive energy. The Dirac picture is useful because it gives the connection to the Lipkin model as used in nuclear physics. The quarks and antiquarks are coupled to gluon pairs with spin zero. The energy of the gluon level is 1.6 GeV [@gluons99], and the energy $\omega_f$ is fixed at the value $\omega_f=0.33$ GeV , which is the effective mass of the constituent quarks.
The basic dynamical constituent blocks of the model are quark-antiquark pairs ${ \mbox{\boldmath $B$} }^\dagger_{\lambda f, SM}$ which are obtained by the coupling of a quark and an antiquark to flavor $\lambda $ ($\lambda = 0,1$) and spin $S$ ($S=0,1$). The index $f$ is a short hand notation for hypercharge $Y$, isospin $I$ and its third component $I_z$. Under complex conjugation the operator obeys the phase rule defined in [@draayer].
The states of the Hilbert space can be classified according to the group chain
$$\begin{aligned}
[1^N] & [h]=[h_1h_2h_3] & [h^{\rm{t}}] \nonumber \\
U(4\Omega ) & \supset U(\frac{\Omega}{3})~~~~~~~~~~ \otimes & U(12) \nonumber \\
& ~~~ \cup ~~~ & \cup \nonumber \\
& (\lambda_C,\mu_C)~SU_C(3) ~~~~ (\lambda_f,\mu_f) & SU_f(3) \otimes SU_S(2) ~S,M ~~~,
\label{group1}\end{aligned}$$
where the irreducible representation (irrep) of the different unitary groups are attached to the symbols of the groups. The irrep of $U(4\Omega)$ is completely antisymmetric (fermions) and the one of $U(\frac{\Omega}{3})$, the color group $U(3)$ for $\Omega =9$, and $U(12)$ are complementary [@hamermesh]. The color irrep $(\lambda_C,\mu_C)$ of the color group $SU_C(3)$ is related to the $h_k$ via $\lambda_C=h_1-h_2$ and $\mu_C=h_2-h_3$. The reduction of the $U(12)$ group to the flavor ($SU_f(3)$) and spin group ($SU_S(2)$) is obtained by using the procedure described in [@ramon; @sergio]. In (\[group1\]) no multiplicity labels are indicated (see (I)).
The classification appearing in (\[group1\]) is useful to determine the dimension and content of the Hilbert space. Instead of working in the fermion space we have introduced a boson mapping [@klein; @hecht]. The quark-antiquark boson operators are mapped to
$$\begin{aligned}
{ \mbox{\boldmath $B$} }_{\lambda f, S M}^{\dagger} & \rightarrow &
{ \mbox{\boldmath $b$} }_{\lambda f, S M}^{\dagger} \nonumber \\
{ \mbox{\boldmath $B$} }_{\lambda f, S M} & \rightarrow &
{ \mbox{\boldmath $b$} }_{\lambda f, S M}
~~~.
\label{bosons}\end{aligned}$$
where the operators on the right hand side satisfy exact boson commutation relations.
The model Hamiltonian is defined completely in the boson space and it is given by
$$\begin{aligned}
{ \mbox{\boldmath $H$} } & = & 2\omega_f { \mbox{\boldmath $n_f$} } + \omega_b \bf{n_b} + \nonumber \\
& & \sum_{\lambda S} V_{\lambda S}
\left\{ \left[ ({ \mbox{\boldmath $b$} }_{\lambda S}^\dagger )^2 +
2{ \mbox{\boldmath $b$} }_{\lambda S}^\dagger { \mbox{\boldmath $b$} }_{\lambda S} + ({ \mbox{\boldmath $b$} }_{\lambda S})^2 \right]
(1-\frac{{ \mbox{\boldmath $n_f$} }}{2\Omega}){ \mbox{\boldmath $b$} } + \right.
\nonumber \\
& & \left. { \mbox{\boldmath $b$} }^\dagger (1-\frac{{ \mbox{\boldmath $n_f$} }}{2\Omega})
\left[ ({ \mbox{\boldmath $b$} }_{\lambda S}^\dagger )^2 +
2{ \mbox{\boldmath $b$} }_{\lambda S}^\dagger { \mbox{\boldmath $b$} }_{\lambda S}+ ({ \mbox{\boldmath $b$} }_{\lambda
S})^2 \right] \right\} ~~~, \label{hamiltonian}\end{aligned}$$
where $({ \mbox{\boldmath $b$} }_{\lambda S}^\dagger )^2$ $=$ $({ \mbox{\boldmath $b$} }_{\lambda
S}^\dagger \cdot { \mbox{\boldmath $b$} }_{\lambda S}^\dagger )$ is a short hand notation for the scalar product. Similarly for $({ \mbox{\boldmath $b$} }_{\lambda
S})^2$ and $({ \mbox{\boldmath $b$} }_{\lambda S}^\dagger { \mbox{\boldmath $b$} }_{\lambda S})$. The factors $(1-\frac{{ \mbox{\boldmath $n_f$} }}{2\Omega})$ simulate the terms which would appear in the exact boson mapping of the quark-antiquark pairs. The ${ \mbox{\boldmath $b$} }^\dagger$ and ${ \mbox{\boldmath $b$} }$ are boson creation and annihilation operators of the gluon pairs with spin $S=0$ and color $\lambda=0$. The interaction describes scattering and vacuum fluctuation terms of fermion and gluon pairs. The strength $V_{\lambda S}$ is the same for each allowed value of $\lambda$ and $S$, due to symmetry reasons, as shown in (I). The matrix elements of the Hamilton operator are calculated in a seniority basis. The interaction does not contain terms which distinguish between states of different hypercharge and isospin. It does not contain flavor mixing terms, either. The procedure used to adjust the four parameters (values of $V_{\lambda S}$), was discussed in detail in (I).
The disadvantage posed by working in the boson space is the appearance of un-physical states. In (I) we have presented a method which is very efficient to eliminate spurious states, as we shall show in this paper. The suitability of the Hamiltonian (\[hamiltonian\]) to describe the gluon pair and quark-antiquark pair contents of mesons has been discussed in details in (I).
As a next step, in this paper, we have introduced temperature and discussed the transition to and from the QGP. As it can be expected, because of the schematic nature of the model, we may attempt to describe only the general trends of the observables. To achieve this goal, further assumptions have to be made respect to the volume of the system because the model, as it has been proposed in (I), has no a priory information about the volume of the particle.
The partition function, some state variables and observables
============================================================
The group classification of the basis (\[group1\]), allows for a complete book-keeping of all possible states belonging to the Hilbert space of (\[hamiltonian\]). The corresponding partition function, which contains the contribution of the quark-antiquark and gluon pairs configurations introduced in the previous section, is given by
$$\begin{aligned}
Z_{qg^2_0} & = & \sum_{[h]}dim(\lambda_C,\mu_C)
\sum_{(\lambda_f,\mu_f)}
\sum_{J} (2J+1) \sum_{P=\pm} \sum_i \nonumber \\
& & mult(E_i) e^{-\beta (E_i - \mu_B B - \mu_s s - \mu_I I_z)}
~~~, \label{partition}\end{aligned}$$
where $\mu_B$, $\mu_s$ and $\mu_I$ are the baryon, strange and isospin chemical potentials, respectively. The sum over $[h]=[h_1h_2h_3]$ denotes all color irreps of $U(3)$ with $\sum_k
h_k =N$, where $N$ is the total number of quarks in the two levels (Dirac’s picture). The transposed Young diagram $[h]^{\rm{t}}$, obtained by interchanging rows and columns, denotes the $U(12)$ irrep. The index “i” refers to all states with the same color, flavor, spin and parity ($P$). These states are obtained after the diagonalization of the Hamiltonian (\[hamiltonian\]). For mesons belonging to the $\pi$-$\eta$ and $\omega$-$\rho$ octet, the mass values entering in (\[partition\]) do not take into account flavor mixing. The eigenvalues $E_i$ are denoted by the eigenvalue index $i$, and they are functions of all the numbers needed to specify the allowed configurations, namely: $s$, $(\lambda_f,\mu_f)$, $P$, $S$, $[h]$ and of the cutoff for the different boson species $[\lambda , S]$ (see (I)). The quantities $B$ and $I_z$ are the baryon number and the third component of the isospin. According to the experimental evidences the value $\mu_I
=0$ is a reasonable approximation, and we have consistently adopted it in our calculations. The dimension corresponding to color configurations is given by $mult(\lambda_f,\mu_f)$ $=$ $\frac{1}{2}(\lambda_f+1)(\mu_f+1)(\lambda_f+\mu_f+2)$ and of the spin by $(2J+1)$.
Since the eigenstates of the Hamiltonian (\[hamiltonian\]) have been calculated after performing a boson mapping, as described in (I), we have consistently fixed the corresponding cutoff-values at the values $2\Omega$, $\Omega$, $\frac{2\Omega}{3}$, and $\frac{\Omega}{3}$ for the boson pair species \[0,0\], \[0,1\], \[1,0\] and \[1,1\] respectively (see (I)). These values are adequate when the fermion (quark-antiquark) configurations entering in the boson states correspond to a full occupation of the fermion lower state ($-\omega_f$). These numbers may be modified when the fermion configurations correspond to states where the upper level is partially occupied and the lower level is partially unoccupied. The distribution of the occupation in the upper and lower fermion levels is fixed for a lowest weight state $|lw>$ of a given $U(12)$ irrep, defined by $B_{\lambda f, S M} |lw>=0$. The irrep of $U(12)$ is given by a Young diagram [@hamermesh] with $m_k$ boxes in the $k$’th row. The lowest weight state is given by $\sum_{k=1}^6 m_k$ quarks in the lower level and $\sum_{k=7}^{12}
m_k$ quarks in the upper level. The highest weight state is obtained by interchanging the occupation. The difference of the number of quarks in the upper level, appearing in the highest and lowest weight states, gives the maximal number of quarks we can excite for a given U(12) irrep. This number is given by
$$2J = \sum_{k=1}^6 m_k - \sum_{k=7}^{12} m_k ~~~.
\label{j}$$
For the case $[3^60^6]$, used in (I), we found $J = \Omega$. Therefore, $2J$ is the maximal number of quarks we can shift to the higher level, i.e. it is equal to the maximal number of quark-antiquark pairs which can be put on top of the lowest weight state of a given $U(12)$ irrep, which is also the state with the lowest energy in absence of interactions.
The total partition function is given by
$$Z = Z_{qg^2_0} Z_{g} ~~~,
\label{ztot}$$
where $Z_{g}$ is the contribution of all gluon states [@gluons99] which do not include contributions of gluon pairs with spin zero. It is written $Z_g= \sum_\alpha exp(-\beta
E_\alpha )$. The values $E_\alpha$ can be deduced by using Eq. (40) of Ref. [@gluons99]. Except for the gluon pairs with spin zero, all other gluon states are treated as spectators because the Hamiltonian in (\[hamiltonian\]) does not contain interactions with these other states. Note, that the interaction between the gluons is taken into account explicitly in the model of Ref. [@gluons99]. As a short hand notation we will abbreviate the partition function by $Z=\sum_i e^{-\beta {\cal E}_i}$, taking into account that ${\cal E}_i$ contains the information about the chemical potential, the contributions of the quarks and gluons.
The observables $<{ \mbox{\boldmath $O$} }>$ are calculated via [@greiner] $$\begin{aligned}
<{ \mbox{\boldmath $O$} }> = \frac{ \sum_i { \mbox{\boldmath $O$} } e^{-\beta {\cal E}_i} }{Z_a} ~~~,
\label{average}\end{aligned}$$ where the index $a$ denotes the color configurations, i.e; $a$=(0,0) when only color zero states are considered and $a=c$ when also states with definite color are allowed. This distinction is needed to investigate the phase where color confinement is effective and the phase where color is allowed over a wide area of space. The quantities to calculate are the internal energy ($<E>$), heat capacity ($<C>$), average baryon number ($<B>$), strangeness ($<s>$) and the expectation value of different particle species ($<n_k>$), where $k$ refers to the quantum numbers of a particular particle and $\sum_{k} <n_{k}>=1$, with the sum over all possible quantum numbers.
The particle expectation values have a simple expression because they select one of the eigenvalues at the time, thus if the state of a given particle is denoted by “$i$” and $E_i$ is its energy, the particle expectation value is given by
$$\begin{aligned}
< n_i > & = & \frac{e^{-\beta (E_i - \mu_B B -\mu_s s)} }{Z_a} ~~~,
\label{n-exp}\end{aligned}$$
(where we have used the value $\mu_I=0$, for the isospin chemical potential ).
At this point we have to make an assumption upon the volume considered. The whole reaction volume can be divided in elementary volumes, and we assume that the elementary volume ($V_{el}$) is of the size of a hadron, corresponding to a sphere with a radius of the order of 1 fm. Later on we shall show that this choice is reasonable, as seen from the calculated thermodynamic properties of the whole system. Another assumption is related to the interaction, which does not take into account confinement. We shall discuss two scenarios, namely: a) no additional interaction related to color is taken into account for temperatures above a critical (de-confinement) value, and b) confinement is operative for temperatures below the critical de-confinement temperature. In the regime (a) the lowest state with color (1,0) lies at the energy $2\omega_f$, and it corresponds to put one quark in the upper fermion level. Although it is a possible configuration, the Hamiltonian (\[hamiltonian\]) cannot act upon it. This is consistent with the fact that above a certain temperature, $T_c$ (de-confinement temperature), only color non-singlet states are allowed. The actual value of $T_c$ will then give us an idea about the regime where hadronization is operative. In the real world hadronization, i.e. confinement, should set in below a critical temperature, as a true phase transition. In our model this will be signaled by a sharp transition from a state where color non-singlet states are still allowed ($T>T_c$) and a state where confinement is effective ($T\leq T_c$).
Description of the high temperature regime: the QGP
===================================================
.
We shall first discuss the case where no additional color interaction is taken into account. The states with energy $E_i$ for a given flavor, spin and parity are obtained from the diagonalization of the model Hamiltonian, now including flavor mixing and the corrections due to the Gel’man-Okubo mass formula for the two lowest meson nonets (one with spin zero and the other with spin 1). In Figure 1 we show the internal energy as a function of the temperature $T$, with and without interactions. The results shown by a dashed line have been obtained by calculating the internal energy in the fermion space [*without*]{} interactions. In this case the Hamiltonian has a simple image in the fermion space and the calculation of the partition function can be performed exactly. The dotted line shows the results obtained by working with the boson mapping, and by enforcing the corresponding cut-offs in the maximal number of bosons as explained above and in (I). The internal energy is a good indicator of the number of active states in the Hilbert space. Note that the results shown by both curves, the doted and dashed lines, practically coincide. This suggests that the number of active states is nearly the same in the boson and fermion spaces for a wide range of temperatures. This does not imply that all un-physical states have disappeared but rather that the approximate method of cutting un-physical states works reasonable well. The curve shown by a solid line, in the same Figure 1, gives the internal energy obtained from the calculation performed in the boson space and in presence of interactions. Although the curve does not show a clear phase transition of first order (e.g: a sharp increase of the energy in a narrow interval around $T_c$ ) the behavior around $T_c=0.170$ GeV is pretty suggestive of it. We have interpreted the observed smearing-out of the curve as follows: for $T=0$ the vacuum state is dominated by pairs of the type \[1,0\] and in this channel a quantum phase transition [@simple] does indeed take place. By this we mean that the pairs \[1,0\] are effectively blocked at high temperature. The other channels contribute less significantly to the ground state (see (I)) and interact weakly than the (\[1,0\]) channels, therefore, they remain in a perturbative regime. Thus, as the temperature increases, an approximate first order phase transition takes place in the channel \[1,0\], a mechanism similar to the one shown in Ref. [@simple] for the \[0,0\] channel, while the other configurations remain un-affected. The superposition of these two mechanism leads to the smearing out of the curve around $T_c$, as shown in Figure 1.
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In Figure 2 the expectation value of the Casimir operator ($C_2=<{ \mbox{\boldmath $C_2$} }>$) of color and its variation ($\Delta C_2 =
\sqrt{<{ \mbox{\boldmath $C_2$} }^2>-<{ \mbox{\boldmath $C_2$} }>^2}$) are shown. The eigenvalue of the Casimir operator, for an irrep with color numbers $(\lambda_C,\mu_C)$, is given by $C_2(\lambda_C,\mu_C)=
\lambda_C^2$ $+$ $\lambda_C\mu_C$ $+$ $\mu_C^2$ $+$ $3(\lambda_C +
\mu_C)$. As a reference, for a color (1,0) irrep $C_2=3$ while the irrep (1,1) has $C_2=9$. We assume that $\mu_B$ and $\mu_s$ are zero. It is interesting to observe that, in the present model, the variation of the color is approximately symmetric around $T=0.170$ GeV. A possible interpretation is the following: at high energy the probability to have a color non-singlet state is large (the variation is not large enough to allow color singlet states) and a QGP is formed where color is effective over a wide range in space. From $T=0.170$ GeV on the probability to find a state in color (0,0) is significantly increased, since the variation is large enough to allow color singlet states. In lowering the temperature the variation is much larger than the average color and the whole QGP dissolves in droplets of color zero. Within the present model, these results, of the average color and its variation, are signals of the transition to the hadronic phase. Accordingly, we assume that it takes place near $T_c=0.170$ GeV, for $\mu_B=\mu_s=0$. We may now calculate the bag pressure and construct the $T-\mu_B$ diagram. At $T=0.170$ GeV the pressure is determined via the expression $p=\frac{Tln(Z)}{V_{el}}$, where $\Phi = -Tln(Z)$ is the grand canonical partition function [@greiner] and $V_{el}$ is the elementary volume $V_{el}=\frac{4\pi}{3}r_{el}$. For $r_{el}$=1 fm we obtain a bag pressure $p^{\frac{1}{4}}$ of about 0.17 GeV, which is in reasonable agreement with standard values. When the chemical potentials $\mu_B$ and $\mu_s$ are different from zero, the temperature dependence of the internal energy changes and also changes the value of the temperature for which the pressure is equal to the bag pressure $p^{\frac{1}{4}}$ $=$ $0.18$ GeV. The results are shown in Figure 3. Assuming that the local strangeness is $<s>=0$, we arrived at a functional relation between $T$, $\mu_B$ and $\mu_s$, i.e. $f(T,\mu_B ,\mu_s )=0$, which fixes $\mu_s$ as a function of $\mu_B$. The results of this functional relation are displayed in Figure 4.
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Once the chemical potential $\mu_s$ is adjusted, by using the results shown in Figures 3 and 4, the chemical potential $\mu_B$ and the transition temperature $T_c$ can be consistently determined.
Up to now we did not take into account an interaction which generates confinement. This has to be done by hand. One possibility is to [*assume*]{} that the transition from the QGP to the hadronic phase takes place within a very small range of temperatures around the critical temperature $T_c$. We require that the partition function above $T_c$ allows any color while for $T<T_c$ it contains only color zero states. Finally, chemical equilibrium connecting both phases, the QGP and the hadron gas, is understood. Figure 5 shows the results of the internal energy, without confinement (upper curve) and with confinement (lower curve). The solid line connecting both curves indicates the values for which confinement vanishes above $T_c$. As seen from the results, the transition is now of first order. Figure 6 shows the heat capacity calculated for the case without confinement (dotted line) and with confinement below $T_c$ (dashed line). The solid line interpolates between them, as in the case of the internal energy (see Figure 5).
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The model can first be tested in the energy region below the transition temperature $T_c$, where the hadron gas should prevail. The confinement is effective and therefore we have to use the partition function $Z_{a=(0,0)}$ in the equations (\[average\]) and (\[n-exp\]). We take, as an example, the measured total production rates of $\pi^+$ at 10 GeV/A, as reported in the SIS-GSI experiment [@gsi-10gev]. The system considered was Au+Au and we assume that all particle participate, i.e. $N_{part}$=394. In Figure 2.3 of Ref. [@gsi-10gev] a temperature of about $T=0.13$ GeV is reported. Assuming local strangeness conservation , $<s>=0$, we obtain a relation of $\mu_s$ versus $T$, which is depicted in Figure 7. From there we obtain, for the reported temperature, $\mu_s \approx 0.128$ GeV and via Figure 4 a value $\mu_B \approx 0.55$ GeV. In Figure 8 we show for a fixed value of $\mu_B$, physically acceptable at temperatures near $T=0.13$ GeV, the resulting total production rate of $\pi^+$ ($N_\pi$) as a function in the temperature $T$. For $T$ = 0.13 GeV the production rate is approximately 180 pions $\pi^+$, which is close to the value 160, which we have obtained by using Figure 2.3 of Ref. [@gsi-10gev]. The good qualitative agreement with the experiment demonstrates that the present model is able, indeed, to describe, approximately, observed QCD features.
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We have also determined ratios of particle production and some absolute production rates. The particle production is calculated for temperatures just below $T_c$, where only color zero states are allowed. This implies that the partition function to use is $Z_{(0,0)}$. We then apply Eq. (\[n-exp\]). Note that in the expression of the particle-production ratios the partition function cancels out and only the dependence on the mass of the particles and the chemical potential remains. Figure 9 shows results for some particle-production ratios for beam energies $\sqrt{s}=130$A GeV. The experimental values are taken from Ref. [@rafelski1], based on the experiment described in Ref. [@exp-rafel] (see also [@braun-m]). For baryons, only the ratios of particle and anti-particle production are shown because these expressions are independent of the mass of the baryon. As noted in (I) the masses of the baryons are not well reproduced because they are considered as consisting of three idealized fermions on top of the meson sea. The interaction to the meson sea is not taken into account yet, but indications about how to do it are given in (I).
The central value of the $K^-/K^+$ production ratio, shown in Figure 9, was reproduced with the values $\mu_s=0.012$ GeV and $\mu_B=0.044$ GeV. The other ratios are predicted by the model. Considering the simplicity of the model, the ratios are found to be in a reasonable agreement with data.
In order to obtain the total yields for kaons and for the $\pi^+$ pions it is necessary to introduce further assumptions about the size of the QGP. The baryon density is given by $\frac{<B>}{V_{el}}$, where $V_{el}$ is the size of the representative volume, as explained before. In order to conserve, on the average, the baryon number we multiply the baryon density by the total volume and require that it must be equal to the total baryon number, given by the number of participants $N_{part}$. This leads to the total volume
$$V_{tot} = \frac{N_{part} V_{el}}{<B>_a} ~~~,
\label{vtot}$$
where the index $a$ refers to, as in the partition function, color ($a=c$) when the average value is calculated in the QGP and $a=(0,0)$ when it is calculated in the hadron gas. In the QGP the average value $<B>_c$ before the transition is smaller than the average value $<B>_{(0,0)}$ in the hadron gas after the transition. This is due to the small value of $Z_{(0,0)}$ at $T_c$, since many other possible color states are excluded from it which do contribute to $Z_c$. As a consequence, for $T=T_c$ the volume of the QGP phase, as a function of $N_{part}$, is much smaller than the one in the hadron gas phase. Assuming a sphere, the radius of the QGP phase is about 8 fm, and it changes to about 20 fm after the transition. This implies volumes of the order of approximately 2 $ 10^3$ fm$^3$ and 3.4 $10^4$ fm$^3$, respectively.
This transition is, as pointed out earlier, assumed to take place suddenly at $T_c$ (probably it should be smeared out but we cannot describe it with the present model). This implies that within the scenario assumed there is a rapid expansion of the volume caused by the transition from the QGP to the hadron phase, which should be observed as a large outward motion. One possible interpretation is that most of the pions are produced during the transition, liberating energy and provoking a rapid expansion of the system. The energy gained is represented by the jump between the lower and upper curves of Figure 5, but its origin cannot be explained by the present model, where confinement was shifted by hand.
Figure 10 shows the total pion yield as a function of the temperature $T$, corresponding to the Au+Au collision. The upper curve describes the total yield when all nucleons participate, while for the lower one we have taken $N_{part}=250$. This value agrees better with the experiment, as seen from the results, and it means that in the collision about 250 nucleons participate in the QGP. In Figure 11 the total kaon production rate is displayed. The upper curve corresponds to $K^+$ and the lower one to $K^-$ absolute production rates, respectively. In both cases $N_{part}=250$ was used, the same value used previously in the calculation of the pion yield. The ratio of the curves was already adjusted at the point corresponding to the $K^+/K^-$ ratio. The absolute production rate and the shape of the curve, however, is a prediction of the model (as far as we can talk about “predictions” within this toy model). Considering the simplicity of the model it is surprising that the absolute production rate is well reproduced. This feature is common to other thermodynamical descriptions of the transition from the QGP to the hadron-gas [@braun-m; @rafelski1].
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Finally, in Figure 12, we show the calculated expectation values of the number of quark and gluon pairs as a function of the temperature T. At $T=0$ GeV the results correspond to the fractions of gluon pairs and fermion (quark-antiquark) pairs in the physical vacuum state. At high temperatures the gluon part increases and takes over the fermion part, which shows saturation. However, at the temperatures of interest, i.e. around the point of the phase transition $T_c \approx 0.16 $ GeV, the gluon number is still suppressed with respect to the fermion pair number. This might be in favor of the ALCOR model [@alcor] which supposes a suppression of gluons in the QGP and takes only constituent quarks and antiquarks into account. Note, that at $T=0.170$ GeV still a sensible amount of gluon pairs are present.
Conclusions
===========
We have presented a toy model of QCD. The model is described in (I) and in this paper we have focused on the thermodynamic properties, at equilibrium, emerging from the model. We have calculated the partition function with and without color, and studied the temperature dependence of some observables, like the internal energy, the heat capacity, and the production rates of particles. The parameters of the model were determined in (I), adjusting the meson spectrum at low energy. Without further parameters the internal energy, the heat capacity and some particle ratios were determined, as explained in the text.
We have applied the model to the case of the Au+Au collision at 10 GeV/A [@gsi-10gev] and shown that it can reproduce qualitatively the absolute production rate of $\pi^+$. At this energy the QGP has not yet formed and, therefore, the results show that the model can be applied to study schematically the thermodynamics of a hadron gas.
Next, we have applied the model to energies where one assumes that the QGP has been already formed. The absolute production rate of $\pi^+$ and kaons were calculated, just below the transition temperature, by taking the number of participant nucleons ($N_{part}$) as an input. The agreement between calculated and experimental values was found to be satisfactory. Also, the resulting production rate was described reasonable well, once the $\mu_s$ chemical potential was fixed to yield the correct (observed central value) $\frac{K^+}{K^-}$ ratio. Some mass-independent baryon-antibaryon ratios, were qualitatively reproduced by the model predictions.
This demonstrates that the model is able to describe the general trend of QCD, in the finite temperature domain, and the transition to and from the quark gluon plasma.
Acknowledgment
==============
We acknowledge financial support through the CONACyT-CONICET agreement under the project name [*Algebraic Methods in Nuclear and Subnuclear Physics*]{} and from CONACyT project number 32729-E. (S.J.) acknowledges financial support from the [*Deutscher Akademischer Austauschdienst*]{} (DAAD) and SRE, (S.L) acknowledges financial support from DGEP-UNAM. Financial help from DGAPA, project number IN119002, is also acknowledged.
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[^1]: e-mail: [email protected]
[^2]: e-mail: [email protected]
[^3]: e-mail: [email protected]
[^4]: e-mail: [email protected]
[^5]: e-mail: [email protected]
|
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abstract: 'We use energy minimization principles to predict the structure of a decagonal quasicrystal - $d$(AlCoNi) - in the Cobalt-rich phase. Monte Carlo methods are then used to explore configurations while relaxation and molecular dynamics are used to obtain a more realistic structure once a low energy configuration has been found. We find five-fold symmetric decagons 12.8Å in diameter as the characteristic formation of this composition, along with smaller pseudo-five-fold symmetric clusters filling the spaces between the decagons. We use our method to make comparisons with a recent experimental approximant structure model from Sugiyama et al (2002).'
address:
- ' Dept. of Physics, Cornell University, Ithaca NY 14853-2501, USA'
- ' Present address: Dept. of Physics, M.I.T., Cambridge MA 02139 \[CORRECT THIS\]'
- '§Permanent address: Institute of Physics, Slovak Academy of Sciences, 84228 Bratislava, Slovakia.'
author:
- 'Nan Gu, M. Mihalkovič§, and C. L. Henley'
title: 'Energy-based Structure Prediction for $d$(Al$_{70}$Co$_{20}$Ni$_{10}$)'
---
\#1 [[\#1]{}]{} \#1 [[**CLH: \#1**]{}]{} \#1 [[**NAN: \#1**]{}]{} \#1 [[**MM: \#1**]{}]{} \#1 [[*\#1*]{}]{} \#1
Introduction
============
This paper reports structural predictions for the decagonal quasicrystal $d$(AlCoNi) in the Cobalt-rich (‘basic Co’) , of approximate composition Al$_{70}$Co$_{20}$Ni$_{10}$ and atomic density 0.068 atoms/Å$^3$, using simulations to minimize energy, using the same methods (and codes) as previous papers by Mihalkovic, Widom, and Henley [@alnico01; @alnico02; @alnico04] on the ‘Ni-rich’ ; this constitutes a first test of the transferability of that approach to other compositions that are described by other tiling geometries.
The Aluminum-Nickel-Cobalt alloy $d$(AlCoNi) is the best studied and perhaps the highest quality of equilibrium decagonal quasicrystals. Decagonal $d$(AlCoNi) has about eight modifications (which we call ‘’ here), each existing in a tiny domain of the phase diagram.[@Ri96b; @Tsai96; @Ed96; @Gr96; @Ri98]. The compositions ‘basic Ni’ (around $\rm Al_{70}Co_{10}Ni_{20}$), and ‘basic Co’ are the best-studied , having the simplest diffraction patterns. (The other show superstructure peaks, indicating modulations.) Several crystal approximants related to ‘basic Co’ are known [@Gr98], one of which has a solved structure [@Su02].
We find that (in contrast to the ‘basic Ni’ ), the framework of ‘basic Co’ at $T=0$ is a network of edge-sharing $12.8$Å diameter decagons (placed like the large atoms in a “binary tiling” quasicrystal). A strong (but not always simple) Co/Ni ordering is found. Our model reproduces most, but not all, of the atomic positions in W(AlCoNi). (A complementary brief account of our results is in Ref. [@Gu-pucker05], and a more complete account is in preparation [@Gu-PRB05].)
Methods
=======
Our main assumption is that the $d$(AlCoNi) structure is well approximated by a stacking of equally spaced two-dimensional tilings built from the Penrose rhombi under periodic boundary conditions. The chief experimentally determined inputs were the values $c = $4.08Å for the stacking period and $a_0 = $2.45Å for quasilattice constant (edge length of small rhombi).
This framework of decorated tilings takes advantage of the fact that all the known atomic structures of decagonals are Penrose-tiling-like. Further, even in random tilings Penrose rhombi admit “inflation” constructions that can relate a tiling to another one using rhombi with edges enlarged by $\tau = {{(1+\sqrt 5)}/{2}} \approx 1.618$ (the golden ratio), which gives a convenient way to build a chain of connections from the atomic level to large tiles so as to describe $d$(AlNiCo) on large scales. Fig. \[fig:tiling\] includes two ways of subdividing a unit cell into a tiling of rhombi (with edges $a_0$ and $\tau a_0$, respectively).
We also assume atomic pair potentials between Al, Co, and Ni as derived ab-initio using Generalised Pseudopotential Theory [@Mo97] (GPT), but modified using results from [*ab initio*]{} calculations to approximate the sums of the omitted many-body interactions when transition metals (TM) are nearest neighbours. \[Fig. 1 of Ref. [[@alnico01]]{} plots these potentials.\] The potentials are cut off by a smooth truncation past 7.0Å.
We know [@alnico01; @alnico02] that the strongest interaction is the Al-TM first potential well. The number of Al-TM neighbours is maximised and, as a corollary, that of TM-TM nearest neighbours is minimised (but TM-TM second neighbours, at $\sim 4$Å separation, are common). However, this scarcely constrains the structure since the weak Al-Al potential allows enormous freedom in placing Al. In practice, one finds in Al-TM quasicrystals that a rather rigid TM-TM network forms, with separations near to the strong second minimum of the TM-TM pair potential; the Al atoms fill the TM interstices in a more variable fashion. Subtle details of the density and composition, as well as the small differences between Co and Ni in the pair potentials, decide which structure optimises the energy. We understand only a few of these factors in detail for the ‘basic Co’ phase, so in this paper we will only describe the atomic structures we found and do not attempt a microscopic rationalization of them.
Our program uses Metropolis Monte Carlo (MMC) simulation to perform atom-atom and atom-vacancy swaps within a site list placed on the Penrose tiles. The first stage simulations include an additional degree of freedom in the form of ‘tile flips.’ These ‘tile flips’ obey the MMC algorithm probabilities and conserve tile number, atomic species number, and the collective outline of tiles rearranged.
Our procedure is to anneal from a high temperature $\beta \approx 4$-10 eV$^{-1}$ to a low temperature at which most degrees of freedom are frozen out, $\beta \approx 20$ eV$^{-1}$, using a step size $\delta \beta \approx 0.5$-1 eV$^{-1}$. (Here $\beta \equiv(k_{B}T)^{-1}$.) At each temperature step, approximately 2000 atomic swaps per site are attempted.
![The 32$\times$23$\times$4 unit cell. The left half is subdivided into $a_0$ scale rhombi (in two ways in part of the cell \[dotted lines\]). The right half of the cell is subdivided into $\tau a_0$ scale rhombi. Insets at top left and top right display the candidate sites on the $a_0$ and $\tau a_0$ scale tiles, respectively. Filled large circles depict atoms in one layer and filled small circles are those in the other layer. Black denotes Nickel, dark gray Cobalt, and light gray Aluminium. Unoccupied sites of the $\tau a_0$ scale tiling are shown by very small empty circles; those of the $a_0$ scale tiling are not shown. []{data-label="fig:tiling"}](tilingBW121305.eps){width="3.1in"}
Multiscale Procedure
--------------------
The strategy of our series of simulations loosely copies that reported previously [@alnico01; @alnico02] on the Ni-rich of $d$(AlNiCo), and can be thought of as a sort of multiscale modelling. It proceeds in the following stages:
- Begin with Monte Carlo simulations allowing atom swaps and ‘tile-flips’ with a tiling of relatively small Penrose rhombi; note prominent features and formations that recur in all the low energy structures from a sufficient number of independent runs.
- Take the observations from the small-scale tiling and promote them to rules: i.e. make the formations inferred in the first stage into fundamental objects of a larger scale tiling. Unused (or underused) degrees of freedom are removed.
- Perform new Monte Carlo simulations on the larger-scale tiling to search for new low energy structures. This larger- scale tiling is generally more efficient at finding lower energies because of the reduced degrees of freedom.
- Verify that the degrees of freedom removed in stage (2) were indeed unnecessary by restoring some of them, judiciously.
- Repeat for larger and larger scale tilings. An eventual goal is to extract a ‘tile Hamiltonian’ [@Mi96a; @Co98a] which is the bridge to modelling long-range order and diffuse scattering. That means ascribing the atom-atom energies to an effective interaction among neighbouring tiles, so that tiles are the only remaining degree of freedom.
It is also necessary to perform relaxations/molecular dynamics to escape the biases which may be introduced by the discrete site list used up through stage (4).
Results {#sec:results}
=======
Small-Scale tiling (edge $a_0=2.45$Å) {#sec:decagon}
-------------------------------------
The first stage of our tiling simulations uses small Penrose rhombi that have $a_0=2.45$Å edges. The unit cell simulations had dimensions $32.01$Å $\times 23.25$Å$ \times 4.08$Å, which we shall call the 32$\times$23$\times$4 cell, shown in Fig. \[fig:tiling\]. The structure is taken to repeat after two layers of the Penrose tiling. At 0.068 atoms/Å$^3$, there are about 200 atoms in the unit cell.
Our low energy configuration generated at this level showed distinct rings of 5 TM and 5 Al atoms surrounding an Al atom and surrounded by an 10 Al atom decagon with an edge length of $a_0$. These 21-atom motifs were evidently energy favourable. Linear regressions of the count of these 21-atom clusters versus energy revealed a noisy, but consistent correlation of $\approx -1.0$ eV per 21-atom cluster.
However, the ideal structure did not simply maximsze the number of 21-atom clusters: though they may be placed adjacent to pack six clusters per cell, that was typically not observed in the low-energy configurations. Instead, further MMC annealing (starting from a low-energy configuration) found that a larger cluster emerged as the characteristic motif at this composition: a decagon with an edge length of $\tau a_0 \approx 4.0$Å (hence a diameter of $\sim 12.8$Å), which we will call ‘[13Å Decagon]{}’ ([13ÅD]{}). This object contains the 21-atom motif at its centre, ten TM atoms at its vertices, and Al atoms irregularly interspersed along and within its edges.
We use the following nomenclature for the components of [13ÅD]{}, from its centre outwards. The centre of the [13ÅD]{} is an Al atom. The 5 TM 5 Al ring is known as the ‘first ring.’ The TM atoms in the first ring are in the same layer as the central Al atom. The 10 Al ring is known as the ‘second ring.’ The 10 TM decagon and the Al atoms along and within the edges are collectively known as the ‘third ring.’ Miscellaneous Al sites occur between the second and third rings, which we designate collectively as the ‘2.5 ring.’
In the [13ÅD]{}, the first ring TM sites are mainly occupied by Co and the third ring (decagon vertex) TM sites also tend to Co occupation, but there is a very strong dependence on the local environment that distinctly favour Ni in certain sites. \[This became clear at the second stage of simulation, Subsec. \[sec:inflated\].\] The Al positions of third-ring and 2.5 ring Al atoms are closely correlated and the rules were not resolved from the 2.45Å tiling simulation.
A second type of motif fills the interstices between the [13ÅDs]{}: it consists of a 10-atom ring and an Al at the centre, much like ring 1 in the [13ÅD]{} cluster, except this kind of motif has mixed Al-TM occupations for the 5 candidate TM sites. In general, about 3 sites out of the 5 candidates are filled with TM atoms, preferably Ni. We name these 11-atom clusters “[Star clusters]{}”; a few of these are contained in Fig. \[fig:tiling\].
Inflated Tiling (${\tau}^{3} a_0 \rightarrow \tau a_0$) {#sec:inflated}
-------------------------------------------------------
In keeping with our ‘multiscale’ methodology, we reassessed the degrees of freedom necessary for the next level of simulation. The dominant geometric object is a decagon with edge $\tau a_0\approx 4$Å, but we found it economical to represent its decoration by candidate sites using a thin 4Å rhombus and the second fat rhombus shown in the right inset of Fig. \[fig:tiling\]; similarly, the [Star cluster]{} decoration is represented using the first fat rhombus.
We assumed that the minimum-energy structure has a maximum density of [13ÅD]{} clusters. A Monte Carlo simulation was carried out (admitting only tile flips) on the ensemble of $\tau a_0$ rhombus tilings using an artificial tile Hamiltonian favouring the creation of nonoverlapping ‘star decagons’ (of edge $\tau a_0$) containing a fivefold star of fat rhombi. [^1] It was found that the ground state configuration was always a ‘binary tiling’ [@La86; @Wi87] of rhombi with edge length ${\tau}^{3} a_0 = $10.4Å with the ‘large atom’ and ‘small atom’ vertices replaced by [13ÅDs]{} and [Star clusters]{}, respectively.
We [**designate**]{} a fixed site list on the $\tau a_0$ rhombi, as shown in the right half of Fig. \[fig:tiling\], designed to match the most frequently occupied sites observed in the low-energy states from out $a_0$ scale simulations. This tiling, unlike the $a_0$ tiling, consists of only [*one*]{} layer of Penrose tiles, each of which has site decorations in [*two*]{} layers spaced 2.04Å in the $c$ direction. Note how the two versions of the fat Penrose rhombus have different site lists. We use the 32$\times$23$\times$4 cell, as before so we may compare the energies for systems with exactly the same density and composition.
At this second stage, our MMC runs had purely lattice gas moves on a fixed tiling (no tile flips). These were able to find lower energy configurations at a much faster rate than the first-stage ($a_0$ scale) simulations at the same composition and atomic density, due to the much decreased site list. Hence, the occupations of many types of sites were discerned with greater accuracy, such as the TM sites in the [13Å Decagon]{} (already described in Subsec. \[sec:decagon\]). Furthermore the tendencies of more variable sites (Al in rings 2.5 and 3, and TM in the [Star cluster]{}), which will be described in a longer paper [@Gu-PRB05], started to come into focus. Finally, despite having [*fewer*]{} allowed configurations, the $\tau a_0$ tiling simulation consistently found [*lower*]{} energies, which serves as our post hoc justification for assuming [13ÅD]{} clusters in the binary tiling geometry and the reduced site list. [^2]
In the prior case of the ‘basic Ni’ , the decoration model [@alnico01] was implemented as a deterministic tiling. A marked contrast in the present ‘basic Co’ case is that we cannot impose a simple rule that fixes the chemical occupation of each site. On certain TM sites, it is difficult to resolve the occupation (Co/Ni); certain Al sites are also variably occupied (Al/vacant) depending on the environment. Presumably, with a sufficiently thorough understanding of our model, we could formulate a deterministic decoration rule on the binary tiling. It is possible, however, that the energy differences among some competing structures are too small to be visible in a reasonable simulation, or to influence the real properties at any accessible temperature.
Cluster orientation {#sec:orientation}
-------------------
The physical accessible candidate sites in the $\tau a_0$ decagons (with our decoration) in fact have a $\overline{10}$ point group symmetry, which is not broken by the binary tiling geometry. The fivefold symmetric [13ÅD]{} cluster breaks this symmetry, by the layer on which the central Al sits and the Al/TM alternation on the first ring. Thus, a major obstacle to writing a deterministic decoration is that the relative orientations of [13ÅDs]{} must be specified, which depends on subtle interaction energies between them. Experimentally, $d$(AlCoNi) with our Co-rich composition was observed to order in a fivefold symmetry, wherein all the Decagons are oriented the same way [@Ri96c; @Li96].
As a test, we compared the lowest-energy configurations resulting from MMC simulations on identical $\tau a_0$ tilings in which neighbouring [13ÅDs]{} were forced either to have always identical or always opposite orientations. \[This is controlled by the orientation of the decagon of $\tau a_0$ rhombi in the tiling that gets decorated.\] we evaluated the minimum-energy configurations from $\sim 60$ independent runs of each type. We used the 32$\times$23$\times$4 unit cell, using a fixed composition Al$_{70}$Co$_{20}$Ni$_{10}$, but repeating the tests for a a series of atom number densities $n$.
The energy difference between these two orientations was density dependent. At $n\approx 0.068$ atoms/ Å$^3$ the two orientational patterns are practically degenerate (though with visibly distinct atom configurations.) From $n=$0.068 Å$^{-3}$ to 0.074 Å$^{-3}$ – a range which includes the most realistic compositions – the pattern with uniform orientations has the lower energy, by up to $4\times 10^{-3}$eV/atom, but this difference disappears again at $n \approx$0.074 Å$^{-3}$.
We find this energy difference arises from different ways of filling the possible TM sites in the respective orientation schemes. In the uniform-orientation scheme, as the TM density is raised (as part of the total density), TM atoms fill the ring 2.5 in the [13Å Decagon]{} before they exceed $\sim {{3}\over{5}}$ filling in the [Star cluster]{} TM sites; whereas in the alternating-orientation case, the ring 2.5 is filled to a lesser degree with TM while the [Star cluster]{} TM sites become overfilled. \[In Ref. [[@Gu-pucker05]]{}, we gave an explanation how these effects could favour [*alternating*]{} orientations.\] But the results just presented are valid only for the (unphysical!) case of atoms confined to hop on fixed ideal sites. When displacements are allowed, so that many atoms “pucker” away from the flat layers (see Sec. \[sec:RMR\]), then the uniform arrangement is preferred more robustly.
Beyond the Discrete Site List {#sec:RMR}
=============================
Up to this point, we reported simulations using fixed atom site positions. We can make our configurations more realistic by subjecting them to relaxation and molecular dynamics (MD). The removal of the tiling and discrete site list leads to effects such as out-of-layer “puckering” and so-called ‘period doubling,’ whereby certain atoms relax into positions that violate the $c$ period of a single unit cell, but instead are periodic with respect to a unit cell with $c'=2c =$ 8.16Å. Exactly such distortions are familiar in the structures of decagonal phases and approximants.
Our standard cycle for such ‘off-site’ studies is a relaxation to a local energy minimum (i.e. to 0 K), followed by an MD cycle beginning at $\sim$600 K and ending at $\sim$50K in increments of 50K. The MD results are then relaxed again to 0K. This protocol will be denoted Relaxation-MD-Relaxation (RMR). Upon relaxation to $T = 0$, we find that the TM atoms are quite immobile and move only slightly. The Al atoms, however are subject to displacements as large as $1.5$. After molecular dynamics and re-relaxation to $T = 0$, we find a few Al atoms be further displaced, but to similar sites so that no systematic difference is apparent in the overall pattern.
Comparison with Experimental W-AlCoNi
-------------------------------------
Major diffraction-based structure models were available for ‘basic Ni’ $d$(AlNiCo), to which the simulation predictions could be compared, but no such model exists for the ‘basic Co’ $d$(AlCoNi). However, the structure of the ‘W phase’ crystal approximant of ‘basic Co’ has been determined by Sugiyama [*et al*]{} [@Su02], and we may apply what we have learned so far – in particular the period doubling and puckering – to predict its structure. \[This study has inspired a more detailed modelling of W(AlCoNi), based on ab-initio energies rather than pair potentials [@Mi05].\]
Using exactly the same tiles and interlayer spacing as in our other simulations, our unit cell is $23.25$$ \times 39.5606$$ \times 8.158$, which differs by less than 1% from the experimentally determined lattice constants. The approximate atom content implied by Sugiyama’s structure solution and Co:Ni ratio is Al$_{385}$Co$_{113}$Ni$_{38}$, with uncertainties arising from the partial and mixed occupations. Adopting this composition as our ideal, we have Al$_{71.8}$Co$_{21.1}$Ni$_{7.1}$ at a density 0.0714 Å$^{-3}$.
We applied our mock Hamiltonian from Sec. \[sec:inflated\] to generate a tiling consistent with the ${\tau}^3 a_0$ binary rhombus tiling. The optimum configuration is unique (modulo symmetries) and has four [13ÅDs]{}, in an arrangement which turns out to be the same as observed in in W(AlCoNi). Cluster orientation comparisons were like performed like those of Subsec. \[sec:orientation\], but using the relaxed energies (without MD). We found that, between the configurations with alternating and uniform cluster orientations, the latter had a lower relaxed energy, in accordance with the experimental W-AlCoNi structure. The atomic locations generated on this unit cell using the 4Å rhombus decoration site list is approximately consistent with the experimentally determined sites.
Our MMC simulations are able to capture the gross features of W-AlCoNi with excellent accuracy. Since this approximant is not far from the ‘basic Co’ decagonal composition, we are not surprised to find [13ÅDs]{} and [Star clusters]{} as the major motifs, arranged in a manner consistent with our binary and $\tau a_0$ tilings, consistent with the observed W-phase. The differences in the exact atomic locations lie mainly within the highly context dependent 2.5th and 3rd rings. We find that the increase in density from our original simulations results in a more dense occupation of these rings. Atomic configurations found after RMR are in even better agreement. Fig. \[fig:WA\] compares one layer from experimental data to our RMR results; the match in the other layers are equally good.
![a) The top figure is one of the mirror layers of W-AlCoNi. b) The bottom figure is the puckered layer of W-AlCoNi. The empty circles are experimentally determined atom sites, the filled circles are those extracted from our simulations. Dark gray filled circles are TM, light gray filled circles are Al. Empty circles with ’teeth’ are TM, smooth empty circles are Al. The large empty circles are experimental atomic sites with mixed Al-TM occupancies. The size of the simulated (filled) atoms may vary due to the visual scheme depicting distance in the $c$-axis dimension.[]{data-label="fig:WA"}](WAC121305.eps){width="3.1in"}
However, a significant number of Al atoms in our RMR picture are in disagreement with the experimental refinement. because certain puckered atoms are not present in our simulations. The top right of Fig. \[fig:WA\] contains a pentagon of displaced Al atoms centred upon the missing Al atom. These extremely puckered Al atoms are perhaps located in energy minima too far away for our MD program to traverse. The existence of such atoms is one of the greatest flaws for beginning with a discrete site simulation.
Conclusion
==========
We find that in the approximate composition (Al$_{70}$Ni$_{10}$Co$_{20}$, the structure of $d$(AlNiCo) is dominated by the formation of the [13ÅD]{} clusters and the [Star clusters]{} which complement them. We find that the gross features are robust under variations of the composition and densities of $\sim 4$%; these will strongly affect a few details of the sites and we have not established the correct answers for those atoms. Our method of tile decorations, and successive reductions of the degrees of freedom, affords an enormous speedup compared to brute-force molecular dynamics, which would get stuck in glassy configurations, at the price of a few plausible assumptions.
By applying our results to the approximant W-AlCoNi, we demonstrated both the validity and the pitfalls of our approach. The pitfall lies in that fact that our explorations are conducted using a fixed site list in two atomic layers, whereas the real structure has four layers and some atoms are strongly puckered out of them. Even though most puckered atoms may be represented as relaxations from ideal sites, the optimal puckered structure might be derived from a fixed-site structure of comparatively high energy that our simulations would pass over. Reliable answers can not be obtained by pure numerical exploration, but seem to demand some physical understanding of the puckering (and other displacements), as we have begun to do in Ref. [[@Gu-pucker05]]{}. The context dependences evident in the ‘basic Co’ show that beyond a point, the errors in defining the pair potentials will surely exceed the energy differences due to some swaps of species, which is a second pitfall that may be alleviated only by checking against experimental phase diagrams.
This work is supported by DOE grant DE-FG02-89ER45405; computer facilities were provided by the Cornell Center for Materials Research under NSF grant DMR-0079992. MM was also supported by grant VEGA-2/5096/25 of the Slovak Academy of Sciences. We thank M. Widom for discussions.
References {#references .unnumbered}
==========
[99]{}
M. Mihalkovič, I. Al-Lehyani, E. Cockayne, C. L. Henley, N. Moghadam, J. A. Moriarty, Y. Wang, and M. Widom, Phys. Rev. B 65, 104205 (2002).
C. L. Henley, M. Mihalkovič, and M. Widom, J. All. Compd. 342 (1-2): 221 (2002). M. Mihalkovič, C. L. Henley, and M. Widom, J. Non-Cyst. Sol. 334: 177 (2004). S. Ritsch, C. Beeli, H. U. Nissen, T. Gödecke, M. Scheffer, and R. Lück, [Phil. Mag. Lett.]{}, [74]{}, (1996) 99-106 K. Edagawa, H. Tamaru, S. Yamaguchi, K. Suzuki, and S. Takeuchi, Phys. Rev. B 50,12413 (1996). B. Grushko, D. Hollard-Moritz, and K. Bickmann, J. All. Comp. 236, 243 (1996)
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B. Grushko, D. Hollard-Moritz, R. Wittmann, and G. Wilde, J. All. Comp. 280, 215 (1998). K. Sugiyama, S. Nishimura, and K. Hiraga, J. Alloy Comp. 342, 65 (2002).
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[^1]: Star decagaons were similarly maximized in Ref. [[@Hen98]]{}, except they could overlap in that case.
[^2]: In additional tests, certain candidate sites that are absent in the inflated tiling were restored systematically to the $\tau a_0$ tiles, and it was verified this had a negligible effect.
|
---
abstract: |
We study front propagation in the reaction diffusion process $\{A\stackrel{\epsilon}\rightarrow2A, A\stackrel
{\epsilon_t}\rightarrow3A\}$ on a one dimensional ($1d$) lattice with hard core interaction between the particles. Using the leading particle picture, velocity of the front in the system is computed using different approximate methods, which is in good agreement with the simulation results. It is observed that in certain ranges of parameters, the front velocity varies as a power law of $\epsilon_t$, which is well captured by our approximate schemes. We also observe that the front dynamics exhibits temporal velocity correlations and these must be taken care of in order to find the exact estimates for the front diffusion coefficient. This correlation changes sign depending upon the sign of $\epsilon_t-D$, where $D$ is the bare diffusion coefficient of $A$ particles. For $\epsilon_t=D$, the leading particle and thus the front moves like an uncorrelated random walker, which is explained through an exact analysis.
address: 'Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India'
author:
- Niraj Kumar and Goutam Tripathy
title: ' Front propagation in A$\rightarrow$2A, A$\rightarrow$3A process in $1d$: velocity, diffusion and velocity correlations. '
---
Introduction
============
Front propagation is an important field of study in nonequilibrium systems. We often encounter these propagating fronts separating different phases in physics, chemistry and biology [@sar]. Here, in this work, we study the dynamics of the front in the reaction-diffusion system $A\rightarrow2A$, $A\rightarrow3A$ in one dimensional lattice. At the macroscopic level, the mean field theory yields the following partial differential equation for the coarse grained concentration $\rho(x,t)$, $$\begin{aligned}
{\label{e1}}
\frac{\partial\rho}{\partial t}=D\frac{\partial^2\rho}{\partial x^2}+
2\epsilon\rho(1-\rho)+\epsilon_t\rho(1-\rho)^2,\end{aligned}$$ where, $D$ is the diffusion coefficient of the particle and $\epsilon$ and $\epsilon_t$ are the rates of single ($A\rightarrow2A$) and twin ($A\rightarrow3A$) offspring production respectively. Equation (\[e1\]) reduces to the well known Fisher equation [@fisher] when $\epsilon_t=0$, which models the reaction diffusion equation $A\rightarrow2A$. The microscopic lattice model for $A\rightarrow2A$ has been studied extensively [@bram][@ker1][@ker2][@ng]. The mean field allows travelling wave solution of the form $\rho(x,t)=\phi(x-vt)$, where the velocity of an initially sharp front between $\rho=1$ ( stable ) and $\rho=0$ ( unstable ) state approaches an asymptotic velocity $V_0=2\sqrt{(
2\epsilon+\epsilon_t)D}$.
Model, Front velocity and diffusion coefficient
===============================================
We consider a $1d$ lattice composed of sites $i=1,2...L$. We start with the step function like distribution where, the left half is filled with $A$ particles while the right half is empty. Each site $i$ can either be empty or occupied by maximum one particle i.e. hard core exclusion is enforced. We update the system random sequentially where $L$ microscopic moves correspond to one Monte Carlo step (MCS). During each updating we randomly select a site and the particle at the site can undergo one of the following three microscopic moves.\
(1) The particle can jump to neighbouring empty site with rate $D$,\
(2) The particle can give birth of one particle at either of the empty neighbouring sites with rate $\epsilon$,\
(3) The particle can generate two new particles at both the neighbouring sites provided both are empty with rate $\epsilon_t$.\
These processes are shown in the Fig. (\[fig:moves\_1d\]).\
As time evolves, these stochastic moves result in the stochastic movement of the front. As has been argued in [@ng], the front may be identified with the rightmost $A$ particle. In this paper, we are interested in the dynamics of front whose evolution may be described by the following master equation[@van]. $$\begin{aligned}
{\label{e4}}
\frac{dP(X,t)}{dt}=(\epsilon+D)P(X-1,t)+\epsilon_tQ_0(X-1,t)+\nonumber\\
DQ_0(X+1,t)-(\epsilon+D)P(X,t)-(\epsilon_t+D)Q_0(X,t)
\end{aligned}$$ Here, $P(X,t)$ is the probability distribution of finding the front particle at position $X$ at a time $t$ and $Q_0(X,t)$ is the joint probability that the front is at $X$ at time $t$ [*[and]{}*]{} the site just behind it is empty. In Eq. (\[e4\]), the first term corresponds to the forward hopping of the front particle from the position $X-1$ to reach $X$ due to birth of a single particle or diffusion. The second term corresponds to twin production at left and right neighbouring sites of site $X-1$ and which results in the front moving from $X-1$ to $X$. The third term corresponds to the backward hopping of the front particle from position $X+1$ due to diffusion, provided site $X$ is empty. Last two terms account for the possible jumps of the front particle from position $X$, which leads to front moving either at $X-1$ or $X+1$. The dynamical properties of the front that we want to study are its velocity $V$ and the diffusion coefficient $D_f$ which are defined as: $$\begin{aligned}
{\label{e5}}
V=\frac{d}{dt}<X(t)>
\end{aligned}$$ $$\begin{aligned}
{\label{e6}}
D_f=\frac{1}{2}\frac{d}{dt}<(X(t)-<X(t)>)^2>
\end{aligned}$$ Where, $<X(t)>=\sum_XXP(X,t)$. Now we use the Eq. (\[e4\]) and normalization $\sum P(X)=1$ and taking $Q_0(X)=(1-\rho_{X-1})P(X)$, where $\rho_{X-1}$ is the probability that site $X-1$ is occupied. Denoting $\rho_{X-1}=\rho_1$, we obtain the following expression for the asymptotic velocity and diffusion coefficient of the front. $$\begin{aligned}
{\label{e7}}
V=\epsilon+\epsilon_t-\rho_1(\epsilon_t-D)
\end{aligned}$$ $$\begin{aligned}
{\label{e8}}
D_f=\frac{1}{2}\{\epsilon+\epsilon_t+2D-\rho_1(\epsilon_t+D)\}
\end{aligned}$$ In order to obtain the velocity and diffusion coefficient we need to know $\rho_1$, which is the density of site just behind the front. In [@bram] for $\epsilon_t=0$, it was shown that front velocity approaches asymptotically the mean field value $V=V_0$ in the limit $\frac{D}{\epsilon}\rightarrow
\infty$, while $V=\epsilon+D$ in the opposite limit $\frac{D}
{\epsilon}\rightarrow 0$. But, when we are in between these two extreme limits we need to know $\rho_1$ and we expect similar features when $\epsilon_t\ne0$. There is no method to find this value exactly. Here we present some approximate analytic estimates for $\rho_1$ and hence the front velocity. In subsection A, we use fixed site representation method, where a truncated master equation is written in the frame moving with the front, as discussed in [@ng]. In subsection B, we apply two particle representation scheme proposed by Kerstein [@ker2] while in C a mixed scheme is proposed which yields better results than either A or B.
Fixed site representation
-------------------------
This method has been proposed in [@ng] for the reaction diffusion process $A\leftrightarrow2A$. Here, we write a truncated master equation in the frame moving with the front. The simplest set of states is:$\{\circ\bullet, \bullet\bullet\}$, which corresponds to the evolution of occupancy at a site just behind the front particle($l=1$). Here the rightmost $\bullet$ in each state corresponds to the front particle. These two states make transitions between each other due to the microscopic processes in the system as shown in Fig. (\[fig:moves\_1d\]).
![Microscopic moves, rightmost $\bullet$ represents the front. (a) Diffusion of the front particle to its right site leading to transition from $\bullet\bullet$ to $\circ\bullet$ with rate $D$. (b) Creation of one particle to the left of the front leads to transition from $\circ\bullet$ to $\bullet\bullet$ with rate $\epsilon$. (c) $\circ\bullet$ changes to $\bullet\bullet$ due to creation of twins at both neighbouring sites of the front with rate $\epsilon_t$. (d) $\circ\bullet
\rightarrow\bullet\bullet$, if the front takes diffusive move to its left and the second site behind the front is occupied. This occurs with rate $D\rho_2$, where $\rho_2$ is the probability of occupancy at the second site behind the front.[]{data-label="fig:moves_1d"}](micro_process.eps){width="2.5in" height="1.50in"}
Considering all such transitions the evolution of probabilities of these two states are given by: $$\begin{aligned}
{\label{e9}}
\frac{dP(\circ\bullet)}{dt}&=&(2D-D\rho_{2})P(\bullet\bullet)
-\{2D\rho_{2}+\epsilon(2+\rho_{2})\nonumber\\
& &+\epsilon_t(1+\rho_{2}(1-\rho_{3}))\}P(\circ\bullet),\nonumber\\
\frac{dP(\bullet\bullet)}{dt}&=&\{2D\rho_{2}+\epsilon(2+
\rho_{2})+\epsilon_t
(1+\rho_{2}(1-\rho_{3}))\nonumber\\& &\}P(\circ\bullet)
-(2D-D\rho_{2})P(\bullet\bullet).
\end{aligned}$$ Here, $\rho_i$ is the density at the $i$th site behind the front and we have neglected the spatial density correlation between consecutive pairs of sites beyond the second site behind the front. Now, using Eq. (\[e9\]) and normalization $P(\circ\bullet)+P(\bullet\bullet)=1$, we obtain the following expression for $\rho_1$: $$\begin{aligned}
{\label{e10}}
\rho_1=\frac{2D\rho_{2}+\epsilon(2+\rho_{2})+\epsilon_t(1+
\rho_{2}(1-\rho_{3}))}
{D\rho_{2}+2D+2\epsilon+\epsilon\rho_{2}+
\epsilon_t(1+\rho_{2}(1-\rho_{3}))}
\end{aligned}$$ From Eq. (\[e10\]), we note that in order to find $\rho_1$ we need to know $\rho_2$ and $\rho_3$. As a crude approximation if we assume that $\rho_{2}=\rho_{3}=\rho^{b}=1$, where $\rho^{b}$ is the bulk density, we get the following value of $\rho_1$. $$\begin{aligned}
{\label{e11}}
\rho_1\simeq\frac{2D+3\epsilon+\epsilon_t}{3D+3\epsilon+\epsilon_t}
\end{aligned}$$ Now using this approximation for $\rho_1$ in Eq. (\[e7\]), we find the estimate for the velocity which is in reasonable agreement with the simulation, as shown in the Fig. (\[fig:velocity\]) . The estimate for $V$ can be improved by including more sites in the truncated representation. For example, for $l=2$ we study the evolution of following set of four states: $\{\circ\circ\bullet, \circ\bullet
\bullet, \bullet\circ\bullet, \bullet\bullet\bullet\}$ and as expected we get improved results as shown in Fig. (\[fig:velocity\]). Here, we notice that for larger values of $\epsilon_t$ the simulation results show nice agreement with that of analytic results. However, as $\epsilon_t$ decreases and approaches zero, we see gradual departure of the simulation data from the analytic one. In fact, $\rho_i$ differs from the bulk density significantly with decreasing value $\epsilon_t$ as shown in the Fig. (\[fig:den\_profile\]). That is, the approximation $\rho_i\approx1$ holds better for larger values of $\epsilon_t$ and hence we get better agreement with the simulation results. The estimate for the velocity can be further improved if we include states with larger number of sites. In Fig. (\[fig:velocity\]) we notice two interesting points: firstly, for $D=\epsilon_t$, the theoretical result matches strikingly with the simulation result, secondly, we observe a power law dependence of the velocity on $\epsilon_t$. In fact, the first point, can be shown to be exact by noting that when $D=\epsilon_t$, the front velocity from Eq.(\[e7\]) is $V=\epsilon+\epsilon_t$, which is independent of $\rho_1$.
Two particle representation
---------------------------
In the following, we try to find the analytic estimates for $\rho_1$ using Kerstein’s two particles representation[@ker2]. In this representation each state of the system is defined by two rightmost particles and thus we have an infinite set of states: $\{11, 101, 1001,
10001, 100001,....\}$. Here, the rightmost ’1’ represents the front particle while the leftmost ’1’ is the second particle behind it and ’0’ stands for empty site. Let us denote by $P_k$ the probability of two particle state with $k$ empty sites between the leading particle and next particle behind it. These states form a closed set under transition due to microscopic processes. We have illustrated few transitions in the Fig. (\[fig:moves\_2P\]). Considering all such transitions and denoting the probability of occupancy of site just behind the second particle by $\rho$, we write the following rate equations for $P_k$. $$\begin{aligned}
{\label{e13}}
\frac{dP_0}{dt}&=&(\epsilon+2D)P_1+\epsilon_t(1-\rho)P_1
+(2\epsilon+\epsilon_t)(1-P_0)\nonumber\\
& &-(2D-D\rho)P_0,\nonumber\\
\frac{dP_k}{dt}&=& (2D-D\rho)P_{k-1}+\{\epsilon+2D+
\epsilon_t(1-\rho)\}P_{k+1}\nonumber\\
&-&(4D-D\rho+3\epsilon+2\epsilon_t-\epsilon_t\rho)P_k,
{\hspace{0.4cm}}k\ge1.
\end{aligned}$$
![Transition between two particle states with rightmost $\bullet$ representing front. (a) Diffusive move of the front particle to its left leading to transition 101$\rightarrow$11 with rate D, (b) When the second particle behind the front jumps to the left, provided it is empty, state changes from 101$\rightarrow$1001 with rate $D(1-\rho)$, (c) Birth of a single particle by the second particle to its left with rate $\epsilon$ leads to transition 1001$\rightarrow$101, (d) 1001$\rightarrow$101 if the second particle gives birth of two particles, provided the the site left to it is empty, with rate $\epsilon_t(1-\rho)$.[]{data-label="fig:moves_2P"}](moves_2P.eps){width="2.2in" height="1.8in"}
In order to solve Eq.(\[e13\]) we need to specify the dependence of $\rho$ on the parameters($\epsilon,
\epsilon_t,D$). Following Kerstein [@ker2], we write $\rho=aP_0-bP_0^2$ and enforcing the condition that $\rho=1$ when $P_0=1$, we write the following expression for $\rho$. $$\begin{aligned}
{\label{e14}}
\rho=(1+\lambda)P_0-\lambda P_0^2
\end{aligned}$$ This equation specifies the dependence of $\rho$ on the parameters implicitly through dependence of $P_0$ on $\epsilon,
\epsilon_t,D$. Here, $\lambda$ is a free parameter to be evaluated as follows. Following Kerstein, using the ansatz $P_k=P_0(1-P_0)^k$ and Eq. (\[e14\]) in Eq.(\[e13\]), we get the following quartic equation in $P_0$. $$\begin{aligned}
{\label{e15}}
\epsilon_t\lambda P_0^4&+&(D\lambda-\epsilon_t-2\epsilon_t
\lambda)P_0^3+(\epsilon+D
+2\epsilon_t+\epsilon_t\lambda\nonumber\\& & -D\lambda )P_0^2
+\epsilon P_0-2\epsilon-\epsilon_t=0
\end{aligned}$$ In order to find $P_0$ we need to fix the value of $\lambda$. For large $D$ and $\epsilon_t=0$, it is known that the front particle moves with its mean field velocity[@bram]. If we assume that this also happens when $\epsilon_t\neq 0$, then equating the mean field front velocity $V_0=2\sqrt{(2\epsilon+\epsilon_t)}D$ with that obtained from Eq. (\[e7\]) i.e. $V\sim DP_0$ when $D$ is very large compared to other parameters, we get $P_0=2\sqrt{\frac{2\epsilon+\epsilon_t}{D}}$. Using this value of $P_0$ in Eq. (\[e15\]) we find $\lambda=3/4$ in the limit $D\rightarrow\infty$. We solve the quartic equation (\[e15\]) to get the value of $\rho_1=P_0$ and hence the front velocity, as shown in the Figs. (\[fig:velocity\]) and (\[fig:rho1\_vel\_et0\]) and marked as $2P$.
Mixed representation
--------------------
Since we are dealing with a multiparticle interacting system it is always desirable to include as many particles as possible while studying the evolution of the system. The simplest extension to the two particle representation is to study the evolution of the following set of states: $\{\circ\bullet\bullet, \bullet\bullet
\bullet , \circ\bullet\circ \bullet, \bullet\bullet\circ \bullet
...\}$, where the rightmost $\bullet$ in each state denotes the front particle. Since in this representation, each state is characterized by two or three particles and hence we name it as mixed representation (MR). The rightmost $\bullet$ in each state denotes the front particle. When viewed in the frame moving with the front, each state contains the location of second particle and the occupancy of the site just behind the second particle. We denote these states as $(k, 0)$ or $(k, 1)$, representing the states having $k$ empty sites between the front and the second particle and the site just after the second particle is empty or occupied respectively. For example, by (0, 0) we mean the state $\circ\bullet\bullet$ and (0, 1) for the state $\bullet\bullet\bullet$. These states are making transitions among each other due to the microscopic processes and form a closed set. We have shown some of the transitions in the Fig. (\[fig:moves\_2\]).
![Transitions between mixed particle states with the rightmost $\bullet$ representing front. (a) Diffusive move of the particle to the right empty site with rate D.This leads to transition from the state (1,1) to (0,0). (b) Birth of a new particle on the right neighbouring empty site with rate $\epsilon$, which changes the state (1,0) to (0,1). (c) Transition from (1,0) to (1,1) with rate D$\rho$, when the third particle jumps to the right neighbouring empty site. (d) (1,0)$\rightarrow$(0,1) when the second particle behind the front in (1,0) realization produces twins at the neighbouring empty sites with rate $\epsilon_t$. []{data-label="fig:moves_2"}](fig_2.eps){width="2.2in" height="1.8in"}
Assuming $\rho$ as the density of site, which is, next nearest neighbour to the second particle, we write the following rate equation for the evolution of probabilities $P(k,0)$ and $P(k,1)$, $k=0,1,..\infty$. $$\begin{aligned}
{\label{16}}
\frac{dP(0,1)}{dt}&=&\{D\rho+\epsilon\rho+2\epsilon+\epsilon_t
\rho(1-\rho)\}P(0,0)\nonumber\\&+&(D+2\epsilon+\epsilon_t)
P(1,1)+(2\epsilon+2\epsilon_t)P(1,0)\nonumber\\&+&
\epsilon_t\{P(2,0)+P(2,1)+P(3,0)+P(3,1)+...\}\nonumber\\
&-&(2D-D\rho)P(0,1),\nonumber\\
\frac{dP(0,0)}{dt}&=&D(1-\rho)P(0,1)+(D+\epsilon)P(1,1)\nonumber\\
&+&(2D+\epsilon)P(1,0)+2\epsilon\{P(2,1)+P(2,0)\nonumber\\
&+&P(3,0)+P(3,1)+...\}-\{2D+2\epsilon+D\rho\nonumber\\
&+&\epsilon\rho+\epsilon_t\rho(1-\rho)\}P(0,0),\nonumber\\
\frac{dP(k,1)}{dt}&=&DP(k-1,1)+D\rho P(k-1,0)\nonumber\\
&+&\{D\rho+\epsilon+\epsilon\rho+\epsilon_t\rho(1-\rho)
\}P(k,0)\nonumber\\&+&(D+\epsilon)P(k+1,1)+(\epsilon+
\epsilon_t)P(k+1,0)\nonumber\\&-&(4D+3\epsilon-D\rho+
\epsilon_t)P(k,1),\nonumber\\
\frac{dP(k,0)}{dt}&=&(D+D(1-\rho))P(k-1,0)+D(1-\rho)P(k,1)
\nonumber\\&+&DP(k+1,1)+2DP(k+1,0)-\{4D+4\epsilon
\nonumber\\&+&D\rho+\epsilon\rho+2\epsilon_t+
\epsilon_t\rho(1-\rho)\}P(k,0).
\end{aligned}$$ In order to find $P_0$, we need to solve the above set of coupled equations. However, one can find the analytic estimate for $P_0$ by solving rate equations for $P(0,0)$ and $P(0,1)$ and assuming $P(1,1)=\rho P_1, P(1,0)=
(1-\rho)P_1$. Using $\displaystyle\sum_{i=0}^{1}P(k,i)=P_k$ and $\displaystyle\sum_{k=0}^{\infty}P_k=1$, we find steady state expression for $P(0,0)$ and $P(0,1)$ in terms of $P_1$ and $\rho$ and then solve the equation: $$\begin{aligned}
{\label{e17}}
P(0,0)+P(0,1)=P_0
\end{aligned}$$ Following Kerstein [@ker2], if we use the ansatz $P_1=P_0(1-P_0)$, we get the following equation. $$\begin{aligned}
{\label{e18}}
\alpha P_0(1-P_0)+\beta P_0+\gamma=0
\end{aligned}$$ Where, $$\begin{aligned}
{\label{e19}}
\alpha &=&(2\epsilon+\epsilon_t+D\rho-\epsilon_t\rho)\{3D+2\epsilon
+\epsilon\rho+\epsilon_t\rho(1-\rho)\}\nonumber\\& &+(2D-D\rho-\epsilon)\{2D
+2\epsilon+\epsilon\rho+\epsilon_t\rho(1-\rho)\},\nonumber\\
\beta &= & \{2D+2\epsilon+\epsilon\rho+D\rho+\epsilon_t\rho
(1-\rho)\}(2D-D\rho)\nonumber\\& &-\{D\rho+2\epsilon+\epsilon\rho+\epsilon_t
\rho(1-\rho)\}(D-D\rho),\nonumber\\
\gamma &=& \epsilon_t(1-P_0)\{3D+2\epsilon+\epsilon\rho+\epsilon_t
\rho(1-\rho)\}\nonumber\\& &+2\epsilon(1-P_0)\{2D+2\epsilon+\epsilon\rho
+\epsilon_t\rho(1-\rho)\}.
\end{aligned}$$
The Eq. (\[e18\]) is in terms of two unknowns $\rho$ and $P_0$ and hence we need to know $\rho$ in order to find $P_0$. We specify the dependence of $\rho$ on $P_0$ similar to what we did in the case of two particle representation. Using the results obtained for $P_0=\rho_1$ in Eq.(\[e7\]) we find the estimates for $V$ as shown in the Figs. (\[fig:velocity\]) and (\[fig:rho1\_vel\_et0\]) marked as MR. We observe good agreement with the simulation results.
![Percentage relative error in $V$, i.e. $\frac{|V^s-V^a|}{V^s}\times100$ ($V^s$ and $V^a$ representing simulation and analytic results respectively), is plotted against $\epsilon_t$ while keeping D=0.25 and $\epsilon=0.025$ fixed. Here stars, open squares, open circles and filled circles correspond to the analytic estimates using $l=2$(two states), $l=3$ (four states), two particle(2P) and mixed representation (MR) respectively. Simulation profile and analytic profile using MR are essentially coincident and hence we notice almost zero error. Inset: log-log plot for the velocity versus $\epsilon_t$, the straight line shows power law $V\approx
\epsilon_t^{0.25}$.[]{data-label="fig:velocity"}](Velocity_3_paper.eps)
![Percentage relative error in $V$ versus $D$, keeping $\epsilon=0.05$ and $\epsilon_t=0$. The top data(filled squares) corresponds to ref.[@ng], where in the frame moving with the front ,the evolution of particles at three sites behind the front was studied and assuming the fourth site at the bulk density. The middle data (open circles) corresponds to Kerstein [@ker2] two particle self consistent representation while the bottom data (filled circles) is the result from the mixed representation.[]{data-label="fig:rho1_vel_et0"}](Rho1_vel_ET_0_paper.eps)
![Density profile behind the front for different values of $\epsilon_t$ while keeping $D=0.25$ and $\epsilon=0.025$ fixed. Here, we notice that as $\epsilon_t$ decreases density profile curve shifts away from the bulk density level.[]{data-label="fig:den_profile"}](rho.eps)
Front diffusion coefficient
===========================
In Fig. (\[fig:diff\_coeff\]), we have shown the simulation results of the front diffusion coefficient and compared it with the results obtained by using the mean field and simulation value of $\rho_1$ in the equation(\[e8\]). Here, we notice the following interesting features: (1) when $D=\epsilon_t$, the analytical value $D_f^{ana}$ matches well with the simulation result $D_f^{sim}$. (2) when $\epsilon_t > D$, the $D_{f}^{sim}
> D_{f}^{ana}$ (3) when $\epsilon_{t}< D $, $D_{f}^{sim}<
D_{f}^{ana}$. The origin of the above discrepancy between the simulation and analytical results can be traced to the master equation (\[e4\]), where we have neglected the temporal velocity correlations. The expression for the asymptotic front diffusion coefficient with temporal correlations in velocity is given as: $$\begin{aligned}
{\label{e20}}
D_f=D_0+\sum_{t=1}^{\infty}C(t)
\end{aligned}$$ where, $D_0$ is the front diffusion coefficient by neglecting correlations as given by Eq. (\[e8\]) and $C(t)$ is the temporal velocity correlation defined through, $$\begin{aligned}
{\label{e21}}
C(t)=<v(t')v(t'+t)> - <v(t')><v(t'+t)>,\nonumber\\
\end{aligned}$$ where, $v(t)$ is the displacement of the front at time $t$.
![Comparison of the front diffusion coefficient obtained analytically (from Eq. (\[e8\]) and using $\rho_1$ corresponding to $l=2$) with the simulation results for different values of $\epsilon_t$ while keeping $D=0.25$, $\epsilon$=0.025 fixed. We note that when $\epsilon_t$=$D=0.25$, the simulation result matches with the analytic one.[]{data-label="fig:diff_coeff"}](Diff_coeff_2.eps){width="3.0in" height="2.5in"}
![Simulation results for velocity correlation of the front with time $t$, for different values of $\epsilon_t$ while keeping $D=0.25$, $\epsilon$=0.025 fixed. We notice that when $\epsilon_t=0.25$, this correlation is zero. Inset: log-normal plot for $C(t)$ versus $t$ with $D=0.05, 0.10, 0.30, 0.40$ from top to bottom.[]{data-label="fig:vel_corr"}](vel_corr_new.eps)
In Fig. (\[fig:vel\_corr\]), we have plotted the temporal velocity correlation $C(t)$ for different values of $\epsilon_t$. For $\epsilon_t>D$, we observe positive correlation while for $\epsilon_t<D$, it is negative and for $\epsilon_t=D$, $C(t)$ seems to vanish for all $t$. Thus, $\epsilon_t=D$ is a special case, where the front particle moves like an uncorrelated random walker.
In the following, we explicitly show that for the special case $\epsilon_t=D$, two consecutive steps of the leading particle are uncorrelated, i.e., $C(1)=0$ in the steady state. Since at most two sites behind the front can be affected in two consecutive steps, we consider 4 states corresponding to $l=2$, namely, $\{001,011,101,111\}$ with the rightmost $'1'$ representing the front. In order to find $C(1)=<v(t)v(t+1)>-<v(t)><v(t+1)>$, we write $<v(t)v(t+1)>=R_{++}-
R_{+-}-R_{-+}+R_{--}$, where $R_{ij}$ denotes the ’flux’ $R^{001}_{ij}+
R^{011}_{ij}+R^{101}_{ij}+R^{111}_{ij}$ for taking two consecutive steps as $i=+/-$ and $j=+/-$. Here, for example, $R^{001}_{--}$ is the flux of two consecutive negative steps starting from the state $001$. The only way it can occur is if the front particle takes two diffusive moves to the left and thus, $R^{001}_{--}=
D^2P_{001}$, where $P_{001}$ is the steady state weight of the configuration $001$. Considering all such two successive moves in the each state, we write the following expression for $R_{++},R_{+-},R_{-+}$ and $R_{--}$, $$\begin{aligned}
{\label{e22}}
R_{++}&=&D^2+2\epsilon D+\epsilon_t D+\epsilon^2+(\epsilon_t D
+\epsilon_t\epsilon)\{P_{001}+P_{101}\},\nonumber\\
R_{+-}&=&D^2,\nonumber\\
R_{-+}&=&(D^2+D\epsilon+D\epsilon_t)P_{001}+(D^2+D\epsilon)P_{101}
,\nonumber\\
R_{--}&=&D^2P_{001}.
\end{aligned}$$ Now using Eqs.(\[e22\]), we have $$\begin{aligned}
{\label{e23}}
<v(t)v(t+1)>&=&\epsilon^2+D\epsilon_t+2\epsilon D
+\{\epsilon_tD+\epsilon_t\epsilon\nonumber\\-D^2-D\epsilon)\}
P_{101}&+&\{\epsilon_t\epsilon-D\epsilon\}P_{001},
\end{aligned}$$ and similarly, $$\begin{aligned}
{\label{e24}}
<v(t)>=<v(t+1)>=\epsilon+\epsilon_t-\{P_{011}+P_{111}\}
(\epsilon_t-D)\nonumber.\\
\end{aligned}$$ Using Eqs.(\[e23\]) and (\[e24\]) we can find the value of $<v(t)v(t+1)>$ or $<v(t)>$ by finding the probabilities of different states, which is harder to compute exactly. However, when $\epsilon_t=D$, we find that $<v(t)v(t+1)>-<v(t)><v(t+1)>$ is independent of all the probabilities and is equal to zero, i. e., two successive steps are uncorrelated as observed in the simulation result (Fig.\[fig:vel\_corr\]). We also note that for this special case the above analysis does not involve any approximation i.e. it is exact. We also notice that when $\epsilon_t\ne D$, $<v(t)v(t+1)>\ne
<v(t)><v(t+1)>$, i.e., the front motion is correlated. Preliminary fits suggest that the the temporal velocity correlation has the form, $C(t)\sim t^\alpha e^{-\beta t}$.
Conclusion
==========
In the present paper, we studied the reaction diffusion system $A\rightarrow 2A, A\rightarrow3A$ in one dimension. Treating the rightmost occupied site as a front we compute the front velocity analytically using different approximate methods. In fixed site representation one can systematically improve upon the estimate by studying the evolution of particles at larger number of sites behind the front. The results from two particle representation and mixed representation show excellent agreement with the simulation results. We also observed that the velocity depends on $\epsilon_t$ as a power law. As far as the computation of front diffusion coefficient is concerned, we notice that one needs to take into account the temporal velocity correlation. In fact the observed temporal correlations in the front dynamics changes sign with sign of $\epsilon_t-D$. For $\epsilon_t>D$ or $\epsilon_t<D$, front moves like a positively or negatively correlated random walk and for $\epsilon_t=D$, the temporal correlations in steps vanish and the front particle moves like a simple uncorrelated random walker. An interesting generalization of the process would be to include the annihilation of particles as well i.e., $2A\rightarrow A$ with rate $W$. The case $\epsilon_t=0$ and $W=D$ is exactly solvable [@bA] and both temporal and spatial correlations vanish. For non-zero $D$, $\epsilon_t$ and $W$, simulations show that temporal correlations vanish on the plane $D=\epsilon_t+W$, although spatial correlations do not. Computation of the previous section can be extended to this case to show that consecutive steps are temporally uncorrelated. A general proof to show that $C(t)$ vanishes for all $t$ remains an interesting open problem.
W.van Saarloos, Phys. Rep. 386, 29 (2003). M. Bramson, P. Calderoni, A. De Masi, P. A. Ferrari, J. L. Lebowitz and R. H. Schonmann , J. Stat. Phys. 45, 905 (1986). A. R. Kerstein J. Stat. Phys. 45, 921 (1986). A. R. Kerstein , J. Stat. Phys. 53,703 (1988). Niraj Kumar and G. Tripathy, Phys. Rev. E 74, 011109 (2006). R. A. Fisher, Ann. Eugenics , 7, 355 (1937); A. Kolmogorov, I. Petrovsky and N. Piscounov, Moscow Uni. Bull. Math. A1, 1 (1937). N. G. van Kampen, Stochastic processes in Physics and Chemistry (North-holland, Amsterdam, 1981). D. ben-Avraham, Phys. Lett. A, 247, 53 (1998).
|
---
abstract: 'The survey presents the well-known Warshall’s algorithm, a generalization and some interesting applications of this.'
title: '**Warshall’s algorithm—survey and applications**'
---
[**Zoltán KÁSA**]{}
Sapientia Hungarian University of Transylvania, Cluj-Napoca
[Department of Mathematics and Informatics,](http://www.ms.sapientia.ro/en)\
Tg. Mureş, Romania
<[email protected]>
Introduction
============
Let $R$ be a binary relation on the set $S=\{s_1, s_2, \ldots, s_n\}$, we write $s_iRs_j$ if $s_i$ is in relation to $s_j$. The relation $R$ can be represented by the so called *relation matrix*, which is $$A=(a_{ij}){\footnotesize_{
\begin{array}{l}
i=\overline{1,n} \\
j=\overline{1,n} \end{array}}
}, {\rm \ where } \ \ a_{ij}= \left\{
\begin{array}{ll}
1 & {\rm if \ } s_i R s_j\\
0 & {\rm otherwise.}\\
\end{array} \right.$$
The transitive closure of the relation $R$ is the binary relation $R^*$ defined as:
$s_i R^* s_j$ if and only if there exists $s_{p_1}$, $s_{p_2}$, $\ldots$, $s_{p_r}, r\ge 2$ such that $s_i=s_{p_1} $, $s_{p_1}R s_{p_2}$, $s_{p_2} R s_{p_3}, \ldots, $ $s_{p_{r-1}} R s_{p_r},$ $s_{p_r}=s_j$. The relation matrix of $R^*$ is $A^*=(a^*_{ij})$.
Let us define the following operations. If $a,b\in \{0,1\}$ then $a+b=0$ for $a=0, b=0$, and $a+b=1$ otherwise. $a\cdot b=1$ for $a=1, b=1$, and $a\cdot b=0$ otherwise. In this case
$$A^*=A+A^2+ \cdots + A^n.$$
The transitive closure of a relation can be computed easily by the Warshall’s algorithm [@warshall], [@baase]:
[Warshall($A,n$)]{} *Input:* the relation matrix $A$; the no. of elements $n$\
*Output:* $W=A^* $\
1 ' $W \leftarrow A$\
2 ' [**for**]{} = $k \leftarrow 1$ [**to**]{} $n$\
3 ' [**do**]{} [**for**]{} = $i \leftarrow 1$ [**to**]{} $n$\
4 ' [**do**]{} [**for**]{} = $j \leftarrow 1$ [**to**]{} $n$\
5 ' [**do**]{} [**if**]{} = $w_{ik}=1$ and $w_{kj}=1$\
6 ' [**then**]{} $w_{ij}\leftarrow 1$\
7 ' [**return**]{} $W$
A binary relation can be represented by a directed graph (i.e. digraph) too. he relation matrix is equal to the adjacency matrix of the corresponding graph. See Fig. \[fig1\] for an example. Fig. \[fig2\] represents the graph of the corresponding transitive closure relation.
![A binary relation represented by a graph with the corresponding adjacency matrix\[fig1\]](fig1.jpg)
$\left(
\begin{array}{ccccc}
0 &1& 0& 0& 0\\
0 &0& 1& 1& 0\\
0 &0 &0 &1 &0 \\
0 &0 &0 &0 &0 \\
0 &1 &0 &0& 0
\end{array}
\right)$
![The transitive closure of the relation in Fig. \[fig1\]\[fig2\]](fig2.jpg)
$\left(
\begin{array}{ccccc}
0& 1& 1& 1& 0\\
0 & 0& 1& 1& 0\\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 0
\end{array}
\right)$
Generalization of Warshall’s algorithm
======================================
Lines 5 and 6 in the Warshall’s algorithm described above can be changed in
$$w_{ij}\leftarrow w_{ij}+ w_{ik}w_{kj}$$
using the operations defined above. If instead of the operations + and $\cdot$ we use two operations $\oplus$ and $\odot$ from a semiring, a generalized Warshall’s algorithm results [@robert]:
[Generalized-Warshall($A,n$)]{} *Input:* the relation matrix $A$; the no. of elements $n$\
*Output:* $W=A^* $\
1 ' $W \leftarrow A$\
2 ' [**for**]{} = $k \leftarrow 1$ [**to**]{} $n$\
3 ' [**do**]{} [**for**]{} = $i \leftarrow 1$ [**to**]{} $n$\
4 ' [**do**]{} [**for**]{} = $j \leftarrow 1$ [**to**]{} $n$\
5 ' [**do**]{} $w_{ij}\leftarrow w_{ij} \oplus (w_{ik}\odot w_{kj})$\
6 ' [**return**]{} $W$
This generalization leads us to a number of interesting applications.
Applications
============
Distances between vertices. The Floyd-Warshall’s algorithm
----------------------------------------------------------
Given a weighted (di)graph with the modified adjacency matrix $D_0=(d^0_{ij})$, we can obtain the distance matrix $D=(d_{ij})$ in which $d_ij$ represents the distance between vertices $v_i$ and $v_j$. The distance between vertices $v_i$ and $v_j$ is the length of the shortest path between them. The modified adjacency matrix $D_0=(d^0_{ij})$ is the following:
$$d^0_{ij}= \left\{
\begin{array}{ll}
0 & \textrm{if } i=j\\
\infty & \textrm{if there is no edge from vertex } v_i \textrm{ to vertex } v_j\\
w_{ij} & \textrm{the weight of the edge from } v_i \textrm{ to } v_j
\end{array}
\right.$$
Choosing for $\oplus$ the $\min$ operation (minimum between two reals), and for $\odot$ the real addition ($+$), we obtain the well-known Floyd-Warshall’s algorithm as a special case of the generalized Warshall’s algorithm [@robert; @vattai] :
[Floyd-Warshall($D_0,n$)]{} *Input:* the adjacency matrix $D_0$; the no. of elements $n$\
*Output:* the distance matrix $D$\
1 ' $D \leftarrow D_0$\
2 ' [**for**]{} = $k \leftarrow 1$ [**to**]{} $n$\
3 ' [**do**]{} [**for**]{} = $i \leftarrow 1$ [**to**]{} $n$\
4 ' [**do**]{} [**for**]{} = $j \leftarrow 1$ [**to**]{} $n$\
5 ' [**do**]{} $d_{ij}\leftarrow \min \{d_{ij}, \ d_{ik}+d_{kj}\}$\
6 ' [**return**]{} $D$
![A weighted digraph \[floyd1\]](floyd1.jpg)
![The corresponding matrices of the Fig. \[floyd1\]\[floyd2\]](floyd2a.jpg "fig:")![The corresponding matrices of the Fig. \[floyd1\]\[floyd2\]](floyd3a.jpg "fig:")
Figures \[floyd1\] and \[floyd2\] contain az example. The shortest paths can be easily obtained if in the description of the algorithm in line 5 we store also the previous vertex $v_k $ on the path. In the case of acyclic digraphs, the algorithm can be easily modified to obtain the longest distances between vertices, and consequently the longest paths.
Number of paths in acyclic digraphs
-----------------------------------
Here by path we understand directed path. In an acyclic digraph the following algorithm count the number of paths between vertices [@kasa; @elzinga]. The operation $\oplus, \odot$ are the classical add and multiply operations for real numbers.
[Warshall-Path($A,n$)]{} *Input:* the adjacency matrix $A$; the no. of elements $n$\
*Output:* $W$ with no. of paths between vertices\
1 ' $W \leftarrow A$\
2 ' [**for**]{} = $k \leftarrow 1$ [**to**]{} $n$\
3 ' [**do**]{} [**for**]{} = $i \leftarrow 1$ [**to**]{} $n$\
4 ' [**do**]{} [**for**]{} = $j \leftarrow 1$ [**to**]{} $n$\
5 ' [**do**]{} $w_{ij}\leftarrow w_{ij}+w_{ik}w_{kj}$\
6 ' [**return**]{} $W$
An example can be seen in Figures \[utak\] and \[utak1\]. For example between vertices 1 and 3 there are 3 paths: (1,2,3); (1,2,5,3) and (1,6,5,3).
![An acyclic digraph \[utak\]](utak.jpg)
$A=\left(
\begin{array}{cccccc}
0& 1& 0& 1& 0& 1\\
0& 0& 1& 0& 1& 0\\
0& 0& 0& 1& 0& 0 \\
0& 0& 0& 0& 0& 0 \\
0& 0& 1& 0& 0& 0 \\
0& 0& 0& 0& 1& 0
\end{array}
\right)$$W=\left(\begin{array}{cccccc}
0& 1& 3& 4& 2& 1\\
0 & 0 & 2 & 2 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 0
\end{array}
\right)$
All paths in digraphs
---------------------
The Warshall’s algorithm combined with the Latin square method can be used to obtain all paths in a (not necessarily acyclic) digraph [@kasa]. A path will be denoted by a string formed by its vertices in there natural order in the path.
Let us consider a matrix ${\cal A}$ with the elements $A_{ij}$ which are set of strings. Initially elements of this matrix are defined as: $$A_{ij}=\left\{ \begin{array}{cl}
\{v_iv_j\}, & \textrm{if there exists an arc from } v_i \textrm{ to } v_j,\\
\emptyset, & \textrm{otherwise},
\end{array} \quad \textrm{ for } \, i,j=1,2,\ldots, n,
\right.$$ If ${\cal A}$ and ${\cal B}$ are sets of strings, ${\cal AB}$ will be formed by the set of concatenation of each string from ${\cal A}$ with each string from ${\cal B}$, if they have no common elements: $${\cal AB} = \big\{ ab \, \big| \, a\in {\cal A}, b\in {\cal B}, \textrm{ if the strings } a \textrm{ and } b \textrm{ have no common elements} \big\}.$$ If $s=s_1s_2\cdots s_p$ is a string, let us denote by $'s$ the string obtained from $s$ by eliminating the first character: $'s=s_2s_3\cdots s_p$. Let us denote by $'{A_{ij}}$ the set ${A_{ij}}$ in which we eliminate from each element the first character. In this case $'{\cal A}$ is a matrix with elements $'A_{ij}.$
Operations are: the set union and set product defined as before.
Starting with the matrix ${\cal A}$ defined as before, the algorithm to obtain all paths is the following:
[Warshall-Latin(${\cal A},n$)]{} *Input:* the adjacency matrix ${\cal A }$; the no. of elements $n$\
*Output:* ${\cal W}$ matrix of the paths between vertices\
1 ' ${\cal W} \leftarrow {\cal A} $\
2 ' [**for**]{} = $k \leftarrow 1$ [**to**]{} $n$\
3 ' [**do**]{} [**for**]{} = $i \leftarrow 1$ [**to**]{} $n$\
4 ' [**do**]{} [**for**]{} = $j \leftarrow 1$ [**to**]{} $n$\
5 ' [**do**]{} [**if**]{} = $W_{ik}\ne \emptyset$ and $W_{kj}\ne \emptyset$\
6 ' [**then**]{} $W_{ij}\leftarrow W_{ij} \cup W_{ik}\, 'W_{kj}$\
7 ' [**return**]{} ${\cal W}$
In Figures \[allpath\] and \[allpathm\] an example is given. For example between vertices $v_1 $ and $v_3 $ there are two paths: $v_1v_3$ and $v_1v_2v_3$.
![An example of digraph for all paths problem[]{data-label="allpath"}](allpaths.jpg)
${\cal A}=\left(
\begin{array}{ccccc}
\emptyset&\{v_1v_2\}&\{v_1v_3\}&\emptyset&\{v_1v_5\}\\
\emptyset&\emptyset&\{v_2v_3 \}&\emptyset&\emptyset\\
\{v_3v_1\}&\emptyset&\emptyset&\emptyset&\emptyset\\
\emptyset&\emptyset&\{v_4v_3\}&\emptyset&\{v_4v_5\}\\
\emptyset&\emptyset&\emptyset&\ \ \emptyset\ \ &\emptyset
\end{array}
\right)
$
${\cal W}=\left(
\begin{array}{ccccc}
\emptyset&\{v_1v_2\}&\{v_1v_3, \,v_1v_2v_3\}&\emptyset&\{v_1v_5\}\\
\{v_2v_3v_1\}&\emptyset&\{v_2v_3 \}&\emptyset&\{v_2v_3v_1v_5\}\\
\{v_3v_1\}&\{v_3v_1v_2\}&\emptyset&\emptyset&\{v_3v_1v_5\}\\
\{v_4v_3v_1\}&\{v_4v_3v_1v_2\}&\{v_4v_3\}&\emptyset&\{v_4v_5\}\\
\emptyset&\emptyset&\emptyset&\ \ \emptyset\ \ &\emptyset
\end{array}
\right)
$
Scattered complexity for rainbow words
--------------------------------------
The application mentioned here can be found in [@kasa]. Let $\Sigma$ be an alphabet, $\Sigma^n$ the set of all length-$n$ words over $\Sigma$, $\Sigma^*$ the set of all finite word over $\Sigma$.
Let $n$ and $s$ be positive integers, $M\subseteq \{1,2,\ldots, n-1\}$ and $u=x_1x_2\ldots x_n\in \Sigma^n$. An **$M$-subword** of length $s$ of $u$ is defined as $v=x_{i_1}x_{i_2}\ldots x_{i_s}$ where
$i_1\ge 1$,
$i_{j+1}-i_j\in M$ for $j=1,2,\ldots, s-1$,
$i_s\le n.$
The number of $M$-subwords of a word $u$ for a given set $M$ is the scattered subword complexity, simply $M$-complexity.
**Examples.** The word $abcd$ has 11 $\{1,3\}$-subwords: $a$, $ab$, $abc$, $abcd$, $ad$, $b$, $bc$, $bcd$, $c$, $cd$, $d$. The $\{2,3\,4,5\}$-subwords of the word $abcdef$ are the following: $a$, $ac$, $ad$, $ae$, $af$, $ace$, $acf$, $adf$, $b$, $bd$, $be$, $bf$, $bdf$, $c$, $ce$, $cf$, $d$, $df$, $e$, $f$.
Words with different letters are called *rainbow words*. The $M$-complexity of a length-$n$ rainbow word does not depend on what letters it contains, and is denoted by $K(n,M)$.
To compute the $M$-complexity of a rainbow word of length $n$ we will use graph theoretical results. Let us consider the rainbow word $a_1a_2\ldots a_n$ and the corresponding digraph $G=(V, E)$, with
$V=\big\{ a_1, a_2, \ldots, a_n \big\}$,
$E=\big\{ (a_i,a_j) \mid j-i\in M, \, i=1,2,\ldots, n, j=1,2,\ldots, n \big\}$.
For $n=6, M=\{2,3,4,5\}$ see Fig. \[fig9\].
(1,2) circle (2pt) (2,2) circle (2pt) (3,2) circle (2pt) (4,2) circle (2pt) (5,2) circle (2pt) (6,2) circle (2pt); (1,2) .. controls (1.5,2.5) and (2.5,2.5) .. (2.95,2.05); (1,2) .. controls (2,2.8) and (3,2.8) .. (3.95,2.05); (1,2) .. controls (2,3.1) and (4,3.1) .. (4.95,2.05); (1,2) .. controls (2,3.4) and (5,3.4) .. (5.95,2.05); (3,2) .. controls (3.5,2.5) and (4.5,2.5) .. (4.95,2.05); (3,2) .. controls (4,2.8) and (5,2.8) .. (5.95,2.05); (4,2) .. controls (4.5,1.4) and (5.5,1.4) .. (5.95,1.95); (2,2) .. controls (2.5,1.4) and (3.5,1.4) .. (3.95,1.95); (2,2) .. controls (3,0.9) and (4,0.9) .. (4.95,1.95); (2,2) .. controls (2.5,0.6) and (5.5,0.6) .. (5.95,1.95); (1) at (1,1.75) [$a$]{}; (2) at (1.95,1.75) [$b$]{}; (3) at (3,1.75) [$c$]{}; (4) at (4,1.75) [$d$]{}; (5) at (5,1.75) [$e$]{}; (6) at (6.05,1.75) [$f$]{};
The adjacency matrix $A=\big(a_{ij}\big)_{\!\!\!\tiny\begin{array}{c}i\!\!=\!\!\overline{1,\!n}\\
j\!\!=\!\!\overline{1,\!n}\end{array}}$ of the graph is defined by:
$$a_{ij}=\left\{ \begin{array}{ll}
1, & \textrm{if } j-i\in M,\\
0, & \textrm{otherwise},
\end{array} \quad \textrm{ for } i=1,2,\ldots, n, j=1,2,\ldots, n.
\right.$$
Because the graph has no directed cycles, the element in row $i$ and column $j$ in $A^k$ (where $A^k=A^{k-1}A$, with $A^1=A$) will represent the number of length-$k$ directed paths from $a_i$ to $a_j$. If $I$ is the identity matrix (with elements equal to 1 only on the first diagonal, and 0 otherwise), let us define the matrix $R= (r_{ij})$: $$R= I+A+A^2+\cdots + A^k, \textrm{ where } A^{k+1}=O \, (\textrm{the null matrix}).$$ The $M$-complexity of a rainbow word is then $$K(n,M)= \sum_{i=1}^{n}{\sum_{j=1}^{n}{r_{ij}}}.$$ Matrix $R$ can be better computed using the <span style="font-variant:small-caps;">Warshall-Path</span> algorithm.
From $W$ we obtain easily $R=I+W$.
For example let us consider the graph in Fig. \[fig9\]. The corresponding adjacency matrix is:
$$A=\left(\begin{array}{cccccc}
0&0&1&1&1&1 \\
0&0&0&1&1&1 \\
0&0&0&0&1&1 \\
0&0&0&0&0&1 \\
0&0&0&0&0&0 \\
0&0&0&0&0&0 \\
\end{array}
\right)$$
After applying the <span style="font-variant:small-caps;">Warshall-Path</span> algorithm: $$W=\left(\begin{array}{cccccc}
0&0&1&1&2&3 \\
0&0&0&1&1&2 \\
0&0&0&0&1&1 \\
0&0&0&0&0&1 \\
0&0&0&0&0&0 \\
0&0&0&0&0&0 \\
\end{array}
\right),
\qquad
R=\left(\begin{array}{cccccc}
1&0&1&1&2&3 \\
0&1&0&1&1&2 \\
0&0&1&0&1&1 \\
0&0&0&1&0&1 \\
0&0&0&0&1&0 \\
0&0&0&0&0&1 \\
\end{array}
\right)$$ and then $K\big(6,\{2,3,4,5\}\big)=20,$ the sum of elements in $R$.
Using the <span style="font-variant:small-caps;">Warshall-Latin</span> algorithm we can obtain all nontrivial (with length at least 2) $M$-subwords of a given length-$n$ rainbow word $a_1a_2\cdots a_n$. Let us consider a matrix ${\cal A}$ with the elements $A_{ij}$ which are set of strings. Initially this matrix is defined as: $$A_{ij}=\left\{ \begin{array}{ll}
\{a_ia_j\}, & \textrm{if } j-i\in M,\\
\emptyset, & \textrm{otherwise},
\end{array} \quad \textrm{ for } \, i=1,2,\ldots, n, \, j=1,2,\ldots, n.
\right.$$
The set of nontrivial $M$-subwords is ${\displaystyle\bigcup_{i,j\in \{ 1,2,\ldots, n \} } W_{ij}}$.
For $n=8$, $M=\{3,4,5,6,7\}$ the initial matrix is:
$\left(\begin{array}{cccccccc}
\emptyset & \emptyset & \emptyset & \{ad\} & \{ae\} &\{af\} &\{ag\} &\{ah\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \{be\}& \{bf\} & \{bg\}& \{bh\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset & \{cf\}& \{cg\} & \{ch\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset & \{dg\} &\{dh\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \{eh\}\\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \emptyset \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \emptyset \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \emptyset \\
\end{array}
\right).
$
The result of the algorithm in this case is:
$\left(\begin{array}{cccccccc}
\emptyset & \emptyset & \emptyset & \{ad\} & \{ae\} &\{af\} &\{ag, adg\} &\{ah,adh,aeh\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \{be\}& \{bf\} & \{bg\}& \{bh,beh\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset & \{cf\}& \{cg\} & \{ch\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset & \{dg\} &\{dh\} \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \{eh\}\\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \emptyset \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \emptyset \\
\emptyset & \emptyset & \emptyset & \emptyset & \emptyset &\emptyset &\emptyset & \emptyset \\
\end{array}
\right).
$
Special paths in finite automata
--------------------------------
Let us consider a finite automaton $A=(Q, \Sigma,\delta, \{q_0\}, F)$, where $Q$ is a finite set of states, $ \Sigma$ the input alphabet, $\delta: Q\times \Sigma \rightarrow Q$ the transition function, $q_0$ the initial state, $F$ the set of finale states. In following we do not need to mark the initial and the finite states.
The transition function can be generalized for words too: $\delta(q,wa)=\delta(\delta(q,w), a)$, where $q\in Q, a \in \Sigma,w\in\Sigma^*$.
We are interesting in finding for each pair $p, q$ of states the letters $a$ for which there exists a natural $k\ge 1$ such that we have the transition $\delta(p,a^k) =q$ [@robert], i.e.: $$w_{pq}=\{a\in \Sigma \mid \exists k\ge 1, \delta(p,a^k) =q\}.$$
Instead of $\oplus$ we use here set union ($\cup$) and instead of $\odot$ set intersection ($\cap$).
[Warshall-Automata($A,n$)]{} *Input:* the adjacency matrix $A$; the no. of elements $n$\
*Output:* $W$ with sets of states\
1 ' $W \leftarrow A$\
2 ' [**for**]{} = $k \leftarrow 1$ [**to**]{} $n$\
3 ' [**do**]{} [**for**]{} = $i \leftarrow 1$ [**to**]{} $n$\
4 ' [**do**]{} [**for**]{} = $j \leftarrow 1$ [**to**]{} $n$\
5 ' [**do**]{} $w_{ij}\leftarrow w_{ij}\cup (w_{ik}\cap w_{kj})$\
6 ' [**return**]{} $W$
![An example of a finite automaton without indicating the initial and finite states\[aut1\]](aut.jpg)
The transition table of the finite automaton in Fig. \[aut1\] is:
$
\begin{array}{c|cccc}
\delta &a &b& c&d \\ \hline
q_1 &\{q_1,q_2\} &\{q_1\} & \emptyset& \{q_5\}\\
q_2 & \{q_1\}& \{q_3\}& \{q_2\}&\{q_3\} \\
q_3 & \emptyset& \{q_4\}& \emptyset& \emptyset\\
q_4 & \emptyset& \{q_5\}& \emptyset& \emptyset\\
q_5 & \emptyset & \{q_2\}& \emptyset& \emptyset
\end{array}
$
Matrices for graph in Fig. \[aut1\] are the following:
$
A=\left(\begin{array}{ccccc}
\{a,b\}&\{a\} &\emptyset& \emptyset&\{d\} \\
\{a\}& \{c\}& \{b,d\}&\emptyset &\emptyset \\
\emptyset& \emptyset&\emptyset &\{b\} &\emptyset \\
\emptyset&\emptyset &\emptyset &\emptyset &\{b\} \\
\emptyset & \{b\} &\emptyset&\emptyset &\emptyset
\end{array}
\right)
$
$
W=\left(\begin{array}{ccccc}
\{a,b\}&\{a\} &\emptyset& \emptyset&\{d\} \\
\{a\}& \{a,b,c\}& \{b,d\}&\{b\} &\{b\} \\
\emptyset& \{b\}&\{b\} &\{b\} &\{b\} \\
\emptyset& \{b\}&\{b\} &\{b\} &\{b\} \\
\emptyset& \{b\}&\{b\} &\{b\} &\{b\} \\
\end{array}
\right)
$
For example $\delta(q_2,bb)=q_4$, $\delta(q_2,bbb)=q_5$, $\delta(q_2,bbbb)=q_2$, $\delta(q_2,c^k)=q_2$ for $k\ge 1$.
[99]{}
S. [Baase]{}, *Computer Algorithms: Introduction to Design and Analysis,* Second edition, Addison–Wesley, 1988.
C. [Elzinga](https://scholar.google.nl/citations?user=0feufd8AAAAJ&hl=nl), H. Wang, [Versatile string kernels](https://www.sciencedirect.com/science/article/pii/S030439751300460X?via%3Dihub), *Theor. Comput. Sci.*, **495,** 1 (2013) 50–65.
Z. [Kása](http://www.ms.sapientia.ro/~kasa), On [scattered subword complexity](http://acta.sapientia.ro/acta-info/C3-1/info31-6.pdf) [**3,**](Acta Universitatis Sapientiae, Informatica,) 1 (2011) 127–136.
P. Robert, J. Ferland, Généralisation de l’algorithme de Warshall, *Revue Française d’Informatique et de Recherche Opérationnelle* **2,** 7 (1968) 71–85. <http://www.numdam.org/item/?id=M2AN_1968__2_1_71_0>
Z. A. [Vattai](http://www.ekt.bme.hu/Szemallo/Gbvattai.shtml): Floyd-Warshall again, <http://www.ekt.bme.hu/Cikkek/54-Vattai_Floyd-Warshall_Again.pdf>
S. [Warshall](https://en.wikipedia.org/wiki/Stephen_Warshall), A [ theorem on boolean matrices](http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Warshall1962.pdf), [*Journal of the ACM*]{}, **9,** 1 (1962) 11–12.
|
---
abstract: 'By embedding a $\cal PT$-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between $\cal{PT}$-symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of $\cal PT$-symmetric systems, with the pre-selected state and its post-selected state resident in the dilated conventional system. Typically in quantum information theory, by projecting out the irrelevant degrees and projecting onto the subspace, even local broken $\cal PT$-symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.'
author:
- Minyi Huang
- 'Ray-Kuang Lee'
- Lijian Zhang
- 'Shao-Ming Fei'
- Junde Wu
title: 'Simulating broken $\cal PT$-symmetric Hamiltonian systems by weak measurement'
---
[*Introduction*]{} Generalizing the conventional Hermitian quantum mechanics, Bender and his colleagues established the Parity ($\cal P$)-time ($\cal T$)-symmetric quantum mechanics in 1998 [@Bender98]. With the additional degree of freedom from a non-conservative Hamiltonian, as well as the existence of exceptional points between unbroken and broken $\cal PT$-symmetries, optical $\cal PT$-symmetric devices have been demonstrated with many useful applications [@El-OL; @Makris-PRL; @Guo-PRL; @Ruter-NP; @Chang-NP; @Tang-NP]. Although calling for more caution on physical interpretations, especially on the consistency problem of local $\mathcal{PT}$-symmetric operation and the no-signaling principle [@Lee14], $\cal PT$-symmetric quantum mechanics has been stimulating our understanding on many interesting problems such as spectral equivalence [@Dorey], quantum brachistochrone [@Gunther-PRA] and Riemann hypothesis [@Bender-17].
Compared with the Dirac inner product in conventional quantum mechanics, $\cal PT$-symmetric quantum theory can be well manifested by the $\eta$-inner product [@Mostafazadeh; @Mostafazadeh-Geom]. In the broken $\cal PT$-symmetry case, the $\eta$-inner product of a state with itself can be negative, which makes the broken $\cal PT$-symmetric quantum systems a complete departure from conventional quantum mechanics. While in the unbroken $\cal PT$-symmetry case, the $\eta$-inner product presents a completely analogous physical interpretation to the Dirac inner product, giving rise to many similar properties between $\cal PT$-symmetric and conventional quantum mechanics. Recent works also show that the $\eta$-inner product is tightly related to the properties of superposition and coherence in conventional quantum mechanics [@JPA].
Despite the original motivation to build a new framework of quantum theory, researchers are aware of the importance of simulating $\cal PT$-symmetric systems with conventional quantum mechanics. It will help explore the properties and physical meaning of $\cal PT$-symmetric quantum systems. On this issue, one should answer the question in what sense a quantum system can be viewed as $\cal PT$-symmetric. One approach, initialized by G[ü]{}nther and Samsonov, is to embed unbroken $\cal PT$-symmetric Hamiltonians into higher dimensional Hermitian Hamiltonians [@Gunther-PRL; @Huang; @Ueda]. By dilating the system to a large Hermitian one and projecting out the ancillary system, this paradigm successfully simulates the evolution of unbroken $\cal PT$-symmetric Hamiltonians. Such a way, inspired by Naimark dilation and typical ideas in quantum simulation, endows direct physical meaning of $\cal PT$-symmetric quantum systems in the sense of open systems. However, the simulation of broken $\cal PT$-symmetric systems is still in suspense, due to its essential distinctions with conventional quantum systems.
In this Letter, we illustrate the simulation for broken $\cal PT$-symmetric systems based on weak measurement [@AAV]. For a system weakly coupled to the apparatus, the pointer state will be shifted by the weak value when a weak measurement is performed. The weak value, tightly related to the non-classical features of quantum mechanics, such as the Hardy’s paradox [@Aharonov], three box paradox [@Resch] and Leggett-Garg inequalities [@Palacios], can take values beyond the expected values of an observable, and even be a complex number. The weak measurement theory has provided new ways to measure geometric phases [@Hosten; @Sjogvist; @Kobayashi; @Lijian] and non-Hermitian systems [@Pati; @Vaidman], as well as to amplify signals as a sensitive estimation of small evolution parameters [@Lundeen; @Starling-2010b; @Brunner]. Our aim is to propose a concrete scenario in which the quantum system can be viewed as $\cal PT$-symmetric by utilizing the weak measurement. Our result reveals the connections between $\cal PT$-symmetry and the weak measurement theory, providing the important missing point for the simulation of broken $\cal {PT}$-symmetric quantum systems.
[*Generalized embedding of $\cal PT$-symmetric systems*]{} Consider $n$-dimensional discrete quantum systems. A linear operator $P$ is said to be a parity operator if ${P}^2=I$, where $I$ denotes the $n\times n$ identity matrix. An anti-linear operator $T$ is said to be a time reversal operator if $T\overline{T}=I$ and $PT=T\overline{P}$, where $\overline{T}$ ($\overline{P}$) stands for the complex conjugation of $T$ ($P$). A Hamiltonian $H$ is said to be $PT$-symmetric if $HPT=PT\overline{H}$ [@note]. $H$ is called unbroken $\cal PT$-symmetric if it is diagonalizable and all of its eigenvalues are real. Otherwise, $H$ is called broken $\cal PT$-symmetric.
In quantum mechanics, a Hamiltonian $H$ gives rise to a unitary evolution of the system. Let $\phi_1$ and $\phi_2$ be two states. On can introduce a Hermitian operator $\eta$ to define the $\eta$-inner product by ${\langle\phi_1|\phi_2\rangle}_{\eta}={\langle\phi_1|\eta|\phi_2\rangle}$. With respect to the $\eta$-inner product, $H$ presents a unitary evolution if and only if ${H}^\dag\eta=\eta H$ [@Mostafazadeh; @Mostafazadeh-Geom; @Deng; @Mannheim; @Horn], where ${H}^\dag$ denotes the conjugation and transpose of $H$. Here, $\eta$ is said to be the metric operator of $H$. Moreover, for a generic $\cal PT$-symmetric operator $H$ and its metric operator $\eta$, there always exist some matrix $\Psi'$ such that $\Psi'^{-1}H\Psi'=J$ and $\Psi'^\dag\eta\Psi'=S$, where $$\label{cano2}
J=diag(
J_{n_1}(\lambda_1,\overline{\lambda}_1),...,J_{n_p}(\lambda_p,\overline{\lambda}_p),J_{n_{p+1}}(\lambda_{p+1}),...,J_r(\lambda_r)),$$ $J_{n_k}(\lambda_k,\overline{\lambda}_k)=
\bpm\begin{smallmatrix} J_{n_k}(\lambda_k)&0\\0&J_{n_k}(\overline{\lambda_k})\end{smallmatrix}\epm$, $J_{n_j}(\lambda_j)$ are the Jordan blocks, $\lambda_1,
\cdots, \lambda_p$ are complex numbers and $\lambda_{p+1},
\cdots, \lambda_r$ are real numbers, $$S=
diag(S_{2n_1},...,S_{2n_p},\epsilon_{n_q} S_{n_q},...,\epsilon_{n_r} S_{n_r}),\label{cano2'}$$ $n_i$ denote the orders of Jordan blocks in Eq. (\[cano2\]), i.e., $S_{k}=\begin{pmatrix}\begin{smallmatrix}&&1\\&\iddots&\\1&&\end{smallmatrix}\end{pmatrix}_{k\times k}$ and $\epsilon_i=\pm 1$ is uniquely determined by $\eta$ [@Huang; @Gohberg]. For convenience, we only consider the situations in which $\epsilon_i=1$. In this case, $S$ is a permutation matrix and $S^2=I$. Note that $S$ can be equal to $I$ if and only if $H$ is unbroken $\cal PT$-symmetric [@Huang]. Henceforth we always assume $S=I$ in the unbroken case. The following theorem gives an important property of $\cal PT$-symmetric Hamiltonians.
\[thm1\] Let $H$ be an $n\times n$ $\cal PT$-symmetric matrix and $\eta$ be the metric matrix of $H$. Let $J$ and $S$ be matrices in Eqs (\[cano2\]) and (\[cano2’\]). Then, there exist $n\times n$ invertible matrices $\Psi$, $\Xi$, $\Sigma$ and a $2n\times 2n$ Hermitian matrix $\tilde{H}$ such that for $\tilde{\Psi}=\bpm\Psi\\ \Xi\epm$ and $\tilde{\Phi}=\bpm\Psi\\ \Sigma\epm$, the following equations hold, $$\begin{aligned}
\tilde{\Phi}^\dag\tilde{\Psi}=S,~~~\tilde{\Phi}^\dag\tilde{H}\tilde{\Psi}=SJ.\label{23}\end{aligned}$$
As was discussed, there exist a matrix $\Psi'$ such that $\Psi'^{-1}H\Psi'=J$ and $\Psi'^\dag\eta\Psi'=S$ [@Huang; @Gohberg]. Since $\Psi'^\dag\Psi'>0$, there always exits a positive number $c$ such that $c^2\Psi'^\dag \Psi' >I$. Set $\Psi=c\Psi'$. Since $\Psi^\dag\Psi>I\geqslant S$, $\Psi^\dag\Psi-S$ is invertible.
Let $\Xi$ be an $n\times n$ invertible matrix. Taking $\Sigma=(\Xi^{-1})^\dag(S-\Psi^\dag\Psi)$, $\eta=(\Psi^{-1})^\dag S\Psi^{-1}$, $H_1=\eta H$, $H_2=(\Psi^\dag)^{-1}(\Xi)^\dag$ and $H_4=-H_2^\dag\Psi\Xi^{-1}-(\Sigma^\dag)^{-1}\Psi^\dag H_2$, one can directly verify that $\tilde{H}=\bpm H_1&H_2\\H_2^\dag&H_4\epm$ is Hermitian and Eq. (\[23\]) holds.
Theorem \[thm1\] actually gives out the inner product structure of $H$ in a subspace. Note that the matrix $\Psi$ in Theorem \[thm1\] can be written as $\Psi=({|\psi_1\rangle},\cdots,{|\psi_n\rangle})$, where the column vectors $\{{|\psi_i\rangle}\}$ form a linear basis of $\mathbb C^n$. Similarly, $\Xi=({|\xi_1\rangle},\cdots,{|\xi_n\rangle})$ and $\Sigma=({|\sigma_1\rangle},\cdots,{|\sigma_n\rangle})$. Correspondingly we have $\tilde{\Psi}=({|\tilde{\psi}_1\rangle},\cdots,{|\tilde{\psi}_n\rangle})$ and $\tilde{\Phi}=({|\tilde{\phi}_1\rangle},\cdots,{|\tilde{\phi}_n\rangle})$, where ${|\tilde{\psi}_i\rangle}=\bpm{|\psi_i\rangle}\\ {|\xi_i\rangle}\epm$ and ${|\tilde{\phi}_i\rangle}=\bpm{|\psi_i\rangle}\\ {|\sigma_i\rangle}\epm$. Moreover, $\tilde{\Phi} S=({|\tilde{\mu}_1\rangle},\cdots,{|\tilde{\mu}_n\rangle})=({|\tilde{\phi}_{s(1)}\rangle},\cdots,{|\tilde{\phi}_{s(n)}\rangle})$, where $S$ is the permutation matrix in Theorem \[thm1\], and $s$ is the permutation induced by $S$. Similarly, we can write $\Psi S=({|\mu_1\rangle},\cdots,{|\mu_n\rangle})$, where ${|\mu_i\rangle}={|\psi_{s(i)}\rangle}$. From the definition of ${|\tilde{\mu}_i\rangle}$, we have ${\langle\tilde{\mu}_i|\tilde{\psi}_j\rangle}=(S\tilde{\Phi}^\dag\tilde{\Psi})_{ij}$ and ${\langle\tilde{\mu}_i|\tilde{H}|\tilde{\psi}_j\rangle}=(S\tilde{\Phi}^\dag\tilde{H}\tilde{\Psi})_{ij}$. According to Eq. (\[23\]), we have $$\begin{aligned}
\label{ijiHj}
{\langle\tilde{\mu}_i|\tilde{\psi}_j\rangle}=\delta_{i,j},~~~{\langle\tilde{\mu}_i|\tilde{H}|\tilde{\psi}_j\rangle}=J_{i,j},\end{aligned}$$ where $J_{i,j}$ is the $(i,j)$-th entry of $J$.
On the other hand, note that the metric matrix $\eta$ of $H$ is $(\Psi^\dag)^{-1}S\Psi^{-1}$. Thus we have the following relations between the Dirac and $\eta$-inner products $$\begin{aligned}
&&{\langle\tilde{\mu}_i|\tilde{\psi}_j\rangle}={\langle\mu_i|\psi_j\rangle}_{\eta},\label{ij'}\\
&&{\langle\tilde{\mu}_i|\tilde{H}|\tilde{\psi}_j\rangle}={\langle\mu_i|H|\psi_j\rangle}_{\eta},\label{iHj'}\end{aligned}$$ where ${\langle\mu_i|H|\psi_j\rangle}_{\eta}={\langle\mu_i|\eta H|\psi_j\rangle}$. The results show that there exist two different basis with the same projections onto the subspace of the $\cal PT$-symmetric system, with respect to the $\eta$-inner product. When confined to the subspace, the Hermitian Hamiltonian $\tilde{H}$ in large space has the same effect as a $\cal PT$-symmetric Hamiltonian $H$, in the sense of this $\eta$-inner product.
[*Simulation of $\cal PT$-symmetric Hamiltonian systems*]{} To infer a quantum system is $\cal PT$-symmetric, it is sufficient to identify the Hamiltonian and its inner product structure. In the weak measurement formalism, one starts by pre-selecting an initial state ${|\varphi_i\rangle}$. The target system is coupled to the measurement apparatus, which is in a pointer state ${|P\rangle}$. Usually, ${|P\rangle}=(2\pi\Delta^2)^{-\frac{1}{4}}exp(-\frac{Q^2}{4\Delta^2})$, a Gaussian state with $\Delta$ its standard deviation. Let $A$ be an observable of the system and $M$ be that of the apparatus, conjugate to $Q$ [@AAV]. The interaction Hamiltonian between the system and apparatus is $H_{int}=f(t)A\otimes M$, with interaction strength $g=\int f(t)dt$. The state evolves as ${|\varphi_i\rangle}\otimes{|P\rangle}\rightarrow e^{-ig A\otimes M}{|\varphi_i\rangle}\otimes{|P\rangle}$. Now if the system satisfies the weak condition that $g /\Delta$ is sufficiently small, then for a post-selected state ${|\varphi_f\rangle}$ that ${\langle\varphi_f|\varphi_i\rangle}\neq 0$, one has ${\langle\varphi_f|}e^{-ig A\otimes M}{|\varphi_i\rangle}{|P\rangle}\approx{\langle\varphi_f|\varphi_i\rangle} e^{-ig{\langleA\rangle}_wM}{|P\rangle}
={\langle\varphi_f|\varphi_i\rangle}(2\pi\Delta^2)^{-\frac{1}{4}}exp(-\frac{(Q-g{\langleA\rangle}_w)^2}{4\Delta^2})$, where ${\langleA\rangle}_w=\frac{{\langle\varphi_f|A|\varphi_i\rangle}}{{\langle\varphi_f|\varphi_i\rangle}}$ is called the weak value. That is, the state is shifted by $g{\langleA\rangle}_w$. Thus the weak value ${\langleA\rangle}_w$ can be read out experimentally, as a generalization of the eigenvalues in Von Neumann measurement [@Dressel].
From Eq. (\[ijiHj\]), we have $\lambda_i=J_{i,i}={\langle\tilde{\mu}_i|\tilde{H}|\tilde{\psi}_i\rangle}=\frac{{\langle\tilde{\mu}_i|\tilde{H}|\tilde{\psi}_i\rangle}}{{\langle\tilde{\mu}_i|\tilde{\psi}_i\rangle}}$. Therefore, the eigenvalues of $H$ can be obtained via a weak measurement, by pre-selecting the vector ${|\tilde{\psi}_i\rangle}$ and post-selecting the vector ${|\tilde{\mu}_i\rangle}$. This observation implies that one can use weak measurement to simulate the measurements on a $\cal PT$-symmetric system.
In conventional quantum mechanics, the expectation value of a Hermitian Hamiltonian $H_0=\sum_i \lambda_i{|u_i\rangle}{\langleu_i|}$ with respect to a sate ${|\psi_0\rangle}=\sum_i d_i{|u_i\rangle}$ is given by the inner product ${\langle\psi_0|H_0|\psi_0\rangle}$. For a $\cal PT$-symmetric Hamiltonian system with the metric matrix $\eta$, the expectation value of a Hamiltonian $H$ with respect to a state ${|u\rangle}=\sum_i a_i{|\psi_i\rangle}$ is instead given by ${\langleu|H|u\rangle}_\eta$. Given two vectors ${|v\rangle}=\sum_i b_i{|\mu_i\rangle}$ and ${|w\rangle}=\sum_i c_i{|\psi_i\rangle}$ of the $\cal PT$-symmetric system. Let ${|\tilde{v}\rangle}=\sum_i b_i{|\tilde{\mu}_i\rangle}$ (unnormalized for convenience) and ${|\tilde{w}\rangle}=\sum_i c_i{|\tilde{\psi}_i\rangle}$ be two vectors in the extended system. It follows from Eq. (\[iHj’\]) that ${\langlev|H|w\rangle}_\eta={\langle\tilde{v}|\tilde{H}|\tilde{w}\rangle}$. Assume that ${|u\rangle}$ satisfies the condition ${\langleu|u\rangle}_\eta=\pm 1$. Now take two states ${|\tilde{u}_1\rangle}=\sum_i a_{s(i)}{|\tilde{\mu}_i\rangle}$ and ${|\tilde{u}_2\rangle}=\sum_i a_i{|\tilde{\psi}_i\rangle}$, whose projections to the $\cal PT$-symmetric subspace are both ${|u\rangle}$. Then we have $$\frac{{\langleu|H|u\rangle}_\eta}{{\langleu|u\rangle}_\eta}=\frac{{\langle\tilde{u}_1|\tilde{H}|\tilde{u}_2\rangle}}{{\langle\tilde{u}_1|\tilde{u}_2\rangle}}.$$ Therefore, confined to the $\cal PT$-symmetric subspace, a weak measurement can completely describe the expectations of $H$.
In conventional quantum mechanics, when an eigenvalue is detected, the measured state collapses to the corresponding eigenstate. However, the problem in $\cal PT$-symmetric system is subtle. According to Eq. (\[ij’\]), ${\langle\psi_i|\psi_i\rangle}_\eta\neq 0$ only if $i=s(i)$. This observation makes it reasonable to assume that for any vector ${|u\rangle}=\sum_i a_i{|\psi_i\rangle}$ satisfying ${\langleu|u\rangle}_\eta\neq 0$, if $a_i\neq 0$, then $a_{s(i)}\neq 0$. That is, if ${\langleu|u\rangle}_\eta\neq 0$, its vector components of ${|\psi_i\rangle}$ and ${|\psi_{s(i)}\rangle}$ take zero or nonzero values simultaneously, while the eigenvalues associated with $\psi_i$ and $\psi_{s(i)}$ are either equal or complex conjugations. In this case, one can generalize the detection of an eigenvalue of $\lambda_i$ in conventional quantum mechanics to the following. For ${|u\rangle}=\sum_i a_i{|\psi_i\rangle}$, if the value of $$\frac{a_i\overline{a_{s(i)}}\lambda_i+\overline{a_i}a_{s(i)}\overline{\lambda}_i}{a_i\overline{a_{s(i)}}+\overline{a_i}a_{s(i)}}$$ is detected [@note; @3], the state ${|u\rangle}$ will collapse to $$\frac{a_i{|\psi_i\rangle}+a_{s(i)}{|\psi_{s(i)}\rangle}}{|a_i\overline{a_{s(i)}}+a_{s(i)}\overline{a_i}|^{1\over 2}}.$$ Apparently, when $i=s(i)$, the state ${|u\rangle}$ will collapse to ${|\psi_i\rangle}$, similar to the case of conventional quantum mechanics. Note that $i=s(i)$ only if the system is unbroken $\cal PT$-symmetric, for which it is analogous to conventional quantum mechanics and such an analogy in state collapse is not unexpected.
By pre- and post-selecting the states, we see that the weak measurements can successfully simulate an arbitrary $\eta$-inner product. Furthermore, when confined to the subspace, the measurement results actually extract the same information as a $\cal PT$-symmetric Hamiltonian system. Such information help us eventually infer that the subsystem is $\cal PT$-symmetric.
[*Discussions and conclusion*]{} We further discuss the mechanism and physical implications related to the weak measurement paradigm, by comparing it with the embedding paradigm [@Gunther-PRL; @Huang]. The essence of the embedding paradigm is to realize the evolution of a $\cal PT$-symmetric Hamiltonian, by evolving the state under the Hermitian Hamiltonian in the large space and then project it to the subspace. The key to this paradigm can be mathematically described as follows [@Huang]: For a given $n\times n$ unbroken $\cal PT$-symmetric Hamiltonian $H$, find a $2n\times 2n$ Hermitian matrix $\tilde{H}$, $n\times n$ invertible matrices $\Psi$, $\Xi$ so that $\tilde{\Psi}^\dag\tilde{\Psi}=I$ and the following equations $$\begin{aligned}
\label{3''4''}
e^{-it\tilde{H}}\tilde{\Psi}=\tilde{\Psi}e^{-itJ},~~~e^{-itH}\Psi=\Psi e^{-itJ}\end{aligned}$$ hold, where $\tilde{\Psi}=\bpm \Psi\\ \Xi \epm$. The equations are actually equivalent to the following conditions [@note; @1]: $$\begin{aligned}
\label{2'3'4'}
\tilde{\Psi}^\dag\tilde{\Psi}=I,~~\tilde{H}\tilde{\Psi}=\tilde{\Psi}J,~~H\Psi=\Psi J.\end{aligned}$$ Equation (\[3”4”\]) ensures that the unitary evolution $\tilde{U}(t)$ gives the evolution $U(t)$ of a $\cal PT$-symmetric Hamiltonian $H$ in a subspace. In this sense, the embedding paradigm gives a natural way of simulation. Nevertheless, in the broken $\cal PT$-symmetric case, the solutions do not exist [@Huang]. In fact, Eq. (\[23\]) is mathematically a generalization of Eq. (\[2’3’4’\]) [@note; @2]. Like the case of the embedding paradigm, it is natural to further require that $\tilde{\Phi}^\dag e^{-it\tilde{H}}\tilde{\Psi}=Se^{-itJ}$, so that $e^{-it\tilde{H}}$ gives the same effect as $e^{-itH}$ in the subspace. However, such a requirement cannot be satisfied for broken $\cal PT$-symmetry, which is obvious from the unboundedness of $Se^{-itJ}$.
However, consider sufficiently small time $t\in [0,\epsilon]$. We have ${|\tilde{u}(t)\rangle}=e^{-it\tilde{H}}{|\tilde{u}\rangle}\approx(I-it\tilde{H}){|\tilde{u}\rangle}$. On the other hand, ${|u(t)\rangle}=e^{-itH}{|u\rangle}\approx(I-itH){|u\rangle}$. Now equations Eqs. (\[ij’\]) and (\[iHj’\]) insure that when confined to the subspace, ${|\tilde{u}(t)\rangle}$ is equivalent to ${|u(t)\rangle}$ in the sense of $\eta$-inner product . This observation implies that $\cal PT$-symmetric quantum systems can be well approximated in a sufficiently small time evolution, by choosing two different sets of basis $\{{|\tilde{\phi}_i\rangle}\}$ and $\{{|\tilde{\psi}_i\rangle}\}$ with the same components in the subspace, which can be realized by weak measurement. Here instead of the small time interval, the weak condition that $g/\Delta$ is sufficiently small ensures the approximation. The weak measurement paradigm can be viewed as a generalization of the embedding paradigm, due to the fact that Eq. (\[2’3’4’\]) is a special case of Eq. (\[23\]) in the $\cal PT$-symmetric unbroken case. Hence, the Hamiltonian $\tilde{H}$ in the embedding paradigm can also be utilized in the weak measurement approach, although the embedding paradigm itself does not work. Comparing our approach with that in [@Pati], where one obtains the expected value of a Hamiltonian in the Dirac inner product by using the polar decomposition, our method lays emphasis on the properties of a $\cal PT$-symmetric Hamiltonian with respect to the $\eta$-inner product.
In summary, we have proposed a weak measurement paradigm to investigate the behaviors of broken $\cal PT$-symmetric Hamiltonian systems. By embedding the $\cal PT$-symmetric system into a large Hermitian system and utilizing weak measurements, we have shown how a $\cal PT$-symmetric Hamiltonian can be simulated. Our paradigm may shine new light on the study of $\cal {PT}$-symmetric quantum mechanics and its physical implications and applications.
This work is supported by National Natural Science Foundation of China (11171301, 11571307, 11690032, 61490711, 61877054 and 11675113), National Key R&D Program of China under Grant No. 2018 YFA0306202 and the NSF of Beijing under Grant No. KZ201810028042.
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Actually, this value is $\frac{{\langlea_i\psi_i+a_{s(i)}\psi_{s(i)}|H|a_i\psi_i+a_{s(i)}\psi_{s(i)}\rangle}_\eta}{a_i\overline{a_{s(i)}}+\overline{a_i}a_{s(i)}}$, which reduces to ${\langle\psi_i|H|\psi_i\rangle}_\eta=\lambda_i$ if $i=s(i)$ (only one vector $\psi_i$ considered).
Equation (\[3”4”\]) is actually the matrix version of the embedding in [@Huang]. Denote $\Xi=\tau \Psi$. Then (\[2’3’4’\]) reduces to $H_1+H_2\tau=H$ and $H_2^\dag+H_4\tau=\tau H$, which gives the equivalent description of the embedding property. A concrete solution to (\[3”4”\]) can also be found in [@Ueda].
When unbroken $\cal PT$-symmetric, it is always possible to take $\Xi=\Sigma$ and $S=I$, Eq. (\[2’3’4’\]) is a special case of Eq. (\[23\]).
Supplemental Material:
======================
An example
----------
To illustrate the validity of our theoretic results, an example is given based on the two dimensional model proposed by Bender et al. [@Bender07]: $$H=\begin{bmatrix}
re^{i\theta} &s\\
s&re^{-i\theta}
\end{bmatrix},~~~~~~P=\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix},
~~~~~~T=\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}.$$ Here, as $HPT=PT\overline{H}$, the Hamiltonian $H$ is $\cal PT$-symmetric. In particular, when $\Delta=s^2-r^2\sin^2\theta< 0$, $H$ is broken $\cal PT$-symmetric. The corresponding eigenvalues and eigenvectors (without normalization) are: $$\lambda_1=r\cos\theta+i\sqrt{-\Delta},~~~\lambda_2=r\cos\theta-i\sqrt{-\Delta}.$$ $$\psi_1=\begin{bmatrix}i(\sqrt{-\Delta}+r\sin\theta)\\s \end{bmatrix}, \psi_2=\begin{bmatrix}-s \\ i(\sqrt{-\Delta}+r\sin\theta)\end{bmatrix}.$$
Then, by denoting the eigenvectors in the matrix form, we have: $$\begin{aligned}
\Psi=[\psi_1,\psi_2]=\begin{bmatrix}i(\sqrt{-\Delta}+r\sin\theta) & -s\\s &i(\sqrt{-\Delta}+r\sin\theta)\end{bmatrix},\end{aligned}$$ It can be verified that $\Psi^{-1}H\Psi=J$ and $\Psi^\dag\eta\Psi=S$, where $$\begin{aligned}
J=\begin{bmatrix}r\cos\theta+i\sqrt{-\Delta}&0\\0& r\cos\theta-i\sqrt{-\Delta} \end{bmatrix},~~~S=\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}.\end{aligned}$$
Now, with the short-handed notations, $$u=\sqrt{-\Delta}+r\sin\theta,~a=2r\sin\theta.$$ we have $$\begin{aligned}
&&\Psi=\begin{bmatrix}iu & -s\\s &iu \end{bmatrix},~\Psi^{-1}=\frac{1}{s^2-u^2}\begin{bmatrix}iu & s\\-s &iu \end{bmatrix},\\
&&S-\Psi^\dag\Psi=\begin{bmatrix}-au&1-2sui\\1+2sui&-au\end{bmatrix},\\
&&det(S-\Psi^\dag\Psi)=-4\Delta u^2-1.\end{aligned}$$ For simplicity, we also assume $-4\Delta u^2-1\neq 0$. Otherwise, as showed in the proof of Theorem $1$, we can take a constant value $c$ such that $S-c^2\Psi^\dag\Psi$ is invertible, i.e., with $c\Psi$ instead of $\Psi$. Now $$(S-\Psi^\dag\Psi)^{-1}=\frac{1}{-4\Delta u^2-1}\begin{bmatrix}-au & -1+2sui \\-1-2sui &-au\end{bmatrix}.$$
To introduce our simulating scenario, we take $\Psi=\Xi$ for convenience, as $\Xi$ is arbitrary. By using the construction in Theorem $1$, one can have $\Sigma=(\Xi^{-1})^\dag(S-\Psi^\dag\Psi)$, $\eta=(\Psi^{-1})^\dag S\Psi^{-1}$, $H_1=\eta H$, $H_2=(\Psi^\dag)^{-1}(\Xi)^\dag$ and $H_4=-H_2^\dag\Psi\Xi^{-1}-(\Sigma^\dag)^{-1}\Psi^\dag H_2
=-I-\Xi(S-\Psi^\dag\Psi)^{-1}\Xi^\dag$.
Then, direct calculations give us $$\begin{aligned}
&&\tilde{H}=\begin{bmatrix}A_1&A_2&1&0\\A_3&A_4&0&1 \\ 1&0& -1-KB_1&-KB_2\\ 0&1&-KB_3&-1-KB_4\end{bmatrix},\\
&&\tilde{\Psi}=\begin{bmatrix}iu&-s\\s&iu\\iu&-s\\s&iu\end{bmatrix},\\
&&\tilde{\Phi}^\dag=\begin{bmatrix}-iu&s&iu-K_2s &iK_2 u-s\\-s&-iu&iK_2 u+s&iu+K_2s\end{bmatrix},\end{aligned}$$ with the notations $$\begin{aligned}
&&K=\frac{1}{-4\Delta u^2-1},~~K_2=\frac{1}{s^2-u^2},\\
&&A_1=\frac{s}{u^2-s^2},~~~A_2=\frac{re^{-i\theta}}{u^2-s^2},\\
&&A_3=\frac{re^{i\theta}}{u^2-s^2},~~~A_4=\frac{s}{u^2-s^2},\\
&&B_1=B_4=-(u^2-s^2)^2,\\
&&B_2=B_3=s^2-u^2.\end{aligned}$$
Based on these solutions, it can be easily verified that $\tilde{\Phi}^\dag\tilde{\Psi}=S,\tilde{\Phi}^\dag\tilde{H}\tilde{\Psi}=SJ$, such that $$\begin{aligned}
{\langle\tilde{\phi}_i,e^{-it\tilde{H}}\tilde{\psi}_j\rangle}\approx \tilde{\phi}_i^\dag(I-it\tilde{H})\tilde{\psi}_j
=\psi_i^\dag\eta(I-itH)\psi_j\approx{\langle\psi_i,e^{-itH}\psi_j\rangle}_\eta.\nonumber \\
\label{approx}\end{aligned}$$
Thus, under the $\eta$-inner product, the reduced system resembles a broken $\cal PT$-symmetric one. In order to illustrate the validity of our simulating paradigm, we introduce four parameters defined below: $$\begin{aligned}
&&Z_{11}=|{\langle\tilde{\phi}_1,e^{-it\tilde{H}}\tilde{\psi}_1\rangle}|,\\
&&Z_{22}=|{\langle\tilde{\phi}_2,e^{-it\tilde{H}}\tilde{\psi}_2\rangle}|,\\
&&Z_{12}=|{\langle\tilde{\phi}_1,e^{-it\tilde{H}}\tilde{\psi}_2\rangle}-{\langle\psi_1,e^{-itH}\psi_2\rangle}_\eta||{\langle\psi_1,e^{-itH}\psi_2\rangle}_\eta|^{-1},\nonumber\\
&&\\
&&Z_{21}=|{\langle\tilde{\phi}_2,e^{-it\tilde{H}}\tilde{\psi}_1\rangle}-{\langle\psi_2,e^{-itH}\psi_1\rangle}_\eta||{\langle\psi_2,e^{-itH}\psi_1\rangle}_\eta|^{-1}.\nonumber\\\end{aligned}$$
The reason $Z_{11}$ an $Z_{22}$ have different forms from $Z_{12}$ and $Z_{21}$ is that ${\langle\psi_1,e^{-itH}\psi_1\rangle}_\eta={\langle\psi_2,e^{-itH}\psi_2\rangle}_\eta=0$, but ${\langle\psi_1,e^{-itH}\psi_2\rangle}_\eta \neq 0, \quad {\langle\psi_2,e^{-itH}\psi_1\rangle}_\eta\neq 0.
$ With the definitions above, apparently, $Z_{ij}$ reflects the difference between ${\langle\tilde{\phi}_i,e^{-it\tilde{H}}\tilde{\psi}_j\rangle}$ and ${\langle\psi_i,e^{-itH}\psi_j\rangle}_\eta$, as shown in FIG. S1.
![Direct calculation on the parameters $Z_{ij}$. The corresponding values of $Z_{ij}$ given in Eqs. (S10-S13) are shown in (a-d), respectively, for the range of $r=\sqrt{2},\theta=\frac{\pi}{4}, t\in[0,0.2],s\in[0,0.2]$.](PT-S1.png){width="16.0cm"}
In the range of $r=\sqrt{2},\theta=\frac{\pi}{4}, t\in[0,0.2],s\in[0,0.2]$, the differences between ${\langle\tilde{\phi}_i,e^{-it\tilde{H}}\tilde{\psi}_i\rangle}$ and ${\langle\psi_i,e^{-itH}\psi_i\rangle}_\eta$ are less than $2 \times 10^{-2}$; while the relative differences between ${\langle\tilde{\phi}_i,e^{-it\tilde{H}}\tilde{\psi}_j\rangle}$ and ${\langle\psi_i,e^{-itH}\psi_j\rangle}_\eta$ are less than $6 \times 10^{-2}$.
In addition, when $t\rightarrow 0$, $Z_{ij}$ and thus ${\langle\tilde{\phi}_i,e^{-it\tilde{H}}\tilde{\psi}_j\rangle}-{\langle\psi_i,e^{-itH}\psi_j\rangle}_\eta$, tend to zero. This means that Eq. (S9) is valid for a sufficiently small time interval $t$, which supports our theoretical conclusion. We want to emphasize that $e^{-it\tilde{H}}$ behaves like a broken $\cal PT$-symmetric evolution under the $\eta$-inner product, but not under the standard Dirac inner product. Hence in this case, the projection of $e^{-it\tilde{H}}\tilde{\psi}_i$ is not expected to be the same as that of $e^{-itH}\psi_i$.
Moreover, our theorem gives the same results for unbroken $\cal PT$-symmetry. When $\cal PT$-symmetry is unbroken, then $S=I$, $\eta=(\Psi^{-1})^{\dag}\Psi^{-1}>0$, $J$ is diagonal, resulting in Eq. (9) being just a special case of our Theorem $1$. Apparently, Eq. (9) implies that the projection of $e^{-it\tilde{H}}\tilde{\psi}_i$ is numerically equal to $e^{-itH}\psi_i$. Hence the embedding paradigms illustrated in Refs. [@Gunther-PRL; @Ueda; @Huang] are also included in our method, although in those papers the $\eta$-inner product and measurements are not considered on purpose.
With the help of the analogy between Dirac inner product and $\eta$-inner product of unbroken $\cal PT$-symmetry, the example illustrated in Ref. [@Gunther-PRL] can be viewed as a proof for our paradigm in the unbroken $\cal PT$-symmetry. Explicitly, one can verify that Eq. (3) holds for the construction given below:
$$\begin{aligned}
&&H=\begin{bmatrix}
E_0+is\sin\theta &s\\
s&E_0-is\sin\theta
\end{bmatrix},~S=\begin{bmatrix}
1 &0\\
0 &1
\end{bmatrix},~
J=\begin{bmatrix}
E_0+s\cos\theta &0\\
0&E_0-s\cos\theta
\end{bmatrix},\\
&&\tilde{H}=
\begin{bmatrix}E_0&s\cos^2\theta&is\cos\theta\sin\theta&0\\s\cos^2\theta&E_0&0&-is\cos\theta\sin\theta
\\ -is\cos\theta\sin\theta&0& E_0&s\cos^2\theta\\ 0&is\cos\theta\sin\theta&s\cos^2\theta&E_0\end{bmatrix},\\
&&\tilde{\Psi}=\tilde{\Phi}=
\begin{bmatrix}
\frac{e^{\frac{i\theta}{2}}}{2}&\frac{ie^{\frac{-i\theta}{2}}}{2}\\
\frac{e^{-\frac{i\theta}{2}}}{2}&-\frac{ie^{\frac{i\theta}{2}}}{2}\\
\frac{e^{-\frac{i\theta}{2}}}{2}&\frac{ie^{\frac{i\theta}{2}}}{2}\\
\frac{e^{\frac{i\theta}{2}}}{2}&-\frac{ie^{-\frac{i\theta}{2}}}{2}
\end{bmatrix}.\end{aligned}$$
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|
---
abstract: 'We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core.'
address: 'F. Bonahon, Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, U.S.A.'
author:
- Francis Bonahon
title: Kleinian groups which are almost fuchsian
---
[^1]
Let $S$ be a surface of finite topological type, obtained by removing finitely many points from a compact surface without boundary, and with negative Euler characteristic. We consider complete hyperbolic metrics on the product $S\times
\left]-\infty,
\infty \right[$.
The simplest ones are the fuchsian metrics defined as follows. Because of our hypothesis that the Euler characteristic of $S$ is negative, $S$ admits a finite area hyperbolic metric, for which $S$ is isometric to the quotient of the hyperbolic plane $\mathbb
H^2$ by a discrete group $\Gamma$ of isometries. The group $\Gamma$ uniquely extends to a group of isometries of the hyperbolic 3–space $\mathbb H^3$ respecting the transverse orientation of $\mathbb H^2 \subset \mathbb
H^3$, for which the quotient $\mathbb H^3/\Gamma$ has a natural identification with $S\times \left]-\infty,
\infty \right[$. A *fuchsian metric* is any metric on $S\times \left]-\infty,
\infty \right[$ obtained in this way. Note that the image of $\mathbb H^2$ in $\mathbb H^3$ provides in this case a totally geodesic surface in $S\times
\left]-\infty,
\infty \right[$, isometric to the original metric on $S$.
These examples can be perturbed to more complex hyperbolic metrics on $S\times \left]-\infty,
\infty \right[$. See for instance [@Thu1][@Mas]. A *quasifuchsian metric* on $S\times \left]-\infty,
\infty \right[$ is one which is obtained by quasi-conformal deformation of a fuchsian metric. Equivalently, a quasifuchsian metric is a geometrically finite hyperbolic metric on $S\times \left]-\infty,
\infty \right[$ whose cusps exactly correspond to the ends of $S$. These also correspond to the interior points in the space of all hyperbolic metrics on $S\times \left]-\infty,
\infty \right[$ for which the ends of $S$ are parabolic [@Mar][@Sul].
If $m$ is a quasifuchsian metric on $S\times \left]-\infty,
\infty \right[$, the totally geodesic copy of $S$ which occurred in the fuchsian case is replaced by the *convex core* $C(m)$, defined as the smallest non-empty closed $m$–convex subset of $S\times \left]-\infty,
\infty \right[$. If $m$ is not fuchsian, $C(m)$ is 3–dimensional and its boundary consists of two copies of $S$, each facing an end of $S\times
\left]-\infty,
\infty \right[$. The geometry of $\partial C(m)$ was investigated by Thurston [@Thu1]; see also [@EpsMar]. The component of $\partial C(m)$ that faces the end $S\times\{+\infty\}$ is a pleated surface, totally geodesic almost everywhere, but bent along a family of simple geodesics; this bending is described and quantified by a measured geodesic lamination $\beta^+(m)$ on $S$. Similarly, the bending of the negative component of $\partial
C(m)$, namely the one facing $S\times\{-\infty\}$, is determined by a measured geodesic lamination $\beta^-(m)$.
If $\mathcal Q(S)$ denotes the space of isotopy classes of quasifuchsian metrics on $S\times \left]-\infty,
\infty \right[$ and if $\mathcal
{ML}(S)$ is the space of measured geodesic laminations on $S$, the rule $m \mapsto \left(
\beta^+(m), \beta^-(m) \right)$ defines a map $\beta: \mathcal
Q(S) \rightarrow \mathcal {ML}(S)
\times \mathcal {ML}(S)$. By definition, $\beta(m)=(0,0)$ is the metric $m$ is fuchsian, in which case the convex core $C(m)$ is just a totally $m$–geodesic copy of $S$. The image $\beta(m)\in \mathcal {ML}(S)^2$, interpreted as a measured geodesic lamination on two copies of $S$, is the *bending measured geodesic lamination* of the quasifuchsian metric $m$.
The space $\mathcal Q(S)$ is a manifold of dimension $2\theta$ where, if $\chi(S)$ is the Euler characteristic of $S$ and $p$ is its number of ends, $\theta=-3\chi(S)-p\geq 0$. It contains the space $\mathcal
F(S)$ as a proper submanifold of dimension $\theta$. To some extend, the bending measured lamination $\beta(m)$ measures how far the metric $m \in
\mathcal Q(S)$ is from being fuchsian. Finally, recall that the space $\mathcal {ML}(S)$ of measured geodesic laminations is a piecewise linear manifold of dimension $\theta$.
Thurston conjectured that the restriction of the bending map to $\beta$ could be used to parametrize $\mathcal Q(S) -
\mathcal F(S)$, namely that it induces a homeomorphism between $\mathcal Q(S) - \mathcal F(S)$ and an open subset of $\mathcal
{ML}(S)^2$. The image of $\beta$ was determined in [@BonOta].
The goal of the current paper is to prove Thurston’s conjecture on a neighborhood of the space of Fuchsian metrics.
\[thm:MainThm\] There exists an open neighborhood $V$ of the fuchsian submanifold $\mathcal
F(S)$ in $\mathcal Q(S)$ such that the bending map $\beta: \mathcal Q(S)
\rightarrow \mathcal{ML}(S)^2$ induces a homeomorphism between $V-\mathcal F(S)$ and its image.
There are well-known restrictions for $(\mu, \nu)\in \mathcal
{ML}(S)^2$ to be in the image of $\beta$; see for instance [@BonOta]. In particular, if $(\mu, \nu)\not = (0,0)$ is the bending measured lamination of some $m\in \mathcal Q(S)$, then the measured geodesic laminations $\mu$ and $\nu$ must *fill up* the surface $S$, in the sense that every non-trivial measured geodesic lamination has non-zero geometric intersection number with at least one of $\mu$, $\nu$. This is equivalent to the condition that every component of $S-\mu \cup
\nu$ is, either a topological disk bounded by the union of finitely many geodesic arcs, or a topological annulus bounded on one side by the union of finitely many geodesic arcs and going to a cusp on the other side.
Let $\mathcal {FML}(S)$ denote the open subset of $\mathcal{ML}(S)^2$ consisting of those $(\mu, \nu)$ where $\mu$ and $\nu$ fill up $S$. Note that $\mathcal
{FML}(S)$ is endowed with an action of $\mathbb R^+$, defined by $t(\mu, \nu)=(t\mu, t\nu)$, which decomposes $\mathcal
{FML}(S)$ as the union of pairwise disjoint rays (=orbits) $\left]0,\infty \right[(\mu,
\nu)$.
\[thm:LocalImageBeta\] In Theorem \[thm:MainThm\], the neighborhood $V$ of $\mathcal
F(S)$ and its image $U=\beta(V)$ can be chosen so that $U-\{(0,0)\}$ is an open subset of $\mathcal {FML}(S)$ which intersects each ray $\left]0,\infty \right[(\mu,
\nu)$ in an interval $\left]0,\varepsilon_{\mu\nu}
\right[(\mu,
\nu)$.
Theorems \[thm:MainThm\] and \[thm:LocalImageBeta\] are proved later as Theorem \[thm:HomeoNoCheck\]. The main idea of the proof is to construct an inverse $\beta^{-1}: U-\{(0,0)\}
\rightarrow V-\mathcal F(S)$ , and splits into two steps: an infinitesimal part, and a transversality argument based on the infinitesimal part. The infinitesimal part is now relatively classical; see for instance [@Ser1]. There are restrictions on which bending data can be realized by an infinitesimal deformation of $m_0\in \mathcal
F(S)$. Through the complex structure of $\mathcal Q(S)$, where multiplication by $\mathrm
i=\sqrt{-1}$ converts bending to shearing, these restrictions can be expressed in a purely 2–dimensional context; see Section \[sect:NecCond\]. The main part of the proof is to show by a transversality argument that any infinitesimal bending data can actually be realized by a deformation. The only significant idea of the paper is to apply the transversality argument, not in $\mathcal Q(S)$ where the necessary hypotheses are not realized, but in the manifold-with-boundary $\check
{\mathcal Q}(S)$ obtained by blowing up $\mathcal Q(S)$ along the fuchsian submanifold $\mathcal F(S)$.
Switching to the blow-up manifold $\check{\mathcal Q}(S)$ actually provides a better understanding of the restriction of $\beta$ on a neighborhood of $\mathcal F(S)$ and of its inverse. See Theorem \[them:HomeoInCheck\] for a precise statement.
**Acknowledgements:** There is strong evidence that the content of this paper has been known to Bill Thurston for about twenty years. I personally learned much of the material of Section \[sect:NecCond\] many years ago from David Epstein who, I believe, had learned it from Thurston. I am also grateful to Pete Storm and Dick Canary for helping me to clarify my thoughts.
The earthquake section {#sect:EarthSec}
======================
We consider the *Teichmüller space* $\mathcal T(S)$, namely the space of isotopy classes of finite area complete hyperbolic metrics on the surface $S$. Recall that $\mathcal T(S)$ is diffeomorphic to $\mathbb R^\theta$, where $\theta=-3\chi(S)-p\geq 0$ if $\chi(S)$ is the Euler characteristic of $S$ and $p$ is its number of ends.
A standard deformation of a metric $m\in
\mathcal T(S)$ is the *left earthquake* $E^\mu_m \in \mathcal
T(S)$ along the measured geodesic lamination $\mu$, as constructed in [@Thu][@Ker][@EpsMar]. We consider the *infinitesimal left earthquake vector* $e^\mu_m=\frac
d{dt}E^{t\mu}_m{}_{|t=0} \in T_m
\mathcal T(S)$. This provides a section $e^\mu : \mathcal T(S)
\rightarrow T\mathcal T(S)$ of the tangent bundle of $\mathcal
T(S)$, defined by $m\mapsto
e^\mu_m$.
There similarly exists a *right earthquake* $E^{-\mu}_m$ along $\mu$; the notation is justified by the fact that $m\mapsto E^{-\mu}_m$ is the inverse of $m\mapsto
E^{\mu}_m$. We can then consider the *infinitesimal right earthquake vector* $e^{-\mu}_m=\frac
d{dt}E^{-t\mu}_m{}_{|t=0}
\in T_m \mathcal T(S)$ and the corresponding section $e^{-\mu} : \mathcal T(S)
\rightarrow T\mathcal T(S)$ of the tangent bundle of $\mathcal
T(S)$. Note that $e^{-\mu}_m=-e^\mu_m$, but it will be convenient to keep a separate notation.
Recall that the measured geodesic laminations $\mu$, $\nu \in
\mathcal {ML}(S)$ *fill up* the surface $S$ if every non-trivial measured geodesic lamination has non-zero geometric intersection number with at least one of $\mu$, $\nu$. This is equivalent to the condition that every component of $S-\mu \cup
\nu$ is, either a topological disk bounded by the union of finitely many geodesic arcs, or a topological annulus bounded on one side by the union of finitely many geodesic arcs and going to a cusp on the other side.
\[prop:TransEarthSectInTgtSpce\] Let $\mu$, $\nu \in \mathcal
{ML}(S)$ be two non-zero measured geodesic laminations. The intersection of the two sections $e^\mu$, $e^{-\nu} : \mathcal T(S)
\rightarrow T\mathcal T(S)$ of the tangent bundle of $\mathcal
T(S)$ is transverse. These sections meet in exactly one point if $\mu$ and $\nu$ fill up the surface $S$, and are otherwise disjoint.
We first translate the problem in terms of the length functions $l_\mu$, $l_\nu: \mathcal T(S)
\rightarrow
\mathbb R$ which to a metric $m\in \mathcal T(S)$ associate the $m$–lengths of the measured geodesic lamination $\mu$ and $\nu$. The Weil-Petersson symplectic form on $\mathcal T(S)$ induces an isomorphism between its tangent bundle $T\mathcal T(S)$ and its cotangent bundle $T^*\mathcal
T(S)$. A celebrated result of Scott Wolpert [@Wol] asserts that this isomorphism sends the section $e^\mu$ of $T\mathcal
T(S)$ to the section $dl_\mu$ of $T^*\mathcal T(S)$. Therefore, Proposition \[prop:TransEarthSectInTgtSpce\] is equivalent to showing that the sections $dl_\mu$ and $-dl_\nu$ transversely meet in 1 or 0 point, according to whether $\mu$ and $\nu$ fill up the surface $S$ or not.
First consider the case where $\mu$ and $\nu$ fill up $S$. The intersection of the section $dl_\mu$ and $-dl_\nu$ of $T^*\mathcal T(S)$ correspond to the points $m\in \mathcal T(S)$ where $d_ml_\mu=-d_ml_\nu$, namely to the critical points of the function $l_\mu +l_\nu
:\mathcal T(S) \rightarrow
\mathbb R$. It is proved in [@Ker][@Ker2] that, because $\mu$ and $\nu$ fill up $S$, the function $l_\mu+l_\nu$ admits a unique critical point $m_0$. In addition, the hessian of $l_\mu+l_\nu$ at $m_0$ is positive definite [@Wol2][@Ker2].
Let $u=d_{m_0}l_\mu =
-d_{m_0}l_\nu \in T^*_{m_0}
\mathcal T(S)$. The intersection of the tangent spaces of the sections $dl_\mu$ and $-dl_\nu$ at $u$ consists of all the vectors of the form $T_{m_0}\left( dl_\mu\right)(v) =
T_{m_0}\left( -dl_\nu \right)(v)$ for some $v \in T_{m_0}\mathcal
T(S)$, where $T_{m_0}\left(
dl_\mu\right)$, $T_{m_0}\left(
-dl_\nu \right):
T_{m_0}\mathcal T(S)
\rightarrow T_u T^* \mathcal
T(S)$ denote the tangent maps of the sections $
dl_\mu$, $
-dl_\nu :
\mathcal T(S)
\rightarrow T^* \mathcal
T(S)$.
Note that, for an arbitrary $w \in T_{m_0}\mathcal
T(S)$, the vectors $T_{m_0}\left( dl_\mu\right)(w)$ and $ T_{m_0}\left( -dl_\nu
\right)(w)$ both project to $w$ through the tangent map of the projection $T^*\mathcal T(S)
\rightarrow \mathcal T(S)$. In particular, the difference $T_{m_0}\left( dl_\mu\right)(w)-
T_{m_0}\left( -dl_\nu
\right)(w)$ is tangent to the fiber, namely is an element of $T^*_{m_0}\mathcal T(S) \subset
T_u T^* \mathcal T(S)$. Using local coordinates, one easily sees that this difference is the image of $w$ under the homomorphism $T_{m_0} \mathcal
T(S) \rightarrow T^*_{m_0}
\mathcal T(S)$ induced by the hessian of the function $l_\mu +
l_\nu$. (Beware of optimistic simplifications, though: for instance, $T_{m_0}\left( -dl_\nu
\right)$ is not the same as $-T_{m_0}\left(dl_\nu
\right)$ since they take their values in different tangent spaces.) Since the hessian is non-degenerate, this homomorphism is actually an isomorphism.
A consequence of this analysis is that, if $T_{m_0}\left( dl_\mu\right)(v) =
T_{m_0}\left( -dl_\nu
\right)(v)$, then $v$ must be 0. Therefore, the intersection of the two sections $dl_\mu$ and $-dl_\nu$ of $T^* \mathcal T(S)$ at the point $u$ is transverse, since these sections have dimension $\theta$ and the total space $T^* \mathcal
T(S)$ has dimension $2\theta$. By Weil-Petersson duality, this proves that the sections $e^\mu$ of $e^{-\nu}$ of $T\mathcal T(S)$ have a transverse intersection consisting of exactly one point.
We now consider the case where $\mu$ and $\nu $ do not fill up the surface. In this case, the function $l_\mu
+l_\nu$ has no critical point [@Ker2] (see also the discussion in [@Ser1 §4.2]). By the above translation, it follows that the sections $e^\mu$, $e^{-\nu}$ are disjoint.
This concludes the proof of Proposition \[prop:TransEarthSectInTgtSpce\].
If the measured geodesic laminations $\mu$, $\nu$ fill up the surface $S$, let $\kappa(\mu,
\nu)$ denote the (unique) critical point of the length function $l_\mu +l_\nu :\mathcal T(S)
\rightarrow
\mathbb R$. As above, $\kappa(\mu, \nu )$ is also the unique $m\in \mathcal T(S)$ such that $e^\mu_m = e^{-\nu}_m$.
\[lem:UniqueInfEarth\] If $\kappa(\mu', \nu)=
\kappa(\mu, \nu)$, then $\mu'=\mu$.
By an (easy) infinitesimal version of [@Thu] (see also [@Ker2]), an infinitesimal left earthquake completely determines the measured geodesic lamination along which it is performed. If $e^{\mu'}_m = e^{-\nu}_m=
e^\mu_m$, it follows that $\mu'=\mu$.
We will need a result similar to Proposition \[prop:TransEarthSectInTgtSpce\] in the unit tangent bundle $T^1\mathcal T(S)$. Recall that the fibers $T_m^1\mathcal T(S)$ of this bundle are the quotient of $T_m\mathcal T(S)-\{0\}$ under the equivalence relation which identifies $v$ to $tv$ when $t\in
\left]0,\infty\right[$. In particular, $T^1\mathcal T(S)$ is a manifold of dimension $2\theta-1$. Let $\overline e{}^\mu$, $\overline
e{}^{-\nu}: \mathcal T(S)
\rightarrow T^1 \mathcal T(S)$ be the sections induced by $e^\mu$ and $e^{-\nu}$.
\[prop:TransEarthSectInUnitTgtSpce\] Let $\mu$, $\nu \in \mathcal
{ML}(S)$ be two non-zero measured geodesic laminations. The intersection of the two sections $\overline e{}^\mu$, $\overline
e{}^{-\nu}: \mathcal T(S)
\rightarrow T^1 \mathcal T(S)$ of the unit tangent bundle of $\mathcal T(S)$ is transverse. If $\mu$ and $\nu$ fill up the surface $S$, these sections meet along a section above a line $K(\mu, \nu)$ properly embedded in $\mathcal T(S)$. If $\mu$, $\nu$ do not fill up $S$, the intersection is empty.
The two sections meet above $m \in \mathcal T(S)$ if $\overline e{}^\mu_m=\overline
e{}^{-\nu}_m$, namely if there is a $t>0$ such that $e^{-\nu}_m=
te^\mu_m = e^{t\mu}_m$. By Proposition \[prop:TransEarthSectInTgtSpce\], this can occur only when $t\mu$ and $\nu$ fill the surface, namely only when $\mu$ and $\nu$ fill the surface. Consequently, the two sections have empty intersection if $\mu$ and $\nu$ do not fill up the surface.
If $\mu$ and $\nu$ fill up the surface then, for every $t>0$, Proposition \[prop:TransEarthSectInTgtSpce\] shows that there is a unique $m=\kappa(t\mu, \nu) \in
\mathcal T(S)$ such that $e^{t\mu}_m = e^{-\nu}_m$. As a consequence, the two sections $\overline e{}^\mu$, $\overline
e{}^{-\nu}: \mathcal T(S)
\rightarrow T^1 \mathcal T(S)$ meet exactly above the image $K(\mu, \nu)$ of the map $\left]0,\infty\right[
\rightarrow \mathcal T(S)$ defined by $t\mapsto \kappa
(t\mu, \nu)$.
If $m=\kappa(t\mu, \nu)$ so that $e^{t\mu}_m = e^{-\nu}_m$, the tangent space $T_{e^{t\mu}_m}T\mathcal T(S)$ is the sum of the tangent spaces $T_m
e^{t\mu} \left ( T_m \mathcal T(S)
\right)$ and $T_m
e^{-\nu} \left ( T_m \mathcal T(S)
\right)$ of the sections $e^{t\mu}$ and $e^{-\nu}$, by the transversality property of Proposition \[prop:TransEarthSectInTgtSpce\]. Therefore, in the unit tangent bundle, the tangent space $T_{\overline
e{}^{\mu}_m}T^1\mathcal T(S)$ is the sum of $T_m \overline
e{}^{\mu}
\left ( T_m \mathcal T(S)
\right)$ and $T_m \overline
e{}^{-\nu} \left ( T_m \mathcal
T(S)
\right)$. As a consequence, the intersection of the two sections $\overline e{}^\mu$ and $\overline
e{}^{-\nu}$ is transverse above the point $m=\kappa(t\mu, \nu)$.
By transversality, the intersection of the sections is a submanifold of the image of $\overline
e{}^\mu$. Its dimension is equal to 1, by consideration of the dimensions of $\mathcal
T(S)$ and $T^1\mathcal T(S)$. Since the projection $\overline e{}^\mu\left( \mathcal
T(S) \right) \rightarrow \mathcal
T(S)$ is a diffeomorphism, it follows that the projection $K(\mu, \nu)$ of this projection is a 1–dimensional submanifold of $\mathcal T(S)$.
By definition, $\kappa(t\mu,
\nu)$ is the unique minimum of the convex function $tdl_\mu+dl_\nu$, which has positive hessian at this minimum. It follows that $\kappa(t\mu,
\nu)$ is a continuous function of $t$. Conversely, $t$ is completely determined by $m=\kappa(t\mu, \nu)$ by Lemma \[lem:UniqueInfEarth\]. As a consequence, if $m=\kappa(t\mu, \nu)$ stays in a bounded subset of $\mathcal
T(S)$, then $t$ stays in a compact subset of $\left]0,\infty
\right[$. In other words, the map $\left]0,\infty\right[
\rightarrow \mathcal T(S)$ defined by $t\mapsto \kappa
(t\mu, \nu)$ is injective, continuous and proper. It follows that its image $K(\mu, \nu)$, which we already know is a 1–dimensional submanifold of $\mathcal T(S)$, is a line properly embedded in $\mathcal T(S)$.
Following the terminology of [@Ser1] (motivated by [@Ker2]), let the *Kerckhoff line* be the proper 1–dimensional submanifold $K(\mu,
\nu)
\subset
\mathcal T(S)$, consisting of all the $\kappa(t\mu, \nu)$ with $t>0$.
Necessary condition for small bending {#sect:NecCond}
=====================================
Let $t\mapsto m_t$, $t\in
\left[0, \varepsilon \right[$, be a small differentiable curve in $\mathcal Q(S)$. If $\beta (m_t)
= \left( \beta^+(t),
\beta^-(t)\right) \in \mathcal
{ML}(S)^2$ is its bending measured geodesic lamination, it is shown in [@Bon98] that the right derivative $\frac d{dt^+} \beta(m_t)_{|t=0}$ exists, as an element of the tangent space of $\mathcal
{ML}(S)^2$ at $\beta(m_0)$. In general, because $\mathcal
{ML}(S)$ is not a differentiable manifold, this tangent space consists of geodesic laminations with a transverse structure which is less regular than a transverse measure [@Bon97]. However, if we assume in addition that the starting point $m_0$ of the curve is fuchsian, the tangent space of $\mathcal
{ML}(S)^2$ at $\beta(m_0)=(0,0)$ is just $\mathcal
{ML}(S)^2$; see [@Bon97].
We can therefore consider the converse problem: Given a fuchsian metric $m_0 \in
\mathcal F(S)$ and a pair $(\mu,
\nu) \in \mathcal {ML}(S)^2$ of measured geodesic laminations, does there exist a small differentiable curve $t\mapsto
m_t\in \mathcal Q(S)$, $t\in
\left[0, \varepsilon \right[$, originating from $m_0$ and such that $\frac
d{dt^+} \beta(m_t)_{|t=0}=(\mu,
\nu)$? The following result shows that $m_0$ is completely determined by $\mu$ and $\nu$.
Note that, by construction, there is a natural identification between the submanifold $\mathcal
F(S)
\subset \mathcal Q(S)$ consisting of all fuchsian metrics and the Teichmüller space $\mathcal T(S)$.
\[prop:NecCondition\] Let $\mu$, $\nu \in \mathcal
{ML}(S)$ be two measured geodesic laminations, and let $t
\mapsto m_t$, $t\in \left[ 0,\varepsilon
\right[$ be a differentiable curve in $\mathcal Q(S)$, originating from a fuchsian metric $m_0$ and such that the derivative $\frac d{dt^+}
\beta(m_t)_{|t=0}$ of the bending measured lamination is equal to $(\mu, \nu)$. Then $\mu$ and $\nu$ fill up the surface $S$, and $m_0
\in
\mathcal F(S) = \mathcal T(S)$ is equal to the minimum $\kappa(\mu,
\nu)$ of the length function $l_\mu+l_\nu:\mathcal T(S)
\rightarrow \mathbb R$.
We consider two other curves in $\mathcal Q(S)$.
The first one is the *pure bending* $t\mapsto
B^{t\mu}_{m_0}$, obtained by bending the surface $S$ along the measured geodesic lamination $t\mu$ while keeping the metric induced on this pleated surface equal to $m_0$. For $t\geq 0$ small enough, $B^{t\mu}_{m_0}$ is a quasifuchsian metric for which the positive side of the boundary $\partial
C(B^{t\mu}_{m_0})$ is a pleated surface with induced metric $m_0$ and with bending measured geodesic lamination $t\mu$. See [@EpsMar §3] or [@Bon96] for the construction of $B^{t\mu}_{m_0}$, and [@Mar §9] to guarantee that it is quasifuchsian for $t$ sufficiently small. In addition, it is proved in [@EpsMar §3.9] [@Bon96] that this curve is differentiable in $\mathcal Q(S)$, and in particular admits a tangent vector $b^{\mu}_{m_0} = \frac
d{dt} B^{t\mu}_{m_0}{}_{|t=0} \in
T_{m_0} \mathcal Q(S)$. This tangent vector $b^\mu_{m_0}$ is the *infinitesimal pure bending* of $m_0\in \mathcal
F(S)$ along the measured geodesic lamination $\mu$.
The second curve will use the shear-bend coordinates associated, as in [@Bon96], to a maximal geodesic lamination $\lambda$ containing the support of $\mu$. These coordinates provide a local parametrization of $\mathcal Q(S)$ in terms of the geometry of a pleated surface with pleating locus $\lambda$. Let $m'_t$ correspond to a pleated surface whose induced metric is equal to the metric $m^+_t \in
\mathcal T(S)$ induced on the positive component of the boundary $\partial C(m_t)$ of the convex core, and whose bending data is equal to $t\mu$. By [@Mar §9], $m_t$ is a quasifuchsian metric for $t$ small.
It is proved in [@Bon98] that, because the curve $t\mapsto m_t$ is differentiable and because $\frac d{dt^+}
\beta^+(m_t)_{|t=0}=\mu$, the right derivative $\dot m^+_0 =
\frac d{dt} m_t^+{}_{|t=0} \in
T_{m_0} \mathcal T(S)$ exists and the two curves $t\mapsto m_t$ and $t\mapsto
m_t'$ have the same tangent vector at $t=0$. In particular, by differentiability of the shear-bend coordinates, the tangent vector $\dot m_0 =
\frac d{dt} m_t{}_{|t=0}=
\frac d{dt} m_t'{}_{|t=0}\in
T_{m_0} \mathcal Q(S)$ is the sum of $b^\mu_{m_0}$ and of $\dot m^+_0 =
\frac d{dt} m_t^+{}_{|t=0}\in
T_{m_0} \mathcal T(S)
=T_{m_0} \mathcal F(S)$.
Similarly, bending $S$ in the negative direction, we can define the infinitesimal pure bending vector $b^{-\nu}_{m_0} = -
b^{\nu}_{m_0}\in T_{m_0}
\mathcal Q(S)$ of $m_0$ along $-\nu$. We can also consider the metric $m^-_t \in \mathcal T(S)$ induced on the negative side of $\partial C(m_t)$. Then, as above, the vector $\dot m_0$ is the sum of $b^{-\nu}_{m_0}$ and of $\dot m^-_0 =
\frac d{dt} m_t^-{}_{|t=0}\in
T_{m_0} \mathcal T(S)
=T_{m_0} \mathcal F(S)$.
Finally, we will use the complex structure of $\mathcal Q(S)$ coming from the fact that the isometry group of $\mathbb H^3$ is $\mathrm{PSL}_2(\mathbb C)$. Indeed, considering the holonomy of hyperbolic metrics embeds $\mathcal Q(S)$ into the space $\mathcal R(S)$ of (conjugacy classes) of representations $\pi_1(S)
\rightarrow
\mathrm{PSL}_2(\mathbb C)$. This representation space $\mathcal
R(S)$ is a complex manifold near the image of $\mathcal Q(S)$, and admits this image as a complex submanifold (an open subset if $S$ is compact). The shear-bend local coordinates are well behaved with respect to this complex structure; see [@Bon96]. In particular, multiplication by $\mathrm i=\sqrt{-1}$ exchanges shearing and bending. We will use the following two consequences of this. First of all, at a fuchsian metric $m_0$, the tangent space $T_{m_0} \mathcal Q(S)$ is the direct sum of $T_{m_0}\mathcal
F(S)$ and of $\mathrm i T_{m_0}
\mathcal F(S)$. In addition, the infinitesimal pure bending vector $b^\mu_{m_0}$ belongs to $\mathrm i T_{m_0}
\mathcal F(S)$ and is equal to $\mathrm i e^\mu_{m_0}$, where $e^\mu_{m_0} \in T_{m_0} \mathcal
T(S) = T_{m_0}\mathcal F(S)$ is the infinitesimal earthquake vector along $\mu$.
Applying the decomposition $T_{m_0} \mathcal
Q(S)=T_{m_0}\mathcal F(S)
\oplus \mathrm i T_{m_0}
\mathcal F(S)$ to the vector $\dot m_0 = \dot m^+_0 +
b^\mu_{m_0} = \dot m^-_0 +
b^{-\nu}_{m_0}$, we conclude that $b^\mu_{m_0} = b^{-\nu}_{m_0}$. Multiplying by $-\mathrm i$, it follows that $e^\mu_{m_0} =
e^{-\nu}_{m_0}$. As in the proof of Proposition \[prop:TransEarthSectInTgtSpce\], this is equivalent to the property that $m_0=\kappa(\mu,
\nu)$.
Realizing small bending
=======================
The goal of this section is to prove a converse to Proposition \[prop:NecCondition\], by constructing in Proposition \[Prop:ConstrSmallBending\] a small curve of quasifuchsian metrics $t \mapsto m_t\in\mathcal Q(S)$, $t\in
\left[0, \varepsilon \right[$, such that $\beta(m_t)=(t\mu, t\nu)$ for every $t$.
For the measured geodesic lamination $\mu
\in
\mathcal{ML}(S)$, let $\mathcal P^+(\mu)$ (resp. $\mathcal P^-(\mu)$) be the space of quasifuchsian metrics $m$ such that the positive (resp. negative) component of the convex core boundary $\partial C(m)$ has bending measured geodesic lamination $t\mu$, for some $t\in\left[0,\infty\right[$.
Recall that $\theta =
-3\chi(S)+p$ denotes the dimension of the Teichmüller space $\mathcal T(S)$.
\[lem:PleatSubmfd\] The space $\mathcal P^\pm (\mu)$ is a submanifold-with-boundary of $\mathcal Q(S)$, with dimension $\theta+1$ and with boundary $\mathcal F(S)$.
We can use the coordinates developed in [@Bon96], and associated to a maximal geodesic lamination $\lambda$ containing the support of $\mu$. These provide an open differentiable embedding $\varphi:\mathcal Q(S)
\rightarrow \mathcal T(S) \times
\mathcal H_0(\lambda; \mathbb
R/ 2\pi \mathbb Z)$. The first component of $\varphi(m)$ is the hyperbolic metric induced on the unique $m$–pleated surface $f_m$ with pleating locus $\lambda$. The second component is the bending transverse cocycle of $f_m$, which belongs to the topological group $\mathcal H_0(\lambda; \mathbb
R/ 2\pi \mathbb Z)
\cong
\left( \mathbb R/ 2\pi \mathbb
Z \right)^\theta \oplus \mathbb
Z/2$ of all $ \left( \mathbb R/ 2\pi \mathbb
Z \right)$–valued transverse cocycles for $\lambda$ that satisfy a certain cusp condition. In general, the bending of the pleated surface $f_m$ is measured by a transverse cocycle and not by a measured geodesic lamination because $f_m$ is not necessarily locally convex.
For notational convenience, it is useful to lift $\varphi$ to an embedding $\psi:\mathcal Q(S)
\rightarrow \mathcal T(S) \times
\mathcal H_0(\lambda; \mathbb
R)$, such that $\psi$ sends $\mathcal F(S) \cong \mathcal
T(S)$ to $\mathcal T(S) \times
\{0\}$ by the identity. Here, $\mathcal H_0(\lambda; \mathbb
R) \cong \mathbb R^\theta$ denotes the space of $\mathbb R$–valued transverse cocycle satisfying the cusp condition. Such a $\psi$ exists and is unique because $\mathcal Q(S)$ is simply connected.
The vector space $\mathcal H_0(\lambda; \mathbb R)$ contains the transverse measure (also denoted by $\mu$) of the measured geodesic lamination $\mu$, and therefore also contains the two rays $
\left[0,\infty
\right[\mu$ and $\left]-\infty, 0
\right]\mu$, consisting of all positive (resp. negative) real multiples of $\mu$.
We claim that $\mathcal
P^+(\mu)$ locally corresponds, under $\psi$, to the intersection of $\psi\left(\mathcal
Q(S)\right)$ with $\mathcal T(S)
\times \left[0,\infty
\right[\mu$. Clearly, $\psi$ sends an element of $\mathcal P^+(\mu)$ to $\mathcal T(S)
\times \left[0,\infty
\right[\mu$. Conversely, it is proved in [@KamTan] that the map $\eta: \mathcal Q(S) \rightarrow
\mathcal T(S) \times
\mathcal {ML}(S)$ which, to a quasifuchsian metric $m \in
\mathcal Q(S)$, associates the induced metric $m^+\in\mathcal
T(S)$ and the bending measured lamination $\beta^+(m)$ of the positive boundary component of the convex core $C(m)$ is a local homeomorphism. In addition, interpreting the ray $\left[0,\infty
\right[\mu$ as a subset of both $\mathcal H_0(\lambda; \mathbb
R)$ and $\mathcal{ML}(S)$, the local inverse for $\eta$ constructed in [@KamTan] coincides with $\psi$ on $\mathcal T(S)
\times \left[0,\infty
\right[\mu$. It follows that, if $m\in\mathcal Q(S)$ is sufficiently close to $m_0\in
\mathcal P^+(\mu)$ and if $\psi(m) \in \mathcal T(S)
\times \left[0,\infty
\right[\mu$, the bending measured lamination $\beta^+(\mu)$ is equal to the second component $t\mu$ of $\psi(m)$. Therefore, a metric $m$ near $m_0\in
\mathcal P^+(\mu)$ is in $
\mathcal P^+(\mu)$ if and only if $\psi(m) \in \mathcal T(S)
\times \left[0,\infty
\right[\mu$.
A similar property holds for $\mathcal P^-(\mu)$ by symmetry. Therefore, under the diffeomorphism $\psi$, $\mathcal P^+(\mu)$, $\mathcal
P^-(\mu)$ and $\mathcal F(S)$ locally correspond to the intersection of $\psi\left(\mathcal
Q(S)\right)$ with $\mathcal T(S)
\times \left[0,\infty
\right[\mu$, $\mathcal T(S)
\times \left]-\infty, 0
\right]\mu$ and $\mathcal T(S)
\times \{0\}$, respectively.
Given two measured geodesic laminations $\mu$, $\nu\in\mathcal{ML}(S)$, we want to consider the intersection of $\mathcal P^+(\mu)$ and $\mathcal P^-(\nu)$. Note that this intersection is far from being transverse, since these two $(\theta+1)$–dimensional submanifolds both contain the $\theta$–dimensional submanifold $\mathcal F(S)$ as their boundary. For this reason, we will consider the manifold-with-boundary $\check
{\mathcal Q}(S)$ obtained by blowing-up ${\mathcal Q}(S)$ along the submanifold $\mathcal F(S)$. Namely, $\check{\mathcal Q}(S)$ is the union of $\mathcal Q(S)-\mathcal
F(S)$ and of the unit normal bundle $N^1\mathcal F(S)$ with the appropriate topology. Recall that the normal bundle $N\mathcal F(S) \rightarrow
\mathcal F(S)$ is intrinsically defined as the bundle whose fiber $N_m\mathcal F(S)$ at $m\in \mathcal F(S)$ is the quotient $T_m\mathcal
Q(S)/T_m\mathcal F(S)$, and that the fiber $N^1_m \mathcal F(S)$ of the unit normal bundle $N^1\mathcal F(S) \rightarrow
\mathcal F(S)$ is the quotient of $N_m\mathcal F(S)-\{0\}$ under the multiplicative action of $\mathbb R^+$. Considering a tubular neighborhood of $\mathcal F(S)$, $\check{\mathcal Q}(S)$ is easily endowed with a natural structure of differentiable manifold with boundary $N^1\mathcal F(S)$.
Exploiting the complex structure of $\mathcal Q(S)$, identify the normal bundle $N\mathcal F(S)$ to $\mathrm i T\mathcal F(S)$. The inclusion map $\mathcal P^+(\mu)-\mathcal F(S)
\rightarrow
\mathcal Q(S)-\mathcal F(S)$ uniquely extends to an embedding $\mathcal P^+(\mu)\rightarrow
\check
{\mathcal Q}(S)$, which to $m\in
\mathcal F(S)= \partial \mathcal
P^+(\mu)$ associates the unit normal vector $\overline b{}_m^\mu \in
N^1\mathcal F(S)$ which is in the direction of the infinitesimal pure bending vector $b_m^\mu$ of $m$ along $+\mu$. We can similarly define an embedding $\mathcal P^-(\nu)\rightarrow
\check
{\mathcal Q}(S)$, by associating to $m\in
\mathcal F(S)$ the unit normal vector $-\overline b{}_m^\nu \in
N^1\mathcal F(S)$ which is in the direction of the infinitesimal pure bending vector $b_m^{-\nu}
=-b_m^\nu$ of $m$ along $-\nu$. Let $\check{\mathcal P}^+(\mu)$ and $\check{\mathcal
P}^-(\nu)\subset\check{\mathcal
Q}(S)$ be the respective images of these embeddings.
The subspaces $\check{\mathcal
P}^+(\mu)$ and $\check{\mathcal
P}^-(\nu)$ are submanifolds of $\check{\mathcal Q}(S)$, with boundary contained in $\partial
\check{\mathcal Q}(S)$.
The embedding $\psi:\mathcal Q(S)
\rightarrow \mathcal T(S) \times
\mathcal H_0(\lambda; \mathbb R)$ of the proof of Lemma \[lem:PleatSubmfd\] sends the vector $b^\mu_m$ to the vector $(0,\mu)$ in the tangent space of $\mathcal T(S) \times
\mathcal H_0(\lambda; \mathbb R)$. The result then immediately follows from the fact that $\psi$ locally identifies $\mathcal
P^+(\mu)$, $\mathcal
P^-(\mu)$ and $\mathcal T(S)$ to $\mathcal T(S)
\times \left[0,\infty
\right[\mu$, $\mathcal T(S)
\times \left]-\infty, 0
\right]\mu$ and $\mathcal T(S)
\times \{0\}$, respectively. This proves Lemma \[lem:PleatSubmfd\].
\[prop:TransInterInBdryBlowUp\] The boundaries $\partial \check{\mathcal
P}^+(\mu)$ and $\partial
\check{\mathcal P}^-(\nu)$ have a non-empty intersection if and only if $\mu$ and $\nu$ fill up the surface $S$. If $\mu$ and $\nu$ fill up $S$, the intersection of $\partial\check{\mathcal
P}^+(\mu)$ and $\partial\check{\mathcal
P}^-(\nu)$ in $\partial\check{
\mathcal Q}(S)$ is transverse, and is equal to the image of the Kerckhoff line $K(\mu, \nu)$ under the section $m\mapsto
\overline b{}_m^\mu
=-\overline b{}_m^\nu\in
\partial\check{\mathcal Q}(S)$.
The diffeomorphism $T^1\mathcal
T(S)=T^1\mathcal F(S)
\rightarrow
\mathrm i T^1
\mathcal F(S)= N^1\mathcal
F(S) =\partial \check {\mathcal
Q}(S)$ defined by multiplication by $\mathrm i$ sends $\overline
e{}^\mu_m$ to $\overline
b{}^\mu_m$ and $\overline
e{}^{-\nu}_m$ to $\overline
b{}^{-\nu}_m$. This translates Proposition \[prop:TransInterInBdryBlowUp\] to a simple rephrasing of Proposition \[prop:TransEarthSectInUnitTgtSpce\].
An immediate consequence of Proposition \[prop:TransInterInBdryBlowUp\] is that the intersection of the two $(\theta+1)$–dimensional submanifolds $\check{\mathcal
P}^+(\mu)$ and $
\check{\mathcal P}^-(\nu)$ in $\check{\mathcal Q}(S)$ is transverse near the boundary $\partial \check{\mathcal Q}(S)$. In particular, the intersection $\check{\mathcal
P}^+(\mu) \cap
\check{\mathcal P}^-(\nu)$ is a 2–dimensional submanifold of $\check {\mathcal Q}(S)$ near $\partial\check {\mathcal
Q}(S)$, with boundary $\partial \check{\mathcal
P}^+(\mu) \cap \partial
\check{\mathcal P}^-(\nu)$ contained in $\partial\check
{\mathcal
Q}(S)$.
By definition, a metric $m \in
{\mathcal P}^+(\mu) \cap
{\mathcal P}^-(\nu)$ has bending measured lamination $\beta(m)=(t\mu, u\nu)$ for some $t$, $u\geq 0$. This gives a differentiable map $\pi: {\mathcal P}^+(\mu) \cap
{\mathcal P}^-(\nu)
\rightarrow \mathbb R^2$, defined by $m
\mapsto (t,u)$.
Let $\check{\mathbb
R}^2$ be obtained by blowing up $\mathbb R^2$ along $\{0\}$. Because $\pi^{-1}(0)=
\mathcal F(S)$, the map $\pi$ lifts to a differentiable map $\check\pi:
\check{\mathcal P}^+(\mu) \cap
\check{\mathcal P}^-(\nu)
\rightarrow \check{\mathbb R}^2$.
\[lem:PiCheckLocalDiffeo\] The map $\check\pi:
\check{\mathcal P}^+(\mu) \cap
\check{\mathcal P}^-(\nu)
\rightarrow \check{\mathbb R}^2$ is a local diffeomorphism near $\partial
\check{\mathcal P}^+(\mu) \cap
\partial
\check{\mathcal P}^-(\nu)$.
We will prove that, at any point $p_0
\in \partial
\check{\mathcal P}^+(\mu) \cap
\partial
\check{\mathcal P}^-(\nu)$, the (linear) tangent map $T_{p_0}\check\pi: T_{p_0}
\check{\mathcal P}^+(\mu)
\cap
T_{p_0} \check{\mathcal P}^-(\nu)
\rightarrow
T_{\check\pi(p_0)}\check{\mathbb
R}^2$ is injective.
Let $v\in
T_{p_0} \check{\mathcal P}^+(\mu)
\cap
T_{p_0} \check{\mathcal
P}^-(\nu)$ be such that $T_{p_0}
\check\pi(v)=0$. Considering the map ${\mathcal P}^+(\mu)
\rightarrow
\left[0,\infty\right[$ which to $m\in \mathcal P^+(\mu)$ associates $\beta^+(m)/\mu\in
\left[0,\infty\right[$ and the induced map $\check{\mathcal P}^+(\mu)
\rightarrow
\left[0,\infty\right[$, we see that $v$ must necessarily be in the tangent space of the boundary $\partial\check{\mathcal
P}^+(\mu)$. Symmetrically, it must be in $T_{p_0} \partial{\check{\mathcal
P}}^-(\nu)$. Therefore, $v$ is tangent to the intersection $\partial
\check{\mathcal P}^+(\mu) \cap
\partial
\check{\mathcal P}^-(\nu)$.
Let us analyze the restriction of $\check \pi$ to the boundary $\partial
\check{\mathcal P}^+(\mu) \cap
\partial
\check{\mathcal P}^-(\nu)$. By Proposition \[prop:TransInterInBdryBlowUp\], $\partial
\check{\mathcal P}^+(\mu) \cap
\partial
\check{\mathcal P}^-(\nu)$ is equal to the image of the Kerckhoff line $K(\mu, \nu)$ under the section $m\mapsto
\overline b{}_m^\mu
=-\overline b{}_m^\nu\in
\partial\check{\mathcal Q}(S)$. Recall that an element of the Kerckhoff line $K(\mu, \nu)$ is of the form $m=\kappa(t\mu, \nu)$ for some $t>0$, which is equivalent to the property that $b{}^{t\mu}_m= -
b{}^{\nu}_m$. We will also need a coordinate chart for $\check{\mathbb R}^2$ near $\check\pi(p_0)$. Noting that the image of $\pi$ is contained in the quadrant $\left[0,\infty\right[^2$, we can use for this the chart $\varphi:
\left]0,\infty\right[ \times
\left [0,\infty \right[
\rightarrow
\check{\mathbb R}^2$ defined on the interior by $(x,y) \mapsto (xy, y)$.
We claim that, if $m=\kappa(t\mu,
\nu)\in K(\mu, \nu)$, then $\varphi^{-1}\circ \check\pi
\left (\,
\overline b{}_m^\mu \right)$ is just equal to $(t,0)$. To see this, choose, in the 2–dimensional manifold $\check{\mathcal P}^+(\mu) \cap
\check{\mathcal P}^-(\nu)$, a small curve $s\mapsto \check m_s$, $s\in \left[0,\varepsilon
\right[$, such that $\check m_0 =
\overline b{}_m^\mu \in
\partial \check{\mathcal P}^+(\mu)
\cap \partial
\check{\mathcal P}^-(\nu)$ and such that $\frac d{ds^+}\check m_s{}_{|t=0}$ is not tangent to the boundary. The curve $s\mapsto
\check m_s$ projects to a differentiable curve $t\mapsto m_t \in \mathcal Q(S)$ with $m_0=m$. By definition of ${\mathcal P}^+(\mu)$ and ${\mathcal P}^-(\nu)$, the bending measured lamination $\beta(m_s)$ is of the form $(t(s)\mu, u(s)\nu)$ for two differentiable functions $t(s)$, and $u(s)$ with $t(0)=u(0)=0$. Since $\frac d{ds^+}\check
m_s{}_{|t=0}$ points away from the boundary, the curve $s\mapsto m_s$ is not tangent to $\mathcal F(S)$ at $s=0$, and it follows that at least one of the derivatives $t'(0)$, $u'(0)$ is non-trivial. If we apply Proposition \[prop:NecCondition\], we conclude that $m=\kappa\left (t'(0)\mu,
u'(0)\nu\right)$. Since $m=\kappa\left (t\mu,
\nu\right)$, it follows that $t'(0)/u'(0)=t$ by Lemma \[lem:UniqueInfEarth\]. In particular, $t(s)/u(s)$ tends to $t$ as $s$ tends to 0. Therefore, $\varphi^{-1}\circ \check\pi
\left (
\overline b{}_m^\mu \right)$, which is the limit of $\varphi^{-1}\circ \check\pi
\left (
\check m_s \right) =
\varphi^{-1}\circ
\pi
\left (
m_s \right)= \varphi^{-1}\circ
\bigl(t(s), u(s)\bigr) = \bigl(
t(s)/u(s), u(s)\bigr)$ as $s>0$ tends to 0, is equal to $(t,0)$.
This computation shows that the restriction of $\check\pi$ to the boundary $\partial
\check{\mathcal P}^+(\mu) \cap
\partial
\check{\mathcal P}^-(\nu)$ is a diffeomorphism onto its image. In particular, if $v\in
T_{p_0} \check{\partial\mathcal
P}^+(\mu)
\cap
T_{p_0} \check{\partial\mathcal
P}^-(\nu)$ is such that $T_{p_0}
\check\pi(v)=0$, then necessarily $v=0$.
This concludes the proof that the tangent map $T_{p_0}\check\pi:
T_{p_0} \check{\mathcal P}^+(\mu)
\cap
T_{p_0} \check{\mathcal P}^-(\nu)
\rightarrow
T_{\check\pi(p_0)}\check{\mathbb
R}^2$ is injective. Since $\check
\pi$ sends the boundary of the 2-dimensional manifold $
\check{\mathcal P}^+(\mu) \cap
\check{\mathcal P}^-(\nu)$ to the boundary of the 2-dimensional manifold $\check{\mathbb
R}^2$, this proves that $\check
\pi:
\check{\mathcal P}^+(\mu) \cap
\check{\mathcal P}^-(\nu)
\rightarrow\check{\mathbb R}^2$ is a local diffeomorphism near $p_0 \in \partial
\check{\mathcal P}^+(\mu) \cap
\partial
\check{\mathcal P}^-(\nu)$.
This immediately gives the following converse to Proposition \[prop:NecCondition\].
\[Prop:ConstrSmallBending\] Let $\mu$, $\nu \in \mathcal
{ML}(S)$ be two measured geodesic laminations which fill up the surface $S$, and let $m_0$ be the minimum $\kappa(\mu, \nu)$ of the length function $l_\mu +l_\nu$. Then there is a small differentiable curve $t\mapsto
m_t\in \mathcal Q(S)$, $t\in
\left[0,\varepsilon\right[$, beginning at $m_0$ and such that the bending measured lamination $\beta(m_t) $ is equal to $(t\mu,
t\nu)$ for every $t$.
Consider the curve $t\mapsto (t,t)$, $t \in \left[
0, \varepsilon \right[$, in $\mathbb R^2$. By Lemma \[lem:PiCheckLocalDiffeo\], for $\varepsilon$ small enough, there is a curve $t\mapsto \check
m_t \in
\check{\mathcal P}^+(\mu) \cap
\check{\mathcal P}^-(\nu)$ such that $t \mapsto \check\pi \left(
\check m_t\right)$ coincides with the lift of $t\mapsto (t,t)$ to $\check{\mathbb R}^2$. By definition of the map $\pi$, this just means that the projection $m_t\in \mathcal Q(S)$ of $\check m_t\in \check{\mathcal
Q}(S)$ is such that $\beta(m_t)=(t\mu, t\nu)$.
Parametrizing quasi-fuchsian groups by their small bending
==========================================================
Recall that $\mathcal {FML}(S)$ denotes the open subset of $\mathcal {ML}(S)^2$ consisting of those pairs $(\mu, \nu)$ such that $\mu$ and $\nu$ fill up the surface $S$.
Let $\check{\mathcal F}
\mathcal{ML}(S)$ be obtained by blowing up $\mathcal {FML}(S)
\cup \{(0,0)\}$ along $\{(0,0)\}$. Namely, $\check{\mathcal F}
\mathcal{ML}(S)$ is formally obtained from $\mathcal {FML}(S)$ by extending each ray $\left]0,\infty\right[(\mu, \nu)$ to a semi-open ray $\left[0,\infty\right[(\mu,
\nu)$, with the obvious topology. Note that the boundary $\partial\check{\mathcal F}
\mathcal{ML}(S)$ is just the quotient space of $\mathcal
{FML}(S)$ under the multiplicative action of $\mathbb R^+$.
For every $(\mu, \nu)\in
\mathcal {FML}(S)$, Proposition \[Prop:ConstrSmallBending\] provides a maximal ray $R_{\mu\nu}=
\left[0,\varepsilon_{\mu\nu}
\right[(\mu, \nu)$ in $\mathcal{FML}(S) \cup
\{(0,0)\}$ and a differentiable map $\Phi_{\mu\nu}: R_{\mu\nu}
\rightarrow Q(S)$ such that $\Phi_{\mu\nu}(\mu', \nu')$ has bending measured lamination $(\mu', \nu')$ for every $(\mu',
\nu') \in R_{\mu\nu}$ and such that $\check{\mathcal P}^+(\mu)$ and $\check{\mathcal P}^-(\nu)$ meet transversely along $\beta(R_{\mu\nu})$. Here, the statement that $R_{\mu\nu}$ is maximal means that $\varepsilon_{\mu\nu} \in
\left]0,\infty\right]$ is maximal for this property.
Note that $R_{\mu\nu}$ and $\varphi_{\mu\nu}$ depend only on the orbit of $(\mu, \nu)$ under the action of $\mathbb R^+$, namely on the corresponding point of $\partial \check{\mathcal
F}\mathcal{ML}(S)$. Let $\check R_{\mu\nu}$ be the lift of $R_{\mu\nu}$ in $\check{\mathcal
F}\mathcal{ML}(S)$, and lift $\Phi_{\mu\nu}$ to $\check
\Phi_{\mu\nu}: \check R_{\mu\nu}
\rightarrow
\check
{\mathcal Q}(S)$. In particular, $\check \Phi_{\mu\nu}$ sends the initial point of $\check
R_{\mu\nu}$ to the bending vector $\overline b{}^\mu_m= \overline
b{}^{-\nu}_m \in N^1 \mathcal F(S)
=
\partial \check{\mathcal Q}(S)$ with $m=\kappa(\mu, \nu)$.
Let $\check U\subset
\check{\mathcal F}\mathcal{ML}(S)$ denote the union of all the $\check R_{\mu\nu}$, and let $\check \Phi : \check U
\rightarrow \mathcal {FML}(S)$ restrict to $\check \Phi_{\mu\nu}$ on each $\check R_{\mu\nu}$. Note that the property that $\Phi_{\mu\nu}(\mu', \nu')$ has bending measured lamination $(\mu', \nu')$ implies that the $\check R_{\mu\nu}$ are pairwise disjoint, so that $\check \Phi$ is well-defined. We want to show that $\check \Phi$ is continuous.
\[lem:ContinuityPleatSbmfd\] As the measured geodesic lamination $\mu$ tends to $\mu_0$ for the topology of $\mathcal
{ML}(S)$, the submanifold $\mathcal P^\pm(\mu)$ tends to $\mathcal P^\pm(\mu_0)$ for the topology of $\mathrm C^\infty$ convergence on compact subsets.
We will use the tools developed in [@Bon96].
Let $\mu_n\in\mathcal {ML}(S)$, $n\in \mathbb N$, be a sequence converging to $\mu_0$. Let $\lambda_n$ be a maximal geodesic lamination containing the support of $\mu_n$. Passing to a subsequence if necessary, we can assume that, for the Hausdorff topology, the geodesic lamination $\lambda_n$ converges to a geodesic lamination $\lambda_0$, which is necessarily maximal and contains the support of $\mu_0$.
The shear-bend coordinates associated to $\lambda_n$ provide an open biholomorphic embedding $\Phi_n: \mathcal Q(S)
\rightarrow \mathcal H_0\left(
\lambda_n;
\mathbb C/2\pi \mathrm i \mathbb
Z \right)$. Here $\mathcal H_0\left(
\lambda_n;
\mathbb C/2\pi \mathrm i \mathbb
Z \right)$ is the topological group of $\mathbb C/2\pi \mathrm i \mathbb
Z $–valued transverse cocycles for the maximal geodesic lamination $\lambda_n$ which satisfy the cusp condition, and is isomorphic to $\left( \mathbb C/2\pi
\mathrm i \mathbb
Z \right)^\theta \oplus \mathbb
Z/2$. For a metric $m\in
\mathcal Q(S)$, the real part of $\Phi_n(m) \in \mathcal H_0\left(
\lambda_n;
\mathbb C/2\pi \mathrm i \mathbb
Z \right) =\mathcal H_0\left(
\lambda_n;
\mathbb R \right) \oplus \mathrm i
\mathcal H_0\left(
\lambda_n;
\mathbb R/2\pi \mathbb
Z \right)$ measures the induced metric of the unique $m$–pleated surface with bending locus $\lambda_n$, and the imaginary part measures its bending. In particular, $\mathcal P^\pm
(\mu_n)$ locally corresponds to $\Phi_n^{-1} \left (
\mathcal H_0\left(
\lambda_n;
\mathbb R \right) \oplus \mathrm i
\mathbb R\mu_n
\right)$, or more precisely to a branch of the immersion which is the composition of the projection $\mathcal H_0\left(
\lambda_n;
\mathbb R \right) \oplus \mathrm i
\mathbb R\mu_n \rightarrow
\mathcal H_0\left(
\lambda_n;
\mathbb C/2\pi \mathrm i \mathbb
Z \right)$ and of $\Phi_n^{-1}$.
To compare the various $\mathcal H_0\left(
\lambda_n;
\mathbb C/2\pi \mathrm i \mathbb
Z \right)$ pick a train track $\tau$ carrying $\lambda_0$. Since $\lambda_n$ converges to $\lambda_0$ for the Hausdorff topology, $\tau$ also carries the $\lambda_n$ for $n$ large enough. Then there is a well-defined isomorphism $\Psi_n: \mathcal
H_0\left(
\lambda_n;
\mathbb C/2\pi \mathrm i \mathbb
Z \right) \rightarrow \mathcal
W\left(
\tau;
\mathbb C/2\pi \mathrm i \mathbb
Z \right)$, where $ \mathcal
W\left(
\tau;
\mathbb C/2\pi \mathrm i \mathbb
Z \right)$ is the group of $\mathbb C/2\pi \mathrm i \mathbb
Z $–valued edge weights for $\tau$ satisfying the switch and cusp relations.
Because $\lambda_n$ converges to $\lambda_0$ for the Hausdorff topology, if follows from the explicit construction of [@Bon96 §5, §8] that $\Phi_n^{-1}\circ
\Psi_n^{-1}$ converges to $\Phi_0^{-1}\circ
\Psi_0^{-1}$, uniformly on compact subsets of the image of $\Psi_0 \circ \Phi_0$. Because these maps are holomorphic, the convergence is actually $\mathrm
C^\infty$. Since $\mu_n$ converges to $\mu_0$ for the topology of $\mathcal {ML}(S)$, the edge weight system $\Psi_n(\mu_n)$ converges to $\Psi_0(\mu_0)$ in $\mathcal W_0(\tau, \mathbb R)$. It follows that $\mathcal P^\pm
(\mu_n)$, which locally corresponds to $\Phi_n^{-1} \circ
\Psi_n^{-1}\left (
\mathcal W_0\left(
\tau;
\mathbb R \right) \oplus \mathrm i
\mathbb R \Psi_n(\mu_n)
\right)$, converges to $\mathcal P^\pm
(\mu_0)$, which locally corresponds to $\Phi_0^{-1} \circ
\Psi_0^{-1}\left (
\mathcal W_0\left(
\tau;
\mathbb R \right) \oplus \mathrm i
\mathbb R \Psi_0(\mu_0)
\right)$, in the topology of $\mathrm
C^\infty$–convergence on compact subsets.
Note that we actually proved real analytic convergence in Lemma \[lem:ContinuityPleatSbmfd\]. However, we will only need $\mathrm C^2$ convergence.
\[them:HomeoInCheck\] The subset $\check U$ is an open neighborhood of $\partial \check{\mathcal
F}\mathcal{ML}(S)$ in $\check{\mathcal
F}\mathcal{ML}(S)$, and $\check\Phi$ is a homeomorphism from $\check U$ to an open neighborhood $\check V$ of $\partial \check{\mathcal Q}(S)$ in $\check{\mathcal Q}(S)$.
The restriction $\check \Phi_{\mu\nu}$ of $\check
\Phi$ to $\check R_{\mu\nu}$ was constructed by considering the transverse intersection of the submanifolds $\check {\mathcal
P}^+(\mu)$ and $\check {\mathcal
P}^-(\nu)$ near the boundary of $\check {\mathcal Q}(S)$. By Lemma \[lem:ContinuityPleatSbmfd\], $\check {\mathcal
P}^+(\mu)$ and $\check {\mathcal
P}^-(\nu)$ depend continuously on $(\mu, \nu)$ for the topology of $\mathrm C^1$ convergence. (Note that one needs the $\mathrm C^2$ continuity of $ {\mathcal
P}^+(\mu)$ and ${\mathcal
P}^-(\nu)$ to guarantee the $\mathrm C^1$ continuity of $\check {\mathcal P}^+(\mu)$ and $\check {\mathcal P}^-(\nu)$ near the boundary.) It follows that the length $\varepsilon_{\mu\nu}$ of $R_{\mu\nu}=\left [0,
\varepsilon_{\mu\nu}
\right[(\mu, \nu)$ is a lower semi-continuous function of $(\mu, \nu)$, and that $\Phi_{\mu\nu}$ depends continuously on $(\mu, \nu)$. This proves that the union $\check
U$ of the $\check R_{\mu\nu}$ is open in $\partial
\check{\mathcal
F}\mathcal{ML}(S)$, and that $\check \Phi$ is continuous.
Let $p:\check {\mathcal Q}(S)
\rightarrow \mathcal Q(S)$ be the natural projection and, as usual, let $\beta: \mathcal Q(S)
\rightarrow \mathcal{ML}(S)^2$ be the bending map. By construction, $\beta \circ p \circ \check \Phi$ is the identity map on $\check
U- \partial \check{\mathcal
F}\mathcal{ML}(S) \subset
\mathcal {ML}(S)^2$. It follows that $\check \Phi$ is injective on $\check U- \partial
\check{\mathcal
F}\mathcal{ML}(S)$, and therefore on all of $\check U$ by Proposition \[prop:NecCondition\].
The two spaces $\check U \subset \check{\mathcal
F}\mathcal{ML}(S)$ and $\check{\mathcal Q}(S)$ are topological manifolds-with-boundary of the same dimension $2\theta$. The map $\check \Phi : \check U
\rightarrow \mathcal {FML}(S)$ is continuous and injective, and sends boundary points to boundary points. By the Theorem of Invariance of the Domain, it follows that its image $\check V
= \check \Phi\bigl ( \check U
\bigr)$ is open in $\check
{\mathcal Q}(S)$, and that $\check \Phi$ restricts to a homeomorphism $\check U
\rightarrow \check V$.
\[thm:HomeoNoCheck\] There exists an open neighborhood $V$ of the fuchsian submanifold $\mathcal
F(S)$ in $\mathcal Q(S)$ such that the bending map $\beta: \mathcal Q(S)
\rightarrow \mathcal{ML}(S)^2$ induces a homeomorphism between $V-\mathcal F(S)$ and its image.
For the canonical identifications between $\check {\mathcal Q}(S)-\partial
\check {\mathcal Q}(S)$ and $
{\mathcal Q}(S)- \mathcal F(S)$, and between $\check{\mathcal
F}\mathcal{ML}(S)
-\partial \check{\mathcal
F}\mathcal{ML}(S)$ and ${\mathcal F}\mathcal{ML}(S)$, the inverse of the restriction of $\check\Phi$ to $\check U -
\partial
\check {\mathcal Q}(S)$ coincides with $\beta$. Therefore, the only part which requires some checking is that the image $V$ of $\check V
= \check \Phi\bigl ( \check U
\bigr)$ under the canonical projection $p:\check {\mathcal Q}(S)
\rightarrow \mathcal Q(S)$ is open in $\mathcal Q(S)$. However, this immediately follows from the fact that the preimages of points under $p$ are all compact, which implies that $p$ is an open map. (Note that this is false for the projection $\check{\mathcal
F}\mathcal{ML}(S)
\rightarrow {\mathcal
F}\mathcal{ML}(S) \cup \{0\}$.)
Theorem \[thm:HomeoNoCheck\] is just Theorem \[thm:MainThm\] stated in the introduction. Theorem \[thm:LocalImageBeta\] immediately follows from the definition of $\check U$ and from the fact that $\check U$ is open (Theorem \[them:HomeoInCheck\]).
We conclude with a few remarks.
A recent result of Caroline Series [@Ser2] shows that we can restrict the neighborhood $V$ of Theorem \[thm:HomeoNoCheck\] so that $\beta^{-1} \left(\beta(V)\right)
=V$.
When $\mu$ and $\nu$ are multicurves, namely when their supports consist of finitely many closed geodesics, it follows from [@HodKer] that the submanifolds $\check {\mathcal
P}^+(\mu)$ and $\check {\mathcal
P}^-(\nu)$ are everywhere transverse, by a doubling argument as in [@BonOta]. Consequently, the open subset $V$ of Theorem \[thm:HomeoNoCheck\] can be chosen so that the image $U=\beta (V) \subset \mathcal
{ML}(S)^2$ contains all rays of the form $\left[ 0, \infty \right[
(\mu, \nu)$ where $\mu$ and $\nu$ are multicurves.
As indicated in the introduction, it is conjectured that we can take $V$ equal to the whole space $\mathcal Q(S)$. See [@BonOta] for a characterization of the image of $\mathcal Q(S)$ under $\beta$.
[EpM]{}
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Steven P. Kerckhoff, *The Nielsen realization problem*, Ann. of Math. **117** (1983), pp. 235–265.
, *Lines of minima in Teichmüller space*, Duke Math. J. **65** (1992), pp. 187–213.
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Caroline M. Series, *On Kerckhoff minima and pleating loci for quasi-Fuchsian groups*, Geom. Dedicata **88** (2001), pp. 211–237.
, *Limits of quasifuchsian groups with small bending*, preprint, 2002.
Dennis P. Sullivan, *Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups*, Acta Math. **155** (1985), pp. 243–260.
William P. Thurston, *The topology and geometry of $3$–manifolds*, Lecture notes, Princeton University, 1976–79.
, *Earthquakes in two-dimensional hyperbolic geometry*, in: *Low-dimensional Topology and Kleinian groups* (D.B.A. Epstein ed.), L.M.S. Lecture Notes Series vol. **112**, 1986, Cambridge University press, pp. 91–112.
Scott A. Wolpert, *On the symplectic geometry of deformations of a hyperbolic surface*, Ann. of Math. **117** (1983), pp. 207–234.
, *Geodesic length functions and the Nielsen problem*, J. Differential Geom. **25** (1987), pp. 275–296.
[^1]: This work was partially supported by grant DMS-0103511 from the National Science Foundation.
|
---
abstract: 'We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivator behind this numerical method is estimating solutions to Adverse Selection problems within a Principal-Agent framework. Problems such as product lines design, optimal taxation, structured derivatives design, etc. can be studied through the scope of these models. We develop a method to estimate their optimal pricing schedules.'
author:
- |
Ivar Ekeland\
Department of Mathematics\
University of British Columbia\
1984 Mathematics Road\
Vancouver, BC, V6T 1Z2\
[email protected]
- |
Santiago Moreno-Bromberg\
Department of Mathematics\
University of British Columbia\
1984 Mathematics Road\
Vancouver, BC, V6T 1Z2\
[email protected]
title: 'A Numerical Approach to the Estimation of Solutions of some Variational Problems with Convexity Constraints [^1]'
---
<span style="font-variant:small-caps;">Preliminary - Comments Welcome</span>
**AMS classification**: 49-04, 49M25, 49M37, 65K10, 91B30, 91B32.
**Keywords**: Variational problems, convexity constraints, adverse selection, non-linear pricing, risk transfer, market screening.
Introduction
============
Arguably, Newton’s problem of the body of minimal resistance is the original variational problem with convexity constraints. It consists of finding the shape of a solid that encounters the least resistance when moving through a fluid. This is equivalent to finding a convex function from a convex domain (originally a disk) in ${\mathbb{R}}^2$ to ${\mathbb{R}}$ that minimizes a certain funtional (see section \[problem\]). Newton’s original “solution" assumed radial symmetry. This turned out to be false, as shown by Brock, Ferone and Kawohl in [@kn:bfk], which sparked new interest to the study of variational problems with convexity constraints. One can also find these kinds of problems in finance and economics. Starting in 1978 with the seminal paper of Mussa and Rosen [@kn:mr], the study of non-linear pricing as a means of market screening under Adverse-Selection has produced a considerable stream of contributions ([@kn:A],[@kn:cet],[@kn:rch],...). In models where goods are described by a single [quality]{} and the set of [agents]{} is differentiated by a single parameter, it is in general possible to find closed form solutions for the [pricing schedule]{}. This is, however, not the case when multidimensional [consumption bundles]{} and agent [types]{} are considered. Although Rochet and Choné [@kn:rch] provided conditions for the existence of an optimal pricing rule and fully characterized the ways in which markets differentiate in a multidimensional setting, they also pointed out that it is only in very special cases that one can expect to find closed form solutions. The same holds true for models where the set of goods lies in an infinite-dimensional space, even when agent types are one-dimensional. This framework was first used, to our knowledge, by Carlier, Ekeland and Touzi [@kn:cet] to price financial derivatives traded “over-the-counter". It was then extended by Horst and Moreno [@kn:HM] to model the actions of a monopolist who has an initial risky position that she evaluates via a coherent risk measure, and who intends to transfer part of her risk to a set of heterogenous agents. In both cases the authors find that only very restrictive examples allow for explicit solutions.
Given that a great variety of problems, such as product lines design, optimal taxation, structured derivatives design, etc. can be studied through the scope of these models, there is a clear need for robust and efficient numerical methods that approximate their optimal pricing schedules. Note that this also provides an approximation of the optimal “products". Most of the papers mentioned above eventually face solving a variational problem under convex constraints. This family of problems have lately been studied under different scopes. Carlier and Lachand-Robert [@kn:CLR] have studied the $C^1$ regularity of minimizers when the functional is elliptic and the admissible functions satisfy a Dirichlet-type boundary condition. Their results can be extended to of our examples. Lachand-Robert and Pelletier [@kn:LRP] characterize the extreme points of a functional depending only on $\nabla f$ over a set of convex functions with uniform convex bounds. In this paper we provide several variants of an algorithm, based on the idea of approximating a convex function by an affine envelope, to solve these types of problems. This deviates from previous work by Choné and Hervé [@kn:ch] and Carlier, Lachand-Robert and Maury [@kn:cl], where the authors use finite element methods. In the former case, a conformal (interior) method is used and a non-convergence result is given. As a consequence, the latter uses an exterior approximation method, which is indeed found to be convergent in the classical projection problem in $H_0^1.$ Lachand-Robert and Oudet present in [@kn:LRO] and algorithm for minimizing functionals within convex bodies that shares some similarities to ours. For a particular problem, they start with an admissible polytope and iteratively modify the normals to the facets in order to find an approximate minimizer.
We estimate the minimizers for several problems with known, closed form solutions as a means of comparing the output of our method to the true solutions. These are taken from [@kn:rch] and [@kn:cet]. Finally, we provide an example in which we approximate the solution to a risk-minimization problem similar to the one presented in [@kn:HM]. This is still based on the affine-envelope idea, but requires some additional methodology, since it involves solving a non-standard variational problem.
The remainder of this paper is organized as follows. In Section \[problem\] we state our problem and provide some classical examples. Our algorithm and a proof of its convergence are presented in Section \[DescriptionConvex\]. In Section \[Examples\] we show the solutions obtained via our algorithm to several problems found in the literature. Since these problems share a common microeconomic motivation, we include a brief discussion on the latter. The examples include the well known “Rochet-Choné" problem, a one dimensional example from Carlier, Ekeland and Touzi and the risk transfer case for a principal who offers call options with type-dependent strikes and evaluates her risk via the “short fall" of her position. This section is followed by our conclusions. Finally a section devoted to technical results and all our codes are included in the appendix.
Setting {#problem}
=======
The aim of this paper is to present a numerical algorithm to approximate the solutions of some variational problems subject to convexity constraints. A classical example of the latter is Newton’s problem of the body of minimal resistance, which, given $\T$ a smooth subset of ${\mathbb{R}}^2,$ consists of minimizing $$I[v]=\int_{\T}\frac{d\t}{1+|\nabla v|^2},$$ over the set of convex functions $\{f:\T\to{\mathbb{R}}\}.$ We use the following notation throughout:
- $\T,Q\subset{\mathbb{R}}^n$ are convex and compact sets,
- $L(\t,z,p)=z-\t\cdot p+C(p),$ where $C$ is strictly convex and $C^1.$
- $\C:=\{f:\T\to{\mathbb{R}}\,\mid\,f\ge 0\,\,{\mbox{is convex, and}}\,\,\nabla f\in Q\,\,{\mbox{a.e}}\},$
- $I[f]:=\int_{\Theta}L(\t, f(\t), \nabla f(\t))d\t.$
Our objective is to (numerically) estimate the solution to $$\label{eq:problem}
{\cal{P}}:=\inf_{f\in C}I[f]$$ We assume $C$ is such that (\[eq:problem\]) has a unique solution (see, for example, [@kn:GH]). Given the properties of $L$ and ${\cal{C}}$ we immediately have the following
\[zerov\] Assume $\v$ solves ${\cal{P}},$ then there is $\t_0$ in $\T$ such that $\v(\t_0)=0.$
Let $\v_0=\min_{\t\in\T}\v(\t)$ (recall $\T$ is compact)and define $\overline{u}(\t):=\v(\t)-\v_0,$ then $$I[\u]=\int_{\T}\u(\t)-\t\cdot\nabla\u(\t)+C(\nabla\u(\t))d\t=I[\v]-\|\T\|\v_0.$$ This would contradict the hypothesis of $\v$ being a minimizer of $I$ over $\C$ unless $\v_0=0.$
It follows from proposition \[zerov\] that we can redefine $\C$ to include only functions that have a root in $\T.$ This, together with the compactness of $Q,$ implies the following proposition, which we will use frequently.
\[zerov2\] There exists $0<K<\infty$ such that $v\le K$ for all $v$ in $\C.$
It follows from Proposition \[zerov2\] and the restriction on the gradients that for each choice of function $C,$ problem ${\cal{P}}$ has a unique solution, since the functional $I$ will be strictly convex, lower semi continuous and the admissible set is bounded (see [@kn:et]).
Our algorithm will still work with more general $L$’s as long as one can prove that the family of feasible minimizers is uniformly bounded.
Description of the Algorithm {#DescriptionConvex}
============================
From this point on, whenever we use supscripts we refer to vectors. For example $V^k=(V^k_1,\ldots, V^k_m).$ On the other hand a subscript indicates a function to be evaluated over some closed, convex subset of ${\mathbb{R}}^k$ of non-empty interior, ie, $\{V_k\}$ is a sequence of functions $V_k:X\to{\mathbb{R}}$ for some $X$ contained in ${\mathbb{R}}^n.$
We will consider $\T=[a,b]^n.$
To find an approximate solution to ${\cal{P}},$ we proceed as follows:
1. We discretize the domain $\T$ in the following way: We partition it into $\Sigma_k,$ which consists of $k^n$ equal cubes of volume $\|\Sigma_k\|:=\left(\frac{b-a}{k}\right)^n.$ The elements of $\Sigma_k$ will be denoted by $\sigma_j^k,$ $1\le j\le k^n.$ Now define $\T_k$ as the set of centers of the $\sigma_j^k$’s. The elements of $\T_k$ will be denoted by $\t_j^k.$ The choice of a uniform partition is done for computational simplicity.
2. We denote $f_i=f\left(\t_i^k\right)$ and associate such weight with $\t_i^k.$
3. We associate to each element $\t_i^k$ of $\T_k$ a non-negative number $v_i^k$ and an n-dimensional vector $D_i^k.$ The former represents the value of $v(\t_i^k)$ and the latter $\nabla v(\t_i^k).$
4. We solve the (non-linear) program
$$\label{eq:ppd}
{\cal{P}}_k:=\inf\|\Sigma_k\|\sum_{i=1}^{k^n}L\left(\t_i, v_i, D_i\right)f_i$$
over the set of all vectors of the form $v=(v_1,\ldots,v_{k^n})$ and all matrices of the form $D=(D_1,\ldots,D_{k^n})$ such that:
1. $v\ge 0$ (non-negativity),
2. $D_i\in Q$ for $i=1,\ldots k^n$ (feasibility),
3. $v_i-v_j+D_i\cdot(\t_j-\t_i)\le 0$ (convexity).
If the problem in hand includes Dirichlet boundary conditions these can be included here as linear constraints that the $D_i$’s corresponding to points on the “boundary" of $\T_k$ must satisfy.
5. Let $(\v^k, \D^k)$ be the solution to ${\cal{P}}_k.$ We define $\v_k(\t):=\max_i p_i(\t),$ where
$$p_i(\t)=\v^k_i+\D^k_i\cdot (\t-\t_i).$$
6. $\v_k$ yields an approximation to the minimizer of ${\cal{P}}.$
\[rem:constraints\] The constraints of the non-linear program determine a convex set.
\[rem:pwafine\] 4 (c) guarantees that $p_i$ is a supporting hyperplane of the convex hull of the points $\{(\t_1, v_1),\ldots,(\t_{k^n}, v_{k^n})\}.$ Note that $\v_k$ is a piecewise affine convex function.
Convergence of the Algorithm {#ConvergenceConvex}
----------------------------
Under the assumptions made on $L,$ the problem ${\cal{P}}_k$ has a unique solution.
The function $$J_k(v^k,D^k):=\sum_{i=1}^{k^n}\left(\t_i\cdot D_i^k-v_i^k-C(D_i^k)\right)\|\Sigma_k\|f_i$$ is strictly convex. It follows from proposition \[zerov2\] that any acceptable vector-matrix pair $(v^k,D^k)$ must lie in $[0,K]^k\times Q^k,$ which together with Remark \[rem:constraints\] implies ${\cal{P}}_k$ consists of minimizing a strictly convex function over a compact and convex set. The result then follows from general theory.
\[pr:conv\] There exists $\v\in{\cal{C}}$ such that:
1. The sequence $\{\v_k\}$ generated by the ${\cal{P}}_k$’s has a subsequence $\{\v_{k_j}\}$that converges uniformly to $\overline{v}.$
2. $\lim_{k_j\to\infty} I[\v_{k_j}]=I[\overline{v}].$
The bounded (Proposition \[zerov2\]) family $\{\v_j\}$ is uniformly equicontinuous, as it consists of convex functions with uniformly bounded subgradients. By the Arzela-Ascoli theorem we have that, passing to a subsequence if necessary, there is a non-negative and convex function $\overline{v}$ such that
$$\v_k\to\v\quad{\mbox{uniformly on}}\,\,\T.$$ By convexity $\nabla\v_k\to\nabla\v$ almost everywhere (lemma \[pr:vprime\]); since $\nabla\v_k(\t)$ belongs to the bounded set $Q,$ the integrands are dominated. Therefore, by Lebesgue Dominated Convergence we have
$$\lim_{k\to\infty}I\left[\v_k\right]=I[\v] .$$
Let $\u$ be the maximizer of $I[\cdot]$ within $C.$ Our aim is to show that $\{\v_k\}$ is a minimizing sequence of problem ${\cal{P}},$ in other words that
$$\lim_{k\to\infty}I\left[\v_k\right]=I\left[\u\right] .$$ We need the following
Let $\u$ be such that $\inf_{u\in{\cal{C}}}I[u]=I[\u].$ Given the lattice $\T_k,$ we define:
1. $\u^k_i:=\overline{u}(\t_i),$
2. $G^k_i:=\nabla {u}(\t_i),$
3. $q_i(\t):=\u^k_i+G^k_i\cdot(\t-\t_i)$ and
4. $\u_k(\t):=\sup_i q_i(\t).$
Notice that $\u_k(\t)$ is also constructed as the convex envelope of a family of affine functions. The inequalities
$$\label{eq:JN}
J_k(\u^k, G^k)\ge J_k(\v^k, \D^k)$$
$$\label{eq:IN}
I[\v_k]\ge I[\u]$$
follow from the definitions of $J_k(\v^k, \D^k),$ $\v_k$ and $\u_k,$ as does the following
\[pr:conv\] Let $\u$ and $\u_k$ be as above, then $\u_k\to\u$ uniformly as $k\to\infty.$
\[pr:conv2\] For each $k$ there exist $\epsilon_1(k)$ and $\epsilon_2(k)$ such that
$$\label{eq:JI1}
\left| J_k(\v^k, \D^k)-I[\v_k]\right|\le\epsilon_1(k)$$
$$\label{eq:JI2}
\left| J_k(\u^k, G^k)-I[\u_k]\right|\le\epsilon_2(k)$$
and $\epsilon_1(k),\epsilon_2(k)\to 0$ as $k\to\infty.$
We will show (\[eq:JI1\]) holds, the proof for (\[eq:JI2\]) is analogous. Define the simple function $$w_k(\t):=L(\t_j^k, v_j^k, D_j^k),\,\, \t\in \sigma_j^k,$$ hence
$$J_k(\v^k, \D^k)=\int_{\T}w_k(\t) d\t.$$
The left-hand side of (\[eq:JI1\]) can be written as
$$\label{eq:JI3}
\left|\int_{\T}w_k(\t)d\t-I[\v_k]\right|$$
It follows from Lemma \[eq:affineconv\] that there exists $\epsilon_1(k),$ such that $$\left|\int_{\T}w_k(\t)d\t-I[v_k]\right|\le\epsilon_1(k)$$ and $$\epsilon_1(k)\to 0\quad k\to\infty.$$
We can now prove the main theorem in this section, namely
The sequence $\{\v_k\}$ is minimizing for problem ${\cal{P}}.$
It follows from Proposition \[pr:conv2\] and equation (\[eq:JN\]) that
$$\label{eq:conv3}
I[\u_k]+\epsilon_2(k)+\epsilon_1(k)\ge I[\v_k]\ge I[\u]$$
Letting $k\to\infty$ in (\[eq:conv3\]) and using Proposition \[pr:conv\] yields the desired result.
Examples {#Examples}
========
In this section we show some results of implementing our algorithm. The first two examples reduce quadratic programs, whereas the third and fourth ones are non-linear optimization programs. All the computer coding has been written in MatLab. However, in both cases supplemental Optimization Toolboxes were used. In the first two examples we used the Mosek 4.0 Optimization Toolbox, wherease in the last two we used Tomlab 6.0. These four examples share a common microeconomic motivation, for which we provide an overview. We refer the interested reader to [@kn:AS] for a comprehensive presentation of Principal-Agent models and Adverse Selection, as well as multiple references.
Some Microeconomic Motivation {#Setup}
-----------------------------
Consider an economy with a single [principal]{} and a continuum of [agents]{}. The latter’s preferences are characterized by n-dimensional vectors. These are called the agents’ [types]{}. The set of all types will be denoted by $\T\subset{\mathbb{R}}^n.$ The individual types $\t$ are private information, but the principal knows their statistical distribution, which has a (non-atomic) density $f(\t).$
Our model takes a [hedonic]{} approach to product differentiation. We assume goods are characterized by (n-dimensional) vectors describing their utility-bearing attributes. The set of [technologically feasible]{} goods that the principal can deliver will be denoted by $Q\subset{\mathbb{R}}_+^n,$ and it will be assumed to be compact and convex. The cost to the principal of producing one unit of product $p$ is denoted by $C(p).$ Products are offered on a take-it-or-leave-it basis, each agent can buy one or zero units of a single product $p$ and it is assumed there is no second-hand market. The (type-dependent) preferences of the agents are represented by the function
$$U:\T\times Q\to{\mathbb{R}}.$$ The (non-linear) price schedule for the technologically feasible goods is represented by
$$\pi:Q\to{\mathbb{R}}.$$ When purchasing good $q$ at a price $\pi(q)$ an agent of type $\t$ has net utility
$$U(\t, q)-\pi(q)$$ Each agent solves the problem
$$\max_{q\in Q}\left\{ U(\t, q)-\pi(q)\right\}.$$ By analyzing the choice of each agent type under a given price schedule $\pi,$ the principal screens the market. Let
$$\label{Max-ut}
v(\t):=U(\t, q(\t))-\pi(q(\t)),$$
where $q(\t)$ belongs to $argmax_{q\in Q} \left\{U(\t,q)-\pi(q)\right\}.$ Notice that for all $q$ in $Q$ we have $$\label{Max-ut2}
v(\t)\ge U(\t, q)-\pi(q)$$ Analogous to the concepts of [subdifferential]{} and [convex conjugate]{} from classical Convex Analysis, we have that the subset of $Q$ where (\[Max-ut2\]) is an equality is called the $U$-[subdifferential]{} of $v$ at $\t$ and $v$ is the $U$-[conjugate]{} of $
\pi$ (see, for example, [@kn:Car]). We write
$$v(\t)=\pi^U(\t)$$ and $$\partial_Uv(\t):=\{q\in Q\,\mid\,\pi^U(\t)+\pi(q)=U(\t, q)\}$$ To simplify notation let $\pi(q(\t))=\pi(\t).$ A single pair $(q(\t), \pi(\t))$ is called a [contract]{}, whereas $\{(q(\t),\pi(\t))\}_{\t\in\T}$ is called a [catalogue]{}. A catalogue is called [incentive compatible]{} if $v(\t)\ge v_0(\t)$ for all $\t\in\T,$ where $v_0(\t)$ is type’s $\t$ non-participation (or reservation) utility. We normalize the reservation utility of all agents to zero, and assume there is always an [outside option]{} $q_0$ that denotes non-participation. Therefore we will only consider functions $v\ge 0.$ The Principal’s aim is to devise a pricing function $\pi:Q\to{\mathbb{R}}$ as to maximize her income
$$\label{eq:profit}
\int_{\T}\left(\pi(\t)-C(q(\t))\right)f(\t)d\t$$
Inserting (\[Max-ut\]) into (\[eq:profit\]) we get the alternate representation
$$\label{eq:pprofit}
\int_{\T}\left(U(\t, q(\t))-v(\t)-C(q(\t))\right)f(\t)d\t.$$
Expression (\[eq:pprofit\]) is to be maximized over all pairs $(v, q)$ such that $v$ is U-convex and non-negative and $q(\t)\in\partial_Uv(\t).$ Characterizing $\partial_Uv(\t)$ in a way that makes the problem tractable can be quite challenging. In the case where $U(\t, q(\t))=\t\cdot q(\t),$ as in [@kn:rch], for a given price schedule $\pi:Q\to{\mathbb{R}},$ the maximal net utility of an agent of type $\t$ is
$$\label{eq:v}
v(\t):=\max_{q\in Q}\left\{ \t\cdot q-\pi(q)\right\}$$
Since $v$ is defined as the supremium of its affine minorants, it is a convex function of the types. It follows from the Envelope Theorem that the maximum in equation (\[eq:v\]) is attained if $q(\t)=\nabla v(\t),$ and we may write
$$\label{v-convex}
v(\t)= \t\cdot\nabla v(\t)-\pi(\nabla v(\t)).$$
The principal’s aggregate surplus is given by
$$\label{eq:surplus}
\int_{\T}\left(\pi (q(\t))-C(q(\t))\right)f(\t)d\t.$$
Inserting (\[v-convex\]) into (\[eq:surplus\]) we get that the principal’s objective is to maximize
$$\label{Ppal-Conv}
I[v]:=\int_{\T}\left(\t\cdot\nabla v(\t)-C(\nabla v(\t))-v(\t)\right)f(\t)d\t$$
over the set
$${\cal{C}}:=\left\{v:\T\to{\mathbb{R}}\,\mid\, v\,{\mbox{convex}},\, v\ge 0,\,\nabla v(\t)\in Q\right\}.$$
The Musa-Rosen Problem in a Square
----------------------------------
The following structures are shared in the first two examples:
- $x=(v^k, D^k),$ this structure will determine any possible candidate for a minimizer to $J_k(\cdot,\cdot)$ in the following way: $v^k$ is a vector of length $k^2$ that will contain the approximate values of the optimal function $\v$ evaluated on the points of the lattice. The vector $D^k$ has length $2*k^2$ and it contains what will be the partial derivatives of $\v$ at the same points
- $h$ is a vector of length $3*k^2.$ The product $hx$ provides the discretization of the integral $\int_{\T}(\t\cdot\nabla {v}-{v}(\t))f(\t)d\t.$
- $B$ is the matrix of constraints. The inequality $Bx\le 0$ imposes the non-negativity of $v$ and $D$ and the convexity of the resulting $\v_k.$
The density $f(\t)$ is “built into" vector $h$ and the cost function $C.$
Let $\T=[1,2]^2,$ $C(q)=\frac{1}{2}\|q\|^2,$ and assume the types are uniformly distributed. This is our the benchmark problem, since the solution to the principal’s problem can be found explicitly [@kn:rch]. In this case we have to solve the quadratic program
$$\sup_x hx-\frac{1}{2}x^tHx$$ subject to $$Bx\le 0$$ $H$ is a $(3*k^2)\times(3*k^2)$ matrix whose first $k^2$ columns are zero, since $v$ does not enter the cost function; the four $k^2\times k^2$ blocks towards its lower right corner form a $(2*k^2)\times (2*k^2)$ identity matrix. Therefore $\frac{1}{2}x^tHx$ is a discretization of $\int_{\T}\|\nabla {v}(\t)\|d\t.$ Figure \[figure:vvv\] was produced using a $17\times 17$-points lattice and a uniform density, whereas Figure \[figure:v1\] shows the traded qualities.
The Musa-Rosen Problem with a Non-uniform Density
-------------------------------------------------
We keep the cost function of the previous example, but now assume the types are distributed according to a bivariate normal distribution with mean $(1.9, 1)$ and variance-covariance matrix
$$\left[\begin{array}{ll}
.3 & .2\\
.2 & .3
\end{array}\right]$$ As noted before, the weight assigned to each to each agent type is built into $h$ and $H,$ so the vector $x$ remains unchanged. We obtain figure \[figure:vvvv\].
It is interesting to see that in this case bunching of the second kind, as described by Roché and Choné in [@kn:rch], appears to be eliminated as a consequence of the skewed distribution of the agents. This can be seen in the non-linear level curves of the optimizing function $v.$ This is also quite evident in the plot of the qualities traded, which is shown below.
The MatLab programs for the two previous examples were run on MatLab 7.0.1.24704 (R14) in a Sun Fire V480 (4$\times$1.2 HGz Ultra III, 16 GB RAM) computer running Solaris 2.10 OS. In the first example 57.7085 seconds of processing time were required. The running time in the second example was 81.7280 seconds.
An example with Non-quadratic Cost
----------------------------------
In this example we approximate a solution to the problem of a principal who is selling over-the-counter financial derivatives to a set of heterogeneous agents. This model is presented by Carlier, Ekeland and Touzi in [@kn:cet]. They start with a standard probability space $(\O, {\cal F}, P),$ and the types of the agents are given by their risk aversion coefficients under the assumption of mean-variance utilities; namely, the set of agent types is $\T=[0,1],$ and the utility of an agent of type $\t$ when facing product $X$ is
$$U(\t, X)=E[X]-\t{\mbox Var}[X]$$
Under the assumptions of a zero risk-free rate and that the principal has access to a complete market, her cost of delivering product $X(\t)$ is given by $-\sqrt{-\xi v'(\t)};$ where $\xi$ is the variance of the Radon-Nikodym derivative of the (unique) martingale measure, and Var$[X(\t)]=-v'(\t).$ The principal’s problem can be written as
$$\sup_{v\in{\cal C}}\int_{\T}\left(\t v'(\t)+\sqrt{-v'(\t)}-v(\t)\right)d\t$$
where ${\cal C}:=\left\{v:\T\to{\mathbb{R}}\,\mid\,v\,\,{\mbox {convex}},\,v\ge 0, v'\le 0\,\, {\mbox{ and Var}}\,\,[X(\t)]=-v'(\t)\right\}.$ Figure \[figure:cet\] shows an approximation of the maximizing $\v$ using 25 agent types.
\[figure:cet\]
Minimizing Risk {#Risk}
---------------
The microeconomic motivation for this section is the model of Horst & Moreno [@kn:HM]. We present an overview for completeness. The principal’s income, which is exposed to non-hedgeable risk factors, is represented by $W \leq 0.$ The latter is a bounded random variable defined on a standard, non-atomic, probability space $\left(\Omega,{\cal{F}}, {\mathbb{P}}\right).$ The principal’s goal is to lay off parts of her risk with the agents whose preferences are mean-variance. The agent types are indexed by their coefficients of risk aversion, which are assumed to lie $\Theta = [a,1]$ [for some ]{} $a>0.$ The principal underwrites call options on her income with type-dependent strikes:
$$X(\t) = (|W|-K(\t))^+ \quad \mbox{with} \quad 0 \leq K(\t)
\leq \|W\|_\infty.$$
If the principal issues the catalogue $\{(X(\t), \pi(\t))\}$, she receives a cash amount of $\int_{\T} \pi \left(\theta \right)d \theta$ and is subject to the additional liability $\int_{\T} X(\theta)
d\t.$ She evaluates the risk associated with her overall position $$W + \int_{\T}(\pi(\t)-X(\t))d\t$$ via the “entropic measure" of her position, i.e. $$\rho\left(W+\int_{\T}(X(\t)-\pi(\t))d\t\right)$$ where $\rho(X)=log E[exp\{-\beta X\}]$ for some $\beta>0.$ The principal’s problem is to devise a catalogue as to minimize her risk exposure. Namely, she chooses a function $v$ and contracts $X$ from the set
$$\{(X,v) \mid v \in {\cal C}, \, v \leq K_1, \, -\textnormal{Var}[(|W|-K(\t))^+] =
v'(\t), \, |v'| \leq K_2, \, 0 \leq K(\t) \leq \|W\|_\infty \},$$ in order to minimize $$\rho\left( W - \int_{\T}\left\{ (|W|-F(v'(\t)))^+ - {\mathbb{E}}[(|W|-F(v'(\t)))^+] \right\} d\right) -
I(v).$$ where $$I(v) = \int_{\T} \left( \t v'(\t) - v(\t) \right) d\t.$$ We assume the set of states of the World is finite with cardinality $m.$ Each possible state $\omega_j$ can occur with probability $p_j.$ The realizations of the principal’s wealth are denoted by $W=(W_1,\ldots,W_m).$ Note that $p$ and $W$ are treated as known data. The objective function of our non-linear program is
$$\begin{aligned}
F(v,v',K) &=& log\left(exp\left\{-\sum_{i=1}^nW_ip_i+\frac{1}{n}\sum_{i=1}^n\left(\sum_{j=1}^n
T(K_j-|W_i|)\right)p_i \right.\right.\\
&-&\left.\left.\frac{1}{n}\sum_{i=1}^n\left(\sum_{j=1}^{n}T(K_j-|W_i|)\right)p_i \right\}\right)\\
&+&
\frac{1}{n}\sum v_i-\t_i v_i'\end{aligned}$$
where $K=(K_1,\ldots,K_n)$ denotes the vector of type dependent strikes. We denote by $ng$ the total number of constraints. The principal’s problem is to find
$$\min_{(v, v', K)} F(v, v', K) \quad \mbox{subject
to} \quad G(v, v', K)\le 0$$ where $G:{\mathbb{R}}^{3n}\to{\mathbb{R}}^{ng}$ determines the constraints that keep $(v,v',K)$ within the set of feasible contracts. Let $(1/6, 2/6,\ldots, 1)$ be the uniformly distributed agent types, and
- $W=4*(-2, -1.7, 1.4,-.7, -.5, 0),$
- $P=(1/10, 1.5/10, 2.5/10, 2.5/10, 1.5/10, 1/10).$
The principal’s initial evaluation of her risk is $1.52$. The following are the plots for the approximating $\v$ and the strikes:
Note that for illustration purposes we have changed the scale for the agent types in the second plot. The interpolates of the approximate to the optimal function $\v$ and the strikes are:
$\v_{1}$ 4.196344
---------- ----------
$\v_{2}$ 3.234565
$\v_{3}$ 2.321529
$\v_{4}$ 1.523532
$\v_{5}$ 0.745045
$\v_{6}$ 0.010025
$K_{1}$ 1.078869
--------- ----------
$K_{2}$ 0.785079
$K_{3}$ 0.733530
$K_{4}$ 0.713309
$K_{5}$ 0.713309
$K_{6}$ 0.713309
The Principal’s valuation of her risk after the exchanges with the agents decreases from $11.49$ to $-3.56.$
Notice the ”bunching" at the bottom.
Conclusions
===========
In this paper we have developed a numerical algorithm to estimate the minimizers of variational problems with convexity constraints, with our main motivation stemming from Economics and Finance. Ours is an [internal]{} method, so at each precision level the approximate minimizers lie within the acceptable set of functions. Our examples are developed over one or two dimensional sets for illustration reasons, but the algorithm can be implemented in higher dimensions. However, it must be mentioned that, as is the case with the other methods found in the related literature, implementing convexity has a high computational cost which increases geometrically with dimension.
Appendix
========
Some technical results {#ConvergenceConvex}
======================
In order to prove convergence of our algorithm we make use of the Convex Analysis results contained in this section. We will work on $C,$ an open and convex subset of ${\mathbb{R}}^n.$
A mapping $F:C\to P({\mathbb{R}}^m)$ (the power set of ${\mathbb{R}}^m$) is said to be [set valued]{} if for each $x$ in $C,$ $F(x)$ is a non-empty subset of ${\mathbb{R}}^m.$
Recall that if $f:C\to{\mathbb{R}}$ is a convex function, then the [subdifferential]{} of $f$ at $x,$ defined as $$\partial f(x):=\{b\in{\mathbb{R}}^n\,\mid\,f(y)-f(x)\ge b\cdot (y-x)\quad{\mbox{for all}}y\in C\},$$ is a non-empty subset of ${\mathbb{R}}^n$ for all $x$ in $C.$ Therefore, the set valued mapping $$x\to\partial f(x)$$ is well defined on $C.$ Notice that if for some $x$ in $C$ we have $\#\partial f(x)=1,$ then $$\partial f(x)=\{\nabla f(x)\}.$$ In such case we say the [subdifferential mapping]{} is single valued at $x$ and we can simply identify it with the gradient of $f$ at $x.$
\[diff\]Let $F:C\to{\mathbb{R}}^m$ be a set valued function. Then we say $F$ is [differentiable]{} at $x_0$ iff there is a linear mapping $L_{x_0}:C\to{\mathbb{R}}^m$ such that for all $\epsilon>0$ there is $\delta>0$ such that if $y_0\in f(x_0),\,y\in f(x)$ and $\|x-x_0\|<\delta$ then $$\frac{\|y-y_0-L_{x_0}(x-x_0)\|}{\|x-x_0\|}\le\epsilon$$
The following theorem is due to Alexandrov ( [@kn:HW])
Let $f:{\mathbb{R}}^n\to{\mathbb{R}}$ be convex, then se set valued function $\partial f$ is differentiable almost everywhere.
Clearly, in the case where $f$ is single valued, definition \[diff\] is equivalent to the regular definition of a Fréchet differentiable function. Moreover if $f$ differentiable at $x_0$ and we choose $y_0,$ $y_1$ in $f(x_0)$ and let $x=x_0$ in definition \[diff\] we get $$\|y_0-y_1\|\le 0,$$ which implies $f$ is single valued at $x_0.$ It follows from Alexandrov’s Theorem and the observation above that for almost all $\t\in\T,$ the set valued mapping $\t\to\partial f(\t)$ can be identified with the single valued assignment $\t\to\nabla f(\t),$ and we have the following
\[lm:ConvCont\] Let $f:C\to{\mathbb{R}}$ be convex . Then the mapping $$\t\to\nabla f(\t)$$ is well defined and continuous almost everywhere.
\[pr:vprime\] Let $A\subset{\mathbb{R}}^n$ be a convex, open set. Assume the sequence of convex functions $\{f_k:A
to{\mathbb{R}}\}$ converges uniformly to $\bar{f},$ then $\nabla f_k\to\nabla\bar{f}$ almost everywhere on $A.$
Denote by $D_if$ the derivative of $f$ in the direction of $e_i.$ The convexity of $f_k$ and $\bar{f}$ implies the existence of a set $B,$ with $\mu(A\setminus B)=0$ such that the partial derivatives of $f_k$ and $\bar{f}$ exist and are continuous in $B.$ Let $x\in B.$ To prove that $D_i f_k(x)\to D_i \bar{f}(x),$ consider $\eta$ such that $x+\eta e_i\in A.$ Since $f_k$ is convex
$$\frac{f_k(x+he_i)-f_k(x)}{h} \ge D_i f_k(x)
\ge\frac{f_k(x-he_i)-f_k(x)}{h}$$ for all $0<h<\eta.$ Hence $$\frac{f_k(x+he_i)-f_k(x)}{h} - D_i\bar{f}(x) \ge D_i f_k(x) -
D_i\bar{f}(x) \ge\frac{f_k(x-he_i)-f_k(x)}{h} - D_i\bar{f}(x).$$ The left-hand side of this inequality is equal to $$\frac{f_k(x+he_i)-\bar{f}(x+he_i)}{h}+\frac{\bar{f}(x)-f_k(x)}{h}+\frac{\bar{f}(x+he_i)-\bar{f}(x)}{h}
- D_i\bar{f}(x).$$ For $\epsilon >0$ let $0<\delta<\eta$ be such that $$\left| \frac{\bar{f}(x+he_i)-\bar{f}(x)}{h} - D_i\bar{f}(x)\right|
<\epsilon$$ for $|h|\le\delta.$ Let $N\in{\mathbb{N}}$ be such that
$$-\epsilon\delta\le f_n(x)-\f(x)\le \epsilon\delta$$ $n\ge N.$ Hence, taking $h=\delta,$ we have that for all $n\ge N,$
$$\dfrac{f_n(x+he_1)-\f(x+he_1)}{h}\le\epsilon\quad{\mbox{and}} \quad\dfrac{\f(x)-f_n(x)}{h}\le\epsilon.$$ Hence $$3\epsilon \ge D_1 f_n(x)-D_1\f(x)$$ for all $x\in B.$ The same argument shows that $$-3\epsilon \le D_1 f_n(x)-D_1\f(x)$$ for all $n\ge N$ and all $x\in B,$ which concludes the proof.
\[convex:C1\] Let $U\subset{\mathbb{R}}^n$ be a convex, compact set and let $g:U\to{\mathbb{R}}$ be a convex function such that for all $x\in U,$ the subdifferentials $\partial g(x)$ are contained in $Q$ for some compact set $Q.$ Then there exists $\{g_j:U\to{\mathbb{R}}\}$ such that $g_j\in C^1(U)$ and $g_j\to g$ uniformly on U.
Fix $\delta>0$ and define $$U_{\delta}:=\{(1+\delta)x\,\mid\,x\in U\}.$$ Extend $g$ to be defined on $U_{\delta}.$ Let $K_{\epsilon}$ be a family of mollifiers (see, for instance [@kn:GH]), then the functions $$h_{\epsilon}:=g*K_{\epsilon}$$ are convex, smooth and they converge uniformly to $g$ on $U$ as long as $\epsilon$ is small enough so that $$U_{\epsilon}:=\{x\in U_{\delta}\,\mid\,d(x,\partial U_{\delta})>\epsilon\}$$ is contained in $U.$ Let $n\in{\mathbb{N}}$ be such that $U_{1/n}\subset U,$ then the sequence $\{g_j:=h_{1/j}\}$ has the required properties.
\[eq:affineconv\] Consider $\phi(\x, z, p)\in C^1\left(\T\times{\mathbb{R}}\times
Q\to{\mathbb{R}}\right),$ where $\T=[a,b]^n$ and $Q$ is a compact convex subset of ${\mathbb{R}}^n.$ Let $\{f_k:\T\to{\mathbb{R}}\}$ be a family of convex functions such that $\partial f_K(\t)\subset Q$ for all $\t\in\T,$ and whose uniform limit is $\f.$ Let $\Sigma_k$ be the uniform partition of $\T$ consisting of $k^n$ cubes of volume $\|\Sigma_k\|:=\left(\frac{b-a}{k}\right)^n.$ Denote by $\sigma_j^k,$ $1\le j\le k^n,$ be the elements of $\Sigma_k$ and let $$\|\Sigma_k\|\sum_{i=1}^{k^n} \phi(\x_j^k, f_k(\x_j^k), \nabla f_k(\x_j^k))$$ be the corresponding Riemann sum approximating $\int_{\T}\phi(x, f_k(\x), \nabla f_k(\x))d\x,$ where $\x_j^k\in\sigma_j^k$ and $\sigma_j^k\in\Sigma_k.$ Then for any $\epsilon>0$ there is $K\in{\mathbb{N}}$ such that
$$\label{eq:2}
\left|\int_{\T}\phi(\x, f_k(\x), \nabla f_k(\x))d\x-\|\Sigma_k\|\sum_{i=1}^{k^n} \phi(\x_j^k, f_k(\x_j^k), \nabla f_k(\x_j^k))\right|\le\epsilon$$
for any $k\ge K.$
By lemma \[convex:C1\], for each $f_k$ there exists a sequence of continuously differentiable convex functions $\{g^k_j\}$ such that $$g^k_j\to f_k\quad{\mbox{uniformly}}.$$ Let $h_k$ be the first element in $\{g^k_j\}$ such that $\|h_k-f_k\|\le\frac{1}{k}$ and $\|\nabla h_k(\t)-\nabla f_k(\t)\|\le\frac{1}{k}$ for all $\t\in\T$ where $\nabla f_k$ is continuous. Then $h_k\to\f$ uniformly, and by Lemma \[pr:vprime\] we have that $\nabla h_k(\t)\to\nabla\f(\t)$ a.e. It follows from Egoroff’s theorem that for every $n\in{\mathbb{N}}$ there exists a set $\Lambda_{n}\subset\T$ such that: $$\mu(\T\setminus\Lambda_{n})<1/n\quad{\mbox{and}}\quad\nabla h_k\to\nabla\f\quad{\mbox {uniformly on}}\quad\Lambda_{n}.$$ Let $\X_{\sigma_j^k}(\cdot)$ be the indicator function of $\sigma_j^k$ and define $$g_k(\t):=\phi(\t, h_k(\t), \nabla k_k(\t))-\sum_{j=1}^{k^n}\X_{\sigma_j^k}(\t)\phi(\t_j^k, h_k(\t_j^k), \nabla h_k(\t_j^k))$$ Fix $n,$ consider $\t_0\in\Lambda_n$ and let $\{\t_{0}^k\}$ be the sequence of $\t_j^k$’s converging to $\t_0$ as the partition is refined. By uniform convergence, $\nabla\f$ is continuous on $\Lambda_n,$ hence
$$\label{eq:pointwise}
\lim_{k\to\infty} h_k(\t_{0}^k)=\f(\t_{0})\quad{\mbox{and}}\quad\lim_{k\to\infty} \nabla h_k(\t_{0}^k)=\nabla\f(\t_{0})$$
It follows from (\[eq:pointwise\]) and the continuity of $\phi$ that $g_k\to 0$ almost everywhere on $\Lambda_n.$ Notice that as a consequence of the compactness of $\T$ and $Q$ and the definition of $h_k$ we have $$\|\phi(\x, f_k(\x), \nabla f_k(\x))\|\le K_1,\quad{\mbox{for al}}\quad\x\in\T\quad{\mbox{and some}}\quad K_1>0.$$ and $$\left| g_k(\t)-\left(\phi(\t, f_k(\t), \nabla f_k(\t))-\sum_{j=1}^{k^n}\X_{\sigma_j^k}(\t)\phi(\t_j^k, f_k(\t_j^k), \nabla f_k(\t_j^k))\right)\right|\le\frac{K_2}{k}$$ for some $K_2>0$ and all $\t\in\T$ where $\nabla f_k$ is continuous. Therefore $$\left|\int_{\T}\phi(\x, f_k(\x), \nabla f_k(\x))d\x-\|\sigma_j^k\|\sum_{i=1}^{k^n} \phi(\x_j^k, f_k(\x_j^k), \nabla f_k(\x_j^k))\right|\le\frac{K_2\|\T\|}{k}+\left|\int_{\T}g_k(\t)d\t\right|$$ By Lebesgue Dominated Convergence $$\lim_{k\to\infty}\int_{\Lambda_{n}}g_k(\t)d\t=0,$$ moreover, the definition of $\Lambda_n$ implies $$\int_{\T\setminus\Lambda_{n}}g_k(\t)d\t\le\frac{2K_1}{n}.$$ Given $\epsilon>0$ take $n\in{\mathbb{N}}$ such that $\frac{2K_1}{n}\le\frac{\epsilon}{2}$ and $K$ such that $$\frac{K_2\|\T\|}{K}+\left|\int_{\Lambda_{n}}g_K(\t)d\t\right|\le\frac{\epsilon}{2}.$$ Then equation (\[eq:2\]) holds for all $k\ge K.$
MatLab code for the examples in section \[Examples\]
====================================================
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[^1]: We thank Guillaume Carlier and Yves Lucet for their thoughtful comments and suggestions.
|
---
author:
- 'Remya Nair,'
- Sanjay Jhingan
title: 'Is dark energy evolving?'
---
Introduction
============
The late time acceleration of the Universe as suggested by type Ia Supernovae (SNeIa) [@exp] observations is now confirmed by various different probes [@DHwein]. Exploring the physics behind the cosmic acceleration is the focus of next generation surveys. The discovery of an accelerating phase of the Universe poses a very pressing question: what is the driving force responsible for this acceleration and what are its properties. There are two possible ways explored by theorists to answer this question: modify gravity on large scales or invoke a non-standard quantity in the framework of general relativity, commonly termed ‘dark energy’ which has negative pressure [@sah]. Many theoretical models have been proposed for dark energy, the cosmological constant $\Lambda$ being the simplest, and evolving dark energy scenario explained by using scalar field models [@samik]. An alternative approach which is complimentary to dark energy model building, is deriving the dark energy properties from the data. The dark energy models are often characterized by the equation of state parameter $w=p/\rho$. For the case of the cosmological constant $w=-1$, and for dynamical models $w$ is a variable. One of the main targets of future Cosmological surveys is to find constraints on the equation of state and also on its evolution (if any).
To quantify the possible dependence of the dark energy properties on redshift, one can either parameterize them or reconstruct them from the data in a non-parametric way. The equation of state is often parameterized as $w = w_0 + w_a(1-a)$. There have been attempts to study the accelerated expansion of the Universe using other kinematic variables like the Hubble parameter $H(z)$, the deceleration parameter $q(z)$, or the jerk parameter $j(z)$ which are all constructed from derivatives of the scale factor $a$ [@kin]. Direct parameterization has an advantage that one can physically interpret the result easily and quantify the dark energy and other kinematical properties with a few numbers [@lindr; @kin]. But this approach can introduce bias in the analysis since the result will always depend on the form of the parameterization chosen [@bruce1]. For this reason non-parametric methods have received a lot of attention in the past few years.
Huterer and Starkman were the first to propose the use of Principal component analysis to obtain information about the properties of dark energy [@hut]. They assumed a fiducial survey with 3000 SNeIa distributed uniformly over a redshift range $0 \leq z \leq
1.7$. Using this simulated data set they derived the best determined weight functions for the equation of state $w(z)$. Later Shapiro and Turner used supernovae measurements to analyse the acceleration history assuming a flat spacetime that is homogeneous and isotropic on large scales. Using principal component analysis they found very strong (5$\sigma$) evidence for a period of acceleration and strong evidence that the acceleration has not been constant [@shap]. Wang and Tegmark proposed a method for measuring the expansion history of the Universe in uncorrelated redshift bins [@wang]. Zunckel and Trotta used a maximum entropy method based on Bayesian framework to reconstruct the equation of state of dark energy [@zunc]. Sarkar et al., constrained the equation of state of dark energy by using uncorrelated binned estimate and showed that more than three independent parameters of the equation of state can be obtained from future dark energy surveys to an accuracy better than 10$\%$ [@sark]. Gaussian process (GP) modelling has also been used to reconstruct the dark energy equation of state [@marina]. Holsclaw et al., used GP modelling and showed that the non-trivial behaviour of $w$ as a function of $z$ can be extracted from future data and they apply their method on SNeIa data to reconstruct the history of the dark energy equation of state out to redshift $z=1.5$ [@hols1]. Ishida et al., used PCA to reconstruct the expansion rate of the universe with SNeIa data [@ishida].
The aim of this work is to find evidence for evolution in the dark energy density. We have used PCA to find constraints on the amplitudes of modes of the dark energy density as a function of redshift [@jason]. The plan of the paper is as follows: in section 2 we briefly review the parameter degeneracy between the curvature and dark energy and we introduce the data sets used in this work and the methodology in section 3. In section 4 we present our main results and discuss our results in section 5.
Parameter constraints and degeneracy
====================================
Finding constraints on the dark energy evolution parameters is complicated by the ‘geometric degeneracy’ between dark energy and curvature (see [@degn] and references therein). It is a common practice to assume spatial flatness to find constraints on dark energy parameters, and a simplified dark energy model is assumed when constraining curvature. For example, if curvature is a free parameter, the equation of state of dark energy is either assumed to be a constant or is parameterized with some simple form: $w(a)=w_0+w_a(1-a)$. The degeneracy arises, as the observed distances depend both on the expansion history of the Universe and the curvature. As a result it is not possible to constrain both the curvature and dark energy parameters. As also discussed by Mortonson [@Mort] (eq.(17) in the paper), if one uses only distance measurements, then for any value of the spatial curvature, one can derive some dark energy evolution to satisfy the observations. Mortonson used growth data to remove this degeneracy, since unlike the distance data, growth data depends only on the expansion rate. In an attempt to demonstrate that including the curvature as a free parameter is imperative to understand the dark energy evolution, Clarkson et al., showed that the assumption of a flat universe leads to large errors in the reconstruction of the dark energy equation of state even if the true cosmic curvature is very small [@clark]. Similarly Shafieloo and Linder analysed the degeneracies that arise in the distance-redshift relation when there is no a priori restriction on the equation of state of dark energy [@shaf] and they found that large variation in the parameters are allowed when using only distance measurements.
In this work we assume a homogeneous and isotropic Universe described by the FLRW metric. We keep $\Omega_k$ as a free parameter along with the dark energy parameters to be constrained by the data.
Methodology and data sets used
==============================
Data sets
---------
The dark energy evolution effects the expansion rate of the Universe and hence the distances on cosmic scales. In this work we have used distance-redshift data from SNeIa and BAO measurements. Listed below are the publicly available data sets used here:
- We use SNeIa Union2.1 sample as described in [@suz] to estimate the luminosity distance. This sample contains 580 supernovae spanning the redshift range $0.015<z<1.414$.
- BAO data from different galaxy cluster surveys - SDSS ($z$=0.2, 0.35), 6dFGS ($z$=0.106), WiggleZ ($z$=0.44, 0.6, 0.73) and BOSS ($z$=0.57) [@bl1; @perc; @beut; @boss1].
In addition to the above data sets we also use the WMAP7 distance priors [@wmap]. The physics at the decoupling epoch ($z_*$) affects the amplitude of the acoustic peaks. The evolution of the Universe between now and $z_*$ effects the angular diameter distance out to decoupling epoch, and hence the locations of the peaks. This information is encoded in the ‘acoustic scale’ $l_A,$ and the ‘shift parameter’ $R$ derived from the power spectrum of cosmic microwave background.
Methodology
-----------
The dark energy contribution to the expansion rate is expressed as a sum of two components. The contribution from the first term is constant across the redshift range. The second component corresponds to a variation in dark energy density and hence its contribution varies across the redshift range. We express the second term as a binned expansion, i.e. we divide the redshift range of the data in bins, and given a complete basis set $\{e_i\}$, the contribution from the second term can be written in terms of the basis vectors. If the redshift range is divided in, say, $N$ number of bins then every element in the $N\times$1 basis vector can be associated with a redshift bin. The continuum limit is reached as $N\rightarrow
\infty$. Thus the dark energy density is expanded as: $$\rho _{\Lambda}(z)=\rho _c \left(\alpha _0+ \sum_{i=1}^N \alpha _i e_i (z)\right).$$ Here $\alpha _0$ specifies the contribution which is constant in redshift and the coefficients $\alpha_i$ (for $1 \leq i \leq N$) specify the evolution in dark energy density. $\alpha$’s thus determine the dark energy density upto an overall constant $\rho _c$ which is the critical energy density today. For $\alpha _0 = \Omega
_\Lambda$ and all other $\alpha _i$’s = 0, we recover the standard $\Lambda$CDM case. The choice of the basis is arbitrary. We chose the basis vectors so that $e_i(z)$=1 in the $i^{th}$ redshift bin and zero otherwise (i.e. we chose the $N$ $\times$ $N$ identity matrix to be the initial basis). In this work the results are produced with $N$=50 bins.
[**SNeIa**]{}: For the standard FLRW metric the luminosity distance is given by $$d_L = \frac{c(1+z)}{\sqrt{|\Omega _k| H_0^2}} ~
S_k\left(\sqrt{|\Omega _k| H_0^2} \int _0 ^z
\frac{dz'}{H(z')}\right),
\label{dl}$$ where $S_k(x)$ is equal to $\sin{x}$, $x$, or $\sinh{x}$ corresponding to closed, flat and open Universe and the expansion rate of the Universe is: $$H(z)=H_0 [\Omega_m (1+z)^3+\Omega_k (1+z)^2+\Omega_r
(1+z)^4+\Omega_\Lambda f(z)]^{1/2}. \label{hz}$$ Here $H_0$ is the value of the Hubble parameter at present and $f(z)$ captures the form of dark energy evolution. $\Omega_i$ is the density parameter, defined as $\Omega_i=\rho_i(z=0)/\rho_c$ and $\Omega_m$, $\Omega_k$, $\Omega_r$ and $\Omega_\Lambda$ refer to the contribution in the energy density at the present epoch from matter, curvature, radiation and dark energy respectively and they add to give unity i.e. $$\Omega_m + \Omega_k + \Omega_r + \Omega_\Lambda =1.$$ The value of $\Omega_r$ can be neglected in the redshift range corresponding to the supernovae and BAO data. The distance modulus $\mu = m-M $, which is obtained from the Union2.1 compilation can be derived from the luminosity distance as $$\mu _{th} = 5 \log _{10} \frac{d_{L}}{Mpc}+25.$$ Here $m$ and $M$ are the apparent and absolute magnitudes respectively. The cosmological parameters are estimated by minimizing the chi-squared merit function: $$\chi_{_{Union2}}^2 =\sum_{i=1}^{580}
\frac{(\mu ^{th}(z_{i},p)-\mu ^{obs}(z_{i}))^2}{\sigma^{2}_{\mu _{i}}},$$ where $p$ is the set of parameters ($\alpha _i , \Omega _k, h$).
[**BAO**]{}: Galaxy cluster surveys provide measurements of an angle-averaged distance $D_{V}$ $$D_{V}=\left(\frac{cz(1+z)^{2}d_{A}^{2}}{H(z)}\right)^{\frac{1}{3}},
\label{dv}$$ or the distilled parameter $d_z = r_{s}(z_d)/D_{V}$. Here $d_{A}$ is the angular diameter distance which is theoretically given by $d_L/(1+z)^2$ and $r_s(z_{d})$ given by $$r_s(z)=\frac{c}{\sqrt{3}} \int _0 ^{1/1+z} \frac{da}{a^2 H(a) \sqrt{1+(3\Omega _b/4\Omega_ {\gamma})a}}$$ is the characteristic scale determined by the comoving sound horizon at an epoch $z_d$ slightly after decoupling. This epoch is measured by CMB anisotropy data. We determine $r_s(z_d)$ using the fitting formula given by Percival et al. [@perc] $$r_s(z_d)=153.5 \left(\frac{\Omega _b h^2}{0.02273} \right)^{-0.134} \left(\frac{\Omega _m h^2}{0.1326} \right)^{-0.255} Mpc.$$ Here $\Omega_b$ is the baryon density and $h$ is defined as $h=H_0/100$ km/s/Mpc. We have 7 BAO points in the redshift range $0.106 \leq z \leq 0.73$. Since some of these points are correlated we use the corresponding covariance matrix $C$ and the chi-squared for this data is: $$\chi_{_{BAO}}^2 = (D)^T C^{-1}D$$ where $D=d_z^{obs}(z)-d_z^{th}(z,p)$ is a column matrix for the seven data points and $p$ is the set of parameters ($\alpha _i , \Omega _k, h$).
[**CMB distance prior**]{}: The ‘acoustic scale’ $l_A$, the CMB ‘shift parameter’ $R$, and the redshift to decoupling $z_*$ mentioned earlier are defined as [@wmap]: $$\begin{aligned}
l_A &\equiv & (1 + z_*) \frac{\pi d_A(z_*)}{r_s (z_*)},\\
R(z_*) &\equiv & \frac{\Omega _m H_0^2}{c} (1 + z_*) d_A(z_*),\\
z_* &=& 1048[1 + 0.00124(\Omega _b h^2)−0.738][1 + g_1(\Omega _m h^2)g_2].\end{aligned}$$ where $g_1$ and $g_2$ are given by $$\begin{aligned}
g_1 &=& \frac{0.0783(\Omega _b h^2)^{−0.238}}{1 + 39.5(\Omega _b h^2)^{0.763}} \;,\\
g_2 &=& \frac{0.560}{1 + 21.1(\Omega _b h^2)^{1.81}} \;,\end{aligned}$$ and the chi-squared can be written as: $$\chi_{_{CMB}}^2 = (D)^T C^{-1}D\;.$$ Here $D$=$(l_A^{th}(p) - l_A^{obs}$, $R^{th}(p)-R^{obs}$, $z_*^{th}(p)-z_*^{obs})^T$, $C$ is the corresponding covariance matrix and $p$ is the set of parameters ($\alpha _i , \Omega _k, h, \Omega _b$).
[**Fisher Matrix and eigenmodes**]{}: We assume that our data set is composed of independent observations and is well approximated by a Gaussian probability density. Now since the measurements are independent one can take the product of their corresponding likelihood functions to obtain the combined likelihood function. One can maximise this likelihood function to find the best fit parameters or one can minimize the corresponding chi-squared merit function. The combined chi-squared for all the data sets will be a sum of chi-squares from individual measurements $$\chi_{_{Total}}^2 = \chi_{_{Union2}}^2+\chi_{_{BAO}}^2+\chi_{_{CMB}}^2.$$ The cosmological parameters in our analysis are as follows: curvature parameter $\Omega_k$, the Hubble parameter $h$, dark energy density parameters $\alpha _i$’s and the baryon density $\Omega _b$. Thus if we have $N$ bins we have $N+$4 unknown parameters. From this chi-square one can construct the Fisher matrix. The Fisher Matrix is defined as the expectation value of the derivatives of the log of the likelihood $L(\propto
e^{-(\chi^2/2)})$, with respect to the parameters: $$F_{ij}=-\left \langle \frac{\partial^2 \ln L}{\partial p_i \partial p_j} \right\rangle \;.$$ Depending upon the data set under consideration we construct the corresponding Fisher matrix. For this, one has to chose a particular point $p_0$ in the parameter space at which the Fisher matrix is evaluated. We chose this point to correspond to the standard $\Lambda$CDM case, $\alpha_0 = \Omega_{\Lambda}$ and $\alpha_i=0$ for $1 \leq i \leq N$. Once we have the Fisher matrix, we marginalize over all the parameters other than $\alpha_i$ ($1 \leq i \leq N$), i.e. $\alpha_0$, $\Omega_k$, $h$ and $\Omega_b$. We are left with a Fisher matrix of the parameters that correspond to the evolution in dark energy density. We diagonalize it so that it can be written as $$F=W\Lambda W^T,$$ where the columns of the orthogonal decorrelation matrix $W$, are the eigenvectors of $F$ and $\Lambda$ is a diagonal matrix with the eigenvalues on its diagonal. We can now find a new set of basis functions which is a linear combination of the old basis function $e_i$. This new basis set has decorrelated vectors and since these basis functions are orthonormal and complete we can write the contribution of dark energy in terms of these uncorrelated basis vectors $c_i$: $$\sum_{i=1}^{N} \alpha_i e_i(z)=\sum_{i=1}^{N} \beta_i c_i(z) \;.$$ The eigenmatrix $W$ is the Jacobian matrix of the transformation of one basis to another, so one can construct these new basis functions $c_i(z)$ as (see [@amara] for details) $$c_i(z)= W e_i(z)$$
Note that in our case the initial basis set is just the identity matrix. The advantage of this new basis is that the $\beta_i$ are uncorrelated, which implies that any pair of coefficients has a non-degenerate error ellipse. These eigenmodes are arranged from the best determined mode to the least determined mode i.e., from the largest to the smallest eigenvalues (the error on these modes goes as $\sigma \propto \lambda^{-1/2}$) and then we choose the first few best determined eigenmodes to reconstruct the dark energy density and carry out the chi-squared minimization. Also note that the constant mode $\alpha_0$ is not decorrelated from the rest of the modes and is unaffected by the diagonalisation of Fisher matrix. We show the four best determined eigenmodes for the different data set combinations in the figures below. A non-zero mode amplitude of these principal components would indicate time evolution in dark energy.
The main advantage of the PCA is dimensionality reduction. Initially our parameter space had $N+4$ parameters. Now, depending on the number of principal components chosen for reconstruction, the parameter space would be reduced. The number of principal components to be used for reconstruction depends on how much information are we willing to discard. If we used all the principal components, we would have 100$\%$ information but the uncertainties in our parameter estimates would be too large to have any meaningful interpretation. The sum of all the eigenvalues $\lambda _i$ quantifies the total variance in the data and if we use the first $M$ principal components then it encloses $r_M$ $\%$ of this variance where $$r_M=100 \frac{\sum_{i=1}^{M} \lambda _i}{\sum_{i=1}^{N} \lambda _i}.
\label{rm}$$
Results
=======
We use Markov chain Monte Carlo (MCMC) method (Metropolis-Hastings MCMC chains) for estimating the parameters and their corresponding errors. We show the best determined eigenmodes for the different dataset combinations. In the plots the different parameter estimates are found by allowing different number of dark energy evolution parameters to vary. First we combine the supernovae measurements with the CMB distance priors and resulting constraints are shown in Figure \[sc1\] and \[sc\]. Then we add the BAO measurements to this data set to see how they effect the parameter constraints. Following [@jason] we employ the following trick when evaluating the Fisher matix after the addition of BAO data: we keep $H(z)$ in $d_z$ as fixed. If we don’t do this we get spike like features in our eigenfunctions. The results after addition of BAO data are shown in figure \[scb1\] and \[scb2\]. For plotting convenience, instead of plotting $\alpha_0$ we show the difference 0.734 - $\alpha_0$ (the WMAP7 constraint on $\Omega_\Lambda$ in the $\Lambda$CDM model is $\Omega_\Lambda \sim 0.734$). We summarise our main results below:
- In the plots above we present the results obtained by varying different number of principal components for each data set. We observe that the constraint on the curvature parameter $\Omega_k$ is robust against variation in the number of principal component chosen for reconstruction. This indicates that the curvature and dark energy evolution parameters have been decorrelated. The results are consistent with a flat Universe.
- The addition of CMB priors is crucial. The analysis using only the distance measurements resulted in a very wide allowed range for $\Omega_k$ and the dark energy parameters.
- The constraints obtained from the supernovae-CMB combined measurements are consistent with a flat $\Lambda$CDM Universe. The first few principal components are consistent with zero. The reconstruction is done using upto 4 principal components. This amounts to incorporating almost 97$\%$ of the information (see \[rm\]).
- The addition of the BAO data to the supernovae-CMB data set improves the parameter estimates but also changes their value. The reconstruction is done using upto 4 principal components. This amounts to incorporating almost 94$\%$ of the information (see \[rm\]). The coefficients of the first two eigenmodes are now slightly shifted from zero.
Discussion
==========
Finding constraints on both the flatness of the Universe and the dark energy parameters, in models that allow dark energy evolution is difficult because of the geometric degeneracy. Weak assumptions on the possible evolution of dark energy can limit the allowed values of $\Omega_k$ and some simple parametrization of the equation of state is often assumed to put simultaneous constraints. But such assumptions/parametrization can introduce significant bias in the analysis and the results obtained. For example Clarkson et al., showed that the assumption of a flat universe leads to large errors in reconstructing the dark energy equation of state even if the true cosmic curvature is very small [@clark].
In this work we have used a non-parametric method: Principal Component Analysis, to look for evidence for evolution in the dark energy density. The dark energy density is expressed as a sum of two terms: a constant term that accounts for the contribution that is redshift independent and an additional term constructed from the non constant density contribution. This later term is formulated using PCA so that all the parameters obtained have uncorrelated errors and a non constant amplitude of these modes would indicate dark energy evolution. The distance-redshift data alone cannot break the degeneracy between curvature and dark energy parameters. One can use growth data to remove this degeneracy [@Mort]. Also since high redshift distances, for example the distance to the last scattering surface, is sensitive to the curvature, one can use this measurement to find simultaneous constraints on $\Omega_k$ and dark energy parameters. In this work we have used the WMAP7 distance priors ($l_A$, $R$ and $z_*$). We used the latest supernovae data along with the CMB distance priors and found that it is consistent with a flat $\Lambda$CDM Universe. Later we incorporated the recent BAO data to see its effect on the parameter estimates. The constraints obtained on the non-constant modes from the addition of BAO data are slightly shifted from zero. The second principal component obtained in this case is not consistent with zero at 1$\sigma$. A possible deviation in dark energy equation of state from -1 was recently shown by the $WMAP$ team (see Fig. 10 [@wmap9]). Also, Gong-Bo Zhao et al., used a new non-parametric Bayesian method for reconstructing the evolution history of the equation-of-state of dark energy and found that the cosmological constant appears consistent with current data, but that a dynamical dark energy model which evolves from $w<-1$ at $z\sim0.25$ to $w > -1$ at higher redshift is mildly favored [@gong]. It is important to note that the result we obtained could be due to some unknown systematic effect and future BAO measurements would play key role in understanding the dark energy dynamics.
Acknowledgement {#acknowledgement .unnumbered}
===============
Authors thank Jason Dick and Deepak Jain for discussions. R.N. acknowledges support under CSIR - SRF scheme (Govt. of India). SJ acknowledges financial support provided by Department of Science and Technology, India under project No. SR/S2/HEP-002/2008.
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---
abstract: 'We extend the Standard Model (SM) by adding a pair of fermionic $SU(2)$-doublets with opposite hypercharge and a fermionic $SU(2)$-triplet with zero hypercharge. We impose a discrete ${Z}_{2}$-symmetry that distinguishes the SM fermions from the new ones. Then, gauge invariance allows for two renormalizable Yukawa couplings between the new fermions and the SM Higgs field, as well as for direct masses for the doublet ($M_{D}$) and the triplet ($M_{T}$). After electroweak symmetry breaking, this model contains, in addition to SM particles, two charged Dirac fermions and a set of three neutral Majorana fermions, the lightest of which contributes to Dark Matter (DM). We consider a case where the lightest neutral fermion is an equal admixture of the two doublets with mass $M_{D}$ close to the $Z$-boson mass. This state remains stable under radiative corrections thanks to a custodial $SU(2)$-symmetry and is consistent with the experimental data from oblique electroweak corrections. Moreover, the amplitudes relevant to spin-dependent or independent nucleus-DM particle scattering cross section [*both*]{} vanish at tree level. They arise at one loop at a level that may be observed in near future DM direct detection experiments. For Yukawa couplings comparable to the top-quark, the DM particle relic abundance is consistent with observation, not relying on co-annihilation or resonant effects and has a mass at the electroweak scale. Furthermore, the heavier fermions decay to the DM particle and to electroweak gauge bosons making this model easily testable at the LHC. In the regime of interest, the charged fermions suppress the Higgs decays to diphoton by 45-75% relative to SM prediction.'
author:
- 'Athanasios Dedes[^1] and Dimitrios Karamitros[^2]'
bibliography:
- 'DarkMatter-Biblio.bib'
title: 'Doublet-Triplet Fermionic Dark Matter'
---
=1
Introduction {#intro}
============
Motivated by astrophysical observations that suggest the existence of Dark Matter [@Bertone:2004pz], we would like to propose a model with a fermionic Weakly Interacting Massive Particle (WIMP) ($\chi_{1}^{0}$) whose mass and couplings are directly associated to electroweak scale providing the universe with the right thermal relic density abundance, not “tuned” by co-annihilation or resonance effects. Today, as opposed to five years ago, attempts of this sort immediately face difficulties due to strong experimental bounds [@XENON100; @LUX][^3] from direct searches on nucleus recoiling energy in WIMP-nucleus scattering processes [@Goodman:1984dc]. As a result, $Z$- and Higgs- boson couplings to $\chi_{1}^{0}$-pairs are strongly constrained and usually come into conflict with values of couplings required from the observed [@Ade:2013zuv] DM relic abundance. We therefore seek for a model at which, at least at tree level, these couplings vanish by a symmetry and at the same time the observed relic density is reproduced. We then discuss further consequences of this idea at Large Hadron Collider (LHC).
We consider a minimal model which realises this situation, hence, in addition to Standard Model (SM) particles, we add a pair of Weyl-fermion doublets $\mathbf{\bar{D}_{1} \sim (1^{c}, 2)_{-1}}$ and $\mathbf{\bar{D}_{2} \sim (1^{c}, 2)_{+1}}$ with opposite hypercharges, and a Weyl-fermion triplet $\mathbf{T\sim (1^{c}, 3)_{0}}$ with zero hypercharge. The new Yukawa interactions allowed by gauge invariance and renormalizability are given by[^4] $$\begin{aligned}
\mathcal{L}_{\mathrm{Yuk}} \ \supset \
Y_{1}\: \mathbf{T\, H\, \tau \, \bar{D}_{1}} \ + \ Y_{2}\: \mathbf{T\, H^{\dagger}\ \tau \, \bar{D}_{2}}
\ - \ M_{D}\: \mathbf{\bar{D}_{1}\, \bar{D}_{2}} \ - \ \frac{1}{2}\, M_{T} \: \mathbf{T \,T} \;,
\label{lag1}\end{aligned}$$ with $\mathbf{\tau}$ being the Pauli matrices. A $Z_{2}$-discrete parity symmetry has been employed to guarantee that the new fermions interact always in pairs. Clearly, $\mathcal{L}_{\mathrm{Yuk}}$ is invariant under the interchange symmetry $\mathbf{H\leftrightarrow H^{\dagger}}$ and $\mathbf{\bar{D}_{1} \leftrightarrow \bar{D}_{2}}$ when $Y_{1} = Y_{2}\equiv Y$. Then, it is very easy to see that in this limit, one eigenvalue with mass $M_{D}$, of the neutral ($3\times 3$) mixing mass matrix, decouples from the two heavier ones and the latter is degenerate with the two eigenvalues of the ($2\times 2$) charged fermion mass matrix. At tree level approximation, except for the lightest neutral fermion ($\chi^{0}_{1}$), all other masses are controlled by the Yukawa coupling $Y$. The state with $m_{\chi_{1}^{0}} = M_{D}$ is our DM candidate particle. This particle state contains an equal admixture of the two doublets but has *no* triplet component, $$\begin{aligned}
{|\chi_{1}^{0}\rangle} = 0 \cdot {|\mathbf{T}\rangle} \ + \ \frac{1}{\sqrt{2}}\: {|\mathbf{\bar{D}_{1}}\rangle} \ + \
\frac{1}{\sqrt{2}}\: {|\mathbf{\bar{D}_{2}}\rangle} \;.\end{aligned}$$ Because the neutral component of the triplet does not participate in ${|\chi_{1}^{0}\rangle}$, the latter does not couple to the Higgs boson at tree level. It does not couple to the $Z$-gauge boson neither because of its equal admixture of neutral particles with opposite weak isospin. The situation here is analogous to the custodial symmetry [@Sikivie:1980hm] imposed in strongly coupled EW scenarios, where the “custodian” new particles are inserted in a similar way to protect certain quark-gauge boson couplings to obtain large radiative corrections [@delAguila:2010es; @Agashe:2006at; @SekharChivukula:2009if; @Carmona:2013cq]. The couplings $h \chi_{1}^{0} \chi_{1}^{0}$ and $Z \chi_{1}^{0} \chi_{1}^{0}$ vanish at tree level, and as a result there are no $s$-channel amplitudes contributing to the annihilation cross section. However, there are off-diagonal interactions such as e.g., $Z \chi_{1}^{0} \chi_{2}^{0}$ that render the $t,u$-channel amplitudes non-zero but yet suppressed enough to obtain the right relic density $\Omega_{\chi}$ for $M_{D} \approx 100$ GeV and $Y \approx 1$. Roughly speaking, the spectrum of the model where this happens is shown schematically in Fig. \[fig:spec\].
![*A sketch for the mass spectrum and decays of the new physical doublet and triplet fermions. The lightest neutral particle, $\chi_{1}^{0}$ is an equal admixture of the two doublets and has mass $M_{D}$. Particles $\chi_{2}^{0}$ ($\chi_{3}^{0}$) and $\chi_{1}^{\pm}$ ($\chi_{2}^{\pm}$) are mass degenerate. For the spectrum masses written to the right we have chosen $M_{D}=110$ GeV, $M_{T}=100$ GeV and $m=Y v=200$ GeV. It provides the correct relic density abundance for dark matter \[see Section \[sec:relic\]\] and is currently about $\sim 10$ times less sensitive to current direct detection searches \[see Section \[sec:direct\]\].*[]{data-label="fig:spec"}](spectrum-1.pdf){height="3in"}
Typically, the lightest stable new particle ($m_{\chi_{1}^{0}} \approx 110$ GeV) is in the vicinity of the EW scale while all other neutral and charged fermions are above $m \equiv Y v$ which is taken around the top quark mass. The splitting of the charged fermions is also controlled by the triplet mass ($M_{T}$). Therefore, the parameters of the model are just three: $M_{D}, M_{T}$ and $m$.
Naively, one may think that this model is similar to the “wino-higgsino” sector of the MSSM [@HK] or it is an extended variant of the singlet-doublet DM model of . Another obvious question is, why does one want to introduce several new fermions, since a single one (for example the triplet, as in minimal DM [@Cirelli:2005uq] models) suffices? The answer to these questions arise from our wish to construct a model with WIMP mass *at the EW scale*, and hides inside the model building details, namely:
1. The off-diagonal entries of the “chargino” or “neutralino” mass matrix contain general Yukawa couplings ($Y_{1}$ and $Y_{2}$) that can be enhanced as opposed to the fixed-value gauge couplings of the MSSM. Evenmore, they can be equal here [*i.e.,* ]{}$Y_{1}=Y_{2}\equiv Y\sim g$, satisfying a custodial symmetry, a realisation which is only phenomenologically allowed in the so called Split-SUSY scenarios [@ArkaniHamed:2004fb; @Giudice:2004tc]. Therefore, this fermionic doublet-triplet DM sector generalises the corresponding DM sector of the Minimal Supersymmetric Standard Model (MSSM).
2. In the region where the common Yukawa coupling is comparable, say, to the top Yukawa coupling there are heavy charged leptons decaying to the lightest new fermion $\chi_{1}^{0}$. This mass pattern, shown in Fig. \[fig:spec\], is different from the singlet-doublet DM model (at least from the minimal version) where the lightest neutral particle is, up to radiative corrections, degenerate with the charged particle a situation which is highly constrained from long lived charged particle searches at LHC [@Chatrchyan:2013oca].
3. In the limit of equal Yukawa couplings ($Y$) in , there is a custodial $SU(2)$-symmetry that guaranties vanishing couplings at tree level between the lightest neutral particle and the $Z$-boson ($Z\chi_{1}^{0} \chi_{1}^{0}$) and also to the Higgs-boson ($h\chi_{1}^{0} \chi_{1}^{0}$). This is a certain “pass” for this model, at least to leading order, from the current strong direct detection experimental contraints [@XENON; @XENON100; @LUX]. Moreover, as we shall see below, $h\chi_{1}^{0} \chi_{1}^{0}$-coupling arises radiatively at one-loop order providing us with certain model predictions. Note that “blind spots” of this kind have been studied in for Split-SUSY and in for the singlet-doublet and singlet-triplet fermionic DM models.
4. Similar to the case here, the dominant annihilation channel in the higgsino DM-phase of MSSM [@Drees:1996pk], is into gauge bosons. But in the higgsino case and due to smallness of the gauge coupling, the lightest charged and neutral fermion states are degenerate so co-annihilation effects [@Griest:1990kh] are very important. It turns out that, that for higgsino mass $\mu\sim 100$ GeV the cross section $<\sigma v> \approx \frac{g^{4}}{16\pi \mu^{2}}$ is large which results in $\Omega_{DM}$ that is too low unless $\mu$ is in the TeV range. In the doublet-triplet fermonic DM model we consider here, the lightest neutral state decouples from the heavy ones, and in the limit of large $m= Y v$ the difference in mass between the lightest neutral fermion and the lightest charged or the second lightest neutral one is normally of the order of 100 GeV (see Fig. \[fig:spec\] for an example). The annihilation cross section now goes through the $t,u$-channels and, relative to higgsino case, is suppressed by a factor $(m_{\chi}/m_{\chi_{j}})^{4}\sim 10-100$ where $m_{\chi_{j}}$ are the heavy fermion masses ($\chi_{2,3}^{0}, \chi_{1,2}^{\pm}$), allowing a WIMP mass, $m_{\chi}$, naturally of the order of 10-100 GeV.[^5]
5. Our attempt here is to find a DM candidate particle consistent with the astrophysical and collider data but with mass around the electroweak scale. Vector-like gauge multiplets that are engaged here have also been used to construct minimal DM Models (MDM) in . It has been found that the masses $M_{D}$ or $M_{T}$ should lie in the few-TeV region. In our scenario, it is the chiral (Dirac) mass terms in that play the most important role. The latter are constrained from perturbativity to be several hundreds of GeV while the lower vector-like masses, $M_{D}$ and $M_{T}$, are protected by an accidental symmetry. Finally, the production and decay phenomenology of the new fermions is very distinct from the ones in MDM models and it is relatively easy to be tested with current and near future LHC data.
Within this framework of doublet-triplet fermionic DM model that we describe in Section \[model\], and in particular in the region where the custodial symmetry is applied, we discuss and check constraints that include:
- An estimate of oblique corrections to electroweak observables ($S,T,U$ parameters) \[Section \[sec:STU\]\].
- DM thermal relic density calculation at tree level \[Section \[sec:relic\]\].
- Direct DM detection prospects through nucleus-DM particle scattering at 1-loop \[Section \[sec:direct\]\]
- Decay rate of the Higgs boson to two photons ($h\to \gamma\gamma$) \[Section \[sec:h2gg\]\]
- Vacuum stability and perturbativity \[Section \[sec:Landau\]\]
- LHC signatures, production and decays of the new fermions.
Our conclusions and various ways to extend this work are discussed in Section \[sec:conclusions\]. An appendix with the explicit one-loop corrections to the $h\chi_{1}^{0}\chi_{1}^{0}$-vertex is given. Beyond the articles we have already mentioned, there is a reach literature regarding minimal DM extensions of the SM. A partial list is given in .
Model Details {#model}
=============
As a result of what we have already mentioned in the introduction, we scan chiral fermion matter extensions of the SM gauge group according to the following, rather obvious, assumptions for the new set of fermions:
1. they must have vectorial electromagnetic interactions,
2. they must be colour singlets with integer charges,
3. their interactions must be gauge (and gravitational) anomaly free,
4. their masses are obtained after $SU(2)_{W}\times U(1)_{Y}$ gauge symmetry breaking, with only the SM Higgs doublet, and if gauge symmetry allows, directly, and
5. there is a parity symmetry, ${\mathbf Z}_{2}$, under which the SM fermions transform as $+1$ while the new fermions as $-1$.
The most minimal model, not containing pure singlet fields,[^6] consists of three fields arranged in colour singlets and representations of $SU(2)_{W}$, with quantum numbers denoted as $\mathbf{(1^{c}, 2I_{W}+1)^{Y}_{L,R}}$, where $\mathbf{I_{W}}$ is the weak $SU(2)_{W}$ isospin and $Y$ is the hypercharge related to the electric charge by $Q=I_{3W} +\frac{Y}{2}$. These new fields are: $$\begin{aligned}
\mathbf{T \sim (1^{c}, 3)^{0}_{L}\;, \qquad D_{1}\sim (1^{c} , 2)_{R}^{+1}\;,
\qquad D_{2}\sim (1^{c},2)^{-1}_{R} \;.}\end{aligned}$$ One can easily check that this is a gauge and gravitational anomaly free set of chiral fermions. They sit in adjacent representations of $SU(2)_{W}$ with weak isospin difference $\Delta I_{W} = \frac{1}{2}$. This matches with the only spinless field of the SM, the Higgs field, with gauge labels $\mathbf{H \sim (1^{c}, 2)_{+1}}$.
It is convenient to represent all fermions, [[*i.e.,* ]{}]{}SM quarks and leptons plus new fermions that belong to the DM sector, with two component, left handed, Weyl fields [@Dreiner:2008tw], namely[^7] $$\begin{aligned}
\mathrm{SM~quarks} :\qquad &
\mathbf{Q = \left ( \begin{array}{c} u \\ d \end{array} \right )\sim (3^{c}, 2)_{+1/3}\;, \quad
\bar{u} \sim (3^{c}, 1)_{-4/3}\;, \quad \bar{d} \sim (3^{c},1)_{+2/3} } \;, \label{SMq}\\[3mm]
\mathrm{SM~leptons} :\qquad &
\mathbf{L = \left ( \begin{array}{c} \nu \\ e \end{array} \right )\sim (1^{c}, 2)_{-1}\;, \quad
\bar{\nu} \sim (1^{c}, 1)_{0}\;, \quad \bar{e} \sim (1^{c},1)_{+2} } \;, \label{SMl} \\[3mm]
\mathrm{DM~fermions} :\qquad &
\mathbf{T = \left ( \begin{array}{c} T_{1} \\ T_{2} \\ T_{3} \end{array} \right )\sim (1^{c}, 3)_{0}\;, }
\nonumber \\[3mm]
&\hspace*{-0.15in} \mathbf{\quad \bar{D}_{1} =
\left( \begin{array}{c} \bar{D}_{1}^{1} \\ \bar{D}_{1}^{2} \end{array} \right )\sim (1^{c}, 2)_{-1}\;,
\quad \bar{D}_{2} =
\left( \begin{array}{c} \bar{D}_{2}^{1} \\ \bar{D}_{2}^{2} \end{array} \right ) \sim (1^{c}, 2)_{+1} }\;\;\;.
\label{DMf} \end{aligned}$$ SM fermions come in three copies of (\[SMq\]) and (\[SMl\]) sets of fields. We have added a left-handed antineutrino Weyl field in the SM field content in order to account for light neutrino masses via the seesaw mechanism. Although there may be interesting links between the neutrino and DM sector fields we shall scarcely refer to neutrinos in this article. We assume only one copy of the DM-sector fields in (\[DMf\]). Of course, we could also add more singlet fermions either in the SM or in the DM-sector but our intention is to keep the model as minimal as possible.
Physical masses are obtained from the gauge invariant form of Yukawa interactions. Under the assumption-5 above, the whole Yukawa Lagrangian of the model is $$\mathscr{L}_{\rm Yuk} = \mathscr{L}_{\rm Yuk}^{\rm SM} \ + \ \mathscr{L}_{\rm Yuk}^{\rm DM} \;,$$ where the SM part reads (flavour indices are suppressed): $$\begin{aligned}
\mathscr{L}_{\rm Yuk}^{\rm SM} &= Y_{u} \epsilon^{ab} H_{a} Q_{b} \bar{u} -
Y_{d} H^{\dagger\, a} Q_{a} \bar{d} - Y_{e} H^{\dagger \, a} L_{a} \bar{e}
\nonumber \\[3mm]
& +Y_{\nu} \epsilon^{ab} H_{a} L_{b} \bar{\nu} - \frac{1}{2} M_{N} \bar{\nu} \bar{\nu} + {\rm H.c.}\;,
\label{LSM}
\end{aligned}$$ and the available DM-sector interactions are $$\begin{aligned}
\mathscr{L}_{\rm Yuk}^{\rm DM} &=
Y_{1}\: \epsilon^{ab}\, T^{A}\, H_{a}\, (\tau^{A})_{b}^{c}\, \bar{D}_{1\, c}
\ - \ Y_{2}\: T^{A}\, H^{\dagger \, a}\, (\tau^{A})_{a}^{c} \, \bar{D}_{2\, c} \nonumber \\[3mm]
&- M_{D}\: \epsilon^{ab} \bar{D}_{1\,a} \bar{D}_{2\, b} \ - \ \frac{1}{2} M_{T}\: T^{A} T^{A}
\ + \ {\rm H.c.} \;.
\label{LDM}\end{aligned}$$ By choosing appropriate field redefinitions and without loss of generality we can make the parameters $Y_{1}, Y_{2}$, and $M_{T}$ real and positive, while leaving $M_{D}$ to be a general complex parameter. This is the only source of $CP$-violation[^8] arising from the DM-sector in this model. If not stated otherwise, we consider real $M_{D}$ values in our numerical results. The parity symmetry assumption-5 removes the following renormalizable operators: $$H^{\dagger}\: \bar{D}_{2}\: \bar{\nu}\;, \quad H\: \bar{D}_{1}\: \bar{\nu}\;, \quad L \: \bar{D}_{2} \;, \quad H\: T\: L \quad {\rm and} \quad H^{\dagger} \: \bar{D}_{1}\: \bar{e} \;.$$ Note that apart from the first two, the rest will not be allowed under the custodial symmetry. Finally, we assume that possible non-renormalizable operators that are allowed by the discrete symmetry are Planck scale suppressed and do not play any particular role in what follows.
The spectrum {#spec}
------------
Since there is no-mixing between the mass terms of the SM fermions and the DM sector ones, we solely concentrate on the non-SM Yukawa interactions of . After electroweak symmetry breaking and the shift of the neutral component of the only Higgs field, $H^{0} = v + h/\sqrt{2}$, we obtain the following mass terms $$\begin{aligned}
\mathscr{L}^{\rm DM}_{\rm Y \, (mass)}
&= - \left ( \mathcal{\tau}_{1}\quad \bar{D}_{2}^{1} \right )^{T}
\: \mathcal{M}_{C} \: \left ( \begin{array}{c} \mathcal{\tau}_{3} \\[2mm] \bar{D}_{1}^{2} \end{array} \right ) - \frac{1}{2} \left ( \mathcal{\tau}_{2}\quad \bar{D}_{1}^{1} \quad \bar{D}_{2}^{2} \right )^{T}
\mathcal{M}_{N} \left ( \begin{array}{c}
\mathcal{\tau}_{2} \\[2mm] \bar{D}_{1}^{1} \\[2mm] \bar{D}_{2}^{2} \end{array}\right )
\ + \ {\rm H.c.}
\nonumber \\
& = - \sum_{i=1}^{2} m_{\chi_{i}^{\pm}} \chi_{i}^{-} \: \chi_{i}^{+} \ - \
\frac{1}{2} \sum_{i=1}^{3} m_{\chi^{0}_{i}}
\chi_{i}^{0} \chi_{i}^{0} \ + \ {\rm H.c.} \;,
\label{Lmass}\end{aligned}$$ where $\mathcal{\tau}_{1} \equiv (T_{1} - i T_{2})/\sqrt{2}$, $\mathcal{\tau}_{3} \equiv (T_{1} + i T_{2})/\sqrt{2}$ and $\mathcal{\tau}_{2} \equiv T_{3}$. The charged ($\mathcal{M}_{C}$) and the neutral ($\mathcal{M}_{N}$) fermion mass matrices in are given by $$\mathcal{M}_{C} = \left(\begin{array}{cc}M_{T} & \sqrt{2}\, m_{1} \\[2mm] \sqrt{2}\, m_{2} & -M_{D}
\end{array}\right) \;, \qquad \mathcal{M}_{N} =
\left(\begin{array}{ccc}M_{T} & m_{1} & -m_{2} \\ m_{1} & 0 & M_{D}
\\ -m_{2} & M_{D} & 0\end{array}\right) \;, \label{mcmn}$$ where $m_{1,2} \equiv Y_{1,2} \: v$. Matrices $\mathcal{M}_{C}$ and $\mathcal{M}_{N}$ are diagonalized following the singular value decomposition and the Takagi factorisation theorems [@Horn] into $m_{\chi^{\pm}} =(2\times 2)$ and $m_{\chi^{0}}= (3\times 3)$ diagonal matrices, $$U_{L}^{T}\: \mathcal{M}_{C} \: U_{R} = m_{\chi^{\pm}}\;,
\quad O^{T}\: \mathcal{M}_{N} \: O = m_{\chi^{0}}\;, \label{omat}$$ respectively, after rotating the current eigenstate fields into their mass eigenstates $\chi^{\pm}_{i} , \chi^{0}_{i}$ with unitary matrices, $U_{L}, U_{R}$ and $O$, as $$\left(\begin{array}{c}\mathcal{\tau}_{3} \\[2mm] \bar{D}_{1}^{2} \end{array}\right) = U_{R} \:
\left(\begin{array}{c}\chi_{1}^{-} \\[2mm] \chi_{2}^{-} \end{array}\right)\;, \quad
\left(\begin{array}{c}\mathcal{\tau}_{1} \\[2mm] \bar{D}_{2}^{1} \end{array}\right) = U_{L} \:
\left(\begin{array}{c}\chi_{1}^{+} \\[2mm] \chi_{2}^{+} \end{array}\right)\;, \qquad
\left(\begin{array}{c}\mathcal{\tau}_{2} \\[2mm] \bar{D}_{1}^{1} \\[2mm]
\bar{D}_{2}^{2} \end{array}\right) =
O \:
\left(\begin{array}{c}\chi_{1}^{0} \\[2mm] \chi_{2}^{0} \\[2mm] \chi_{3}^{0} \end{array}\right)\;.
\label{rots}$$ Therefore the spectrum of this model contains, apart from the SM masses for quarks and leptons, two additional charged Dirac fermions and three neutral Majorana particles. It is the lightest Majorana particle $\chi_{1}^{0}$ with mass $m_{\chi_{1}^{0}}$, that, perhaps, supplies the universe with cold Dark Matter.
It is crucial for what follows and also enlightening, to discuss the decoupling of the $M_{D}$-eigenvalue from the particle spectrum. First, $\mathcal{M}_{N}$, is a real symmetric matrix, under the assumption of real $M_{D}$. Then, consider the following unitary matrix $\Sigma$, having as columns orthonormal vectors, $$\begin{aligned}
\Sigma = \frac{1}{\sqrt{2}}\:
\left(\begin{array}{ccc}\sqrt{2} & 0 & 0 \\0 & 1 & 1 \\0 & -1 & 1\end{array}\right)\;,\end{aligned}$$ which by a similarity transformation, brings the lower right $2\times 2 $ sub-block of $\mathcal{M}_{N}$ into a diagonal form, $$\begin{aligned}
\mathcal{M}_{N}^{\prime} \ = \Sigma^{\dagger}\: \mathcal{M}_{N}\: \Sigma =
\left(\begin{array}{ccc}M_T & (m_1 + m_2)/\sqrt{2} & (m_1 - m_2)/\sqrt{2} \\
(m_1 + m_2)/\sqrt{2} & -M_D & 0 \\ (m_1 - m_2)/\sqrt{2} & 0 & M_D\end{array}\right) \;.
\label{mnp}\end{aligned}$$ Note that since $\Sigma$ is unitary matrix, the eigenvalues of $\mathcal{M}_{N}$ and $\mathcal{M}_{N}^{\prime}$ are equal. We therefore obtain, that for $m_{1} = m_{2}$ the charged fermion mass matrix $\mathcal{M}_{C}$ becomes the upper-left sub-block of the $\mathcal{M}_{N}^{\prime}$ in . Therefore the eigenvalue, $M_{D}$, decouples from the neutral fermion mass matrix [[*i.e.,* ]{}]{}it is independent of any mixing and therefore any v.e.v, while the rest of eigenvalues of both matrices, $\mathcal{M}_{C}$ and $\mathcal{M}_{N}$, are one to one degenerate.
The interactions {#sec:interactions}
----------------
We now turn to the interactions between the new fermions and the SM gauge-bosons or the SM Higgs-boson. The latter can be read from after rotating fields by exploiting the relations in (\[rots\]). After a little bit of algebra we obtain[^9] $$\mathscr{L}_{\rm Y (int)}^{\rm DM} = - Y^{h\chi_{i}^{-} \chi_{j}^{+}} \: h \: \chi_{i}^{-} \: \chi_{j}^{+}
\ - \ \frac{1}{2}\: Y^{h\chi_{i}^{0} \chi_{j}^{0}} \: h \: \chi_{i}^{0} \: \chi_{j}^{0} \ + \ {\rm H.c.} \;,
\label{lyint}$$ where $$\begin{aligned}
Y^{h\chi_{i}^{-} \chi_{j}^{+}} & \equiv \frac{1}{v} \left ( m_{1}\: U_{R\, 2i} \:U_{L\, 1j} +
m_{2}\: U_{R\, 1i} \: U_{L\, 2j} \right ) \;, \label{yhpp}\\[2mm]
Y^{h\chi_{i}^{0} \chi_{j}^{0}} & \equiv \frac{O_{1i}}{\sqrt{2}\: v}\: \left (
m_{1} \: O_{2j} - m_{2}\: O_{3j} \right ) \ + \ (i \leftrightarrow j)\;.\label{yhxx}
\end{aligned}$$ For completeness and especially for loop calculations, we append here the interactions between Goldstone bosons and the new fermions: $$\begin{aligned}
\mathscr{L}_{\rm G\chi\chi } &= -\frac{i\: O_{1i}}{\sqrt{2} v} \,
(m_{1}\: O_{2j} + m_{2} \: O_{3j})\, G^{0} \chi_{i}^{0} \chi_{j}^{0}
-\frac{i}{v}\, (m_{1}\: U_{R\,2i} \:U_{L\, 1j} - m_{2} \: U_{R\, 1i} \: U_{L\, 2j})\,
G^{0} \chi_{i}^{-} \chi_{j}^{+}
\nonumber \\[2mm]
&+ \frac{m_{1}}{v} (\sqrt{2} \: U_{R\,1i}\: O_{2j} - U_{R\,2i} \: O_{1j})\, G^{+} \chi^{-}_{i} \chi_{j}^{0}
-\frac{m_{2}}{v} \: (\sqrt{2} \: U_{L\, 1i}\: O_{3j} + U_{L\, 2i}\: O_{1j})\, G^{-} \chi_{i}^{+} \chi_{j}^{0}
\nonumber \\[2mm] &+ \mathrm{H.c} \;.\label{LGB}
\end{aligned}$$ Interactions among the new fermions and gauge bosons arise from the respective fermion kinetic terms. Interactions between $\chi^{\pm}$ and the photon are purely vectorial, $$\mathscr{L}_{\rm KIN (int)}^{\gamma - \chi^{\pm}} = -
(+e) \:(\chi_{i}^{+})^{\dagger} \bar{\sigma}^{\mu}
\chi_{i}^{+} \: A_{\mu} - (-e)\: (\chi_{i}^{-})^{\dagger} \bar{\sigma}^{\mu}
\chi_{i}^{-} \: A_{\mu} \;,$$ where $A_{\mu}$ is the photon field and $(-e)$ the electron electric charge. The $Z$-gauge boson couplings to both charged and neutral fermions can be read from[^10], $$\mathscr{L}_{\rm KIN (int)}^{Z - \chi} = \frac{g}{c_{W}} O_{ij}^{\prime \, L} \: (\chi_{i}^{+})^{\dagger}
\: \bar{\sigma}^{\mu}\: \chi_{j}^{+} \:Z_{\mu} -
\frac{g}{c_{W}} O_{ij}^{\prime \, R} \: (\chi_{j}^{-})^{\dagger}
\: \bar{\sigma}^{\mu}\: \chi_{i}^{-} \:Z_{\mu} +
\frac{g}{c_{W}} O_{ij}^{\prime\prime \, L} \: (\chi_{i}^{0})^{\dagger}
\: \bar{\sigma}^{\mu}\: \chi_{j}^{0} \:Z_{\mu} \;,$$ where $$\begin{aligned}
O_{ij}^{\prime\, L} &= -U^{*}_{L1i} \: U_{L1j} - \frac{1}{2}\: U_{L 2i}^{*}\: U_{L2j} +
s_{W}^{2} \delta_{ij}\;, \label{eq:OLp}\\
O_{ij}^{\prime\, R} &= -U_{R1i} \: U^{*}_{R1j} - \frac{1}{2} \: U_{R 2i}\: U_{R2j}^{*} +
s_{W}^{2} \delta_{ij}\;, \label{eq:ORp} \\
O_{ij}^{\prime\prime\, L} &= \frac{1}{2}\: \left ( O_{3i}^{*} \: O_{3j} - O_{2i}^{*} \: O_{2j} \right ) \;.
\label{gZxx}\end{aligned}$$ with $s_{W}, c_{W}$ the $\sin$ and $\cos$ of the weak mixing angle and $g$ the $SU(2)_{W}$ gauge coupling. Finally, interactions between $\chi$’s and $W$-bosons are described by the terms $$\begin{aligned}
\mathscr{L}_{\rm KIN (int)}^{W^{\pm}-\chi^{0}-\chi^{\mp}} \ &= \
g\: O_{ij}^{L} \: (\chi_{i}^{0})^{\dagger} \: \bar{\sigma}^{\mu} \: \chi^{+}_{j} \: W_{\mu}^{-}
-g \: O_{ij}^{R} \: (\chi_{j}^{-})^{\dagger} \: \bar{\sigma}^{\mu} \: \chi^{0}_{i} \: W_{\mu}^{-}
\nonumber \\[2mm]
& + g \: O_{ij}^{L*} \: (\chi_{j}^{+})^{\dagger} \: \bar{\sigma}^{\mu}\: \chi^{0}_{i}\: W_{\mu}^{+}
-g\: O_{ij}^{R*} \: (\chi_{i}^{0})^{\dagger} \: \bar{\sigma}^{\mu}\: \chi^{-}_{j} \: W_{\mu}^{+} \;,\end{aligned}$$ where the mixing matrices $O^{L}$ and $O^{R}$ are given by
$$\begin{aligned}
O_{ij}^{L} &= O_{1i}^{*} \: U_{L1j} \ - \ \frac{1}{\sqrt{2}} \: O_{3i}^{*} \: U_{L2j} \;,
\\[2mm]
O_{ij}^{R} &= O_{1i} \: U_{R1j}^{*} \ + \ \frac{1}{\sqrt{2}} \: O_{2i} \: U_{R2j}^{*} \;.\end{aligned}$$
\[eq:OLOR\]
We open a parenthesis here to discuss a comparison with MSSM: mass matrices for neutral and charged fermion in remind those of neutralinos and charginos in the MSSM. It is of course trivially understood why this happens: the doublet and the triplet fields possess the same gauge quantum numbers as the higgsino and wino fields, respectively. However, there are two crucial differences: first there is no restriction to add a bino singlet and therefore the minimal $\mathcal{M}_{N}$ is a $3\times 3$, instead of $4\times 4$, simpler matrix and second, and more important, the off-diagonal entries in $\mathcal{M}_{N}$ and $\mathcal{M}_{C}$, are not proportional to gauge couplings but to, Yukawa couplings, $Y_{1}$ and $Y_{2}$. The latter entries ($\sim Y v$) can be substantially bigger than the corresponding ones ($\sim g v$) in the neutralino mass matrix of MSSM. Furthermore, since $\tan\beta = 1$ is not, in general, a phenomenologically viable case in MSSM, there should always be a factor of hierarchy between the off diagonal entries. This is not necessarily the case here. In fact, the $\tan\beta=1$ “blind spot” [@Cheung:2012qy], is a point in parameter space protected by a custodial symmetry.
A custodial symmetry {#sec:symmetry}
--------------------
It is well known that the Higgs sector in the SM obeys, in addition to the standard electroweak gauge symmetry, a custodial $SU(2)_{R}$ global symmetry. This symmetry is broken explicitly by the hypercharge gauge coupling $g'$, and by the difference between the top- and bottom-quark Yukawa couplings. Similarly, the fermionic DM sector, described by , obeys also such a symmetry if $Y_{1}=Y_{2}\equiv Y$. More explicitly, can be written in a $SU(2)_{L}\times SU(2)_{R}\times U(1)_{X}$ invariant form as $$\mathscr{L}_{\rm Yuk}^{\rm DM} = -Y \: T^{A} \: \mathscr{H}^{x,a} \: (\tau^{A})_{a}^{b} \:\bar{\mathscr{D}}_{x,b} - \frac{1}{2}\: M_{D} \:\epsilon^{xy}\: \epsilon^{ab} \:
\bar{\mathscr{D}}_{x,a} \: \bar{\mathscr{D}}_{y,b} - \frac{1}{2}\: M_{T} \: T^{A} \: T^{A}
\ + \ {\rm H.c.}\;,
\label{LDMsym}$$ where $x,y$ denote $SU(2)_{R}$ group indices and $$\mathscr{H}^{x,a} = \left(\begin{array}{c} H^{a} \\ H^{\dagger\, a}\end{array}\right)
\;, \qquad
\bar{\mathscr{D}}_{x,a} = \left(\begin{array}{c} \bar{D}_{1a} \\
\bar{D}_{2a} \end{array}\right) \;, \label{eq:sym}$$ with $H^{a} = \epsilon^{ab} H_{b}$. This extra global symmetry stands for the rotations between $H \leftrightarrow H^{\dagger}$ and $\bar{D}_{1} \leftrightarrow \bar{D}_{2}$. Although this symmetry is broken by the hypercharge gauge symmetry, it is natural to study interactions among extra fermions $(\bar{\mathscr{D}}, T)$ and SM-bosons under the assumption that $SU(2)_{R}$ is approximately preserved in the DM sector, that is, $$Y_{1}\ = \ Y_{2} \ \Rightarrow \ m_{1}\ = \ m_{2} \;. \label{ass}$$ In addition, is invariant under a global $U(1)_{X}$ fermion number symmetry, under which only $\bar{\mathscr{D}}$ and $T$ fields are charged with $[\bar{\mathscr{D}}]= [D_{1}] = [D_{2}]=-[T]=1$. In that case $M_{D}$ and $M_{T}$ are not allowed. We therefore conclude that the limit where $Y\equiv Y_{1}=Y_{2}$ and $M_{D} = M_{T} \to 0$ is radiatively stable and this fact motivates us to study it in more detail. Note again that, both $SU(2)_{R}$ and $U(1)_{X}$ symmetries are broken explicitly by hypercharge symmetry.
Lightest Neutral fermion interactions under the symmetry {#sec:LP}
--------------------------------------------------------
Let’s introduce the mass difference, $\Delta m \equiv m_{1} -m_{2}$, between the chiral masses (or between Yukawa couplings, $Y_{1}$ and $Y_{2}$, if you wish). If $SU(2)_{R}$ symmetry is approximately preserved, [[*i.e.,* ]{}]{} approximately holds, $\Delta m$ must be treated as perturbation compared to $m_{1}$ or $m_{2}$ masses, which collectively denoted by $m=m_{1}$, [[*i.e.,* ]{}]{}$\Delta m\ll m$. We can then write the neutral fermion mass matrix in a suggestive perturbative form $$\mathcal{M}_{N} \ = \ \mathcal{M}_{N}^{(0)} + Q \;,$$ where $$\mathcal{M}_{N}^{(0)}=
\left(\begin{array}{ccc}M_T & m & -m \\m & 0 & M_D \\-m & M_D & 0\end{array}\right)
\;, \qquad Q =
\left(\begin{array}{ccc}0 & 0 & \Delta m \\0 & 0 & 0 \\\Delta m & 0 & 0\end{array}\right)\;.$$ The zeroth order eigenvalues of $\mathcal{M}_{N}^{(0)}$ read
$$\begin{aligned}
m_{\chi_{1}^{0}} &= M_{D}\;, \label{eig1}\\
m_{\chi_{2}^{0}} &= \frac{1}{2} \left [ M_{T} - M_{D} -
\sqrt{8 m^{2} + (M_{T} + M_{D})^{2}} \right ]\;, \label{eig2}\\
m_{\chi_{3}^{0}} &= \frac{1}{2} \left [ M_{T} - M_{D} +
\sqrt{8 m^{2} + (M_{T} + M_{D})^{2}} \right ]\;, \label{eig3}\end{aligned}$$
\[eigen\]
while the corresponding eigenvectors are $${|1\rangle}^{(0)} = \frac{1}{\sqrt{2}}\: \left(\begin{array}{c}0 \\1 \\ 1 \end{array}\right)\;,
\quad
{|2\rangle}^{(0)} = \frac{-1}{\sqrt{2+a^{2}}} \: \left(\begin{array}{c}a \\1 \\ -1\end{array}\right)
\;, \quad {|3\rangle}^{(0)} = \frac{1}{\sqrt{2+a^{2}}} \:
\left(\begin{array}{c}\sqrt{2} \\-\frac{a}{\sqrt{2}} \\\frac{a}{\sqrt{2}}\end{array}\right) \;,
\label{vecs}$$ where the parameter $a$ is given by $$a = \frac{m_{\chi_{1}^{0}} + m_{\chi_{2}^{0}}}{m} \;. \label{eq:a}$$ The parameter $a$, varies in the interval $[-\sqrt{2} , 0]$ for positive $M_{D}$. A little examination of the eigenvalues show that unless, $M_{D} \gg M_{T}>0$ where the Lightest Particle (LP) becomes the triplet, in the rest of the parameter space the LP is a “very well tempered” mixed doublet fermion, ${|\chi_{1}^{0}\rangle} = \frac{1}{\sqrt{2}}
({|\bar{D}_{1}^{1}\rangle} + {|\bar{D}_{2}^{2}\rangle})$, with mass $m_{\chi_{1}^{0}} = M_{D}$.[^11] The DM particle $(\chi_{1}^{0})$ has then vanishing coupling to the Higgs boson because in it is $O_{11}=0$. Note that, every neutral fermion has always vanishing diagonal couplings to $Z$-gauge boson, $|O_{2i}|=|O_{3i}|$, since the two doublets, $\bar{D}_{1}$ and $\bar{D}_{2}$ couple to $Z$ with opposite weak isospin. It is therefore worth examining how eigenvalues and eigenvectors are corrected after switching on to $\Delta m \ne 0$.
Obviously, in order to find how $\chi^{0}_{1}$ couples to $Z$ or $h$ non-trivially, [[*i.e.,* ]{}]{}to find the couplings $Y^{h\chi_{1}^{0}\chi_{1}^{0}}$ and $g^{Z\chi_{1}^{0}\chi_{1}^{0}} = g O_{11}^{\prime\prime \, L}/c_{W}$ in , respectively, we need to know the $O(\Delta m)$ corrections, in eigenvector the $O_{i1}$. The corrected eigenvector, ${|1\rangle} = {|1\rangle}^{(0)} + {|1\rangle}^{(1)} + O[(\Delta m)^{2}]$, which is nothing else but the first column of the matrix $O$ in is found to be, $$O_{i1}={|1\rangle} =\frac{1}{\sqrt{2}}
\: \left(\begin{array}{c}x \: \Delta m \\1+y\: \Delta m \\1-y \: \Delta m\end{array}\right) +
O[(\Delta m)^{2}] \;, \label{O1icor}$$ where $$\begin{aligned}
x &\equiv \frac{1}{(2+ a^{2})} \left [\frac{a^{2}}{m_{\chi_{1}^{0}} - m_{\chi_{2}^{0}}}
+ \frac{2}{m_{\chi_{1}^{0}} - m_{\chi_{3}^{0}}} \right ]\;, \\[2mm]
y &\equiv \frac{a}{(2+ a^{2})} \left [\frac{1}{m_{\chi_{1}^{0}} - m_{\chi_{2}^{0}}} -
\frac{1}{{m_{\chi_{1}^{0}} - m_{\chi_{3}^{0}}}} \right ] \;.\label{eq:y}\end{aligned}$$ Simple substitution of into gives $$\begin{aligned}
Y^{h\chi_{1}^{0}\chi_{1}^{0}} &= \frac{(\Delta m)^{2}}{\sqrt{2} v} \: x \: (1+ 2 m y) +
\mathrm{O}[(\Delta m)^{2}/m^{2}]\;, \label{eq:Yhxx}\\[3mm]
g^{Z\chi_{1}^{0}\chi_{1}^{0}} &\equiv \frac{g}{c_{W}} O_{11}^{\prime\prime \, L} =
-\frac{g}{c_{W}} \: y \: \Delta m + \mathrm{O}[(\Delta m)^{2}/m^{2}]\;.\label{eq:Zxx}
\end{aligned}$$ Obviously, for sufficiently small mass difference $\Delta m$, the Spin-Independent (SI) coupling $(Y^{h\chi_{1}^{0}\chi_{1}^{0}})$ is suppressed by $(\Delta m)^{2}/m^{2}$ while the Spin-Dependent (SD) one $(g^{Z\chi_{1}^{0}\chi_{1}^{0}})$ is suppressed by $\Delta m/m$ relative to their values away from the $SU(2)_{R}$-symmetric limit. This maybe the reason why we have not detected DM-nucleon interactions so far. A question arises immediately about the stability of $\Delta m$ under radiative corrections. A quick RGE analysis [@Giudice:2011cg; @ArkaniHamed:2012kq] shows that the $\beta$-function for $\Delta m$ at 1-loop is $$\frac{d \Delta m}{d \ln(Q)} \ = \ \frac{\Delta m}{16 \pi^{2}} \,
\left [ \frac{29}{4} Y^{2} + 3 Y_{t}^{2} - \frac{9}{20} g_{1}^{2} - \frac{33}{4} g_{2}^{2} \right ] \;,
\label{eq:dmrge}$$ where $Y_{t}$ is the top-Yukawa coupling, $Y\equiv Y_{1} \simeq Y_{2}$, and $g_{1,2}$ the hypercharge and weak gauge couplings, respectively. means that $\Delta m$ is only multiplicatively renormalized. Therefore, setting $\Delta m$ to zero at tree level stays zero at 1-loop and possibly at higher orders[^12] because this is a parameter point protected by the global symmetry. From we conclude that for $\Delta m=0$, only finite (threshold) and calculable quantum corrections will affect the couplings $Y^{h\chi_{1}^{0}\chi_{1}^{0}}$ and $g^{Z\chi_{1}^{0}\chi_{1}^{0}}$ which are relevant to Direct DM searches. We confirm this consequence with a direct calculation of $\delta Y^{h\chi_{1}^{0}\chi_{1}^{0}}$ in section \[sec:direct\] and in \[sec:appA\].
Note that $x$ vanishes in the limit $M_{D} \to 0$ while $(1+ 2 m y)$ vanishes at both $M_{D}\to 0$ and $M_{D} \to M_{T}$ limits. However, is not accurate since $(\Delta m)^{2}/m^{2}$-terms are missing in our perturbative expansion. It turns out that the $M_{D}\to M_{T}$ limit is violated by those and higher terms, but the limit $M_{D}\to 0$ is protected because of the $U(1)_{X}$-symmetry that we discussed in section \[sec:symmetry\]. In contrast, is within 1% of its exact numerical outcome. It is also worth noticing that in the case where the Majorana masses are dominant, $M_{D},M_{T}\gg m$, then $y\to 0$ and therefore $g^{Z\chi_{1}^{0}\chi_{1}^{0}} \to 0$, up to higher order terms.
It will be useful for the discussion, especially on the relic density, to show the mass difference between the next-to-lightest ($|m_{\chi_{2}^{0}}|$) and the lightest ($|m_{\chi_{1}^{0}}|$) neutral fermion states. This is depicted as contour lines in Fig. \[fig:masses\](a,b) on the $M_{D} - M_{T}$ plane (left plot) and on the $M_{D} - m$ plane with $M_{T} = M_{D}$ (right plot). Note that $M_{D}$ coincides with the LP mass [[*i.e.,* ]{}]{} $M_{D} = m_{\chi_{1}^{0}}$, everywhere in these graphs. For $m=200$ GeV, the mass difference is nowhere smaller than approximately 80 GeV, and typically, it is as large as the parameter $m$ with the maximum value at $M_{D}=M_{T}$. Subsequently, in Fig. \[fig:masses\]b, we plot the maximum values of the mass difference on $M_{D}-m$ plane. Alternatively, it is easy to read from Fig. \[fig:masses\], the parameter $a$ defined in , because for the $M_{D}$ values taken throughout, it is $a = - (|m_{\chi_{2}^{0}}| - |m_{\chi_{1}^{0}}|)/m$. For instance, in the plots shown, this parameter varies, approximately, in the region, $a \in [-1,-0.3]$.
Analytical expressions for the new interactions under the symmetry {#sec:a}
------------------------------------------------------------------
As we have already discussed in section \[spec\], in the symmetric $SU(2)_{R}$ limit of (\[ass\]), two of the eigenvalues from the charged fermion mass matrix are degenerate respectively with those of the neutral fermion masses given in , $$m_{\chi_{1}^{\pm}} = m_{\chi_{2}^{0}}\;, \quad m_{\chi_{2}^{\pm}} = m_{\chi_{3}^{0}}\;.
\label{cusmass}$$ In addition, it is useful for further reference to present analytical expressions for all new interactions appear in the model. All these new interactions can be simply written in matrix forms containing (at most) one parameter, the real parameter $a$ of . For example, rotation matrices defined in read $$\begin{aligned}
U = U_{L} = U_{R}= \frac{1}{\sqrt{2+ a^{2}}} \: \left(\begin{array}{cc}a & -\sqrt{2} \\\sqrt{2} & a\end{array}\right) \;, \quad
O = \left(
\begin{array}{ccc}
0 & -\frac{a}{\sqrt{2+a^2}} & \frac{\sqrt{2}}{\sqrt{2+a^2}} \\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2+a^2}} & -\frac{a}{\sqrt{2} \sqrt{2+a^2}} \\
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2+a^2}} & \frac{a}{\sqrt{2} \sqrt{2+a^2}}
\end{array}
\right) \;.\end{aligned}$$ The couplings between $\chi_{1}^{0}$, $W$ and $\chi^{\pm}$ given in become explicitly: $$\begin{aligned}
O_{1j}^{L} = - O_{1j}^{R\, *}\;, \quad
O^{L} = \left(
\begin{array}{cc}
-\frac{1}{\sqrt{2} \sqrt{2+a^2}} & -\frac{a}{2 \sqrt{2+a^2}} \\
-\frac{1+a^2}{2+a^2} & \frac{a}{\sqrt{2} \left(2+a^2\right)} \\
\frac{a}{\sqrt{2} \left(2+a^2\right)} & -\frac{4+a^2}{4+2 a^2}
\end{array}
\right) \;, \quad
O^{R} = \left(
\begin{array}{cc}
\frac{1}{\sqrt{2} \sqrt{2+a^2}} & \frac{a}{2 \sqrt{2+a^2}} \\
-\frac{1+a^2}{2+a^2} & \frac{a}{\sqrt{2} \left(2+a^2\right)} \\
\frac{a}{\sqrt{2} \left(2+a^2\right)} & -\frac{4+a^2}{4+2 a^2}
\end{array}
\right) \;,\end{aligned}$$ while those in , $$\begin{aligned}
O^{\prime\, L(R)} =\left(
\begin{array}{cc}
\frac{-1-a^2+\left(2+a^2\right) s_W^2}{2+a^2} & \frac{a}{\sqrt{2} \left(2+a^2\right)} \\
\frac{a}{\sqrt{2} \left(2+a^2\right)} & -\frac{4+a^2-2 \left(2+a^2\right) s_W^2}{2 \left(2+a^2\right)}
\end{array}
\right)\;, \quad
O^{\prime\prime\, L} = \left(
\begin{array}{ccc}
0 & \frac{1}{\sqrt{2} \sqrt{2+a^2}} & \frac{a}{2 \sqrt{2+a^2}} \\
\frac{1}{\sqrt{2} \sqrt{2+a^2}} & 0 & 0 \\
\frac{a}{2 \sqrt{2+a^2}} & 0 & 0
\end{array}
\right)\;. \label{eq:anOLpp}\end{aligned}$$ Finally, the Higgs couplings to neutral and charged fermions in are respectively: $$\begin{aligned}
Y^{h\chi^{0}\chi^{0}} = \frac{m}{v} \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & \frac{2 \sqrt{2} a }{\left(2+a^2\right) } & \frac{\left(-2+a^2\right) }{\left(2+a^2\right) } \\
0 & \frac{\left(-2+a^2\right) }{\left(2+a^2\right) } & -\frac{2 \sqrt{2} a }{\left(2+a^2\right) }
\end{array}
\right)\;, \qquad
Y^{h\chi^{-}\chi^{+}} = \frac{m}{v} \left(
\begin{array}{cc}
\frac{2 \sqrt{2} a }{\left(2+a^2\right) } & \frac{\left(-2+a^2\right) }{\left(2+a^2\right) } \\
\frac{\left(-2+a^2\right) }{\left(2+a^2\right) } & -\frac{2 \sqrt{2} a }{\left(2+a^2\right) }
\end{array}
\right)\;, \label{eq:hx0x0}\end{aligned}$$ while those to Goldstone bosons given in , can now be simply written as $$\begin{aligned}
Y^{G^{0}\chi^{0}\chi^{0}} = \frac{i m}{v} \left(
\begin{array}{ccc}
0 &- \frac{a}{\sqrt{2+a^2}} & \frac{\sqrt{2}}{\sqrt{2+a^2}} \\
-\frac{a}{\sqrt{2+a^2}} & 0 & 0 \\
\frac{\sqrt{2}}{\sqrt{2+a^2}} & 0 & 0
\end{array}
\right)\;, \qquad
Y^{G^{0}\chi^{-}\chi^{+}}= \frac{ i m}{v} \left(
\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}
\right)\, \forall{a}\;,\end{aligned}$$ and $$\begin{aligned}
Y^{G^{+}\chi^{-}\chi^{0}} = \frac{m}{v}\left(
\begin{array}{ccc}
\frac{a}{\sqrt{2+a^2}} & 0 & -1 \\
-\frac{\sqrt{2}}{\sqrt{2+a^2}} & 1 & 0
\end{array}
\right)\;, \qquad
Y^{G^{-}\chi^{+}\chi^{0}} = \frac{m}{v}
\left(
\begin{array}{ccc}
-\frac{a}{\sqrt{2+a^2}} & 0 & -1 \\
\frac{\sqrt{2}}{\sqrt{2+a^2}} & 1 & 0
\end{array}
\right)\;.\end{aligned}$$ Depending on whether the chiral mass $m$ or the vectorial masses $M_{D}$ and $M_{T}$ are dominant, and for $M_{D}>0$, there are two extreme limits for the model at hand $$\begin{aligned}
\mathrm{``Majorana~ dominance'' } &: M_{T} \approx M_{D} \gg m \Rightarrow a \approx 0 \;, \quad
m_{\chi_{1}^{0}}^{2} \approx m_{\chi_{2}^{0}}^{2} \approx M_{D}^{2}\;,
\quad m^{2}_{\chi_{3}^{0}} \approx M^{2}_{T}\;.\\[3mm]
\mathrm{``Dirac~ dominance '' } &: M_{T} \approx M_{D} \ll m \Rightarrow a \approx -\sqrt{2} \;, \quad
m_{\chi_{2}^{0}}^{2} \approx m_{\chi^{0}_{3}}^{2} \approx M_{D}^{2}+2 m^{2} \;.
\label{eq:dd}\end{aligned}$$ The “Majorana dominance” limit corresponds more or less to the “higgsino-wino” scenario of the MSSM where the first two neutral particle masses are degenerate, while the “Dirac dominance” limit is the imprint of a large Yuakawa coupling in . It is the latter case that in addition to $SU(2)_{R}$-symmetry, it is protected by the global $U(1)_{X}$ symmetry. For example, plugging in $a=-\sqrt{2}$ into , we immediately see that the Higgs couplings to new fermions become diagonal resulting in a vanishing, as long as $M_{D}\to 0$, one loop corrections to the $h-\chi_{1}^{0}-\chi_{1}^{0}$ vertex, as we qualitatively confirmed in section \[sec:LP\] below , and as we shall see below in section \[sec:direct\].
Composition of the lightest Neutral Fermion
-------------------------------------------
As we showed in , in the symmetric limit $m_{1}=m_{2}$, the neutral fermion mass matrix $\mathcal{M}_{N}$, can be diagonalized analytically into three mass eigenstates $${|\chi_{i}^{0}\rangle} \ = \ O_{i1}\, {|1\rangle} + O_{i2}\, {|2\rangle} + O_{i3}\, {|3\rangle} \;.$$ Following conventional MSSM nomenclature [@Jungman:1995df], lets define the “Doublet” composition of the $\chi_{i}^{0}$ as $$F_{D}^{i} = |O_{i2}|^{2} + |O_{i3}|^{2}\;.\label{eq:fd}$$ Then we say that a state of $\chi_{i}^{0}$ is (D)oublet-like if $F_{D}^{i} > 0.99$, it is (T)riplet like if $F_{D}^{i} < 0.01$ and it is (M)ixed state if $0.01<F_{D}^{i} <0.99$.
![*The composition of the WIMP in terms of (D)oublet, (T)riplet and (M)ixed states following the definition given in the paragraph below , on a $M_{D}$ vs. $M_{T}$ plane and for fixed (common) Yukawa coupling, $Y=m/v\simeq 200/174\simeq 1.15$.* []{data-label="fig:comp"}](Composition-Plot-2.pdf){width="3in"}
In Fig. \[fig:comp\] we present the composition of the DM candidate particle $\chi_{1}^{0}$ on a $M_{D}$ vs. $M_{T}$ plane for fixed mass, $m=200$ GeV. Both $Z$ and Higgs-boson couplings to pairs of $\chi_{1}^{0}$’s vanish at tree level only in the region denoted by (D) (for Doublet) where $M_{D}$ is (most of the time) positive and equal to or less than $M_{T}$. It is mostly in this region we are focusing on in this article, because in this region the model evades, without further tweaks, direct DM detection experimental bounds. Note also that for light $M_{D} = \chi_{1}^{0}\lesssim 150~\mathrm{GeV} << m$, the WIMP composition satisfies (D) condition for every value of $M_{T}$. For negative values of $M_{D}$, $\chi_{1}^{0}$ is a pure doublet only in the region $|M_{D}| \le m$ but shrinks down to unacceptably small $M_{D}$ for large values of $M_{T}$; otherwise it is a mixed state everywhere in Fig. \[fig:comp\]. For large $M_{D} \gg M_{T}$, the $\chi_{1}^{0}$-composition consists of mainly a triplet.
Note that when the lightest state is pure (D)oublet the heavier states are exactly an equal admixture of doublets and the triplet [[*i.e.,* ]{}]{}$F_{D}^{2,3} = 0.5$.
Estimate of Electroweak Corrections {#sec:STU}
===================================
In the limit of large Yukawa couplings, $Y = Y_{1} = Y_{2} \simeq 1$, we generally expect large contributions from the new fermions, $\chi^{0},\chi^{\pm}$, to $(Z,W)$-gauge boson self-energy one-loop diagrams. In this section we investigate constraints on the doublet-triplet fermion model parameter space, $\{M_{D}, M_{T}, m\}$, from the oblique electroweak parameters $S,T$ and $U$ [@Peskin:1991sw].
Due to ${Z}_{2}$-parity symmetry, at one-loop level, there is no mixing between the extra fermions, $\chi^{0},\chi^{\pm}$, and the SM leptons. Therefore corrections to electroweak precision observables involving light fermions arise only from gauge bosons vacuum polarisation Feynman diagrams [[*i.e.,* ]{}]{}there are only oblique electroweak corrections. In order to estimate these corrections it is convenient to calculate the $S, T$ and $U$ parameters, in the limit where $m_{\chi^{0}}, m_{\chi^{\pm}} \gtrsim m_{Z}$. This is true when the doublet mass $M_{D},$ is greater than $m_{Z}$ and $m$ is much greater than $m_{Z}$ (see Fig. \[fig:masses\]). We shall not consider the case of a light dark matter particle, $m_{\chi_{1}^{0}} \lesssim m_{Z}$.
Following closely the notation by Peskin and Takeuchi in , we write,
$$\begin{aligned}
\alpha \, S &\equiv 4 e^{2} \: \frac{d}{dp^{2}} \left [\Pi_{33}(p^{2}) - \Pi_{3Q}(p^{2})
\right ] \biggl |_{p^{2}=0} \;, \\
\alpha \, T &\equiv \frac{e^{2}}{s_{W}^{2} c_{W}^{2} m_{Z}^{2}}\:
\left [\Pi_{11}(0) - \Pi_{33}(0) \right ]\;, \\
\alpha \, U &\equiv 4 e^{2} \: \frac{d}{dp^{2}} \left [\Pi_{11}(p^{2}) - \Pi_{33}(p^{2})
\right ]\biggl |_{p^{2}=0} \;, \end{aligned}$$
\[eq:STU\]
where $\alpha = e^{2}/4\pi$. In numerics we use input parameters from , the bare value at lowest order $s_{W}^{2} = g^{'2}/(g^{'2}+g^{2}) \simeq 0.2312$ and the $Z$-pole mass $m_{Z}=91.1874$ GeV. We calculate corrections arising only from the extra fermions, $\chi_{i=1..3}^{0}, \chi_{i=1..2}^{\pm}$, to the $g^{\mu\nu}$ part of the gauge boson self energy amplitudes, $\Pi_{IJ}\equiv \Pi_{IJ}(p^{2})$, where $I$ and $J$ may be photon ($\gamma)$, $W$ or $Z$,
$$\begin{aligned}
\Pi_{\gamma\gamma} &= e^{2}\: \Pi_{QQ} \;, \\
\Pi_{Z\gamma} &= \frac{e^{2}}{c_{W}s_{W}} \: \left ( \Pi_{3Q} - s^{2} \Pi_{QQ} \right )\;,\\
\Pi_{ZZ} &= \frac{e^{2}}{c_{W}^{2} s_{W}^{2}} \:
\left (\Pi_{33} - 2 s^{2} \Pi_{3Q} + s^{4} \Pi_{QQ}\right )\;,\\
\Pi_{WW} &= \frac{e^{2}}{s_{W}^{2}} \: \Pi_{11} \;,\end{aligned}$$
where $s_{W}=\sin\theta_{W}, c_{W}=\cos\theta_{W}$. We find,
$$\begin{aligned}
\Pi_{QQ} &= - \frac{p^{2}}{8 \pi^{2}} \sum_{i=1}^{2} \: \left [ \frac{2}{3}\, E -
4 \,b_{2}(p^{2}, m_{\chi_{i}^{\pm}}^{2},m_{\chi^{\pm}_{i}}^{2}) \right ]\;, \\
\Pi_{3Q} &= \frac{p^{2}}{16\pi^{2}}\: \sum_{i=1}^{2} (Z_{ii}^{L} + Z_{ii}^{R}) \:
\left [ \frac{2}{3}\, E - 4\, b_{2}(p^{2}, m_{\chi_{i}^{\pm}}^{2},m_{\chi^{\pm}_{i}}^{2}) \right ]\;, \\
\Pi_{33} &= \frac{1}{16\pi^{2}} \: \sum_{i,j=1}^{2}
\left [ (Z_{ij}^{L} Z_{ji}^{L} + Z_{ij}^{R} Z_{ji}^{R}) \:
G(p^{2}, m_{\chi_{i}^{\pm}}^{2},m_{\chi^{\pm}_{j}}^{2}) -2 \, Z_{ij}^{L} Z_{ji}^{R}\:
m_{\chi^{\pm}_{i}}m_{\chi^{\pm}_{j}} \: I(p^{2}, m_{\chi_{i}^{\pm}}^{2},m_{\chi^{\pm}_{j}}^{2}) \right ]
\nonumber \\
&+ \frac{1}{16\pi^{2}}\: \sum_{i,j=1}^{3}\left [ O_{ij}^{\prime\prime \, L} O_{ji}^{\prime\prime \, L} \:
G(p^{2}, m_{\chi_{i}^{0}}^{2},m_{\chi^{0}_{j}}^{2}) + (O_{ij}^{\prime\prime \, L})^{2} \:
m_{\chi^{0}_{i}}m_{\chi^{0}_{j}} \: I(p^{2}, m_{\chi_{i}^{0}}^{2},m_{\chi^{0}_{j}}^{2}) \right ]\;, \\
\Pi_{11} &= \frac{1}{16\pi^{2}} \: \sum_{i=1}^{3} \sum_{j=1}^{2} \left [ (|O_{ij}^{L}|^{2} +
|O_{ij}^{R}|^{2} ) \: G(p^{2}, m_{\chi_{i}^{0}}^{2},m_{\chi^{\pm}_{j}}^{2}) -
2 \Re e(O_{ij}^{L\, *} O_{ij}^{R}) \, m_{\chi_{i}^{0}} m_{\chi^{\pm}_{j}}\: I(p^{2}, m_{\chi_{i}^{0}}^{2},m_{\chi^{\pm}_{j}}^{2}) \right ] \;,
$$
\[eq:pij\]
where $Z_{ij}^{L(R)} \equiv O_{ij}^{\prime \, L(R)} -s_{W}^{2} \delta_{ij}$. In addition, $E\equiv \frac{2}{\epsilon} -\gamma+ \log 4\pi - \log Q^{2}$ is the infinite part of loop diagrams. The various one-loop functions in eqs. (\[eq:pij\]) are given by $$\begin{aligned}
G(p^{2},x,y) &= -\frac{2}{3} p^{2} \: E + (x+y) \: E + 4 \: p^{2}\: b_{2}(p^{2},x,y)
- 2 \left [ y\: b_{1}(p^{2},x,y) + x\: b_{1}(p^{2},y,x) \right ] \;, \\
I(p^{2},x,y) &= 2 \: E - 2\: b_{0}(p^{2},x,y) \;, \\
b_{0}(p^{2},x,y) &= \int_{0}^{1} dt \log \frac{\Delta}{Q^{2}} \;, \quad
b_{1}(p^{2},x,y) = \int_{0}^{1} dt \, t \,\log \frac{\Delta}{Q^{2}} \;, \\
b_{2}(p^{2},x,y) &= \int_{0}^{1} dt \, t\, (1-t)\,\log \frac{\Delta}{Q^{2}} \;, \quad
\Delta = t y + (1-t) x - t(1-t)\, p^{2} - i\epsilon\;.\end{aligned}$$ There are numerous useful identities, $$\begin{aligned}
b_{0}(p^{2},x,y) &= b_{0}(p^{2},y,x)\;, \quad b_{2}(p^{2},x,y) = b_{2}(p^{2},y,x)\;, \\
G(p^{2},x,y) &= G(p^{2},y,x) \;, \quad I(p^{2},x,y) = I(p^{2},y,x) \;, \\
b_{1}(p^{2}, x,y) &= b_{0}(p^{2},y,x) - b_{1}(p^{2},y,x)\;, \quad
b_{1}(p^{2},x,x) =\frac{b_{0}(p^{2},x,x)}{2}\;,\end{aligned}$$ that will help us to simplify our expressions below.
Furthermore, in the exact $SU(2)_{R}$ limit where $m_{1} = m_{2}$, there is no isospin breaking in $\bar{D}$-components and therefore $T=0$, while the $S$-parameter receives non-zero, non-decoupled, contributions due to the enlarged particle number of the $SU(2)$-sector. Specifically, in the limit where $M_{D} = M_{T} \ll m=m_{1}=m_{2}$, there is a light neutral fermion $(m_{\chi_{1}^{0}})$ and heavy degenerate other four (two neutral and two charged) fermions, with squared mass $x$, resulting in
$$\begin{aligned}
\Pi_{3Q}^{\prime} (0) &\approx \frac{1}{16\pi^{2}} \: \biggl [ - 2 E + 2 \ln(\frac{x}{Q^{2}} )\biggr ] \;, \\
\Pi_{33}^{\prime} (0) &= \Pi_{11}^{\prime}(0)\approx \frac{1}{16\pi^{2}} \: \biggl [ - 2 E +
2 \ln(\frac{x}{Q^{2}} )+
\frac{1}{18} \biggr ] \;, \label{p33p}\\
\Pi_{33}(0) &= \Pi_{11}(0) \approx \frac{1}{16\pi^{2}}\: \biggl [ \frac{3 x}{2} E -
\frac{3 x}{2} \ln (\frac{x}{Q^{2}}) +
\frac{x}{4} \biggr ]\;.
$$
\[eq:appSTU\]
Plugging in eqs. (\[eq:appSTU\]) into eqs. (\[eq:STU\]) we arrive at the approximate values expressions $$S \ \approx \ \frac{1}{18\pi}\;, \quad T \approx U \approx 0 \;.
\label{appresSTU}$$ This result is also confirmed numerically in Fig. \[fig:STU\] where we draw contours of the $S$-parameter on $M_{D}$ vs. $M_{T}$ plane (left plot) and on $M_{D}$ vs. $m$ plane (right plot). As it is shown, for large $m$ we obtain $S \to 1/18\pi \simeq 0.0177$ while for $m\to 0$ we obtain $S\to 0$, as expected because in this case only vector-like masses will exist in $\mathscr{L}_{\rm Yuk}^{\rm DM}$ of , that make no contribution to parameter $S$. Experimentally, we know [@PDG] that when the $U$-parameter is zero, the parameters $S$ and $T$ which fit the electroweak data are constrained to be
$$\begin{aligned}
S &= 0.04 \pm 0.09\;, \label{PDG-S}\\
T &= 0.07 \pm 0.08 \;.\end{aligned}$$
Predictions for the $S$-parameter shown in Fig. \[fig:STU\]a,b comfortably fall within the bound of (\[PDG-S\]). In addition, even though it is not shown, the $T,U$-parameters are always negligibly small.
The Thermal Relic Dark Matter Abundance {#sec:relic}
=======================================
![*Lower level Feynman diagrams contributing to annihilation cross section for the process $\chi+\chi \to V +V$ for $V=W,Z$.*[]{data-label="fig:xxtoVV"}](xxtoVV.png){width="3in"}
As we have seen, $V\chi_{1}^{0}\chi_{1}^{0}$ with $V=W,Z$ and $h\chi_{1}^{0}\chi_{1}^{0}$ are forbidden at tree level if $\chi_{1}^{0}$ is a pure doublet [[*i.e.,* ]{}]{} $m_{\chi_{1}^{0}} = M_{D}$, in the exact $SU(2)_{R}$-limit. Therefore, the annihilation cross section for the lightest neutral fermion results solely from the following $t-$ and $u-$channel tree level Feynman diagrams, shown in Fig. \[fig:xxtoVV\], with neutral or charged fermion exchange, collectively shown as $\chi_{i}$, with axial-vector interactions
$$\begin{aligned}
\chi_{1}^{0} \ +\ \chi_{1}^{0} & \rightarrow W^{+} \ + \ W^{-} \;, \\
\chi_{1}^{0} \ +\ \chi_{1}^{0} & \rightarrow Z \ + \ Z \;.
$$
All other processes vanish at tree level. This can easily be understood by looking at the matrix forms of $O^{\prime\prime\,L}$ and $Y^{h\chi^{0}\chi^{0}}$ in . Before presenting our results for the annihilation cross section it is helpful to (order of magnitude) estimate the thermal dark matter relic density for $\chi_{1}^{0}$s. Consequently, by expanding the total cross section as $\sigma_{Ann} v = a_{V} + b_{V} v^{2} +\, ...$ [@Drees:1992am; @Jungman:1995df] and keeping only the zero-relative-velocity $a$-terms we find (for $M_{D} = M_{T}$):
$$\begin{aligned}
a_{W} &= \frac{g^{4}\, \beta_{W}^{3}}{32 \pi} \:
\frac{m_{\chi}^{2}}{(m_{\chi}^{2} + m_{\chi_{j}}^{2} - m_{W}^{2})^{2}}
\ \xrightarrow[m \gg M_{D}]{m_{\chi_{j}} \gg m_{\chi}} \ \frac{g^{4}\,
\beta_{W}^{3}}{32 \pi} \: \left(\frac{m_{\chi}}{m_{\chi_{j}}} \right )^{4} \: \frac{1}{m_{\chi}^{2}}
\;, \label{eq:aW} \\[2mm]
a_{Z} &= \frac{g^{4}\, \beta_{Z}^{3}}{64 \pi \, c_{W}^{4}} \:
\frac{m_{\chi}^{2}}{(m_{\chi}^{2} + m_{\chi_{j}}^{2} - m_{W}^{2})^{2}} \
\xrightarrow[m \gg M_{D}]{m_{\chi_{j}} \gg m_{\chi}} \
\frac{g^{4}\, \beta_{Z}^{3}}{64 \pi \, c_{W}^{4}} \:
\left(\frac{m_{\chi}}{m_{\chi_{j}}} \right )^{4} \: \frac{1}{m_{\chi}^{2}}
\;,
\label{eq:aZ}\end{aligned}$$
\[eq:as\]
where $g\approx 0.65$ is the electroweak coupling, $\beta_{V}= \sqrt{1-m_{V}^{2}/m_{\chi}^{2}}$ for $V=W,Z$, and in order to simplify notation, we take $m_{\chi}\equiv m_{\chi_{1}^{0}}$ to denote the DM particle mass and $m_{\chi_{j}} \equiv m_{\chi_{j}^{0}} = m_{\chi_{j-1}^{\pm}} \ge m_{\chi}$ for $j=2,3$ \[see \] the heavier neutral and charged fermions of the DM-sector. In the case where $M_{D}=M_{T}$, the heavier fermions are degenerate with mass, $m^{2}_{\chi_{j}} = 2 m^{2} + M_{D}^{2}$, and the mass spectrum pattern is similar to the one shown in Fig. \[fig:spec\]. Following this pattern in we have taken the limit of $m\gg M_{D}$ or alternatively, $m_{\chi_{j}} \gg m_{\chi}$.
Obviously, viewed as functions of $M_{D}$, exhibit a maximum extremum since both $a$’s vanish in the limits of $M_{D} \to 0$ and $M_{D} \to \infty$ and, in addition, they are positive definite. The maximum cross section is obtained approximately at $M_{D} \approx \sqrt{2} m$. The situation is clearly sketched in Fig. \[fig:xsec\].
![Sketch of the resulting annihilation cross section. []{data-label="fig:xsec"}](Xsection-sketch-2.pdf){width="3in"}
Once again, we assume that particle-$\chi$ is a cold thermal relic, and that its mass is about few tens bigger than its freeze-out temperature. Then, universe’s critical density times the Hubble constant squared (in units of 100 km/s/Mpc, $h^{2}\simeq 0.5$) for $\chi$’s is $$\Omega_{\chi}\, h^{2} \sim 0.1\; \frac{10^{-8} \; \mathrm{GeV}^{-2}}{\sigma v} \;.
\label{ome}$$ Therefore, if the correct cross section, $\sigma v \approx 10^{-8}~\mathrm{GeV}^{-2}$, that produces the right relic density, $\Omega_{\chi}\, h^{2} \sim 0.1$, happens to be below the maximum of $\sigma v$ in Fig. \[fig:xsec\] then there are two of its points crossing the observed relic density: one for low $M_{D}$ and one for high $M_{D}$ with the single crossing point being at $M_{D} \approx \sqrt{2} m$. The mass spectrum of new fermions with high $M_{D}$ exhibits nearly degeneracy in the first two states [[*i.e.,* ]{}]{}$m_{\chi} = m_{\chi_{2}} \simeq M_{D}$. This shares similarities with the MSSM (or more precisely with the Split SUSY with $\tan\beta =1$ “wino-higgsino” scenario) for higgsino Dark Matter which is well studied and we are not going to pursue further. The other case, on the other hand, with low $M_{D} \lesssim m$, exhibits a mass hierarchy between the DM candidate particle ($\chi$) and all the rest ($\chi_{j}$) particles. It is the suppression factor $(m_{\chi}/m_{\chi_{j}})$ to the fourth power in that prohibits the cross section from taking on very large values. It is therefore evident that this low $M_{D}$ scenario can provide the SM with a DM candidate particle with mass $M_{D}$ that lies “naturally” at the EW scale as this suggested by the observation $\sigma \approx 10^{-8}~\mathrm{GeV}^{-2}$, and is accompanied by heavy fermions few to several times heavier (depending on the value of $m$) than $M_{D}$.
Before proceeding further, it is worth looking back at Fig. \[fig:masses\], the mass difference between the first two neutral states. For $m \gtrsim 100$ GeV the mass difference is always more than 50% than the lightest mass $m_{\chi}$. This in turn suggests that *no significant* contributions to $\Omega_{\chi} h^{2}$ are anticipated from co-annihilation effects [@Griest:1990kh].
In the end, we have calculated the today’s relic density of the neutral, stable, and therefore, DM-candidate particle $\chi$. Our calculation is a tree level one; see however comments below. Within the context of the (spatially flat) six-parameters standard cosmological model, [*Planck*]{} experiment [@Ade:2013zuv] reports a density for cold, non-baryonic, dark matter, that is $$\begin{aligned}
\Omega \, h^{2} = 0.1199 \pm 0.0027 \;. \label{eq:omobs}\end{aligned}$$ The 2-$\sigma$ value is satisfied only in the area between the two lines in both plots in Fig. \[figos\]. This happens for rather low $m_{\chi}=M_{D}$ in the region $92 \lesssim m_{\chi_{1}^{0}} \lesssim 110$ GeV and for $M_{T} \lesssim 420$ GeV on the $M_{D}-M_{T}$ plane with fixed $m=200$ GeV, in Fig. \[figos\]a.[^13] We also observe that the result for $\Omega_{\chi} h^{2}$ is not very sensitive to the triplet mass, $M_{T}$. Even vanishing $M_{T}$-values are in accordance with the observed $\Omega_{\chi} h^{2}$, with mass values $m_{\chi}$ laying nearby the EW scale. (If $M_{D}$ is in the region $m_{W} < M_{D} < m_{Z}$, and if we neglect three body decays, then the cross section becomes about half the one for $M_{D}>m_{Z}$. This means that $\Omega h^{2}$ is doubled and therefore larger $M_{T}$ (about twice as large) masses may be consistent with the observed $\Omega h^{2}$ values in .)
We also consider the effect on $\Omega_{\chi} h^{2}$ from varying $m$ and $M_{D}$, with $M_{D}=M_{T}$, in Fig. \[figos\]b. Obviously, the lower the $m$ is the lower the $M_{D}$ should be. For $m_{\chi} \simeq 91$ GeV the correct density is obtained for $m \simeq 140$ GeV. As we move to heavier values [[*i.e.,* ]{}]{}$m\approx 300$ GeV, $M_{D}$ (which is equal to $m_{\chi}$), is required to be heavier, but not much heavier, than $M_{Z}$. However, as we shall discuss in section \[sec:Landau\], those heavy values of $m$ are not accepted by the vacuum stability constraint without modifying the model.
![Same as Fig. \[figos\]a but for negative values of $M_{D}$.[]{data-label="figosn"}](contours-Omega-NegativeMD.pdf){width="90mm"}
Consistent $\Omega_{\chi} h^{2}$ with observation is also achieved for negative values of $M_{D}$ in the same region as for positive $M_{D}$ as it is shown in Fig. \[figosn\]. (This is the small area for negative $M_{D}$ shown in Fig. \[fig:comp\] where $\chi_{1}^{0}$ is Doublet). The $M_{T}$ values where this happens are limited in the mass region smaller than about 120 GeV. The EW $S$ parameter in this region is slightly moved upwards but is still consistent with . However, as we shall see below, the $M_{D}<0$ region suffers from huge suppression relative to SM in the $h\to \gamma\gamma$ decay rate.
One loop corrections to the annihilation cross section contribute only to the $b_{V}$-parameter [[*i.e.,* ]{}]{}they are $p$-wave suppressed, if $m_{\chi} \lesssim (m_{Z} + m_{h})/2$. Our estimate, using the crude formula of below, shows that one loop induced $b_{V}$-terms are, numerically, about 10 times smaller than the tree level ones. However, if the above limit is not hold, then ($s$-wave) $a$-terms are coming into the final $\sigma_{Ann} v$. These terms could be of the same order as for the tree level $b$-terms and, in principle, for a precise $\Omega_{\chi} h^{2}$ prediction, they have to be included in the calculation.
We therefore conclude that, DM particle mass around the EW scale is possible and this requires large couplings of the heavy fermions to the Higgs boson [*i.e.,*]{} large $m=Y v$ with $Y\approx 1$. and secondarily, relatively low values of triplet mass [*i.e.,*]{} $M_{T} \simeq M_{D}$. This scenario can be hinted or completely excluded at the LHC because the couplings of the heavy new fermions (both neutral and charged) to the Higgs and gauge bosons are, in general, not suppressed in the symmetry limit \[see discussion in Section \[sec:LHC\]\].
Direct DM Detection {#sec:direct}
===================
Following the notation of Drees and Nojiri in , the Higgs boson mediated part of the effective Lagrangian for light quark $(u,d,s)$ - WIMP ([[*i.e.,* ]{}]{}the neutral fermion $\chi_{1}^{0}$) interaction is given by $$\mathcal{L}_{\mathrm{scalar}} = f^{(h)}_{q}\, \bar{\chi}_{1}^{0} \chi_{1}^{0}\, \bar{q} q \;.
\label{effxxqq}$$ Note that in this model there are no tensor contributions (at 1-loop level) since $\chi_{1}^{0}$ does not interact directly with coloured particles (as opposed to supersymmetric neutralino for example). The next step is to form the nucleonic matrix elements for the $\bar{q}q$ operator in and we write $${\langle n|}{m_{q} \bar{q} q}{|n\rangle} = m_{n} f_{Tq}^{(n)} \;,$$ where $m_{n}=0.94$ GeV, is the nucleon mass. The form factors $f_{Tq}^{(n)}$ are obtained within chiral perturbation theory and the experimental measurements of pion-nucleon interaction term, and they are subject to significant uncertainties. $f_{Tq}^{(n)}$ for $q=u,d$ [@Crivellin:2013ipa] are generically small by, say, a factor of $O(10)$ compared to $f_{Ts}=0.14$ obtained from value which we adopt into our numerical findings here. However, bear in mind that $f_{Ts}$ is subject to large theoretical errors [@Jungman:1995df; @Crivellin:2013ipa]. For instance, the average value quoted from lattice calculations [@Junnarkar:2013ac] is $0.043\pm 0.011$, which is smaller by a factor of three from the one obtained from chiral perturbation theory. This will result in, at least, a factor of $\mathcal{O}(10)$ reduction in the WIMP-nucleon cross section results, presented in Fig. \[fig:1loophiggs\], below.
The Higgs boson couples to quarks and then to gluons through the one-loop triangle diagram. Subsequently, the gluons ($G$) couple to the heavy quark current through the heavy quarks ($Q=c,b,t$) in loop. The analogous ($q\to Q$) matrix element in for $m_{Q} \bar{Q} Q$ can be replaced by the trace anomaly operator $-(\alpha_{s}/12\pi) G\cdot G$ to obtain $${\langle n|}{m_{Q} \bar{Q} Q}{|n\rangle} = \frac{2}{27} m_{n} \biggl [
1 - \sum_{q=u,d,s} f^{n}_{Tq} \biggr ] \equiv \frac{2}{27} m_{n} f_{TG} \;.$$ We are ready now to write down the effective couplings of $\chi_{1}^{0}$ to nucleons ($n=p,n$): $$\frac{f_{n}}{m_{n}} = \sum_{q} \frac{f_{q}^{(h)}}{m_{q}} f^{(n)}_{Tq} \ + \ \frac{2}{27}
\sum_{Q} \frac{f_{Q}^{(h)}}{m_{Q}} f_{TG} \;. \label{fnfacs}$$ Note that the bigger the $f_{Ts}$ is, the bigger the $f_{n}$ becomes. Also note that $f_{q}^{(h)} \propto m_{q}$. Furthermore, for $f_{Ts} \simeq 0.14$ the second term in , which is formally a two loop contribution to $f_{n}$, is about a factor of two smaller than the first one. Under the above assumption for the $f_{Ts}$ dominance we obtain $f_{p} = f_{n}$. In this case, the Spin Independent (SI) elastic scattering cross section at zero momentum transfer, of the WIMP $\chi_{1}^{0}$ scattering off a given target nucleus with mass $m_{N}$ in terms of the coupling $f_{p}$ is $$\sigma_{0 \mathrm{(scalar)}} = \frac{4}{\pi} \,
\frac{ m_{\chi_{1}^{0}}^{2} m_{N}^{4} }{(m_{\chi_{1}^{0}} + m_{N} )^{2} } \,
\left (\frac{f_{p}}{m_{n}} \right )^{2} \;.$$ The perturbative dynamics of the model is contained in the factor $f_{p}$ and therefore, from , in $f_{q}^{(h)}$ and $f_{Q}^{(h)}$. In this particular model the form factor $f_{q}^{(h)}$ reads, $$\frac{f_{q}^{(h)}}{m_{q}} \ = \ \frac{g\: [\Re e (Y^{h \chi^{0}_1 \chi^{0}_1 }) -
\delta Y^{h \chi^{0}_1 \chi^{0}_1 }]}{4 \, m_{W} \, m_{h}^{2}} \;.$$ The Higgs coupling to lightest neutral fermions is given in . In particular, under the custodial symmetry consideration we adopt here, it is obvious from , that $Y^{h \chi^{0}_1 \chi^{0}_1 }=0$, at tree level. Generic one-loop corrections will be proportional to, $g^{2} Y/4\pi \approx 0.03$, which can easily fall in the experimental exclusion region from current direct experimental DM searches for large $Y\sim 1$ coupling (see for instance eq.(3) in ). We therefore need to calculate the one loop corrections, $\delta Y^{h \chi^{0}_1 \chi^{0}_1 } \equiv \delta Y$ to the $h \chi^{0}_1 \chi^{0}_1$-vertex.
There is a fairly quick way to get an order of magnitude reliable calculation of $\delta Y$ through the Low Energy Higgs Theorem (LEHT) [@Ellis:1975ap; @Vainshtein:1980ea; @Kniehl:1995tn; @Pilaftsis:1997fe]. Application of LEHT in the region of our interest i.e., $m_{\chi_{1}^{0}} \approx m_{W} \approx m_{h} \ll m_{\chi_{i}^{\pm}}$ or $M_{D} \approx M_{T} \approx m_{W} \ll m$, and considering only Goldstone boson contributions to $\chi_{1}^{0}$ one-loop self energy diagrams, results in $$\delta Y \ =\ \frac{\partial}{\partial v}\delta M_{D}(v) \ \approx \ \frac{Y^{3}}{4 \pi^{2}} \,\frac{ M_{D} \, m}{M_{D}^{2} + 2 m^{2}} \;, \quad M_{D} \approx M_{T} \approx m_{W} \approx m_{h} \ll m \;.\label{eq:LET}$$ Let’s inspect . First, the middle term explains trivially why the Higgs coupling is zero at tree level: the lightest eigenvalue of the neutral mass matrix is $M_{D}$ which is independent on any vacuum expectation value (v.e.v.). Then because at one loop, the $\chi_{1}^{0}$ self-energies involve only the heavy fermion masses (both charged and neutral) which depend on the v.e.v through $m=Y v$ or through $m_{W}, m_{Z}$ in the propagators of $\chi^{\pm}_{i}, \chi^{0}_{i=2,3}$ and $W,Z$ respectively, the one loop correction $\delta Y$ *does not* in general vanish. Second, the third term of the equality shows that the effect increases by the third power of the Yukawa coupling $Y$ \[recall \] and vanishes when $M_{D} \rightarrow 0$ \[the $U(1)_{X}$ symmetry limit\]. As for the numerical approximation, is always less than 20% of the exact calculation (see below) even though we have completely neglected the non-Goldstone diagrams that are proportional to gauge couplings. It is however a crude approximation which is only relevant when the new heavy fermions are far heavier than the $Z,W,h$-bosons as well as from the lightest neutral fermion.
![*Feynman diagrams (in unitary gauge) related to spin independent (SI) elastic cross section $\chi_{1}^{0} + q \rightarrow \chi_{1}^{0} + q$ where $q=u,d,s$ – light quarks. Particle $V$ represents $W$ or $Z$ and $\chi$ represents $\chi_{i=1..2}^{\pm}$ or $\chi_{i=1..3}^{0}$, respectively. One loop self energy corrections are absent in the particular scenario we have chosen.* []{data-label="fig:1loophiggs"}](FeynGraphs-Direct-1.png){width="5in"}
In \[sec:appA\], we calculate the exact one-loop amplitude for the vertex $h-\chi_{1}^{0}-\chi_{1}^{0}$ with physical external $\chi_{1}^{0}$ particles at a zero Higgs-boson momentum transfer. A similar calculation has been carried out in for the MSSM and in for minimal DM models. However, due to peculiarities of this model that have been stressed out in the introduction with respect to the aforementioned models, a general calculation is needed. The one-loop corrected vertex amplitude arises from (a) and (b) diagrams[^14] depicted in Fig. \[fig:1loophiggs\] involving vector bosons ($W$ or $Z$) and new charged ($\chi_{i=1,2}^{\pm}$) or neutral ($\chi_{i=1..3}^{0}$) fermions, as $$i \,\delta Y = \sum_{j=(a),(b)} (i \,\delta Y_{j}^{\chi^{\pm}} + i\, \delta Y_{j}^{\chi^{0}})\;.$$ Detailed forms, not resorting to $CP$-conservation, for $\delta Y$’s are given in \[sec:appA\]. We have proven both analytically and numerically that when the external particles $\chi_{1}^{0}$ are on-shell, infinities cancel in the sum of the two vertex diagrams in Fig. \[fig:1loophiggs\]a,b without the need of any renormalization prescription, and the resulting amplitude - $i \,\delta Y$ - is finite and renormalization scale invariant.
We have also carried out the one-loop calculation of the box diagrams in Fig. \[fig:1loophiggs\]c. The effective operators for box diagrams consist of scalar, $f_{q}^{\mathrm{(box)}}$ \[like the $f_{q}$ in \] and twist operators, $g_{q}^{(1)}$ and $g_{q}^{(2)}$ written explicitly for example in . In the parameter space of our interest where $M_{D} \ll m$, the $f_{q}^{\mathrm{(box)}}$ contributions to $f_{q}^{(h)}$ in , are in general two orders of magnitude smaller than the vertex ones arising from Fig. \[fig:1loophiggs\](a,b), and they are only important in the case where the latter cancel out among each other. Moreover, it has recently been shown in that, the full two-loop gluonic contributions are relevant for a correct order of magnitude estimate of the cross section in the heavy WIMP mass limit, especially when adopting the “lattice” value for $f_{T_{s}}$. We are not aware, however, of any study dealing with those corrections and WIMP mass around the electroweak scale which is the case of our interest. Such a calculation is quite involved and is beyond the scope of the present article.
In Fig. \[fig:directX\] we present our numerical results for the SI nucleon-WIMP cross section. The current LUX [@LUX] (XENON100 [@XENON100]) experimental bounds for a 100 GeV WIMP mass is $\sigma_{0}^{(SI)} \lesssim 1 (2)\times 10^{-45}~\mathrm{cm}^{2}$ at 90% C.L.
From the left panel of Fig. \[fig:directX\] we observe that in the region where $M_{T}\ll M_{D} \ll m$ the cross section is by one to two orders of magnitude smaller than the current experimental bound. More specifically, in the region where we obtain the right relic density \[see Fig. \[figos\]a\] the prediction for the $\sigma_{0}^{\mathrm(SI)}$ is about to be observed only for large values of $M_{T}$ ($M_{T}\approx 500$ GeV), while it is by an order of magnitude smaller for low values of $M_{T}$ ($M_{T} \lesssim 100$ GeV). There is a region, around $M_{T} \approx 25$ GeV, where box corrections, that arise from the diagram in Fig. \[fig:1loophiggs\]c, on scalar and twist-2 operators become important because the vertex corrections mutually cancel out. However, in this region the cross-section becomes two to four orders of magnitude smaller than the current experimental sensitivity. We also remark that $\sigma_{0}^{\mathrm(SI)}$ reaches a maximum value, indicated by the closed contour line in the upper left corner of Fig. \[fig:directX\]a, and then starts decreasing for larger $M_{T}$ and $M_{D}$ values, a situation that looks like following the Appelquist-Carazzone decoupling theorem [@Appelquist:1974tg]. However, even at very large masses, $M_{D}$ and $M_{T}$, not shown in Fig. \[fig:directX\], there is a constant piece of $\delta Y$, and hence of $\sigma_{0}^{\mathrm(SI)}$, that does not decouple. This can be traced respectively in the second and the first terms of integrals $I_{4}^{V}$, and $I_{5}^{V}$ of , in the limit $M_{D}=M_{T}\to \infty$. This non-decoupling can also been seen in the heavy particle, effective field theory analysis of and also in . We have also checked numerically that $\sigma_{0}^{\mathrm(SI)}$ vanishes at $M_{D}\to 0$ as expected from and from the $U(1)_{X}$-symmetry.[^15]
In Fig. \[fig:directX\]b, we also plot predictions for the doublet-triplet fermionic model on SI cross section $\sigma_{0}^{\mathrm(SI)}$ on $M_{D}$ vs. $m$ plane for $M_{T}=M_{D}$. As we recall from , the cross section increases with $m$ (or $Y$) as $m^{2}\propto Y^{2}$. It becomes within current experimental sensitivity reach for $m \gtrsim 400$ GeV while for low $m\approx 100$ GeV, $\sigma_{0}^{\mathrm(SI)}$ is about 100 times smaller. Besides, for heavy $M_{D}$ and $m$ (upper right corner), $\sigma_{0}^{\mathrm(SI)}$ becomes excluded by current searches although vacuum stability bounds hit first. If we compare with the corresponding plot for the relic density in Fig. \[figos\]b, we see that the observed $\Omega_{\chi} h^{2}$ is allowed by current experimental searches on $\sigma_{0}^{\mathrm(SI)}$ but it will certainly be under scrutiny in the forthcoming experiments [@Panci:2014gga].
Finally, for negative values of $M_{D}$ consistent with the observed density depicted in Fig. \[figosn\], it turns out that $\sigma_{0}^{\mathrm(SI)}$ is by a factor of about $\sim 10$ bigger than the corresponding parameter space for $M_{D}>0$ given in Fig. \[fig:directX\]a. In fact, the region of 1-loop cancellations happened for $M_{T} \approx 20$ GeV, do not take place for $M_{D}<0$. However, within errors discussed at the beginning of this section, this is still consistent with current experimental bounds.
Higgs boson decays to two photons {#sec:h2gg}
=================================
In the doublet-triplet fermionic model there are two pairs of electromagnetically charged fermions and antifermions, namely, $\chi_{1}^{\pm},\chi_{2}^{\pm}$. They have electromagnetic interactions with charge $Q=\pm1$ and interactions with the Higgs boson, $Y^{h\chi^{-}\chi^{+}}$, given in general by , or in particular, in the symmetry limit, by . These latter interactions are of similar size as of the top-quark-antiquark pairs with the Higgs boson [[*i.e.,* ]{}]{}$Y\sim 1$. Hence, we expect a substantial modification of the decay rate, $\Gamma(h \to \gamma\gamma)$ relative to the SM one[^16] $\Gamma(h \to \gamma\gamma)_{SM}$, through the famous triangle graph [@Ellis:1975ap],
involving $W$-gauge bosons, the top-quark ($t$) and the new fermions $\chi_{i}^{\pm}$. Under the assumption of real $M_{D}$, $Y^{h\chi_{i}^{-}\chi_{i}^{+}}$ is also real, and we obtain: $$R\equiv \frac{\Gamma(h\to \gamma\gamma)}{\Gamma(h\to \gamma\gamma)_{\mathrm{(SM)}}} \ = \
\: \biggl |\: 1 \ + \
\frac{1}{A_{\mathrm{SM}}} \: \sum_{i= \chi_{1}^{\pm}, \chi_{2}^{\pm}} \,
\sqrt{2} \: \frac{Y^{h\chi_{i}^{-}\chi_{i}^{+}}\, v}{m_{\chi_{i}^{+}}} \: A_{1/2}(\tau_{i})\: \biggr |^{2}\;,
\label{hi2g}$$ where $A_{\mathrm{SM}} \simeq -6.5$ for $m_{h}=125$ GeV, is the SM result dominated by the $W$-loop [@Dedes:2012hf], with $\tau_{i} = m_{h}^{2}/4 m_{i}^{2}$ and $A_{1/2}$ is the well known function given for example in [^17]. The $\chi_{i}^{\pm}$-fermion contribution ($Q=1,N_{c}=1$), is also positive because the ratio, ${Y^{h\chi_{i}^{-}\chi_{i}^{+}}}/{m_{\chi_{i}^{+}}}$, is always positive when $m_{\chi_{1}^{0}} = M_{D}$, as can be seen by inspecting . After using the simplified (by symmetry) with $a \approx -\sqrt{2}$, we approximately obtain $$\sum_{i} \frac{\sqrt{2}\, m}{m_{\chi_{i}^{+}}} \: A_{1/2}(\tau_{\chi_{i}^{+}}) \approx + \frac{8}{3}\;,$$ which means that $\Gamma(h\to \gamma\gamma)$ is smaller than the SM expectation. But how much smaller? In Fig. \[fig:h2gg\] we plot contours of the ratio $R\equiv \Gamma(h\to \gamma\gamma)/\Gamma(h\to \gamma\gamma)_{(\mathrm{SM})}$ on $(M_{D}$ vs. $M_{T})$-plane for $m=200$ GeV (Fig \[fig:h2gg\]a) and $M_{D}$ vs. $m$-plane for $(M_{T}=M_{D})$ (Fig \[fig:h2gg\]b). Our numerical results plotted in Fig. \[fig:h2gg\] are exact at one-loop. We observe that the new charged fermions render the ratio less than unity $$R \lesssim 1 \;,$$ everywhere in the parameter space considered. Let’s look at this in a more detail. The contribution of fermions $\chi_{i}^{\pm}$ in , depends on the quantity[^18] $$\begin{aligned}
\sim \:\frac{2\, m^{2}}{2\, m^{2} + M_{D}\, M_{T}}\;,
\label{eq:supap}\end{aligned}$$ which is always positive for $M_{D}, M_{T}>0$ [[*i.e.,* ]{}]{}it adds to the top-quark contribution and subtracts from the large and negative $W$-boson one resulting in a suppressed $R$-ratio. If instead we choose $M_{D}<0$, then for $|M_{D} M_{T}| > \sqrt{2} |m|$, one can obtain $R\gtrsim 1$, a situation which is explored in . As can be seen from Fig. \[fig:comp\] however, in this case the DM candidate particle $(\chi_{1}^{0})$ is not a pure doublet. It is instead a mixed state. (In fact the states ${|1\rangle}$ and ${|2\rangle}$ are interchanged in ). As a consequence, there is a non-zero (and generically large) $h\chi_{1}^{0}\chi_{1}^{0}$-coupling already present at tree level, and, bear in mind fine tuning, it is excluded by direct DM search bounds. By comparing areas with the observed relic density in Figs.\[figos\](a,b) we see that, the results for $0.35\lesssim R \lesssim 0.5$ shown in Figs. \[fig:h2gg\](a,b) are within $1\sigma$-error compatible with current central values of CMS measurements [@Chatrchyan:2012ufa] $(0.78 \pm 0.27)$ but are highly “disfavoured” by those from ATLAS [@Aad:2012tfa] ones, $1.65 \pm 0.24(stat)^{+0.25}_{-0.18}(syst)$. The forthcoming second LHC run will be decidable in favour or against this outcomes here.
Fig. \[fig:h2gg\](a) or , shows also that when $M_{T}$ becomes heavy the ratio $R$ approaches the current CMS central value. This happens because one of the two charged fermion eigenvalues becomes very heavy, $m_{\chi_{2}^{+}} \approx M_{T}$, and therefore it is decoupled from the ratio. As we discussed in section \[sec:relic\], large $M_{T}\sim 1$ TeV values, may be consistent with the observed $\Omega_{\chi} h^{2}$ for $m_{W} < M_{D} < m_{Z}$. We have found that even in this case, $R$ is always smaller than $0.65$.
If we assume that $M_{D}<0$ and $\chi_{1}^{0}$ pure doublet as shown in Fig. \[fig:comp\], then it is always $R<1$. In fact, using the input values from Fig. \[figosn\] for the correct relic density, the suppression of $R$ is even higher, $0.25 \lesssim R \lesssim 0.35$. Alternatively, if we assume that $M_{D}$ is a general complex parameter, then the coupling, $Y^{h\chi^{-}_{1}\chi^{+}_{1}}$, is complex too. In this case one has to add the CP-odd Higgs contribution into which is always positive definite. For large phases relatively large $M_{T}$ the ratio $R$ may be greater than one, however, again the direct detection bounds are violated by a factor of more than 10-1000.
Of course, if we increase $M_{D}$, the parameter space may be compatible with the observed relic density seen in the right side of “heavy” $M_{D}$-branch in Fig. \[fig:xsec\]. However, following our motivation for “*only* EW scale DM” we do not discuss this region further which is anyhow very well known from MSSM studies. We therefore conclude that in the doublet triplet fermionic model *thermal DM relic abundance for low DM particle mass $m_{\chi_{1}^{0}} \approx M_{Z}$, consistent with observation [@Ade:2013zuv] and with direct DM searches [@LUX; @XENON100] leads to a substantial suppression (45-75%) for the rate $\Gamma(h\to \gamma\gamma)$ relative to the SM expectation.*
We have also calculated the ratio $R$ for the Higgs boson decay into $Z\gamma$. The results are similar to the case of $R(h \to \gamma\gamma)$. In particular, in the parameter space explored in Fig. \[fig:h2gg\](a), we observe exactly the same shape of lines with a ratio slightly shifted upwards in the region, $0.4 \lesssim R(h\to Z\gamma) \lesssim 0.7$. This suppression is due to the same reason discussed in the paragraph below .
Vacuum Stability {#sec:Landau}
================
The stability of the Standard Model vacuum is an important issue, so we need to find an energy scale ($\Lambda_{UV}$) where new physics is needed, in order to make the vacuum stable or a metastable (unstable with lifetime larger than the age of the universe). To make an estimate about the $\Lambda_{UV}$ of the theory, one needs to calculate the tunnelling rate between the false and the true vacuum and impose that the SM vacuum has survived until today[^19].
![*The vacuum stability plot: $\Lambda_{UV}$ against $m=Y \upsilon$.*[]{data-label="LambdaUV"}](luv){width="0.55\linewidth"}
Following , we can see that the bound for the Higgs self coupling, $\lambda$, becomes[^20]: $$\label{luv}
\lambda(\Lambda_{UV}) = \frac{4 \pi ^2}{ 3 \; \ln{\left( \dfrac{H}{\Lambda_{UV}}\right) }},$$ where $\Lambda_{UV}$ is the cut off scale and $H$ is the Hubble constant $H=1.5 \times 10^{-42}$ GeV. In order to impose the contstraint (\[luv\]), we also need to find the running parameter $\lambda$ by solving the renormalization group equations. The one-loop beta functions for the model at hand are given in [^21], and we solve this set of differential equations using as initial input parameters: $$\begin{aligned}
\label{in_con}
\alpha_3( M_Z ) &= 0.1184 \;, \quad
\alpha_2 ( M_Z ) = 0.0337\;, \quad
\alpha_1 ( M_Z ) = 0.0168 \;, \\
\lambda ( M_Z ) & = 0.1303\;, \quad
y_t ( M_Z ) = 0.9948 \;, \quad
M_Z = 91.1876 \;\; \mathrm{GeV}.\end{aligned}$$ The result for the cut off scale as a function of $m=Y\upsilon$ is given in Fig. \[LambdaUV\]. As we can see, $\Lambda_{UV} \approx 600$ GeV for $m \approx 200$ GeV which is quite small while $\Lambda \approx 20$ TeV for $m\approx 130$ GeV. The result for $\Lambda_{UV}$ in Fig. \[LambdaUV\] is only approximate. Threshold effects, from the physical masses of the doublet, triplet and even the top-quark, together with comparable two-loop corrections to $\beta$-functions, which can be found for example in , are missing in Fig. \[LambdaUV\]. These effects may change the outcome for $\Lambda_{UV}$ by a factor of two or so but they will not change the conclusion, that extra new physics is required already nearby the TeV-scale. The form of new physics will probably be in terms of new scalar fields since extra new fermions will make $\Lambda_{UV}$ even smaller. These scalars may be well within reach at the second run of the LHC [@ArkaniHamed:2012kq] but it is our assumption here that they do not intervene with the DM sector. As far as the (1-loop) perturbativity of the Yukawa couplings $Y\sim 1.2$ (for $m=200$) and $Y_{t}$, is concerned, these exceed the value $4\pi$ at around the respective scales, $10^{9}$ and $10^{10}$ GeV. Given the modifications of the model that must be performed at $\Lambda_{UV} \sim \mathrm{TeV}$ scale, the perturbativity bound is of secondary importance here.
Heavy fermion production and decays {#sec:LHC}
===================================
The unknown new fermions that have been introduced into this model to accompany the DM mechanism can be searched for at the LHC in a similar fashion as for charginos and neutralinos of the MSSM. Multilepton final states associated with missing energy may arise in three different ways from the decays of new fermion pairs: $\chi^{+}_{i}\chi^{-}_{j}$, $\chi^{\pm}_{i} \chi^{0}_{j}$, and $\chi_{i}^{0} \chi_{j}^{0}$.
Production
----------
![*Contours of the production cross section for the new fermions, $\sigma(pp\to \chi_{1}^{\pm} \chi_{2}^{0})$ \[in $pb$\], on $M_{D}$ vs. $M_{T}$ plane, at LHC with $\sqrt{s} = 8$ TeV.*[]{data-label="fig:ppxx"}](xsection-ppxx.pdf){width="80mm"}
A recent study at LHC [@ATLAS:2013rla; @CMS:2013dea] has presented upper limits in the signal production cross sections for charginos and neutralinos, in the process $$p \ + \ p \ \rightarrow \ W^{*} \ \rightarrow \ \chi^{+}_{1} \ + \ \chi_{2}^{0} \;,$$ which is mediated by the $W$-gauge boson. One can use Fig. 9b from to set limits to the cross section and therefore to constrain the parameter space. This figure fits perfectly into our study since it assumes a) 100% branching ratio for the $\chi^{+}_{1}$ and $\chi_{2}^{0}$ decays as it is the case here \[see section \[dec\] below\] b) degenerate masses for $\chi^{+}_{1}$ and $\chi_{2}^{0}$ as it is exactly the case here as shown in . The production cross section has been calculated in also including next to leading order QCD corrections. The parton-level, tree level, result is $$\frac{d \hat{\sigma}}{d\hat{t}}(u + d^{\dagger} \rightarrow W^{*}
\rightarrow \chi^{+}_{i} + \chi_{j}^{0} ) \ = \ \frac{1}{16 \pi \hat{s}^{2}} \, \left ( \frac{1}{3\cdot 4}
\sum_{\rm spins} |\mathcal{M}|^{2} \right ) \;, \label{xsec}$$ where the factors $1/3$ and $1/4$ arise from colour and spin average of initial states, $\hat{s},\hat{t},\hat{u}$ are the Mandelstam variables at the parton level, and $$\begin{aligned}
\sum_{\rm spins} |\mathcal{M}|^{2} = |c_{1}|^{2} (\hat{u} - m_{\chi_{i}^{+}}^{2})
(\hat{u} - m_{\chi_{j}^{0}}^{2}) + |c_{2}|^{2} (\hat{t} - m_{\chi_{i}^{+}}^{2})
(\hat{t} - m_{\chi_{j}^{0}}^{2})
+ 2 \Re e [c_{1} c_{2}^{*} ] m_{\chi_{i}^{+}} m_{\chi_{j}^{0}} \, \hat{s} \;,
\label{cs}
\end{aligned}$$ with the coefficients $c_{i}$ being $$c_{1} = -\frac{\sqrt{2} \, g^{2} }{\hat{s}-m_{W}^{2}} \, O^{L\,*}_{ji}\;, \qquad
c_{2} = - \frac{\sqrt{2} \, g^{2} }{\hat{s}-m_{W}^{2}} \, O^{R\,*}_{ji} \;.
\nonumber$$ We let the indices $i=1,2$ and $j=1,2,3$ free as there is a situation of a complete mass degeneracy between the heavy neutral and charged fermions when $M_{D} = M_{T}$. Our result in are in agreement with .
By convoluting with the proton’s pdfs and integrating over phase space we obtain in Fig. \[fig:ppxx\], the production cross section for $\sigma(pp\to \chi_{1}^{\pm} \chi_{2}^{0})$ \[in $pb$\]. In the region with correct DM relic density, we obtain typical values varying in the interval $(0.07 - 0.2) pb$ for $\sqrt{s}=8$ TeV. This is about 1400-4000 events at LHC before any experimental cuts assuming $20 fb^{-1}$ of accumulated luminosity. This is within current sensitivity search and analysis has been performed by ATLAS [@ATLAS:2013rla] and CMS [@CMS:2013dea] for simplified supersymmetric models. Looking for example in Fig. 9b in ATLAS [@ATLAS:2013rla], for the same parameter space as in our Fig. \[fig:ppxx\], the observed upper limit on the signal cross section varies in the interval (0.14-1.2) $pb$. In the region where $M_{D}=M_{T}$, all heavy fermions are mass degenerate. In this case the total cross section is the sum of all possible production modes $\chi_{1,2}^{\pm} \chi_{2,3}^{0}$, and the total cross section is about 0.15 $pb$ which is on the spot of current LHC sensitivity ($0.14$ pb) [@ATLAS:2013rla].
Decays {#dec}
------
Just by looking at a typical spectrum of the model in Fig. \[fig:spec\], we see that the heavy fermions can decay on-shell to two final states with a gauge boson and the lightest neutral stable particle. Therefore, the lightest charged and the next to lightest neutral fermions decay like
$$\begin{aligned}
\chi^{\pm}_{1} \ &\rightarrow \ \chi^{0}_{1}\ + \ W^{\pm} \;, \label{rw}\\
\chi^{0}_{2} \ &\rightarrow \ \chi^{0}_{1} \ + \ Z \label{rz}\;.\end{aligned}$$
In our case where $\chi_{1}^{0}$ is a “well tempered doublet” there are no-off diagonal couplings to the Higgs boson, like for example $h\chi_{1}^{0}\chi_{2}^{0}$. Therefore, particles $\chi_{1}^{\pm}$ and $\chi_{2}^{0}$ decay purely to final states following (\[rw\]) and (\[rz\]) with 100% branching fractions. The signature at hadron colliders is the well know from SUSY searches, trileptons plus missing energy.
Analytically we find the decay widths [@Gunion:1987yh; @Dreiner:2008tw]: $$\begin{aligned}
\Gamma(\chi^{+}_{i} \to \chi^{0}_{j} + W^{+}) &= \frac{g^{2} \, m_{\chi^{+}_{i}}}{32 \pi}\,
\lambda^{1/2}(1,r_{W}, r_{j}) \, \biggl \{ (|O^{L}_{ji}|^{2} + |O^{R}_{ji}|^{2})\,
\biggl [1 + r_{j} - 2 r_{W} + (1-r_{j})^{2}/r_{W} \biggr ]
\nonumber \\[2mm] &- 12\, \sqrt{r_{j}} \, \Re e(O^{L\,*}_{ji} O^{R}_{ji}) \biggr \}\;,
\nonumber \\[2mm]
\Gamma(\chi^{0}_{i} \to \chi^{0}_{j} + Z) &= \frac{g^{2} \, m_{\chi^{0}_{i}}}{16 \pi c_{W}^{2}}\,
\lambda^{1/2}(1,r_{Z}, r_{j}) \, \biggl \{ |O^{\prime\prime \, L}_{ij}|^{2}\,
\biggl [1 + r_{j}^{0} - 2 r_{Z} + (1-r_{j}^{0})^{2}/r_{Z} \biggr ]
\nonumber \\[2mm] &+ 6\, \sqrt{r_{j}^{0}} \, \Re e[(O^{\prime\prime\, L}_{ij})^{2}] \biggr \}\;,\end{aligned}$$ where
$$\begin{aligned}
r_{W} \equiv m_{W}^{2}/m_{\chi_{i}^{+}}^{2}\;, & \quad r_{Z} \equiv m_{Z}^{2}/m_{\chi_{i}^{0}}^{2}\;,
\quad r_{j} \equiv m_{\chi_{j}^{0}}^{2}/m_{\chi_{i}^{+}}^{2} \;,
\quad r_{j}^{0} \equiv m_{\chi_{j}^{0}}^{2}/m_{\chi_{i}^{0}}^{2} \\[1mm]
& \lambda(x,y,z) \equiv x^{2} + y^{2} + z^{2} -2 x y - 2 x z - 2 y z \;.\end{aligned}$$
Numerical results for the decay widths for the processes (\[rw\]) and (\[rz\]) in the area of interest are depicted in Fig. \[fig:chidecays\](a) and (b), respectively.
Both decay widths behave similarly. In the area $M_{D} \approx M_{T} \approx 100$ GeV we observe maximum values $\Gamma \approx 3$ GeV. As $M_{T}$ is increases or decreases, the widths get smaller than $1$ GeV. This is easily understood if we look back at the mass difference $|m_{\chi_{2}^{0}}|-|m_{\chi_{1}^{0}}|$ in Fig. \[fig:masses\](a) and recall that for the parameter considered in Fig. \[fig:chidecays\], it is $m_{\chi_{1}^{0}} = M_{D}$ and $m_{\chi_{2}^{0}} = m_{\chi_{1}^{\pm}}$.
For heavier charged fermions, new decay channels include
$$\begin{aligned}
\chi_{2}^{+} &\rightarrow \chi_{1}^{+} + Z\;, \\
\chi_{2}^{+} &\rightarrow \chi_{1}^{+} + h \;,\end{aligned}$$
that are mostly kinematically allowed in the low $M_{D} \approx 100$ GeV but high $M_{T} \gtrsim 220$ GeV regime. For the heavier neutral particles, if kinematically allowed they would decay to $W,Z$-gauge bosons and/or the Higgs boson,
$$\begin{aligned}
\chi_{3}^{0} &\rightarrow \chi^{\pm}_{1} + W^{\mp} \;, \\
\chi_{3}^{0} &\rightarrow \chi_{2}^{0} + Z \;, \\
\chi_{3}^{0} &\rightarrow \chi_{2}^{0} + h \;.\end{aligned}$$
Conclusions and Future Directions {#sec:conclusions}
=================================
Our motivation for writing this paper is to import a simple DM sector in the SM with particles in the vicinity of the electroweak scale responsible for the observed DM relic abundance, preferably not relying on co-annihilations or resonant effects, and capable of escaping current detection from nucleon-recoil experiments. Meanwhile, we study consequences of this model in EW observables and Higgs boson decays ($h\to \gamma\gamma, Z\gamma$) and other possible signatures at LHC. This SM extension consists of two fermionic $SU(2)_{W}$-doublets with opposite hypercharges and a fermionic $SU(2)_{W}$-triplet with zero hypercharge. The new interaction Lagrangian is given in , and contains both Yukawa trilinear terms together with explicit mass terms for the doublets and triplet fields. Under the assumption of a certain global $SU(2)_{R}$-symmetry, discussed in section \[sec:symmetry\], that rotates $H$ to $H^{\dagger}$ and $\bar{D}_{1}$ to $\bar{D}_{2}$, the two Yukawa couplings become equal with certain consequences that capture our interest throughout this work. After electroweak symmetry breaking this sector widens the SM with two charged Dirac fermions and three neutral Majorana fermions, the lightest ($\chi_{1}^{0}$) of which plays the role of the DM particle. Under the symmetry assumption and for Yukawa couplings comparable to top-quark, the lightest neutral particle ($\chi_{1}^{0}$) may have mass equal to the vector-like mass of the doublets, $M_{D}$, and its field composition contains only an equal amount of the two doublets \[see Fig. \[fig:comp\]\]. As a result, the couplings of the Higgs and the $Z$ bosons to the lightest neutral fermion pair vanish at tree level.
Within this framework we observe in Fig. \[figos\], that $\Omega_{\chi} h^{2}$, is in accordance with observation \[\] provided that the parameters of the model, $M_{D}, M_{T}$ and $m$, lie naturally at the EW scale [[*i.e.,* ]{}]{}without the need for resonant or co-annihilation effects. Moreover, the $\chi_{1}^{0}$-nucleon SI cross section appears at one-loop, turns out to be around 1-100 times smaller than the current experimental sensitivity from LUX and XENON1T as it is shown in Fig. \[fig:directX\]. In addition, we find that the oblique electroweak parameters $S,T$ and $U$ are all compatible with EW data fits as it is shown in Fig. \[fig:STU\], a result which is partly a consequence of the global symmetry exploited.
We also look for direct implications at the LHC. We find that the existence of the extra charged fermions reduces substantially the ratios of the Higgs decay to di-photon (see Fig. \[fig:h2gg\]) and to $Z\gamma$ w.r.t the SM. This is a certain prediction of this scenario that cannot be avoided by changing the parameter space. For very large Yukawa coupling, this reduction maybe of up to 65% relative to the SM expectation as we obtain from Fig. \[fig:h2gg\]. Furthermore, the production and decays of those new charged/neutral fermion states, is within current and forthcoming LHC reach. Decay rates for some of these states are shown in Fig. \[fig:chidecays\].
We should notice here that the minimality of the Higgs sector together with the $Z_{2}$-parity symmetry preserves the appearance of new flavour changing or CP-violating effects beyond those of the SM, for up to two-loop order (for a nice discussion of effects on EDMs from the charged fermions, see ).
On top of collider/astrophysical constraints, we made an estimate of the consequences of the new states to vacuum stability of the model. The 1-loop result for the UV cutt off scale, above which the model needs some completion, is given in Fig. \[LambdaUV\]. We see that for the parameter space of interest, new physics, probably in the form of new, supersymmetric, scalars is needed already nearby the TeV or multi-TeV scale to cancel fermionic contributions in the quartic Higgs coupling. For example, this solution may take the form of an MSSM extension with $\bar{D}_{1,2}$ and $T$ superfield (extensions with a triplet superfield have been explored in ).
In summary, in this work we basically studied the synergy between three observables: $\Omega_{\chi} h^{2}, \sigma^{SI}_{0}$, and $R(h\to \gamma\gamma)$, in a simple fermionic DM model. If charged fermion states are discovered at the second run of LHC and are compatible with $\Omega_{\chi} h^{2}$ with $m_{\chi}\sim m_{Z}$, then $R(h\to \gamma\gamma)$ has to be suppressed [[*i.e.,* ]{}]{}$R$ will turn towards the CMS central value. If instead $R(h\to \gamma\gamma)\gtrsim 1$ is enhanced, then the DM particle is heavy, $m_{\chi}\sim 1$ TeV, or otherwise excluded by direct DM detection bounds. If $R\sim 1$, then one has to go to large $M_{T}$ values where, however, $\Omega_{\chi} h^{2}$ is only barely compatible with $m_{\chi} \simeq m_{Z}$. In this latter case, the mass of the DM particle may be below the EW gauge boson masses. However, in this case an entire new analysis is required.
Apart from studying the regime with mass $m_{\chi}$ lower than $M_{Z}$, this work can be extended in several ways as for example, to investigate the role of $CP$-violating phases of $M_{D}$ on baryogenesis. Indirect DM searches could be also an interesting avenue together with extensions of the Higgs sector. We postpone all these interesting phenomena for future study.
Acknowledgements {#acknowledgements .unnumbered}
================
[*We are grateful to Susanne Westhoff for helping us fiinding a mistake in our formulae in the Appendix A. Our results for nucleon-WIMP cross section, depicted in Fig. 10, are now in good agreement with .*]{} A. D. would like to thank, M. Drees for useful comments, A. Barucha for discussions on the ongoing LHC chargino searches, C. Wagner for drawing our attention to , and F. Goertz for discussions on custodians. We are grateful to J. Rosiek for letting us using his code for squaring matrix elements and to compare with our analytical results. Our numerical routines for matrix diagonalization and for loop functions follow closely those in , respectively.
This research Project is co-financed by the European Union - European Social Fund (ESF) and National Sources, in the framework of the program “THALIS" of the “Operational Program Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) 2007-2013. D.K. acknowledges full financial support from the research program “THALIS".
{#sec:appA}
The 1-loop corrected vertex amplitude arises from (a) and (b) diagrams depicted in Fig. \[fig:1loophiggs\] involving vector bosons ($W$ or $Z$) and new charged ($\chi_{i=1,2}^{\pm}$) or neutral ($\chi_{i=1..3}^{0}$) fermions. It can be written as, $$i \,\delta Y = \sum_{j=(a),(b)} (i \,\delta Y_{j}^{\chi^{\pm}} + i\, \delta Y_{j}^{\chi^{0}})\;,$$ where
$$\begin{aligned}
i\, \delta Y_{(a)}^{\chi^{\pm}} &=&
- {g^2} \sum_{i,j=1}^{2} \biggr \{ \left( O_{1j} ^R \, O_{1i} ^{L*}\,
Y^{h \chi^{-}_j \chi^{+}_i } \ + \ O_{1i}^R \, O_{1j}^{L*}\, Y^{h \chi^{-}_i \chi^{+}_j} \right)\,
I^{Wij} _1 \label{M1xc} \nonumber \\[1mm]
& + & m_{\chi_i ^{+}}\: m_{\chi_j ^{+}}\, \left( O_{1j}^R\, O_{1i}^{L*}\,
Y^{h \chi^{-}_i \chi^{+}_j *} \ + \
O_{1i}^R\, O_{1j}^{L*}\, Y^{h \chi^{-}_j \chi^{+}_i * } \right)\, I^{Wij} _2 \\[1mm]
&+& \left[ O_{1j}^L\, O_{1i}^{L*} \left( m_{\chi_i ^{+}}\, Y^{h \chi^{-}_i \chi^{+}_j *} +
m_{\chi_j ^{+}}\,Y^{h \chi^{-}_j \chi^{+}_i } \right ) \right. \nonumber \\[1mm]
& + & \left. O_{1i}^R \, O_{1j}^{R*} \,
\left( m_{\chi_i^{+}} Y^{h \chi^{-}_j \chi^{+}_i *}
+ m_{\chi_j^{+}}Y^{h \chi^{-}_i \chi^{+}_j } \right) \right ]\, I^{Wij} _3 \biggr \},
\nonumber \\[3mm]
i\, \delta Y_{(a)}^{\chi^{0}}
& = & \dfrac{g^2}{c_W ^2 }\, \sum_{i,j=1}^{3} \biggr \{
O_{j1}^{\prime \prime L}\, O_{i1}^{\prime \prime L } \,Y^{h \chi^{0}_i \chi^{0}_j }
\, I^{Zij} _1
+ m_{\chi_i ^{0}}\, m_{\chi_j ^{0}}\, \ O_{i1}^{\prime \prime L}\, O_{j1}^{\prime \prime L}
\, Y^{h \chi^{0}_i \chi^{0}_j *} \, I^{Zij} _2
\nonumber \\[1mm]
&-& O_{1j}^{\prime \prime L}\, O_{i1}^{\prime \prime L} \left( m_{\chi_i ^{0}}
Y^{h \chi^{0}_i \chi^{0}_j *} + m_{\chi_j^{0}} Y^{h \chi^{0}_i \chi^{0}_j } \right)
\, I^{Zij} _3 \biggr \},
\label{M1x0} \\[3mm]
i\, \delta Y_{(b)}^{\chi^{\pm}} &=&
- \dfrac{\sqrt{2}\, g^2 m_W ^2 }{v} \sum_{i=1}^{2}
\biggr[ \left( |O_{1i}^L|^{2} \ + \ |O_{1i}^R|^{2} \right)\, I_4 ^{Wi}
\ + \ 2 \, m_{\chi_i ^{+}} \, O_{1i}^{L*}\, O_{1i}^{R}
\, I_5 ^{Wi} \biggr], \label{M2xc}
\\[3mm]
i\, \delta Y_{(b)} ^{\chi^{0}} &=& - \dfrac{\sqrt{2} g^2 m_Z^2 }{c_W^2 \,v }
\sum_{i=1}^{3} \biggr \{ O_{i1}^{\prime \prime L} \, O_{1i}^{\prime \prime L} \, I_4^{Zi} \ - \
m_{\chi_i ^{0}} \, (O_{i1}^{\prime \prime L})^{2} \, I_5 ^{Zi} \biggr \} \;, \label{M2x0}\end{aligned}$$
where the integrals, $I_{1...5}^{V}$, are defined in terms of Passarino-Veltman (PV) functions [@Passarino:1978jh] as, $$\begin{aligned}
I_1 ^{Vij} &=&
(D-1)\, m_i^2 \, C_{0}(-p,p,m_i,m_V,m_j) - \dfrac{m_i ^2}{m_V^2} \, B_0 (0,m_i,m_j) \label{I1f}
\nonumber \\[2mm]
&+& (D-1) \, B_0 (p,m_V,m_j) - \dfrac{1}{m_V^2} \, A_0 (m_j) \;,
\end{aligned}$$ $$\begin{aligned}
I_2 ^{Vij} = (D-1)\, C_{0}(-p,p,m_i,m_V,m_j) - \dfrac{1}{m_V^2}\, B_0 (0,m_i,m_j),\label{I2f} \end{aligned}$$ $$\begin{aligned}
I_3 ^{Vij} &=&
\left( D - 2 +\dfrac{m_i^2}{m_{V}^2} - \dfrac{m_{\chi_1 ^0}^2}{m_V^2} \right)\,
m_{\chi_1^0} \, [C_{11}(-p,p,m_i,m_V,m_j) - C_{12}(-p,p,m_i,m_V,m_j)]
\nonumber \\[1mm]
&+&
\left ( 1+\dfrac{m_i ^2}{m_{V}^2} - \dfrac{m_{\chi_1 ^0}^2}{m_V^2} \right)\,m_{\chi_1 ^0}\,
C_{0}(-p,p,m_i,m_V,m_j)
-\dfrac{m_{\chi_1 ^0}}{m_V^2}\,
B_1 (p,m_V,m_j)
\nonumber \\[1mm]
& + & \dfrac{m_{\chi_1 ^0}}{m_V^2}\, B_0 (0,m_i,m_j)\;,
\end{aligned}$$ $$\begin{aligned}
I_4 ^{Vi} &=& \left( 2-D -\dfrac{m_i ^2}{m_{V}^2} + \dfrac{m_{\chi_1 ^0}^2}{m_V^2}\,
\right) m_{\chi_1 ^0} \left [C_{11}(p,-p,m_V,m_i,m_V) - C_{12}(p,-p,m_V,m_i,m_V) \right ]
\label{I4f} \nonumber \\
& - & (D-3) \, m_{\chi_1 ^0}\, C_{0}(p,-p,m_V,m_i,m_V) \ + \ \dfrac{m_{\chi_1 ^0}}{m_V ^4}\, (m_i ^2 - m_{\chi_1 ^0}^2)\, B_1 (p,m_V,m_i) \nonumber \\ &-& \dfrac{m_{\chi_1 ^0}}{m_V^4} \, A_0 (m_i)\;,\end{aligned}$$ $$\begin{aligned}
I_5 ^{Vi} &=& (D-1) \, C_{0}(p,-p,m_V, m_i, m_V) \ + \ \dfrac{1}{m_V^4}A_0 (m_i) \;, \label{I5f}\end{aligned}$$ where $D\equiv 4- 2 \,\epsilon \, \delta_{\overline{\mathrm{MS}}}$ and $\delta_{\overline{\mathrm{MS}}} =1$ for $\overline{\mathrm{MS}}$ and $\delta_{\overline{\mathrm{MS}}}=0$ for $\overline{\mathrm{DR}}$ scheme. All external particles (i.e., $\chi_{1}^{0}$) are taken on-shell and $m_i = m_{\chi_i^0} $ for $V = Z$ and $m_i = m_{\chi_i^{\pm}} $ for $V = W$. Our notation for PV-functions $A,B,C$, follows closely the one defined in the Appendix of . Functions $A_{0},B_{0},B_{1}$ contain both infinite and finite parts while $C_{0}, C_{11}, C_{12}$ - functions are purely finite. Our calculation has been done in unitary and (for a cross check) in Feynman gauge. The result for - $i \,\delta Y$ - is both renormalization scale invariant and finite.
[^1]: email: [[email protected]]{}
[^2]: email: [[email protected]]{}
[^3]: There are of course tantalising hints from DAMA, CoGeNT, CRESST-II and CDMS-Si experiments but these face stringent constraints from recent null result experiments like XENON100 and LUX making puzzling any theoretical interpretation of them all. For a recent review, see .
[^4]: All gauge group indices are suppressed in this equation. Its detailed form is given below in .
[^5]: In this article, we are only interested in DM mass of the order of the electroweak scale.
[^6]: However, see comments below.
[^7]: The bar symbol over the Weyl fields is part of their names.
[^8]: Electron and Neutron EDMs will arise first at two-loop level. Similarly for the anomalous magnetic moments of SM leptons. See relevant discussion in .
[^9]: We use Weyl notation for fermions [@Dreiner:2008tw] throughout.
[^10]: Our notation resembles closely the one in Appendix E of [[*i.e.,* ]{}]{} $U\to U_{L}^{\dagger}$, $V\to U_{R}^{\dagger}$ and $N\to O^{\dagger}$.
[^11]: It is easy to show that since $^{(0)}{\langle 1|}Q{|1\rangle}^{(0)} = 0$, there is no correction, up to $(\Delta m)^{2}$, on $m_{\chi_{1}^{0}} = M_{D}$ LP mass.
[^12]: We confirm that this result remains unchanged at two-loops.
[^13]: We have not considered the case $M_{D} < M_{Z}$ as this would require further three body decay analysis which is beyond the scope of this paper.
[^14]: Note that, implies that there are no self energy contributions to - $i \,\delta Y$ - at one-loop.
[^15]: Because only $\bar{D}_{1,2}$ are charged under $U(1)_{X}$ (not the Higgs boson), and $\chi_{1}^{0}$ is a linear combination of only $\bar{D}$’s.
[^16]: The Higgs boson production cross section is the same with the SM because the new fermions are uncoloured.
[^17]: The Higgs-fermion vertex is parametrized here as $\mathcal{L} \supset - Y\, h\: \bar{f}\: f +\mathrm{h.c.}$ and therefore for the top-quark Yukawa we obtain $Y_{i} \to Y_{t}/\sqrt{2}$ from while for the new charged fermions $Y_{i} \to Y^{h\chi_{i}^{-}\chi_{i}^{+}}$ from .
[^18]: This quantity is obtained also by using the low energy Higgs theorem as in for the singlet-doublet DM-case.
[^19]: The probability of the tunnelling has been calculated at tree level in .
[^20]: This bound can also be found in .
[^21]: We need to make the substitutions $\tilde{g}_{2d} \rightarrow -Y_{1}$ and $\tilde{g}_{2u} \rightarrow - Y_{2}$ because of different conventions with .
|
---
abstract: 'We measure the fractal dimension of an African plant that is widely cultivated as ornamental, the [*Asparagus plumosus*]{}. This plant presents self-similarity, remarkable in at least two different scalings. In the following, we present the results obtained by analyzing this plant via the box counting method for three different scalings. We show in a quantitatively way that this species is a fractal.'
author:
- |
J. R. Castrejón Pita,$^1$ A. Sarmiento Galán,$^2$ and R. Castrejón Garc[í]{}a.$^3$\
\
[$^1$ Centro de Investigación en Energ[í]{}a UNAM, Ap. Postal 34, 62580 Temixco, Morelos, México]{}\
[$^2$ Instituto de Matemáticas UNAM, Av. Universidad s/n, 62200 Chamilpa, Morelos, México]{}\
[$^3$ Instituto de Investigaciones Eléctricas, Av. Reforma 113, 62490 Temixco, Morelos, México]{}\
title: |
Fractal dimension and self-similarity in\
[*Asparagus plumosus*]{}\
---
[1.]{}
Running head: Fractal dimension in [*Asparagus plumosus*]{}
[1.5]{}
Introduction
============
Nowadays it is frequent to use computational algorithms in order to produce images of plants and trees that resemble their natural counterparts. These visualizations, which present several symmetric bifurcations [@Mandelbrot], encouraged us to analyze the [*Asparagus plumosus*]{} [@Catalogue]. This plant is a native of Africa, but often cultivated in the rest of the world as an ornament. The plant can be easily identified: it is semi-climbing, has a typical height of 2 m, its main branches measure from 25 to 50 cm, and all branches have philiform divisions; its flowers are white and have six petals each, their fruits are purple spheres, 7 mm in diameter. Observed in some detail (Figs. 1-3), the ‘leaves’ of this plant, consist of repeated bifurcations from the main stem, showing a high degree of both, symmetry and scaling; these branching can also be observed even at the smallest scale. Two other peculiar characteristics of the ‘leaves’ of this plant are their flatness and their uniform green color. Although the branches may be dramatically different in shape (actually, Fig. 2 shows an atypical branch), we will show that their fractal dimension is the same.
Method
======
The method of box counting is widely known [@Chaos]. Briefly, the box counting technique consists of counting the number of boxes in a grid that intersect any part of an image that has been placed over it. In order to calculate the fractal dimension of the image, denoted by $D$, using a square grid of side size given by $\varepsilon $, one needs to analyze the changes in the number of boxes required to cover the image, $N$, as the size of the grid is reduced, [*i. e.*]{},
$$D=\lim_{\varepsilon \rightarrow 0}\frac{\log N(\varepsilon )}{\log 1/\varepsilon }.$$
[We have applied this method to the three branches shown in Figs. 1-3 at three different levels: the three different scales at which symmetry is observed. In Figs. 3-4, we visually exemplify the application of the box counting method. The ‘leaves’ we have designated as medium-size branches correspond to the ramifications at the lower right corner of the branches in Figs. 1-3, and those called small-size branches were selected from the medium-size ones following the same criteria; Fig. 4 exemplifies the selection for the main branch in Fig. 3. All the images were obtained by positioning the corresponding branch directly on a scanner ($640$ x $460$ resolution, bit map images), and since the leaves are objects immersed in a two dimensional space, it was not necessary to use any kind of projection. The digital scanning was made in black & white, and in real scale. The side size of the square grid was varied from $1$ to $200$ pixels, by steps of 1 pixel. The original size of the main branch in Fig. 3 is $428.8 $ x $492.0$ mm, $127.5$ x $220.6$ mm for the medium-size branch and $23.0$ x $66.9$ mm for the small one. Since our images all have well defined borders, there is no need to analyze the contour threshold [@Fluids].]{}
Results
=======
The values of $N$, obtained varying the grid size from $1$ to $200$ pixels, are shown in Figs. 5-7. This pixel range allows for a direct comparison in real scale of the results for the three levels at which similarity is observable. A bigger side-size box is not used because the width of the smallest branches (at the base) is 200 pixels, and therefore, a bigger side-size box would mean that a single box would almost cover the whole branch. Since the relations are linear over a wide range of $\varepsilon $ values, the fractal dimension $D$ is then given by the slope of the corresponding line, see Figs. 5-7. Finally, the values obtained for the fractal dimension of the three branches and at the three different scales, are shown in the Table together with the uncertainty in the slope ($\Delta D$) and the correlation of the linear regression ($R$).
Conclusion
==========
From the previous analysis, where we have shown that the fractal dimension of the three branches is practically the same, we can conclude that the shape of a branch of [*Asparagus plumosus*]{} is independent for the determination of its fractal dimension. The very small uncertainties in these values ($\Delta D/D < 3$ x $10^{-3}$) can be easily interpreted in terms of the high linear correlations shown in the Table. Accordingly, we can confirm the fractal dimension in this species, a new type of natural fractal being added to the extensive already well know gallery (for a recent, man-made example, see [@Tokyo]). Additionally, since the value of the fractal dimension obtained from the analysis of the two bigger scales is indeed very similar, we can conclude that there is the same level of complexity at these two scales: the plant is self-similar. Unfortunately, we do not seem to find the same self-similarity at the smallest scale.
[9]{}
B. B. Mandelbrot, [*The Fractal Geometry of Nature*]{} (Freeman, San Francisco, 1989).
M. Mart[í]{}nez, [*Catálogo de Nombres Vulgares y Cient[í]{}ficos de Plantas Mexicanas*]{} (Fondo de Cultura Económica, México, 1979).
K. T. Alligood, T. D. Sauer and J. A. Yorke, [*Chaos: An Introduction to Dynamical Systems*]{}, (Springer Press: New York, 1996), pp 172-180.
R. R. Prasad and K. R. Sreenivasan, [*Phys. Fluids A*]{}, [**2**]{}, 5 (1990).
V. Rodin and E. Rodina, [*Fractals*]{}, [**8**]{}, 4 (2000).
[.5]{} [^1]
Table
[c]{} Fractal dimension ($D$) Uncertainty ($\Delta D$) Linear regression ($R$)\
-----------
Branch
Fig. 1
Fig. 2
Fig. 3
-----------
------- ------- ---------- ------- ------- ------- --------- ------- -------
Main Med Small Main Med Small Main Med Small
1.742 1.712 1.825 0.003 0.003 0.005 0.999 0.999 0.999
1.787 1.765 1.869 0.002 0.002 0.005 0.999 0.999 0.999
1.760 1.722 1.819 0.002 0.003 0.006 0.999 0.999 0.998
------- ------- ---------- ------- ------- ------- --------- ------- -------
\
Fig. 1 A typical example of a main branch of *Asparagus plumosus*.
Fig. 2 An atypical main branch of *Asparagus plumosus*, note the differences in shape with respect to the usual branches in Figs. 1 and 3
Fig. 3 Third example of a main branch; a grid of boxes with a side-length of $60$ pixels is also shown (those boxes that have an intersection with the image are shaded in gray).
Fig. 4 Medium- and small-size branches lying on a square grid with side-length of $20$ pixels.
Fig. 5 Symbols represent the results of applying the box counting method to the main branches in Figs. 1-3. Straight lines show linear regressions performed for each data set.
Fig. 6 Analogue of Fig. 5 for medium-size branches.
Fig. 7 Analogue of Fig. 5 for small-size branches.
[^1]: Martin Nezadal and Oldrich Zmeskal (Institute of Physical and Applied Chemistry at Brno, Czech Republic) are greatefully aknowledged for their HarFA program. This work has been partially supported by DGAPA-UNAM (IN101100), and UC-MEXUS.
|
---
abstract: 'BICEP2 has observed a primordial gravitational wave corresponding to the tensor-to-scalar ratio of 0.16. It seems to require a super-Planckian inflationary model. In this paper, we propose a double hybrid inflation model, where the inflaton potential dynamically changes with the evolution of the inflaton fields. During the first phase of inflation over 7 e-folds, the power spectrum can be almost constant by a large linear term in the hybrid potential, which is responsible also for the large tensor-to-scalar ratio. In the second phase of 50 e-folds, the dominant potential becomes dynamically changed to the logarithmic form as in the ordinary supersymmetric hybrid inflation, which is performed by the second inflaton field. In this model, the sub-Planckian field values ($\sim 0.9~{M_P}$) can still yield the correct cosmic observations with the sufficient e-folds.'
author:
- 'Ki-Young Choi$^{(a)}$[^1] and Bumseok Kyae$^{(b)}$[^2]'
title: |
**Primordial gravitational wave of BICEP2\
from dynamical double hybrid inflation**
---
[ ]{}
Introduction
============
The generation of the large scale structures and the anisotropy in the temperature of the cosmic microwave background (CMB) suggests that there were already small inhomogeneities in the early Universe, a few Hubble times before the observable scale enters the horizon [@review]. The time-independent curvature perturbation $\zeta$ sets the initial conditions for such inhomogeneity and the subsequent evolution of the scalar perturbation. After the first observation by Penzias and Wilson (1965) fifty years ago, the precise observations of the CMB [@COBE; @Komatsu:2010fb; @Ade:2013zuv] found that the primordial power spectrum is Gaussian with the size of ${\cal P}_\zeta \approx 2.43\times 10^{-9}$ and is almost scale-independent with the spectral index $n_\zeta \approx 0.96$.
The inflation models not only explain the problems of the standard big bang cosmology such as the flatness, horizon and monopole problems but also predicts the cosmological perturbations in the matter density and spatial curvature, which explain well the primordial power spectrum [@InflationFluctuation]. Those have arisen naturally from the vacuum fluctuations of light scalar field(s) during inflation, and been promoted to classical one around the time of the horizon exit. As well as the scalar perturbation, the tensor perturbation is also generated during inflation and shows particular features in the B-mode of the CMB polarization data. This B-mode polarization from the primordial tensor spectrum has been searched for a long time as a signature of the primordial inflation.
Recently, BICEP2 [@Ade:2014xna] has announced that they have measured the B-mode from the primordial gravitational wave as well as that from the gravitational lensing effect. The observation prefers to the non-zero tensor spectrum with the tensor-to-scalar ratio, [$$\begin{split}
r=0.2^{+0.07}_{-0.05}.
\end{split}$$]{} After foreground subtraction with the best dust model, however, the tensor-to-scalar ratio shifts down to [@Ade:2014xna] [$$\begin{split}
r=0.16^{+0.06}_{-0.05}.
\end{split}$$]{} Such a large gravitational wave has profound implications for inflation models. The tensor power spectrum comes from the expansion of the Universe during inflation [$$\begin{split}
{\cal P}_T = \frac{8H_*^2}{4\pi^2},
\end{split}$$]{} where $H_*$ is the expansion rate at the horizon exit, and thus the tensor-to-scalar ratio is given by [@Liddle:1992wi] [$$\begin{split}
r =\frac{{{\cal P}}_T}{{{\cal P}}_{\zeta}}=\frac{8{{\cal P}}_*}{{M_P}^2{{\cal P}}_{\zeta}} .
\end{split}$$]{} Here $M_P$ denotes the Planck mass ($\approx 2.4\times 10^{18} {\,\textrm{GeV}}$). Combining with the observed power spectrum [@Ade:2013zuv] [$$\begin{split}
{\cal P}_\zeta = (2.198\pm0.056) \times 10^{-9},
\end{split}$$]{} the observed large tensor spectrum corresponds to the Hubble expansion parameter [$$\begin{split}
H_*\approx 1.0\times 10^{14}{\,\textrm{GeV}},
\end{split}$$]{} or to the potential energy during slow-roll inflation [$$\begin{split}
V^{1/4}\approx 2.08\times 10^{16}{\,\textrm{GeV}}.
\end{split}$$]{}
However, the slow-roll condition during inflation gives the relation between the field variation and the tensor spectrum known as Lyth bound [@Lyth:1996im], [$$\begin{split}
\frac{\Delta \phi}{{M_P}} \gtrsim \mathcal{O} (1) \times {{\left(\frac{r}{0.1} \right) }}^{1/2} .
\end{split}$$]{} Thus, a large tensor is possible only for a large field variation, which is usually larger than the Planck scale. More accurate bounds were studied in [@lyth2; @Choudhury:2013iaa] for the single field inflation. The problem of sub-Planckian inflation with $\epsilon \approx 0.01$ is that the e-folding number is connected to the field variation as [$$\begin{split}
\Delta N \approx \frac{1}{{M_P}} \int \frac{d\phi}{\sqrt{2\epsilon}} \approx 7 ~{{\left(\frac{\Delta \phi}{{M_P}} \right) }} \sqrt{\frac{0.01}{\epsilon}},
\end{split}$$]{} and so only $\Delta N \sim 7$ is maximally obtained for $\Delta \phi\sim M_P$. In order to achieve a large enough e-foldings, hence, $\epsilon$ should somehow be made decreasing after about 7 e-folds. To be consistent with the observation of CMB, moreover, the power spectrum should be maintained as almost a constant even under such a large field variation for the first 7 e-folds [@Bringmann:2011ut] corresponding to the observable scales by CMB, $10 ~\rm{Mpc} \lesssim k^{-1} \lesssim 10^4 ~\rm{Mpc}$ [@Ade:2013zuv]. There are some ways suggested to accommodate the large tensor-to-scalar ratio in the sub-Planckian inflation models by non-monotonic evolution [@Hotchkiss:2008sa; @BenDayan:2009kv; @Shafi:2010jr; @Rehman:2010wm; @Okada:2011en; @Civiletti:2011qg; @Hotchkiss:2011gz; @Choudhury:2013jya; @Antusch:2014cpa] in the single field models or in the assisted inflation [@Liddle:1998jc; @Kim:2006ys].
In the inflation with multiple scalar fields [@Polarski:1992dq; @Bassett:2005xm; @Wands:2007bd], however, the simple relation in the single field inflation is modified due to the quite different inflationary dynamics. The curvature perturbation continues the evolution until the non-adiabatic perturbation is converted to the adiabatic one [@Komatsu:2008hk; @Choi:2008et]. Even the condition ending the inflation can generate the power spectrum [@Lyth:2005qk; @Lyth:2006nx; @Sasaki:2008uc; @Naruko:2008sq; @Huang:2009xa; @Clesse; @Byrnes:2008zy; @Huang:2009vk; @Yokoyama:2008xw; @Emami:2011yi; @Choi:2012hea] and, therefore, changes the tensor-to-scalar ratio. However, the B-mode observation requires that the inflaton perturbation must account for much more than 10% of the primordial curvature perturbation for the slow-roll hypothesis [@Lyth:2014yya].
“Hybrid inflation” [@Linde:1993cn] was suggested with two scalar fields, where one is the inflaton and the other, called the waterfall field, is to terminate inflation when it becomes tachyonic. The advantage of it is that the inflaton’s field value is small compared to the Planck scale, and thus it is legitimate to use it as a low energy effective theory. In the supersymmetric (SUSY) version of the hybrid inflation [@FtermInf2; @FtermInf], the potential can be made flat enough, avoiding the eta-problem: fortunately the Hubble induced mass term is accidentally canceled out with the minimal K$\ddot{\rm a}$hler potential and the Polonyi type superpotential during inflation. The specific form of the superpotential can be guaranteed by the introduced U(1)$_R$ symmetry.
By the logarithmic quantum correction to the scalar potential, the inflaton can be drawn to the true minimum, leading to reheating of the universe by the waterfall fields. Moreover, thanks to such a logarithmic correction, the vacuum expectation values (VEVs) of the waterfall fields can be determined with the CMB anisotropy [@FtermInf]. The VEVs turn out to be tantalizingly close to the scale of the grand unified theory (GUT). Accordingly, the waterfall fields can be regarded as GUT breaking Higgs fields in this class of models [@3221; @422; @FlippedSU(5); @SO(10)]. This inflationary model predicts a red-tilted power spectrum [@FtermInf] around $$n_\zeta \approx 1+2\eta\approx 1-\frac{1}{N}\approx 0.98$$ for $N=50-60$ e-folds. It is too large compared to the present bound on the spectral index. At the same time, the tensor spectrum is accordingly too small to detect. In the SUSY hybrid inflation models with a single inflaton field, it was found that the tensor-to-scalar ratio is $r\lesssim 0.03$ [@Shafi:2010jr; @Rehman:2010wm; @Okada:2011en; @Civiletti:2011qg].
In this paper, we study a dynamical two field hybrid inflation model [@twofieldhybrid]. The dominant potential changes dynamically due to the evolution of another hybrid inflaton field. In the first phase of inflation for around $7$ e-foldings, two inflaton fields are active and generate the power spectrum. When the first waterfall fields are effective, one inflaton falls down to the minimum and the second phase of hybrid inflation starts. Since the vacuum energy and $\epsilon$ are almost constant during the first phase of inflation, we can obtain an almost constant power spectrum in this model. In the second phase of inflation, the potential has the usual shape of the logarithmic one and gives a sufficient e-folding number until the second waterfall fields are effective and the whole inflation ends. Since $\epsilon$ can be made much smaller than $0.01$ in the second phase, we can achieve a large enough e-foldings. Recent studies on the hybrid inflation after BICEP2, one can refer to Refs. [@Carrillo-Gonzalez:2014tia; @Buchmuller:2014epa; @Kobayashi:2014rla].
This paper is organized as follows. In Section \[double\], we briefly explain our setup and in Section \[sugra\], we set up a SUSY model and show the the spectrum and its index for both scalar and tensor perturbations. We conclude in Section \[sec:conclusion\].
Two field inflation {#double}
===================
In this section, we briefly review a general two field inflation model with a potential separable by sum [@Choi:2007su], [$$\begin{split}
W(\phi,\chi)=U(\phi)+V(\chi). \label{sum}
\end{split}$$]{} During the slow-roll inflation, the fields must satisfy the equations of motion, [$$\begin{split}
3H\dot{\phi} + \frac{\partial W }{\partial \phi} = 0,\qquad 3H\dot{\chi} + \frac{\partial W }{\partial \chi} = 0,
\label{EOM}
\end{split}$$]{} respectively, and hence the fields satisfy [$$\begin{split}
\int \frac{d \phi }{\partial W/ \partial \phi } = \int \frac{d \chi }{\partial W/ \partial \chi }.
\end{split}$$]{} along the trajectory. The number of e-foldings during the inflation is given by [$$\begin{split}
N = \int H dt ,
\end{split}$$]{} which can be expressed in terms of the fields using the field equations in [Eq. (\[EOM\])]{}.
For the separable potential in [Eq. (\[sum\])]{} of two fields, the slow-roll parameters are given by [$$\begin{split}
&\epsilon_\phi
= \frac{{M_P}^2}{2}\left( \frac{U_\phi}{W} \right)^2 \,,~~~
\epsilon_\chi
= \frac{{M_P}^2}{2}\left( \frac{V_\chi}{W} \right)^2 \,,
\\
& \eta_{\phi}= {M_P}^2 \frac{U_{\phi\phi}}{W} , \,\,~~~~~~~
\eta_{\chi}= {M_P}^2 \frac{V_{\chi\chi}}{W}\,,
\end{split}$$]{} where the subscripts in $U$ and $V$ stand for the partial derivatives with respect to the corresponding fields. Using these, the cosmological observables, the power spectrum (${\cal P}_\zeta$), scalar spectral index ($n_\zeta$), tensor-to-scalar ratio ($r$), and its spectral index ($n_r$) can be expressed in terms of the slow-roll parameters as follows [@GarciaBellido:1995qq; @VW; @Choi:2007su]: &&[P]{}\_= (+) = (1+) , \[power\]\
&&n\_-1 = -2(\_\^\*+\_\^\*) +2 , \[spectral\]\
&&r= =,\[t/s\]\
&&n\_r = -2 . \[nr\] In the above equations, $u$, $v$, and $\hat{r}$ are defined as [$$\begin{split}
u\equiv\frac{U_*+\widetilde{Z}^c}{W_*} ~,\qquad v\equiv\frac{V_*-\widetilde{Z}^c}{W_*} ~, \qquad \hat{r}\equiv\frac{v^2}{u^2}\frac{\epsilon_\phi^*}{\epsilon_\chi^*} ~,
\end{split}$$]{} where [$$\begin{split}
\widetilde{Z}^c\equiv \frac{V_c\epsilon_\phi^c-U_c\epsilon_\chi^cR^{-1}}{\epsilon_\phi^c+\epsilon_\chi^cR^{-1}} ~, \qquad
R^{-1}\equiv \frac{\partial_{\phi_c} U_c}{\partial_{\phi_c} F_c}~\frac{\partial_{\chi_c} G_c}{\partial_{\chi_c} V_c} .
\end{split}$$]{} The super- or subscripts, “$*$” and “$c$” denote the values evaluated at a few Hubble times after horizon exit and the end of (the first phase of) inflation, respectively. Here $u$ and $v$ parametrize the end effect of inflation, satisfying $u+v=1$ [@Choi:2007su]. $R$ shows the deviation between a hypersurface of end of inflation, $F_c(\phi_c)+G_c(\chi_c)={\rm constant}$ and an equi-potential hyper surface, $U(\phi_c)+V(\chi_c)={\rm constant}$. $R$ is generically of order unity. However, it can be very large or small (even negative) depending on how the inflation ends [@Lyth:2005qk; @Sasaki:2008uc; @Naruko:2008sq; @Byrnes:2008zy; @Huang:2009xa; @Alabidi:2010ba; @Choi:2012hea]. From the constraint $u+v=1$, we find easily a maximum of $r$, $r\leq 16(\epsilon_\phi^* + \epsilon_\chi^*)\equiv 16\epsilon^*$ [@ss1].
For $\epsilon_\phi^c\gg\epsilon_\chi^c$, $\widetilde{Z}^c$ is approximated to $\widetilde{Z}^c\approx V_c-W_cR^{-1}(\epsilon_\chi^c/\epsilon_\phi^c)$. If $U$ and $V$ are almost constant during inflation, then, $v$ and $\hat{r}$ are approximately given by $v\approx R^{-1}(\epsilon_\chi^c/\epsilon_\phi^c)$ and $\hat{r}\approx R^{-2}(\epsilon_\chi^c/\epsilon_\phi^c)^2(\epsilon_\phi^*/\epsilon_\chi^*)$, respectively.
The double hybrid inflation {#sugra}
============================
Let us consider the following form of the superpotential, [$$\begin{split} \label{superPot}
W=\kappa_1S_1\left(M_1^2-\psi_1\overline{\psi}_1\right)
+\kappa_2S_2\left(M_2^2-\psi_2\overline{\psi}_2\right)+mS_1S_2,
\end{split}$$]{} The superpotential $W$ contains the inflaton fields $S_{1,2}$ and the waterfall fields, $\{\psi_{1,2},\overline{\psi}_{1,2}\}$. While $\{S_1,S_2\}$ carry the U(1)$_R$ charges of $2$, the other superfields are neutral. The last term in [Eq. (\[superPot\])]{} breaks the U(1)$_R$ symmetry softly, assuming $m\ll M_{1,2}$. We suppose that it is the dominant U(1)$_R$ breaking term. In fact, $S_1\psi_2\overline{\psi}_2$ and $S_2\psi_1\overline{\psi}_1$ are also allowed in $W$. For simplicity of discussion, however, let us assume that their couplings are small enough. Then the derived potential is [$$\begin{split} \label{V}
V&=\left|\kappa_1(M_1^2-\psi_1\overline{\psi}_1)+ mS_2 \right|^2
+\left|\kappa_2(M_2^2-\psi_2\overline{\psi}_2)+ mS_1 \right|^2
\\
&~~+ \kappa_1^2|S_1|^2\left(|\psi_1|^2+|\overline{\psi}_1|^2\right)
+ \kappa_2^2|S_2|^2\left(|\psi_2|^2+|\overline{\psi}_2|^2\right).
\end{split}$$]{} For $|S_1|^2\gtrsim M_1^2$ and $|S_2|^2\gtrsim M_2^2$, the waterfall fields become stuck to the origin, $\psi_{1,2}=\overline{\psi}_{1,2}=0$, and the potential becomes dominated by a constant energy: [$$\begin{split}
V_{I}&=\kappa_1^2M_1^4+\kappa_2^2M_2^4+\sqrt{2}\kappa_1M_1^2m\phi_2 +\frac{m^2}{2}\phi_2^2
+\sqrt{2}\kappa_2M_2^2m\phi_1 +\frac{m^2}{2}\phi_1^2,
\\
&\equiv \mu^4 +A_1^3\phi_2 +\frac{m^2}{2}\phi_2^2 +A_2^3\phi_1 +\frac{m^2}{2}\phi_1^2,
\label{V1}
\end{split}$$]{} where $\phi_{1,2}$ denote the real components of $S_{1,2}$ ($\equiv {\rm Re}(S_{1,2}/\sqrt{2}))$, and we defined $\mu^4\equiv \kappa_1^2M_1^4+\kappa_2^2M_2^4$ and $A_{1,2}^3\equiv \sqrt{2}\kappa_{1,2}M_{1,2}^2m$ for simple notations. Since SUSY is broken by the positive vacuum energy, the non-zero logarithmic potential can be generated [@FtermInf2; @FtermInf]. We will ignore it for the first phase of inflation because of its relative smallness.
During the first period of inflation, the two fields drive inflation with the following slow-roll parameters: [$$\begin{split} \label{SR}
&\epsilon_{\phi_1}
=\frac{M_P^2A_2^6}{2\mu^8}\left(1+\frac{m^2\phi_1}{A_2^3}\right)^2 ,
\qquad
\epsilon_{\phi_2}=\frac{M_P^2}{2}\left(\frac{A_1^3+m^2\phi_2}{\mu^4}\right)^2,
\\
&\eta_{\phi_1}=\eta_{\phi_2} \equiv \eta=\frac{M_P^2m^2}{\mu^4}.
\end{split}$$]{} We assume that $M_2^2\gg M_1^2$ and so $A_2^3\gg A_1^3$. If $A_2^3 \gg m^2\phi_{1,2}$, then, the almost constant $\epsilon_{\phi_1}$ is dominant over $\epsilon_{\phi_2}$ for this period. In this case, the total $\epsilon$ is approximated by [$$\begin{split}
\epsilon\equiv \epsilon_{\phi_1}+ \epsilon_{\phi_2} \approx \epsilon_{\phi_1} \approx \frac{M_P^2A_2^6}{2\mu^8}.
\end{split}$$]{} As will be explained later, the large $A_2^3$ is necessary for the large tensor-to-scalar ratio and the almost constant power spectrum during the first 7 e-folds.
The first phase of inflation continues until the field $\phi_2$ arrives at $\phi_{2}^{c} \approx \sqrt{2} M_2$. The e-folding number for this phase ($\equiv N_I$) is given in terms of the $\phi_2$ field as [$$\begin{split} \label{N_I1}
N_I = \frac{1}{M_P^2}\int^{\phi_{2}^{*}}_{\phi_{2}^{c}} d\phi_2\frac{ \mu^4}{A_1^3+m^2\phi_2}
=\frac{1}{\eta} ~{\rm log}\left(\frac{A_1^3+m^2\phi_2^*}{A_1^3+\sqrt{2}m^2M_2}\right) .
\end{split}$$]{} During the first phase, $\phi_1$ evolves as [$$\begin{split} \label{N_I2}
N_I= \frac{1}{M_P^2}\int^{\phi_{1}^{*}}_{\phi_{1}^{c}} d\phi_1\frac{ \mu^4}{A_2^3+m^2\phi_1}
= \frac{1}{\eta} ~{\rm log}\left(\frac{A_2^3+m^2\phi_1^*}{A_2^3+m^2\phi_1^c}\right)
\approx \frac{1}{\sqrt{2\epsilon_{\phi_1}}}\left(\frac{\phi_1^*-\phi_1^c}{M_P}\right) ~,
\end{split}$$]{} where $\phi_1^c$ denotes the field value of $\phi_1$ at the end of the first phase. Here we assumed that $A_2^3 \gg m^2\phi_1$. In Eqs. (\[N\_I1\]) and (\[N\_I2\]), $\eta$ and $\epsilon$ were defined in [Eq. (\[SR\])]{}. As seen in [Eq. (\[N\_I2\])]{}, $N_I$ cannot be large enough, if $\phi_1^*$ should be sub-Planckian. It is because of the large constant $A_2^3$, suppressing the logarithmic part in [Eq. (\[N\_I2\])]{}. Hence, the $A_2^3$ needs to be turned-off in the second phase of inflation for a large enough e-folds.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Contour plot of tensor-to-scalar ratio $r$ and the spectral index $n_\zeta$ in the plane of model parameters $m$ and $A_2$ at the cosmologically relevant scale. $\phi_1^*$ is adjusted to be around $N_I=7$ in [Eq. (\[N\_I2\])]{}. The red lines are the contour of $r=0.05,0.1,0.16,0.2,0.3$ from the below, the blue lines are for $n_\zeta=1,0.96,0.9$ from the below and the dashed lines denote $\phi_1^*=0.5{M_P}$ and $0.9{M_P}$ respectively as denoted in the figure.[]{data-label="fig:cont1"}](Contour1.eps "fig:"){width="60.00000%"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
During this period, the power spectrum is determined by the two fields $\phi_1$ and $\phi_2$ as in [Eq. (\[power\])]{}. From the CMB observation, the power spectrum needs to be maintained as almost a constant for the first 7 e-folds corresponding to the scales $10 ~\rm{Mpc} \lesssim k^{-1} \lesssim 10^4 ~\rm{Mpc}$ [@Ade:2013zuv]. We will identify the first 7 e-folds as $N_I$. Assuming $R^{-1}\ll 1$ and so $\hat{r}\ll 1$, we can take $u^2/\epsilon_1^* \gg v^2/\epsilon_2^*$ at the observational scale so that [$$\begin{split}
{{\cal P}}_\zeta &
\approx \frac{\mu^4}{24\pi^2{M_P}^4\epsilon_{\phi_1}^*} , \\
r &\approx 16\epsilon_{\phi_1}^*.
\end{split}$$]{} From the observation of tensor-to-scalar ratio $r=0.16$, we can determine the scale of $\mu$: [$$\begin{split}
\mu \approx \sqrt{\kappa_2}M_2\approx 2.08\times 10^{16} {\,\textrm{GeV}}\end{split}$$]{} with $\epsilon_{\phi_1}^*\approx 0.01$. Since $\epsilon_{\phi_1}^*\gg\epsilon_{\phi_2}^*$ and $\hat{r}\ll 1$, the spectral index is given by [$$\begin{split}
n_\zeta \approx 1 -6\epsilon_{\phi_1}^* +2\eta_{\phi_1}^* .
\end{split}$$]{} Hence, $\eta^*_{\phi_1}=0.01$ is required for $n_\zeta\approx 0.96$. It determines $m\approx 1.8\times 10^{13} {\,\textrm{GeV}}$ from [Eq. (\[SR\])]{}, and $A_2\approx 2.2\times 10^{15} {\,\textrm{GeV}}$. From $\epsilon\approx \epsilon_{\phi_1}^*\approx 0.01$ and $N_I\approx 7$ in [Eq. (\[N\_I2\])]{}, we can obtain the minimum value of $\phi_1^*$, $\phi_1^*\approx 0.9 M_P$ for $\phi_1^*\gg\phi_1^c$. On the other hand, [Eq. (\[N\_I1\])]{} is easily satisfied with ${\rm log}[(A_1^3+m^2\phi_2^*)/(A_1^3+\sqrt{2}m^2M_2)]\approx 0.07$ or $\phi_2^*\gtrsim\sqrt{2}M_2$.
In figure \[fig:cont1\], we show the contour plot of the tensor-to-scalar ratio $r$ and the spectral index $n_\zeta$ in the plane of model parameters $m$ and $A_2$ at the cosmologically relevant scale. $\phi_1^*$ is adjusted to be around $N_I=7$ from [Eq. (\[N\_I2\])]{}. The red lines are the contour of $r=(0.05,0.1,0.16,0.2,0.3)$ from the below, the blue lines are for $n_\zeta=(1,0.96,0.9)$ from the below and the dashed lines denote $\phi_1^*=0.5{M_P}$ and $0.9{M_P}$ respectively as denoted in the figure.
When $\phi_2$ reaches $\sqrt{2}M_2$, the first waterfall fields $\{\psi_2,\overline{\psi}_2\}$ become heavy and rapidly fall down to the near minima acquiring VEVs. $\phi_2$ also becomes heavy by the VEVs of $\{\psi_2,\overline{\psi}_2\}$ and so decoupled from the inflation.[^3] As a result, $\phi_2$, $\kappa_2M_2^2$, and $A_2$ effectively disappear in [Eq. (\[V1\])]{}. Since $mS_1S_2$ term in the superpotential should also be dropped, the inflation is driven only by $\phi_1$ with $V_{\rm inf}=\kappa_1^2M_1^4$ after $N\approx 7$. In this case, we need to consider the logarithmic piece in the potential, $V_{\rm inf}\approx \kappa_1^2M_1^4\alpha {\rm log}\frac{\phi_1}{\Lambda}$, which has been neglected so far because of its smallness. In the second stage of inflation, thus, the potential becomes [$$\begin{split}
V_{II}&=\kappa_1^2M_1^4 \left(1+ \alpha {\rm log}\frac{\phi_1}{\Lambda} \right) ,
\label{V2}
\end{split}$$]{} where $\alpha\approx \kappa_1^2/8\pi^2$. It is just the inflaton potential in the ordinary SUSY hybrid inflation [@FtermInf2; @FtermInf]. During the second phase of inflation with the slow-roll parameters, [$$\begin{split}
\epsilon_{II} = \frac{\alpha^2{M_P}^2}{2\phi_1^2},\qquad \eta_{II} = -\frac{\alpha {M_P}^2 }{\phi_1^2} ,
\end{split}$$]{} which are only relevant to smaller scales and not observable in CMB. The second stage of inflation continues from $\phi_1^c$ to $\phi_1^e \approx\sqrt{2} M_1$. The corresponding e-folding number is [$$\begin{split}
N_{II}= \frac{1}{\alpha {M_P}^2 } \left[(\phi_1^c)^2 - (\phi_1^e)^2\right].
\end{split}$$]{} With a small value of $\alpha$, therefore, we can have a sufficient e-folding number ($\sim 50$).
So far we have not considered supergravity (SUGRA) corrections. Finally, we propose one example of the setups, which can protect above our discussions against SUGRA corrections. We suppose a logarithmic K${\rm \ddot{a}}$hler potential with a “modulus” $T$ and an exponential type superpotential for stabilization of $T$: [$$\begin{split}
K=-{\rm log}\bigg|T+T^*-\sum_i|z_i|^2\bigg|+K_X
~~{\rm and}~~
W=W_0+W_T + W_X ,
\end{split}$$]{} where $W_T=m_{3/2}Te^{-T/f}$. $m_{3/2}$ and $f$ are mass parameters of order TeV and $M_P$, respectively. Here we set $M_P=1$ for simplicity. While $z_i$ ($=S_{1,2}$) and $W_0$ ($=\kappa_1M_1^2S_1+\kappa_2M_2^2S_2+mS_1S_2$) are the fields and the superpotential [*during inflation*]{} considered before, $K_X$ and $W_X$ denote other contributions (which have not been discussed so far) to the K${\rm \ddot{a}}$hler and superpotential, respectively. Then, the $F$-term scalar potential in SUGRA is given by $$\begin{aligned}
\label{V_sugra}
V_{F}&=&e^{K_X}\bigg[\sum_i\left|\frac{\partial W_0}{\partial z_i}\right|^2
+\left|\frac{\partial W_T}{\partial T}\right|^2(T+T^*)
-\left\{\frac{\partial W_T}{\partial T}\left[W^*_T+W^*_X-m^*S_1^*S_2^*\right]+{\rm h.c.}\right\}
\nonumber \\
&& +\frac{1}{T+T^*-\sum_i|z_i|^2}\bigg\{\sum_{I,J}(\partial_{X_I}\partial_{X_J^*}K_X)^{-1}(D_{X_I}W)(D_{X_J}W)^*-2|W|^2\bigg\}\bigg] ,\end{aligned}$$ where $(\partial_{X_I}\partial_{X_J^*}K_X)^{-1}$ means the inverse K${\rm \ddot{a}}$hler metric by $K_X$, and $D_{X_I}W$ is the covariant derivative in SUGRA ($=\partial W/\partial X_I+W\partial K/\partial X_I$). As discussed above, $S_2$ ($S_1$) is decoupled after the first (second) phase of inflation. The first term, $\sum_i\left|\partial W_0/\partial z_i\right|^2$ exactly reproduces [Eq. (\[V1\])]{} \[or [Eq. (\[V\])]{} for $\{\psi_{1,2},\overline{\psi}_{1,2}\}\subset \{z_i\}$\]. It decouples from $T$ unlike the no-scale type SUGRA. It is because we take $-1$ as the coefficient of the logarithmic piece of the K${\rm \ddot{a}}$hler potential. Only if $e^{K_X}\approx 1$, thus, the SUGRA corrections leave intact [Eq. (\[V1\])]{} \[or (\[V\])\].
From the last term \[and also $(\partial_{X_I}\partial_{X_J^*}K_X)^{-1}(D_{X_I}W)(D_{X_J}W)^*$\] of [Eq. (\[V\_sugra\])]{}, the inflaton fields potentially get Hubble scale masses during inflation. However, they could be smaller for $T+T^*\gtrsim 1$. Moreover, only if we have more fields and so e.g. $W= \kappa_XM_X^2X+\kappa_1M_1^2S_1+\kappa_2M_2^2S_2+mS_1S_2$ with $\kappa_XM_X^2\gtrsim \kappa_{1,2}M_{1,2}^2$, then $|W|^2$ provides just a mass term of $\kappa_XM_X^2X+\kappa_1M_1^2S_1+\kappa_2M_2^2S_2$: its orthogonal components, $S_1-(\kappa_1M_1^2/\kappa_XM_X^2)X$ and $S_2-(\kappa_2M_2^2/\kappa_XM_X^2)X$, which are approximately $S_1$ and $S_2$, respectively, still remain light enough. We will not discuss the dynamics of $X$ here. It would be closely associated with the complete forms of $K_X$ and $W_X$, but not directly related to our observations.
Conclusion {#sec:conclusion}
==========
The observation of B-mode polarization by BICEP2 provides hints on inflation models. The hybrid inflation with a single inflaton field might be difficult to accommodate all the observations within the sub-Planckian regime. In this paper, we proposed a double hybrid inflation model, in which the inflaton potential dynamically changes with the evolution of the inflaton fields. During the first phase of inflation over 7 e-folds, the power spectrum remains almost invariant. The large tensor-to-scalar ratio and the constant power spectrum during the first inflationary phase are possible by a large linear term in the inflaton potential. In the second phase of 50 e-folds, the dominant potential becomes dynamically replaced by the logarithmic term as in the ordinary SUSY hybrid inflation. Such a change in the inflaton potential is performed by the second inflaton field. In this model, the sub-Planckian field values ($\sim 0.9 ~M_P$) can still admit the correct cosmic observations with the sufficient e-folds.
K.-Y.C. appreciates Asia Pacific Center for Theoretical Physics for the support to the Topical Research Program. This research is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Grant No. 2011-0011083 (K-Y.C.) and No. 2013R1A1A2006904 (B.K.). B.K acknowledges the partial support by Korea Institute for Advanced Study (KIAS) grant funded by the Korea government.
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[^1]: email: [email protected]
[^2]: email: [email protected]
[^3]: After end of the first stage of inflation, the heavy fields might oscillate and affect the power spectrum as studied in Refs. [@Gao:2012uq; @Konieczka:2014zja]. In this model, however, the relevant scale is outside that can be observed by CMB and LSS. Thus, they do not affect out result.
|
---
abstract: 'The ability to perform computations on encrypted data is a powerful tool for protecting a client’s privacy, especially in today’s era of cloud and distributed computing. In terms of privacy, the best solutions that classical techniques can achieve are unfortunately not unconditionally secure in the sense that they are dependent on a hacker’s computational power. Here we theoretically investigate, and experimentally demonstrate with Gaussian displacement and squeezing operations, a quantum solution that achieves the unconditional security of a user’s privacy using the practical technology of continuous variables. We demonstrate losses of up to 10 km both ways between the client and the server and show that security can still be achieved. Our approach offers a number of practical benefits, which can ultimately allow for the potential widespread adoption of this quantum technology in future cloud-based computing networks.'
author:
- Kevin Marshall
- 'Christian S. Jacobsen'
- Clemens Schäfermeier
- Tobias Gehring
- Christian Weedbrook
- 'Ulrik L. Andersen'
bibliography:
- 'refs.bib'
title: Practical quantum computing on encrypted data
---
[^1]
[^2]
Introduction
============
Incredibly, 2.5 quintillion bytes of data are produced in the world every single day. In fact, it has been estimated that over 90% of the world’s data was created in the last two years [@IBM]. Most of this data is stored around the world in data centers and accessed remotely via the cloud. Because cloud computing is operated by third parties (e.g., Amazon, Facebook), one of the outcomes of this acceleration in information is the need to better protect our privacy. In principle, the cloud contains various types of data for which security and privacy are essential. For example, an individual’s personal data (such as medical records and credit card information), the trade secrets and intellectual property of multinational corporations, and sensitive government information (e.g., the CIA bought cloud space from Amazon). Therefore securing a client’s privacy in the cloud is one of the core security challenges we face today.
One of the current solutions to this challenge is homomorphic encryption [@Naehrig11]. The requirement for such a solution was first identified in the 1970’s by Rivest and colleagues [@Rivest78]. It was not until over 30 years later that IBM researcher Craig Gentry discovered fully homomorphic encryption [@Gentry09]. Although there has been much progress in recent years, the best known implementations of fully homomorphic encryption are impractical for today’s computers [@Naehrig11; @Brakerski11; @Brakerski12; @Gentry15].
It is interesting when one considers the generalization of homomorphic encryption to the quantum realm. If one restricts the class of quantum operations to be implemented, then it was shown one can hide up to a constant fraction, which can be made arbitrarily close to unity, of the encrypted information while only requiring polynomial overhead [@Tan14]. Unfortunately, it has been shown that, perfectly secure, deterministic fully homomorphic quantum computation is only possible at the expense of an exponential overhead [@Yu14]. One can relax the requirements of quantum homomorphic encryption by allowing further rounds of interaction between the client and server. Such a scheme was first studied by Childs [@Childs05] where he outlined a protocol to allow an individual of limited quantum ability (a client) to delegate a quantum computation to another person (a server) who is in possession of a fully-fledged quantum computer. The idea here being that such a computer is only initially available to a select few. Progress towards this direction was demonstrated recently by IBM who made available a small (5 qubit) quantum computer for access in the cloud [@IBM2]. The crux of the issue in developing such a protocol lies in the fact that the client wants to hide some subset of: her input, the quantum program, and the final result. That such a scheme is even possible is astounding from a classical viewpoint. Seminal work by Broadbent [@Broadbent09] built upon this idea, in the cluster-state framework [@Briegel09], to develop the notion of universal blind quantum computing; a protocol which fulfils all three of the above criteria and requires only that the client is able to prepare and send single qubits from a finite set.
In this paper, we offer a new and novel approach to quantum computing on encrypted data, that does not require challenging single photon sources or single photon detectors and is based on a different type of substrate, known as continuous variables [@Braunstein05; @Weedbrook12]. Continuous variables (CVs) offer a number of practical advantages over its qubit counterpart: deterministic gate implementation, low-cost and affordability of components (such as laser sources and detectors), high detection efficiencies, high-rate of information transfer, and the ability to be fully integrated within current telecommunication infrastructures. Here, we answer the questions of how limited the client’s quantum capacity can be, how much classical communication is required between the client and server, and how many classical/quantum operations are needed per gate. Furthermore, we provide proof-of-principle experimental results which highlight the effect of loss over 10 km of effective loss.
![**Protocol for quantum computing on encrypted data.** Input: A displaced vacuum state is prepared. Encryption: A random displacement is applied to the initial state as an encryption procedure. Channel: The state is transmitted over a Gaussian lossy channel to the server (transmission $t$). Gate: The server applies the desired (Gaussian displacement or squeezing) unitary. Channel: The state is sent back over the Gaussian lossy channel to the client. Decryption: The client applies a decryption operation to retrieve the final output state.[]{data-label="fig:Scheme"}](ComputingScheme.pdf){width="8.5cm"}
Our protocol for computing on encrypted CVs consists of three stages (cf. ): an encryption stage, a program stage and a decryption stage. We will now elaborate in more detail. First, the client performs an encryption operation on their desired input to limit the amount of information the server can obtain about the initial state. The state is then sent to the server, who performs a predetermined set of gates known to both parties (corresponding to the program needed to be performed). Finally, the state is sent back to the client who is able to perform a decryption operation which recovers the output of the desired computation.
To discuss the encryption operation we first define the Heisenberg-Weyl operators [@Weedbrook12] $X(Q)=\exp(-iQ\hat p)$ and $Z(P)=\exp(iP\hat q)$, as well as the displacement operator $D(\alpha)=\exp(\alpha \hat a^\dagger - \alpha^* \hat a)$ where $\hat q,\hat p$ are the canonical amplitude and phase operators, respectively, which obey Heisenberg’s uncertainty relation $[\hat q,\hat p]=i$. The annihilation and creation operators are denoted by $\hat a,\hat a^\dagger$ respectively, and are defined by $\hat a=(\hat q+i\hat p)/\sqrt 2$ and its adjoint. Consider the action of applying a random displacement in phase-space to the input state; intuitively speaking, if the displacement is chosen randomly then the state will look mixed and smeared out over all of phase-space to somebody unaware of the displacement parameters. This can be made rigorous by invoking the fact that $\frac{1}{\pi}\int D(\alpha)\ket\psi\bra\psi D^{\dagger}(\alpha)d^2\alpha =\mathbbm{1}$ holds for any normalized state $\ket\psi$. This shows us that averaging the displacements applied to any normalized state, over the entire complex plane, results in a quantity proportional to the identity. In reality, a uniform distribution of displacements over $\mathbb R^2$ is unphysical and we must replace this with a function which dies off sufficiently fast in order to adhere to some energy threshold. For a fixed energy, a Gaussian distribution will maximize the entropy of the resulting state [@Holevo99] and thus we restrict ourselves to Gaussian distributions. This will potentially give the server the capability of extracting some information about the input state, but the amount will be bounded based on the width of the Gaussian. A formal security analysis of our protocol, in the limit of finite squeezing and displacements, remains an open problem.
We now turn our attention to the server who is asked to perform some known algorithm on the encrypted data, where in principle, the algorithm corresponds to a universal quantum computation. To show that the server can perform such a computation, it suffices to show that we can implement a universal set of gates. Namely, we need to show that the set $\mathcal G=\{X(Q),Z(P),U_2(T),U_3(T),F,C_Z\}$ where $U_k(T)=\exp(iT\hat q^k)$, $F=\exp\left[\frac{i\pi}{4}\left(\hat q^2+\hat p^2\right)\right]$, and $C_Z=\exp(i\hat q_1\otimes\hat q_2)$ can be implemented and decrypted with an appropriately chosen operation. Note that of the gates in this set only the $U_3(T)$ gate [@Marek11; @Marshall15] is not Gaussian, and we only require this one non-Gaussian gate in order to achieve universal quantum computation [@Menicucci06]. However, it is prudent to note that non-Gaussian operations are very challenging to implement and that there remain many challenges in devising a robust quantum computer based on continuous variables [@Menicucci06].
Gate Correction
---------- --------------------------------------------------------
$Z(T)$ $X(-Q)Z(-P)$
$X(T)$ $X(-Q)Z(-P)$
$U_2(T)$ $X(-Q)Z(-2QT-P)$
$U_3(T)$ $X(-Q)Z(3Q^2T-P)U_2(-3QT)$
$F$ $X(P)Z(-Q)$
$C_Z$ $X_1(-Q_1)Z_1(-Q_2-P_1)\otimes X_2(-Q_2)Z_2(-Q_1-P_2)$
: The decryption operations corresponding to each gate, up to a phase, for the encryption operation $D(Q,P)$ for single mode gates and $D_1(Q_1,P_1)D_2(Q_2,P_2)$ for two-mode gates.[]{data-label="tab:unlocking"}
Except for $U_3$, all of these operations have decryption operators that correspond to displacements (cf. Table \[tab:unlocking\]), and this allows for the straightforward composition of gates. To make it clearer we consider a simple example, namely the $Z(S)$ gate. The encryption operation $D(Q,P)$ consists of a translation in both the amplitude and phase quadratures and it can be decomposed as a sequence of an $X(Q)$ as well as a $Z(P)$ gate, the latter commutes with $Z(S)$ and so will simply slide through the gate. Consider the application of the $X(Q)$ gate; this gate slides through the $Z(S)$ gate up to a phase as $X(Q)Z(S)=e^{-iQS}Z(S)X(Q)$. Thus we can construct a decryption operation as $$\begin{aligned}
Z(S)D(Q,P)&=C^\dagger(Q,P,S)Z(S),\end{aligned}$$ where $C(Q,P,S)=\exp[i(QP/2-QS)]X(-Q)Z(-P)$ is the decryption gate which, when applied, will undo the effect of the initial encryption operation. However, the $U_2$ decryption gate present in the $U_3$ operation does not easily slide through the Fourier gate $F$, and thus we must have the server correct for this on-the-fly; this is possible in a manner similar to the discrete-variable protocol presented in Ref. [@Fisher14]. To perform $U_3(T)$ the client instead sends the server two modes, the first of which is the encrypted state and the second being the state $U_2(A)Z(Q')\ket 0_p$, where $A, Q'$ are chosen randomly and $\ket 0_p$ denotes a momentum eigenstate; finite squeezing does not present any issues other than the ones normally associated with teleportation, namely the introduction of extra Gaussian noise [@Braunstein98]. The server is then able to implement $U_3(T)$ as shown in the circuit below. $$\begin{aligned}
\Qcircuit @C=1em @R=1em {
\push{\rule{6em}{0em}} & \lstick{D(Q,P)\ket\psi} &\gate{F^\dagger U_3(T)}& \ctrl{1} & \qw &\measureD{\hat p=m_1}\\
\push{\rule{6em}{0em}} & \lstick{U_2(A)Z(Q')\ket{0}_p} &\qw& \ctrl{-1} & \gate{D(Q'',P'')}& \gate{U_2(B)},
}\end{aligned}$$ After the client sends both modes, and the value of $B$, the server performs the desired $U_3(T)$ gate, after application of the inverse Fourier gate, before interacting the two modes with a controlled phase gate, indicated by the vertical line. The server then measures the first mode and after performing the $U_2(B)$ gate obtains the desired state $U_3(T)\ket\psi$ on the remaining mode, up to displacement corrections, provided that $A+B=-3QT$. Note that these additional corrections depend on the value $m_1A$ and so the server must communicate the value of $m_1$ obtained to the client, thus requiring one round of classical communication for the full implementation.
We discuss the implementation of this protocol in more detail in the supplementary material, including a discussion of the decryption operations for each gate in the universal set, how to compose gates, the effects of transmission and imperfect encryption, an entanglement-based analogue, the use of channel estimation and limitations on squeezing.
Experiment
==========
The CV quantum gates associated with linear phase space displacements and squeezing transformations allow for an experimental test of quantum computing on encrypted data solely based on Gaussian quantum states and Gaussian operations. Such operations can be performed with high fidelity within the field of CV quantum optics. In the following, we thus use Gaussian displacement and squeezing operations to test the basic principles of quantum computing on encrypted data, i.e. we implement a server performing firstly $Z$ and $X$ gates and secondly a squeezing gate, related to the $U_2$ gate.
We start by testing the effectiveness of the encryption operation by using the experimental setup shown in Fig. \[fig:MutualSetup\]. Quantum information at the location of the client was generated in the form of a coherent state of light $\ket\phi = \ket\alpha$. To test the protocol for many different coherent state excitations simultaneously, we produced an ensemble of coherent states by means of a set of electro-optical modulators (EOMs), thereby preparing the Gaussian ensemble $\rho=\int G_\text{in}(Q,P)D(Q,P)\ket 0\bra 0 D^\dagger (Q,P) dQdP$ where $G_\text{in}(Q,P)$ is a Gaussian probability density function with variance $V_\text{in}$. This information was then encrypted by applying a randomized phase space displacement onto the coherent state ensemble using the same two EOMs driven by two independent Gaussian white noise sources with equal variances $V_Q=V_P=V_\text{enc}$ for the amplitude ($Q$) and phase ($P$) quadratures. This encryption noise results in an encrypted state $\rho=\int G_\text{tot}(Q,P) D(Q,P)\ket 0\bra 0 D(Q,P)^\dagger dQdP$ where $G_\text{tot}(Q,P)$ is a Gaussian distribution with a total variance of $V_\text{in}+V_\text{enc}$. We measured the encrypted quantum states with homodyne detection, and recorded the correlations between the measurements and the input signal. Using these correlations we calculate the mutual information as plotted in Fig. \[fig:Mutual\]. The solid line is a theoretical prediction given by $$I(\text{server}_\text{enc}\!:\!\text{client}_\text{in}) = \dfrac{1}{2} \ln \left(1+ \dfrac{V_\text{in}}{V_\text{enc}} \right)\ .
\label{eq:mutualinformation}$$
\
For efficient encryption, the quadrature correlations and, thus, the mutual information between the encrypted state received by the server and the input state prepared by the client should be vanishingly small. From the plot, we clearly see the effects of a finite encryption variance.
We first implement the $Z$ and $X$ displacement gates as illustrated in Fig. \[fig:SetupDisplacements\]. The protocol was performed with a Gaussian alphabet of coherent states with variance $V_\text{in}=0.28$ shot-noise units (SNU) embedded in encryption noise of $V_\text{enc}=31$ SNU. For this particular encryption the mutual information is $I=\SI{0.005}{bits/use}$. The $Z$, $X$ gates were tested for a symmetric Gaussian distribution of displacements with variance $V_\text{gate} = 0.6$ SNU. This results in the state $\rho$ after the computation having a total variance of $V_\text{gate}+V_\text{in}+V_\text{enc}$.
Finally, the state is sent back through a lossy channel to the client who is decrypting the state using two EOMs driven by noise which was optimally anti-correlated with the noise used for encryption. The final state is then ideally given by $D(Q,P) \ket\phi $, but due to imperfections there is some residual noise from the encryption protocol and thus we must consider the output state $\rho$ with a variance of $V_\text{in}+V_\text{gate}+V_\text{res}$. For the presented measurement the residual noise was $V_\text{res} = 0.072$ SNU. To visualize the evolution of the information content at different stages of the scheme, in Fig. \[fig:CoherentSNR\], we plot the signal-to-noise-ratios (SNRs) of a single quadrature after each stage of the protocol. It is clear from these numbers that the amount of information in the encrypted state is close to zero, and that the decryption operation is almost ideal. For further quantification we show in Fig. \[fig:FidelitiesDisplacements\] the fidelity between the ensembles of output states of the quantum computation using encrypted states and plain-text states. The fidelities are above $97\,\%$ for all measured transmission values.
We now turn to the implementation of the squeezing gate which is defined as $S(r)=\exp(r(\hat{a}^2 - \hat{a}^{\dagger 2})/2)$, where $r$ is the squeezing parameter. It is directly related to the $U_2(T)$ gate by two additional phase shifts and a suitable transformation between $T$ and $r$ [@Miyata2014], see the supplementary material for a full justification. Figure \[fig:SetupSqueezing\] shows the experimental setup. In contrast to the implementation of the displacement gates we used a single coherent excitation rather than ensembles of coherent input states. The squeezed-light source was based on parametric down-conversion in a potassium titanyl phosphate crystal placed in a linear cavity. In our realization of the gate we directly squeezed the (encrypted) input state in the squeezed-light source. We note that this constitutes the first demonstration of an in-line squeezing transformation of quantum information. Previous demonstrations have relied on off-line squeezed states [@Miyata2014]. A full Wigner function illustration is presented in Figs. \[fig:squeezing\]c-f measured by homodyne detection after each of the four steps. The state after the squeezing gate is shown in Fig. \[fig:squeezing\]e, and it is clear that the squeezing operation is hardly visible as a result of the encryption noise. Finally, the transformed state is decrypted through displacements at the client, thereby revealing the output state of the server which is displayed in Fig. \[fig:squeezing\]f. The squeezing transformations of the first and second moments of the state are now clearly visible.
\
We quantify the performance of the squeezing protocol on encrypted data by computing the fidelity between the state retrieved by the client after decrypting the computed input state and the state received by the client when no encryption was used. This comparison is carried out under the variation of the amount of encryption noise for both no transmission loss and loss corresponding to 10km fiber propagation at a telecom wavelength, equivalent to 5 dB transmission loss. This loss is implemented with a half-wave plate and polarizing beam splitter combination. The results are presented in Fig. \[fig:squeezing\]b. The variation of the fidelities, as was also observed for the displacement gates, comes mainly from systematic errors in the fine tuning of the decryption and system drifts. The deviation from unity fidelity is mainly caused by imperfect reconstruction of the Wigner functions, system drifts as well as non-ideal decryption due to a small amount of decorrelation between the encryption and decryption noise. Despite these imperfections, the fidelity is close to unity and stays above 98.5% for all parameters. In general the fidelities for the loss case are a bit higher due to the fact that the loss forces the decrypted states closer to the vacuum state. The measured high fidelities show that performing quantum computing on encrypted states rather than on plain-text states is indeed feasible.
To quantify the performance of the implemented remote gate itself, we display in Fig. \[fig:squeezing\]g the SNRs of the $Q$ and $P$ quadratures for the different steps of the protocol. Ideally, i.e. without channel loss, without loss in the gate and with perfect decryption, the squeezing gate will preserve the SNR as indicated by the two dashed lines. For the output state we display the SNRs for both with and without encryption, showing a small decrease in SNR if encryption is used. The remaining reduction of the SNR in comparison to the ideal gate is mainly due to optical loss, i.e. channel loss of 5dB in each direction and about 2.2dB for the gate, including the squeezer and the supply optics. A more detailed loss analysis can be found in the supplementary material.
Conclusion
==========
We have developed a continuous-variable protocol for quantum computing on encrypted variables where we required a baseline of only two uses of a quantum channel; one use for the input and another for the output. We required one additional round of classical communication in each direction and one additional use of the quantum channel to implement a cubic phase gate $U_3(T)$, while Gaussian gates can be implemented with no communication cost. The client need only be capable of performing displacements to both encrypt and decrypt, except when they perform a $U_3(T)$ gate the client must be capable of implementing a $U_2(A)$ gate as well. Alternatively, one could run an equivalent entanglement-based version of this protocol which relies on teleportation [@Pirandola15]. To achieve high fidelity teleportation, one could use a hybrid teleportation scheme such as that of Ref. [@Andersen13].
In this paper, we also studied how to compose gates for encryption and decryption, the effects of transmission and imperfect encryption, an entanglement-based analogue, the use of channel estimation as well as the limitations on squeezing in the cubic phase gate. We have experimentally demonstrated our scheme in performing both displacements and online squeezing operations on an alphabet of coherent states and studied the resulting performance in terms of the fidelity. Finally, to the best of our knowledge, this is the first time quantum computing on encrypted data has been generalized to continuous variables, as well as the first proof-of-principle demonstration of any form (qubit or qumode) of secure delegated quantum computing over a lossy channel. Extending this protocol to long-distances will inevitably require quantum-repeater technologies to preserve the encrypted states [@Sangouard11]. We hope our results will help lay the ground work for future theoretical explorations and experimental demonstrations like those for quantum key distribution, such as free-space and field demonstrations.
Gate Decryption Operators
=========================
In this section we first define a set of continuous-variable gates as well as the following conventions. $$\begin{aligned}
[\hat q,\hat p]&=i\\
X(Q)&=\exp(-iQ\hat p)\\
Z(P)&=\exp(iP\hat q)\\
U_k(T)&=\exp(iT\hat q^k)\\
F&=\exp\left[\frac{i\pi}{4}\left(\hat q^2+\hat p^2\right)\right]\\
C_{Z,12}&=\exp(i\hat q_1\otimes\hat q_2)\\
D(\alpha)&=\exp(\alpha \hat a^\dagger - \alpha^* \hat a)\\
D(Q,P)&=e^{-\frac{i}{2}QP}Z(P)X(Q) \text{ for } \alpha=\frac{1}{\sqrt 2}(Q+iP)\\
\ket\psi_{L}&=D(Q,P)\ket\psi\end{aligned}$$ Identities: $$\begin{aligned}
e^Xe^Y&=\exp\{Y+\sum_{n=1}^\infty\frac{1}{n!}[X,Y]_{(n)}\}e^X\\
[X,Y]_{(n)}&=[X,[X,[X,\ldots,[X,Y\underbrace{],\ldots,]]]}_{n-\text{brackets}}\end{aligned}$$ We define a encryption operation, $D(Q,P)$, consisting of a random displacement in phase space. We consider a universal set of gates $G\in \{X(Q),Z(P),U_2(T),U_3(T),F,C_{Z,12}\}$. We wish to demonstrate the existence of a correction operator, $C(Q,P,G)$, which depends on the encryption parameters and $G$ such that $C(Q,P,G)GD(Q,P)\ket\psi=G\ket\psi$.
$Z(S)$
------
$$\begin{aligned}
X(Q)Z(S)X^\dagger(Q)&=\exp(-iQ\hat p)\exp(iS\hat q)\exp(iQ\hat p)\\
&=\exp(iS\hat q + [-iQ\hat p,iS\hat q]+\ldots)\\
&=\exp(iS\hat q - iQS)\\
\label{eq:XZ}
\Rightarrow X(Q)Z(S)&=e^{-iQS}Z(S)X(Q)\end{aligned}$$
Correction: $$\begin{aligned}
Z(S)D(Q,P)&=Z(S)e^{-\frac{i}{2}QP}Z(P)X(Q)\\
&=e^{-\frac{i}{2}QP}Z(P)Z(S)X(Q)\\
&=e^{-\frac{i}{2}QP}Z(P)e^{iQS}X(Q)Z(S)\\
&=C^\dagger(Q,P,S)Z(S)\\
C(Q,P,S)&=\exp[i(QP/2-QS)]X(-Q)Z(-P)\end{aligned}$$
$X(S)$
------
Correction: $$\begin{aligned}
X(S)D(Q,P)&=X(S)e^{-\frac{i}{2}QP}Z(P)X(Q)\\
&=e^{-\frac{i}{2}QP}e^{-iPS}Z(P)X(S)X(Q) \quad \text{By {(\ref{eq:XZ})}}\\
&=C^\dagger(Q,P,S) X(S)\\
C(Q,P,S)&=\exp[i(QP/2+PS)]X(-Q)Z(-P)\end{aligned}$$
$U_2(T)$
--------
$$\begin{aligned}
X(Q)U_2(T)X^\dagger(Q)&=\exp(-iQ\hat p)\exp(iT\hat q^2)\exp(iQ\hat p)\\
&=\exp(iT\hat q^2 + [-iQ\hat p, iT\hat q^2]+\ldots)\\
&=\exp(iT\hat q^2 -2iQT\hat q +\frac{1}{2!}[-iQ\hat p,-2iQT\hat q]+\ldots)\\
&=\exp(iT\hat q^2 -2iQT\hat q +\frac{1}{2}(2iQ^2T)+\ldots)\\
&=\exp(iT\hat q^2 -2iQT\hat q +iQ^2T)\\
\Rightarrow X(Q)U_2(T)&=\exp(iT\hat q^2 -2iQT\hat q +iQ^2T)X(Q)\\
&=e^{iQ^2T}Z(-2QT)U_2(T)X(Q)\end{aligned}$$
Correction: $$\begin{aligned}
U_2(T)D(Q,P)&=U_2(T)e^{-\frac{i}{2}QP}Z(P)X(Q)\\
&=e^{-\frac{i}{2}QP}Z(P)U_2(T)X(Q)\\
&=e^{-\frac{i}{2}QP}Z(P)e^{-iQ^2T}Z(2QT)X(Q)U_2(T)\\
&=C^\dagger(Q,P,T)U_2(T)\\
\Rightarrow C(Q,P,T)&=\exp[i(QP/2+Q^2T)]X(-Q)Z(-2QT)Z(-P)\end{aligned}$$
$U_3(T)$
--------
$$\begin{aligned}
X(Q)U_3(T)X^\dagger(Q)&=\exp(-iQ\hat p)\exp(iT\hat q^3)\exp(iQ\hat p)\\
&=\exp(iT\hat q^3 +[-iQ\hat p,iT\hat q^3]+\ldots)\\
&=\exp(iT\hat q^3 -3iQT\hat q^2+\frac{1}{2!}[-iQ\hat p, -3iQT\hat q^2]+\ldots)\\
&=\exp(iT\hat q^3 -3iQT\hat q^2+\frac{1}{2!}(6iQ^2T\hat q)+\frac{1}{3!}[-iQ\hat p,6iQ^2T\hat q]+\ldots)\\
&=\exp(iT\hat q^3 -3iQT\hat q^2+3iQ^2T\hat q+\frac{1}{6}(-6iQ^3T)+\ldots)\\
&=\exp(iT\hat q^3 -3iQT\hat q^2+3iQ^2T\hat q-iQ^3T)\\
\Rightarrow X(Q)U_3(T)&=e^{-iQ^3T}Z(3Q^2T)U_2(-3QT)U_3(T)X(Q)\end{aligned}$$
Correction: $$\begin{aligned}
U_3(T)D(Q,P)&=U_3(T)e^{-\frac{i}{2}QP}Z(P)X(Q)\\
&=e^{-\frac{i}{2}QP}Z(P)U_3(T)X(Q)\\
&=e^{-\frac{i}{2}QP}Z(P) e^{iQ^3T}U_2(3QT)Z(-3Q^2T)X(Q)U_3(T)\\
&=C^\dagger(Q,P,T)U_3(T)\\
C(Q,P,T)&=\exp[i(QP/2-Q^3T)]X(-Q)Z(3Q^2T)U_2(-3QT)Z(-P)\end{aligned}$$ However, $U_2(T)$ does not slide nicely through $F$ so we wish to have the server handle this correction on the fly. $$\begin{aligned}
U_3(T)D(Q,P)&=\exp[i(-QP/2+Q^3T)]U_2(3QT)Z(P-3Q^2T)X(Q)U_3(T)\\\end{aligned}$$ $$\begin{aligned}
\Qcircuit @C=1em @R=2em {
\lstick{\ket{\phi}} & \ctrl{1} & \meter & \rstick{\hat p = m_1} \qw \\
\lstick{\ket{0}_p} & \ctrl{-1} & \qw& \rstick{X(m_1)F\ket\phi} \qw
}\end{aligned}$$ Let $\ket\phi\rightarrow F^\dagger U_3(T) D(Q,P)\ket\psi$ so that the output of the teleportation is given by $$\begin{aligned}
&X(m_1)U_3(T) D(Q,P)\ket\psi\\
&=\exp[i(-QP/2+Q^3T)]X(m_1)U_2(3QT)Z(P-3Q^2T)X(Q)U_3(T)\ket\psi.\end{aligned}$$ It is clear that applying the correction operator $U_2(-3QT)X(-m_1)$ to this state would eliminate the appearance of $U_2$, however we cannot divulge the value of $Q$ as this would compromise the security. Consider the circuit given by $$\begin{aligned}
\Qcircuit @C=1em @R=2em {
\lstick{F^\dagger U_3(T) D(Q,P)\ket\psi} & \ctrl{1} & \meter & \rstick{\hat p = m_1} \qw \\
\lstick{\ket{0}_p} & \ctrl{-1} & \gate{X(-m_1)}& \gate{U_2(A)}& \gate{U_2(B)}& \rstick{Z(P-3Q^2T)X(Q)U_3(T)\ket\psi} \qw,
}\end{aligned}$$ where $A+B=-3QT$. We can then slide $U_2(A)$ all the way to the left as $$\begin{aligned}
\Qcircuit @C=1em @R=2em {
\lstick{D(Q,P)\ket\psi} &\gate{F^\dagger U_3(T)}& \ctrl{1} & \meter & \rstick{\hat p = m_1} \qw \\
\lstick{U_2(A)Z(Q_2)\ket{0}_p} &\qw& \ctrl{-1} & \gate{Z(2m_1A)}&\gate{X(-m_1)}& \gate{U_2(B)}& \ \qw,
}\end{aligned}$$ where the output is given by $Z(Q_2)Z(2m_1A)Z(P-3Q^2T)X(Q)U_3(T)\ket\psi$ and we also include another $Z$ gate as part of the preparation. If we choose $A$ at ‘random’ then we are giving the server negligible information about the encryption parameter $Q$ by telling them the value of $B$. Note that the server will also have to inform the client of $m_1$ obtained so that they can correct for the additional $Z(2m_1A)$ gate, and thus implementing this gate requires a single round of communication. Furthermore the density operator corresponding to the state $U_2(A)Z(Q_2)\ket{0}_p$ when averaged over $Q_2$ is proportional to the identity, and thus no information about $Q$ can be attained from the state itself. This can be seen from the fact that $$\begin{aligned}
\int dQ_2~ P(Q_2)~Z(Q_2)\ket{0}_p\bra{0}_p Z^\dagger(Q_2)&=\int dQ_2~ P(Q_2)~\ket{Q_2}_p\bra{Q_2}_p\\
\rightarrow \int dQ_2 ~\ket{Q_2}_p\bra{Q_2}_p &\propto \mathbbm{1},\end{aligned}$$ if we choose $Q_2$ according to some Gaussian probability density function, $P(Q_2)$, whose variance tends towards infinity.
$C_{Z,12}$
----------
$$\begin{aligned}
C_{Z,12}[X(Q)\otimes \mathbbm{1}]C^\dagger_{Z,12}&=\exp(i\hat q_1\otimes \hat q_2)[\exp(-iQ\hat p_1)\otimes \mathbbm{1}]\exp(-i\hat q_1\otimes \hat q_2)\\
&=\exp(i\hat q_1 \hat q_2)[\exp(-iQ\hat p_1)]\exp(-i\hat q_1 \hat q_2)\quad\text{Tensor structure implicit in subscripts}\\
&=\exp(-iQ\hat p_1 + [i\hat q_1\hat q_2,-iQ\hat p_1]+\ldots)\\
&=\exp(-iQ\hat p_1 + iQ\hat q_2)\\
&=X_1(Q)Z_2(Q)\end{aligned}$$
$$\begin{aligned}
C_{Z,12}[X_1(Q_1)\otimes X_2(Q_2)]C^\dagger_{Z,12}&=C_{Z,12}[(X_1(Q_1)\otimes \mathbbm{1}_2)(\mathbbm{1}_1\otimes X_2(Q_2)]C^\dagger_{Z,12}\\
&=C_{Z,12}[(X_1(Q_1)\otimes \mathbbm{1}_2)C^\dagger_{Z,12}C_{Z,12}(\mathbbm{1}_1\otimes X_2(Q_2)]C^\dagger_{Z,12}\\
&=[X_1(Q_1)\otimes Z_2(Q_1)][Z_1(Q_2)\otimes X_2(Q_2)]\\
&=X_1(Q_1)Z_1(Q_2)\otimes Z_2(Q_1)X_2(Q_2)\\
\Rightarrow C_{Z,12}[X_1(Q_1)\otimes X_2(Q_2)]&=[X_1(Q_1)Z_1(Q_2)\otimes Z_2(Q_1)X_2(Q_2)]C_{Z,12}\end{aligned}$$
Correction: $$\begin{aligned}
C_{Z,12}[D_1(Q_1,P_1)\otimes D_2(Q_2,P_2)]&=C_{Z,12}e^{-\frac{i}{2}(Q_1P_1+Q_2P_2)}[Z_1(P_1)X_1(Q_1)\otimes Z_2(P_2)X_2(Q_2)]\\
&=e^{-\frac{i}{2}(Q_1P_1+Q_2P_2)}[Z_1(P_1)\otimes Z_2(P_2)]C_{Z,12}[X_1(Q_1)\otimes X_2(Q_2)]\\
&=e^{-\frac{i}{2}(Q_1P_1+Q_2P_2)}[Z_1(P_1)\otimes Z_2(P_2)][X_1(Q_1)Z_1(Q_2)\otimes Z_2(Q_1)X_2(Q_2)]C_{Z,12}\\
&=e^{-\frac{i}{2}(Q_1P_1+Q_2P_2)}[Z_1(P_1)X_1(Q_1)Z_1(Q_2)\otimes Z_2(P_2)Z_2(Q_1)X_2(Q_2)]C_{Z,12}\\
&=C^\dagger(Q_1,P_1,Q_2,P_2)C_{Z,12}\\
C(Q_1,P_1,Q_2,P_2)&=e^{\frac{i}{2}(Q_1P_1+Q_2P_2)}[Z_1(-Q_2)X_1(-Q_1)Z_1(-P_1)\otimes X_2(-Q_2)Z_2(-Q_1)Z_2(-P_2)]\end{aligned}$$
$F$
---
$$\begin{aligned}
F Z(P) F^\dagger&=X(-P)\\
F X(Q) F^\dagger &= Z(Q)\\
F D(Q,P)&=F e^{-\frac{i}{2}QP}Z(P)X(Q)\\
&=e^{-\frac{i}{2}QP} F Z(P) F^\dagger F X(Q) F^\dagger F\\
&=e^{-\frac{i}{2}QP} X(-P)Z(Q) F\\
&= C^\dagger(Q,P) F\\
C(Q,P)&=e^{\frac{i}{2}QP} Z(-Q)X(P)\end{aligned}$$
Equivalence of the Squeezing Gate
=================================
To demonstrate more than simply shifts in phase space, we implemented a squeezing operation, $S(r)=\exp(r(\hat{a}^2 - \hat{a}^{\dagger 2})/2)$, in the experiment. To see why this squeezing operation is as challenging as the $U_2(T)$ gate which is required for universal computation, simply notice that it is equivalent up to rotations $U_2(T)=R(\theta)S(r)R(\phi)$ [@Miyata2014] where $R(\theta)=\exp(i\theta\hat a^\dagger\hat a)$ is generated by the free Hamiltonian. To compose gates, these additional phase shifts must be performed by the server, however to demonstrate a single $U_2(T)$ gate one can notice the following $$\begin{aligned}
C^\dagger(\alpha,T)U_2(T)D(\alpha)\ket \psi&=C^\dagger(\alpha,T)R(\theta)S(r)R(\phi)D(\alpha)\ket \psi\\
&=\tilde C^\dagger(\alpha,T,\theta) S(r) R(\phi)\exp(\alpha \hat a^\dagger-\alpha^*\hat a)R^\dagger(\phi)R(\phi)\ket \psi\\
&=\tilde C^\dagger(\alpha,T,\theta) S(r) \exp(\alpha e^{i\phi} \hat a^\dagger-\alpha^* e^{-i\phi}\hat a) R(\phi)\ket\psi\\
&=\tilde C^\dagger(\alpha,T,\theta) S(r) D(\tilde \alpha) \left[ R(\phi) \ket \psi\right].\end{aligned}$$ We have defined a modified decryption operator $\tilde C^\dagger(\alpha,T,\theta)=C^\dagger(\alpha,T) R(\theta)$, reinterpreted the phase reference for the input state $\ket\psi\rightarrow R(\phi) \ket \psi$, and the encryption parameter has simply changed via $\tilde \alpha=\alpha e^{i\phi}$. A similar trick can be used to shift the rotation $R(\theta)$ through the decryption operator, which is only some displacement, to show that $\tilde C^\dagger(\alpha,T,\theta)=R(\theta)\mathcal C^\dagger(\alpha,T,\theta)$ for some displacement $\mathcal C$. Thus up to an appropriate redefinition of the initial and final phase references, implementation of a squeezing operation $S(r)$ is sufficient to demonstrate a $U_2(T)$ gate.
Compositions of Gates
=====================
To implement a single gate from the universal set defined above, we have demonstrated that there exists a correction (decryption) operator, $C_G$, that satisfies $$\begin{aligned}
\hat C_G \hat G \hat D(Q,P)\ket\psi&=\hat G\ket\psi,\end{aligned}$$ and similarly for the two-qumode $C_{Z,12}$ gate. We would like the server to be able to compose gates without needing the client to decrypt (or correct) between applications. Namely we wish to demonstrate that $$\begin{aligned}
\hat{C'}\hat G_2\hat G_1\hat D(Q,P)\ket\psi=\hat G_2\hat G_1\ket\psi,\end{aligned}$$ or that the client can update their correction operator while only requiring possibly classical communication with the server. Suppose that $$\begin{aligned}
\hat C_1\hat G_1\hat D(Q,P)\ket\psi=\hat G_1\ket\psi\\
\hat C_2\hat G_2\hat D(Q,P)\ket\psi=\hat G_2\ket\psi,\end{aligned}$$ can we construct a $\hat C'$ from our knowledge of $\hat C_{1,2}$? Notice that for all of the gates in the universal set the decryption operators, $C$, involve (up to a phase) only products of operators from the set $C\in \{X(Q),Z(P)\}$; aside from the special case of $U_3(T)$ discussed above. Suppose $G\in \{X(Q),Z(P),U_2(T),U_3(T),F,C_{Z,12}\}$ we would like to show that there exists a modified decryption operator $C'$ such that $$\begin{aligned}
C' G &= G D(Q,P) C\end{aligned}$$ so that we can *slide* a new correction through the gate as $$\begin{aligned}
C'_1 G_2 G_1 D(Q,P)\ket\psi&= [G_2 D(Q,P) C_1]G_1 D(Q,P)\ket\psi\\
&=G_2 D(Q,P) [C_1G_1 D(Q,P)]\ket\psi\\
&=G_2 D(Q,P) G_1\ket\psi\\
\Rightarrow C_2 C_1' G_2 G_1 D(Q,P)&= C_2 G_2 D(Q,P) G_1\ket\psi\\
&= G_2G_1\ket\psi.\end{aligned}$$ Notice first that gates, $G$, with Hamiltonians involving only powers of $\hat q$ slide trivially through each other. The only correction operator, in $\{X(Q),Z(P)\}$, that does not involve powers of only $\hat q$ is precisely $X(Q)$. But we have already shown how to rewrite $$\begin{aligned}
C'G&=GX(Q),\end{aligned}$$ it is easy to insert the extra displacement $D(Q,P)$ between the two terms in the RHS of the above equation since $Z(P),X(Q)$ easily slide through all elements $G$ as illustrated in the above work. Also note for the case where $G=X(Q)$ and $C$ involving powers of $\hat q$ we can apply the same identities by shifting the extra terms to the other side. By recursively applying the identities to swap orderings by applying new corrections we can not only slide the *gate* through the *encryption* operation, we can also slide *corrections* through other *gates*. In this manner we can build up more complicated gates, with no communication between the client and server for Gaussian gates and one additional round of communication for $U_3$.
Effect of Transmission
======================
To consider the action of each step of the protocol on our state we move to the Wigner function representation; this is convenient as all operations of interest are linear transformations in the canonical operations $\hat q,\hat p$ and thus are simple transformations of the associated Wigner function. We model the loss as a beamsplitter $B_{12}(t)$ $$\begin{aligned}
B_{12}(t)&=\left(\begin{array}{cc}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{array} \right),\end{aligned}$$ where the transmission (reflection) factor is given by $t=\cos\theta$ ($r=\sin\theta$). This is associated with the input-output relations $$\begin{aligned}
\left(\begin{array}{c}
q_1'\\
q_2'
\end{array}\right)&=B_{12}(t)
\left(\begin{array}{c}
q_1\\
q_2
\end{array}\right),\end{aligned}$$ and similarly for $p_{1,2}$. When subjecting our mode of interest to loss we inject the vacuum state, $W_{vac}(q,p)=1/\pi\exp(-q^2-p^2)$, in the spare port of the beam splitter. We can study the effect of loss in comparison to the ideal case by integrating over the loss modes and calculating the fidelity between the resulting Wigner function and the one that would be obtained in the absence of loss. If the initial state is a coherent state, $\ket\alpha$, then these steps lead to the transformation $\ket\alpha\rightarrow\ket{t^2\alpha+t\beta}$ where $\beta$ is the displacement provided by the server, the general case is presented in \[tab:procedure\]. The symmetric case, where the encryption and the server’s displacement are distributed according to a Gaussian probability distribution function with zero mean and variance $\Delta^2$, is plotted in .
![Average fidelity between $|\langle \alpha+\beta|t^2\alpha+t\beta\rangle|^2$ for an initial coherent state subject to the protocol with and without loss, which corresponds to a channel with transmission factor $t$. Both $\alpha$ and $\beta$ are distributed according to Gaussians with zero mean and variance $\Delta^2$.[]{data-label="fig:fidelity"}](fidelity.pdf)
Step Operation Transformation
------------ -------------------- -------------------------------------------------------------------------------------------------
Initial $W(q_1,p_1)$
Encryption $D_1(Q,P)$ $W(q_1-Q,p_1-P)$
Loss $W(q_1-Q,p_1-P)W_{vac}(q_2,p_2)$
$B_{12}(t)$ $W(t(q_1-Q)+rq_2,t(p_1-P)+rp_2)\times$
$W_{vac}(-r(q_1-Q)+tq_2,-r(p_1-P)+tp_2)$
Server $D_1(A,B)$ $W(q_1'-A,p_1'-B)W_{vac}(q_2',p_2')$
Loss $W(q_1'-A,p_1'-B)W_{vac}(q_2',p_2')W_{vac}(q_3,p_3)$
$B_{13}(t)$ $W(t(q_1'-A)+rq_3,t(p_1'-B)+rp_3) W_{vac}(q_2',p_2') W_{vac}(-r(q_1'-A)+tq_3,-r(p_1-A')+tp_3)$
Decryption $D_1(-t^2Q,-t^2P)$ $W(t^2q_1-tA+trq_2+rq_3, t^2p_1-tB+trp_2+rp_3) W_{vac}(q_2',p_2') W_{vac}(q_3',p_3')$
Imperfect Encryption
====================
To encrypt a state we used the fact that randomly shifting the initial state in phase space results in the maximally mixed state $$\begin{aligned}
X(\ket\psi\bra\psi)=\frac{1}{\pi}\int D(\alpha)\ket\psi\bra\psi D^\dagger(\alpha)d^2\alpha &=\mathbbm{1}.\end{aligned}$$ To see why this is true, notice that $D(\beta)$ provides an irreducible representation, up to a phase, of the Heisenberg-Weyl group. Furthermore, the operator $X$ commutes with $D(\beta)$ and hence by Schur’s Lemma, for unitary groups, $X\propto \mathbbm 1$. Using the fact that $\ket\psi$ is normalized it is easy to show that the constant of proportionality is $1$, and thus we have the relationship. From a physical point of view it is not possible to apply a random displacement over $\mathbb R^2$ as this would require infinite energy; from a mathematical perspective it’s not clear what a ‘random’ displacement over $\mathbb R^2$ even means. Instead we consider applying a displacement that is Gaussian distributed with zero mean and variance $\Delta^2$ which follows the mapping $$\begin{aligned}
X(\ket\psi\bra\psi)&=\frac{1}{2\pi\Delta^2}\int D(\alpha)\ket\psi\bra\psi D^\dagger(\alpha)e^{-\frac{|\alpha|^2}{2\Delta^2}}d^2\alpha.\end{aligned}$$ The finite width of the Gaussian will lead to this state differing from the maximally mixed state; in general the resulting state will depend on the initial state. A simple figure of merit we can study is the purity of the state $\text{Tr}\rho^2$, in discrete variable systems the purity is always greater than $1/d$ but in continuous variable systems the purity can be zero, as is the case for the maximally mixed state. We can find the purity of an arbitrary initial state, $\rho\rightarrow W(\alpha)$, subject to the map $X(\ket\psi\bra\psi)$ by working in the Wigner function formalism $$\begin{aligned}
W(\alpha)\rightarrow W(\alpha')&=\frac{1}{2\pi\Delta^2}\int W(\alpha-\beta)e^{-\frac{|\beta|^2}{2\Delta^2}}d^2\beta\\
\text{Tr}\rho^2&=2\pi\int W(\alpha')^2d^2\alpha'.\end{aligned}$$ Note that this transformation is also just the convolution of our initial Wigner function with a Gaussian probability distribution. We can then use the purity to characterize how well our encryption operation simulates the ideal encryption procedure, given a particular class of input states. Consider the case where our initial state is also a coherent state, given by $\ket\alpha$ $$\begin{aligned}
X(\ket\alpha\bra\alpha)&=\frac{1}{2\pi\Delta^2}\int D(\beta)\ket\alpha\bra\alpha D^\dagger(\beta)e^{-\frac{|\beta|^2}{2\Delta^2}}d^2\beta\\
&=\frac{1}{2\pi\Delta^2}\int \ket{\alpha+\beta}\bra{\alpha+\beta} e^{-\frac{|\beta|^2}{2\Delta^2}}d^2\beta\\
&=\frac{1}{2\pi\Delta^2}\int \ket{\gamma}\bra{\gamma} e^{-\frac{|\gamma-\alpha|^2}{2\Delta^2}}d^2\gamma.\end{aligned}$$ Note that a thermal state $\rho_{th}(\bar n)$ of average excitation number $\bar n$ has Glauber-Sudarshan representation given by $$\begin{aligned}
\rho_{th}(\bar n)&=\frac{1}{\pi\bar n}\int\ket\alpha\bra\alpha e^{-\frac{|\alpha|^2}{\bar n}}d^2\alpha.\end{aligned}$$ Thus we have that $X(\ket\alpha\bra\alpha)$ corresponds to a thermal state with $\bar n=2\Delta^2$ which is displaced by an amount $\alpha$. For a fixed amount of energy, thermal states maximize their Von Neumann entropy $S(\rho)=-\text{Tr}(\rho\log\rho)$ which is also related to purity; $S(\rho)$ is maximized by the maximally mixed state and is zero for pure states. The purity of $\rho_{th}(\bar n)$ can be calculated as $\text{Tr}(\rho_{th}(\bar n)^2)=1/(1+2\bar n)$ so that the purity of $X(\ket\alpha\bra\alpha)$ is given by $1/(1+4\Delta^2)$.
In general one can use the Wigner function to evaluate the action of the encryption and to determine how mixed the resulting state is. In we show the action of imperfect encryption on the vacuum state and in we show the same for a squeezed vacuum state. In general, we want the Gaussian envelope to be large enough to smear out any details about the set of possible initial states.
![(a) The Wigner function corresponding to the vacuum state. (b) The Wigner function after the vacuum is encrypted where $\Delta=2$; this corresponds to a thermal state with $\bar n=8$.[]{data-label="fig:coherent"}](figure2.pdf)
![(a) The Wigner function for a squeezed vacuum state with squeezing parameter $r=\ln2$. (b,c,d) The Wigner function after the squeezed state is encrypted with parameter $\Delta=1,2,3$ respectively. Notice that for small $\Delta$ the encrypted state is asymmetric, which gives us some information about the initial state, but for higher values the plot more closely resembles a thermal state. The purity of the resulting states (a-d) are $\approx 1,0.27,0.10,0.05$ respectively.[]{data-label="fig:squeezed"}](figure3.pdf)
Entanglement-Based Analogy
==========================
One can exploit the well-known equivalence between preparing a coherent state, chosen according to a Gaussian distribution, and performing Heterodyne detection on half of an EPR pair to show an analogous entanglement-based encryption scheme. In conventional CV teleportation [@Braunstein98], where the parties share a perfectly squeezed EPR pair, the final correction operator is precisely a displacement. Using the principle of deferred measurement, where we simply wait to perform the two homodyne measurements until after the state has been returned from the server, it is clear that the server could not have known of the measurement outcomes and hence the encryption parameters. This analogy works only in the limit of infinite squeezing; in the realistic scenario of finite squeezing there is excess noise introduced onto the mode of interest and the entanglement-based scheme is not a perfect analogy. However, in the limit of large squeezing, or equivalently a Gaussian distribution with large variance, the excess noise tends to zero and this is a good approximation to the actual implementation.
One can also show that the interactive $U_3(T)$ gate has an entanglement-based equivalent. First note that,
$$\begin{aligned}
U_{2,b}(A)\ket{EPR}&\propto U_{2,b}(A)\int dq \ket q_a\ket q_b\\
&\propto \int dq e^{iA q^2} \int dp e^{-iqp} \ket q_a\ket p_b.\end{aligned}$$
If we now measure the second mode and obtain an outcome $P$ we have the following, $$\begin{aligned}
&\rightarrow \int dq e^{iA q^2} e^{-iqP}\ket q_a\\
&= U_{2,a}(A) \int dq e^{-iqP}\ket q_a\\
&\propto U_{2}(A) Z(-P)\ket 0_p.\end{aligned}$$ Thus we can use the above procedure to modify the implementation of $U_3(T)$ and defer the final measurement to construct an entanglement-based version. Explicitly, the client can implement $U_2(A)$ on half of an EPR pair, shared with the server, before measuring the $\hat p$ quadrature; this will have the same effect as the previous scheme. As before, this holds in an exact sense only in the unphysical limit of perfect squeezing.
Channel Estimation
==================
When accounting for loss, as seen in \[tab:procedure\], we require the displacement $D(-t^2Q,-t^2P)$ as part of the correction. To perform this displacement we must know the transmission coefficient $t$, however we could forgo this stipulation and assume that $t\approx 1$, but doing so will result in a reduced fidelity. In order to partially correct for loss it is necessary to do parameter estimation on the channel. This can be done, for example, by periodically sending random coherent states and asking the server to apply some random displacement, after which one can measure the state in order to learn information about the loss parameter. Generally, there will also be additional noise that one can not correct for as well, but if desired one can estimate the magnitude of this noise by sending an appropriate ensemble of initial states. By doing this frequently enough one can ensure that the channel is not varying appreciably with time, and if so to adjust the necessary correction.
This process ensures that the server is not maliciously altering the channel parameters, in fact we can take this notion one step further and check that the server is indeed performing the desired displacement thus verifying the $X,Z$ operations in our full calculation. In the original proposal for blind quantum computation, the ability to verify that the server is performing the desired calculation arises from the capability of hiding the gates that one is performing. Through doing so one is able to periodically perform a check by doing an easy computation of which one already knows the answer. Since the server is unable to discriminate this check from the actual computation it is unable to pass the check without operating faithfully. In our case the server is fully aware of all gates being performed and could for example perform all operations except $U_3(T)$ as intended. Fortunately, there is a large class of desirable computations for which there exist efficient classical verification methods and so one is able to detect an incorrect answer with classical post-processing.
We show the advantage of accounting for the loss in the channel by comparing the fidelities to the ideal state both with and without taking into account the loss parameter in .
![(a) Average fidelity between $|\langle \alpha+\beta|t^2\alpha+t\beta\rangle|^2$ for an initial coherent state subject to the protocol with and without loss, which corresponds to a channel with transmission factor $t$. (b) The same plot as in part (a) but where the final displacement does not take into account the channel loss $t$, given by $|\langle \alpha+\beta|t^2\alpha+t\beta\rangle+(t^2-1)\gamma|^2$, where $\gamma$ is the encryption parameter. All of $\alpha,\beta$ and $\gamma$ are distributed according to Gaussians with zero mean and variance $\Delta^2$.[]{data-label="fig:channelestimate"}](channelestimate.pdf)
Limitations on $U_2(T)$
=======================
Suppose we wish to decompose $U_2(X_\text{tot})=U_2(X_\text{in})U_2(X_\text{enc})$ as the sum of two parts $X_\text{tot}=X_\text{in}+X_\text{enc}$, as in the case where we split the operator $U_2(-3QT)$ between the client and server in order to implement $U_3(T)$. Furthermore, assume that we do not want the knowledge of $X_\text{enc}$ to reveal the value of $X_\text{tot}$. To attempt to hide the value of $X_\text{tot}$, we choose $X_\text{in}$ to be a random variable of zero mean and variance $V_\text{in}$. One method of characterizing the amount of information that a random variable carries about a parameter upon which its probability distribution depends on is with the Fisher information, given by $$\begin{aligned}
\mathcal I_X(\theta)&=\mathbb E\left[\left(\frac{\partial}{\partial\theta}\log P(X;\theta)\right)^2\Bigg | \theta\right].\end{aligned}$$ Explicitly, in the case where $X_\text{tot}=-3QT$, the Fisher information for $X_\text{enc}$ has the property that $\mathcal I(Q)\propto T^2/V_\text{in}$. Notice that as $V_\text{in}\rightarrow 0$ the Fisher information increases without bound indicating that $X_\text{enc}$ allows one to make a very good estimate of the encryption parameter $Q$. However, when $V_\text{in}\rightarrow \infty$ then Fisher information approaches zero and the server learns no information of the encryption parameter.
Measuring encryption efficiency
===============================
A shot-noise limited laser at 1064 nm generated a highly coherent beam, which was split into two. One part was designated as a local oscillator (LO), the other as the signal beam. A sketch of the setup is shown in Fig. \[fig:SetupEncryptionEffectivenessSupp\].
![A schematic representation of the encryption efficiency experiment. PD: Photo diode. DAQ: Data acquisition.[]{data-label="fig:SetupEncryptionEffectivenessSupp"}](SetupEncryptionEffectivenessExtended.pdf){width="40.00000%"}
The local oscillator was sent directly to the detection stage, while the signal beam was passed through a set of optical modulators, to be modulated in the phase and amplitude quadratures. The electronic input to the modulators in either quadrature was an addition of two signals, one representing the alphabet of the different input states of the client, the other the client’s encryption noise. The noise was determined to be white within the measurement bandwidth of 1 MHz. The alphabet was recorded by the data acquisition to establish the size of the correlations. The homodyne detector output was demodulated at 10.5 MHz, with a 1 MHz lowpass filter to set the measurement bandwidth. The sampling rate was 5 MHz, with 14 bit resolution. The variance of the Gaussian distributed input state alphabet was $V_{\text{in}} = 0.6$ SNU. From the measured data we determined the variances and covariances which give an estimate for the mutual information by the formula,
$$I(\text{server}_{\text{enc}}:\text{client}_{\text{in}}) = \dfrac{1}{2} \log_2 \left(\dfrac{V_{\text{in}}}{V_{\text{in}} - \frac{C^2}{V}} \right),$$
where $C$ is the covariance between the recorded alphabet and the signal, and $V = V_\text{in} + V_\text{enc}$, i.e. the variance of the signal and the encryption.
Displacement gates
------------------
A shot-noise limited laser at 1064 nm generated a highly coherent beam, which was split into two. One part was designated as a local oscillator (LO), the other as the signal beam. A sketch of the setup is shown in Fig. \[fig:SetupDisplacementsSupp\].
The local oscillator was sent directly to the detection stage, while the signal beam was passed through a set of optical modulators, to be modulated in the phase and amplitude quadratures. The electronic input to the modulators in either quadrature was an addition of two noise signals, one representing the alphabet of the different input states of the client, the other the client’s encryption noise. The noise signals were determined to be white within the measurement bandwidth of 1 MHz. The state then passed through a half-wave plate and polarizing beam splitter combination which simulated an attenuation $t$ from the client to the server.
Then the state was further displaced by the server, with another signal representing the alphabet of a simple linear gate. Like the previous modulations the displacement signal was also white within the measurement bandwidth. The state was then attenuated with another polarizing beam splitter and a half-wave plate, with a setting identical to the first. This was done to simulate a state experiencing the same loss on the return trip from the server to the client. Following this final loss simulation, the state was modulated again by the client, to eliminate the encryption noise. This elimination was possible down to less than 0.2 SNU in both quadratures. This was in part made possible due to a custom-made noise generator with extremely well correlated outputs at the relevant frequencies. To obtain the right phase between the noise modulation and the cancelling modulation, a controllable phase delay between the modulations was also introduced, by using a DB64 Coax Delay Box from Stanford Research Systems from the noise generator to the modulator. The state was then interfered with the local oscillator at a 50/50 beam splitter, with the output modes subsequently detected by PIN photo diodes with a quantum efficiency of 90 %. The visibility of the interference was 92 %, due to the distortion of the spatial beam profile by the many modulators. Obtaining this visibility was made easier by introducing a cavity to which both the signal and the local oscillator were matched, to ensure efficient overlap despite the spatial distortions.
![A schematic representation of the displacement gate experiment.[]{data-label="fig:SetupDisplacementsSupp"}](SetupDisplacementsExtended.pdf){width="40.00000%"}
Two types of measurements were made. The first type was where the relative phase of the local oscillator with the signal was continuously scanned by a slow function generator at 5 Hz. The outputs of the two photo detectors were subtracted and mixed down to DC using a strong electronic local oscillator at 10.5 MHz. The output of the mixer was amplified and low-pass filtered at 1 MHz before being sent to a data acquisition card (DAC) where the voltages were sampled with a rate of 5 MHz. Digitally a 10 kHz high-pass filter was implemented to dampen a strong 50 Hz modulation originating from the AC line, which was detrimental to the quality of the state reconstruction. The measurements were used to reconstruct the density matrices and Wigner functions of the input and output states using a maximum likelihood algorithm [@Lvovsky2009] and the Python module QuTiP [@Johansson2013].
The first measurements were used to estimate the fidelity of the decryption operation relative to an unencrypted state going through the gate and the channel, applying the definition of fidelity [@Weedbrook12],
$$\label{eq:Fidelity}
F = \text{Tr} \left(\sqrt{\sqrt{\rho_0} \rho_1 \sqrt{\rho_0}} \right),$$
to the reconstructed density matrices. The different Wigner functions are shown in Fig. \[fig:DispGateResultsSupp\]a and b.
The second type of measurement had the relative phase locked such that the phase quadrature was measured. Here the subtracted photo detector signals were sent to a spectrum analyzer (SA), which monitored the signal strength using a zero span trace at 10.5 MHz. This was done to easily monitor the variance of the applied Gaussian noise modulations. Though this variance was only monitored in the phase quadrature, the state reconstruction was done on-line to ensure generation of states with approximately symmetric variances. The estimation of the residual noise can be seen in Figure \[fig:DispGateResultsSupp\]c.
\
Squeezing gate
--------------
A shot-noise limited laser at 1064 nm generated a highly coherent beam, which was split into three. One part was designated as a local oscillator (LO), the second as the signal beam, and the third as the decryption beam. A sketch of the setup is shown in Fig. \[fig:SetupSqueezingSupp\].
![A schematic representation of the squeezing gate experiment.[]{data-label="fig:SetupSqueezingSupp"}](SetupSqueezingExtended.pdf){width="40.00000%"}
The signal beam was modulated with a single displacement in $Q$ and $P$, and in addition a noise modulation generated by a noise generator. Another output of each of the noise generators, highly correlated with the first, was connected to the corresponding modulators in the decryption path. The signal beam then encountered a half-wave plate and polarizing beam splitter combination to simulate the channel, and was then forwarded through a Faraday rotator and another half-wave plate to ensure that the light was mostly in the s-polarization before entering the squeezing cavity. The light entered the resonant linear semi-monolithic cavity containing the $1 \times 2 \times 10\,\text{mm}^3$ periodically poled potassium titanyl phosphate crystal pumped with a 532nm pump beam of 7mW through the coupling mirror. The outer crystal face was coated to have a high reflectivity of 99.95% for both wavelengths, while a curved mirror with a radius of curvature of 20mm, a reflectivity of 90% for the fundamental wavelength at 1064nm and a reflectivity of 20% for the pump wavelength served as a piezo-tunable coupling mirror. The crystal was kept at a phase matching temperature of $\SI{36.2}{\celsius}$ , and most of the signal returned from where it entered, towards the Faraday rotator, but having been squeezed by performing round trips in the cavity. Before hitting the Faraday rotator, a beam sampler redirected 2% of the light for generating an error signal for cavity and pump phase locking. Because the squeezing cavity was birefringent, the small p-polarization component was used to generate an error signal with the Hänsch-Couillaud locking technique [@Hansch1980]. The phase of the pump beam with respect to the signal beam was locked using a phase modulation of the signal beam at 36.7MHz generated using the phase modulator used for encrypted input state generation.
After entering the Faraday rotator from the other direction, the beam was now reflected, rather than transmitted, on the polarizing beam splitter. It then went through a half-wave plate and polarizing beam splitter combination with the same setting as the initial one, to simulate identical channels to and from the server. It was then interfered with the decryption beam on another beam sampler, to minimize loss to the signal. The two beams were locked to destructive interference by another sideband lock, using the same 36.7 MHz sideband, with a photodiode monitoring the interference fringes in the secondary output port of the beam sampler. The offset on this error signal allowed for optimizing the relative phase between the beams, and this, in addition to the adjustable gain settings for the correlated outputs, made it possible to cancel the encryption modulation very accurately. Following the decryption operation the signal was interfered with the local oscillator for scanned homodyne detection for density matrix reconstruction. Data acquisition was the same as for the displacement gate experiment. To compute the fidelity the definition in Equation was used. To reconstruct the encrypted states, which are highly thermal with a slowly decaying photon distribution, a maximum likelihood algorithm which requires a truncation of the Fock space of the density matrix is not feasible. Instead the Wigner function was obtained directly from the data with the help of the inverse Radon transform and the filtered back-projection algorithm [@Lvovsky2009]. For states of smaller magnitude where one can justify the truncation of the density matrix, the maximum likelihood algorithm is far superior [@Lvovsky2009]. A plot of the scanned squeezing, obtained by homodyne detection, is shown in Fig. \[fig:SqueezingSupp\].
We here list the factors contributing to optical loss in the system. Firstly, the mode matching of the input beam into the squeezer was 97 %. This was partly because of a very high sensitivity to the focus of the incoming beam, but also because, as mentioned, the Hänsch-Couillaud lock required some deviation from the ideal polarization. This intentional misalignment directly translated into loss of squeezing. Secondly, the reflected squeezed beam encountered a beam sampler, which directly produced 2 % of transmission loss, but was necessary for the locking scheme, as the pump phase error signal was not sufficiently strong in the transmission of the squeezing cavity. Further, the Hänsch-Couillaud lock clearly only works in reflection. The Faraday rotator induced 3 % loss, and the beam sampler for interfering the signal with the decryption beam produced another 2 % of loss. Lastly, the visibility of the homodyne detection was above 98 % and the photodiodes in the homodyne detector both had a quantum efficiency of 99 %.
![Variance of the squeezed state used for the quantum gate, with a scanned local oscillator. The green trace represents shotnoise.[]{data-label="fig:SqueezingSupp"}](SqueezingSupp.pdf){width="40.00000%"}
[^1]: These two authors contributed equally to this work
[^2]: These two authors contributed equally to this work
|
---
abstract: 'Thick Gas Electron Multipliers (THGEMs) have the potential of constituting thin, robust sampling elements in Digital Hadron Calorimetry (DHCAL) in future colliders. We report on recent beam studies of new single- and double-THGEM-like structures; the multiplier is a Segmented Resistive WELL (SRWELL) - a single-faced THGEM in contact with a segmented resistive layer inductively coupled to readout pads. Several 10$\times$10 cm$^2$ configurations with a total thickness of 5-6 mm (excluding electronics) with 1 cm$^2$ pads coupled to APV-SRS readout were investigated with muons and pions. Detection efficiencies in the 98$\%$ range were recorded with average pad-multiplicity of $\sim$1.1. The resistive anode resulted in efficient discharge damping, with potential drops of a few volts; discharge probabilities were $\sim$10$^{-7}$ for muons and $\sim$10$^{-6}$ for pions in the double-stage configuration, at rates of a few kHz/cm$^2$. Further optimization work and research on larger detectors are underway.'
address:
- 'Weizmann Institute of Science, Rehovot, Israel'
- 'I3N - Physics Department, University of Aveiro, 3810-193 Aveiro, Portugal'
- 'University of Coimbra, Coimbra, Portugal'
- 'CERN, Geneva, Switzerland'
- 'University of Texas, Arlington, USA'
author:
- Lior Arazi
- Carlos Davide Rocha Azevedo
- Amos Breskin
- Shikma Bressler
- Luca Moleri
- Hugo Natal da Luz
- Eraldo Oliveri
- Michael Pitt
- Adam Rubin
- Joaquim Marques Ferreira dos Santos
- João Filipe Calapez Albuquerque Veloso
- Andrew Paul White
title: |
Beam Studies of the Segmented Resistive WELL: a Potential Thin Sampling Element for Digital Hadron Calorimetry\
Presented at the $13^{th}$ Vienna Conference on Instrumentation, February 2013
---
Micropattern gaseous detectors (MPGD) ,THGEM ,SRWELL ,Digital hadron calorimetry (DHCAL) ,Resistive electrode ,SRS ,ILC ,CLIC
Introduction
============
The Thick Gas Electron Multiplier (THGEM) \[1\] is a simple and robust electrode suitable for large area detectors, which can be economically produced by industrial Printed Circuit Board (PCB) methods. Its properties and potential applications are reviewed in \[2,3\]; recent progress can be found in \[4-7\]. One possible application of THGEM-based detectors is in Digital Hadronic Calorimeters (DHCAL), of the kind proposed for the ILC/CLIC-SiD experiment \[8,9\]. In this project, the calorimeter design dictates very narrow sampling elements, in the sub-centimeter range, with a lateral pixel size of 1$\times$1 cm$^2$. Additional requirements are high detection efficiency ($>$95$\%$) and minimum pad multiplicity (number of pads activated per particle).
RPCs presently constitute the baseline technology for the SiD DHCAL, with 94$\%$ efficiency and pad multiplicity of 1.6 \[10\]; other solutions have been investigated, e.g. MICROMEGAS with 98$\%$ efficiency and multiplicity of 1.1 \[11\] and Double-GEMs, with 95$\%$ efficiency and pad multiplicity of 1.3 \[12\] (all results are for muons).
Recently, THGEM-based sampling elements were proposed; they were investigated with muons and pions, primarily in single- and double-THGEM configurations with direct charge collection on readout pads, separated by a 2 mm induction gap from the multiplier \[13\]. The potential value of this concept for DHCAL was demonstrated, leaving room for further optimization, in terms of stability in hadronic beams, efficiency, multiplicity and overall thickness.
We report here on the results of our latest beam study, conducted at the CERN SPS/H4 RD51 beam-line with 150 GeV/c muons and pions; further substantial progress was made with a novel THGEM-like configuration, the Segmented Resistive WELL (SRWELL). More details can be found elsewhere \[14\].
Experimental setup and methodology
==================================
The SRWELL, first suggested in \[13\], is shown schematically in figures 1 and 2; it is a THGEM that is copper-clad on its top side only, whose bottom is closed by a resistive anode. The anode consists of a 0.1 mm thick FR4 sheet patterned with a square grid of narrow copper lines, with the entire area coated with a resistive film (e.g. graphite mixed with epoxy \[15\]). Avalanche-induced signals are recorded inductively on a pad array located below the FR4 sheet. The grid lines on the resistive anode correspond to the inter-pad boundaries; they serve to prevent charge spreading across neighboring pads by allowing for rapid draining of the avalanche electrons diffusing across the resistive layer. The resistive layer itself (of $\sim$10-20 M$\Omega$/square) serves to significantly reduce the energy of occasional discharges. The closed-bottom geometry, similar in its field shape to the WELL \[16\] and C.A.T. \[17\], reduces the total thickness of the detector; it also results in attaining higher gain at a given applied voltage, compared to a standard THGEM with an induction gap \[13\]. The SRWELL has a segmented square hole-pattern with “blind" copper strips above the grid lines; these prevent more energetic discharges in holes located above the metal grid.
![The three layers comprising the SRWELL. Bottom: readout pad array (here 4$\times$4); middle: resistive layer on top of a copper grid (on FR4 sheet); top: segmented single-faced THGEM. The layers are assembled one on top of the other in direct contact (see Fig. 2).[]{data-label="fig:SRWELL_layers"}](figure1.pdf){width="\columnwidth"}
Two basic detector configurations were investigated (figure 2): one comprising a single-stage SRWELL and the other a double-stage structure with a standard THGEM followed by an SRWELL. In both cases the electrodes were 10$\times$10 cm$^2$ in size. Based on previous experience with neon-based gas mixtures, which allow for high-gain operation at relatively low voltages \[4\], the detectors were operated in Ne/5$\%$CH$_4$ at 1 atm, in a typical flow of a few l/h; a minimally ionizing particle (MIP) passing through this gas mixture generates, on the average, $\sim$60 electron-ion pairs per cm along its track \[18\].
In the single-stage detector the SRWELL was either 0.4 or 0.8 mm thick, with corresponding drift gaps of 5.5 and 5 mm. In the double-stage configuration, both the THGEM and SRWELL were 0.4 mm thick; the transfer gap between them was 1.5 mm wide and the drift gap was 2.5, 3 or 4 mm wide. The total thickness of the detector from the resistive anode to the drift electrode was thus between 4.8 and 6.3 mm. The THGEM and SRWELL electrodes were manufactured by Print Electronics Israel \[19\] by mechanical drilling of 0.5 mm holes in FR4 plates, Cu-clad on one or two sides, followed by chemical etching of 0.1 mm wide rims around each hole. In the double-stage detectors the THGEM had an hexagonal hole pattern with a pitch of 1 mm; the SRWELL square hole pattern had a pitch of 0.96 mm, with 0.86 mm wide “blind" strips above the grid lines (1.36 mm between the centers of holes on the opposite sides of the strip). The resistive layers had a surface resistivity of 10-20 M$\Omega$/square; the FR4 sheet serving as the base of the resistive anode was 0.1 mm thick. The grid patterned on the FR4 sheet comprised 0.1 mm wide copper lines, defining an array of 8$\times$8 squares, 1 cm$^2$ each, matching the 8$\times$8 readout pad array patterned here on a 3.2 mm thick FR4 plate located below the anode.
![The two detector configurations investigated in this work. Left: single-stage SRWELL; right: double-stage detector with a standard THGEM multiplier followed by an SRWELL.[]{data-label="fig:single_and_double_schemes"}](figure2.pdf){width="\columnwidth"}
Data acquisition was done using the new CERN-RD51-SRS readout electronics \[20\], with the 64-pad array read by a single SRS analog 128-channel APV25 chip \[21\]. External triggering and tracking were done using the RD51 tracker/telescope setup \[22\], comprising three 10$\times$10 cm$^2$ scintillators in coincidence with three MICROMEGAS tracking units, each equipped with two APV25 chips. The three tracker detectors and the investigated detector shared the same external trigger and front-end card (FEC), enabling event-by-event matching and track reconstruction. This permitted measuring both the global average values of the detection efficiency and pad multiplicity and their local, position-dependent values (e.g. its variation at the pad boundaries). The low-noise electronics enabled operating the detectors at relatively modest gas gains of $\sim$2000-3000.
The detector electrodes were biased individually through a CAEN SY2527 HV system. The voltage and current on each HV channel were monitored and stored using the RD51 slow-control system \[23\], allowing for measuring the rate and magnitude of occasional discharges (e.g. momentary voltage drops, accompanied by current pulses).
The detectors were investigated in a broad low-rate (10-20 Hz/cm$^2$) muon beam, and in narrow, $\sim$1cm$^2$, pion beams; the pion rates were varied between $\sim$0.5 kHz/cm$^2$ to $\sim$70 kHz/cm$^2$, with the majority of the data taken at rates of up to a few kHz/cm$^2$.
Average and local values of the detector efficiency and pad multiplicity were studied using selected tracker events. Pads were considered as activated if their signal was above a pad-specific threshold (set according to its noise level). The detector efficiency was defined as the fraction of tracks where a corresponding cluster of pads was found with its calculated center of gravity not more than 10 mm away from the track projection on the detector. These same tracks were used to calculate the average pad multiplicity by counting the number of pads activated per event. For more details see \[14\].
Results
=======
Studies on single-stage detectors included two configurations: one with a 0.4 mm thick SRWELL and a 5.5 mm drift gap and the other with a 0.8 mm thick SRWELL and a 5 mm drift gap. In a muon beam, the former reached 97$\%$ global efficiency at an average pad multiplicity of 1.2, and the latter (0.8 mm thick SRWELL) displayed 98$\%$ global efficiency already at 1.1 multiplicity. The measured Landau pulse-height distributions were well above noise at gains of $\sim$1500-2000. Discharge probabilities with muons were of the order of 10$^{-6}$ for both configurations. However, with pions both configurations displayed a gain drop by a factor of $\sim$2 at the above operating conditions, with a $\sim$5-10 fold increase in discharge probability; this resulted in lower detection efficiencies with pions in both cases. The observed discharges could be divided, in both detector configurations, into two distinct groups: (a) a vast majority of micro-discharges, involving small ($\sim$10-15 V) voltage drops with a typical recovery time of $\sim$2 seconds; (b) a small fraction of discharges involving large voltage drops ($\sim$100-200 V), with longer recovery times (a few seconds, depending on the size of the voltage drop). Of the two detectors, the 0.8 mm thick SRWELL appeared to be more stable, but this requires further study and more precise quantification.
Studies of the double-stage detectors were done with 0.4 mm thick THGEM and SRWELL electrodes. The transfer gap was kept at 1.5 mm and the drift gap was varied between 2.5 and 4 mm. The efficiencies recorded with muons were similar to those obtained with the single-stage detectors, albeit shifted to slightly higher multiplicities. For example, the 4 mm drift, double-stage detector reached 97$\%$ global efficiency at an average multiplicity of 1.15; the 3 mm drift, double-stage detector reached 94$\%$ efficiency at a multiplicity of 1.2. Discharge probabilities with muons were extremely low, of the order of 10$^{-7}$, for the 4 mm drift double-stage. Figure 3 shows the global efficiency vs. average pad multiplicity for the 0.8 mm thick single-stage SRWELL and the double-stage THGEM/SRWELL with 4 mm drift. Measurement details are provided in \[14\].
![Global detection efficiency vs. average pad multiplicity of the 0.8 mm thick single-stage SRWELL with 5 mm drift gap and the double-stage THGEM/SRWELL with 4 mm and 1.5 mm drift and transfer gaps.[]{data-label="fig:efficiency_vs_multiplicity"}](figure3.pdf){width="\columnwidth"}
Unlike the single-stage detectors, no gain drop was observed when switching from muons to pions in the double-stage detectors; figure 4 compares the pulse-height distributions measured for both particles with the double-stage detector of a 4 mm drift gap, under the same operation voltages. Although occasional discharges occurred with pions for the double-stage 4 mm drift detector, their probability, at rates of a few kHz/cm$^2$, was of the order of 10$^{-6}$, and the voltage drops (on the SRWELL top) were all minute, limited to $\sim$3 V, with a recovery time of $\sim$1 second (no large discharges were observed). The efficiency for pions was similar to that obtained with muons ($\sim$95$\%$ and above). A comparison between runs with and without these micro-discharges showed that their effect is negligible in terms of the detection efficiency. Moreover, the presence of micro-discharges had no effect on the data acquisition system, which operated stably even in high rate pion beams.
![Pulse height (Landau) distributions for muons (left) and pions (right) measured for the double-stage detector of 4 mm drift gap. The parameters and operation conditions are given in the figures. No gain drop was observed with pions in this configuration.[]{data-label="fig:Landaus_double"}](figure4.pdf){width="\columnwidth"}
The ability to accurately match events between the tracker and investigated detectors permitted studying the dependence of the local efficiency and pad multiplicity on the track position relative to the pad boundary. The results are shown in figure 5; essentially no drop in local efficiency occurred above the “blind" SRWELL strips in both configurations; the local increase in pad multiplicity above the inter-pad boundary resulted from charge sharing between holes on the opposite sides of the copper strip (see Fig. 1).
![Local detection efficiency (left) and pad multiplicity (right) as a function of the muon-hit distance from the pad edge for the single-stage 0.8 SRWELL and the 4 mm drift double-stage detectors.[]{data-label="fig:Local_efficiency_and_multiplicity"}](figure5.pdf){width="\columnwidth"}
Summary and discussion
======================
The beam tests described in this work investigated, for the first time, structures based on the Segmented Resistive WELL (SRWELL) concept. This new THGEM-variant has several key advantages which make it a promising candidate for Digital Hadronic Calorimetry: (1) By removing the “standard" induction gap, it allows for a significant reduction in width - a critical feature in applications such as the SiD experiment; the total thickness of the detector configurations studied in this test (excluding readout electronics) was 5-6 mm; (2) The resistive anode effectively quenches occasional discharges, whose magnitude, in the double-stage configuration, was limited to $\sim$3 V with $\sim$1 s recovery time with no effect on the detection efficiency or stability of the electronic readout system; (3) The copper grid underneath the resistive layer significantly reduces the cross-talk between neighboring pads, limiting the multiplicity to $\sim$1.1-1.15; the latter is mostly due to particles inducing avalanches on more than one hole; (4) The detection efficiency with muons is exceptionally high: 98$\%$ at a multiplicity of 1.1 for the single-stage 0.8 mm SRWELL and 97$\%$ at a multiplicity of 1.15 for the 4 mm drift double-stage THGEM/SRWELL. Finally - the SRWELL, like the standard THGEM, is a robust electrode which is essentially immune to spark damage and which can be readily and economically produced over large areas by industrial methods. The combination of the above properties make the SRWELL-based detectors highly competitive compared to the other technologies considered for the SiD-DHCAL.
Single-stage detectors are obviously advantageous in terms of cost when considering large-area applications such as the DHCAL. While their efficiency and multiplicity figures with muons are very convincing, the gain drop in the single-stage SRWELL under pions - not observed for the double-stage detectors - is intriguing, and should be clarified (and mitigated) in additional laboratory tests.
The detector thickness limitation imposed by the SiD experiment calls for the use of extremely thin front-end electronics (a requirement which is, at present, not met by the SRS system). Two alternative readout systems may be suitable for this application: SLAC’s KPiX board \[24\], already beam-tested with THGEM-based detectors \[13\], and the MICROROC chip developed by the LAL/Omega group and LAPP \[11\], which was extensively tested with MICROMEGAS detectors; its investigations with THGEM-based detectors is already underway.
Optimization studies on SRWELL detectors (single and double stage), as well as work on larger detectors, are underway. One attractive option is the return to argon-based gas mixtures, implying 2-3 fold higher MIP-induced ionization electrons, though at the cost of higher operation potentials \[25\]. Modern low-noise electronics may allow for lower-gain operation, so this might not be of a problem.
Acknowledgements
================
We are indebted to Hans Muller, Sorin Martoiu, Marcin Byszewski and Konstantinos Karakostas for their assistance with the SRS electronics and associated software. We thank Victor Revivo and Amnon Cohen of Print Electronics Israel for producing the detector electrodes. We thank Rui de Oliveira of CERN for helpful discussions and assistance with the detector electrodes, for producing the readout pad array and for his on-site support during the beam test. We thank Leszek Ropelewski and his team at CERN’s Gas Detector Development group, in particular George Glonti, for their kind assistance during the tests. This work was supported in part by the Israel-USA Binational Science Foundation (Grant 2008246), by the Benozyio Foundation and by the FCT Projects PTDC/FIS/113005/2009 and CERN/FP/123614/2011. The research was done within the CERN RD51 collaboration. H. Natal da Luz is supported by FCT grant SFRH/BPD/66737/2009. C. D. R. Azevedo is supported by the SFRH/BPD/79163/2011 grant. A. Breskin is the W.P. Reuther Professor of Research in the Peaceful use of Atomic Energy.
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|
---
author:
- Andrei Gaponenko
bibliography:
- 'unfolding.bib'
title: A practical way to regularize unfolding of sharply varying spectra with low data statistics
---
\[sec:into\]Introduction
========================
The procedure of extracting a “truth level” physics distribution that can be directly compared to a theoretical model from measured quantities affected by finite detector resolution is called unfolding [@Cowan1998]. The mathematical problem of unfolding is known to be ill-posed: truth level spectra that are significantly different from each other can map into detector distributions that have only infinitesimally small differences [@Blobel:1984ku; @Cowan1998; @Cowan:2002in; @Prosper:2011zz]. The best possible unbiased solution of an unfolding problem would have an unacceptably large variance [@Cowan1998]. It has been shown that approximate solutions to unfolding problems can be obtained by using a regularization procedure [@Tikhonov1963a; @Tikhonov1963b; @Phillips1962], which reduces the variance of the result at the price of introducing a bias. Implementations of unfolding algorithms for particle physics applications, such as RUN/TRUEE [@Milke:2012ve] and TUnfold [@Schmitt:2012kp] exist. However they are based on the Gaussian approximation of the log-likelihood function, and regularized unfolding using the complete Poisson likelihood is still listed in the “ideas” section in this year’s conference talk [@Schmitt:phystat2019].
The current work was performed in the context of measuring momentum spectrum of charged particles emitted in the process of negative muon capture on atomic nuclei at rest [@twist-mucapture]. The median number of data entries in non-empty bins of a reconstructed 2-dimensional distribution was about 10, necessitating the use of Poisson likelihood in the analysis. The spectrum varied by more than an order of magnitude in the unfolding region. A straightforward application of standard regularization techniques, introduced in section \[sec:notation\] to a toy model, defined in section \[sec:toy\], yielded unfolded spectra with undesirable artifacts, as described in section \[sec:spline\] below. Section \[sec:soe\] presents modifications to the unfolding procedure that allowed us extract the result without unphysical features. Section \[sec:lcurve\] discusses the choice of regularization strength, and \[sec:conclusion\] summarizes the findings.
\[sec:notation\]Regularized unfolding
=====================================
The formulation of the unfolding problem involves an experimental observable $x$, truth level variable $y$ with unknown distribution $f(y)$, which we would like to determine, and detector response $R$. Both experimental observables and truth level variables are in general multidimensional. For example, in the capture measurement [@twist-mucapture] truth level information comprises particle species and its true momentum, while experimental observables include measured track momentum and its range in the detector.
We consider the case when the experimental spectrum is binned. Detector response $R_{i}(y)$ is the expectation value of the number of reconstructed events in bin $i$ given a true event occurring at $y$. It describes all the detector effects: acceptance, efficiency, and resolution—but is independent of the physics spectrum that is being measured. Detector response is usually determined from a Monte-Carlo simulation, which forces a discretization in the $y$ space: $\int{R_i(y)f(y)\,dy}\longrightarrow \sum_j R_{ij} f_j$ where $f_i$ is the integral of $f(y)$ over bin $j$. The bin size in the $y$ space has to be much smaller than the experimental resolution in order for the simulation-derived $R_{ij}$ to be independent of the particular truth level spectrum shape used in the simulation. Small bin size in $y$ leads to a large number of unknowns $f_j$. This large number of unknowns is purely technical and is not related to the number of effective degrees of freedom of the problem, which scales with the size of the dataset [@Panaretos:2011bxp]. However it can make non-linear numerical minimization not feasible. To reduce the number of degrees of freedom to a physically appropriate value one can approximate the unknown functions with splines [@Blobel:1984ku], as is illustrated later in this paper.
The expected number of data events in bin $i$, $\mu_i$, can be written as $$\mu_i = N_{\text{true}} \sum_{j} R_{ij} f_{j} + b_i
\label{eq:predicted}$$ where $N_{\text{true}}$ is the true number of events of interest in the dataset, and $b_i$ is the background contribution. A maximum likelihood estimator for $f_{j}$ is formed by minimizing $$\label{eq:Poisson}
-\log{\mathcal{L}}(d|\mu\{f\}) = -\sum_{i} ( d_i \log\mu_i - \mu_i )$$ where $d_i$ is the observed number of data events in bin $i$. However the unfolding problem is ill-posed and must be regularized to obtain a useful solution. Regularized unfolding can be performed by minimizing a combination of the log likelihood of data and a regularization functional $S\{f\}$ [@Tikhonov1963a; @Tikhonov1963b; @Phillips1962]. $$\label{eq:regularization}
{\mathcal{F}} = -\log{\mathcal{L}}(d|\mu\{f\}) - \alpha S\{f\}$$ where $\alpha$ is the regularization parameter.
A widely used Tikhonov [@Cowan1998; @Tikhonov1963a; @Tikhonov1963b; @Phillips1962] regularization imposes a “smoothness” requirement on the spectrum by penalizing the second derivative of the solution. It therefore biases the result towards a linear function. Another well established regularization, the maximum entropy (or “MaxEnt”) approach [@Cowan1998], is based on the entropy of a probability distribution [@Shannon:1948zz]: $$\label{eq:reg-maxent}
S_{\text{MaxEnt}} = -\sum_j q_{j} \ln(q_{j}), \qquad q_{j}\equiv f_{j}/\sum_k f_{k}$$ It biases unfolding result towards a constant.
Unfolding with Tikhonov regularization can be implemented in a computationally efficient way when $\chi^2$ minimization is used. However this advantage is lost when Poisson likelihood is needed. On the other hand, MaxEnt guarantees that the unfolded spectrum is positive, as is required for a particle emission spectrum, whereas Tikhonov with a large regularization strength $\alpha$ pulls the solution towards a straight line, which can cause some of $f_j$ to be negative. The present work uses the MaxEnt regularization term.
\[sec:toy\]Toy model
====================
 Toy model momentum spectrum $f(p)$ and a random distribution of events drawn from it (initial sample), the distribution modified by detector acceptance times efficiency, and the final distribution after smearing. See text for more details. (b) Toy model detector acceptance times efficiency vs momentum.](toymodel.eps){width="12cm"}
Unfolding issues will be illustrated using a one-dimensional toy model that demonstrates some features first observed in the real life application of the technique. The model is based on the spectrum of protons ejected in the process of negative muon nuclear capture. The spectrum is known to follow an exponential distribution in kinetic energy for large proton energies, and to have a low energy threshold due to the Coulomb barrier [@Measday:2001yr]. We use the empirical functional shape and parameters proposed in [@Hungerford:1999], and convert the distribution from kinetic energy to momentum space: $$f(p) = C \frac{p}{\sqrt{p^2+m^2}}\times\left(1-\frac{1.40{\ensuremath{\text{~MeV}}}}{T(p)}\right)^{1.3279} \times\exp\left\{-\frac{T(p)}{3.1{\ensuremath{\text{~MeV}}}}\right\}
\label{eq:toy-truthp}$$ where $m=938.27{\ensuremath{\text{~MeV}}}/c^2$ is the proton mass, $C$ is a normalization constant, and $T(p)=\sqrt{p^2+m^2}-m$ is the kinetic energy of the proton. The distribution is shown in Fig. \[fig:toymodel\](a). Detector efficiency times acceptance is modeled as $$\epsilon(p)=\begin{cases}
0, & p <= p_0\\
\left(\displaystyle\frac{p}{p_0} - 1\right)^{0.2} \times\left(\displaystyle\frac{p}{p_0}\right)^2,
& p > p_0
\end{cases}$$ with $p_0=80{\ensuremath{\text{~MeV}/\text{c}}}$, illustrated in Fig. \[fig:toymodel\](b). The momentum resolution of the toy detector model as a Gaussian with $\sigma=10{\ensuremath{\text{~MeV}/\text{c}}}$. A sample of 5000 momentum values was drawn from the $f(p)$ distribution (the “initial sample” in Fig. \[fig:toymodel\](a)). Some of the “events” were randomly dropped following the $\epsilon(p)$ curve, then each remaining momentum smeared with the Gaussian resolution to form the “final sample” of 1569 events used for the unfolding tests below. The response matrix for the tests was computed analytically and contains no statistical fluctuations, corresponding to the limit of infinite MC statistics. The toy model contains no background.
\[sec:spline\]Example of application
====================================
 A set of B-splines. (b) $f(p)$ approximated by a linear combination of the splines.](genfit.eps){width="\textwidth"}
To implement the approach outlined in section \[sec:notation\] we need to define an unfolding interval and select a set of splines on that interval to approximate the distribution being unfolded. Cubic $B$-splines [@Boor1978b] provide a convenient basis for modeling smooth continuous physics distributions. Figure \[fig:genfit\](a) shows a set of cubic $B$-splines obtained by placing 3 internal knots that split the interval $80{\ensuremath{\text{~MeV}/\text{c}}}<p<230{\ensuremath{\text{~MeV}/\text{c}}}$ into 4 equal parts, and locating all other necessary knots at the end points [@Boor1978b]. Figure \[fig:genfit\](b) illustrates how a linear combination of these splines can approximate the function $f(p)$ from Eq. \[eq:toy-truthp\]: $$\label{eq:splinesum}
f(p)\approx\sum w_i B_i(p).$$ where $B_i(p)$ are the basis splines, and the $w_i$ are coefficients.
![\[fig:result-spline\]Unfolding results.](unfolding-spline.eps){width="\textwidth"}
The unfolding is performed by minimizing Eq. \[eq:regularization\] with respect to $w_i$ for a fixed value of $\alpha$. The choice of a starting point is critical for the success of a nonlinear multi-dimensional minimization. Our implementation starts with $f_j=\text{const}$ being an exact minimum of (\[eq:regularization\]) for $\alpha\to\infty$, and minimizes the target functional for a large finite value of $\alpha$. Then $\log\alpha$ is reduced by a small amount, and the minimization is re-run by using the previous minimum as the starting point. As $\log\alpha$ is further reduced, each new minimization starts at a point that is linearly interpolated from the two previously found mimima. The process is repeated until the desired value of regularization strength is reached.
Figure \[fig:result-spline\] shows results for several settings of regularization strength $\alpha$. As it is reduced, the solution changes from an almost constant function for $\alpha=10^6$, dominated by the entropy term $S\{f\}$, to curves that are influenced by the likelihood of “data” $\log{\mathcal{L}}(d|\mu\{f\})$. The $f(p)$ spectrum used to produce the toy MC sample is also shown figure \[fig:result-spline\]. One can see that $\alpha=5\times10^2$ is still too large, and the corresponding curve does not reach $f(p)$ in both its peak and tail regions. On the other hand, it already develops a unphysical rising behavior at the end of the unfolding range. Using a lower value $\alpha=52$ produces a spectrum that oscillates about the ideal result and has a pronounced rise at the end of the range.
The toy model example illustrates a typical behavior observed in a real life applications of the unfolding technique. In some cases the procedure does not yield a satisfactory result for any value of $\alpha$. The result spikes at the end, and if one moves the upper boundary of the unfolding interval the spike moves with it. There are two effects that “pull up” the distribution at the end of the unfolding region: the $S\{f\}$ regularization term, and the effect of “overflows” (i.e. reconstructed events that originated outside of the unfolding interval). The regularization term bias is exacerbated due to the fact that a constant is not a good approximation for the rapidly falling true distribution function. A generalization of the MaxEnt approach, cross entropy regularization [@Schmelling:1993cd; @Cowan1998], allows to bias to an arbitrary reference distribution instead of a constant. The distortion due to overflow events can be addressed by treating the part of the signal distribution outside of the unfolding region as a fixed shape background, as is done in e.g. [@Schmitt:2012kp]. In that approach the model of the signal distribution is not continuous, because the resulting distribution in the unfolding region does not generally match the a priori “background” distribution at the interval ends. Instead of trying to guess the steepness of the “true” distribution for the cross entropy and overflow background priors, we suggest to fit it from data, as is detailed below.
\[sec:soe\]An improved technique
================================
![\[fig:result-soe\]Results for the improved technique.](unfolding-soe.eps){width="\textwidth"}
The main ideas to improve on the results of the previous section are:
- Inside the unfolding region, bias towards a physically motivated function instead of a constant, with parameters of the function included in the fit. For the spectrum of protons from muon capture example an exponential in kinetic energy was chosen, because the spectrum is know to approach this shape at high energies. Note that the true distribution in the toy model (Eq. \[eq:toy-truthp\]) is not a simple exponential, however an exponential is a much better approximation for it in the unfolding interval than a constant.
- Include the “overflow” region in the minimization, and fit not just the normalization but also the exponential slope in that region.
- Require that the distribution is continuous and has two continuous derivatives. This requirement connects the unfolded distribution to the overflow tail in a way that prevents the unphysical spike at the boundary.
Specifically, we represent $$f(p) = A \frac{p}{\sqrt{p^2+m^2}}\exp\{-\gamma T(p)\}
\times
\begin{cases}
1 + \phi(p) & p_{\text{min}} < p \le p_{\text{max}} \\
1 & p_{\text{max}} < p
\end{cases}$$ where $p_{\text{min}}$ and $p_{\text{max}}$ determine the limit of the unfolding region, $m$ is the mass of the particle and $T(p)$ its kinetic energy, $A$ and $\gamma$ are parameters pertaining to the exponential behavior of the spectrum, and $\phi(p)$ is an arbitrary function to be determined from the unfolding. The regularization term has the form (\[eq:reg-maxent\]) but now acts on $1+\phi$ instead of $f$: $$\label{eq:reg-maxent-final}
{S}_{\text{MaxEnt}} = -\sum_j \tilde{q}_{j} \ln(\tilde{q}_{j}),
\qquad \tilde{q}_{j}\equiv (1+ \phi_{j})/\sum_k (1+\phi_{k})$$
The function $\phi(p)$ is approximated by a linear combinations of cubic basis splines $B_l$ [@Boor1978b] $$\phi(p) = \sum_{l}^{n} w_{l} B_{l}(p),
\qquad p_{\text{min}}<p\le p_{\text{max}}$$ Here $w_{l}$ are the spline coefficients determined from the unfolding process. We require that the resulting spectrum has a continuous second derivative, leading to $\phi(p_{\text{max}})=\phi'(p_{\text{max}})=\phi''(p_{\text{max}})=0$, which is provided by having a single-fold spline knot at the endpoint $p_{\text{max}}$. There are no continuity constraints at $p_{\text{min}}$, therefore a 4-fold knot should be used at that point to support the most general cubic spline shape.
To illustrate the modified technique, we use the same unfolding interval $80{\ensuremath{\text{~MeV}/\text{c}}}<p<230{\ensuremath{\text{~MeV}/\text{c}}}$ as in section \[sec:spline\] and the same set of internal knots. The resulting splines are $B_1$ to $B_4$ shown in Fig. \[fig:genfit\]. Splines $B_5$ to $B_7$ would violate the continuity condition and must not be included. Like before, we start with the maximally regularized solution and reduce $\log\alpha$ in small steps. The resulting curves for several values of $\alpha$ are shown in Fig. \[fig:result-soe\]. Note that the starting solution ($\alpha=1\times10^6$) is now close to an exponential, not a constant, and that some of the resulting curves closely follow the original $f(p)$ from Eq. \[eq:toy-truthp\].
\[sec:lcurve\]Choice of the regularization strength
===================================================
![\[fig:lcurve-soe\]L-curve for the improved technique.](lcurve-soe.eps){width="\textwidth"}
The regularization strength $\alpha$ in Eq. (\[eq:regularization\]) should be chosen to provide an optimal balance between the variance and the bias of the result. The L-curve [@Hansen:1992; @Hansen:1993] provides a way to visualize a transition from strongly biased, regularization term dominated solutions for large $\alpha$, to noise dominated ones. For a given $\alpha$ the minimization of ${\mathcal{F}}$ in Eq. (\[eq:regularization\]) yields particular values of $\log{\mathcal{L}}$ and $S$. Our code minimized a binned likelihood ratio, so this is what we will use below instead of the “bare” $\log\mathcal{L}$. Following [@Hansen:1993], we define parametric functions $\rho(\alpha)=-\log\left({\mathcal{L}}(d|\mu)/{\mathcal{L}}(d|d)\right)$ and $\eta(\alpha)=S$, and consider the curve $-\log\eta(\alpha)$ vs $\log\rho(\alpha)$. The choice of signs in the definition of the $\eta$ term provides the conventional orientation of the “L”. A plot of the curve is shown in Fig. \[fig:lcurve-soe\]. As $\alpha$ is initially reduced from $\alpha=1\times10^6$, the curve is almost horizontal, with quality of fit to data improving while not significantly affecting the regularization term. For small $\alpha$ the regularization penalty grows sharply without much improvement in the data fit. The optimal value of $\alpha$ lies in the transition region, and can be defined as the point of the maximum curvature on the L-curve [@Hansen:1993].
In our example, the maximum curvature point is at $\alpha=37$. The corresponding unfolded spectrum is shown as the solid line in Fig. \[fig:result-soe\]. It is indeed a reasonable fit: the curves for smaller $\alpha$ are farther away from the correct solution for $p>200{\ensuremath{\text{~MeV}/\text{c}}}$ and $150<p<170{\ensuremath{\text{~MeV}/\text{c}}}$, while the curve for a larger $\alpha=3.6\times10^2$ deviates more in the peak region $p\approx80{\ensuremath{\text{~MeV}/\text{c}}}$.
\[sec:conclusion\]Conclusion
============================
The proposed method combines unfolding to an arbitrary function shape in a phase space region with sufficient data statistics and a parametric fit in the low statistics tail. The whole distribution is required to be twice continuously differentiable, which guarantees a physically reasonable behavior of the result. Factoring out the exponential part of a sharply varying spectrum and applying the regularization to just the deviation from the pure exponent reduces the bias. The use of the L-curve approach for finding the optimal regularization strength has been demonstrated for Poisson likelihood fit to data with the MaxEnt regularization term.
The author thanks Richard Mischke, Art Olin, Glen Marshall, Alexander Grossheim, and Anthony Hillairet, who worked with me on the muon capture analysis and provided encouragement vital for the completion of this study. In addition, Richard and Art provided valuable feedback on the text of this article.
The numerical minimization code used for the study utilized the GNU Scientific Library [@GSL]. The figures were prepared with Asymptote [@Asymptote].
This document was prepared by the author using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
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abstract: 'promises to overcome *vendor lock-in* by enabling a multi-vendor hardware and software ecosystem in operator networks. However, we observe that this is currently not happening. A framework allowing to compose applications combining different frameworks can help revert the trend. In this paper, we analyze the challenges in the current landscape and then present the multi-controller framework developed by the project. Our architecture supports different southbound protocols and we have implemented a proof of concept using the protocol, which has given us a greater insight on its shortcomings.'
author:
- 'Pedro A. Aranda Gutiérrez, Roberto Doriguzzi-Corin and Elisa Rojas[^1][^2][^3]'
bibliography:
- 'netide.bib'
title: |
Lessons learnt from the NetIDE project:\
Taking SDN programming to the next level
---
[Aranda Gutiérrez : Lessons learnt from the NetIDE project: Taking SDN programming to the next level]{}
Software-Defined Networking, Portability, Conflict resolution, OpenFlow
Acknowledgments {#acknowledgments .unnumbered}
===============
The work presented in this paper has been partially sponsored by the European Union through the FP7 project NetIDE, grant agreement 619543.
\[last-page\]
[^1]: Pedro A. Aranda Gutiérrez is with Universidad Carlos III, Spain. e-mail: [email protected]
[^2]: Roberto Doriguzzi-Corin is with FBK CREATE-NET, Italy. e-mail: [email protected]
[^3]: Elisa Rojas is with Telcaria Ideas S.L., Spain. e-mail: [email protected]
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abstract: 'In this paper we obtain time uniform propagation estimates for systems of interacting diffusion processes. Using a well defined metric function $h$ , our result guarantees a time-uniform estimates for the convergence of a class of interacting stochastic differential equations towards their mean field equation, and this for a general model, satisfying various conditions ensuring that the decay associated to the internal dynamics term dominates the interaction and noise terms. Our result should have diverse applications, particularly in neuroscience, and allows for models more elaborate than the one of Wilson and Cowan, not requiring the internal dynamics to be of linear decay. An example is given at the end of this work as an illustration of the interest of this result.'
author:
- 'Jamil Salhi, James MacLaurin and Salwa Toumi'
title: On Uniform Propagation of Chaos and application
---
[^1]
Stochastic differential equation, Mean Fields, McKean-Vlasov equations, Interacting Diffusion, Uniform Propagation of Chaos, Neural Network.
60K35, 60J60, 60J65, 92B20
Introduction
============
In this paper we obtain time-uniform estimates for the convergence of a class of interacting diffusion stochastic differential equations towards the associated mean field equation. The propagation of chaos resulting from this convergence when the number of particles $N$ tends to infinity is uniform in time which means that not only the particles are independent of each other, but also this independence is reached uniformly in time.
The $N$-particle interacting diffusion model is of the following form $$X^j_t = x_{ini} + \int_0^t b_0(X^j_s) + \frac{1}{N}\sum_{k=1}^N b_1(X^j_s,X^k_s) ds + \int_0^t b_2(X^j_s)dW^j_s.\label{eqn base system}$$ Here $(W^j)_{j\in{\mathbb{Z}}^+}$ are independent Wiener Processes, $x_{ini}$ is a constant and $b_0,b_2: {{\mathbb R}}\to {{\mathbb R}}$, $b_1: {{\mathbb R}}\times{{\mathbb R}}\to {{\mathbb R}}$, are measurable functions. We will explain further below our reasons for studying this type of model. For a probability measure on ${{\mathbb R}}$, $\gamma$, write $\bar{b}_1(x,\gamma) = \int_{{{\mathbb R}}}b_1(x,y)d\gamma(y)$. The limiting processes $(\bar{X}^j_t)$ are defined to be $$\bar{X}^j_t = x_{ini} + \int_0^t b_0(\bar{X}^j_s) + \bar{b}_1(\bar{X}^j_s,\bar{\mu}_s)ds + \int_0^t b_2(\bar{X}^j_s)dW_s\label{eqn limit system},$$ where $\bar{\mu}_s$ is the law of $\bar{X}^j_s$. The classical propagation of chaos result states that, under suitable conditions on $b_0,b_1$ and $b_2$, the probability law of $X^j_t$ over some fixed time interval $[0,T]$ (this being a probability law on $C([0,T];{{\mathbb R}})$), converges weakly to the probability law of $\bar{X}^j_t$. Refer to [@sznitman:91; @baladron-fasoli-etal:12b; @bossy-faugeras-etal:15] for more details. We briefly consider the following toy model to motivate our problem. Consider for the moment the system $$Y^j_t = y_{ini} + \frac{1}{N}\int_0^t \sum_{k=1}^N b_1(Y^j_s,Y^k_s)ds + W^j_t,$$ where $b_1$ has Lipschitz constant $\tilde{b}^{Lip}_N$. Define $$\bar{Y}^j_t = Y_{ini} +\frac{1}{N} \int_0^t \int_{{{\mathbb R}}}b_1(\bar{Y}^j_s,y)d\tilde{\mu}_s(y)ds + W^j_t,$$ where $\tilde{\mu}_s$ is the law of $\bar{Y}^j_s$. Assume that both of the above equations have strong solutions. Using Gronwall’s Inequality and the Cauchy-Schwartz inequality, [@sznitman:91] obtained a bound of the form $${\mathbb E}\left[ \sup_{t\in [0,T]}|Y^j_t - \bar{Y}^j_t|\right] \leq\exp\left( 2 T \tilde{b}^{Lip}_N\right)\sup_{s\in [0,T]}{\mathbb E}\left[b_1(\bar{Y}^j_s,\bar{Y}^k_s)^2\right]^{\frac{1}{2}}.$$ It is clear from the above that $\bar{Y}^j_t$ is a good approximation to $Y^j_t$ when $NT\tilde{b}^{Lip}_N \ll 1$. It is also clear that as $T\to \infty$, this bound becomes very poor, particularly due to the exponentiation. In much modeling of interacting diffusions, such as neuroscience, it is difficult to assume that $T$ is small: indeed, often it is difficult to properly model the ‘start’ of a system. It is therefore desirable to obtain convergence results which are uniform in time. This is the focus of this paper.
For $x,y\in {{\mathbb R}}$, let $$h(x,y) = g(x)g(y)f(x-y),$$ for some functions $f\geq 0$ and $g\geq 1$ described further below. We expect (but do not require) $f$ to be of the form $f(z) = f_{const} z^{2k}$ where $k$ is a positive integer. $f$ modulates the rate of convergence for when $X^j_t$ is ‘close’ to $\bar{X}^j_t$. $g\geq 1$ is a weight function which modulates the behavior for when $|X^j_t|$ or $|\bar{X}^j_t|$ asymptote to $\infty$. If $h$ is a metric, then this result guarantees that the Wasserstein Distance (with respect to $h$) between the laws of $X^j_t$ and $\bar{X}^j_t$ converges to zero as $N\to\infty$, with a rate which is uniform in $t$. As a consequence of Theorem \[theorem major result\], and since $h$ is a metric, the result guarantees that the joint law of any finite set of neurons(or particles in the general case) converges to a tensor product of iid processes, each with law given by the SDE in (\[eqn limit system\]). It is easily verified as explained in Corollary \[corollary1\]. To the best of our knowledge, the first work on uniform propagation of chaos was [@moral-miclo:00] when approximating Feynman Kac Formula for non linear filtering. Other authors applied Log-Sobolev inequalities and concentration inequalities [@malrieu:01; @carrillo-mccann-etal:03; @veretennikov:06; @cattiaux-guillin-etal:08; @moral-rio:11; @bolley-gentil-etal:13; @moral-tugaut:14]. Most of the previously cited works assume that the interaction term is of the form $b_1(x,y) = \nabla F(x-y)$ and the local term is of the form $\hat{b}_0 = \nabla V$ for some $F,V$ satisfying certain convexity properties. This work is essentially a generalization of [@veretennikov:06].
We are motivated in particular by the application of these models to neuroscience (see for instance [@wilson-cowan:72; @hansel-sompolinsky:96; @gerstner-kistler:02b; @deco-jirsa-etal:08; @destexhe-sejnowski:09; @coombes:10; @touboul-ermentrout:11; @bressloff:12; @touboul:14]) although we expect in fact that these results are applicable in other domains such as agent-based modeling in finance, insect-swarms, granular models and various other applications of statistical physics. We have been able to weaken some of the requirements in [@veretennikov:06] and other works, so that the results may be applied in arguably more biologically realistic contexts. We do not assume that the interaction term $b_1(x,y)$ is a function of $x-y$, as in many of the previously cited works. The uniform propagation of chaos result is essentially due to the stabilizing effect of the internal dynamics ($b_0$ term) outweighing the destabilizing effect of the inputs from other neurons ($b_1$ term) and the noise ($b_2$ term). In [@veretennikov:06], it was assumed that the gradient of $b_0$ is always negative, and is at least linear. However it is not clear (at least in the context of neuroscience) that the decay resulting from the internal dynamics term is always this strong for large values of $|X^j_t|$. Neuroscientific models are only experimentally validated over a finite parameter range, and therefore it is not certain how to model the dynamics for when the state variable $X^j_t$ is very large or small. Our more abstract setup does not require the decay to be linear (as in for example the Wilson-Cowan model) for large values of $|X^j_t|$: indeed the decay could be sub-linear or super-linear; all that is required is that in the asymptotic limit the decay from $b_0$ dominates the destabilizing effects of $b_1$ and $b_2$. Another improvement of our model over [@veretennikov:06] is that we consider multiplicative noise (i.e. $b_2 \neq 1$). This is more realistic because we expect the noise term $ \int_0^t b_2(X^j_s)dW^j_s$ to be of decreasing influence as $|X^j_t|$ gets large. This is because one would expect in general that the system is less responsive to the noise when its activity is greatly elevated, since the system should be stable. The point is that experimentalists should have some liberty in fitting our model to experimental data; all that is required is that in the asymptotic limit the decay from $b_0$ dominates the destabilizing effects of $b_1$ and $b_2$.
We do not delve into the details of existence and uniqueness of solutions, and so throughout we assume that
\[assumption one\] There exist unique strong solutions to and .
Our major result is the following uniform convergence property.
\[theorem major result\] If Assumption \[assumption one\] and the assumptions in Section \[Sect assumptions\] hold, then there exists a constant $K$ such that for all $t\geq 0$ $${\mathbb E}\left[ h(X^j_t,\bar{X}^j_t)\right] \leq K N^{-\frac{a}{q(a-1)}},$$ for integers $a>1$ and $q \geq 1$.
It is easy to show existence and uniqueness if, for example, $b_0,b_1$ and $b_2$ are each globally Lipschitz. In the case of existence and uniqueness of , [@mao:08 Theorem 3.6] provides a useful general criterion. Refer to [@bossy-faugeras-etal:15] for a discussion of how to treat the existence and uniqueness of in a more general case.
Our paper is structured as follows. In Section \[Sect assumptions\] we outline the assumptions of our model, in Section \[Section Proof\] we prove Theorem \[theorem major result\] and in Section \[application\] we outline an example of a system satisfying the assumptions of Section \[Sect assumptions\].
Assumptions {#Sect assumptions}
===========
The requirements outlined below might seem quite tedious. However in the next section we consider an application which allows us to simplify many of them. We split ${{\mathbb R}}$ into two domains ${\mathcal{D}}$ and ${\mathcal{D}}^c$. ${\mathcal{D}}\subset
{{\mathbb R}}$ is a closed compact interval which we expect the system to be most of the time. Over ${\mathcal{D}}$, we require that the natural convexity of $b_0$ dominates that of $b_1$ and $b_2$. In ${\mathcal{D}}^c$ we require bounds for when the absolute values of the variables are asymptotically large.
Assume that $f \geq 0$, that $f(z) = f(-z)$, $g\geq 1$ and $f(z) = 0$ if and only if $z=0$. Suppose that for $z\in{\mathcal{D}}$, $g(z) = 1$ and clearly $g'(z) = 0$. Write $\hat{b}_0 = - b_0$. Assume that for all $x,y\in {{\mathbb R}}$, $$f'(x-y)\left(\hat{b}_0(x) - \hat{b}_0(y)\right) -\frac{1}{2}f''(x-y)(b_2(x) - b_2(y))^2\geq 0.\label{eq I1 positive}$$ Assume that for all $x,y\in {\mathcal{D}}$, there exists a constant $c_0 > 0$ such that $$f'(x-y)\left(\hat{b}_0(x) - \hat{b}_0(y)\right) -\frac{1}{2}f''(x-y)(b_2(x) - b_2(y))^2\geq c_0 f(x-y).\label{eq I1 uniform bound}$$ Assume that for all $z\in {{\mathbb R}}$, there is some $a>1$ such that $$f'(z)^a \leq f(z).\label{eq f differential bound}$$ Assume that there exists a constant $a_0\in {{\mathbb R}}$ such that for all $x\notin {\mathcal{D}}$, $$\left| \frac{g'(x)}{g(x)}b_2(x)\right| \leq a_0.\label{eq a0 bound}$$ Assume that for $x\notin {\mathcal{D}}$, for all probability measures $\gamma$ and all $y\in{{\mathbb R}}$, $$\frac{g'(x)}{g(x)}\left(\hat{b}_0(x)-\bar{b}_1(x,\gamma)-\frac{f'(x-y)}{f(x-y)}b_2(x)(b_2(x)-b_2(y))-\frac{1}{2}a_0 b_2(x)\right) \geq c_0.\label{large x g bound}$$ $$\frac{g''(x)}{g(x)}b_2(x)^2\leq c_2.\label{eq c2 bound}$$ Assume that there exist constants $\breve{c}_1,\grave{c}_1\in {{\mathbb R}}$ such that for all $x,y_1,y_2 \in {{\mathbb R}}$, $$\begin{aligned}
g(x)^{\frac{2(a-1)}{a}}(b_1(x,y_1)-b_1(x,y_2)) \leq & \breve{c}_1 g(y_1)^{\frac{a-1}{a}} g(y_2)^{\frac{a-1}{a}}
f(y_1-y_2)^{\frac{a-1}{a}}.\label{eq c1 bound 1}\\
\left| b_1(y_1,x) - b_1(y_2,x)\right| \leq& \grave{c}_1f(y_1-y_2)^{\frac{a-1}{a}}.\label{eq c1 bound 2}\end{aligned}$$ Assumption might seems a little strange. If $g(x) \to \infty$ as $x\to \infty$, in the context of neuroscience it would mean that the relative influence of neuron $k$ on neuron $j$ decreases as $X^j_t \to \infty$. This seems biologically reasonable. We assume that $c_0$ dominates the other terms, i.e. $$c := c_0 - \breve{c}_1 -\grave{c}_1 - c_2 > 0.\label{eq c definition}$$ For some positive integer $q>2$, we require that there exists a constant $C_2$ such that for all $s > 0$, $$\begin{aligned}
{\mathbb E}\left[ \bar{b}_1(\bar{X}_s,\bar{\mu}_s)^q\right] \leq C_2.\label{eq b1 q bound}\end{aligned}$$ Assume that there exists a constant $C_1 > 0$ such that for all $s>0$ and for all $N$, $${\mathbb E}\left[ g(\bar{X}_s)^{\frac{2(a-1)q}{aq-a-q}}\right],{\mathbb E}\left[ g(X_s)^{\frac{2(a-1)q}{aq-a-q}}\right] \leq C_1.\label{eq g bound}$$
Proof of Theorem \[theorem major result\] {#Section Proof}
=========================================
We now outline the proof of Theorem \[theorem major result\].
We will prove that there exists a constant $C$ such that $$\label{eq final C }
{\mathbb E}\left[ h(X^j_t,\bar{X}^j_t)\right] \leq \int_0^t -c {\mathbb E}\left[ h(X^j_s,\bar{X}^j_s)\right] + CN^{-\frac{1}{q}}{\mathbb E}\left[ h(X^j_s,\bar{X}^j_s)\right]^{\frac{1}{a}}ds.$$ The theorem will then follow from the application of Lemma \[lemma ut\] to the above result.
We observe using Ito’s Lemma that $$\label{Ito}
h(X^j_t,\bar{X}^j_t) = I_1 + I_2' + I_2'' + I_3 + I_4 + I_5 + \int_0^t \left( \frac{\partial h}{\partial x}b_2(X^j_s) + \frac{\partial h}{\partial y}b_2(\bar{X}^j_s)\right)dW^j_s.$$ The $I_j$ are $$\begin{aligned}
I_1 = &\int_0^t -g(X^j_s)g(\bar{X}^j_s)f'(X^j_s - \bar{X}^j_s)\left(\hat{b}_0(X^j_s) - \hat{b}_0(\bar{X}^j_s)\right)\nonumber\\
&+ \frac{1}{2} g(X^j_s)g(\bar{X}^j_s)f''\left(X^j_s - \bar{X}^j_s\right)(b_2(X^j_s) - b_2(\bar{X}^j_s))^2 ds \label{eq I1}\\
I'_2 = &\int_0^t f(X^j_s - \bar{X}^j_s)g(X^j_s)g'(\bar{X}^j_s)(b_1(\bar{X}^j_s,\bar{\mu}_s) -\hat{b}_0(\bar{X}^j_s))\nonumber\\ &
-f'(X^j_s - \bar{X}^j_s)g(X^j_s)g'(\bar{X}^j_s)b_2(\bar{X}^j_s)(b_2(\bar{X}^j_s) - b_2(X_s^j))ds\\
I_2'' =&\int_0^t f(X^j_s - \bar{X}^j_s)g'(X^j_s)g(\bar{X}^j_s)\left(\bar{b}_1(X^j_s,\hat{\mu}_s) - \hat{b}_0(X^j_s)\right)\nonumber \\
&+f'(X^j_s - \bar{X}^j_s)g'(X^j_s)g(\bar{X}^j_s)b_2(X^j_s)(b_2(X^j_s) - b_2(\bar{X}_s^j)) ds\label{eq I2} \\
I_3 =&\int_0^t g(X^j_s)g(\bar{X}^j_s)f'(X^j_s - \bar{X}^j_s)\left(\bar{b}_1(X^j_s,\hat{\mu}) - \bar{b}_1(\bar{X}^j_s,\bar{\mu}_s)\right)ds \label{eq I3}\\
I_4 = &\int_0^t f(X^j_s - \bar{X}^j_s)g'(X^j_s)g'(\bar{X}^j_s)b_2(X^j_s)b_2(\bar{X}^j_s)ds \label{eq I4}\\
I_5 =&\frac{1}{2}\int_0^t f(X^j_s - \bar{X}^j_s)\left( g''(X^j_s)g(\bar{X}^j_s) b_2(X^j_s)^2 + g''(\bar{X}^j_s)g(X^j_s) b_2(\bar{X}^j_s)^2\right) ds \label{eq I5}\end{aligned}$$ We start by establishing that $$\label{I1 bound}
E\left[I_1+I'_2+I''_2 + I_4 \right] \leq -c_0 \int_0^t {\mathbb E}\left[ h(X^j_s,\bar{X}^j_s)\right]ds.$$ We prove that the sum of the integrands of $I_1$,$I_2'$,$I_2''$ and $I_4$ is less than or equal to $-c_0 h(X^j_s,\bar{X}^j_s)$. Suppose firstly that $X^j_s,\bar{X}^j_s \in {\mathcal{D}}$. Then the integrands of $I_2',I_2''$ and $I_4$ are all zero. Furthermore, using , the integrand of $I_1$ satisfies the bound $$\begin{gathered}
-g(X^j_s)g(\bar{X}^j_s)f'(X^j_s - \bar{X}^j_s)\left(\hat{b}_0(X^j_s) - \hat{b}_0(\bar{X}^j_s)\right)\\
+ \frac{1}{2} g(X^j_s)g(\bar{X}^j_s)f''\left(X^j_s - \bar{X}^j_s\right)(b_2(X^j_s) - b_2(\bar{X}^j_s))^2
\leq -c_0 h(X^j_s,\bar{X}^j_s).\end{gathered}$$ Now suppose that $\bar{X}^j_s\notin{\mathcal{D}}$. The integrand of $I_1$ is less than or equal to zero because of . Through , $$\begin{gathered}
g'(\bar{X}^j_s)\bigg[ \big(\hat{b}_0(\bar{X}^j_s) - \bar{b}_1(\bar{X}^j_s,\hat{\mu}_s)\big)f(\bar{X}^j_s - X^j_s) - f'(\bar{X}^j_s - X^j_s)b_2(\bar{X}^j_s)\big(b_2(\bar{X}^j_s) - b_2(X^j_s)\big) \\- \frac{a_0}{2}b_2(\bar{X}^j_s)f(\bar{X}^j_s - X^j_s)\bigg] \geq g(\bar{X}^j_s)c_0 f(\bar{X}^j_s - X^j_s).\end{gathered}$$ Since $f(-z) = f(z)$ and $f'(-z) = -f(z)$, upon multiplying the above identity by $-g(X^j_s)$, $$\begin{gathered}
g(X^j_s)g'(\bar{X}^j_s)\left[f(X^j_s - \bar{X}^j_s)(\bar{b}_1(\bar{X}^j_s,\hat{\mu}_s) -\hat{b}_0(\bar{X}^j_s)) +\right.\\ \left.f'(X^j_s - \bar{X}^j_s)b_2(\bar{X}^j_s)(b_2(X^j_s) - b_2(\bar{X}_s^j))\right] +\frac{1}{2} f(X^j_s - \bar{X}^j_s)g'(X^j_s)g'(\bar{X}^j_s)b_2(X^j_s)b_2(\bar{X}^j_s)\\
\leq -c_0 h(X^j_s,\bar{X}^j_s)-\frac{1}{2}a_0g(X^j_s)g'(\bar{X}^j_s)b_2(\bar{X}^j_s) f(X^j_s - \bar{X}^j_s)+\\ \frac{1}{2} f(X^j_s - \bar{X}^j_s)g'(X^j_s)g'(\bar{X}^j_s)b_2(X^j_s)b_2(\bar{X}^j_s)
\leq - c_0 h(X^j_s,\bar{X}^j_s),\label{int temp 3}\end{gathered}$$ since by , $$\frac{1}{2}a_0g(X^j_s)g'(\bar{X}^j_s)b_2(\bar{X}^j_s) f(X^j_s - \bar{X}^j_s)-\frac{1}{2} f(X^j_s - \bar{X}^j_s)g'(X^j_s)g'(\bar{X}^j_s)b_2(X^j_s)b_2(\bar{X}^j_s) \geq 0.$$
Notice that the left hand side of is the sum of the integrand of $I_2'$ and half of the integrand of $I_4$. Similarly if $X^j_s\notin{\mathcal{D}}$, the integrand of $I_1$ is less than or equal to zero, and through and , $$\begin{gathered}
g'(X^j_s)g(\bar{X}^j_s)\left[f(X^j_s - \bar{X}^j_s)(\bar{b}_1(X^j_s,\hat{\mu}_s) -\hat{b}_0(X^j_s)) +\right.\\ \left.f'(X^j_s - \bar{X}^j_s)b_2(X^j_s)(b_2(X^j_s) - b_2(\bar{X}_s^j))\right] +\frac{1}{2} f(X^j_s - \bar{X}^j_s)g'(X^j_s)g'(\bar{X}^j_s)b_2(X^j_s)b_2(\bar{X}^j_s)\\
\leq -c_0 h(X^j_s,\bar{X}^j_s)-\frac{1}{2}a_0g'(X^j_s)g(\bar{X}^j_s)b_2(X^j_s) f(X^j_s - \bar{X}^j_s)+\\ \frac{1}{2} f(X^j_s - \bar{X}^j_s)g'(X^j_s)g'(\bar{X}^j_s)b_2(X^j_s)b_2(\bar{X}^j_s)
\leq - c_0 h(X^j_s,\bar{X}^j_s).\label{int temp 4}\end{gathered}$$ The left hand side of the above is equal to the integrand of $I_2''$ and half of the integrand of $I_4$. Observe that if $\bar{X}^j_s \in {\mathcal{D}}$, then the left hand side of is zero because $g'$ is zero in ${\mathcal{D}}$. Similarly if $X^j_s \in {\mathcal{D}}$, then the left hand side of is zero because $g'$ is zero in ${\mathcal{D}}$. These considerations yield the bound .
It follows from that $${\mathbb E}\left[ I_5\right] \leq c_2 \int_0^t {\mathbb E}\left[ h(X^j_s,\bar{X}^j_s)\right]ds.$$
We finish by bounding the $I_3$ term. Suppose that $g(X^j_s) \geq g(\bar{X}^j_s)$. Then using , - and the triangular inequality $$\begin{gathered}
\left| f'(X^j_s - \bar{X}^j_s)g(X^j_s)g(\bar{X}^j_s)\left(b_1(X^j_s,X^k_s) - b_1(\bar{X}^j_s,\bar{X}^k_s)\right)\right| \leq\\
\left| f'(X^j_s - \bar{X}^j_s)\right| g(X^j_s)g(\bar{X}^j_s)\left|b_1(X^j_s,\bar{X}^k_s)- b_1(\bar{X}^j_s,\bar{X}^k_s)\right|\\+
\left| f'(X^j_s - \bar{X}^j_s)\right| g(X^j_s)g(\bar{X}^j_s)\left| b_1(X^j_s,X^k_s) - b_1(X^j_s,\bar{X}^k_s)\right|\\
\leq \grave{c}_1 \left| f'(X^j_s - \bar{X}^j_s)\right| g(X^j_s)g(\bar{X}^j_s)f(X^j_s - \bar{X}^j_s)^{\frac{a-1}{a}}\\+\left| f'(X^j_s - \bar{X}^j_s)\right| g(X^j_s)^{\frac{1}{a}}g(\bar{X}^j_s)^{\frac{1}{a}}g(X^j_s)^{2-\frac{2}{a}}\left| b_1(X^j_s,X^k_s) - b_1(X^j_s,\bar{X}^k_s)\right|\\
\leq \grave{c}_1 g(X^j_s)g(\bar{X}^j_s)f(X^j_s - \bar{X}^j_s) +\\ g(X^j_s)^{\frac{1}{a}}g(\bar{X}^j_S)^{\frac{1}{a}}f(X^j_s-\bar{X}^j_s)^{\frac{1}{a}} g(X^j_s)^{2-\frac{2}{a}}\left| b_1(X^j_s,X^k_s) - b_1(X^j_s,\bar{X}^k_s)\right| \\
\leq \grave{c}_1 g(X^j_s)g(\bar{X}^j_s)f(X^j_s - \bar{X}^j_s) +\\ \breve{c}_1f(X^j_s - \bar{X}^j_s)^{\frac{1}{a}}g(X^j_s)^{\frac{1}{a}}g(\bar{X}^j_s)^{\frac{1}{a}}g(X^k_s)^{\frac{a-1}{a}}g(\bar{X}^k_s)^{\frac{a-1}{a}}f(X^k_s - \bar{X}^k_s)^{\frac{a-1}{a}}.\end{gathered}$$ We obtain the same inequality when $g(\bar{X}^j_s) \geq g(X^j_s)$. That is, $$\begin{gathered}
\left| f'(X^j_s - \bar{X}^j_s)g(X^j_s)g(\bar{X}^j_s)\left(b_1(X^j_s,X^k_s) - b_1(\bar{X}^j_s,\bar{X}^k_s)\right)\right| \leq\\
\left| f'(X^j_s - \bar{X}^j_s)\right| g(X^j_s)g(\bar{X}^j_s)\left|b_1(\bar{X}^j_s,X^k_s)- b_1(\bar{X}^j_s,\bar{X}^k_s)\right|\\+
\left| f'(X^j_s - \bar{X}^j_s)\right| g(X^j_s)g(\bar{X}^j_s)\left| b_1(X^j_s,X^k_s) - b_1(\bar{X}^j_s,X^k_s)\right|\\
\leq \grave{c}_1 g(X^j_s)g(\bar{X}^j_s)f(X^j_s - \bar{X}^j_s) +\\ \breve{c}_1f(X^j_s - \bar{X}^j_s)^{\frac{1}{a}}g(X^j_s)^{\frac{1}{a}}g(\bar{X}^j_s)^{\frac{1}{a}}g(X^k_s)^{\frac{a-1}{a}}g(\bar{X}^k_s)^{\frac{a-1}{a}}f(X^k_s - \bar{X}^k_s)^{\frac{a-1}{a}}.\end{gathered}$$ Applying Holder’s Inequality to the above, $$\begin{gathered}
{\mathbb E}\left[f'(X^j_s - \bar{X}^j_s)g(X^j_s)g(\bar{X}^j_s)\left(b_1(X^j_s,X^k_s) - b_1(\bar{X}^j_s,\bar{X}^k_s)\right)\right] \leq\\
\grave{c}_1 {\mathbb E}\left[g(X^j_s)g(\bar{X}^j_s)f(X^j_s - \bar{X}^j_s)\right] +\\ \breve{c}_1 {\mathbb E}\left[g(X^j_s)g(\bar{X}^j_s)f(X^j_s-\bar{X}^j_s)\right]^{\frac{1}{a}}{\mathbb E}\left[g(X^k_s)g(\bar{X}^k_s)f(X^k_s - \bar{X}^k_s)\right]^{\frac{a-1}{a}} \\
=(\grave{c}_1 + \breve{c}_1){\mathbb E}\left[g(X^j_s)g(\bar{X}^j_s)f(X^j_s - \bar{X}^j_s)\right] \end{gathered}$$ We use Holder’s Inequality to see that $$\begin{gathered}
{\mathbb E}\left[ f'(X^j_s - \bar{X}^j_s)g(X^j_s)g(\bar{X}^j_s)\left(\sum_{k=1}^N b_1(\bar{X}^j_s,\bar{X}^k_s) - \bar{b}_1(\bar{X}^j_s,\bar{\mu}_s)\right)\right] \leq \\
{\mathbb E}\left[ f'(X^j_s - \bar{X}^j_s)^a g(X^j_s)g(\bar{X}^j_s)\right]^{\frac{1}{a}} {\mathbb E}\left[ g(\bar{X}_s^j)^{\frac{a-1}{a}\times\frac{2aq}{aq-a-q}}\right]^{\frac{aq-a-q}{2aq}}\times\\ {\mathbb E}\left[ g(X_s^j)^{\frac{a-1}{a}\times\frac{2aq}{aq-a-q}}\right]^{\frac{aq-a-q}{2aq}}\times {\mathbb E}\left[\left(\sum_{k=1}^N b_1(\bar{X}^j_s,\bar{X}^k_s) - \bar{b}_1(\bar{X}^j_s,\bar{\mu}_s)\right)^q\right]^{\frac{1}{q}}.\end{gathered}$$ where $q$ is the integer that appears in assumption .\
By Assumption , $${\mathbb E}\left[ g(\bar{X}_s^j)^{\frac{2(a-1)q}{aq-a-q}}\right]^{\frac{aq-a-q}{aq}}\times {\mathbb E}\left[ g(X_s^j)^{\frac{2(a-1)q}{aq-a-q}}\right]^{\frac{aq-a-q}{aq}}$$ is uniformly bounded for all $s$. Furthermore through Assumption and Lemma \[lemma bound polynomial\], ${\mathbb E}\left[\left(\sum_{k=1}^N b_1(\bar{X}^j_s,\bar{X}^k_s) - \bar{b}_1(\bar{X}^j_s,\bar{\mu}_s)
\right)^q\right]^{\frac{1}{q}}$ is bounded by $\mathfrak{C} N^{\frac{q-1}{q}}$. Finally, using Assumption , $${\mathbb E}\left[ f'(X^j_s - \bar{X}^j_s)^a g(X^j_s)g(\bar{X}^j_s)\right]^{\frac{1}{a}} \leq {\mathbb E}\left[h(X^j_s,\bar{X}^j_s)
\right]^{\frac{1}{a}}.$$ We thus find that for some constant $C$, $${\mathbb E}\left[ I_3\right] \leq C\int_0^t N^{-\frac{1}{q}}{\mathbb E}\left[h(X^j_s,\bar{X}^j_s)\right]^{\frac{1}{a}}ds.$$ In summary, noting the assumption , we now have all the ingredients for .
\[corollary1\] Let $l \in \mathbb{N}^{*}$ and fix $l$ neurons $(i_{1},...,i_{l})\in \mathbb{N}^{*}$. Under the assumptions of Theorem 1, the law of $(X^{i_{1}}_{t},...,X_{t}^{i_{l}} )$, converges toward $\mu_{t}^{\otimes l}$ for all $t\geq 0$.
$${\mathbb E}\left[ \left|(X^{i_{1}}_{t},...,X^{i_{l}}_{t})-(\bar{X}^{i_{1}}_{t},...,\bar{X}^{i_{l}}_{t})\right|^2\right]
\leq \sum^{l}_{k=1} {\mathbb E}\left[ \left|X^{i_{k}}_{t}-\bar{X}^{i_{k}}_{t}\right|^2\right]
\leq l K N^{-\frac{a}{q(a-1)}},$$ Hence $\forall t \geq 0$ the law of $( X^{i_{1}}_t,...,X^{i_{l}}_t ) $ converges when $N$ tends to infinity to the law of $ (\bar{X}^{i_{1}}_t,...,\bar{X}^{i_{l}}_t)$ , whose law is equal to $\mu_{t}^{\otimes l}$ by definition.
We present now the lemmas used in the proof of Theorem \[theorem major result\].
\[lemma bound polynomial\] Suppose that $(e^j)_{j=1}^\infty$ are independent identically-distributed ${{\mathbb R}}$-valued random variables such that ${\mathbb E}\left[ (e^j)^q\right] < \infty$ and ${\mathbb E}[e^j] = 0$. Then there exists a constant $\mathfrak{C}$ such that for all $N$ $${\mathbb E}\left[ \left(\sum_{j=1}^N e^j\right)^q\right] < \mathfrak{C} N^{q-1}.$$
Consider the binomial expansion of $ \left(\sum_{j=1}^N e^j\right)^q$. There are $N^q$ terms in total. The expectation of at least $N\times (N-1)\times (N-2)\times \ldots (N-q+1)$ of these must be zero, as the constituent factors are all independent. Let $(j_i)_{i=1}^q$, $1\leq j_i \leq N$, be an arbitrary set of indices. Then through Holder’s Inequality, $${\mathbb E}\left[ \prod_{p=1}^q e^{j_p}\right] \leq {\mathbb E}\left[ (e^1)^q\right].$$ Thus $$\begin{gathered}
{\mathbb E}\left[ \left(\sum_{j=1}^N e^j\right)^q\right] \leq\\ {\mathbb E}\left[ (e^1)^q\right]\times\left( N^q - N\times (N-1)\times (N-2)\times \ldots (N-q+1)\right) \\
\leq {\mathbb E}\left[ (e^1)^q\right]\times \left( N^q - (N-q+1)^q\right)\\
= {\mathbb E}\left[ (e^1)^q\right]N^{q-1}\left( N - N\left( 1 - \frac{q-1}{N}\right)^{q-1}\right)
\leq {\mathbb E}\left[ (e^1)^q\right]N^{q-1}(q-1)(q-2).\end{gathered}$$
The following lemma is an easy generalization of a result in [@veretennikov:06].
\[lemma ut\] Suppose that $u$ is continuous and satisfies, for some constants $\mathcal{C},c > 0$ and positive integer $a > 1$, for all $t< T$, $$u(T) - u(t) \leq \int_t^T -cu(s)+\mathcal{C}u(s)^{\frac{1}{a}}ds.$$ Furthermore $u(0) = 0$. Then for all $t\geq 0$ $$u(t) \leq \left(\frac{\mathcal{C}}{c}\right)^{\frac{a}{a-1}}.$$
It may be seen that $u$ is differentiable, with the derivative satisfying $$\dot{u}(t) \leq - c u(t) + \mathcal{C}u(t)^{\frac{1}{a}}.$$ Let $v(t) = u(t)\exp(ct)$. Then $$\dot{v}(t) \leq \mathcal{C}v(t)^{\frac{1}{a}}v\exp\left( \frac{(a-1)ct}{a}\right).$$ If $v(t)=0$ then there is nothing to show. Thus we may assume that for all $t>0$, $v(t) > 0$. Hence $$\dot{v}(t) v(t)^{-\frac{1}{a}} \leq \mathcal{C}\exp\left( \frac{ct(a-1)}{a}\right).$$ Upon integration, $$\frac{a}{a-1} v(t)^{\frac{a-1}{a}} \leq \frac{a}{a-1}\frac{\mathcal{C}}{c}\left( \exp\left(\frac{ct(a-1)}{a}\right) - 1 \right)\leq\frac{a}{a-1}\frac{\mathcal{C}}{c}\exp\left(\frac{ct(a-1)}{a}\right).$$ Thus $$v(t) \leq \left(\frac{\mathcal{C}}{c}\right)^{\frac{a}{a-1}}\exp(ct).$$
Application
===========
In this section we are going to provide an example of a system satisfying the requirements of Section \[Sect assumptions\], so that the result of Theorem \[theorem major result\] will apply. We start by defining the following functions.\
For all $x\in {{\mathbb R}}$, let $f(x):=\frac{1}{4}x^2.$ Let ${\mathcal{D}}=[-A,A]$ for some $A\gg 0$.We take $a=2$ and $q=3$. Define the sigmoid function $S(x):=\frac{1}{1+exp(-x)}$, it is clear that $S$ is of class $C^\infty$ , $0<S(x)<1$ and its derivative is bounded and positive. Using this, we define $$g(x)=\left\{\begin{array}{lll}
1 \qquad\mbox{if}\qquad x\in{\mathcal{D}}\\
S(-A-x)+\frac{1}{2} \qquad\mbox{if}\qquad x<-A \\
S(-A+x)+\frac{1}{2} \qquad\mbox{if}\qquad x>A
\end{array}\right.$$
The function $g$ is continuous on ${{\mathbb R}}$, $1\leq g(x) \leq \frac{3}{2}$, its derivative $g'$ is bounded, negative for $x<-A$ and positive for $x>A$.
We consider a population of $N$ neurons, with evolution equation $$dV^j_t = (-\frac{1}{\tau}V^j_t + \frac{1}{N}\sum_{k=1}^N J(V_t^j,V_t^k)S(V_t^k)+I(t))dt +
\sigma dW^j_t, \label{eqn example1}$$ where $V_t^j$ is the membrane potential of neuron $j$, $I(t)$ is the deterministic input current. $J(V_t^j,V_t^k)$ denotes the synaptic weight from neuron $k$ to neuron $j$. The function $J: {{\mathbb R}}\times{{\mathbb R}}\rightarrow{{\mathbb R}}$ is assumed to be of class $C^1$ in both variables, such that both it and its derivative are bounded.\
The above assumptions are sufficient for the requirements of Section \[Sect assumptions\] to be satisfied. In particular, using the Mean Value Theorem, one can easily verify the bounds \[eq c1 bound 1\] and \[eq c1 bound 2\]. Morever, one can refer to [@bolley-gentil-etal:13] and verify that assumption \[assumption one\] is satisfied. It then follows, using Theorem \[theorem major result\], that for all $t>0$ $${\mathbb E}\left[ (V^j_t-\bar{V}^j_t)^2\right] \leq 4K N^{-\frac{2}{3}},$$ In other words, the law of an individual neuron converges to its limit as $N\to \infty$ at the time-uniform rate given above.
[12]{} J. Baladron, D. Fasoli, O. Faugeras, and J. Touboul, [*Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons*]{} , The journal Of Mathematical Neuroscience, 2 (2012). F. Bolley, I. Gentil, and A. Guillin, [*Uniform convergence to equilibrium for granular media*]{}, Archive for Rational Mechanics and Analysis, 208 (2013), pp. 429-445. M. Bossy, O. Faugeras, and D. Talay, [*Clarification and complement to “mean-field description and propagation of chaos in networks of hodgkin-huxley and fitzhugh-nagumo neurons*]{} , tech. report, HAL INRIA, 2015. P.C. Bressloff, [*Spatiotemporal dynamics of continuum neural fields*]{}, Journal of Physics A: Mathematical and Theoretical, 45 (2012). J. A. Carillo, R. J. Mccann, and C. Villani,[*Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates*]{}, Revista Matematica Iberoamericana, 19 (2003), pp. 971-1018. P. Cattiaux, A. Guillin, and F. Malrieu, [*Probabilistic approach for granular media equa- tions in the non-uniformly convex case*]{}, Probability Theory and Related Fields, (2008). S. Coombes, [*Large-scale neural dynamics: Simple and complex*]{}, Neurolmage, 52 (2010),pp. 731-739. G. DECO,V. K. Jirsa, P. A. Robinson, M. Breakspear, and K. Friston, [*The dynamic brain: From spiking neurons to neuralmasses and cortical fields*]{}, PloS Comput. Biol., 4 (2008). A. Destexhe and T. J. Sejnowski,[*The Wilson-Cowan model*]{}, 36 years later, Biological Cybernetics, 101 (2009), pp. 1-2. W. Gerstner and W. Kistler, [*Spiking Neuron Models*]{}, Cambridge University Press, 2002. D. Hansel and H. Sompolinsky, [*Chaos and synchrony in a model of a hypercolumn in visual cortex*]{}, Journal of Computational Neuroscience, 3 (1996), pp. 7-34. F. Malrieu, [*Logarithmic sobolev inequalities for some nonlinear pde’s*]{}, Stochastic Processes and their Applications, 95 (2001), pp. 109-132. X. Mao, [*Stochastic differential equations and applications*]{}, Horwood, 2008, 2nd Edition. P. Del Moral and L. Miclo, [*Branching and interacting particle systems approximations of feynman-kac formulae with applications to non-linear filtering,in Séminaire de Probabilités XXXIV*]{} , J. Azéma, M. Emery, M. Ledoux, and M. Yor, eds., vol. 1729, Springer- Verlag Berlin, 2000. P. Del Moral and E. Rio, [*Concentration inequalities for mean field particle models*]{} , Annals of Applied Probability, (2011). P. Del Moral and J. Tugaut, [*Uniform propagation of chaos for a class of inhomogeneous diffusions*]{}, tech. Report, HAL INRIA, 2014. A. Sznitman, [*Topics in propagation of chaos, in Ecole d’été de Probabilités de Saint-Flour XIX-1989, Donald Burkholder,Etienne Pardoux, and Alain-Sol Sznitman, eds*]{}, vol. 1464 of Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1991, pp. 165-251. 10.1007/BFb0085169. J. Touboul, [*the propagation of chaos in neural fields*]{}, The Annals of Applied Probability, 24 (2014). visual cortex, Journal of Computational Neuroscience, 3 (1996), pp. 7-34. J. Touboul and B. Ermentrout, [*Finite-size and correlation-induced effects in mean-field dynamics*]{}, J Comput Neurosci, 31 (2011), pp.453-484. A. YU VERETENNIKOV,[*On ergodic measures for mckean-vlasov stochastic equations*]{}, Monte-Carlo and Quasi-Monte-Carlo Methods, (2006), pp. 471-486. H.R. Wilson and J.D. Cowan,[*Excitatory and inhibitory interactions in localized polulations of model neurons*]{}, Biophys. J., 12 (1972),pp. 1-24.
[^1]: Contact: [email protected], [email protected], [email protected]
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abstract: |
Phase transition of strongly excited granular materials in 2D pipe is investigated numerically. By changing the ratio between width of the pipe and the height of the granular bed, we observe the transition between the 1 dimensional like state and the actual 2 dimensional state. Moreover, it is found that the character of the transition changes when the magnitude of dissipation passes through a critical value.\
PACS number(s):\
address: |
Department of Mathematical Sciences\
Osaka Prefecture University, Sakai 599-8531 Japan.
author:
- 'Akinori Awazu[^1]\'
title: 'Effective Dimension Transition of the Dynamics of Granular Materials in the Pipe.'
---
Granular materials exhibit some complex phenomena[@re1; @re2]. These complex phenomena strongly depend on not only the character of external forces but also that of boundary condition. When granular materials in slim pipes with fluid (air, water, and so on) flow down or are fluidized by inducing fluid from the bottom, slugging is observed[@tr9; @tr12] . In most of the investigations of such slugging in a slim pipe, the system’s effective dimension is regarded as 1[@tr10; @tr11; @tr12]. On the other hand, two or more dimensional phenomena, bubbling and channeling, occur when granular materials are in a wider vessel[@fh1]. The similar tendency is also observed for granular materials in a vibrating vessel[@con0; @con00; @con1; @con2]. When the width of the vessel is small, most of particles construct solid structure in bulk and only particles near side walls flow down to the bottom[@re1; @con00]. However, in a wide vessel, most of particles are strongly fluidized and construct multi-roll structure like Bernard convection[@con2] From these facts, the behaviors of granular materials are extremely different between that in a slim vessel and that in a wide vessel. Then, an important problems appear; what quantity determines ’slim’ (one dimensional system) or ’wide’ (two or three dimensional system) for a given system? The purpose of this paper is to make a clear view on this problem.
Here, we simulate following simplified situation[@setu]. The system consists of two dimensional particles of mass 1 and diameter $d$ in a two-dimensional box under uniform gravity. The width of the box is $a$, and the height of the box is infinite. We employ the following particle model which is one of the simplest model of granular materials. The equation of the motion of the $i$th particle is $$\ddot{{\bf x}_{i}}=-\sum_{j=1}^{N}\theta(d-|{\bf x}_{i}-{\bf x}_{j}|)\{{\bf \nabla}U(d-|{\bf x}_{i}-{\bf x}_{j}|)+\eta({\bf v}_{i}-{\bf v}_{j})\}+{\bf g}$$ $$U(d-|{\bf x}_{i}-{\bf x}_{j}|)=\frac{k}{2}(d-|{\bf x}_{i}-{\bf x}_{j}|)^{2}$$ here, $\theta$ is Heviside function, $N$ is the total number of particles, $k$ and $\eta$ are respectively the elastic constant and the viscosity coefficient[@keta], and ${\bf x}_{i}(x_{i},y_{i})$, ${\bf v}_{i}(v_{x_{i}}, v_{y{i}})$, and ${\bf g}=(0,-g)$ are, respectively, the position, the velocity and the gravity of $i$th particles. In this model, the effect of particles’ rotation is neglected. The system is driven by a simple energy source at the bottom of the box; a particle hitting the bottom with velocity $(v_{x}, v_{y})$ bounces back with the velocity $(v_{x}, V (V>0))$. We regard the bottom of box as $x$ axis ($y=0$), and $x=0$ as the center of box. The side walls of the box are put along $x=a/2$ and $x=-a/2$, and the viscosity which works between these walls and particles is zero. At the initial condition, we put particles bed with height $b$ on the bottom of box. We simulate this system with some combinations of parameters $(\eta, a, b)$, where $a$ and $b$ are enough large compared to the particle’s diameter $d$. The above equations are calculated with the Euler’s scheme. The time step $\delta t$ is set enough small such that $\delta x$, the displacement of the $i$th particle during $\delta t$, does not exceed a given value. In this paper, we set $(-g,V)$ so that the average height of the center of mass of the system $CM=<(\sum y_{i})/N>_{t}$ keeps enough large compared to $b$.
Figure 1 shows typical snapshots of the system for respectively, (a) $a<<b$ with $\eta>\eta_{*}$, (b) $a>>b$ with $\eta>\eta_{*}$, (c) $a<<b$ with $\eta<\eta_{*}$, and (d) $a>>b$ with $\eta<\eta_{*}$. Here, we fixed $b$, whereas $(-g,V)$ of (a) is same as that of (b), and $(-g,V)$ of (c) is same as that of (d). In cases of (a) and (c), only the particles distribution in the horizontal direction is symmetric, and the center of mass of this system moves a little only in the vertical direction. On the other hand, the particles distribution is non-uniform in vertical and horizontal direction, and one convection appears in cases of (b) and (d). In order to characterize the system, we introduce the order parameter $L(t)=\sum_{i}|x_{i}(t)v_{y{i}}(t)|/N$ which indicates the strength of the convection of the system. Figure 2 (a) and (b) are typical probability distributions of $L(t)$ which is given by one time series for respectively the cases of $a<<b$ and those of $a>>b$. In Fig.2 (a), the peak of the probability distribution of $L(t)$ appears at $L(t)=0$ which means there are no convection for the case $a<<b$. In these cases, the effective dimension of this system can be regarded as 1. We name such states as the ’1D state’. On the contrary the peak of the probability distribution of $L(t)$ appears at $L(t)>0$ in Fig.2 (b) which means a convection with finite magnitude appears in the system. In these cases, the system is actually 2 dimensional system. We name such states with a convention as the ’2D state’. Such a transition between the 1D state and the 2D state which depends on the relation between $a$ and $b$ is observed in a simple system.
In our simulation, particles are strongly excited by the energy source at the bottom of the box. Then, if we set a improper $g$ for a small $\eta$ or a small $b$, the height of particles diverge. In order to clear off such cases, we need to control $g$ for several $\eta$ and $b$. By the simulation, we found the fact that $CM$ have almost same values independent of $a$ with $a<<b$ for a fixed set of $g$, $\eta$, $b$ and $V$. Hereafter, we fix $V$ and determine the gravity $g$ for a given set $(\eta,b)$ independent of $a$. In this paper, we set $V=2.0$ and $g(\eta,b)$ with which $CM \sim 60d$ is realized for all cases of $a<<b$. When we set $V=3.0$ or $V=4.0$, we can get qualitatively same results as those of following discussions with $V=2.0$.
Before the discussion of the transition width at which the 1D-2D transition takes place or the character of the transition, we discuss the dependency of $\eta$ for the 1D states. Figure 3 (a) shows the time averaged packing fraction profile of the $y$ direction as the function of $y-y_{CM}$ for several $\eta$ with $a=9d$ and $b=30d$. Here, $y_{CM}$ is the $y$ component of center of masses of particles, and we define the packing fraction as followings. We divide the space by a lattice with the lattice constant $d$, and we define the number of centers of particle within each $d \times d$ square as the packing fraction in the square. The packing fraction of $y$ direction is given as the average of them through the $x$ direction for each $y$. For large $\eta$ ($\eta=0.6 , 1.0$), each profile includes a flat region with large packing fraction near the center of mass of this system. These flat regions mean the existence of the solid structure in which particles are almost completely packed. The length of this flat region decreases with decreasing $\eta$, and this length becomes $0$ for $\eta=\eta^{*}\sim 0.38$. If $\eta<\eta^{*}$ ($\eta=0.25 , 0.35$), each packing fraction profile includes no flat region, and maximum packing fraction is smaller than that of $\eta>\eta^{*}$. It means that no solid structures are created for such small $\eta$. Thus, a transition between a state which includes a solid structure and the other state which include no solid structure occurs at the critical value $\eta=\eta^{*}$. Similar results are obtained in following two cases, $b=20d$ and $b=40d$. Now we introduce following no dimensional values: $a'=a/d$, $b'=b/d$, and $e'=1-e$ where $e=exp(\pi
\eta/(k-\eta^{2})^{\frac{1}{2}}))$, and the length of flat regions $h(e')=h'(e')d$. ($h'(e')$ has no dimensions.) Here, $e$ indicates the coefficient of restitution for head-on collisions between two particles[@con0]. The relation between two rescaled values, $e' b'$ and $\frac{h'(e')}{b'^{2}}$, is obtained as shown in Fig.4(a). The rescaled critical point $(e' b')^{*}$ is determined independent of the system size, and the profile of this relation fits with $$\frac{h'(e')}{b'^{2}}=h'_{o}|e' b' - (e'b')^{*}|^{0.3}$$ near $(e'b')^{*} \sim 0.282$.($h'_{o}$ is constant.) Moreover, we introduce the cluster length $\sigma(e')=\sigma'(e') d$ ($\sigma'(e')$ has no dimensions.) which is defined as the distance between two nearest inflection points from the maximum point of the packing fraction profile. Thus, the relation between two rescaled values, $e' b'$ and $\sigma'(e') /b'$, are obtained as shown in Fig.4 (b). The profile of this relation fits with $$\frac{\sigma'(e')}{b'} = \frac{0.0255}{(e' b')^{2}}-\frac{0.105}{e'
b'}+0.98.$$ We expect $\sigma'(e') /b' \to \infty$ for $e'b' \to 0$, and $\sigma'(e') /b' \sim 1$ for $e'b' \to \infty$.
Now, we discuss the character of transition between the 1D state and the 2D state. Here, we define this transition width $a^{*}$ as the maximum width that the 2D state cannot be observed in the system. Figure 5 shows the typical probability distributions of $L(t)$ for several width $a$ around $a=\sigma(e')$. Here, (a) indicates for the case $e'<e'^{*}$ ($\eta<\eta^{*}$) and (b) indicates for the case $e'>e'^{*}$ ($\eta>\eta^{*}$)both for $b=20d$. In Fig.5 (a), each profile of probability distribution of $L(t)$ includes only one peak. The position of peak is at $L(t)=0$ for $a<\sigma(e')$. For $a>\sigma(e')$, however, the peak appears at $L(t)>0$ and this peak moves with the width of the system. Thus the continuous transition between the 1D state and the 2D state appears for the case $e'<e'^{*}$ ($\eta<\eta^{*}$), and the transition width is given as $a^{*} \sim \sigma(e')$. In Fig.5 (b), on the contrary, probability distributions of $L(t)$ for $a$, which is a little larger than $\sigma(e')$, include two peaks at $L(t)=0$ and $L(t)>0$. This means that two locally stable states, one is the 1D state and the other is the 2D state, coexist and they appear periodically for the case of $e'>e'^{*}$ ($\eta>\eta^{*}$). In such cases, the system includes a solid structure when 1D state is realized. In this solid structure, the friction between particles are strong because particles are densely packed. Hence, the solid structure is break-proof, and this originates the stability of the 1D state for $a>\sigma(e')$. Figure 6 shows the semi-log scale profiles of probability distributions of $L(t)$ for respectively $a<\sigma(e')$, $a \sim \sigma(e')$ and $a>\sigma(e')$. When $a'<\sigma'(e')$, the profile is proportion to $exp(-\gamma_{0}
L^{2})$, which means that fluctuations of $L(t)$ are so small that they can be neglected. However, a profile which proportion to $exp(-\gamma_{1} L)$ is obtained when $a$ is close to $\sigma(e')$, which means that the fluctuation from $L(t)=0$ become large. Moreover, the profile includes two peaks at $L(t)=0$ and $L(t)>0$ when $a \geq \sigma(e')$. Then, the transition width $a^{*}$ is regarded as $a^{*} \sim \sigma(e')$ for also $e'>e'^{*}$ ($\eta>\eta^{*}$). These results mean following two facts. I) The critical width which the 2D state can appear, is equal to the cluster length. II) The magnitude of dissipation separates the type of the transition between the 1D state or the 2D state. We obtain the similar results for the case $b'=30$. Then, by using rescaled parameter $e' b'$ and $a'/b'$, the phase diagram is obtained as shown in Fig.7.
In this paper, we simulated strongly excited granular materials in a 2 dimensional pipe. When the width of the system is small, a characteristic state, in which the effective dimension of particles’ dynamics is regarded as 1, appears. We named this state as the 1D state. However, when the width of the system is larger than a critical width, the actual 2 dimensional state in which convection appears. We named this state as the 2D state. Moreover, we found that the critical width is equal to the length of the cluster which appears in the 1D state. The character of the 1D state depends on the magnitude of dissipation as followings. When the magnitude of dissipation is larger than the critical value, the system include solid structure. On the contrary, such solid structure disappears when the magnitude of dissipation is smaller than the critical value. According to the differences of the magnitude of dissipation, following two types of behaviors appears in the system near the critical width. When the magnitude of dissipation is smaller than the critical value, the continuous transition between the 1D state and the 2D state appears. On the contrary, when the magnitude of dissipation is larger than the critical value, two meta-stable states, the 1D state and the 2D state, appear periodically for a little over the critical width. By another simulation, the existence of such a critical magnitude of dissipation is reported[@fh7; @fh8]. The analytical derivation of this critical value is one of the most important issue for the research of granular materials. Moreover, simulations of more highly excited systems, larger systems, and analytical study of the critical width for several magnitude of dissipation are important future issues.
The author is grateful to H.Nishimori for useful discussions. This research was supported in part by the Ibaraki University SVBL and Grant-in-Aid for JSPS Felows 10376.
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R.Ramirez, D.Risso and P.Cordero cond-matt/0002433
The elastic constant $k$ and the viscosity coefficient $\eta$ are related with the coefficient of restitution $e$ and the collision time $t_{col}$, time period during collision \[2\].
[^1]: E-mail: [email protected]
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abstract: 'In this manuscript, it is shown that the group of $K_1$-zero-cycles on the second generalized Severi-Brauer variety of an algebra $A$ of index 4 is given by elements of the group $K_1(A)$ together with a square-root of their reduced norm. Utilizing results of Krashen concerning exceptional isomorphisms, we translate our problem to the computation of cycles on involution varieties. Work of Chernousov and Merkurjev then gives a means of describing such cycles in terms of Clifford and spin groups and corresponding $R$-equivalence classes. We complete our computation by giving an explicit description of these algebraic groups.'
address: 'Department of Mathematics, University of South Carolina, Columbia, SC 29208'
author:
- 'Patrick K. McFaddin'
title: 'The Group of $K_1$-Zero-Cycles on the Second Generalized Severi-Brauer Variety of an Algebra of Index 4'
---
[^1]
Introduction
============
The theory of algebraic cycles on homogenous varieties has seen many useful applications to the study of central simple algebras, quadratic forms, and Galois cohomology. Significant results include the Merkurjev-Suslin Theorem, the Milnor and Bloch-Kato Conjectures (now theorems of Orlov-Vishik-Voevodksy and Voevodsky-Rost-Weibel) [@Voe], and Suslin’s Conjecture, recently proven by Merkurjev [@MerkSusConj]. Despite these successes, a general description of Chow groups (with coefficients) remains elusive, and computations of these groups are done in various cases, e.g., [@ChernMerkSpin; @ChernMerk; @KrashZeroCycles; @Mer95; @MerkRat; @MerkSus; @PSZ; @Swan].
The impetus for the investigation presented here is the result of Merkurjev and Suslin [@MerkSus], showing that the group of $K_1$-zero-cycles of the Severi-Brauer variety of a central simple algebra $A$ is given by $K_1(A)$. In this paper, we show that the group of $K_1$-zero-cycles on the second generalized Severi-Brauer variety of an algebra $A$ of index 4 is given by elements of $K_1(A)$ together with a square root of their reduced norm. More precisely, the main result is the following:
$\operatorname{(\ref{thm:mainthm})}$ Let $A$ be a central simple algebra of index 4 and arbitrary degree over a field $F$ of characteristic not 2, and let $X$ be the second generalized Severi-Brauer variety of $A$. The group $A_0(X, K_1)$ can be identified as the group of pairs $(x, \alpha) \in K_1(A) \times F^{\times}$ satisfying $\operatorname{Nrd}_A(x) = \alpha ^2$.
The paper is organized as follows. In Section 2, we recall some definitions and facts about central simple algebras, involutions, and Severi-Brauer and involution varieties. We then define cycle modules and $K$-cohomology.
In Section 3, we introduce certain algebraic groups arising from algebras with involution. These include various unitary groups, the Clifford group, and spin group. We also define the notion of $R$-equivalence and its use in relating these groups to $K$-theory and $K$-cohomology, using results of Voskresenskiĭ and Chernousov-Merkurjev.
In Section 4, we prove Lemma \[lem:red\], which gives a reduction to algebras of square degree. Given an algebra $A$ of index 4, we reduce our computation of the group of $K_1$-zero-cycles on the second generalized Severi-Brauer variety of $A$ to that of the second generalized Severi-Brauer variety of $D$, where $A = M_n(D)$. The fact that $D$ has degree 4 allows one to utilize a result of Krashen relating this variety to a certain involution variety.
In Section 5, we prove our main result. After transferring our computation into the realm of involution varieties, we utilize work of Chernousov-Merkurjev expressing the group of $K_1$-zero-cycles in terms of the Clifford and spin groups of a certain algebra with involution. An analysis of these groups completes the proof.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The material presented in this manuscript was extracted from my thesis. I would like to express my sincere gratitude to my advisor, Danny Krashen, for posing this problem to me and for all of the helpful suggestions he has provided.
Preliminaries and Notation
==========================
Throughout, $F$ denotes an infinite perfect field of characteristic unequal to 2. While this assumption on the characteristic is only necessary for the proofs of Proposition \[prop:deg4\] and Theorem \[thm:mainthm\], it aids in adding clarity to the exposition, eliminating the need for the use of quadratic pairs. If $L/F$ is a field extension and $A$ is an $F$-algebra, we write $A_L:= A \otimes _F L$. By a *scheme* we will mean a separated scheme of finite type over the field $F$. A variety is an integral scheme. If $X$ and $Y$ are $F$-varieties, we write $X_Y : = X\times _{{\operatorname{Spec}}F} Y$.
A *central simple algebra* over $F$ is a finite-dimensional $F$-algebra with no two-sided ideals other than $(0)$ and $(1)$ and whose center is precisely $F$. Unless otherwise stated, all algebras will be central simple over $F$. Recall that the dimension of a central simple algebra $A$ is a square, and we define the *degree* of $A$ to be $\text{deg}(A) = \sqrt{\dim A}$. One may write $A = M_n(D)$ for a division algebra $D$, unique up to isomorphism, and we define the *index* of $A$ to be $\text{ind}(A) = \text{deg}(D).$ We say two central simple algebras over $F$ are *Brauer-equivalent* if their underlying division algebras are isomorphic.
By an *algebra with involution* we will mean a pair $(A,\sigma)$ where $A$ is a central simple algebra and $\sigma: A \to A$ is an anti-automorphism satisfying $\sigma ^2 = {\operatorname{id}}_A$. An involution *of the first kind* satisfies $\sigma |_F = {\operatorname{id}}_F$, while an involution *of the second kind* induces a nontrivial degree 2 automorphism of $F$. We refer to involutions of the second kind as *unitary involutions*. An involution of the the first kind which is a twisted form of a symmetric bilinear form is *orthogonal*. Otherwise, it is *symplectic*. The assumption that $\text{char}(F) \neq 2$ implies that algebras with quadratic pair [@MerkBook §5.B] are simply algebras with orthogonal involution.
It will be useful to extend the notion of “unitary involution" to include semi-simple $F$-algebras of the form $A_1 \times A_2$, where each $A_i$ is a central simple over $F$. The center $L$ of an algebra with unitary involution $(A, \sigma)$ will generally be an étale quadratic extension of $F$, i.e., either $L \simeq F \times F$ or $L/F$ is a separable quadratic field extension. In the first case, $A \simeq A_1 \times A_2$ as above, and in the second case $A$ is a central simple algebra over $L$. We will refer to such an algebra as a *central simple algebra with unitary involution*, even though the algebra is not necessarily simple and its center is not $F$ (see introduction to [@MerkBook §2.B]).
Severi-Brauer and Involution Varieties
--------------------------------------
The following may be found in [@Blanch; @MerkBook; @KrashZeroCycles; @Tao]. Let $A$ be a central simple algebra of degree $n$. For any integer $1 \leq k \leq n$, the $k^{\text{th}}$ generalized Severi-Brauer variety ${\operatorname{SB}}_k(A)$ of $A$ is the variety of right ideals of dimension $nk$ in $A$. Such a varietiy is a twisted form of the Grassmannian ${\operatorname{Gr}}(k, n)$ of $k$-dimensional subspaces of an $n$-dimensional vector space. The variety ${\operatorname{SB}}_1(A) = {\operatorname{SB}}(A)$ is the usual Severi-Brauer variety of $A$, which is a twisted form of projective space. For a field extension $L/F$, the variety ${\operatorname{SB}}_k(A)$ has an $L$-rational point if and only if $\text{ind}(A_L) \mid k$ [@MerkBook Prop. 1.17]. We concern ourselves with the case $k = 2$, the second generalized Severi-Brauer variety of $A$.
Let $(A, \sigma)$ be an algebra with orthogonal involution and let $I$ be a right ideal of $A$. The *orthogonal ideal* of $I$ with respect to $\sigma$ is given by $$I^{\perp} = \{ x \in A \mid \sigma(x)y = 0 \text{ for } y \in I\}.$$ We say that an ideal $I$ is *isotropic* if $I \subset I ^{\perp}$. Let $\text{IV}(A,\sigma)$ denote the collection of isotropic ideals in $A$ of dimension $n$. It is a subvariety of ${\operatorname{SB}}(A)$, with inclusion morphism defined by forgetting the isotropy condition. Just as ${\operatorname{SB}}(A)$ is a twisted form of projective space, $\text{IV}(A, \sigma)$ is a twisted form of a projective quadric.
Cycle Modules
-------------
Cycle modules were first introduced by M. Rost in [@Rost] and good references are [@GMS] and [@EKM] for the case of Milnor $K$-theory.
A *cycle module* $M$ over $F$ is a function assigning to every field extension $L/F$ a graded abelian group $M(L) = M_*(L)$, which is a graded module over the Milnor $K$-theory ring $K_*^M(F)$ satisfying some data and compatibility axioms. This data includes
1. For each field homomorphism $L \to E$ over $F$, there is a degree 0 homomorphism $r_{E/L}: M(L) \to M(E)$ called *restriction*.
2. For each field homomorphism $L \to E$ over $F$, there is a degree 0 homomorphism $c_{E/L}: M(E) \to M(L)$ called *corestriction* (or norm).
3. For each extension $L/F$ and each (rank 1) discrete valuation $v$ on $L$, there is a degree $-1$ homomorphism $\partial _{v}: M(L) \to M(\kappa(v))$ called the *residue homomorphism*, where $\kappa(v)$ is the residue field of $v$.
These homomorphisms are compatible with the corresponding maps in Milnor $K$-theory. See D1-D4, R1a-R3e, FD, and C of [@Rost Def. 1.1, Def. 2.1].
Let $X$ be an $F$-variety and $M$ a cycle module over $F$. Utilizing this axiomatic framework, we set $$C_p(X, M_q) = \coprod _{x \in X_{(p)}} M_{p+q}(F(x)),$$ where $X_{(p)}$ denotes the collection of points of $X$ of dimension $p$, to obtain a complex $$\cdots \xrightarrow{} C_{p+1}(X, M_{q}) \xrightarrow{d_X} C_p(X M_q) \xrightarrow{d_X} C_{p-1}(X, M_{q}) \xrightarrow{} \cdots$$ with differentials $d_X$ induced by the homomorphisms $\partial _v$ associated to valuations of codimension 1 subvarieties. This is often referred to as the *Rost complex* for general cycle modules and the *Gersten complex* for the cycle module given by Quillen $K$-theory. We denote the homology group at the middle term by $A_p(X, M_q)$. Our main focus will be the group $$A_0(X, K_1) = \operatorname{coker} \left( \coprod _{x\in X_{(1)}} K_2(F(x))\xrightarrow{d_X} \coprod _{x \in X_{(0)} } K_1(F(x)) \right)$$ of $K_1$-zero-cycles, where $K_*$ is the cycle module given by Quillen $K$-theory. More concretely, we wish to describe the collection of equivalences classes of formal sums $\sum (\alpha, x)$, where $x$ is a closed point on $X$ and $\alpha \in F(x)^{\times}$.
Involutions, Algebraic Groups, and $R$-Equivalence
==================================================
We define certain algebraic groups associated to central simple algebras with involution. These groups are twisted forms of their more classical counterparts. Their importance comes from the fact that algebras with involution of type $A_3$ are closely related to Clifford and spin groups associated to algebras with involution of type $D_3$, coming from the exceptional identification of the $A_3$ and $D_3$ Dynkin diagrams (this is made precise in Proposition \[prop:excid\]). Furthermore, utilizing a result of Chernousov and Merkurjev stated below, these groups can be used to compute $K_1$-zero-cycles for involution varieties associated to algebras with involution which have index no more than 2.
An *algebraic group* is a smooth affine group scheme. We follow the same notational convention as [@ChernMerkSpin], utilizing $\text{GL}_1(A)$, $\text{SL}_1(A)$, $\text{Spin}(A, \sigma)$, etc., to denote the groups of $F$-points of the corresponding algebraic groups $\textbf{GL}_1(A),$ $\textbf{SL}_1(A)$, $\textbf{Spin}(A, \sigma)$, etc. We suppress reference to the scheme structure and focus only on the collections of $F$-points, although we continue to refer to them as “algebraic groups."
Involutions and Algebraic Groups
--------------------------------
A good reference for the following material is [@MerkBook]. Let $A$ be a central simple algebra over $F$, and consider the algebraic group $\text{GL}_1(A) = A^{\times}$ of invertible elements in $A$, called the *general linear group* of $A$. The kernel of the reduced norm homomorphism $\text{Nrd}_A: \text{GL}_1(A) \to F^{\times}$ [@MerkBook $\S$1.6] is denoted $\text{SL}_1(A)$, called the *special linear group* of $A$.
Let $(A, \sigma)$ be a central simple algebra with involution. A *similitude* of $(A, \sigma)$ is an element $g \in A$ satisfying $\sigma(g) g \in F^{\times}$. The collection of all similitudes of $(A, \sigma)$ is denoted $\text{Sim}(A, \sigma)$. The scalar $\mu(g) := \sigma(g) g$ is called the *multiplier* of $g$. The association $g \mapsto \mu(g)$ defines a group homomorphism $\mu: \text{Sim}(A, \sigma) \to F^{\times}$. If the involution $\sigma$ is of unitary type, we denote $\text{Sim}(A, \sigma)$ by $\text{GU}(A, \sigma)$ and call it the *general unitary group* of $(A, \sigma)$.
Let $(A, \sigma)$ be an algebra with unitary involution, with $\text{deg}(A) = 2m$. Define the *special general unitary* and *special unitary* groups $$\text{SGU}(A, \sigma) =\{g \in \text{GU}(A, \sigma) \mid \text{Nrd}_A(g) = \mu(g) ^ m\}$$ $$\text{SU}(A, \sigma) = \{ u \in \text{GU}(A, \sigma)\mid \text{Nrd}_A(u) = 1\}.$$
In the case where $A$ has center $L \simeq F \times F$, there is a central simple algebra $E$ over $F$ such that $(A, \sigma) \simeq (E \times E^{\text{op}}, \varepsilon)$, where $\varepsilon$ is the involution which switches factors (the *exchange* involution) [@MerkBook Prop. 2.14]. The general, special general, and special unitary groups of $(A, \sigma)$ are then given by [@MerkBook §14.2] $$\text{GU}(E \times E^{\text{op}}, \varepsilon) = \{(x, \alpha(x^{-1})^{\text{op}}) \mid \alpha \in F^{\times}, x \in E^{\times}\} \simeq E^{\times} \times F^{\times}$$ $$\text{SGU}(E \times E^{\text{op}}, \varepsilon) = \{(x, \alpha) \in E^{\times} \times F^{\times} \mid \text{Nrd}_E(x) = \alpha ^m\}$$ $$\text{SU}(E\times E^{\text{op}}, \varepsilon) = \{x \in E^{\times} \mid \text{Nrd}_E (x) = 1\} = \text{SL}(E).$$
Clifford and Spin Groups
------------------------
Given an algebra with orthogonal involution $(A, \sigma)$, the *Clifford algebra* $C(A, \sigma)$ is an $F$-algebra which is a quotient of the tensor algebra of $A$. Its multiplication is defined in terms of the involution $\sigma$ and it is a twisted form of the even Clifford algebra associated to a quadratic space. Together with its canonical involution, the Clifford algebra enjoys the structure of a central simple algebra with unitary, orthogonal, or symplectic involution, depending on its degree and the characteristic of $F$ [@MerkBook Prop. 8.12]. We will be interested in cases where this canonical involution is unitary, so the center of the Clifford algebra is an étale quadratic extension of $F$. The multiplicative group of $C(A, \sigma)$ contains a group $\Gamma(A, \sigma)$, called the *Clifford group*, whose action on $C(A, \sigma)$ fixes $A \subset C(A, \sigma)$. One defines a certain *multiplier map* $\underline{\mu}: \Gamma(A, \sigma) \to F^{\times}/ F^{\times 2}$ whose kernel is called the *spin group* of $(A, \sigma)$, denoted $\text{Spin}(A, \sigma)$. We refer the reader to [@MerkBook $\S\S$ 8, 13] for all pertinent definitions.
If $(A, \sigma)$ is an algebra with unitary involution of degree $n = 2m$, the *discriminant algebra* of $(A, \sigma)$, denoted $D(A, \sigma)$, is a central simple algebra of degree ${ n \choose m}$ which plays the role of the the exterior algebra. If $F^s$ denotes a separable closure of $F$, then $A_{F^s} \simeq \text{End}_{F^s} (V)$ and $D(A, \sigma)_{F^s} = \text{End}_{F^s}(\wedge ^m V)$. The discriminant algebra comes equipped with a so-called canonical orthogonal involution $\underline{\sigma}$ induced by the involution $\sigma$ (see [@MerkBook $\S$10.E]).
Let us make precise the relationship between the Clifford, spin, and unitary groups arising from the exceptional identifications of algebras of types $A_3$ and $D_3$. We summarize the discussion found in [@MerkBook $\S$15.D].
Let $\mathsf{A}_3$ denote the groupoid of central simple algebras with unitary involution of degree 4, $\mathsf{D}_3$ the groupoid of central simple algebras $F$-algebras with orthogonal involution of degree 6. There are functors $\delta: \mathsf{A}_3 \to \mathsf{D}_3$ and $\gamma: \mathsf{D}_3 \to \mathsf{A}_3$, where $\gamma$ defined by taking an algebra with orthogonal involution to its Clifford algebra with canonical involution, and $\delta$ is defined by taking an algebra with unitary involution to its discriminant algebra with canonical orthogonal involution. These functors define an equivalence of groupoids [@MerkBook Theorem 15.24].
\[[@MerkBook], Prop.15.27\]\[prop:excid\] Let $(A, \sigma)$ and $(B, \tau)$ correspond to one another under the groupoid equivalence of $\mathsf{A}_3$ and $\mathsf{D}_3$. Then we have identifications $\Gamma(A, \sigma) = \operatorname{SGU}(B, \tau)$ and $\operatorname{Spin}(A, \sigma) = \operatorname{SU}(B, \tau).$
$R$-Equivalence
---------------
See [@ChernMerkSpin]. Let $G$ be an algebraic group over $F$. A point $x \in G(F)$ is called $R$-*trivial* if there is a rational morphism $f: {{\mathbb P}^1}\dashrightarrow G$, defined at 0 and 1, and with $f(0) = 1$ and $f(1) = x$. The collection of all $R$-trivial elements of $G(F)$ is denoted $RG(F)$ and is a normal subgroup of $G(F)$. If $H$ is a normal closed subgroup of $G$ then $RH(F)$ is a normal subgroup of $G(F)$ [@ChernMerkSU Lemma 1.2].
The notion of $R$-equivalence is related to algebraic $K$-groups and $K$-cohomology groups, as we see in the following results.\
**Example.**\[K1Ex\] See [@ChernMerkSpin; @Vos]. For a central simple algebra $A$, the abelianization map $ \text{GL}_1(A) = A^{\times} \to A^{\times}_{\text{ab}} = K_1(A)$ induces an isomorphism $$\text{GL}_1(A)/R\text{SL}_1(A) \simeq K_1(A).$$
\[thm:ChernMerkSpin\] Let $A$ be a central simple algebra over $F$ of even dimension and index at most 2 with quadratic pair $(\sigma, f)$, $X$ be the corresponding involution variety. Then there is a canonical isomorphism $$\Gamma(A, \sigma, f) / R \operatorname{Spin}(A, \sigma, f) \simeq A_0(X, K_1).$$
Reduction to Algebras of Square Degree
======================================
Given a prime integer $p$ and an algebra $A$ of index $p^2$, we reduce the computation of $K_1$-zero-cycles of ${\operatorname{SB}}_p(A)$ to that of ${\operatorname{SB}}_p(D)$, where $D$ is the underlying division algebra of $A$. For $p =2$, this reduction to algebras of degree 4 will allow the use of involution varieties in the proof of the main theorem.
For $J$ a right ideal of $A$, define ${\operatorname{SB}}_k(J)$ as the collection of right ideals of $A$ of reduced dimension $k$ which are contained in $J$ [@KrashZeroCycles Def. 4.7].
\[lem:red\] Let $p$ be a prime integer and let $A = M_n(D)$ be a central simple algebra of index $p^2$. Let $X = {\operatorname{SB}}_p(A)$, and $Y = {\operatorname{SB}}_p(D)$. There is an isomorphism $A_0(X, K_1) \simeq A_0(Y, K_1)$.
Fix an ideal $J$ of $ A$ of reduced dimension $p^2$, the existence of which follows from $\text{ind}(A) = p^2$, and let $e_J$ be the corresponding idempotent element of $A$ [@MerkBook §12]. Define a rational map $\varphi _J: {\operatorname{SB}}_p(A) \dashrightarrow {\operatorname{SB}}_p(J)$ by the association $I \mapsto e_J I \subset J$, for any ideal $I $ of reduced dimension $p$. The map $\varphi _J$ is defined on the open locus consisting of ideals $I$ satisfying $\text{rdim}(e_J I) = p.$
By [@KrashZeroCycles Thm. 4.8], the algebra $D:= e_J A e_J$ is degree $p^2$, Brauer-equivalent to $A$ and satisfies ${\operatorname{SB}}_p(J) = {\operatorname{SB}}_p(D)$. Since $\text{ind}(A)= p^2$, the algebra $D$ is division and $A = M_n(D)$. We denote the resulting map ${\operatorname{SB}}_p(A) \dashrightarrow {\operatorname{SB}}_p(D)$ also by $\varphi _J$.
Let $\eta$ be the generic point of ${\operatorname{SB}}_p(D)$ and take $L = F(\eta)$. Let $\mathfrak{f}$ be the generic fiber of $\varphi _J$, i.e., $\mathfrak{f} = {\operatorname{SB}}_p(A)_{L} = {\operatorname{SB}}_p(A_{L})$ is the scheme-theoretic fiber over $\eta$. We first show that $\mathfrak{f}$ is a rational $L$-variety. The field $L$ satisfies $\text{ind} (D_{L}) \mid p$. Since $D$ is Brauer-equivalent to $A$, we have $\text{ind}(A_{L}) \mid p$, so that ${\operatorname{SB}}_p(A)$ has an $L$-rational point. By [@Blanch Prop. 3], the function field of ${\operatorname{SB}}_p(A)_{L}$ is purely transcendental over $L$, so that $\mathfrak{f}= {\operatorname{SB}}_p(A)_{L}$ is rational. Therefore, ${\operatorname{SB}}_p(D)_{\mathfrak{f}}$ is birational to ${\operatorname{SB}}_p(D) \times \mathbb{P}_L ^{\dim \mathfrak{f}}$.
The group of $K_1$-zero-cycles is an invariant of smooth projective varieties [@KM Cor. RC.13], so that we have an isomorphism $$A_0({\operatorname{SB}}_p(D)_{\mathfrak{f}}, K_1) \simeq A_0({\operatorname{SB}}_p(D) \times \mathbb{P}_L ^{\dim \mathfrak{f}}, K_1).$$ This isomorphism in conjunction with the Projective Bundle Theorem [@EKM Theorem 53.10], yields $$A_0({\operatorname{SB}}_p(D) _{\mathfrak{f}}, K_1) \simeq A_0({\operatorname{SB}}_p(D), K_1).$$ The variety ${\operatorname{SB}}_p(A)$ is birational to ${\operatorname{SB}}_p(D)_{ \mathfrak{f}}$, isomorphic along the open locus of definition of $\varphi _J$. Again using that $A_0(-, K_1)$ is a birational invariant, and combining this with the above isomorphisms gives $A_0({\operatorname{SB}}_p(A), K_1) \simeq A_0({\operatorname{SB}}_p(D), K_1).$
Main Result
===========
Having reduced our computation to the case of an algebra of degree 4, we utilize a result of Krashen [@KrashZeroCycles] to transfer the computation of zero-cycles of the second generalized Severi-Brauer variety to that of an involution variety. This result also guarantees that the involution variety of interest comes from an algebra of index no greater than 2. We then make use of Theorem \[thm:ChernMerkSpin\] to translate this computation into an analysis of those algebraic groups defined in $\S 3$.
\[prop:deg4\] Let $A$ be a central simple algebra of degree 4 over a field $F$ and let $X$ be the second generalized Severi-Brauer variety of $A$. The group $A_0(X, K_1)$ can be identified as the group of pairs $(x, \alpha) \in K_1(A) \times F^{\times}$ satisfying $\operatorname{Nrd}_A(x) = \alpha ^2$.
By [@KrashZeroCycles Lem. 6.5], ${\operatorname{SB}}_2(A)$ is isomorphic to the involution variety $\operatorname{IV}(B, \sigma)$ of a degree 6 algebra $B$ with orthogonal involution $\sigma$. In particular, $$A_0(\text{SB}_2(A), K_1) = A_0(\text{IV}(B, \sigma), K_1).$$ Moreover, $B$ is Brauer-equivalent to $A^{\otimes 2}$, so that $\operatorname{ind }(B) \leq 2$. The involution $\sigma$ is obtained from the bilinear form $$\wedge ^2 V \times \wedge ^2 V \to \wedge ^4 V \simeq F$$ by descent. Consider the algebra $(A\times A^{\text{op}}, \varepsilon)$ over $F \times F$ with unitary involution defined by exchanging factors. Its discriminant algebra $D(A \times A^{\text{op}}, \varepsilon)$ is given by $\lambda ^2 A$ [@MerkBook 10.31], which has degree 6 and is Brauer-equivalent to $A ^{\otimes 2}$ [@MerkBook 10.4]. The canonical involution $\underline{\varepsilon}:= \varepsilon^{\wedge 2}$ [@MerkBook 10.31] on $\lambda ^2 A$ is also induced by the bilinear form $\wedge ^2 V \times \wedge ^2 V \to \wedge ^4 V \simeq F$ [@MerkBook Def. 10.11], yielding an isomorphism $(B, \sigma) \simeq (D(A \times A^{\text{op}}, \varepsilon), \underline{\varepsilon})$ of algebras with orthogonal involution. Thus, $(B, \sigma)$ and $(A\times A^{\text{op}}, \varepsilon)$ correspond to one another under the groupoid equivalence of $\mathsf{A}_3 $ and $\mathsf{D}_3$.
Since $\text{ind}(B) \leq 2$, Theorem \[thm:ChernMerkSpin\] yields a canonical isomorphism $$A_0(\text{IV}(B, \sigma), K_1) \simeq \Gamma(B,\sigma)/R\operatorname{Spin}(B, \sigma).$$\[eq:CliffSpin\] As $(B, \sigma) \in \mathsf{D}_3$ corresponds to $(A \times A^{\text{op}}, \varepsilon) \in \mathsf{A}_3$, there are exceptional identifications $\Gamma(B, \sigma) = \text{SGU}(A \times A^{\text{op}}, \varepsilon)$ and $ \text{Spin}(B, \sigma) = \text{SU}(A \times A^{\text{op}}, \varepsilon)$, as noted in Proposition \[prop:excid\]. Furthermore, the discussion in $\S$3.1 gives $$\text{SGU}(A \times A^{\text{op}}, \varepsilon) = \{(x, \alpha) \in A^{\times} \times F^{\times} \mid \text{Nrd}_A(x) = \alpha ^2\}$$ $$\text{SU}(A \times A^{\text{op}}, \varepsilon)= \text{SL}_1(A)$$ with the inclusion of the latter given by inclusion into the first factor $x \mapsto (x, 1)$. The quotient in \[eq:CliffSpin\].1 can then be identified as $$\{(x, \alpha) \in A ^{\times} \times F^{\times} \mid \text{Nrd}_A(x) = \alpha ^2\}/ R\text{SL}_1(A),$$ and therefore consists of elements $ x \in A^{\times}/R\text{SL}_1(A) = \text{GL}_1(A)/R\text{SL}_1(A) = K_1(A)$ together with a square-root $\alpha$ of $\text{Nrd}_A(x)$ in $F^{\times}$.
\[thm:mainthm\] Let $A$ be a central simple algebra of index 4 over a field $F$, and let $X$ be the second generalized Severi-Brauer variety of $A$. The group $A_0(X, K_1)$ can be identified as the group of pairs $(x, \alpha) \in K_1(A) \times F^{\times}$ satisfying $\operatorname{Nrd}_A(x) = \alpha ^2$.
The reduced norm respects the canonical isomorphism $K_1(A) = K_1(M_n(D)) = K_1(D)$. Combining the isomorphisms of Lemma \[lem:red\] and Proposition \[prop:deg4\] yields the desired result.
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[^1]: Partially supported by NSF Grant DMS-1151252
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abstract: 'The (un)weighted stability for some matrices is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weight. Inverse-closed Banach subalgebras of matrices with certain off-diagonal decay can be informally interpreted as localization preservation under inversion, which is of great importance in many mathematical and engineering fields. Let ${\mathcal B}(\ell^p_w)$ be the Banach algebra of bounded operators on the weighted sequence space $\ell^p_w$ on a simple graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a simple graph are inverse-closed in ${\mathcal B}(\ell^p_w)$ for all $1\le p<\infty$ and Muckenhoupt $A_p$-weights $w$, and the Beurling norm of the inversion of a matrix $A$ is bounded by a bivariate polynomial of the Beurling norm of the matrix $A$ and the operator norm of its inverse $A^{-1}$ in ${\mathcal B}(\ell^p_w)$.'
address:
- 'Qiquan Fang: Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, Zhejiang, 310023, China. Email: [email protected]'
- 'Chang Eon Shin: Department of Mathematics, Sogang University, Seoul, 04109, Korea. Email: [email protected] '
- 'Qiyu Sun: Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA. Email: [email protected]'
author:
- 'Qiquan Fang, Chang Eon Shin and Qiyu Sun'
title: Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs
---
[^1]
Introduction
============
Let ${\mathcal G}:=(V,E)$ be a connected simple graph with the vertex set $V$ and edge set $E$. Our illustrative examples are (i) the $d$-dimensional lattice graph ${\mathcal Z}^d:=({{\mathbb Z}}^d, E^d)$ where there exists an edge between $k$ and $l\in {{\mathbb Z}}^d$, i.e., $(k, l)\in E^d$, if the Euclidean distance between $k$ and $l$ is one; (ii) the (in)finite circulant graph ${\mathcal C}_G= (V_G, E_G)$ associated with an abelian group $$V_G=\Big\{ \prod_{i=1}^k g_i^{n_i}, \ n_1, \ldots, n_k\in {{\mathbb Z}}\Big\}$$ generated by $G=\{g_1, \ldots, g_k\}$, where $(\lambda, \lambda')\in E_G$ if and only if either $\lambda ({\lambda'})^{-1}$ or $\lambda' \lambda^{-1}\in G$ [@BE14; @BE12; @LO01; @samei19]; and (iii) the communication graph of a spatially distributed network (SDN) whose agents have limited sensing, data processing, and communication capacity for data transmission, where agents are used as elements in the vertex set and direct communication links between two agents as edges between two vertices [@akyildiz02; @CJS18; @chong02; @shinsun19].
For $1\le p<\infty$ and a weight $w=(w(\lambda))_{\lambda\in V}$ on the graph ${\mathcal G}$, let $\ell^p_w:= \ell_w^p({\mathcal G})$ be the Banach space of all weighted $p$-summable sequences $c=(c(\lambda))_{\lambda\in V}$ equipped with the standard norm $$\|c\|_{p,w}=\Big(\sum_{\lambda \in V} |c(\lambda)|^p w(\lambda) \Big)^{1/p}.$$ For the trivial weight $w_0=(w_0(\lambda))_{\lambda\in V}$, we will use the simplified notation $\ell^p$ and $\|\cdot\|_p$ instead of $\ell^p_w$ and $\|\cdot\|_{p, w}$, where $w_0(\lambda)=1$ for all $\lambda\in V$. We say that a matrix $$\label{def-matrix}
A:=\big(a(\lambda, \lambda') \big)_{\lambda, \lambda' \in V}$$ on the graph ${\mathcal G}$ has [*$\ell_w^p$-stability*]{} if there exist two positive constants $B_1$ and $B_2$ such that $$\label{stability-cond}
B_1 \|c\|_{p,w} \le \|A c\|_{p,w}\le B_2 \|c\|_{p,w}, \ \ c\in \ell^p_w$$ [@akramjfa09; @shinsun19; @shincjfa09; @sunxian14; @tesserajfa10]. We call the maximal constant $B_1$ for the weighted stability inequality to hold as the [*optimal lower $\ell^p_w$-stability bound*]{} of the matrix $A$ and denote by $\beta_{p, w}(A)$. The (un)weighted stability for matrices is an essential hypothesis in time-frequency analysis, applied harmonic analysis, and many other mathematical and engineering fields [@aldroubisiamreview01; @christensenbook; @grochenigbook; @moteesun17; @sunsiam06].
In practical sampling and reconstruction on an SDN of large size, signals and noises are usually contained in some range. For robust signal reconstruction and noise reduction, the sensing matrix on the SDN is required to have stability on $\ell^\infty$ [@CJS18], however there are some difficulties to verify $\ell^p$-stability of a matrix in a distributed manner for $p\ne 2$ [@moteesun19; @sunpams10]. For a matrix $A$ on a [*finite*]{} graph ${\mathcal G}=(V, E)$, its weighted $\ell^p_w$-stability are equivalent to each other for different exponents $1\le p\le\infty$ and weights $w$, since $\ell^p_w$ is isomorphic to $\ell^2$ for any exponent $1\le p\le\infty$ and weight $w$. In particular, for the unweighted case one may verify that the optimal lower stability bounds of a matrix $A$ for different exponents are comparable, $$\label{Mstability}
\frac{\beta_{p, w_0}(A)}{\beta_{q, w_0}(A)}\le M^{|1/p-1/q|}, \ 1\le p, q\le\infty,$$ where $M=\# V$ is the number of vertices of the graph ${\mathcal G}$. The above estimation on optimal lower stability bounds for different exponents is unfavorable for matrices of large size, but it can be improved only if the matrix $A$ has some additional property, such as off-diagonal decay. For an [*infinite*]{} matrix $A=(a(i,j) )_{i,j\in{\mathbb Z}^d}$ in the Baskakov-Gohberg-Sjöstrand algebra, it is proved in [@akramjfa09; @shincjfa09; @tesserajfa10] that its unweighted stabilities are equivalent to each other for all exponents, i.e., for all $1\le p, q<\infty$, $$\beta_{q, w_0}(A)>0 \ {\rm if \ and \ only \ if} \ \beta_{p, w_0}(A)>0.
$$ In [@sunca11], Beurling algebras of infinite matrices $A=(a(i,j) )_{i,j\in{\mathbb Z}^d}$ are introduced. Comparing with the Baskakov-Gohberg-Sjöstrand algebras, matrices in the Baskakov-Gohberg-Sjöstrand algebra (resp. the Beurling algebra) are dominated by a bi-infinite Toeplitz matrix associated with a (resp. radially decreasing) sequence with certain decay, and they are bounded linear operators on unweighted sequence spaces $\ell^p_{w_0}$ (resp. on weighted spaces $\ell^p_w$ for all Mukenhoupt $A_p$-weights $w$). For an infinite matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are established in [@sunca11], $$\beta_{p, w}(A)>0 \ {\rm if \ and \ only \ if} \ \beta_{q, w'}(A)>0
$$ where $1\le p, q<\infty$ and $w, w'$ are Muckenhoupt $A_p$- and $A_q$-weights respectively. Obviously, the lattice ${{\mathbb Z}}^d$ is the vertex set of the lattice graph ${\mathcal Z}^d$. Inspired by the above observation, Beurling algebras ${\mathcal B}_{r,\alpha}({\mathcal G})$ of matrices $A=(a(\lambda, \lambda') )_{\lambda, \lambda' \in V}$ on an arbitrary simple graph ${\mathcal G}=(V, E)$ are introduced in [@shinsun19], where $1\le r\le \infty$ and $\alpha\ge 0$. In [@shinsun19], unweighted stabilities of a matrix $A\in {\mathcal B}_{r,\alpha}({\mathcal G})$ for different exponents are shown to be equivalent to each other, where $1\le r\le \infty, \alpha>d_{\mathcal G} (1-1/r)$ and $d_{\mathcal G}$ is the Beurling dimension of the graph ${\mathcal G}$. Moreover we have the following polynomial control on its optimal lower stability bounds for different exponents, $$\label{polynomialunweighted.eq}
\frac{\beta_{p, w_0}(A)}{\beta_{q, w_0}(A)}
\le D_1
\Big(\frac{\|A\|_{{\mathcal B}_{r, \alpha}}}{\beta_{p, w_0}(A)}\Big)^{D_0|1/p-1/q|},\ \ 1\le p, q<\infty,$$ where $D_0, D_1$ are absolute constants independent of matrices $A$ and the size $M$ of the graph ${\mathcal G}$. In the first part of this paper, we establish a polynomial control property for a matrix $A\in {\mathcal B}_{r, \alpha}({\mathcal G})$ on the optimal lower weighted stability bounds for different exponents and Muckenhoupt weights, see Theorem \[mainthm\] and Remark \[mainthm.rem\] in Section \[lowerbound.section\],
Let ${\mathcal B}(\ell^p_w)$ be the Banach algebra of all matrices $A$ which are bounded operators on the weighted vector space $\ell^p_w$ and denote the norm of $A\in {\mathcal B}(\ell^p_w)$ by $\|A\|_{{\mathcal B}(\ell^p_w)}$. The weighted $\ell^p_w$-stability of a matrix $A$ is usually considered as a weak notion of its invertibility, since $$\beta_{p, w}(A) \ge \big(\|A^{-1} \|_{{\mathcal B}(\ell^p_w)}\big)^{-1}$$ when the matrix $A$ is invertible in $\ell^p_w$. However for a matrix $A$ in a Beurling algebra, we discover that its weighted stability in $\ell^{p}_w$ implies the existence of its “inverse" $B=(b(\lambda, \lambda'))_{\lambda, \lambda'\in V}$ in the same Beurling algebra such that $$\label{crucialestimate}
|c(\lambda)|\le \sum_{\lambda'\in V} |b(\lambda, \lambda')| (Ac)(\lambda')|, \ \lambda\in V,$$ hold for all vectors $c=(c(\lambda))_{\lambda\in V}\in \ell^{p}_w$, see Lemma \[tech.lem2\]. The above estimate is crucial for us to discuss polynomial control on optimal lower weighted stability bounds for different exponents and Muckenhoupt weights, and also to establish norm-controlled inversion of Beurling algebras in ${\mathcal B}(\ell^p_w)$ in the second topic of this paper.
Given two Banach algebras ${\mathcal A}$ and ${\mathcal B}$ with common identity such that ${\mathcal A}$ is a Banach subalgebra of ${\mathcal B}$, we say that ${\mathcal A}$ is [*inverse-closed*]{} in ${\mathcal B}$ if $A\in {\mathcal A}$ and $A^{-1} \in {\mathcal B}$ implies $A^{-1}\in {\mathcal A}$ [@baskakov90; @gltams06; @jaffard90; @sjostrand94; @sunca11; @suntams07; @suncasp05; @wiener32]. An equivalent condition for the inverse-closedness of ${\mathcal A}$ in ${\mathcal B}$ is that given an $A\in {\mathcal A}$, its spectral sets $\sigma_{\mathcal A}(A)$ and $\sigma_{\mathcal B}(A)$ in Banach algebras ${\mathcal A}$ and ${\mathcal B}$ are the same, i.e., $$\sigma_{\mathcal A}(A)=\sigma_{\mathcal B}(A)\ \ {\rm for \ all} \ A\in {\mathcal A}.$$ In this paper, we also call the inverse-closed property for a Banach subalgebra as Wiener’s lemma for that subalgebra [@sunca11; @suntams07; @suncasp05; @wiener32]. For algebras of matrices with certain off-diagonal decay, Wiener’s lemma can be informally interpreted as localization preservation under inversion. Such a localization preservation is of great importance in applied harmonic analysis, numerical analysis, and many mathematical and engineering fields, see the survey papers [@grochenig10; @Krishtal11; @shinsun13] and references therein for historical remarks. We remark that Wiener’s lemma does not provide a norm estimate for the inversion, which is essential for some mathematical and engineering applications.
We say that a Banach subalgebra ${\mathcal A}$ of ${\mathcal B}$ admits [*norm-controlled inversion*]{} in ${\mathcal B}$ if there exists a continuous function $h$ from $[0, \infty)\times [0, \infty)$ to $[0, \infty)$ such that $$\label{normcontrol}
\|A^{-1}\|_{\mathcal A}\le h\big(\|A\|_{\mathcal A}, \|A^{-1}\|_{\mathcal B}\big)$$ for all $A\in {\mathcal A}$ being invertible in ${\mathcal B}$ [@gkII; @gkI; @nikolski99; @samei19; @shinsun19]. By the norm-controlled inversion , we have the following estimate for the resolvent of $A\in {\mathcal A}$, $$\label{normcontrolresolvent}
\|(\lambda I-A)^{-1}\|_{\mathcal A}\le h\big(\|\lambda I-A\|_{\mathcal A}, \|(\lambda I-A)^{-1}\|_{\mathcal B}\big), \ \ \lambda\not\in \sigma_{\mathcal B}(A)=\sigma_{\mathcal A}(A),$$ where $I$ is the common identity of Banach algebras ${\mathcal A}$ and ${\mathcal B}$. The norm-controlled inversion is a strong version of Wiener’s lemma. The classical Wiener algebra of periodic functions with summable Fourier coefficients is an inverse-closed subalgebra of the Banach algebra of all periodic continuous functions [@wiener32], however it does not have norm-controlled inversion [@belinskiijfaa97; @nikolski99]. We say that ${\mathcal A}$ is a [*differential subalgebra of order $\theta\in (0, 1]$*]{} in ${\mathcal B}$ if there exists a positive constant $D:=D({\mathcal A}, {\mathcal B}, \theta)$ such that $$\label{differentialnorm.def}
\|AB\|_{\mathcal A}\le D\|A\|_{\mathcal A} \|B\|_{\mathcal A} \Big (\Big(\frac{\|A\|_{\mathcal B}}{\|A\|_{\mathcal A}}\Big)^\theta +
\Big(\frac{\|B\|_{\mathcal B}}{\|B\|_{\mathcal A}}\Big)^\theta
\Big)
\quad {\rm for \ all} \ A, B \in {\mathcal A}.$$ The concept of differential subalgebras of order $\theta$ was introduced in [@blackadarcuntz91; @kissin94; @rieffel10] for $\theta=1$ and [@christ88; @gkI; @shinsun19] for $\theta\in (0, 1)$. It has been proved that a differential $*$-subalgebra ${\mathcal A}$ of a symmetric $*$-algebra ${\mathcal B}$ has norm-controlled inversion in ${\mathcal B}$ [@gkII; @gkI; @samei19; @shinsun20; @suncasp05]. A crucial step in the proof is to introduce $B:=I- \|A^* A\|_{\mathcal B}^{-1} A^* A$ for any $A\in {\mathcal A}$ being invertible in ${\mathcal B}$, whose spectrum is contained in an interval on the positive real axis. The above reduction depends on the requirements that ${\mathcal B}$ is symmetric and both ${\mathcal A}$ and ${\mathcal B}$ are $*$-algebras with common identity and involution $*$.
Several algebras of localized matrices with certain off-diagonal decay, including the Gröchenig-Schur algbera, Baskakov-Gohberg-Sjöstrand algebra, Beurling algebra and Jaffard algebra, have been shown to be differential $*$-subalgebras of the symmetric $*$-algebra ${\mathcal B}(\ell^2)$, and hence they admit norm-controlled inversion in ${\mathcal B}(\ell^2)$ [@gkII; @gkI; @grochenigklotz10; @jaffard90; @rssun12; @samei19; @shinsun19; @sunca11; @suntams07; @suncasp05]. In [@gkII; @gkI; @shinsun19], the authors show that for the Baskakov-Gohberg-Sjöstrand algebra, Jaffard algebra, and Beurling algebra of matrices, a bivariate polynomial can be selected to be the norm-control function $h$ in .
For applications in some mathematical and engineering fields, the widely-used algebras ${\mathcal B}$ of infinite matrices are the operator algebras ${\mathcal B}(\ell^p_w), 1\le p\le \infty$, which are symmetric only when $p=2$. To our knowledge, there is no literature on norm-controlled inversion in a nonsymmetric algebra. In this paper, we prove that Beurling algebras of localized matrices admit norm-controlled inversion in ${\mathcal B}(\ell^p_w)$ for all exponent $1\le p<\infty$ and Muckenhoupt $A_p$-weights $w$, and that the Beurling algebra norm of the inversion of a matrix $A$ is bounded by a bivariate polynomial of its Beurling algebra norm of the matrix $A$ and the operator norm of its inverse $A^{-1}$ in ${\mathcal B}(\ell^p_w)$, see Theorem \[norm-contro-inversion.thm\] and Remark \[norm-contro-inversion.thm.remark\].
The paper is organized as follows. In Section \[Preliminaries.section\], we recall some preliminary results on a connected simple graph ${\mathcal G}$, Beurling algebras of matrices on the graph ${\mathcal G}$ and on its maximal disjoint sets, and weighted norm inequalities for matrices in a Beurling algebra. For matrices in a Beurling algebra, we consider the equivalence of their weighted stability for different exponents $1\le p< \infty$ and Muckenhoupt $A_p$-weights $w$ in Section \[lowerbound.section\], and their norm-controlled inversion in ${\mathcal B}(\ell^p_w)$ in Section \[normedinversion.section\]. All proofs, except the proof of Theorem \[mainthm\] in Section \[lowerbound.section\], are collected in Section \[proofs.section\].
Notation: For a real number $t$, we use the standard notation $\lfloor t\rfloor$ and $\lceil t\rceil$ to denote its floor and ceiling, respectively. For two terms $A$ and $B$, we write $A\lesssim B$ if $A\le CB$ for some absolute constant $C$, and $A\approx B$ if $A\lesssim B$ and $B\lesssim A$.
Preliminaries {#Preliminaries.section}
=============
In Section \[graph.subsection\], we recall the doubling property for the counting measure $\mu$ on a connected simple graph ${\mathcal G}$ [@CJS18; @shinsun19; @YYH13book], show that the counting measure $\mu$ has the strong polynomial growth property , and then define generalized Beurling dimension of the graph ${\mathcal G}$. In Sections \[beurling.subsection\] and \[beurlingmaximal.subsection\], we recall the definition of two closely-related Beurling algebras of matrices on the graph ${\mathcal G}$ and on its maximal disjoint sets [@beurling49; @shinsun19; @sunca11], and provide some algebraic and approximation properties of those two Banach algebras of matrices. In Section \[weighted.subsection\], we prove that any matrix in a Beurling algebra is a bounded linear operator on weighted vector spaces $\ell^p_w$ for all $1\le p<\infty$ and Muckenhoupt $A_p$-weights $w$.
Generalized Beurling dimension of a connected simple graph {#graph.subsection}
----------------------------------------------------------
Let $\rho$ be the [*geodesic distance*]{} on the connected simple graph ${\mathcal G}$, which is the nonnegative function on $V\times V$ such that $\rho(\lambda,\lambda)=0, \lambda\in V$, and $\rho(\lambda,\lambda')$ is the number of edges in a shortest path connecting distinct vertices $\lambda, \lambda'\in V$ [@chungbook]. This geodesic distance $\rho$ is a metric on $V$ of a connected simple graph ${\mathcal G}$. For the lattice graph ${\mathcal Z^d}$, one may verify that its geodesic distance between two points $k=(k_1,..., k_d)$ and $\ell=(\ell_1,..., \ell_d)$ is given by $\rho(k, \ell):=\sum_{j=1}^d|k_j-\ell_j|$; for the circulant graph ${\mathcal C}_G$ generated by $G=\{g_1, \ldots, g_k\}$, we have $$\rho(\lambda, \lambda')= \inf \Big\{\sum_{i=1}^k |n_i|, \ \lambda' \lambda^{-1}=\prod_{i=1}^k g_i^{n_i}, n_1, \ldots, n_k\in {{\mathbb Z}}\Big\};$$ and for the communication graph of an SDN, $\rho(\lambda, \lambda')$ is the time delay of data transmission between two agents $\lambda$ and $\lambda'$. Using the geodesic distance $\rho$, we define the closed ball with center $\lambda\in V$ and radius $r>0$ by $$B(\lambda, r)= \{\lambda'\in V, \ \rho(\lambda, \lambda')\le r\},$$ which contains all $r$-neighboring vertices of $\lambda\in V$.
Let $\mu$ be the counting measure on the vertex set $V$, i.e., $\mu(F)$ is the number of vertices in $F\subset V$. In this paper, we always assume that the counting measure $\mu$ has [*doubling property*]{}, i.e., there exists a positive constant $D$ such that $$\label{doubling}
\mu\big(B(\lambda, 2r)\big) \le D \mu\big(B(\lambda, r)\big) \ \ \text{for all } \lambda \in V \text{ and } r > 0$$ [@CJS18; @shinsun19; @YYH13book]. We denote the minimal constant $D$ in the doubling property by $D(\mu)$, which is also known as the [*doubling constant*]{} of the measure $\mu$. Applying the doubling property repeatedly, we have $$\label{doubling2}
\mu(B(\lambda, r))\le \mu\big(B(\lambda, 2^{\lceil \log_2 (r/r')\rceil} r')\big) \le D(\mu) (r/r')^{\log_2 D(\mu)}\mu(B(\lambda, r')), \ r\ge r'> 0.$$ Taking $r'=1-\epsilon$ in for sufficiently small $\epsilon>0$, we conclude that the counting measure $\mu$ has [*polynomial growth*]{} in the sense that $$\label{polynomial-growth}
\mu(B(\lambda, r))\le D_1 (r+1)^{d_1}\ \ \text{for all } \ \lambda \in V \text{ and } r\ge 0,$$ where $D_1$ and $d_1$ are positive constants. The notion of polynomial growth for the counting measure $\mu$ is introduced in [@CJS18], where the minimal constants $d_1$ and $D_1$ in , to be denoted by $d_{\mathcal G}$ and $D_{\mathcal G}$, are known as the [*Beurling dimension*]{} and [*density*]{} of the graph ${\mathcal G}$ respectively.
Let $N\ge 0$. We say that a set $V_N\subset V$ of fusion vertices is [*maximal $N$-disjoint*]{} if $$\label{maximum1}
B(\lambda, N)\cap \big(\cup_{\lambda_m \in{V_N}} B(\lambda_m, N)\big) \ne \emptyset
\ \text{ for all } \lambda \in V$$ and $$\label{maximum2}
B(\lambda_m, N)\cap B(\lambda_n, N) =\emptyset \ \text{ for all distinct } \lambda_m, \lambda_n\in V_N.$$ For $N=0$, one may verify that the whole set $V$ is the only maximal $N$-disjoint set $V_N$, i.e., $$\label{V1.pro}
V_N=V \ {\rm if} \ N=0,$$ while for $N\ge 1$, one may construct many maximal $N$-disjoint sets $V_N$. For example, we can construct a maximal $N$-disjoint set $V_N = \{\lambda_m, m\ge 1\}$ by taking a vertex $\lambda_1\in V$ and defining vertices $\lambda_m, m\ge 2$, recursively by $\lambda_m= {\rm arg \ min}_{\lambda\in A_m} \rho(\lambda, \lambda_1),$ where $A_m=\{\lambda\in V, B(\lambda, N)\cap \cup_{m'=1}^{m-1}
B(\lambda_{m'}, N)=\emptyset\}$ [@CJS18]. For a maximal $N$-disjoint set $V_N$ of fusion vertices, it is observed in [@CJS18; @shinsun19] that for any $N'\ge 2N$, $B(\lambda_m, N'), \lambda_m\in V_N$, form a finite covering of the whole set $V$, and $$\label{overlap-counting}
1 \le \inf_{\lambda \in V} \sum_{\lambda_m \in V_N}\chi_{B(\lambda_m, N')}(\lambda)
\le \sup_{\lambda \in V} \sum_{\lambda_m \in V_N}\chi_{B(\lambda_m, N')}(\lambda)
\le \big(D(\mu)\big)^{\lceil \log_2(2N'/N+1)\rceil}.$$ For $\lambda\in V$ and $R\ge 0$, set $$\label{AR.def}
A_R(\lambda, N):=\big\{ \lambda_m \in V_N :\ \rho(\lambda_m, \lambda) \le (N+1)R \big\},$$ and let $\lambda_{m_0}\in A_R(\lambda, N)$ be so chosen that $$\label{lambdam0.def0}
\mu(B(\lambda_{m_0}, N))= \inf_{\lambda_m\in A_R(\lambda, N)} \mu(B(\lambda_m, N)).$$ Then we obtain from , and that $$\begin{aligned}
\label{counting}
\mu(A_R(\lambda, N))
& \le & \frac{ \sum_{\lambda_m\in A_R(\lambda, N)} \mu(B(\lambda_m, N))}{\mu(B(\lambda_{m_0}, N))}
= \frac{ \mu\big( \cup_{\lambda_m\in A_R(\lambda, N)} B(\lambda_m, N)\big)}{\mu(B(\lambda_{m_0}, N))}
\nonumber\\
& \le &
\frac{ \mu(B(\lambda_{m_0}, N+2(N+1)R)}{\mu(B(\lambda_{m_0}, N))}
\le (D(\mu))^3 (R+1)^{\log_2 D(\mu)}.\end{aligned}$$ Therefore the counting measure $\mu$ on the graph ${\mathcal G}$ has [*strong polynomial growth*]{} since there exist two positive constants $D$ and $d$ such that $$\label{strongpolynomial-growth}
\sup_{\lambda\in V} \mu\big(\big\{ \lambda_m \in V_N : \ \rho(\lambda_m, \lambda) \le (N+1)R \big\}\big) \le D (R+1)^{d}$$ hold for all $R, N\ge 0$ and maximal $N$-disjoint set $V_N$ of fusion vertices. Recall that the whole set $V$ is the only maximal $N$-disjoint set $V_N$ for $N=0$. So in this paper the minimal constants $d$ and $D$ in , to be denoted by $\tilde d_{\mathcal G}$ and $\tilde D_{\mathcal G}$, are considered as [*generalized Beurling dimension*]{} and [*density*]{} respectively. Moreover it follows from and that $$d_{\mathcal G}\le \tilde d_{\mathcal G}
\le \log_2 D(\mu)$$ where $d_{\mathcal G}$ is the Beurling dimension of the graph ${\mathcal G}$.
We say that the counting measure $\mu$ on the graph ${\mathcal G}$ is [*Ahlfors $d_0$-regular*]{} if there exist positive constants $B_3$ and $B_4$ such that $$\label{ahlfors.def}
B_3 (r+1)^{d_0}\le \mu \big(B(\lambda, r)\big)\le B_4 (r+1)^{d_0}$$ hold for all balls $B(\lambda, r)$ with center $\lambda\in V$ and radius $0\le r \le {\rm diam}\ {\mathcal G}$, where ${\rm diam}\ {\mathcal G}$ denotes the diameter of the graph ${\mathcal G}$ [@Keith2008; @YYH13book]. Clearly for a graph ${\mathcal G}$ with its counting measure $\mu$ being Ahlfors $d_0$-regular, its Beurling dimension $d_{\mathcal G}$ is equal to $d_0$. In the following proposition, we show that the generalized Beurling dimension $\tilde d_{\mathcal G}$ is also equal to $d_0$, see Section \[regular.prop.pfsection\] for the proof.
\[regular.prop\] Let ${\mathcal G}$ be a connected simple graph. If the counting measure $\mu$ is Ahlfors $d_0$-regular, then $\tilde d_{\mathcal G}=d_0$.
Beurling algebras of matrices on graphs {#beurling.subsection}
---------------------------------------
Let ${\mathcal G}:=(V, E)$ be a connected simple graph with its counting measure $\mu$ satisfying the doubling property . For $1 \le r \le \infty$ and $\alpha \ge 0$, we define the Beurling algebra ${\mathcal B}_{r, \alpha}:={\mathcal B}_{r,\alpha}({\mathcal G})$ by $$\label{beurling.def}
{\mathcal B}_{r, \alpha}({\mathcal G}):=\Big\{A=\big(a(\lambda, \lambda') \big)_{\lambda, \lambda' \in V}: \ \ \|A\|_{{\mathcal B}_{r, \alpha}}<\infty\Big\},$$ where $d_{\mathcal G}$ is the Beurling dimension of the graph ${\mathcal G}$, $h_A(n)=\sup_{\rho(\lambda, \lambda')\ge n} |a(\lambda, \lambda')|, n\ge 0$, and $$\label{bralpha.norm}
\|A\|_{{\mathcal B}_{r, \alpha}}:=\left\{\begin{array}{ll}
\big(\sum_{n=0}^\infty h_A(n)^r (n+1)^{\alpha r+d_{\mathcal G}-1} \big)^{1/r} & {\rm if} \ 1\le r<\infty\\[5pt]
\sup_{n\ge 0} h_A(n) (n+1)^\alpha
& {\rm if} \ r=\infty.
\end{array}
\right.$$ The Beurling algebra ${\mathcal B}_{r,\alpha}({\mathcal G})$ is introduced in [@sunca11] for the lattice graph ${\mathcal Z}^d$ and for an arbitrary simple graph ${\mathcal G}$ in [@shinsun19]. For a matrix $A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}$ in the Beurling algebra $ {\mathcal B}_{r,\alpha}({\mathcal G})$, we define approximation matrices $A_K, \ K\ge 1$, with finite bandwidth by $$\label{defAN}
A_K:=\big( a(\lambda, \lambda')
\chi_{[0,1]}(\rho(\lambda, \lambda')/K)\big)_{\lambda, \lambda'\in V}.$$ For the Beurling algebra ${\mathcal B}_{r, \alpha}({\mathcal G})$, we recall some elementary properties where the first four conclusions have been established in [@shinsun19], see Section \[beurling.prop.pfsection\] for the proof.
\[beurling.prop\] Let ${\mathcal G}:=(V, E)$ be a connected simple graph such that its counting measure $\mu$ satisfies the doubling property with the doubling constant $D(\mu)$. Then the following statements hold.
- ${\mathcal B}_{r, \alpha}({\mathcal G})$ with $1\le r\le \infty$ and $\alpha\ge 0$ are solid in the sense that $$\label{solid.eq}
\|A\|_{{\mathcal B}_{r, \alpha}}\le \|B\|_{{\mathcal B}_{r, \alpha}}$$ hold for all $A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}$ and $B=(b(\lambda, \lambda'))_{\lambda, \lambda'\in V}$ satisfying $|a(\lambda, \lambda')|\le |b(\lambda, \lambda')|$ for all $\lambda, \lambda'\in V.$
- ${\mathcal B}_{1, 0}({\mathcal G})$ is a Banach algebra, and $$\|AB\|_{{\mathcal B}_{1, 0}}\le d_{\mathcal G}
D_{\mathcal G} 2^{d_{\mathcal G}+1} \|A\|_{{\mathcal B}_{1, 0}} \|B\|_{{\mathcal B}_{1, 0}} \ {\rm for \ all}\ A, B\in {\mathcal B}_{1, 0}({\mathcal G}).$$
- ${\mathcal B}_{r, \alpha}({\mathcal G})$ with $1\le r\le \infty$ and $\alpha>d_{\mathcal G}(1-1/r)$ are Banach algebras, and $$\label{beurling.prop.eq3}
\|AB\|_{{\mathcal B}_{r, \alpha}} \le d_{\mathcal G} D _{\mathcal G} 2^{\alpha +1+d_{\mathcal G}/r} \Big(\frac{\alpha-(d_{\mathcal G}-1)(1-1/r)}{\alpha-d_{\mathcal G}(1-1/r)}\Big)^{1-1/r}
\|A\|_{{\mathcal B}_{r, \alpha}}\|B\|_{{\mathcal B}_{r, \alpha}}\ \ {\rm for \ all} \ A, B\in {\mathcal B}_{r, \alpha}({\mathcal G}).$$
- ${\mathcal B}_{r, \alpha}({\mathcal G})$ with $1\le r\le \infty$ and $\alpha>d_{\mathcal G}(1-1/r)$ are Banach subalgebras of ${\mathcal B}_{1, 0}({\mathcal G})$, and $$\label{beurling.prop.eq1}
\|A\|_{{\mathcal B}_{1,0}}
\le \Big( \frac{\alpha -(d_{\mathcal G}-1)(1-1/r)}{\alpha -d_{\mathcal G}(1-1/r)}\Big)^{1-1/r}
\|A\|_{{\mathcal B}_{r,\alpha}} \ {\rm for \ all} \ A\in {\mathcal B}_{r, \alpha}({\mathcal G}).$$
- A matrix $A$ in ${\mathcal B}_{r, \alpha}({\mathcal G})$ with $1\le r\le \infty$ and $\alpha>d_{\mathcal G}(1-1/r)$ is well approximated by its truncation $A_K, K\ge 1$, in the norm $\|\cdot\|_{{\mathcal B}_{1, 0}}$, $$\label{AN-appr}
\|A-A_K\|_{{\mathcal B}_{1, 0}} \le C_0 \|A\|_{{\mathcal B}_{r,\alpha}} K^{-\alpha +d_{\mathcal G}(1-1/r)},$$ where $$C\_0= {
[ll]{} 2\^[+1 ]{} & [if]{} r=1\
& [if]{} r>1.
. $$
Beurling algebras of matrices on a maximal disjoint set of fusion vertices {#beurlingmaximal.subsection}
--------------------------------------------------------------------------
Given $1\le r\le \infty, \tilde \alpha\ge 0$ and a maximal $N$-disjoint set $V_N$ of fusion vertices, we define Beurling algebras of matrices $B:=\big(b(\lambda_m, \lambda_k)\big)_{\lambda_m, \lambda_k\in V_N} $ on $V_N$ by $$\mathcal{B}_{r,\tilde \alpha ;N}(V_N):=\big\{B, \
\|B\|_{\mathcal{B}_{r,\tilde \alpha;N}}<\infty\big\}$$ where $$\label{N-norm}
\|B\|_{\mathcal{B}_{r,\tilde \alpha;N}}:= \left\{\begin{array}{ll}
\Big(\sum_{n=0}^\infty (n+1)^{\tilde \alpha r+\tilde d_{\mathcal G}-1} \Big(\sup_{\rho(\lambda_m, \lambda_k)\ge n (N+1)}
|b(\lambda_m, \lambda_k )|\Big)^r\Big)^{1/r} & {\rm if} \ 1\le r<\infty\\
\sup_{n\ge 0} (n+1)^{\tilde \alpha } \big(\sup_{\rho(\lambda_m, \lambda_k)\ge n (N+1)}
|b(\lambda_m, \lambda_k )| \big) & {\rm if} \ r=\infty.\end{array}
\right.$$ The Banach algebra $\mathcal{B}_{r,\tilde \alpha;N}(V_N)$ is introduced in [@shinsun19], where the counting measure $\mu$ is assumed to be Ahlfors regular in which the generalized Beurling dimension $\tilde d_{\mathcal G}$ and the Beurling dimension $d_{\mathcal G}$ coincides by Proposition \[regular.prop\]. Following the argument used in the proof of Proposition \[beurling.prop\] with the polynomial growth property replaced by the strong polynomial growth property , we have the following properties for Banach algebras ${\mathcal B}_{r, \tilde \alpha; N}(V_N)$ of matrices on $V_N$.
\[VNbanach.prop\] Let ${\mathcal G}:=(V, E)$ be a connected simple graph such that its counting measure $\mu$ satisfies the doubling property , and $V_N$ be a maximal $N$-disjoint set of fusion vertices. Then the following statements hold.
- $\mathcal{B}_{1,0;N}(V_N)$ is a Banach algebra and $$\label{banach-algebra}
\|AB\|_{\mathcal{B}_{1,0;N}}\le \tilde d_{\mathcal G} \tilde D_{\mathcal G} 2^{3\tilde d_{\mathcal G}+1} \|A\|_{\mathcal{B}_{1,0;N}}\|B\|_{\mathcal{B}_{1,0;N}}, \ A, B\in {\mathcal B}_{1, 0; N}(V_N).$$
- ${\mathcal B}_{r, \tilde \alpha; N}(V_N)$ with $1\le r\le \infty$ and $\tilde \alpha > \tilde d_{\mathcal G} (1-1/r)$ are Banach subalgebras of ${\mathcal B}_{1, 0; N}(V_N)$, and $$\label{banach-algebra+2}
\|A\|_{\mathcal{B}_{1,0 ;N}}\le
\Big( \frac{\tilde \alpha -(\tilde d_{\mathcal G}-1)(1-1/r)}{\tilde \alpha -
\tilde d_{\mathcal G}(1-1/r)}\Big)^{1-1/r}
\|A\|_{\mathcal{B}_{r,\tilde \alpha;N}}, A\in {\mathcal B}_{r, \tilde \alpha; N}(V_N).$$
- ${\mathcal B}_{r, \tilde \alpha ;N}(V_N)$ with $1\le r\le \infty$ and $\tilde \alpha> \tilde d_{\mathcal G} (1-1/r)$ are Banach algebras, and $$\begin{aligned}
\label{banach-algebra+1}
\|AB\|_{\mathcal{B}_{r,\tilde \alpha ;N}}&\le &
\tilde d_{\mathcal G}
\tilde D_{\mathcal G}
2^{\tilde \alpha+ \tilde d_{\mathcal G}(2+1/r)+2}\Big( \frac{\tilde \alpha -(\tilde d_{\mathcal G}-1)(1-1/r)}{\tilde\alpha -
\tilde d_{\mathcal G}(1-1/r)}\Big)^{1-1/r}
\nonumber \\& &
\times \|A\|_{\mathcal{B}_{r,\tilde \alpha;N}}\|B\|_{\mathcal{B}_{r, \tilde \alpha;N}},
\ \ A, B\in
{\mathcal B}_{r, \tilde \alpha; N}(V_N).\end{aligned}$$
Beurling algebra on the graph ${\mathcal G}$ and on its maximal $N$-disjoint set $V_N$ of fusion vertices are closely related. For $N=0$, we have $${\mathcal B}_{r, \tilde \alpha; 0}(V_0)={\mathcal B}_{r, \tilde \alpha+ (\tilde d_{\mathcal G}-d_{\mathcal G})/r}({\mathcal G})$$ as the only maximal $0$-disjoint set $V_0$ is the whole vertex set $V$. For $N\ge 1$, we have the following results about Beurling algebras on a graph and its maximal disjoint sets, which will be used in our proofs to establish the equivalence of weighted stability for different exponents and weights and also the norm-controlled inversion. The detailed proof will be given in Section \[beurlingonmaximalsets.pr.pfsection\].
\[beurlingonmaximalsets.pr\] Let $1\le r\le \infty$, ${\mathcal G}:=(V, E)$ be a connected simple graph such that its counting measure $\mu$ satisfies the doubling property , and $V_N, N\ge 1$, be a maximal $N$-disjoint set of fusion vertices. Then the following statements hold.
- If $A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}\in {\mathcal B}_{r, \alpha} ({\mathcal G}), \alpha \ge 0$, then its submatrix $B= (a(\lambda_m, \lambda_k))_{\lambda_m, \lambda_k\in V_N}$ belongs to ${\mathcal B}_{r, \alpha - (\tilde d_{\mathcal G}-d_{\mathcal G})/r; N}$, and $$\label{beurlingonmaximalsets.pr.eq1}
\|B\|_{{\mathcal B}_{r, \alpha - (\tilde d_{\mathcal G}-d_{\mathcal G})/r; N}}\le \|A\|_{{\mathcal B}_{r, \alpha}}.$$
- If $B= (b(\lambda_m, \lambda_k))_{\lambda_m, \lambda_k\in V_N}\in {\mathcal B}_{r, \alpha; N}, \alpha\ge 0$, the matrix $$\label{beurlingonmaximalsets.pr.eq2}
A=\Big(\sum_{\lambda_m\in B(\lambda, 2N)} \sum_{\lambda_k\in B(\lambda', 4N)} b(\lambda_m, \lambda_k)\Big)_{\lambda, \lambda'\in V}$$ on the graph ${\mathcal G}$ belongs to $B_{r, \alpha+ (\tilde d_{\mathcal G}-d_{\mathcal G})/r}({\mathcal G})$, and $$\label{beurlingonmaximalsets.pr.eq3}
\|A\|_{{\mathcal B}_{r, \alpha+ (\tilde d_{\mathcal G}-d_{\mathcal G})/r}}\le 8^{\alpha +\tilde d_{\mathcal G}/r} (D(\mu))^{7} N^{\alpha +\tilde d_{\mathcal G}/r}
\|B\|_{{\mathcal B}_{r, \alpha; N}}.$$
- If $A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}\in {\mathcal B}_{r, \alpha} ({\mathcal G})$ for some $\alpha> d_{\mathcal G}(1-1/r)$, then the matrix $$S_{A,N}=\big(S_{A, N}(\lambda_m, \lambda_k)\big)_{\lambda_m, \lambda_k\in V_N},$$ belongs to ${\mathcal B}_{r, \alpha-(\tilde d_{\mathcal G}-d_{\mathcal G})/r; N}$, and $$\label{SAN.estimate}
\|S_{A, N}\|_{\mathcal{B}_{r, \alpha-(\tilde d_{\mathcal G}-d_{\mathcal G})/r;N}}\le
\tilde C_0
\|{A}\|_{{\mathcal B}_{r,\alpha}}\times \left\{\begin{array}{ll} N^{-\min(1,\alpha-d_{\mathcal G}/r')} & {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
N^{-1} (\ln (N+1))^{1/r'} & {\rm if} \ \alpha= d_{\mathcal G}/r'+1,
\end{array}\right.$$ where $1/r'=1-1/r$, $h_A(n)=\sup_{\rho(\lambda, \lambda')\ge n} |a(\lambda, \lambda')|, n\ge 0$, $$\tilde C_0= 2^{4\alpha+4d_{\mathcal G}/r+2} \times
\left\{
\begin{array}{ll}
\big(\frac{1+|\alpha -1-d_{\mathcal G}/r'|}{|\alpha-1-d_{\mathcal G}/r'|}\big)^{1/r'}
& {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1
\\
1 & {\rm if} \ \alpha= d_{\mathcal G}/r'+1,
\end{array}\right.$$ and for $\lambda_m, \lambda_k\in V_N$, $$\label{VAN-1}
S_{ A, N}(\lambda_m, \lambda_k)= \left\{
\begin{array}{ll}
N^{d_{\mathcal G}} h_{ A}\big(\rho(\lambda_m, \lambda_k)/2\big)
&{\rm if }\ \rho(\lambda_m, \lambda_k)> 12(N+1)
\\
N^{-1} \sum_{n=0}^{2N} h_A(n) (n+1)^{d_{\mathcal G}}
&
{\rm if }\ \rho(\lambda_m, \lambda_k)\le 12(N+1).
\end{array}
\right.$$
Weighted norm inequalities {#weighted.subsection}
--------------------------
Let ${\mathcal G}:=(V, E)$ be a connected simple graph with its counting measure $\mu$ satisfying the doubling property . For $1\le p<\infty$, a positive function $w=(w(\lambda))_{\lambda\in V}$ on the vertex set $V$ is a [*Muckenhoupt $A_p$-weight*]{} if there exists a positive constant $C$ such that $$\label{Aq-weight1}
\Big( \frac{1}{\mu(B)} \sum_{\lambda \in B} w(\lambda)\Big)
\Big( \frac{1}{\mu(B)} \sum_{\lambda \in B} \big(w(\lambda)\big)^{-1/(p-1)}\Big)^{p-1}
\le C$$ for $1<p<\infty$, and a Muckenhoupt $A_1$-weight if $$\label{Aq-weight2}
\frac{1}{\mu(B)} \sum_{\lambda \in B} w(\lambda) \le C \inf_{\lambda \in B} w(\lambda)$$ for any ball $B\subset V$ [@garciabook]. The smallest constant $C$ for which holds for $1<p<\infty$, and holds for $p=1$, respectively is known as the $A_p$-bound of the weight $w$ and is denoted by $A_p(w)$. An equivalent definition of a Muckenhoupt $A_p$-weight $w:=(w(\lambda))_{\lambda\in V}$ is that $$\label{Aq-char}
\Big( \frac{1}{\mu(B)} \sum_{\lambda \in B} |c(\lambda)|\Big)^p
\Big( \frac{1}{\mu(B)} \sum_{\lambda \in B} w(\lambda)\Big)
\le \frac{A_p(w)} {\mu(B)} \sum_{\lambda \in B} |c(\lambda)|^p w(\lambda)$$ holds for all balls $B\subset V$ and sequences $c:=\big(c(\lambda)\big)_{\lambda\in V}\in \ell^p_w$, where $A_p(w)$ is $A_p$-bound of the weight $w$. For $\lambda\in V$ and $r\ge 0$, set $$w(B(\lambda, r))=\sum_{\lambda'\in B(\lambda, r)} w(\lambda').$$ It is well known that a Muckenhoupt $A_p$-weight $w$ is a doubling measure. In fact, replacing the ball $B$ and the sequence $c$ by $B(\lambda, 2^jr), 1\le j\in {{\mathbb Z}}$ and the index sequence on $B(\lambda, r)$ in and using the doubling condition for the counting measure $\mu$, we obtain that $$\label{weighteddoubling}
w(B(\lambda, 2^jr))\le A_p(w) \Big(\frac{\mu(B(\lambda, 2^jr)}{\mu(B(\lambda, r)}\Big)^p w(B(\lambda, r))\le
(D(\mu))^{jp} A_p(w) w(B(\lambda, r))$$ hold for all $\lambda\in V, r\ge 0$ and positive integers $j$.
Weighted norm inequalities of linear operators are an important topic in harmonic analysis, see [@garciabook] and references therein for historical remarks. In the following proposition, we show that the Banach algebra ${\mathcal B}_{1, 0}({\mathcal G})$ is a Banach subalgebra of ${\mathcal B}(\ell^p_w)$, see Section \[weighted.prop.pfsection\] for the proof.
\[weighted.prop\] Let ${\mathcal G}:=(V, E)$ be a connected simple graph such that its counting measure $\mu$ satisfies the doubling property . Then ${\mathcal B}_{1,0}({\mathcal G})$ is a subalgebra of ${\mathcal B}(\ell_w^p)$ for any $1\le p<\infty$ and Muckenhoupt $A_p$-weight $w$, and $$\label{continuity}
\|Ac\|_{p,w} \le 2^{3d_{\mathcal G}} D_{\mathcal G} (A_p(w))^{1/p}
\|A\|_{{\mathcal B}_{1,0}} \|c\|_{p, w}
\ \ {\rm for \ all} \ A\in {\mathcal B}_{1, 0}({\mathcal G})\ {\rm and}\ c \in \ell_w^p.$$
By Propositions \[beurling.prop\] and \[weighted.prop\], we conclude that ${\mathcal B}_{r, \alpha}({\mathcal G})$ with $1\le r\le \infty$ and $\alpha>d_{\mathcal G}(1-1/r)$ are Banach subalgebras of ${\mathcal B}(\ell^p_w)$ too. We remark that the subalgebra property in Proposition \[weighted.prop\] was established in [@CJS18; @shinsun19] for the unweighted case and in [@sunca11] for the weighted case on the lattice graph ${\mathcal Z}^d$.
Polynomial control on optimal lower stability bounds {#lowerbound.section}
====================================================
In this section, we show that weighted stabilities of matrices in a Beurling algebra for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of the optimal lower stability bound for one exponent and weight is dominated by a polynomial of reciprocal of the optimal lower stability bound for another exponent and weight.
\[mainthm\] Let $ 1 \le r \le \infty, \ 1\le p, q<\infty$, ${\mathcal G}:=(V,E)$ be a connected simple graph satisfying the doubling property , $w, \, w'$ be Muckenhoupt $A_p$-weight and $A_q$-weight respectively, and let $A \in {{\mathcal B}_{r,\alpha}}({\mathcal G})$ for some $\alpha >\tilde d_{\mathcal G}- d_{\mathcal G}/r$, where $d_{\mathcal G}$ and $\tilde d_{\mathcal G}$ are the Beurling and generalized Beurling dimension of the graph ${\mathcal G}$ respectively. If $A$ has $\ell^p_w$-stability with the optimal lower stability bound $\beta_{p, w}(A)$, $$\label{mainthm.eq-1}
\beta_{p, w}(A) \|c\|_{p,w} \le \|A c\|_{p,w} \ \ {\rm for \ all} \ c\in \ell^p_w,$$ then $A$ has $\ell^{q}_{w'}$-stability with the optimal lower stability bound denoted by $\beta_{q, w'}(A)$, $$\label{mainthm.eq0}
\beta_{q, w'}(A) \|c\|_{q,w'} \le \|A c\|_{q,w'} \ \ {\rm for \ all}\ c\in \ell^{q}_{w'}.$$ Moreover, there exists an absolute constant $C$, independent of matrices $A\in {{\mathcal B}_{r,\alpha}}({\mathcal G})$ and weights $w, \, w'$, such that $$\begin{aligned}
\label{lower-bound}
\frac{ \beta_{p, w}(A)}{ \beta_{q,w'}(A)} & \le & C
\big(A_{q}(w')\big)^{1/q} \big(A_p(w)\big)^{1/p}
\Big( \frac{\big(A_p(w)\big)^{2/p} \|A\|_{{\mathcal B}_{r,\alpha}}}{\beta_{p, w}(A)}\Big)^{E(\alpha, r, d_{\mathcal G})
}\nonumber\\
& & \times
\left\{
\begin{array}{ll} 1
& {\rm if} \ \alpha\ne 1+d_{\mathcal G}/r'\\
\Big(\ln\Big(\frac{(A_p(w))^{2/p}
\|A\|_{{\mathcal B}_{r,\alpha}}}{\beta_{p,w}(A)}\Big)\Big)^{(2 d_{\mathcal G}+1)/r'}
& {\rm if} \ \alpha=1+d_{\mathcal G}/r'
\end{array}\right.\end{aligned}$$ where $1/r'=1-1/r$ and $$E(\alpha, r, d_{\mathcal G})=\frac{\tilde d_{\mathcal G}+d_{\mathcal G}+1}{\min\big(1,\alpha-d_{\mathcal G}/r'\big)}.$$
\[mainthm.rem\] [ The equivalence of unweighted stabilities for different exponents is discussed for matrices in Baskakov-Gohberg-Sjöstrand algebras, Jaffard algebras and Beurling algebras [@akramjfa09; @shincjfa09; @sunca11; @tesserajfa10], for convolution operators [@barnes90], and for localized integral operators of non-convolution type [@fang18; @fang17; @rssun12; @shincjfa09]. For a matrix $A$ in the Beurling algebra ${\mathcal B}_{r, \alpha}$ with $1\le r\le \infty$ and $\alpha>d_{\mathcal G}(1-1/r)$, Shin and Sun use the boot-strap argument in [@shinsun19] to prove that reciprocal of its optimal lower unweighted stability bound for one exponent is dominated by a polynomial of reciprocal of its optimal lower unweighted stability bound for another exponent, $$\label{stability.oldthm.eq2}
\frac{\|A\|_{{\mathcal B}_{r, \alpha}}}{\beta_{q, w_0}}
\le C \left\{\begin{array}
{ll} \Big(\frac{\|A\|_{{\mathcal B}_{r, \alpha}}}{\beta_{p, w_0}}\Big)^{(1+\theta(p, q))^{K_0}}
& {\rm if} \ \alpha\ne 1+d_{\mathcal G}/r'\\
\Big(\frac{\|A\|_{{\mathcal B}_{r, \alpha}}}{\beta_{p, w_0}}
\ln \big(1+\frac{\|A\|_{{\mathcal B}_{r, \alpha}}}{\beta_{p, w_0}}\big)\Big)^{(1+\theta(p, q))^{K_0}}
& {\rm if} \ \alpha=1+d_{\mathcal G}/r',
\end{array}\right.$$ where $C$ is an absolute constant, $K_0$ is a positive integer satisfying $K_0> \frac{d_{\mathcal G}}{\min(\alpha-d_{\mathcal G}/r', 1)}$, and $$\theta(p, q)=\frac{d_{\mathcal G}|1/p-1/q|}{K_0 \min(\alpha-d_{\mathcal G}/r', 1)-d_{\mathcal G}|1/p-1/q|}.$$ Given $1\le p<\infty$, we remark that for an exponent $q$ close to $p$, the conclusion provides a better estimate to the optimal lower unweighted stability bound $\beta_{q, w_0}(A)$ than the one in with $w=w'=w_0$, while the conclusion with $w=w'=w_0$ gives a tighter estimate to the optimal lower unweighted stability bound $\beta_{q, w_0}(A)$ than the one in when $q$ is close to one or infinity. ]{}
For $ 1\le N\in{{\mathbb Z}}$ and $ \lambda \in V$, we introduce a truncation operator $\chi_\lambda^N$ and its smooth version $\Psi_\lambda^N$ by $$\label{def-trunc}
\chi_{\lambda}^N:\big(c(\lambda)\big)_{{\lambda\in V}} \longmapsto
\big( \chi_{[0,N]} \big(\rho(\lambda,\lambda') \big) c(\lambda')
\big)_{{\lambda'\in V}}$$ and $$\label{def-smooth-trunc}
\Psi_{\lambda}^N:\big(c(\lambda)\big)_{{\lambda\in V}}\longmapsto
\big( \psi_0 \big(\rho(\lambda,\lambda')/N \big) c(\lambda')
\big)_{{\lambda'\in V}},$$ where $\psi_0(t)=\max\{0,\min(1,3-2|t|)\}$ is the trapezoid function. The operators $\chi_{\lambda}^N$ and $\Psi_{\lambda}^N$ localize a sequence to a neighborhood of $\lambda$ and they can be considered as diagonal matrices with entries $\chi_{B(\lambda,N)}(\lambda') $ and $\psi_0(\rho(\lambda,\lambda')/N), \lambda'\in V$ respectively.
Let $ V_{N}$ be a maximal $N$-disjoint set of fusion vertices. To prove Theorem \[mainthm\], we start from an estimate to the weighted terms $\big(w\big(B(\lambda_m, 4N)\big))^{-1/p}
\|\Psi_{\lambda_m}^{2N} c\|_{p,w},\lambda_m\in V_N$, for sufficiently large $N$, which is established in [@shinsun19] for the trivial weight $w_0$.
\[tech.lem\] Let $1\le p<\infty, 1\le r\le \infty, \alpha>d_{\mathcal G}(1-1/r)$, $w$ be a Muckenhoupt $A_p$-weight, and $A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}\in {\mathcal B}_{r, \alpha}({\mathcal G})$ have $\ell^p_w$-stability. Assume that $N\ge 1$ is a positive integer such that $$\label{tech.lem.eq1}
2C_1 (A_p(w))^{1/p}
\|A\|_{{\mathcal B}_{r,\alpha}} N^{-\alpha +d_{\mathcal G}(1-1/r)}\le \beta_{p, w}(A),$$ where $\beta_{p, w}(A)$ is the optimal lower $\ell^p_w$-stability bound, $C_0$ is the constant in and $C_1= 2^{3d_{\mathcal G}} C_0 D_{\mathcal G}$. Then for all maximal $N$-disjoint sets $ V_{N}$ of fusion vertices and weighted sequences $c\in \ell^p_w$, we have $$\begin{aligned}
\label{tech.lem.eq2}
& & \beta_{p, w}(A) \big(w\big(B(\lambda_m, 4N)\big))^{-1/p}
\|\Psi_{\lambda_m}^{2N} c\|_{p,w} \le 2 \big(w\big(B(\lambda_m, 4N)\big))^{-1/p}
\|\Psi_{\lambda_m}^{2N} {A}c\|_{p,w} \nonumber\\
& & \qquad\qquad
+ C_2 \big(A_p(w)\big)^{2/p}
\sum_{\lambda_k\in V_N} S_{A, N}(\lambda_m, \lambda_k) \big(w\big(B(\lambda_k, 4N)\big))^{-1/p} \|\Psi_{\lambda_k}^{2N} c\|_{p,w},
\ \lambda_m\in V_N,
$$ where the smooth truncation operators $\Psi_{\lambda_m}^{2N}, \lambda_m\in V_N$, are defined in , the matrix $S_{A, N}=(S_{A, N}(\lambda_m, \lambda_k))_{\lambda_m, \lambda_k\in V_N} $ is given in , and $C_2\ge 2$ is an absolute constant.
Let $[\Psi_{\lambda_m}^{2N}, {A}]:=\Psi_{\lambda_m}^{2N}{A}- {A} \Psi_{\lambda_m}^{2N}$ be the commutator between the smooth truncation operator $\Psi_{\lambda_m}^{2N}$ and the matrix ${A}$ [@shinsun19; @shincjfa09; @sjostrand94]. A crucial step in the proof of Lemma \[tech.lem\] is the following estimate to the commutator $[\Psi_{\lambda_m}^{2N}, {A}]$, $$\|\chi_{\lambda_m}^{4N } [\Psi_{\lambda_m}^{2N}, {A}]\chi_{\lambda_k}^{3N}\|_{{\mathcal B}_{1, 0}}\lesssim S_{A, N}(\lambda_m, \lambda_k),
\ \lambda_m, \lambda_k\in V_N,$$ see Section \[tech.lem.pfsection\] for the detailed argument.
By Propositions \[VNbanach.prop\] and \[beurlingonmaximalsets.pr\], there exists an absolute constant $C_3$ such that $$\label{norm-power}
\big\|\big(S_{A, N}\big)^{\ell}\big\|_{\mathcal{B}_{r, \alpha-(\tilde d_{\mathcal G}-d_{\mathcal G})/r;N}} \le
(C_3\|{A}\|_{{\mathcal B}_{r,\alpha}})^l \times \left\{\begin{array}{ll} N^{-\min(1,\alpha-d_{\mathcal G}/r')l } & {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
N^{-l} (\ln (N+1))^{l/r'} & {\rm if} \ \alpha= d_{\mathcal G}/r'+1
\end{array}\right.$$ hold for all $l\ge 1$, where $1/r'=1-1/r$. Applying and repeatedly, we have the following crucial estimates and , see Section \[tech.lem2.pfsection\] for the proof.
\[tech.lem2\] Let $1\le p<\infty, w$ be a Muckenhoupt $A_p$-weight, $A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}\in {\mathcal B}_{r, \alpha}({\mathcal G})$ for some $1\le r\le \infty$ and $\alpha>\tilde d_{\mathcal G}-d_{\mathcal G}/r$. Assume that $A$ has $\ell^p_w$-stability with the optimal lower stability bound $\beta_{p, w}(A)$ and that $1\le N\in {{\mathbb Z}}$ satisfies $$\label{tech.lem2.eq1}
\beta_{p, w} (A) \ge 2 \max(C_1, C_2 C_3) \big(A_p(w)\big)^{2/p}
\|{A}\|_{{\mathcal B}_{r,\alpha}}
\times
\left\{\begin{array}{ll} N^{-\min(1,\alpha-d_{\mathcal G}/r') } & {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
N^{-1} (\ln (N+1))^{1/r'} & {\rm if} \ \alpha=d_{\mathcal G}/r'+1,
\end{array}\right.$$ where $C_1, C_2, C_3$ are absolute constants in , and respectively. Then there exist a matrix $H_{A,N}=(H_{A,N}(\lambda, \lambda'))_{\lambda, \lambda'\in V}$ and two absolute constants $C_4$ and $C_5$ such that $$\label{tech.lem2.eq2}
\|H_{A, N}\|_{{\mathcal B}_{r, \alpha}}\le C_4 N^{\alpha+d_{\mathcal G}/r}
$$ and $$\label{tech.lem2.eq3}
|c(\lambda)|\le C_5
(A_p(w))^{1/p} ( \beta_{p, w}(A))^{-1}
N^{d_{\mathcal G}} \sum_{\lambda'\in V}
H_{A, N}(\lambda, \lambda')|Ac(\lambda')|, \ c\in \ell^p_w.$$
Now we are ready to finish the proof of Theorem \[mainthm\].
As for $\alpha'\ge \alpha$, ${\mathcal B}_{r, \alpha'}({\mathcal G})$ is a Banach subalgebra of ${\mathcal B}_{r, \alpha}({\mathcal G})$. Then it suffices to prove for all $\alpha$ satisfying $$\label{alpha.res}
d_{\mathcal G}/r'\le \tilde d_{\mathcal G}-d_{\mathcal G}/r<\alpha\le \tilde d_{\mathcal G}-d_{\mathcal G}/r+1.$$ Define $$\label{mainthm.pf.eq1}
N_0=
\left\{\begin{array}{ll}
\tilde N_0
& {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
2 \tilde N_0 (\ln (\tilde N_0+1))^{1-1/r}
& {\rm if} \ \alpha = d_{\mathcal G}/r'+1,
\end{array}\right.$$ where $$\tilde N_0=\left\lfloor \Bigg (\frac{ 2 \max(C_1, C_2 C_3) \big(A_p(w)\big)^{2/p}
\|{A}\|_{{\mathcal B}_{r,\alpha}}} {\beta_{p, w}(A)} \Bigg)^{1/\min(1,\alpha-d_{\mathcal G}/r')}\right\rfloor +2.$$ Then one may verify that is satisfied for $N=N_0$. Applying Lemma \[tech.lem2\] with $N$ replaced by $N_0$ and also Proposition \[weighted.prop\], we have $$\label{mainthm.pf.eq2}
\|c\|_{q,w'} \lesssim
( A_{q}(w'))^{1/q} ( A_p(w))^{1/p} (\beta_{p, w}(A))^{-1}
N_{0}^{d_{\mathcal G}}
\|H_{A, N_{0}}\|_{\mathcal{B}_{1,0}}\|{A}c\|_{q,w'}, \ c\in \ell_w^p\cap \ell^{q}_{w'},$$ where $w'$ is an $A_{q}$-weight with $1\le q<\infty$ and the matrix $H_{A, N}$ is given in Lemma \[tech.lem2\]. This together with and the density of $\ell_w^p\cap \ell^{q}_{w'}$ in $\ell^{q}_{w'}$ implies that $$\|c\|_{q, w'}\lesssim
(A_{q}(w'))^{1/q} (A_p(w))^{1/p} (\beta_{p, w}(A))^{-1}N_{0}^{\alpha+d_{\mathcal G}(1+1/r)}
\|Ac\|_{q,w'}\ {\rm for \ all} \ c\in \ell^{q}_{w'}.$$ Therefore $$\frac{
\|{A}\|_{{\mathcal B}_{r,\alpha}}} {\beta_{q, w'}(A)}
\lesssim \frac{\big(A_{q}(w')\big)^{1/q}}{\big(A_p(w)\big)^{1/p}}
\frac{
\big(A_{p}(w)\big)^{2/p}
\|{A}\|_{{\mathcal B}_{r,\alpha}}} {\beta_{p, w}(A)} N_{0}^{\alpha+d_{\mathcal G}(1+1/r)},$$ where $\beta_{q, w'}(A)$ is the optimal lower $\ell^{q}_{w'}$-stability bound of the matrix $A$. This together with and completes the proof of Theorem \[mainthm\].
Norm-controlled inversion {#normedinversion.section}
=========================
In this section, we show that Banach algebras ${{\mathcal B}_{r,\alpha}}({\mathcal G})$ with $ 1 \le r \le \infty$ and $\alpha >\tilde d_{\mathcal G}-d_{\mathcal G}/r$ admit a polynomial norm-controlled inversion in $\mathcal B(\ell^p_w)$ for all $1\le p<\infty$ and Muckenhoupt $A_p$-weights, see Section \[norm-contro-inversion.thm.pfsection\] for the proof.
\[norm-contro-inversion.thm\] Let $ 1 \le r \le \infty, \ 1\le p<\infty$, ${\mathcal G}:=(V,E)$ be a connected simple graph satisfying the doubling property , and $w$ be a Muckenhoupt $A_p$-weight. If $A$ belongs to ${{\mathcal B}_{r,\alpha}}({\mathcal G})$ for some $\alpha>\tilde d_{\mathcal G}-d_{\mathcal G}/r$ and it is invertible in $\mathcal B(\ell^p_{w})$, then $A^{-1}\in {\mathcal B}_{r, \alpha}$. Moreover, there exists an absolute constant $C$ such that $$\begin{aligned}
\label{norm-inversion}
\|A^{-1}\|_{{\mathcal B}_{r, \alpha}}&\le C(A_{p}(w))^{1/p}\|A^{-1}\|_{\mathcal B(\ell^p_{w})}
\Big(\big(A_p(w)\big)^{2/p}\|A^{-1}\|_{\mathcal B(\ell^p_{w})} \|A\|_{{\mathcal B}_{r,\alpha}}\Big)^{(\alpha+d_{\mathcal G}(1+1/r))/\min(\alpha-d_{\mathcal G}/r', 1)}
\nonumber \\&
\quad\times
\left\{
\begin{array}{ll}
1
&{\rm if }\ \alpha\neq 1+d_{\mathcal G}/r'
\\
\Big(\ln\big((A_p(w))^{2/p}\|A^{-1}\|_{\mathcal B(\ell^p_{w})} \|A\|_{{\mathcal B}_{r,\alpha}}+1\big) \Big)^{(2 d_{\mathcal G}+1)/r'} &
{\rm if }\ \alpha= 1+d_{\mathcal G}/r',
\end{array}
\right.\end{aligned}$$ where $1/r'=1-1/r$.
\[norm-contro-inversion.thm.remark\] [Under the assumption that the counting measure $\mu$ is Ahlfors regular in which $\tilde d_{\mathcal G}=d_{\mathcal G}$ by Proposition \[regular.prop\], the authors in [@shinsun19] show that Beurling algebras ${{\mathcal B}_{r,\alpha}}({\mathcal G})$ for some $1\le r\le \infty$ and $\alpha>d_{\mathcal G}(1-/r)$ admit norm-controlled inversion in the symmetric $*$-algebra $\mathcal B(\ell^2)=\mathcal B(\ell^2_{w_0})$. Moreover for any matrix $A\in {\mathcal B}_{r,\alpha}({\mathcal G})$ being invertible in $\mathcal B(\ell^2)$, $$\begin{aligned}
\label{norm-inversion.old}
\|A^{-1}\|_{{\mathcal B}_{r, \alpha}}&\le C\|A^{-1}\|_{\mathcal B(\ell^2)}
\big(\|A^{-1}\|_{\mathcal B(\ell^2)} \|A\|_{{\mathcal B}_{r,\alpha}}\big)^{(\alpha+d_{\mathcal G}/r)/\min(\alpha-d_{\mathcal G}/r', 1)}
\nonumber \\&
\quad\times
\left\{
\begin{array}{ll}
1
&{\rm if }\ \alpha\neq 1+d_{\mathcal G}/r'
\\
\big(\ln\big(\|A^{-1}\|_{\mathcal B(\ell^2)} \|A\|_{{\mathcal B}_{r,\alpha}}+1\big) \big)^{(d_{\mathcal G}+1)/r'} &
{\rm if }\ \alpha= 1+d_{\mathcal G}/r',
\end{array}
\right.\end{aligned}$$ where $1/r'=1-1/r$ and $C$ is an absolute constant. Therefore under the assumption that the counting measure $\mu$ is Ahlfors regular, the conclusion provides a better upper bound estimate to $\|A^{-1}\|_{{\mathcal B}_{r, \alpha}}$ than the one in with the exponent $p$ and Muckenhoupt $A_p$-weight $w$ replaced by $2$ and the trivial weight $w_0$ respectively. ]{}
We conclude this section with a family of matrices on the lattice graph ${\mathcal Z}$ to demonstrate the almost optimality of the norm estimate for the inversion.
\[exam\]
Let $A_{\kappa}:=(a_{\kappa}(n-n'))_{n,n'\in {{\mathbb Z}}}$, where $\kappa\in (0, 1)$ is a constant sufficiently close to one, and $$a_\kappa(n)= \left\{\begin{array}{ll} 1 & {\rm if} \ n=0\\
-\kappa & {\rm if} \ n=1\\
0 & {\rm otherwise}.
\end{array}
\right.$$ Then its inverse $(A_\kappa)^{-1}=(\check a_{\kappa}(n-n'))_{n, n'\in {{\mathbb Z}}}$ is given by $$\check a_\kappa(n)= \left\{\begin{array}{ll} \kappa^n & {\rm if} \ n\ge 0\\
0 & {\rm otherwise}.
\end{array}
\right.$$ For $1\le r\le \infty$, we have $$\label{Agamma-bound}
\|A_{\kappa}\|_{{\mathcal B}_{r,\alpha}} \approx 1 \ \ {\rm and} \ \
\|A_\kappa^{-1}\|_{{\mathcal B}_{r, \alpha}} \approx (1-\kappa)^{-\alpha-1/r}.$$
Let $w_\theta=((|n|+1)^\theta)_{n\in {{\mathbb Z}}}, -1<\theta<p-1$. Then $w_\theta$ is a Muckenhoupt $A_p$-weight and $$\label{weighted.ex1}
\|A_\kappa^{-1}\|_{{\mathcal B}({\ell^p_{w_\theta}})}\lesssim \|A_\kappa^{-1}\|_{{\mathcal B}_{1,0}}\lesssim (1-\kappa)^{-1}.$$ Take $c_0=(c_0(n))_{n\in {{\mathbb Z}}}$, where $$c_0(n):=
\left\{
\begin{array}{ll} \kappa^n
& {\rm if } \ n\ge 0 \\
0
& {\rm otherwise. }
\end{array}
\right.$$ Therefore $$\label{weighted.ex2}
\|c_0\|_{p, w_\theta}= \Big(\sum_{n=0}^\infty \kappa^{np} (n+1)^\theta\Big)^{1/p}
\approx (1-\kappa)^{-(\theta+1)/p}$$ and $$\label{weighted.ex3}
\|A_\kappa^{-1} c_0\|_{p, w}=
\Big(\sum_{n=0}^\infty \kappa^{nq} (n+1)^p (n+1)^\theta\Big)^{1/p}\approx (1-\kappa)^{-(\theta+p+1)/p}.$$ By , and , we have $$\label{weighted.ex4}
\|A_\kappa^{-1}\|_{{\mathcal B}(\ell^p_{w_\theta})}\approx (1-\kappa)^{-1}.$$ Combining and yields $$\|A_\kappa^{-1}\|_{{\mathcal B}_{r, \alpha}} \approx \|A_\kappa^{-1}\|_{\mathcal B(\ell^p_{w_\theta})}
\big(\|A_\kappa^{-1}\|_{\mathcal B(\ell^p_{w_\theta})} \|A_\kappa\|_{{\mathcal B}_{r,\alpha}}\big)^{(\alpha+d_{\mathcal G}/r-d_{\mathcal G})},$$ while the estimate in Theorem \[norm-contro-inversion.thm\] for $\alpha>1+d_{\mathcal G}(1-1/r)$ is $$\|A_\kappa^{-1}\|_{{\mathcal B}_{r, \alpha}} \lesssim \|A_\kappa^{-1}\|_{\mathcal B(\ell^p_{w})}
\big(\|A_\kappa^{-1}\|_{\mathcal B(\ell^p_{w})} \|A_\kappa\|_{{\mathcal B}_{r,\alpha}}\big)^{(\alpha+d_{\mathcal G}/r+d_{\mathcal G})}.$$
Proofs {#proofs.section}
======
In this section, we collect the proofs of Propositions \[regular.prop\], \[beurling.prop\], \[beurlingonmaximalsets.pr\] and \[weighted.prop\], Lemmas \[tech.lem\] and \[tech.lem2\], and Theorem \[norm-contro-inversion.thm\].
Proof of Proposition \[regular.prop\] {#regular.prop.pfsection}
-------------------------------------
By , it suffices to establish for $N\ge 1$ and $R\ge 3$. Let $V_N$ be a maximal $N$-disjoint set, and define $A_R(\lambda, N)$ as in . Then we obtain from and that $$\begin{aligned}
\mu\big(A_R(\lambda, N)\big) & \le &
\frac{\sum_{\lambda_m\in A_R(\lambda, N)} \mu(B(\lambda_m, N))}
{\inf_{\lambda_{m'} \in A_R(\lambda, N)} \mu(B(\lambda_{m'}, N))}\le
B_3^{-1} N^{-d_0} \mu \big(\cup_{\lambda_m\in A_R(\lambda, N)} B(\lambda_m, N)\big)\nonumber\\
& \le & B_3^{-1} N^{-d_0} \mu\big(B(\lambda, N+ (N+1)R\big)
\le 2^{d_0} B_3^{-1} B_4 (R+1)^{d_0},\end{aligned}$$ which implies that $$\label{regular.prop.eq1}
\tilde d_{\mathcal G}\le d_0.$$ Similarly by and , we get $$\begin{aligned}
\mu\big(A_R(\lambda, N)\big) & \ge &
\frac{\sum_{\lambda_m\in A_R(\lambda, N)} \mu(B(\lambda_m, 2N))}
{\max_{\lambda_m \in A_R(\lambda, N)} \mu(B(\lambda_m, 2N))}\ge
B_4^{-1} (2N+1)^{-d_0} \mu \big(\cup_{\lambda_m\in A_R(\lambda, N)} B(\lambda_m, 2N)\big)\nonumber\\
& \ge & 3^{-d_0}B_4^{-1} N^{-d_0} \mu(B(\lambda, N (R-2) )
\ge 2^{-2d_0}3^{-d_0}B_4^{-1} B_3 (R+1)^{d_0}, R\ge 3,\end{aligned}$$ where the third inequality holds as $B(\lambda_m, 2N))\cap B(\lambda, N(R-2))=\emptyset$ for all $\lambda_m\not\in A_R(\lambda, N)$. This show that $$\label{regular.prop.eq2}
\tilde d_{\mathcal G}\ge d_0.$$ Combining and completes the proof.
Proof of Proposition \[beurling.prop\] {#beurling.prop.pfsection}
--------------------------------------
The conclusion (i) is obvious and the conclusions in (ii), (iii) and (iv) are presented in [@shinsun19 Propositions 3.3 and 3.4]. Now we prove the conclusion (v). Write $A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}$ and set $h_A(n)=\sup_{\rho(\lambda, \lambda')\ge n} |a(\lambda, \lambda')|, n\ge 0$. Then for $K\ge 1$ and $1<r\le \infty$, we have
$$\begin{aligned}
\|A-A_K\|_{{\mathcal B}_{1, 0}} &
\le & 2 \sum_{n=\lceil (K+1)/2\rceil}^\infty h_A(n)(n+1)^{d_{\mathcal G}-1}\le 2 \|A\|_{{\mathcal B}_{r,\alpha}} \Big(\sum_{n=\lceil (K+1)/2\rceil}^\infty (n+1)^{-\alpha r'+d_{\mathcal G}-1}\Big)^{1/r'}
\nonumber\\
& \le &
2 \|A\|_{{\mathcal B}_{r, \alpha}}
\Big(\int_{\lceil (K+1)/2\rceil}^\infty x^{d_{\mathcal G}-1-\alpha r'} dx
\Big)^{1/r'}
\le
\frac{2^{\alpha -d_{\mathcal G}/r'+1}}{(\alpha r' -d_{\mathcal G})^{1/r'}} \|A\|_{{\mathcal B}_{r,\alpha}}
K^{-\alpha +d_{\mathcal G}/r'},\end{aligned}$$
where $r'=r/(r-1)$. This proves for $1<r\le \infty$. Similarly we can prove for $r=1$.
Proof of Proposition \[beurlingonmaximalsets.pr\] {#beurlingonmaximalsets.pr.pfsection}
-------------------------------------------------
The conclusion (i) follows from the definition of Beurling algebras on the graph ${\mathcal G}$ and on its maximal disjoint set $V_N$.
Now we prove the conclusion (ii). Set $\check\alpha= \alpha+ (\tilde d_{\mathcal G}-d_{\mathcal G})/r$. For $1\le r<\infty$, we obtain $$\begin{aligned}
\label{Bra.eqN1}
\|A\|_{{\mathcal B}_{r, \check\alpha}}^r
& \le & \sum_{n'=0}^\infty \sum_{n=n'(N+1)}^{(n'+1)(N+1)-1} (n+1)^{\alpha r +\tilde d_{\mathcal G}-1} \Big(\sup_{\rho(\lambda, \lambda')\ge n}
\sum_{\lambda_m\in B(\lambda, 2N), \lambda_k\in B(\lambda', 4N)} |b(\lambda_m, \lambda_k)|\Big)^r\nonumber\\
& \le & (N+1)^{\alpha r+\tilde d_{\mathcal G}} \sum_{n'=0}^\infty (n'+1)^{\alpha r+\tilde d_{\mathcal G}-1}
\Big(\sup_{\rho(\lambda, \lambda')\ge n'(N+1)}
\sum_{\lambda_m\in B(\lambda, 2N), \lambda_k\in B(\lambda', 4N)} |b(\lambda_m, \lambda_k)|\Big)^r\nonumber\\
& \le & (D(\mu))^{7r}
(N+1)^{\alpha r+\tilde d_{\mathcal G}} \sum_{n'=0}^\infty (n'+1)^{ \alpha r+\tilde d_{\mathcal G}-1}
\Big(\sup_{\rho(\lambda_m, \lambda_k)\ge \max(n'-6, 0)(N+1)}
|b(\lambda_m, \lambda_k)|\Big)^r\nonumber\\
& \le & 8^{\alpha r+\tilde d_{\mathcal G}} (D(\mu))^{7r} N^{\alpha r+\tilde d_{\mathcal G}} \|B\|_{{\mathcal B}_{r, \alpha; N}}^r,
\end{aligned}$$ where the third inequality follows from . This proves and the conclusion (ii) for $1\le r<\infty$. Similarly for $r=\infty$, we have $$\begin{aligned}
\label{Bra.eqN2}
\|A\|_{{\mathcal B}_{\infty, \check\alpha}}& \le & \sup_{\lambda, \lambda'\in V}
\sum_{\lambda_m\in B(\lambda, 2N), \lambda_k\in B(\lambda', 4N)} |b(\lambda_m, \lambda_k)|
(\rho(\lambda, \lambda')+1)^{\alpha} \nonumber\\
& \le & \|B\|_{\infty, \alpha; N}
\sup_{\lambda, \lambda'\in V}
\sum_{\lambda_m\in B(\lambda, 2N),\lambda_k\in B(\lambda', 4N)} (\lfloor\rho(\lambda_m, \lambda_k)/N\rfloor+1)^{-\alpha}
(\rho(\lambda, \lambda')+1)^{ \alpha}\nonumber\\
& \le & 8^\alpha (D(\mu))^7 N^{\alpha} \|B\|_{\infty, \alpha; N}.
\end{aligned}$$ Combining and proves and the conclusion (ii).
Finally we prove the conclusion (iii). Set $\tilde \alpha= \alpha- (\tilde d_{\mathcal G}-d_{\mathcal G})/r$ and $1/r'=1-1/r$. Then for $1< r< \infty$, we have $$\begin{aligned}
\label{beurlingonmaximalsets.pr.pf.eq5}
\|S_{A, N}\|_{\mathcal{B}_{r,\tilde \alpha;N}} &\le &
N^{d_{\mathcal G} } \Big( \sum_{n=13}^\infty \big(h_A(n(N+1)/2)\big)^r (n+1)^{\alpha r + d_{\mathcal G}-1}\Big)^{1/r}\nonumber\\
& & +
N^{-1} \Big(\sum_{n=0}^{2N} h_A(n) (n+1)^{d_{\mathcal G}}\Big) \times \Big(\sum_{n=0}^{12} (n+1)^{ \alpha r+ d_{\mathcal G}-1}\Big)^{1/r}
\nonumber \\ & \le & 2^{2\alpha +(2 d_{\mathcal G}-1)/r} N^{-\alpha +d_{\mathcal G}/r'}
\Big(\sum_{m=4N}^\infty ( h_A(m))^{r} (m+1)^{\alpha r+d_{\mathcal G}-1}\Big)^{1/r}\nonumber\\
& & +13^{\alpha +d_{\mathcal G}/r} N^{-1} \sum_{n=0}^{2N} h_A(n) (n+1)^{d_{\mathcal G}} \nonumber\\
& \le & 2^{2\alpha +(2 d_{\mathcal G}-1)/r} N^{-\alpha +d_{\mathcal G}/r'}
\|A\|_{{\mathcal B}_{r, \alpha}} \nonumber\\
& &
+ 13^{\alpha +d_{\mathcal G}/r}\|{ A}\|_{{\mathcal B}_{r,\alpha}} N^{-1} \Big(\sum_{n=0}^{2N} (n+1)^{-(\alpha-1) r'+ d_{\mathcal G}-1}\Big)^{1/r'} \nonumber\\
& \le & 2^{2\alpha +(2 d_{\mathcal G}-1)/r} N^{-\alpha +d_{\mathcal G}/r'}\|{A}\|_{{\mathcal B}_{r,\alpha}}
+
13^{\alpha +d_{\mathcal G}/r} \|{A}\|_{{\mathcal B}_{r,\alpha}}
\nonumber\\
& & \times \left\{\begin{array}{ll}
2 \Big(\frac{1+|\alpha-1-d_{\mathcal G}/r'|}{|\alpha-1-d_{\mathcal G}/r'|}\Big)^{1/r'} N^{-\min(1,\alpha-d_{\mathcal G}/r')} & {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
3^{1/r'} N^{-1} (\ln (N+1))^{1/r'} & {\rm if} \ \alpha= d_{\mathcal G}/r'+1.
\end{array}\right. $$ This proves for $1<r<\infty$. Using similar argument, we can prove for $r=1, \infty$.
Proof of Proposition \[weighted.prop\] {#weighted.prop.pfsection}
--------------------------------------
Write $ A=(a(\lambda, \lambda'))_{\lambda, \lambda'\in V}$, and set $h_A(n)=\sup_{\rho(\lambda, \lambda')\ge n} |a(\lambda, \lambda')|$, $n\ge 0$. Then for any $c\in \ell^p_w$ with $1<p<\infty$, we have $$\begin{aligned}
\label{h-sum2}
\|Ac \|_{p,w} &\le &
\Big( \sum_{\lambda \in V} \Big(
\sum_{\lambda' \in V} h_A(\rho(\lambda, \lambda'))|c(\lambda')|\Big)^p w(\lambda)\Big)^{1/p}
\nonumber \\
&
\le &
h_A(0) \|c\|_{p,w}
+\Bigg(\sum_{\lambda \in V}
\Big( \sum_{l=1}^\infty h_A(2^{l-1}) \sum_{ 2^{l-1} \le \rho(\lambda, \lambda')<2^l}
|c(\lambda')|\Big)^p w(\lambda) \Bigg)^{1/p}
\nonumber \\
&\le & h_A(0) \|c\|_{p,w}+ \Big( \sum_{l=1}^\infty h_A(2^{l-1})2^{ld_{\mathcal G}}\Big)^{1-1/p}\nonumber\\
& & \times
\Bigg(\sum_{l=1}^\infty h_A(2^{l-1})2^{-(p-1)l d_{\mathcal G} }
\sum_{\lambda \in V} \Big(\sum_{ 2^{l-1} \le \rho(\lambda, \lambda')<2^l}
|c(\lambda')|\Big)^{p} w(\lambda) \Bigg)^{1/p}.\end{aligned}$$ By the equivalent definition of the Muckenhoupt $A_p$-weight $w$ and the polynomial property of the counting measure $\mu$, we obtain $$\begin{aligned}
\label{h-sum2+}
& &
\sum_{\lambda \in V} \Big(\sum_{ \rho(\lambda, \lambda')<2^l}
|c(\lambda')|\Big)^{p} w(\lambda) \nonumber\\
& \le &
\sum_{\lambda \in V}
\Big(\sum_{ \rho(\lambda, \lambda')<2^l}
|c(\lambda')|^p w(\lambda')\Big) \Big(\sum_{ \rho(\lambda, \lambda^{\prime\prime})<2^l} ( w(\lambda^{\prime\prime}))^{-1/(p-1)}\Big)^{p-1} w(\lambda)
\nonumber\\
&\le & A_p(w)
\sum_{\lambda\in V} w(\lambda)
\Big( \sum_{\rho(\lambda', \lambda)<2^l}
|c(\lambda')|^p w(\lambda')\Big)
\times
\frac{ \big(\mu(B(\lambda, 2^{l+1}-2))\big)^p} {\sum_{\rho(\lambda, \lambda^{\prime\prime})\le 2^{l+1}-2} w(\lambda^{\prime\prime})}
\nonumber\\
&\le & A_p(w) (D_{\mathcal G})^p 2^{p(l+1) d_{\mathcal G} }
\sum_{\lambda'\in V}
|c(\lambda')|^p w(\lambda') \times \sum_{\rho(\lambda, \lambda')<2^l}
\frac{ w(\lambda)} {\sum_{\rho(\lambda', \lambda^{\prime\prime})\le 2^{l}-1} w(\lambda^{\prime\prime})}\nonumber\\
& = & A_p(w) (D_{\mathcal G})^p 2^{p(l+1) d_{\mathcal G} } \|c\|_{p, w}^p.
\end{aligned}$$ This together with and the following estimate $$\label{hA.eq00}
h_A(0)+\sum_{l=1}^\infty h(2^{l-1}) 2^{ld_{\mathcal G}}\le h_A(0)+ 2^{2d_{\mathcal G}} \sum_{l=1}^\infty \sum_{2^{l-2}<n\le 2^{l-1}} h_A(n) (n+1)^{d_{\mathcal G}-1}\le 2^{2d_{\mathcal G}} \|A\|_{{\mathcal B}_{1, 0}}$$ proves for $1<p<\infty$.
Applying a similar argument as above, we can verify for $p=1$.
Proof of Lemma \[tech.lem\] {#tech.lem.pfsection}
---------------------------
We follow the procedure used in [@shinsun19], where a similar result is established for the unweighted case. Take $\lambda_m\in V_N$. Denote the commutator between the smooth truncation operator $\Psi_{\lambda_m}^{2N}$ and the matrix ${A}$ by $[\Psi_{\lambda_m}^{2N}, {A}]:=\Psi_{\lambda_m}^{2N}{A}- {A} \Psi_{\lambda_m}^{2N}$, and set $\Phi^{2N}:=\big(\sum_{\lambda_k\in V_N}\Psi_{\lambda_k}^{2N}\big)^{-1}$. Replacing $c$ in by $\Psi_{\lambda_m}^{2N} c$ and applying Proposition \[weighted.prop\], we have $$\begin{aligned}
\label{lowerbound1}
\beta_{p, w}(A) \|\Psi_{\lambda_m}^{2N} c\|_{p,w}
& \le & \| A\Psi_{\lambda_m}^{2N} c\|_{p,w}
\le \|\Psi_{\lambda_m}^{2N} A c\|_{p,w}
+
\|[\Psi_{\lambda_m}^{2N}, A] c\|_{p,w}\nonumber\\
& \le & \|\Psi_{\lambda_m}^{2N} A c\|_{p,w}+
\|\chi_{\lambda_m}^{4N}
[\Psi_{\lambda_m}^{2N}, {A}]c \|_{p,w}
+ \|(I-\chi_{\lambda_m}^{4N}) {A} \chi_{\lambda_m}^{3N}
\Psi_{\lambda_m}^{2N}c\|_{p,w}
\nonumber \\
& \le & \|\Psi_{\lambda_m}^{2N} A c\|_{p,w}+
\sum_{\lambda_k \in V_{N}}
\|\chi_{\lambda_m}^{4N}[\Psi_{\lambda_m}^{2N}, {A}] \chi_{\lambda_k}^{3N}
\Phi^{2N}\Psi_{\lambda_k}^{2N}c\|_{p,w}\nonumber\\
& & + 2^{3d_{\mathcal G}} D_{\mathcal G} \big(A_p(w)\big)^{1/p}
\|(I-\chi_{\lambda_m}^{4N}) {A} \chi_{\lambda_m}^{3N}\|_{{\mathcal B}_{1, 0}}
\|\Psi_{\lambda_m}^{2N}c\|_{p,w},\end{aligned}$$ where the second inequality holds, as $(I-\chi_{\lambda_m}^{4N}) [\Psi_{\lambda_m}^{2N}, {A}]=
(I-\chi_{\lambda_m}^{4N}) A \Psi_{\lambda_m}^{2N}= (I-\chi_{\lambda_m}^{4N}) A \chi_{\lambda_m}^{3N}\Psi_{\lambda_m}^{2N}$ by the supporting properties for $\chi_{\lambda_m}^{3N}, \chi_{\lambda_m}^{4N}$ and $\Psi_{\lambda_m}^{2N}$. From and in Proposition \[beurling.prop\], we obtain $$\label{lowerbound1+}
\|(I-\chi_{\lambda_m}^{4N}) {A} \chi_{\lambda_m}^{3N}\|_{{\mathcal B}_{1, 0}}
\le \|A-{A}_{N}\|_{{\mathcal B}_{1, 0}}\le C_0 \|A\|_{{\mathcal B}_{r,\alpha}} N^{-\alpha +d_{\mathcal G}(1-1/r)}.$$ Combining and yields $$\begin{aligned}
\beta_{p, w}(A) \|\Psi_{\lambda_m}^{2N} c\|_{p,w}
& \le & \|\Psi_{\lambda_m}^{2N} A c\|_{p,w}+
\sum_{\lambda_k \in V_{N}}
\|\chi_{\lambda_m}^{4N}[\Psi_{\lambda_m}^{2N}, {A}] \chi_{\lambda_k}^{3N}
\Phi^{2N}\Psi_{\lambda_k}^{2N}c\|_{p,w}\nonumber\\
& & + 2^{3d_{\mathcal G}} C_0 D_{\mathcal G} (A_p(w))^{1/p}
\|A\|_{{\mathcal B}_{r,\alpha}} N^{-\alpha +d_{\mathcal G}(1-1/r)}
\|\Psi_{\lambda_m}^{2N}c\|_{p,w}.\end{aligned}$$ This together with proves that $$\label{lowerbound1++}
\beta_{p, w}(A) \|\Psi_{\lambda_m}^{2N} c\|_{p,w}
\le 2 \|\Psi_{\lambda_m}^{2N} A c\|_{p,w}+
2 \sum_{\lambda_k \in V_{N}}
\|\chi_{\lambda_m}^{4N}[\Psi_{\lambda_m}^{2N}, {A}] \chi_{\lambda_k}^{3N}
\Phi^{2N}\Psi_{\lambda_k}^{2N}c\|_{p,w}.$$
For $\lambda_k\in V_{N}$ with $\rho(\lambda_m, \lambda_k)> 12(N+1)$, we obtain from the finite covering property for the maximal $N$-disjoint set $V_N$, the equivalent definition of the weight $w$, the polynomial growth property of the counting measure $\mu$, and the monotonicity of $h_A(n), n\ge 0$, that $$\begin{aligned}
\label{lowerbound4}
& & \|\chi_{\lambda_m}^{4N } [\Psi_{\lambda_m}^{2N}, {A}]\chi_{\lambda_k}^{3N}\Phi^{2N}
\Psi_{\lambda_k}^{2N} c\|_{p,w}
= \|\Psi_{\lambda_m}^{2N} {A}\chi_{\lambda_k}^{3N}\Phi^{2N}\Psi_{\lambda_k}^{2N} c\|_{p,w}
\nonumber \\&
\le &
h_{ A}\Big(\frac{\rho(\lambda_m, \lambda_k)}{2}\Big)
\Bigg(\sum_{\lambda\in B(\lambda_m, 4N)}
\Big(\sum_{\lambda'\in B(\lambda_k, 4N)}
|\Psi_{\lambda_k}^{2N} c(\lambda')|\Big)^p w(\lambda)
\Bigg)^{1/p}
\nonumber \\&
\lesssim & (A_p(w))^{1/p} S_{A, N}(\lambda_m, \lambda_k)
\|\Psi_{\lambda_k}^{2N}c \|_{p,w}
\Bigg( \frac{w\big(B(\lambda_m, 4N)\big)}{w\big(B(\lambda_k, 4N)\big)}\Bigg)^{1/p}.
$$
Set $\tilde A_M=\big(|a(\lambda, \lambda')| \rho(\lambda, \lambda')\chi_{[0, M]}(\rho(\lambda, \lambda'))\big)_{\lambda, \lambda'\in V},\ M\ge 0$. For $\lambda_k \in V_{N}$ with $\rho(\lambda_m, \lambda_k)< 12(N+1)$, we have $$\begin{aligned}
\label{8N-bound}
\|\chi_{\lambda_m}^{4N}
[\Psi_{\lambda_m}^{2N}, {A}]\chi_{\lambda_k}^{3N}\Phi^{2N}\Psi_{\lambda_k}^{2N} c\|_{p,w}
& \lesssim & \big( A_p(w)\big)^{1/p}
\|\chi_{\lambda_m}^{4N}[\Psi_{\lambda_m}^{2N}, {A}]\chi_{\lambda_k}^{3N}\|_{{\mathcal B}_{1, 0}}
\|\Phi^{2N}\Psi_{\lambda_k}^{2N} c\|_{p,w}\nonumber\\
&\lesssim&
\big(A_p(w)\big)^{1/p} N^{-1}
\|\tilde A_{19N+12}\|_{{\mathcal B}_{1, 0}}
\|\Psi_{\lambda_k}^{2N} c\|_{p,w}, $$ where the first inequality follows from the weighted norm inequality in Proposition \[weighted.prop\], and the second one holds by the solidness of the Banach algebra ${\mathcal B}_{1, 0}({\mathcal G})$ in Proposition \[beurling.prop\] and the Lipschitz property for the trapezoid function $\psi_0$. Observe that $$\label{8N-bound+1}
w\big(B(\lambda_k, 4N)\big)\le w\big(B(\lambda_m, 19N+12)\big)\le (D(\mu))^{3p} A_p(w) w\big(B(\lambda_m, 4N)\big)$$ by the double property for the $A_p$-weight $w$, and $$\begin{aligned}
\label{8N-bound+2} \|\tilde A_{19N+12}\|_{{\mathcal B}_{1, 0}}
& = & \sum_{n=0}^{19N+12} (n+1)^{d_{\mathcal G}-1} \sup_{n\le \rho(\lambda, \lambda')\le 19N+12} |a(\lambda, \lambda')| \rho(\lambda, \lambda')\nonumber\\
& \le & 2 \sum_{n=0}^{19N+12} (n+1)^{d_{\mathcal G}-1} \sum_{n/2\le m\le 19N+12} h_{A} (m)
\lesssim \sum_{n=0}^{2N} h_A(n) (n+1)^{d_{\mathcal G}}
$$ by the monotonicity of $h_A(n), n\ge 0$. Combining , and , we get $$\label{8N-bound+3}
\|\chi_{\lambda_m}^{4N}
[\Psi_{\lambda_m}^{2N}, {A}]\chi_{\lambda_k}^{3N}\Phi^{2N}\Psi_{\lambda_k}^{2N} c\|_{p,w}
\lesssim \big(A_p(w)\big)^{2/p} S_{A, N}(\lambda_m, \lambda_k)
\|\Psi_{\lambda_k}^{2N}c \|_{p,w}
\Bigg( \frac{w\big(B(\lambda_m, 4N)\big)}{w\big(B(\lambda_k, 4N)\big)}\Bigg)^{1/p}$$ if $\rho(\lambda_m, \lambda_k)\le 12 (N+1)$.
Combining , and proves .
Proof of Lemma \[tech.lem2\] {#tech.lem2.pfsection}
----------------------------
Set $\alpha_{\lambda_m}:= w\big(B(\lambda_m, 4N)\big), \lambda_m\in V_N$, and write $$\big(S_{A, N}\big)^l:=\big(S_{A, N; l}(\lambda_m, \lambda_k)\big)_{\lambda_m, \lambda_k\in V_N}, l\ge 1.$$ By , the integer $N$ satisfies and hence holds by Lemma \[tech.lem\]. Applying repeatedly, we get $$\begin{aligned}
\label{tech.lem.eq2++}
& & \big(\alpha_{\lambda_m}\big)^{-1/p}
\|\Psi_{\lambda_m}^{2N} c\|_{p,w} \nonumber\\
& \le & 2 (\beta_{p, w}(A))^{-1} \big(\alpha_{\lambda_m}\big)^{-1/p}
\|\Psi_{\lambda_m}^{2N} {A}c\|_{p,w} \nonumber\\
& &
+ C_2 \big(A_p(w)\big)^{2/p} (\beta_{p, w}(A))^{-1}
\sum_{\lambda_k\in V_N} S_{A, N}(\lambda_m, \lambda_k) \big(\alpha_{\lambda_k}\big)^{-1/p} \|\Psi_{\lambda_k}^{2N} c\|_{p,w}\nonumber\\
& \le & \cdots\nonumber\\
& \le & 2 (\beta_{p, w}(A))^{-1} \big(\alpha_{\lambda_m}\big)^{-1/p}
\|\Psi_{\lambda_m}^{2N} {A}c\|_{p,w}+ 2 (\beta_{p, w}(A))^{-1}\nonumber\\
& & \times \sum_{l=1}^{L-1}
\big(C_2 \big(A_p(w)\big)^{2/p} (\beta_{p, w}(A))^{-1}\big)^l
\sum_{\lambda_k\in V_N} S_{A, N; l}(\lambda_m, \lambda_k) \big(\alpha_{\lambda_k}\big)^{-1/p} \|\Psi_{\lambda_k}^{2N} Ac\|_{p,w}\nonumber\\
& & + (C_2 \big(A_p(w)\big)^{2/p} (\beta_{p, w}(A))^{-1}\big)^L
\sum_{\lambda_k\in V_N} S_{A, N; L}(\lambda_m, \lambda_k) \big(\alpha_{\lambda_k}\big)^{-1/p} \|\Psi_{\lambda_k}^{2N} c\|_{p,w},
$$ where $L\ge 2$. Define $$\label{wan.def0}
W_{A, N}= 2 I+2 \sum_{l=1}^\infty \big(C_2 \big(A_p(w)\big)^{2/p} (\beta_{p, w}(A))^{-1}\big)^l (S_{A, N})^l.$$ Then by and , we have $$\begin{aligned}
\label{wan.estimate++}
\|W_{A, N}\|_{{\mathcal B}_{r, \alpha- (\tilde d_{\mathcal G}-d_{\mathcal G})/r; N}} & \le &
2+2\sum_{l=1}^\infty \big(C_2 C_3 \big(A_p(w)\big)^{2/p} (\beta_{p, w}(A))^{-1} \|{A}\|_{{\mathcal B}_{r,\alpha}}\big)^l\nonumber\\
& & \quad \times \left\{\begin{array}{ll} N^{-\min(1,\alpha-d_{\mathcal G}/r')l } & {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
N^{-l} (\ln (N+1))^{l/r'} & {\rm if} \ \alpha= d_{\mathcal G}/r'+1,
\end{array}\right. \nonumber\\
& \le & 2+2\sum_{l=1}^\infty 2^{-l} =4.\end{aligned}$$
Following the argument used in the proof of Proposition \[weighted.prop\], we obtain $$\begin{aligned}
& & \Big( \sum_{\lambda_m\in V_N} \Big|
\sum_{\lambda_k\in V_N} S_{A, N; L}(\lambda_m, \lambda_k) \big(\alpha_{\lambda_k}\big)^{-1/p} \|\Psi_{\lambda_k}^{2N} c\|_{p,w}\Big|^p
\alpha_{\lambda_m}\Big)^{1/p}\nonumber\\
& \lesssim & \big \| (S_{A, N})^L\big\|_{{\mathcal B}_{1, 0; N}}
\Big(\sum_{\lambda_k\in V_N} \Big|\big(\alpha_{\lambda_k}\big)^{-1/p} \|\Psi_{\lambda_k}^{2N} c\|_{p,w}\Big|^p \alpha_{\lambda_k}\Big)^{1/p}\nonumber\\
& \le & C_6 \big\| (S_{A, N})^L\big\|_{{\mathcal B}_{r, \alpha- (\tilde d_{\mathcal G}-d_{\mathcal G})/r; N}} \|c\|_{p, w},\end{aligned}$$ where $C_6$ is an absolute constant. This together with and implies that $$\begin{aligned}
\label{wan.estimate++2}
& & \Big(\sum_{\lambda_m\in V_N} \Big|
\sum_{\lambda_k\in V_N} S_{A, N; L}(\lambda_m, \lambda_k) \big(\alpha_{\lambda_k}\big)^{-1/p} \|\Psi_{\lambda_k}^{2N} c\|_{p,w}\Big|^p
\alpha_{\lambda_m}\Big)^{1/p}\nonumber\\
& &\quad \times \Big(C_2 \big(A_p(w)\big)^{2/p} (\beta_{p, w}(A))^{-1}\Big)^L
\le C_6 2^{-L} \|c\|_{p, w}\to 0\ {\rm as} \ L\to \infty.\end{aligned}$$ Taking limit $L\to \infty$ in and applying and , we obtain $$\begin{aligned}
\label{wan.estimate++3}
& & \beta_{p, w}(A) \big(\alpha_{\lambda_m}\big)^{-1/p}
\|\Psi_{\lambda_m}^{2N} c\|_{p,w} \le \sum_{\lambda_k\in V_N} W_{A,N}(\lambda_m, \lambda_k)
\big(\alpha_{\lambda_k}\big)^{-1/p}
\|\Psi_{\lambda_k}^{2N} Ac\|_{p,w}, \ \lambda_m\in V_N,\end{aligned}$$ where $c\in \ell^p_w$.
Define $H_{A, N}:=\big(H_{A, N}(\lambda, \lambda')\big)_{\lambda, \lambda'\in V}$ by $$\label{def-H}
H_{A, N}(\lambda, \lambda'):=\sum_{\lambda_m\in B(\lambda, 2N)}
\sum_{\lambda_k \in B(\lambda', 4N)}
W_{{A}, N}(\lambda_m, \lambda_k ).$$ Then the desired norm estimate follows from , and .
Let $\lambda\in V$ and select $\lambda_m\in V_{N}$ such that $\lambda\in B(\lambda_m, 2N)$. Such a fusion vertex $\lambda_m$ exists by the covering property . Replacing the vector $(c(\lambda'))_{\lambda'\in V}$ and the ball $B$ by the delta vector $(\delta_0(\lambda, \lambda'))_{\lambda'\in V}$ and $B(\lambda_m, 4N)$ in , respectively, we get $$\label{est-c}
\alpha_{\lambda_m} \lesssim A_p(w) N^{pd_{\mathcal G}}
w(\lambda).$$ Combining , and , we obtain $$\begin{aligned}
\label{preresult}
|c(\lambda)|&\lesssim &
( A_p(w))^{1/p} N^{d_{\mathcal G}} \sum_{\lambda_m\in B(\lambda, 2N)} \alpha_{\lambda_m}^{-1/p}
\|\Psi_{\lambda_m}^{2N}c \|_{p,w}
\nonumber \\
& \lesssim & ( A_p(w))^{1/p} (\beta_{p, w}(A))^{-1}
N^{d_{\mathcal G}} \sum_{\lambda_m\in B(\lambda, 2N)} \sum_{\lambda_k \in V_{N}}
W_{A, N}(\lambda_m, \lambda_k )\alpha_{\lambda_k}^{-1/p}
\|\Psi_{\lambda_k}^{2N} {A}c\|_{p,w}
\nonumber \\
&
\lesssim &
(A_p(w))^{1/p} (\beta_{p, w}(A))^{-1}
N^{d_{\mathcal G}} \sum_{\lambda'\in V}
H_{{A}, N}(\lambda, \lambda')|{A}c(\lambda')|\ \ {\rm for \ all} \ c\in \ell^p_w,\end{aligned}$$ where the last inequality holds as $\alpha_{\lambda_k}^{-1/p}
\|\Psi_{\lambda_k}^{2N} {A}c\|_{p,w}\le \|\Psi_{\lambda_k}^{2N} {A}c\|_{p,w_0}\le \|\Psi_{\lambda_k}^{2N} {A}c\|_{1,w_0}.$ This proves .
Proof of Theorem \[norm-contro-inversion.thm\] {#norm-contro-inversion.thm.pfsection}
----------------------------------------------
By the invertibility assumption of the matrix $A$ in $\ell^p_w$, it has the $\ell^p_w$-stability and its optimal lower stability bound $\beta_{p, w}(A)$ satisfies $$\label{wiener.pff.eq1}
\beta_{p, w}(A) \ge \big(\|A^{-1} \|_{{\mathcal B}(\ell^p_w)}\big)^{-1}.$$ Let $r'$ be the conjugate exponent of $r$, i.e., $1/r+1/r'=1$, $N\ge 2$ be an integer satisfying $$\label{wiener.pff.eq3}
\big(\|A^{-1} \|_{{\mathcal B}(\ell^p_w)}\big)^{-1}\ge 2 \max(C_2 C_3, C_1) \big(A_p(w)\big)^{2/p}
\|{A}\|_{{\mathcal B}_{r,\alpha}}
\times
\left\{\begin{array}{ll} N^{-\min(1,\alpha-d_{\mathcal G}/r') } & {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
N^{-1} (\ln (N+1))^{1/r'} & {\rm if} \ \alpha=d_{\mathcal G}/r'+1,
\end{array}\right.$$ and $H_{A, N}=(H_{A, N}(\lambda, \lambda'))_{\lambda, \lambda'\in V}$ be as in except replacing $\beta_{p, w}(A)$ by $(\|A^{-1} \|_{{\mathcal B}(\ell^p_w)})^{-1}$. Following the argument used in the proof of Lemma \[tech.lem2\], we obtain $$\label{wiener.pff.eq2-}
\|H_{A,N}\|_{{\mathcal B}_{r, \alpha}}\lesssim N^{\alpha+d_{\mathcal G}/r}$$ and $$\label{wiener.pff.eq2}
|c(\lambda)|
\lesssim
(A_p(w))^{1/p} \|A^{-1} \|_{{\mathcal B}(\ell^p_w)}
N^{d_{\mathcal G}} \sum_{\lambda'\in V}
H_{A, N}(\lambda, \lambda')|(Ac)(\lambda')|, \ c=(c(\lambda))_{\lambda\in V}\in \ell^p_w.$$
Write $A^{-1}:=(\check{a}(\lambda', \lambda))_{\lambda', \lambda\in V}$ and denote $\check{a}_{\lambda}:=(\check{a}(\lambda', \lambda))_{\lambda'\in V}, \ \lambda\in V$. Then $\check{a}_{\lambda}\in \ell^p_w$ by and the invertibility of the matrix $A$. Replacing $c$ in by $\check{a}_{\lambda}$, we get $$\begin{aligned}
\label{wiener.pff.eq4}
|\check{a}(\lambda', \lambda)|
& \lesssim &
(A_p(w))^{1/p} \|A^{-1} \|_{{\mathcal B}(\ell^p_w)}
N^{d_{\mathcal G}} \sum_{\lambda^{\prime\prime}\in V}
H_{{A}, N}(\lambda', \lambda^{\prime\prime})|({A}\check a_{\lambda}) (\lambda^{\prime\prime})|\nonumber\\
& = & (A_p(w))^{1/p} \|A^{-1} \|_{{\mathcal B}(\ell^p_w)} N^{d_{\mathcal G}} H_{{A}, N}(\lambda', \lambda)\ \ {\rm for \ all} \ \lambda, \lambda'\in V.\end{aligned}$$ This together with and the solidness of the Beurling algebra ${\mathcal B}_{r, \alpha}({\mathcal G})$ in Proposition \[beurling.prop\] implies that $$\label{wiener.pff.eq5}
\|A^{-1}\|_{{\mathcal B}_{r, \alpha}} \lesssim
(A_p(w))^{1/p} \|A^{-1} \|_{{\mathcal B}(\ell^p_w)} N^{d_{\mathcal G}} \|H_{{A}, N}\|_{{\mathcal B}_{r, \alpha}}
\lesssim (A_p(w))^{1/p} \|A^{-1} \|_{{\mathcal B}(\ell^p_w)}N^{\alpha+d_{\mathcal G}(1+1/r)}.$$
Define $$\label{tech.lem4.N1}
N_1=
\left\{\begin{array}{ll}
\tilde N_1
& {\rm if} \ \alpha\ne d_{\mathcal G}/r'+1\\
2 \tilde N_1 (\ln (\tilde N_1+1))^{1/r'}
& {\rm if} \ \alpha= d_{\mathcal G}/r'+1,
\end{array}\right.$$ where $$\tilde N_1=\left\lfloor \Big (2 \max(C_1, C_2 C_3) \big(A_p(w)\big)^{2/p}\|A^{-1}\|_{\mathcal B(\ell^p_{w})}
\|{A}\|_{{\mathcal B}_{r,\alpha}}\Big)^{1/\min(1,\alpha-d_{\mathcal G}/r')}\right\rfloor +2$$ and $C_1, C_2, C_3$ are absolute constants in , and respectively. One may verify that $N_1$ satisfies . Then replacing $N$ in by the above integer $N_1$ completes the proof.
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[^1]: The project is partially supported by NSF of China (Grant Nos.11701513, 11771399, 11571306), the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2019R1F1A1051712) and the National Science Foundation (DMS-1816313)
|
---
abstract: 'We have developed a mathematical theory of the topological vertex—a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds.'
address:
- 'Department of Mathematics, Stanford University, Stanford, CA 94305, USA'
- 'Department of Mathematics, Northwestern University, Evanston, IL 60208, USA'
- 'Center of Mathematical Sciences, Zhejiang University, Hangzhou, China; Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555, USA '
- 'Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China'
author:
- Jun Li
- 'Chiu-Chu Melissa Liu'
- Kefeng Liu
- Jian Zhou
title: A mathematical theory of the topological vertex
---
sec1.tex
sec2.tex
sec3.tex
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sec5.tex
sec6.tex
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sec8.tex
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|
---
abstract: 'It is shown that the Einstein–Podolski–Rosen type experiments are the natural consequence of the groupoid approach to noncommutative unification of general relativity and quantum mechanics. The geometry of this model is determined by the noncommutative algebra ${\mbox{${\cal A}$}}= {\mbox{$C^{\infty}_{c}(G,{\bf C})$}}$ of complex valued, compactly supported, functions (with convolution as multiplication) on the groupoid $G = E \times \Gamma $. In the model considered in the present paper $E$ is the total space of the frame bundle over space-time and $\Gamma $ is the Lorentz group. The correlations of the EPR type should be regarded as remnants of the totally non-local physics below the Planck threshold which is modelled by a noncommutative geometry.'
author:
- |
Michael Heller\
Vatican Observatory, V-12000 Vatican City State[^1]
- |
Wies[ł]{}aw Sasin\
Institute of Mathematics, Warsaw University of Technology\
Plac Politechniki 1, 00-661 Warsaw, Poland
title: 'Einstein-Podolski-Rosen Experiment from Noncommutative Quantum Gravity'
---
Introduction
============
One of the greatest challenges of contemporary physics is to explain the non-local effects of quantum mechanics theoretically predicted (in the form of a gedanken experiment) by Einstein, Podolski and Rosen (EPR, for short) [@EPR] and experimentally verified by Aspect et al. [@Aspect1; @Aspect2; @Aspect3] (for a comprehensive review see [@Redhead]). Although non-local effects of this type logically follow from the postulates of quantum mechanics, it seems strange and against our “realistic common sense” to accept that two particles separated in space could be so strongly correlated (provided they once interacted with each other) that they seem to “know” about each other irrespectively of the distance separating them. In spite of long lasting discussions, so far no satisfactory explanation of this effect has been offered. In the present paper we shall argue that effects of the EPR type are remnants of the totally non-local physics of the fundamental level (below the Planck threshold). We substantiate our argument by explaining the EPR experiment in terms of a quantum gravity model, based on a noncommutative geometry, proposed by us in [@HSL] (see also [@HS1]), although the explanation itself does not depend on particulars of the model.
The main physical idea underlying our model is that below the Planck threshold (we shall speak also on the “fundamental level”) there is no space-time but only a kind of pregeometry which is modeled by a suitable noncommutative space, and that space-time emerges only in the transition process to the classical gravity regime. Accordingly, we start our construction not from a space-time manifold $M$, but rather from a groupoid $G = E \times \Gamma $ where $E$ is a certain abstract space and $\Gamma $ a suitable group of “fundamental symmetries”. In the present paper, for the sake of simplicity, we shall assume that $E$ is the total space of the frame bundle over space-time $M$ and $\Gamma = SO(3,1)$. We define, in terms of this geometry, the noncommutative algebra ${\mbox{${\cal A}$}}= {\mbox{$C^{\infty}_{c}(G,{\bf C})$}}$ of smooth, compactly supported, complex-valued functions on the groupoid $G$ with the usual addition and convolution as multiplication. We develop a noncommutative differential geometry basing on this algebra, and define a noncommutative version of Einstein’s equation (in the operator form). The algebra can be completed to become a -algebra, and this subalgebra of which satisfies the generalized Einstein’s equation is called [*Einstein - algebra*]{}, denoted by (for details see [@HSL]). And now quantization is performed in the standard algebraic way. Since the explanation of the EPR type experiments depends on the noncommutative structure of the groupoid $G$ rather than on details of our field equations and the quantization procedure, we shall not review them here; the reader interested in the particulars of our model should consult the original paper [@HSL].
It can be shown that the subalgebra (elements of are called [*projectible*]{}) of functions which are constant on suitable equivalence classes of fibres $\pi_E^{-1}(p)$, $\pi_E$ being the projection $G=E\times
\Gamma \rightarrow \Gamma $ and $p \in E$, is isomorphic to the algebra $C^{\infty }(M)$ of smooth functions on $M$. Consequently, by making the restriction of to we recover the ordinary space-time geometry and the standard general relativity. In our model, to simplify calculations, we have assumed that the noncommutative differential geometry is determined by the submodule $V$ of the module of all derivations of , and that $V$ has the structure adapted to the product structure of the groupoid $G=E\times \Gamma $, i. e., $V = V_E \oplus V_{\Gamma }$, where $V_E$ and $V_{\Gamma }$ are “parts” parallel to $E$ and $\Gamma $, correspondingly. It can be seen that in our model the geometry “parallel” to $E$ is responsible for generally relativistic effects, and that “parallel” to $\Gamma $ for quantum effects. In general, “mixed terms” should appear, and then one would obtain stronger interaction between general relativity and quantum physics. This remains to be elaborated in the future.
The crucial point is that the geometry as determined by the noncommutative algebra is non-local, i. e., there are no maximal ideals in which could determine points and their neighborhoods in the corresponding space, and consequently neither space points nor time instants can be defined in terms of . Physical states of a quantum gravitational system are identified with states on the algebra , i. e., with the set of positive linear functionals (normed to unity) on , and pure states in the mathematical sense are identified with pure states in the physical sense. Let $a \in \cal A $ be a quantum gravitational observable, i. e., a projectible and Hermitian element of ($a$ must be projectible to leave traces in the macroscopic world), and $\varphi $ a state on . Then $\varphi (a)$ is the expectation value of the observable $a$ when the system is in the state $\varphi $. The fact that $a$ is an element of a “non-local” (noncommutative) algebra implies that when $a$ is projected to the space-time $M$ it becomes a real-valued (since $a$ is Hermitian) function on $M$, and the results of a measurement corresponding to $a$ are values of this function. Consequently, one should expect correlations between various measurement results even if they are performed at distant points of space-time $M$. We shall see that this is indeed the case.
The organization of our material is the following. In Section 2, we consider the eigenvalue equation for quantum gravitational observables. In Section 3, we show that correlations of the EPR type between distant events in space-time are consequences of non-local (noncommutative) physics of the quantum gravitational regime, and in Section 4 we present details of the EPR experiment in terms of the noncommutative approach. Section 5 contains concluding remarks.
Measurement on Quantum Gravitational System
===========================================
Let $\varphi : {\mbox{${\cal A}$}}\rightarrow {\bf C}$ be a state on the algebra , i. e., $\varphi({\bf 1})=1$ and $\varphi (aa^*)
\geq 0$ for every $a \in {\mbox{${\cal A}$}}$. It can be easily seen that $\varphi|{\mbox{${\cal A}$}}_{proj}: {\mbox{${\cal A}$}}_{proj} \rightarrow {\bf C}$ is a state on the subalgebra ${\mbox{${\cal A}$}}_{proj}$.
Let now $a \in {\mbox{${\cal A}$}}_{proj} $ be Hermitian, then there exists a function $\bar{a} \in C^{\infty }(M)$ with $ \bar{a} \circ pr =
a$, where $pr: G \rightarrow M$ is the projection, and the state $\bar{\varphi }: C^{\infty }(M) \rightarrow {\bf R}$ on the algebra $C^{\infty }(M)$, such that $\varphi (a) = \bar{\varphi
}(\bar{a})$. Since the algebras ${\mbox{${\cal A}$}}_{proj}$ and $C^{\infty
}(M)$ are isomorphic, the spaces of states of these algebras are isomorphic as well.
To make a contact with the standard formulation of quantum mechanics we represent the noncommutative algebra in a Hilbert space by defining, for each $p\in E$, the representation $$\pi_p: {\mbox{${\cal A}$}}\rightarrow {\mbox{${\cal B}$}}({\mbox{${\cal H}$}}),$$ where ${\mbox{${\cal B}$}}({\mbox{${\cal H}$}})$ is the algebra of operators on the Hilbert space ${\mbox{${\cal H}$}}= L^2(G_p)$ of square integrable functions on the fibre $G_p = \pi_E^{-1}(p)$, $\pi_E: G \rightarrow E$ being the natural projection, with the help of the formula $$\pi_p(a)\psi = \pi_p(a)*\psi$$ or more explicitly $$(\pi_p(a)\psi)(\gamma ) = \int_{G_p}a(\gamma_1)\psi(\gamma_1^{-
1}\gamma),$$ $a \in {\mbox{${\cal A}$}},\, \gamma = \gamma_1 \circ \gamma_2,\; \gamma ,
\gamma_1, \gamma_2 \in G_p$, $\psi \in L^2(G_p)$, and the integration is with respect to the Haar measure. This representation is called the [*Connes representation*]{} (see [@Connes p.102], [@HSL]).
Now, let us suppose that $a$ is an observable, i. e., $a \in
{\mbox{${\cal A}$}}_{proj} $, and we perform a measurement of the quantity corresponding to this observable. The eigenvalue equation for $a$ is $$\int_{G_P}a(\gamma_1)\psi(\gamma_1^{-1}\gamma ) = r_p\psi
(\gamma) \label{eq1}$$ where the eigenvalue $r_q$ is the expected result of the measurement when the system is in the state $\psi $. Here we must additionally assume that the “wave function” $\psi $ is constant on fibres of $G$ to guarantee for equation (\[eq1\]) to have its usual meaning in the non-quantum gravity limit. If this is the case, equation (\[eq1\]) can be written in the form $$\psi (\gamma_1^{-1}\gamma ) \int_{g_p}a(\gamma_1) = r_p\psi
(\gamma )$$ and consequently $$r_p = \int_{G_p}a(\gamma_1).$$ Let us notice that the measurement result is a measure in the mathematical sense.
It is obvious that if we define the “total phase space” of our quantum gravitational system $$L^2(G) := \bigoplus_{p\in E}L^2(G_p)$$ and the operator $$\pi(a) := (\pi_p(a))_{p\in E}$$ acting on $L^2(G)$ then the eigenvalue equation becomes $$\pi (a) \psi = r\psi$$ where $r: M \rightarrow {\bf R}$ is a function on space-time $M$ given by $$r(x) = \int_{G_p}a(\gamma_1)
\label{eq2}$$ where $x$ is a point in $M$ to which the frame $p$ is attached. Let us notice that the function $r$ is equal to the function $\bar{a}: M \rightarrow {\bf R}$ (see the beginning of the present Section). Let us now consider a composed quantum system the state of which is described by the single vector in the Hilbert space, and let us perform a measurement on its parts when they are at a great distance from each other. Formula (\[eq2\]) asserts that in such a case the results of the measurement are not independent but are the values of the same function defined on space-time. This can be regarded as a “shadow” of a non-local character of the observable $a$ projected down to space-time $M$.
EPR Non-Locality
================
So far we were mainly concerned with what happens when we project the algebra onto the “horizontal component” $E$ of the groupoid $G$. This, of course, gives us the transition to the classical space-time geometry (general relativity). In the present Section, we shall be interested in projecting onto the “vertical component” $\Gamma $ of $G$. This gives us quantum effects of our model.
Let us consider functions [*projectible*]{} to $\Gamma $. We define $${\mbox{${\cal A}_{\Gamma }$}}:= \{f \circ pr_{\Gamma }: f\in C^{\infty }_c(\Gamma,
{\bf C}) \} \subset {\mbox{${\cal A}$}}.$$ The reasoning similar to that in the beginning of the present section shows that if $s \in {\mbox{${\cal A}_{\Gamma }$}}$ and $\psi : {\mbox{${\cal A}_{\Gamma }$}}\rightarrow {\bf C}$ is a state on then $\psi (s) =
\underline{\psi }(\underline{s})$, where $s
= \underline{s} \circ pr_{\Gamma }, \; pr_{\Gamma }: G
\rightarrow \Gamma $ is the projection, and $\underline{\psi }:
C^{\infty }_c(\Gamma, {\bf C}) \rightarrow {\bf C}$ is a state on $C^{\infty }_c(\Gamma, {\bf C})$.
Let now $\Phi $ be a state on $C^{\infty }_c(\Gamma, {\bf C})$. We say that the state $\varphi: {\mbox{${\cal A}$}}\rightarrow {\bf C}$ is [*$\Gamma $-invariant associated to $\Phi $*]{} on if $$\varphi (s) = \left\{
\begin{array}{cr}
\Phi(\underline{s}), & \mbox{if $s \in {\mbox{${\cal A}_{\Gamma }$}}$},\\
0, & \mbox{if $s \not\in {\mbox{${\cal A}_{\Gamma }$}}$ }.
\end{array}
\right.$$ Since all fibres of $G_p, \, p \in E$, of $G$ are isomorphic, the number $\varphi (s) = \Phi (\underline{s})$, for $s \in
{\mbox{${\cal A}_{\Gamma }$}}$, is the same in each fibre $G_p$. If additionally $s$ is a Hermitian element of , and if a measurement performed at a certain point of space-time $M$ gives the number $\varphi (s)$ as its result, then this result is immediately “known” at all other fibres $G_p$, $p\in E$, of $G$, and consequently at all other points of space-time $x = \pi_M(p) \in
M$, where $\pi_M: E \rightarrow M$ is the canonical projection.
This can be transparently seen if we consider the problem in the Hilbert space by using the Connes representation of the algebra . Let $a \in {\mbox{${\mbox{${\cal A}$}}_{\Gamma }$}}$, and let us consider the following Connes representations $$\pi_p(a)(\xi_p) = a_p * \xi_p,
\label{Connesa}$$ and $$\pi_q(a)(\xi_q) = a_q * \xi_q,
\label{Connesb}$$ where $\xi_p \in L^2(G_p), \, \xi_q \in L^2(G_q), \, p,q \in E,
\, p \neq q$. Since $G_p$ and $G_q$ are isomorphic, we can choose $\xi_p$ and $\xi_q$ to be isomorphic with each other, which implies that $\pi_p(a)$ and $\pi_q(a)$ are isomorphic as well. We have the following important
[*Lemma.*]{} If $a \in {\mbox{${\mbox{${\cal A}$}}_{\Gamma }$}}$ then its image under the Connes representation $\pi_p$ does not depend of the choice of $p \in
E$ (up to isomorphism).
Since $p \in E$ projects down to the space-time point $\pi_M(p)
\in M$, $\pi_M: E \rightarrow M$, the above result should be interpreted as stating that all points of $M$ “know” what happens in the fiber $G_g, \, g \in \Gamma $. This, together with the fact that vectors $\xi_p$, upon which the observable $\pi_p(a)$, $a \in {\mbox{${\cal A}$}}_{\Gamma }$ acts, also do not depend of $p$, in principle, explains the EPR type experiments. However, let us go deeper into details.
EPR Experiment in Terms of Noncommutative Geometry
==================================================
In this section we consider a group $\Gamma $ such that $\Gamma_0= SU(2)$ is its compact subgroup. We look for an element $s\in
{\mbox{${\mbox{${\cal A}$}}_{\Gamma }$}}$ such that $$\pi_p(s): L^2(\Gamma_0) \rightarrow L^2(\Gamma_0).$$ Of course, ${\bf C}^2 \subset L^2(\Gamma_0)$. We define two linearly independent functions on the group $\Gamma_0$, for instance the constant function $${\bf 1}: \Gamma_0 \rightarrow {\bf C},$$ and $${\rm det}: \Gamma_0 \rightarrow {\bf C},$$ which span the linear space ${\bf C}^2$, i. e., ${\bf C}^2 =
\langle {\bf 1}, {\rm det}\rangle_{{\bf C}}$. Let $\hat{S}_z =
\pi_p(s)|_{{\bf C}^2}$ be the usual z-component spin operator. We have $$\pi_p(s)\psi = \hat{S}_z\psi ,$$ for $\psi \in {\bf C}^2$ or, by using the Connes representation and the fact that $\hat{S}_z\psi = \pm \frac{\hbar }{2}\psi $, $$\int_{\Gamma_0}s_p(\gamma_1)\psi (\gamma_1^{-1}\gamma) = \pm
\frac{\hbar}{2}\psi .$$ Since $s_p = $const, one obtains $$\int_{\Gamma_0 }\psi (\gamma_1^{-1} \gamma ) \sim
\psi (\gamma ).$$ One of the solutions of this equation is $\psi = {\bf
1}_{\Gamma_0}$. Therefore $$\frac{\hbar }{2} = \pm \int_{\Gamma_0}s_p(\gamma_1).$$ Hence $$(s_p)_1 = +\frac{\hbar }{2} \frac{1}{{\rm vol}\Gamma_0},$$ $$(s_p)_2 = -\frac{\hbar }{2} \frac{1}{{\rm vol}\Gamma_0},$$ and consequently $$\pi_p((s_p)_1)\psi = + \frac{\hbar }{2}\psi \;\; {\rm for}\,
\psi\in {\bf C}^+,$$ $$\pi_p((s_p)_2)\psi = - \frac{\hbar }{2}\psi \;\; {\rm for}\,
\psi\in {\bf C}^-,$$ where ${\bf C}^+ := {\bf C}\times \{0\}$, and ${\bf C}^- :=
\{0\} \times {\bf C}$. To summarize these results we can define $$\hat{S}_z \psi = \pi_p(s_1,s_2)\psi :=
\left\{
\begin{array}{cr}
(s_{1})_p*\psi & \mbox{{\rm if} $\psi \in {\bf C}^+$,} \\
(s_{2})_p*\psi & \mbox{{\rm if} $\psi \in {\bf C}^-$.}
\end{array}
\right.$$
Now, the analysis of the “EPR paradox” proceeds in the same way as in the standard textbooks on quantum mechanics (see for instance [@Isham pp. 179-181]. An observer $A$, situated at $\pi_M(p) = x_A \in M$, measures the z-spin component of the one of the electrons[^2], i. e., he applies the operator $\hat{S}_z
\otimes {\bf 1}|_{{\bf C}^2}$ to the vector $\xi =
\frac{1}{\sqrt{2}}(\psi \otimes
\varphi - \varphi \otimes \psi )$ where $\psi \in {\bf C}^+$ and $\varphi \in {\bf C}^-$. Let us suppose that the result of the measurement is $\frac{\hbar }{2}$. This means that the state vector $\xi = \frac{1}{\sqrt{2}}(\psi \otimes \varphi -
\varphi \otimes \psi ) \in {{\bf C}}^2 \otimes {{\bf C}}^2 \subset
L^2(G_r) \otimes L^2(G_r)$, $r \in E$, has collapsed to $\xi_0
=\frac{1}{\sqrt{2}}(\psi \otimes \varphi )$, and that immediately after the measurement the system is in the state $\xi_0$ which is the same (up to isomorphism) for all fibres $G_r$ whatever $r \in E$, and consequently it does not depend of the point in space-time to which $r$ is attached (see formulae (\[Connesa\]) and (\[Connesb\]) which are obviously valid also for tensor products). In particular, the vector $\xi_0$ is the same for the fibres $G_p$ and $G_q$ where $p$ is such that $\pi_M(p) = x_A$ and $q$ is such that $\pi_M(q) = x_B \; (x_A
\neq x_B)$. It is now obvious that if an observer $B$, situated a $x_B$ measures the z-spin component of the second electron, i. e., if he applies the operator $ {\bf 1}|_{{\bf C}^2} \otimes
\hat{S}_z$ to the vector $\xi_0$, he will obtain the value $-\frac{\hbar }{2}$ as the result of his measurement.
Concluding Remarks
==================
To conclude our analysis it seems suitable to make the following remarks.
It should be emphasized that our scheme for noncommutative quantum gravity does not “explain” quantum mechanical postulates. In the very construction of our scheme it has been assumed that the known postulates which, in the standard formulation of quantum mechanics are valid for the algebra of observables, can be extended to a more general noncommutative algebra. However, the very fact that these postulates are valid in the conceptual framework of noncommutative geometry gives them a new flavour. For instance, since in the noncommutative regime there is no time in the usual sense, the sharp distinction between the continuous unitary evolution and the non-continuous process of measurement (“collapse of the wave function”) disappears. This distinction becomes manifest only when time emerges (see [@Emergence]) in changing from the noncommutative regime to the usual space-time geometry.
What our approach does explain is the fact that some quantum effects are strongly correlated even if they occur at great distances from each other. These effects are “projections” from the fundamental level at which all concepts have purely global meanings.
This explanation does not depend on “details” of our model, such as some particulars of the construction of noncommutative differential geometry, the concrete form of generalized Einstein’s equation, or the dynamical equation for quantum gravity. However, it does depend on (or even more, it is deeply rooted in) the noncommutative character of the algebra ${\mbox{${\cal A}$}}=
{\mbox{$C^{\infty}_{c}(G,{\bf C})$}}$ and the product structure of the groupoid $G = E
\times \Gamma $.
[99]{} A. Einstein, B. Podolski and N. Rosen, “Can quantum description of physical reality be considered complete?” [*Phys. Rev.*]{} [**47**]{}, 777–780 (1935). A. Aspect, P. Grangier and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem”, [*Phys. Rev. Lett.*]{}, [**47**]{}, 460–463 (1981). A. Aspect, P. Grangier and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bells inequalities”, [*Phys. Rev. Lett.*]{}, [**49**]{}, 91–94 (1982). A. Aspect, J. Dalibard and G. Roger, “Experimental tests of Bell inequalities using time-varying analyzers”, [*Phys. Rev. Lett.*]{}, [**49**]{}, 1804–1807 (1982). M.L.G. Redhead, [*Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics*]{}, Clarendon Press, Oxford, 1987. M. Heller, W. Sasin and D. Lambert, “Groupoid approach to noncommutative quantization of gravity, [*J. Math. Phys.*]{}, [**38**]{}, 5840–5853 (1997). M. Heller, and W. Sasin, ”Towards noncommutative quantization of gravity, gr-qc/9712009. A. Connes, [*Noncommutative Geometry*]{}, Academic Press, San Diego-New York, 1994. C. J. Isham, [*Lectures on Quantum Theory*]{}, Imperial College Press, London, 1995. M. Heller and W. Sasin, “Emergence of Time”, gr-qc/9711051.
[^1]: Correspondence address: ul. Powstańców Warszawy 13/94, 33-110 Tarnów, Poland. E-mail: [email protected]
[^2]: Let us notice that when $A$ measures the spin of the electron, he simultaneously determines the position of the electron (at least roughly), i. e., the position $x_A$ at which he himself is situated (spin and position operators commute).
|
---
abstract: |
The Mallows model on $S_n$ is a probability distribution on permutations, $q^{d(\pi,e)}/P_n(q)$, where $d(\pi,e)$ is the distance between $\pi$ and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs $(i,j)$ where $1\leq i<j\leq n$, but $\pi_i>\pi_j$. Analyzing the normalization $P_n(q)$, Diaconis and Ram calculated the mean and variance of $d(\pi,e)$ in the Mallows model, which suggests the appropriate $n \to \infty$ limit has $q_n$ scaling as $1-\beta/n$. We calculate the distribution of the empirical measure in this limit, $u(x,y)\, dx\, dy = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} \delta_{(i,\pi_i)}$. Treating it as a mean-field problem, analogous to the Curie-Weiss model, the self-consistent mean-field equations are $\frac{\partial^2}{\partial x \partial y} \ln u(x,y) = 2 \beta u(x,y)$, which is an integrable PDE, known as the hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process, and the ground state of the $\mathcal{U}_q(\mathfrak{sl}_2)$-symmetric XXZ ferromagnet.
[**Keywords:**]{} Mallows model, random permutation, Liouville equation, ASEP, XXZ model. .2 cm [**MCS numbers: 82B05, 82B10, 60B15**]{}
address: |
Department of Mathematics\
Hylan Building\
University of Rochester\
Rochester, NY 14627
author:
- Shannon Starr
date: 'March 2, 2009'
title: 'Thermodynamic Limit for the Mallows Model on $S_n$'
---
Introduction and Main Results
=============================
The Coxeter generators of the symmetric group $S_n$ are the transpositions $(1,2)$, $(2,3)$, …, $(n-1,n)$. The height of a permutation is defined distance to the identity element $e$, $$d(\pi,e)\, =\, \min\{k \geq 0\, :\, \exists\, \tau_1,\dots,\tau_k \in \{(1,2),\dots,(n-1,n)\}\ \text{ such that }\ \pi = \tau_1 \cdots \tau_k\}\, .$$ More generally, $d(\pi_1,\pi_2) = d(\pi_2^{-1} \pi_1,e)$. It is easy to see that $d(\pi,e) = d(\pi^{-1},e)$. In fact, another formula is $$d(\pi,e)\, =\, |\{(i,j) \in {\mathbb{Z}}^2\, :\, 1\leq i<j\leq n\ \text{ and }\ \pi_i>\pi_j\}|\, .$$ The pairs $(i,j)$ are called inversions of $\pi$. In [@DiaconisRam], Diaconis and Ram studied the Mallows measure, which is a probability measure on $S_n$ given by $$\mathbbm{P}_{n}^{q}(\pi)\, =\, \frac{q^{d(\pi,e)}}{P_n(q)}\, ,$$ with $P_n(q)$ being a normalization constant. Actually, Diaconis and Ram studied a Markov chain on $S_n$ for which the Mallows model gives the limiting distribution. This was followed up by another paper on a related topic by Benjamini, Berger, Hoffman and Mossel (BBHM) [@BBHM] who related the biased shuffle and the Mallows model to the asymmetric exclusion process and the “blocking” measures (of Liggett, see [@Liggett], Chapter VIII, especially Example 2.8 and the end of Section 3). They did this using Wilson’s height functions [@Wilson]. We will discuss this more in Section \[sec:Applications\]. For now, let it suffice that Diaconis and Ram identified the explicit formula for the normalization which they remarked is the “Poincaré polynomial”: $$P_n(q)\, =\, \prod_{i=1}^{n} \left(\frac{q^{i}-1}{q-1}\right)\, =\, [n]_q!\, =\, [n]_q \cdots [1]_q\, ,\quad \text{where}\quad
[n]_q\, =\, \frac{q^n-1}{q-1}\, .$$ Further references for the statistical applications[^1] of the Mallows model can be found in their paper.
Note that, physically speaking, one would define a Hamiltonian energy function $H_n : S_n \to {\mathbb{R}}$ as $$H_n(\pi)\, =\, \frac{1}{n-1}\, \sum_{1\leq i<j\leq n} \mathbbm{1}_{(0,\infty)}(\pi_i-\pi_j)\, .$$ In this case, one thinks of $\pi = (\pi_1,\dots,\pi_n)$ as some type of constrained spin system, where each of the components $\pi_1,\dots,\pi_n$ are spins in $\{1,\dots,n\}$ as in a Potts model. The choice of the normalization of the Hamiltonian is then standard for mean-field models. One would be most interested in the free energy $$f_n(\beta)\, =\, -\frac{1}{\beta n}\, \ln \sum_{\pi \in S_n} e^{-\beta H_n(\pi)}\, .$$ For our purposes, we prefer to consider the mathematically simpler “pressure” $$p_n(\beta)\, =\, \frac{1}{n}\, \ln \left(\frac{1}{n!}\, \sum_{\pi \in S_n} e^{-\beta H_n(\pi)}\right)\, .$$ (Note that, contrary to the usual conventions of statistical physics, we have divided the partition function, which is $Z_n(\beta) = \sum_{\pi in S_n} e^{-\beta H_n(\pi)}$, by the infinite-temperature partition function $Z_n(0)=n!$, which is equivalent to starting with a normalized [*a priori*]{} measure rather than counting measure on $S_n$.) It is trivial to see that this is given precisely by the Poincaré polynomial described by Diaconis and Ram: $$p_n(\beta)\, =\, \frac{1}{n} \ln\frac{P_n(e^{-\beta/(n-1)})}{n!}\, =\, \frac{1}{n} \ln \frac{[n]_{e^{-\beta/(n-1)}}!}{n!}\, .$$ With this scaling, it is also easy to calculate the limit: $$p(\beta)\, =\, \lim_{n \to \infty} p_n(\beta)\, =\, \int_0^1 \ln\left(\frac{1-e^{-\beta x}}{\beta x}\right)\, dx\, ,$$ (which can be solved explicitly using the polylogarithm function). From this one can calculate the mean and variance in the limiting Gibbs measure. For instance, one can calculate ${\mathbb{E}}(d(\pi,e) - n(n-1)/4)^2 \sim n^3/72$ in the uniform measure on $S_n$.
To go beyond the statistics of $d(\pi,e)$ it seems worthwhile to study the empirical measure of $\pi$: $$\frac{1}{n} \sum_{i=1}^{n} \delta_{(i,\pi_i)}\, ,$$ which is a normalized measure on $\{1,\dots,n\} \times \{1,\dots,n\}$. More specifically, this is a random measure. Rescaling the discrete cube $\{1,\dots,n\} \times \{1,\dots,n\}$ to $[0,1] \times [0,1]$, it is easy to see that the random empirical measure converges, in probability, to the non-random Lebesgue measure, when $\beta=0$ (the uniform case). Our main theorem generalizes this result.
\[thm:main\] For any $\beta \in {\mathbb{R}}$, $$\lim_{\epsilon \downarrow 0} \lim_{n \to \infty}
\mathbbm{P}_n^{1-\beta/n}\left\{\left|\frac{1}{n} \sum_{i=1}^{n} f(i/n,\pi_i/n) - \int_{[0,1] \times [0,1]} f(x,y) u(x,y)\, dx\, dy\right| > \epsilon\right\}\, =\, 0\, ,$$ for every continuous function $f : [0,1] \times [0,1] \to {\mathbb{R}}$, where $$u(x,y)\, =\, \frac{(\beta/2) \sinh(\beta/2)}{\big(e^{\beta/4} \cosh(\beta[x-y]/2) - e^{-\beta/4} \cosh(\beta[x+y-1]/2)\big)^2}\, .$$
Note that (one can show) the limit $\beta \to 0$ gives $1$. The proof of Theorem \[thm:main\] uses a rigorous version of mean-field theory, as in the solution of the Curie-Weiss model. An interesting feature is that the self-consistent mean-field equation leads us to the characterization of $u$ as the solution of an integrable PDE $$\frac{\partial^2}{\partial x \partial y} \ln u(x,y)\, =\, 2 \beta u(x,y)\, .$$ It is not unusual for mean-field problems to lead to integrable PDE’s. We demonstrate this briefly in the next section with the ubiquitous toy model, the Curie-Weiss ferromagnet.
Toy Model: The Curie-Weiss Ferromagnet
======================================
We include this section merely to point out that mean-field problems often do lead to integrable PDE. However the issue is serious: in fact there is a recent paper by Genovese and Barra which we recommend for more details [@GenoveseBarra]. Our approach merely summarizes their results (in our own words) as well as the earlier paper by Barra, himself [@Barra]. The configuration space of the CW model is $\Omega_N = \{+1,-1\}^N = \{\sigma = (\sigma_1,\dots,\sigma_n)\, :\, {\sigma}_1,\dots,{\sigma}_n = \pm 1\}$. For technical reasons, we choose the Hamiltonian as $$H_N({\sigma},t,x)\, =\, - \frac{t}{2N}\, \sum_{i,j=1}^{N} {\sigma}_i {\sigma}_j - x \sum_{i=1}^{N} {\sigma}_i\, ,$$ we assume $t\geq 0$ and $x \in {\mathbb{R}}$. Defining $m_N({\sigma}) = N^{-1} \sum_{i=1}^{N} {\sigma}_i$, which takes values in $[-1,1]$, we see that $$H_N\, =\, - N \left(\frac{t m_N^2}{2} + x m_N\right)\, .$$ Therefore, defining $$p_N(t,x)\, =\, \frac{1}{N} \ln \sum_{{\sigma}\in \Omega_N} e^{-H_N({\sigma},t,x)}\, ,$$ we easily see that $$\frac{\partial}{\partial t} p_N(t,x)\, =\, \frac{1}{2} \langle m_N^2 \rangle_{N,t,x}\, ,$$ and $$\frac{\partial^2}{\partial x^2} p_N(t,x)\, =\, N \left( \langle m_N^2 \rangle - \langle m_N\rangle^2\right)\, ,$$ where $$\langle f \rangle\, =\, \langle f\rangle_{N,t,x}\, =\, \frac{\sum_{{\sigma}\in \Omega_N} f(\sigma) e^{-H_N({\sigma},t,x)}}{\sum_{{\sigma}\in \Omega_N} e^{-H_N({\sigma},t,x)}}\, .$$ Actually it is easier to consider the “order parameter,” $$u_N(t,x)\, =\, \langle m_N \rangle_{N,t,x}\, =\, \frac{\partial}{\partial x} p_N(t,x)\, ,$$ from which $p_N(t,x)$ can be calculated by solving the ODE: $$\begin{cases}
\frac{\partial}{\partial x} p_N(t,x)\, =\, u_N(t,x) & \text{ for $x \in {\mathbb{R}}$,}\\
p_N(t,x) - |x| \to \frac{1}{2} t^2 & \text{ as $x \to \pm \infty$.}
\end{cases}$$ Then we see that $u_N(t,x)$ satisfies the viscous Burgers equation (with velocity equal to the negative amplitude): $$\begin{cases}
\frac{\partial}{\partial t}\, u_N(t,x)\, =\, u_N(t,x)\, \frac{\partial}{\partial x}\, u_N(t,x) + \frac{1}{2N} \cdot \frac{\partial^2}{\partial x^2} u_N(t,x)
& \text{ for $t>0$ and $x \in {\mathbb{R}}$,}\\
u_N(0,x)\, =\, \tanh(x)
& \text{ for $x \in {\mathbb{R}}$.}
\end{cases}$$ This is an integrable PDE, using the Cole-Hopf transform. See, for instance, Chapter 4 of Whitham, [@Whitham]. Actually, this leads to a solution in terms of Gaussian integrals. The analogous transform in spin-configuration notation is the Hubbard-Stratonovich transform: $$e^{N t m^2/2}\, =\, \int_{-\infty}^{\infty} \frac{e^{Nt (mx - x^2/2)}}{\sqrt{2\pi/t}}\, dx\, ,$$ which “linearizes” the dependence of the Hamiltonian on $m_N$, in the exponential. This trick is used to solve the Curie-Weiss model. See, for example, Thompson [@Thompson].
Note that in the $N \to \infty$ limit, one obtains $u(t,x) = \lim_{N \to \infty} u_N(t,x)$ being the vanishing-viscosity solution of the inviscid Burgers equation. Shocks correspond to phase transitions. The Lax-Oleinik variational formula for solutions of hyperbolic conservation laws applies. See for example, Section 3.4.2 of Evans [@Evans]. In this context we claim that this is equivalent to the Gibbs variational formula, in the mean-field limit. We review this next.
Gibbs Variational Formula {#sec:SCMFE}
=========================
Let us begin by considering a general problem in classical statistical mechanics. Suppose that $\mathcal{X}$ is a compact metric space, and suppose that there is a two-body interaction $$h : \mathcal{X} \times \mathcal{X} \to {\mathbb{R}}\cup \{+\infty\}\, .$$ We assume that $h$ is bounded below. Then for each $N \geq 2$, one can consider the mean-field Hamiltonian $H_N : \mathcal{X}^N \to {\mathbb{R}}\cup \{+\infty\}$ $$H_N(x_1,\dots,x_N) = \frac{1}{N-1}\, \sum_{1\leq i<j\leq N} h(x_i,x_j)\, .$$ Suppose that there is also an [*a priori*]{} measure $\mu_0$ on $\mathcal{X}$, which we assume is normalized so that $$\int_{\mathcal{X}} d\mu_0(x)\, =\, 1\, .$$ Then the thermodynamic quantities are the partition function, $$Z_N(\beta)\,
=\, \int_{\mathcal{X}^N} e^{-\beta H_N(x_1,\dots,x_N)}\, d\mu_0(x_1) \cdots d\mu_0(x_N)\, ,$$ the pressure, $$p_N(\beta)\,
=\, \frac{1}{N}\, \ln Z_N(\beta)\, ,$$ and the Boltzmann-Gibbs measure $$d\mu_N^{\beta}(x_1,\dots,x_N)\,
=\, \frac{e^{-\beta H_N(x_1,\dots,x_N)}}{Z_N(\beta)}\, d\mu_0(x_1) \cdots d\mu_0(x_N)\, .$$ Physically, it is more correct to consider the free energy rather than the pressure, $f_N(\beta) = - \frac{1}{\beta} p_N(\beta)$. But we will consider $p_N(\beta)$, which seems slightly easier to handle, mathematically.
We will write $\mu_0^N$ for the measure $d\mu_0^N(x_1,\dots,x_N) = d\mu_0(x_1) \cdots d\mu_0(x_N)$. Also, if $f$ is a function, then we use the short-hand $\mu(f)$ for $\int f d\mu$. Then, according to the Gibbs variational principle, we have $$\label{eq:GibbsVariational}
p_N(\beta)\, =\, \max_{\mu_N \in \mathcal{M}_{+,1}(\mathcal{X}^N)} \frac{1}{N} \left[S_N(\mu_N, \mu_0^{\otimes N}) - \beta \mu_N(H_N)\right]\, ,$$ where $S_N(\mu_N, \mu_0^{\otimes N})$ is the relative entropy (and $\mathcal{M}_{+,1}(\mathcal{X}^N)$ denotes all Borel probability measures on $\mathcal{X}^N$) $$S_N(\mu_N, \mu_0^{\otimes N})\, =\, \begin{cases} \mu_0^N(\phi(d\mu_N/d\mu_0^N)) & \text{ if $\mu_N$ is absolutely continuous with respect to $\mu_0^N$,}\\
-\infty & \text{ otherwise,}
\end{cases}$$ and $\phi(x) = - x \ln(x)$, which is $0$ if $x=0$. Also, the unique $\mu_N$ maximizing the Gibbs variational formula (the “arg-max”) is the Boltzmann-Gibbs measure $\mu_N^{\beta}$.
A natural ansatz for the optimizing measure is $\mu_N = \mu^N$, for some measure $\mu \in \mathcal{M}_{+,1}(\mathcal{X})$. Probabilistically, this means that all the $x_1,\dots,x_N$ are independent and identically distributed. Technically, this cannot usually be exact for finite $N$. But it leads to a simpler formula because $$S_N(\mu^N,\mu_0^N)\, =\, N S_1(\mu,\mu_0)\quad \text{ and }\quad
\mu^N(H_N)\, =\, \frac{N}{2}\, \mu^2(h)\, ,$$ and one hopes that the formula may become exact in the thermodynamic limit. Mark Fannes, Herbert Spohn and Andre Verbeure proved that this approach is rigorous in the $N \to \infty$ limit [@FSV]:
\[prop:FSV\] The limiting pressure exists, $p(\beta) = \lim_{N \to \infty} p_N(\beta)$, and solves the variational problem $$p(\beta)\, =\, \max_{\mu \in \mathcal{M}_{+,1}(\mathcal{X})} [S_1(\mu,\mu_0) - \frac{\beta}{2} \mu^2(h)]\, .$$ Moreover, any subsequential limit of the sequence $(\mu_N^\beta)$ is a mixture of infinite product measures $\mu^{\infty}$, for $\mu$’s maximizing the right-hand-side of the formula above.
Note that the Gibbs variational principle (\[eq:GibbsVariational\]) is true in general for all Hamiltonians whether they are mean-field or not. (See, for instance, Lemma II.3.1 from Israel’s monograph [@Israel], or any other textbook on mathematical statistical mechanics, for a rigorous proof which also applies directly in the thermodynamic limit.) But the product ansatz which seems to yield the formula from the proposition is not generally valid, since there are nontrivial correlations in the true Boltzmann-Gibbs state. Nevertheless Fannes, Spohn and Verbeure proved the mean-field limit in the $N\to\infty$ limit, using de Finetti’s theorem (which states that all infinitely exchangeable measures are mixtures of product states) and properties of the relative entropy.
Because one has $\mu^2 = \mu \times \mu$, one replaces the linear form $\mu_N(H_N)$ by the nonlinear one $\mu^2(h)$. Fannes, Spohn and Verbeure actually proved their theorem more generally for quantum statistical mechanics models, such as the Dicke maser, but it also applies to classical models. For the quantum models, one replaces de Finetti’s theorem by the non-commutative analogue, Störmer’s theorem. (See [@Aldous] and references therein for a detailed survey of de Finetti’s theorem, and refer to Fannes, Spohn and Verbeure’s paper and references therein for the noncommutative analogue, which we will not need.) With Eugene Kritchevski, we tried to find a simpler proof of the specialization of Proposition \[prop:FSV\] to the classical case. But there were several errors in our proof, which have been brought to my attention by Alex Opaku, to whom I am grateful. Fortunately, Fannes, Spohn and Verbeure’s original paper definitely does also apply to classical models.
Application to the Mallows model {#sec:Mallows,}
================================
\[sec:Liouville\]
We take for $\mathcal{X}$ the unit square $[0,1] \times [0,1]$. Suppose that $f,g : [0,1] \to {\mathbb{R}}$ are probability densities: $f,g\geq 0$ and $\int_0^1 f(x)\, dx = \int_0^1 g(y)\, dy = 1$. For simplicity, later on, we also assume that there are constants $0<c<C<\infty$ such that $c \leq f,g\leq C$. Then we take the [*a priori*]{} measure to be $$d\mu_0(x,y)\, =\, f(x) g(y)\, dx\, dy\, .$$ We take the interaction to be $$h((x_1,y_1),(x_2,y_2))\, =\, \theta(x_1-x_2) \theta(y_2-y_1)
+ \theta(x_2-x_1) \theta(y_1-y_2)\, ,$$ where $\theta : {\mathbb{R}}\to {\mathbb{R}}$ is the Heaviside function, $$\theta(x)\, =\, \begin{cases} 1 & \text{ if $x>0$,}\\
0 & \text{ if $x<0$.}
\end{cases}$$ Since $\mu_0$ is absolutely continuous with respect to Lebesgue measure, all $x_1,\dots,x_N$ and $y_1,\dots,y_N$ are distinct, with probability 1. (This is why we do not bother to specify $\theta$ at the discontinuity point $0$.)
Let $X_1<\dots<X_N$ and $Y_1<\dots<Y_N$ be any points. Then for any $\sigma, \tau \in S_N$, the symmetric group, we have $$\frac{d\mu_N^{\beta}}{d\mu_0^N}((X_{\sigma_1},Y_{\tau_1}),\cdots,(X_{\sigma_N},Y_{\tau_N}))\,
=\, \mathbbm{P}_N^{\exp(-\beta/(N-1))}(\sigma^{-1} \tau)\, ,$$ where $\mathbbm{P}_N^q$ is the Mallows measure on $S_N$. So studying the limit of the $\mu_N^{\beta}$’s gives us direct information on the limit of $\mathbbm{P}^{1-\beta/N}_N$. For any fixed $\sigma \in S_N$, the permutation $\sigma^{-1} \tau$ is uniform on $S_N$, if $\tau$ is. Because of this, we have the following result for the marginal of $\mu_N^{\beta}$ on $(x_1,\dots,x_N)$, $$\int_{\mathcal{X}^N} U(x_1,\dots,x_N)\, d\mu_N^{\beta}((x_1,y_1),\dots,(x_N,y_N))\,
=\, \int_{[0,1]^N} U(x_1,\dots,x_N) f(x_1)\cdots f(x_N)\, dx_1\cdots dx_N\, ,$$ and the marginal on $(y_1,\dots,y_N)$, $$\int_{\mathcal{X}^N} U(y_1,\dots,y_N)\, d\mu_N^{\beta}((x_1,y_1),\dots,(x_N,y_N))\,
=\, \int_{[0,1]^N} U(y_1,\dots,y_N) g(y_1)\cdots g(y_N)\, dy_1\cdots dy_N\, ,$$ for all bounded, continuous functions $U : (0,1)^N \to {\mathbb{R}}$. Enforcing these conditions on the marginals, Proposition \[prop:FSV\] yields the following: $$\label{eq:FSV}
p(\beta)\, =\, \max_{\mu \in \mathcal{M}_{+,1}(f,g)} [S_1(\mu,\mu_0) - \beta \mu^2(h)]\, ,$$ where $\mathcal{M}_{+,1}(f,g)$ is the set of all probability measures $\mu \in \mathcal{M}_{+,1}(\mathcal{X})$ such that $d\mu(x,y)$ has marginals $f(x) dx$ and $g(y) dy$.
Suppose that $\mu$ is any arg-max of the right-hand-side of (\[eq:FSV\]). Since we have chosen $\mu_0$ to be absolutely continuous with respect to Lebesgue measure on $\mathcal{X}$, the same must be true of $\mu$. Otherwise the relative entropy would be $-\infty$. So we can write $$d\mu(x,y)\, =\, u(x,y)\, dx\, dy\, .$$ Then it is easy to see that the Euler-Lagrange equations for (\[eq:FSV\]) are $$\label{eq:EulerLagrange}
\ln u(x,y)\, =\, \ln f(x) + \ln g(y) + C - \beta \int_{\mathcal{X}} u(x',y') [\theta(x-x') \theta(y'-y) + \theta(x'-x) \theta(y-y')]\, dx' dy'\, ,$$ for some constant, $C<\infty$. Therefore, $u$ solves the equation $$\label{eq:EL}
\begin{cases}
u(x,y)\, =\, \frac{1}{\mathcal{Z}} f(x) g(y) e^{- \beta \int_{\mathcal{X}} h((x,y),(x',y')) u(x',y')\, dx'\, dy'}
& \text{ for $(x,y) \in \mathcal{X}$,}\\
\int_0^1 u(x,y)\, dy\, =\, f(x) & \text{ for $x \in [0,1]$,}\\
\int_0^1 u(x,y)\, dx\, =\, g(y) & \text{ for $y \in [0,1]$,}
\end{cases}$$ where $\mathcal{Z}$ is a normalization constant.
Since $u(x,y)$ solves an integral equation it can be differentiated both with respect to $x$ and $y$. Doing so yields the partial differential equation $$\label{eq:LiouvillePDE}
\frac{\partial^2}{\partial x\partial y} \ln u(x,y)\, =\, 2 \beta u(x,y)\, ,$$ known as the hyperbolic Liouville equation. This equation arises naturally in differential geometry, related to the problem of choosing a metric on a given manifold. I am very grateful to S.G. Rajeev for important information regarding this PDE. One of the facts he imparted is the symmetry of the differential equation under the following general transformation: $$\label{eq:symmetry}
\begin{split}
v(x,y)\, =\, F'(x) G'(y) u(F(x),G(y)) \qquad&\\
\Rightarrow
\frac{\partial^2}{\partial x \partial y} \ln v(x,y)\,
&=\, \frac{\partial^2}{\partial x \partial y} u(F(x),G(y)) + \frac{\partial}{\partial y}\left(\frac{F''(x)}{F'(x)}\right)
+ \frac{\partial}{\partial x} \left(\frac{G''(y)}{G'(y)}\right)\\
&=\, \frac{\partial^2}{\partial x \partial y} u(F(x),G(y))\\
&=\, 2 \beta F'(x) G'(y) u(F(x),G(y))\\
&=\, 2 \beta v(x,y)\, .
\end{split}$$ So, if $\frac{\partial^2}{\partial x\partial y} \ln u = \beta u$ then the same is true for $v(x,y) = F'(x) G'(y) u(F(x),G(y))$.
Our real goal is to solve the Euler-Lagrange equation (\[eq:EL\]). But as a first step, we want to consider the Cauchy problem for (\[eq:LiouvillePDE\]). In other words, we want to consider the problem $$\label{eq:Liouville}
\begin{cases}
\frac{\partial^2}{\partial x \partial y} \ln u(x,y)\, =\, 2 \beta u(x,y)
& \text{ for $(x,y) \in [0,L_1] \times [0,L_2]$,}\\
u(x,0)\, =\, \phi(x) & \text{ for $x \in [0,L_1]$,}\\
u(0,y)\, =\, \psi(y) & \text{ for $y \in [0,L_2]$,}
\end{cases}$$ for some $L_1,L_2 > 0$ and $\phi : [0,L_1] \to {\mathbb{R}}$, $\psi : [0,L_2] \to {\mathbb{R}}$ both positive and continuous.
Note that $\frac{\partial^2}{\partial x \partial y}$ is a wave operator, with characteristics directed along $x$ and $y$. Specifically, defining $\xi = (x+y)/\sqrt{2}$ and $\zeta = (x-y)/\sqrt{2}$, we have $\frac{\partial^2}{\partial x \partial y} = \frac{1}{2} (\frac{\partial^2}{\partial \xi^2} - \frac{\partial^2}{\partial \zeta^2})$, the usual wave operator. Therefore, D’Alembert’s formula for solutions of the wave equation allow us to reformulate (\[eq:Liouville\]) as an integral equation, $$\label{eq:CauchyIntegral}
\ln u(x,y)\, =\, \ln \phi(x) + \ln \psi(y) - \ln \alpha + 2 \beta \int_{[0,x] \times [0,y]} u(x',y')\, dx'\, dy'\, ,$$ which we prefer. This equation is supposed to be solved for all $(x,y) \in [0,L_1] \times [0,L_2]$. We have introduced the number $\alpha = \phi(0)$, which we also assumed equals $\psi(0)$, for consistency since both are supposed to give $u(0,0)$. (Note that the initial surface, $([0,L_1] \times \{0\}) \cup (\{0\} \times [0,L_2])$, is [*not*]{} a non-characteristic surface. This is the reason that our Cauchy problem does not require initial data for the tangential derivative of $u$ even though the wave equation is second order.) We refer to Evans textbook for PDE’s, (especially Section 2.4 on the wave equation and Section 4.6 on the Cauchy-Kovalevskaya theorem).
As we will see, the symmetry (\[eq:symmetry\]) is the key to solving both the Euler-Lagrange equation (\[eq:EL\]) and the Cauchy problem (\[eq:CauchyIntegral\]).
The Cauchy Problem
==================
We start with uniqueness for the Cauchy problem.
\[lem:IVP\] For any $L_1,L_2>0$, the Cauchy problem (\[eq:CauchyIntegral\]) having $\phi=\psi=\alpha=1$ has at most one solution in the class of nonnegative integrable functions.
Since $\phi=\psi=\alpha=1$, equation (\[eq:CauchyIntegral\]) simplifies to $$\ln u(x,y)\, =\, 2 \beta \int_{[0,x] \times [0,y]} u(x',y')\, dx'\, dy'\, .$$ Assuming that $u$ is nonnegative and integrable, this implies that $\ln u$ is bounded and continuous. Then, using these properties in the right-hand-side of the equation again (similarly as one does to prove elliptic regularity) we deduce that $\ln u$ is continuously differentiable and globally Lipschitz. In particular, it is continuous up to the boundary.
Now suppose that there are two solutions $u$ and $v$. Letting $z = \ln u - \ln v$, we have $$z(x,y)\, =\, 2 \beta \int_0^x \int_0^y [1-e^{-z(x',y')}] u(x',y')\, dx'\, dy'\, .$$ Since both $\ln u$ and $\ln v$ are bounded, we see that $z$ is as well. Therefore, there exists a constant $K<\infty$ such that $|1 - e^{-z}| \leq K |z|$ for all values of $z$ in the range. So we have $$|z(x,y)|\, \leq\, \beta K \|u\|_{\infty} \int_0^x \int_0^y |z(x',y')|\, dx'\, dy'\, .$$ A version of Gronwall’s lemma then implies that $z \equiv 0$. We outline this now, although our argument can probably be improved.
Let $Z(t) = \sup\{ |z(x,y)|\, :\, (x,y) \in (0,L_1) \times (0,L_2)\, ,\ xy\leq t\}$. Then we obtain, after making the change of variables $(x,y) \mapsto (x,t)$ where $t= xy$, and using Fubini-Tonelli to integrate over $x$ first, $$Z(t)\, \leq\,\beta K \|u\|_{\infty} \int_0^t \ln(t/t') Z(t')\, dt'\, .$$ We rewrite this as $$Z(t)\, \leq\, \beta K \|u\|_{\infty} \int_0^t [\ln(t/L_1L_2) - \ln(t'/L_1L_2)] Z(t')\, dt'\, .$$ Since $\ln(t/L_1L_2)\leq 0$ for $t\leq L_1L_2$, and since $Z\geq 0$, we can drop the term $\ln(t/L_1L_2) Z(t')$ in the integrand to obtain $$Z(t)\, \leq\, \beta K \|u\|_{\infty} \int_0^t |\ln(t'/L_1L_2)| Z(t')\, dt'\, .$$ Finally, setting $\zeta(t) = \int_0^t |\ln(t'/L_1L_2)| Z(t')\, dt'$, this leads to $$\zeta'(t)\, \leq\, \beta K \|u\|_{\infty} |\ln(t/L_1L_2)| \zeta(t)\, .$$ By Gronwall’s inequality (see for example Appendix B of Evans [@Evans]), we obtain $$\zeta(t)\, =\, e^{\beta K \|u\|_{\infty} \int_0^t |\ln(t'/L_1L_2)|\, dt'} \zeta(0)\, =\, e^{\beta K \|u\|_{\infty} (1+|\ln(t/L_1L_2)|) t/L_1L_2} \zeta(0)\, .$$ But $\zeta(0) = 0$. Hence $\zeta(t)=0$ for all $t$. This implies $Z(t)=0$ for all $t$ which implies $z(x,y) = 0$ for all $x,y$.
Next we derive the explicit solution of (\[eq:CauchyIntegral\]), for the case $\phi=\psi=\alpha=1$.
\[cor:Cauchy\] Suppose that $L_1,L_2>0$ and either $\beta\leq 0$ or $L_1L_2 < 1/\beta$. Then the unique solution of the Cauchy problem (\[eq:CauchyIntegral\]) with $\phi=\psi=\alpha=1$ is $$u(x,y)\, =\, (1 - \beta xy)^{-2}\, .$$
Uniqueness was proved in Lemma \[lem:IVP\], and it is trivial to check that this solves the PDE (\[eq:LiouvillePDE\]). Therefore, assuming that $u$ is integrable on $[0,L_1] \times [0,L_2]$, we may derive D’Alembert’s formula by standard calculus: $$\begin{aligned}
\ln u(x,y)\,
&=\, \int_0^x \frac{\partial}{\partial x} \ln u(x',y)\, dx' + \ln \psi(y)\\
&=\, \int_{(0,x) \times (0,y)} \frac{\partial^2}{\partial x \partial y} \ln u(x',y')\, dx'\, dy'
+ \int_0^x \frac{\partial}{\partial x} \ln \phi(x')\, dx' + \ln \psi(y)\\
&=\, 2 \beta \int_{(0,x) \times (0,y)} u(x',y')\, dx'\, dy' + \ln \psi(y) + \ln \phi(x) - \ln \alpha\, .\end{aligned}$$ The only issue is to check integrability, which amounts to checking $\inf_{(x,y) \in [0,L_1] \times [0,L_2]} 1-\beta xy>0$. This holds if and only if $\beta\leq 0$ or $L_1 L_2 < 1/\beta$.
Let us briefly explain one approach to deriving this formula. For nonlinear PDE’s one always first guesses a scaling solution, in hopes of finding an explicit formula. Because of the hyperbolic nature it makes sense to look for a solution $u(x,y) = U(xy)$ for some $U(z)$. This leads to the ODE $$\frac{d}{dz} \ln U(z) + z \frac{d^2}{dz^2} \ln U(z)\, =\, 2 \beta U(z)\, ,$$ which can also be expressed as $$\frac{d}{dz} \left(z\, \frac{d}{dz} \ln U(z)\right)\, =\, 2 \beta U(z)\, .$$ The idea of using a power law solution is natural because the derivative of the logarithm results in a power law, itself. Trying $U(z) = (1+cz)^p$ leads to $$\ln U(z)\, =\, p \ln(1+cz)\quad \Rightarrow\quad
z \frac{d}{dz} \ln \phi(z)\, =\, \frac{cpz}{1+cz}\, =\, p - \frac{p}{1+cz}\quad
\Rightarrow \quad
\frac{d}{dz} \left(z\, \frac{d}{dz} \ln U(z)\right)\, =\, \frac{cp}{(1+cz)^2}\, .$$ So, taking $p=-2$ and $c=-\beta$, this solves the equation, and gives $U(z) = (1-\beta z)^{-2} \Rightarrow u(x,y) = (1-\beta x y)^{-2}$. Finally, we are led to the solution of the general Cauchy problem.
\[cor:GenCauchy\] Suppose that $\phi, \psi : [0,1] \to {\mathbb{R}}$ are continuous and satisfy $c\leq \phi,\psi\leq C$, for some constants $0<c<C<1$. Also suppose that $\phi(0) = \psi(0) = \alpha$ for some $\alpha$. Then the Cauchy problem (\[eq:CauchyIntegral\]) has a solution if and only if $\beta \leq 0$ or $\int_0^{1} \phi(x)\, dx \int_0^{1} \psi(y)\, dy < \alpha/\beta$. In case a solution exists, it is unique and equals $$\label{eq:GenSol}
u(x,y)\, =\, \frac{\alpha \phi(x) \psi(y)}{(\alpha-\beta \Phi(x) \Psi(y))^{2}}\, ,$$ where $\Phi(x) = \int_0^x \phi(x')\, dx'$ and $\Psi(y) = \int_0^y \psi(y')\, dy'$.
Suppose that $u$ is any solution of (\[eq:CauchyIntegral\]). Let $v$ be given by $$v(x,y)\, =\, \frac{u(\Phi^{-1}(\alpha^{1/2} x),\Psi^{-1}(\alpha^{1/2} y))}{\alpha \Phi'(\Phi^{-1}(\alpha^{1/2} x))\Psi'(\Psi^{-1}(\alpha^{1/2} y))}\, .$$ Then, by the symmetry (\[eq:symmetry\]), $v$ is a solution of Liouville’s PDE (\[eq:LiouvillePDE\]) on the domain $[0,\alpha^{-1/2} \Phi(1)]\times [0,\alpha^{-1/2} \Psi(1)]$. But $v(x,0) = v(0,y) = 1$ because $\Phi' = \phi$ and $\Psi'=\psi$. So uniqueness, the conditions for existence, and the formula for the solution all follow from Lemma \[lem:IVP\] and Corollary \[cor:Cauchy\].
Solving the Euler-Lagrange Equation
===================================
By general principles, we know that a solution of (\[eq:EL\]) always exists: specifically, the optimizer in Proposition \[prop:FSV\]. Next we calculate it, and prove uniqueness.
\[lem:marginal\] If $f=g=1$, then the unique solution of (\[eq:EL\]) is given by (\[eq:GenSol\]) for $$\phi(z)\, =\, \psi(z)\, =\, \frac{\beta e^{-\beta z}}{1-e^{-\beta}}\, ,\quad
\Phi(z)\, =\, \Psi(z)\, =\, \frac{1-e^{-\beta z}}{1-e^{-\beta}}\quad \text{and}\quad
\alpha\, =\, \frac{\beta}{1-e^{-\beta}}\, .$$
Suppose $u$ solves (\[eq:EL\]). Note that $\lim_{x \to 0} h((x,y),(x',y')) = \theta(y-y')$ for all $(x',y') \in \mathcal{X}$. By the dominated convergence theorem, this implies $$\lim_{x \to 0} \int_{\mathcal{X}} h((x,y),(x',y')) u(x',y')\, dx'\, dy'\,
=\, \int_0^1 \int_0^1 \theta(y-y') u(x',y')\, dx'\, dy'\,
=\, \int_0^1 \theta(y-y')\, dy'\, =\, y\, ,$$ where we used the fact that $\int_0^1 u(x',y')\, dx' = 1$ for all $y'$. So $$\psi(y)\, =\, \lim_{x \to 0} u(x,y)\, =\, \frac{1}{\mathcal{Z}} e^{-\beta y}\, .$$ Similar arguments lead to $\phi(x)\, =\, \lim_{y \to 0} u(x,y)\, =\, \frac{1}{\mathcal{Z}} e^{-\beta x}$. Since $\int_0^1 u(x,y)\, dy = 1$ for all $x \in [0,1]$, it again follows from the dominated converge theorem, taking the limit $x \to 0$, that $\int_0^1 \psi(y)\, dy$ must also be $1$. So $\mathcal{Z} = (1-e^{-\beta})/\beta$. Checking, the reader will easily see that this gives the stated value for $\phi$, $\psi$ and $\alpha$. Integrating, it also leads to $\Phi$ and $\Psi$.
Uniqueness follows from uniqueness of the Cauchy problem, Corollary \[cor:GenCauchy\]. Since this is the only possible solution, and since a solution exists, this must be it.
Substituting in, and simplifying leads to the formula $$\label{eq:solution}
u(x,y)\, =\, \frac{(\beta/2) \sinh(\beta/2)}{\big(e^{\beta/4} \cosh(\beta[x-y]/2) - e^{-\beta/4} \cosh(\beta[x+y-1]/2)\big)^2}\, .$$ Therefore, we arrive at the final formula.
As long as $c\leq f,g\leq C$ for some $0<c<C<1$, and $\int_0^1 f(x)\, dx = \int_0^1 g(y)\, dy = 1$, the unique solution of (\[eq:EL\]) is $$u(x,y)\, =\, \frac{(\beta/2) \sinh(\beta/2) f(x) g(y)}{\big(e^{\beta/4} \cosh(\beta[F(x)-G(y)]/2) - e^{-\beta/4} \cosh(\beta[F(x)+G(y)-1]/2)\big)^2}\, ,$$ where $F(x) = \int_0^x f(x')\, dx'$ and $G(y) = \int_0^y g(y')\, dy'$.
Suppose that $u$ is a solution of (\[eq:EL\]) under the conditions stated. Define $$v(x,y)\, =\, \frac{u(F^{-1}(x),G^{-1}(y))}{f(F^{-1}(x)) g(G^{-1}(y))}\, ,$$ analogously to the proof of Corollary \[cor:GenCauchy\]. Note that $F$ and $G$ are continuously, strictly increasing bijections of $[0,1]$. Using (\[eq:EL\]), we see that $$\begin{aligned}
\ln v(x,y)\,
&=\, \ln u(F^{-1}(x),G^{-1}(y)) - \ln f(F^{-1}(x)) - \ln g(G^{-1}(y))\\
&=\, -\ln \mathcal{Z} - \beta \int_{\mathcal{X}} h((F^{-1}(x),G^{-1}(y)),(x',y')) u(x',y')\, dx'\, dy'\, .\end{aligned}$$ Making the change-of-variables $x'' = F(x')$ and $y'' = F(y')$, we see that $dx' = dx''/f(F^{-1}(x''))$ and $dy' = dy''/g(G^{-1}(y''))$. So we have $$\ln v(x,y)\, =\, - \ln \mathcal{Z} - \beta \int_{\mathcal{X}} h((F^{-1}(x),G^{-1}(y)),(F^{-1}(x''),G^{-1}(y''))) v(x'',y'')\, dx''\, dy''\, .$$ But the Heaviside function satisfies $\theta(F(x)-F(x')) = \theta(x-x')$ for any continuous, strictly increasing function $F$. For this reason, $$h((F^{-1}(x),G^{-1}(y)),(F^{-1}(x''),G^{-1}(y'')))\,
=\, h((x,y),(x'',y''))\, .$$ In other words, $v$ also solves (\[eq:EL\]), except that $$\int_0^1 v(x,y)\, dy\, =\, \int_0^1 v(x,y)\, dx\, =\, 1\, ,$$ using the change-of-variables formula, again. So uniqueness and the formula follows from Lemma \[lem:marginal\].
Proof of Main Result
====================
We now explain the minor details needed to go from Proposition \[prop:FSV\] to a proof of Theorem \[thm:main\]. According to Fannes, Spohn and Verbeure’s result, $\mu_{N}^{\beta}$ must converge weakly to a mixture of i.i.d., product measures, each of whose 1-particle marginal optimizes $S_1(\mu,\mu_0) - \frac{\beta}{2} \mu^2(h)$. But $\mu_N^{\beta}$ has marginals on $(x_1,\dots,x_N)$ and $(y_1,\dots,y_N)$ equal to the product measures of $f(x)\, dx$ and $g(y)\, dy$, respectively. Therefore, according to the weak law of large numbers (WLLN), we know that all the $\mu$’s in the support of the directing measure for the limit of $\mu_N^{\beta}$, must have $x$ marginal equal to $f(x)\, dx$ and $y$ maginal $g(y)\, dy$. Hence, this constraint can be imposed when looking for an optimizer. This is actually a relevant comment because all optimizers, for all choices of [*a priori*]{} measure $\mu_0$, have the same value/pressure: that due to the Mallows measure on $S_n$. For concreteness, we will now take $f=g=1$.
Now suppose that $\mu$ optimizes the Gibbs formula. It must be absolutely continuous with respect to $\mu_0$ in order to not have the relative entropy equal to $-\infty$. So we can write $$d\mu(x,y)\, =\, u(x,y)\, dx\, dy\, ,$$ where $u(x,y)$ is absolutely continuous. Choosing any continuous function $\phi : [0,1] \times [0,1] \to {\mathbb{R}}$, with $$\int_{\mathcal{X}} u(x,y) \phi(x,y)\, dx\, dy\, =\, 0\, ,$$ we can take $$u_{\epsilon}(x,y)\, =\, (1+\epsilon \phi(x,y)) u(x,y)\, .$$ For $|\epsilon| < 1/\|\phi\|_{\infty}$, we have that $u_{\epsilon}$ is a probability measure. It is easy to see that $$S_1(\mu_{\epsilon},\mu_0)\, =\, S_1(\mu,\mu_0) - \epsilon \int \phi u \ln u
- \int [1+\epsilon \phi] \ln[1+\epsilon \phi] u\, .$$ Since $1+\epsilon \phi$ is bounded away from $0$ (and infinity) for the $\epsilon$ we are considering, it is clear that both integrals above are well-defined. Moreover, it is clear that $$\int [1+\epsilon \phi] \ln[1+\epsilon \phi] u\, =\, o(\epsilon)\, ,$$ because $\ln[1+\epsilon \phi] = \epsilon \phi + o(\epsilon)$ and $\int \phi u = 0$. A similar calculation also shows that $$\mu_{\epsilon}^2(h)\, =\, \mu^2(h) + \epsilon \mu^2([\phi(x,y)+\phi(x',y')] h) + O(\epsilon^2)\, .$$ Since $\mu$ is supposed to be the optimizer, the terms linear in $\epsilon$ must vanish: $$\int \phi u \ln u\, =\, \frac{\beta}{2}\, \mu^2([\phi(x,y) + \phi(x',y')] h)\, .$$ Since $h$ is symmetric, by varying over all $\phi$ orthogonal to $u$, we deduce that $$u(x,y) \ln u(x,y)\, =\, \beta u(x,y) \int_{\mathcal{X}} h((x,y),(x',y')) u(x,y)\, dx'\, dy' + C u(x,y)\, ,$$ for some constant $C$. (The reason we cannot assume $C=0$ is because we left out one direction for $\phi$, namely the direction parallel to $u$, so that there is an indeterminacy in this direction, as seen using the Riesz representation theorem.) In other words, we have just deduced equation (\[eq:EulerLagrange\]). On the other hand, we have also proved that this equation has a unique solution given by (\[eq:solution\]). Therefore, $\mu_N^{\beta}$ does converge weakly to the i.i.d., product measure of $\mu$, where $d\mu(x,y) = u(x,y)\, dx\, dy$.
Because of all this, if we take the empirical measure with respect to $\mu_N^{\beta}$, $$\frac{1}{N}\, \sum_{i=1}^{N} \delta_{(x_i,y_i)}\, ,$$ then this does satisfy just the type of convergence claimed in Theorem \[thm:main\]. But, taking the order statistics $X_1<\dots<X_N$ and $Y_1<\dots<Y_N$, we do have $(x_i,y_i) = (X_{\sigma_i},Y_{\tau_i})$ for some permutations $\sigma,\tau \in S_N$. Moreover (by commutativity of addition) $$\frac{1}{N}\, \sum_{i=1}^{N} f(x_i,y_i)\, =\,
\frac{1}{N}\, \sum_{i=1}^{N} f(X_i,Y_{\pi_i})\, ,$$ where $\pi = \tau \sigma^{-1}$. As noted before, $(X_1,\dots,X_N)$ and $(Y_1,\dots,Y_N)$ are distributed as the order statistics coming from Lebesgue measure, the effect of the Hamiltonian is only present in the Mallow model $\mathbbm{P}_N^{\exp(-\beta/(N-1))}$-measure of $\pi$. By the WLLN for the order statistics, we see that, defining $$g_N(x,y)\, =\, \sum_{i,j=1}^{N} f(X_i,Y_j) \mathbbm{1}_{((i-1)/N,i/N]}(x) \mathbbm{1}_{((j-1)/N,j/N]}(y)\, ,$$ we have that the random function $g_N$ converges in probability to $f$, everywhere in $(0,1]\times (0,1]$. Therefore, since $$\frac{1}{N}\, \sum_{i=1}^{N} f(x_i,y_i)\, =\, \frac{1}{N}\, \sum_{i=1}^{N} g_N(i/N,\pi_i/N)\, ,$$ we do deduce the theorem from the corresponding result for $\mu_N^{\beta}$. Finally note that taking $\exp(-\beta/(n-1))$ versus $1-\beta/n$ in the theorem does not matter, since the probability measures are continuous with respect to $\beta$, and $\exp(-\beta/(n-1)) = 1 - \beta(1+o(1))/n$.
Applications {#sec:Applications}
============
The ground state of the $\mathcal{U}_q(\mathfrak{sl}_2)$-symmetric XXZ quantum spin system, and the invariant measures of the asymmetric exclusion process on an interval can be obtained from $\mathbbm{P}_N^{q}$. See Koma and Nachtergaele’s paper [@KomaNachtergaele] and Gottstein and Werner’s paper [@GottsteinWerner] for information about the XXZ model. For information about the blocking measures and the asymmetric exclusion process, we find it convenient to refer to Benjamini, Berger, Hoffman and Mossel (BBHM), [@BBHM]. The reader can easily deduce information for the XXZ model, since there is a perfect dictionary between these two. An excellent reference for this is Caputo’s review [@Caputo].
An interesting perspective on the ground state of the quantum XXZ ferromagnet was discovered by Bolina, Contucci and Nachtergaele in [@BCN]. They viewed the ground state of the quantum spin system as a thermal Boltzmann-Gibbs state for a classical model at inverse temperature $\beta = \ln(q^{-2})$. The state space they considered was the set of all up-right paths from $(0,0)$ to $(m,n) \in \mathbb{Z}^2$ (with $m,n\geq 0$). The Hamiltonian energy function for such a path is the energy under the path, and above the $x$-axis. Note that the Hamiltonian for the Mallows model also has a graphical representations as the number of “crossings” of the permutation. Using their representation, they explained some symmetries of the ground state of the XXZ model, and obtained estimates which were later useful in their follow-up paper, [@BCN2]. The two models are related, but only the Mallows model is manifestly a mean-field model.
We consider the (nearest neighbor) asymmetric exclusion process on $\{1,\dots,N\}$, with hopping rate to the left $p$ and hopping rate to the right $1-p$, and $q = (1-p)/p$. We no longer use $p$ or $p_N$ for the pressure, instead we use it for the hopping rate as expressed above. As BBHM explain, the invariant measure of the ASEP is a push-forward of $\mathbbm{P}_N^q$. Given a permutation $\pi \in S_N$ and a particle configuration $\eta = (\eta_1,\dots,\eta_N) \in \{0,1\}^N$, let $\pi \eta = (\eta_{\pi_1},\dots,\eta_{\pi_N})$. Let $(1^k,0^{N-k}) = (1,\dots,1,0,\dots,0)$ with $k$ $1$’s and $N-k$ $0$’s. Then, taking a random permutation $\pi$, distributed according to $\mathbbm{P}_N^q$, and letting $$\eta^{(k,N-k)} = \pi (1^k,0^{N-k})\, ,$$ the law of $\eta^{(k,N-k)}$ is the invariant measure for the ASEP, with $k$ particles and $N-k$ holes. As BBHM explain, this is an instance of Wilson’s general height function approach to tiling and shuffling [@Wilson][^2].
The question we can answer is the non-random limiting density of $\eta^{(k,N-k)}$ in the scaling limit, $N \to \infty$, $p_N = \frac{1}{2} + \beta/4N$, $k_N = \lfloor{y N}\rfloor$. (Note that this corresponds to $q_N = 1- \beta(1+o(1))/N$.) Namely, for a continuous function $f : [0,1] \to {\mathbb{R}}$, we have $$\lim_{\epsilon \downarrow 0} \lim_{N \to \infty} \mathbbm{P}^{1-\beta/N}_{N}\left\{
\left| \frac{1}{N}\, \sum_{i=1}^N f(i/N) \eta^{(\lfloor{y N}\rfloor,\lceil{(1-y)N}\rceil)}_i - \int_0^1 f(x) \rho(x;y)\, dx\right| > \epsilon\right\}\, =\, 0\, ,$$ for all $y \in [0,1]$, where $$\rho(x;y)\, =\, \int_0^y u(x,y')\, dy'\, .$$ The scaling $p_N = \frac{1}{2} + \beta/4N$ is the regime typically called “weakly asymmetric.” See, for example, Enaud and Derrida’s paper [@EnaudDerrida], following the matrix method used, for example by Derrida, Lebowitz and Speer [@DerridaLebowitzSpeer]. Note that while they considered the nonequilibrium case, we consider the particle conserving, equilibrium case. On the other hand, we are sure that the formula above is known.
The integral for $\rho(x;y)$ is readily evaluated. Setting $\phi$, $\psi$, $\Phi$, $\Psi$ and $\alpha$ as in Lemma \[lem:marginal\], $$\begin{aligned}
\rho(x;y)\,
&=\, \int_0^{y} \frac{\alpha \phi(x) \psi(y')}{(\alpha - \beta \Phi(x) \Psi(y'))^2}\, dy'\\
&=\, \frac{\alpha \phi(x)}{\beta \Phi(x) (\alpha - \beta \Phi(x) \Psi(y'))} \bigg|_0^y\\
&=\, \frac{\phi(x) \Psi(y)}{\alpha - \beta \Phi(x) \Psi(y)}\, .\end{aligned}$$ Substituting in, and doing minor algebraic simplifications, we obtain $$\rho(x;y)\, =\, \frac{(1-e^{-\beta y}) e^{-\beta x}}{(1-e^{-\beta}) - (1-e^{-\beta x})(1-e^{-\beta y})}\, .$$ From this formula it is obvious that the $\beta \to 0$ limit recovers $\rho(x;y) \equiv y$, as it should (for the symmetric case). Also, after further “simplifications,” we obtain $$\rho(x;y)\, =\, \frac{e^{\beta(\frac{1}{2}-x)/2} \sinh(\beta y/2)}{e^{\beta/4} \cosh(\beta [x-y]/2) - e^{-\beta/4} \cosh(\beta[x+y-1]/2)}\, .$$ In particular, one can observe that the particle-hole/reflection symmetry is manifest in this formula due to the invariance under the transformation $(\beta,x) \mapsto (-\beta,1-x)$.
Finally, we note that we can partially undo the scaling limit by taking $\beta \to \infty$ with $x=y+t/\beta$ (assuming $0<y<1$). Approximating $\sinh(\beta y/2) \approx \frac{1}{2} e^{\beta y/2}$ and noting that $e^{-\beta/2} \cosh(\beta [x+y-1]/2) \to 0$ since $|x+y-1|<1$, we obtain $$\rho(x;y) \to \frac{1}{1+e^t}\, .$$ This is not correctly normalized due to the fact that $dx = dt/\beta$, and $\beta \to \infty$. On the other hand, this does recover the actual lattice scaling limit for the density (modulo a reflection), as has been previously calculated for the XXZ model by Dijkgraaf, Orlando and Reffert in Appendix A of [@DijkgraafOrlandoReffert].
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported in part by a U.S. National Science Foundation grant, DMS-0706927. I am very grateful to the following people for useful discussions and suggestions: S. G. Rajeev, Alex Opaku, Carl Mueller, Bruno Nachtergaele, Wolfgang Spitzer and Pierluigi Contucci. I also thank the anonymous referees for their useful suggestions for improvement.
[10]{}
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B. Derrida, J. L. Lebowitz, and E. R. Speer. Free energy functional for nonequilibrium systems: an exactly solvable case. (2001), no. 15, 150601, 4 pp. <http://arxiv.org/abs/cond-mat/0105110>.
P. Diaconis and A. Ram. Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques. Dedicated to William Fulton on the occasion of his 60th birthday. (2000), 157–190.
R. Dijkgraaf, D. Orlando and S. Reffert. Quantum crystals and spin chains. (2009), no. 3, 463–490. <http://arxiv.org/abs/0803.1927>.
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[^1]: Independently, a similar $q$-deformed combinatorial formula was explained for a problem in quantum statistical mechanics, the ground state of the ferromagnetic $\mathcal{U}_q(\mathfrak{sl}_2)$-symmetric XXZ quantum spin chain, by Bolina, Contucci and Nachtergaele [@BCN]. We will comment more on this in Section 8.
[^2]: Because of this, let us note that the ground state of the XXZ model is also a projection, or marginal, of the Mallows model for permutations (using the correspondence between the ASEP and the XXZ model [@Caputo]). This raises an interesting point for further consideration: are other integrable models projections of mean-field models?
|
---
abstract: 'Interdependent networks are more fragile under random attacks than simplex networks, because interlayer dependencies lead to cascading failures and finally to a sudden collapse. This is a hybrid phase transition (HPT), meaning that at the transition point the order parameter has a jump but there are also critical phenomena related to it. Here we study these phenomena on the Erdős–Rényi and the two dimensional interdependent networks and show that the hybrid percolation transition exhibits two kinds of critical behaviors: divergence of the fluctuations of the order parameter and power-law size distribution of finite avalanches at a transition point. At the transition point global or “infinite" avalanches occur while the finite ones have a power law size distribution; thus the avalanche statistics also has the nature of a HPT. The exponent $\beta_m$ of the order parameter is $1/2$ under general conditions, while the value of the exponent $\gamma_m$ characterizing the fluctuations of the order parameter depends on the system. The critical behavior of the finite avalanches can be described by another set of exponents, $\beta_a$ and $\gamma_a$. These two critical behaviors are coupled by a scaling law: $1-\beta_m=\gamma_a$.'
author:
- Deokjae Lee
- 'S. Choi'
- 'M. Stippinger'
- 'J. Kertész'
- 'B. Kahng'
title: 'Hybrid Phase Transition into an Absorbing State: Percolation and Avalanches'
---
Introduction
============
Hybrid phase transitions (HPTs) in complex networks have attracted substantial attention. In these transitions, the order parameter $m(z)$ exhibits behaviors of both first-order and second-order transitions simultaneously as $$m(z)=\left\{
\begin{array}{lr}
0 & ~{\rm for}~~ z < z_c, \\
m_0+r(z-z_c)^{\beta_m} & ~{\rm for}~~ z \ge z_c,
\end{array}
\right.
\label{eq:order}$$ where $m_0$ and $r$ are constants and $\beta_m$ is the critical exponent of the order parameter, and $z$ is a control parameter. Examples include the $k$-core percolation [@kcore1; @kcore2; @kcore_prx], generalized epidemic spreading [@dodds; @janssen; @grassberger_epi], and synchronization [@pazo; @moreno; @mendes_sync].
Percolation in the cascading failure (CF) model [@buldyrev; @baxter; @bashan; @bianconi; @zhou; @makse; @boccaletti; @kivela] on interdependent multi-layer random, Erdős-Rényi (ER) networks is another example. In this CF model the process is controlled by the mean degree $z$ of the networks [@model_comment]. When a node on one layer fails and is deleted, it leads to another failure of the conterpart node in the other layer of the network. Subsequently, links connected to the deleted nodes are also deleted from the networks. This process continues back and forth, always eliminating the possibly separated finite clusters until a giant mutually connected component (MCC) remains or the giant component gets entirely destroyed as a result of the cascades [@baxter]. As nodes are deleted in such a way, the behavior is similar to that at a second-order phase transition until the transition point $z_c$ is reached from above. Beyond that, as $z$ is further decreased infinitesimally, the percolation order parameter drops suddenly to zero indicating a first-order phase transition. Thus, a HPT occurs at $z=z_c$. This transition may be regarded as a transition to an absorbing state [@absorbing].
In the CF model one has to distinguish between clusters and avalanches. Clusters are MCCs [@mcc]. Avalanches consist of MCCs separated from the giant component as a consequence of a triggering removed node and the subsequent cascade [@baxter]. The avalanche sizes depend on the control parameter, the triggering nodes and on the network configurations. We call global avalanches with size equal to the order parameter “infinite", the others are the “finite” avalanches. The size distribution of finite avalanches follows power law at $z_c$ [@baxter], provided the infinite avalanche is discarded. This fact suggests that the avalanche dynamics at $z_c$ exhibits a critical pattern. The finite MCCs at $z_c$ mostly consist of one or two nodes [@buldyrev; @grassberger], which is in discord with the power-law behavior of the cluster size distribution at a transition point characteristic of the conventional second-order percolation transition [@stauffer; @christensen]. As the avalanches show critical behavior but the clusters do not, an important challange emerges: How to relate the critical behavior of the order parameter to the avalanche dynamics in a single theoretical framework. Further fundamental questions have been still open, such as how the fluctuations of the order parameter behave at the critical point, whether the scaling relation holds between critical exponents of the order parameter exponent, the susceptibility exponent and the correlation size exponent and whether the hyperscaling relation is valid. These questions are not limited to the CF model, but are also relevant to other systems undergoing HPT driven by avalanche dynamics, for instance, $k$-core percolation model [@kcore_prx].
One of the main difficulties in answering those questions has been the need for major computational capacity. Thanks to the efficient algorithm introduced recently by our group [@hwang], we are now able to address those important unsolved problems. In this paper, we report about large scale simulations and analytical results on the CF model of interdependent networks. Based on them we have constructed a theoretical framework connecting the critical behaviors of the order parameter and the avalanche dynamics and have understood the nature of the hybrid percolation transition.
In this paper we study the interdependent CF model for coupled ER networks and two-dimensional square lattices (2D). The control parameter for ER (2D) interdependent networks is the average degree $z$ of a node (the fraction $q$ of original nodes kept in a layer); the order paramater $m$ is the size of the giant mutually connected cluster per node. To describe the HPT, we introduce two sets of critical exponents. The set $\{\beta_m, \gamma_m, {\bar{\nu}}_m \}$ is associated with the order parameter and its related quantities, and the other set $\{\tau_a, \sigma_a, \gamma_a, {\bar{\nu}}_a \}$ is associated with the avalanche size distribution and its related ones. The subscripts $m$ and $a$ refer to the order parameter and avalanche dynamics respectively: The exponent $\beta_m$ is defined by the behavior of the order parameter (Eq. \[eq:order\]), and $\gamma_m$ is the exponent of the susceptibility $\chi \equiv N(\langle m^2\rangle -\langle m \rangle^2) \sim (z - z_c)^{-\gamma_m}$ where $N$ is the system size. The exponent $\bar{\nu}_m$ is defined by the finite size scaling behavior of the order parameter: $m - m_0 \sim N^{-\beta_m/\bar{\nu}_m}$ at $z = z_c$. The exponents $\tau_a$, $\sigma_a$, and $\bar{\nu}_a$ characterize the avalanche size distribution $p_s \sim s_a^{-\tau_a} f(s_a / s_a^*)$, where $s_a$ denotes the avalanche size and $f$ is a scaling function. Here $s_a^*$ is the characteristic avalanche size, which behaves as $\sim (z - z_c)^{-1/\sigma_a}$ for $N\to \infty$ and $s_a^* \sim N^{1/\sigma_a \bar{\nu}_a}$ is its finite size scaling at $z_c$. The exponent $\gamma_a$ determines the scaling of the mean size of finite avalanches $\left< s_a \right> \sim (z - z_c)^{-\gamma_a}$. One may think naively that the exponents $\bar{\nu}_m$ and $\bar{\nu}_a$ would be the same and $\gamma_m$ and $\gamma_a$ are as well. However, it reveals that those pairs of exponents differ from each other. However, we will show that they are related to each other.
Main results {#sec:main_result}
============
The numerically estimated values of the critical exponents for the ER case are listed in Table I, together with those of the 2D case. For the ER and 2D cases, the hyperscaling relation ${\bar{\nu}}_m=2\beta_m+\gamma_m$ holds even though data collapsing for the 2D case is not as satisfactory as for the ER case. The relation $\sigma_a {\bar{\nu}}_a=\tau_a$ does not hold (Sec. \[sec:num\_result\_ER\] and \[sec:num\_result\_2D\]). The few analytic results related to CF model have been limited so far to locally tree-like graphs where the exponent of the order parameter was found to be $\beta_m=1/2$ and the exponent $\tau_a$ of the avalanche size distribution $p_s$ is $\tau_a=3/2$, with the definition $p_s \sim s^{-\tau_a}$ at $z_c$. We show that $\beta_m=1/2$ is valid not only for tree-like networks but generally for interdependent networks with random dependency links (Sec. \[sec:beta-proof\]). Moreover, we also show that the two sets of critical exponents $\{\beta_m, \gamma_m, {\bar{\nu}}_m \}$ and $\{ \tau_a, \sigma_a, \gamma_a, {\bar{\nu}}_a \}$ are not independent of each other. They are coupled through the relation $m(z)+\int_z^{z_0} \langle s_a(z) \rangle \mathrm{d}z =1$, where $z_0$ is the mean degree at the beginning of cascading processes. This leads to $\mathrm{d}m(z)/\mathrm{d}z= \langle s_a(z)\rangle$ and yields $1-\beta_m=\gamma_a$ (Sec. \[sec:sum-rule\]). Our numerical values support this relation.
We classify avalanches in the critical region as finite and infinite avalanches. Infinite avalanche means that the avalanche size is as large as the order parameter. Thus, when it occurs, the GMCC completely collapses, and the system falls into an absorbing state. We find that the mean number of hopping steps denoted as $\langle t \rangle$ between the two layers in avalanche processes depends on the system size $N$ in different ways for the different types of avalanches: $\langle t \rangle \sim \ln N$ for finite avalanches, and $\sim N^{1/3}$ for infinite avalanches on the ER interdependent network (Sec. \[sec:hop-number\]).
Simulation method {#method}
=================
The numerical test of the relevant quantities had been a challenging task. Recently, however, efficient algorithms have been developed [@hwang], in which the sizes of not only a GMCC but also other MCCs can be measured with computational time of $O(N^{1.2})$, as compared to the earlier $O(N^2)$ complexity. (For other algorithms see Refs. [@grassberger]) and [@herrmann_algo]. Now we can investigate critical properties of the hybrid percolation transition of the CF model thoroughly by measuring various critical exponents including susceptibility and correlation size that were missing in previous studies [@grassberger] for both the ER and two dimensional (2D) lattice interdependent networks.
Simulation results on the ER interdependent networks {#sec:num_result_ER}
====================================================
We first describe the simulation results on the double-layer ER random networks. On each layer, an ER network is constructed, with $N$ nodes in both, which are kept fixed. Each node in one layered network has a one-to-one partner node in the other network. The number of occupied edges $M$ in each layer is controlled. The control parameter $z$ is defined as the mean degree $z =2M/N$. Using the algorithm [@hwang], we measure the size of GMCC as a function of $z$. The order parameter $m(z)$ defined as the size of the GMCC per node, which behaves according to Eq. (\[eq:order\]). To trigger an avalanche and to measure its size, we remove a randomly chosen node in one layer and measure the subsequent decrements of the GMCC size, which sum up to the avalanche size. Then we recover the removed nodes and repeat the above process to obtain a reliable statistics of the avanlanche size distribution for a given point $z$. We simulate $10^4$ network configurations for each system size $N/10^5=4, 16, 64$ and $256$, and $10^3$ configurations for $N/10^5=1024$. We obtain $10^{-4}N$ different avalanche samples for each configuration.
Critical behavior of GMCC {#ss:er_crit_order_parameter}
-------------------------
For the double-layer ER network model, the numerical values of $m_0$ and $z_c$ were obtained in Ref. [@son_grassberger] with high precision as $m_0=0.511700\dots$ and $z_c=2.45540749\dots$. We use these values to evaluate our simulation data. We first check whether Eq. (\[eq:order\]) is consistent with the theoretical value $\beta_m=1/2$ [@baxter]. In Fig.\[fig:order\](a), we plot $(m-m_0)N^{\beta_m/{\bar{\nu}}_m}$ versus $\Delta zN^{1/{\bar{\nu}}_m}$ in scaling form for different system sizes $N$, where $\Delta z\equiv z-z_c(\infty)$. We confirm the exponent to be $\beta_m=0.5\pm 0.01$ from the $\Delta z$ region in which the finite-size effect is negligible. Performing finite-size scaling analysis in Fig. \[fig:order\](a), we obtain the correlation size exponent defined as $z_m^*(N)-z_c(\infty)\sim N^{-1/{\bar{\nu}}_m}$ to be ${\bar{\nu}}_m \approx 2.10 \pm 0.02$.
![ (Color online) (a) Scaling plot of the rescaled order parameter $(m-m_0)N^{\beta_m/{\bar{\nu}}_m}$ vs. $\Delta z N^{1/{\bar{\nu}}_m}$. With $\beta_m=0.5$ and ${\bar{\nu}}_m=2.10$ data are well collapsed onto a single curve. (b) Scaling plot of $(\langle m^2 \rangle - \langle m \rangle^2)N^{1-\gamma_m/{\bar{\nu}}_m}$ for different $N$ versus $\Delta z N^{1/{\bar{\nu}}_m}$, where $\gamma_m=1.05$ is used.[]{data-label="fig:order"}](Fig2.pdf){width=".95\linewidth"}
Next, we consider the susceptibility $\chi(z)$ as the fluctuations of the order parameter over the ensemble. This quantity is expected to exhibit critical behavior $\chi \sim (z-z_c)^{-\gamma_m}$ for $z > z_c$. In Fig. \[fig:order\](b), we plot a rescaled quantity $(\langle m^2 \rangle-\langle m \rangle^2)N^{1-\gamma_m/{\bar{\nu}}_m}$ versus $\Delta z N^{1/{\bar{\nu}}_m}$. We find that for the critical $\Delta z$ region, the data decay in a power-law manner with the exponent $\gamma_m \approx 1.05\pm 0.05$. Moreover, with the choice of ${\bar{\nu}}_m = 2.1$, the data are well collapsed onto a single curve. The obtained exponents $\beta_m\approx 0.5\pm 0.01$, $\gamma_m\approx 1.05\pm 0.05$ and ${\bar{\nu}}_m \approx 2.1 \pm 0.02$ satisfy the hyperscaling relation ${\bar{\nu}}_m=2\beta_m+\gamma_m$ reasonably well.
We also study the probability to contain nonzero GMCC at a certain point $z$, denoted as $R_N(z)$ [@grassberger]. We find that $R_N$ approaches a step function in a form that scales as $R_N([z-z_c(N)]N^{1/2})$ (see Fig \[fig:RN\]). Thus, the slope $\mathrm{d}R_N(z)/\mathrm{d}z$ exhibits a peak at $z_c(N)$, where its value increases as $N^{1/2}$. This means, the probability that the collapse of GMCC occurs at $z_c(N)$ increases with the rate $N^{1/2}$. Finite size scaling theory suggests the interpretation that ${\bar{\nu}}_m=2$, which is compatible with the result we obtained earlier from Fig. \[fig:order\]. One can introduce the order parameter $S(z)$ averaged over all configurations as $S(z)=m(z)R_N(z)$ behaves similarly to the one obtained previously in Fig. 1 of Ref. [@son_grassberger].
The probability $R_N(z)$ is the basic quantity for large cell renormalization group transformation in percolation theory [@RSK; @herrmann_rg]. To proceed, we rescale the control parameter as $p=z/z_0$, where $z_0$ is the mean degree at the beginning of the cascading processes and taken as $z_0=2z_c(\infty)$ for convenience; then $1-p$ is the fraction of nodes removed. Let us define ${\tilde p}=R_N(p)$, where $\tilde p$ can be interpreted as the probability that a node is occupied in a coarse-grained system scaled by $N$. Using the renormalization group idea, once we find the fixed point $p^*(N)$ satisfying $p^*=R_N(p^*)$ and take the slope $\lambda=\mathrm{d}R_N(p)/\mathrm{d}p$ at $p^*(N)$. Then, we can obtain ${\bar{\nu}}_m=\ln N/\ln \lambda$. Numerically we obtain that $\lambda\sim N^{0.51\pm 0.02}$ and thus ${\bar{\nu}}_m$ is obtained to be ${\bar{\nu}}_m\approx 1.96 \pm 0.07$ (Fig. \[fig:Rslope\]). This value is close to the one previously obtained by data collapse method.
![ (Color online) The probability $R_N(p)$ that the giant cluster exists. Here $p = z / 2 z_c$ is the node occupation probability of the site percolation in ER networks with mean degree $2 z_c$, which corresponds to the mean degree $z$ of the bond percolation in ER networks. The critical point $p = 0.5$ corresponds to $z = z_c$ in this convention. The inset is the plot of $R_N$ vs. $(z - z_c(N)) N^{1/2}$. This data collapse requires $z_c(N)$ instead of $z_c(\infty)$. []{data-label="fig:RN"}](Fig3.pdf){width=".95\linewidth"}
![ (Color online) The slope of $R_N(p)$ at the fixed point $p^*$, for which $p^* = R_N(p^*)$, as a function of the system size $N$. We measure the slope of the right three data points using the least-square-fit method to be $0.51 \pm 0.02$. Thus, ${\bar{\nu}}_m \approx 1.96\pm 0.07$. Solid line is a guideline with a slope $0.51$ []{data-label="fig:Rslope"}](Fig4.pdf){width=".95\linewidth"}
Interestingly, we measure $p_c-p^*(N)\sim N^{-1/1.5}$ yielding $z_c(\infty)-z^*(N)\sim N^{-1/1.5}$. Similarly, from direct simulations we obtain $z_c(\infty)-z_c(N)\sim N^{-1/1.5}$, where $z_c(N)$ is the average finite size transition point (Fig \[fig:fssfig\]). In a conventional second-order transition, we would expect that these quantities scale with $N$ as $N^{-1/{\bar{\nu}}_m}$. The difference to ${\bar{\nu}}_m\approx 2$ indicates either an additional diverging scale or extraordinarily large corrections. But as we have seen previously, the standard definition of the exponent yields ${\bar{\nu}}_m\approx 2$. This is confirmed by the inset of \[fig:fssfig\] which shows $\sqrt{\mathrm{Var}(z_c(N))}\sim N^{-0.5}$, where $\mathrm{Var}$ stands for the variance. We conclude ${\bar{\nu}}_m\approx 2$ which is also consistent with the value we obtained using the renormalization group transformation eigenvalue.
![(Color online) $z_c(N)$ is the mean position of the order parameter discontinuity when the system size is $N$. It approaches to $z_c(\infty) = 2.45540749\cdots$ as $N$ increases, and the difference scales as $N^{-1/1.5}$. The inset is the standard deviation of $z_c(N)$, which decreases as $N^{-0.5}$. []{data-label="fig:fssfig"}](Fig5.pdf){width=".95\linewidth"}
Critical behavior of avalanche dynamics
---------------------------------------
To characterize the avalanche processes, we count the avalanche size defined as the number of nodes removed in each layer during the cascading processes, denoted as $s_a(z)$. The distribution of those avalanche sizes collected from different triggering nodes and configurations is denoted as ${p_{s}}(z)$. In Ref. [@baxter] analytically ${p_{s}}(z_c) \sim s_a^{-\tau_a}$ with $\tau_a=3/2$ was obtained for locally tree like graphs. We confirm this exponent value in Fig. \[fig:aval\](a). Avalanches in the region $z < z_a^*(N)$ need to be classified as finite or infinite avalanches; the latter locate separately in Fig. \[fig:aval\](a). Infinite avalanche means the avalanche size is as large as $m(z)$, i.e., the GMCC completely collapses, and the system falls into an absorbing state. The infinite avalanche begins to appear at $z=z_a^*(N)$. Fig. \[fig:aval\](a) shows the scaling behavior of the [avalanche size distribution]{}in form of ${p_{s}}N^{\tau_a/\sigma_a {\bar{\nu}_a}}$ versus $s_aN^{-1/\sigma_a {\bar{\nu}_a}}$ at $z_c$. The data from different system sizes are well collapsed onto a single curve by the choices of $\tau_a=3/2$ and $\sigma_a{\bar{\nu}_a}\approx 1.85$. This result suggests that there exists a characteristic size $s_a^*\sim N^{1/\sigma_a {\bar{\nu}_a}}$ with $\sigma_a{\bar{\nu}_a}\approx 1.85 \pm 0.02$ for finite avalanches. These values indicate that the hyperscaling relation $\sigma_a{\bar{\nu}_a}=\tau_a$ does not hold for the avalanche dynamics. For infinite avalanches, $s_{a,\infty}^*\sim O(N)$.
For $z > z_c$, we examine the avalanche size distribution versus $s_a$ for different $\Delta z$, and find that it behaves as ${p_{s}}\sim s_a^{-\tau_a} f(s_a/s_a^*)$ where $f$ is a scaling function. Following conventional percolation theory [@stauffer], we assume $s_a^{*}\sim \Delta z^{-1/\sigma_a}$. The exponent $\sigma_a$ is obtained from the scaling plot of ${p_{s}}(z)\Delta z^{-\tau_a/\sigma_a}$ vs $s_a\Delta z^{1/\sigma_a}$ in Fig. \[fig:aval\](b). The data are well collapsed with $\sigma_a\approx 1.0$, leading to ${\bar{\nu}}_a\approx 1.85$. This is different from $\bar{\nu}_m$ and indicates that there exists another divergent scale.
![(Color online) (a) Scaling plot of $p_s(z_c)N^{\tau_a/\sigma_a {\bar{\nu}_a}}$ vs $s_a/N^{1/\sigma_a{\bar{\nu}_a}}$ for different system sizes, with $\tau_a=1.5$ and $\sigma_a {\bar{\nu}_a}\approx 1.85$. Note that infinite avalanche sizes for different $N$ do not collapse onto a single dot, because they depend on $N$ as $s_{a,\infty}^* \sim N$. (b) Scaling plot of $p_s(z)\Delta z^{-\tau_a/\sigma_a}$ vs $s_a\Delta z^{1/\sigma_a}$ for different $\Delta z$ but a fixed system size $N = 2.56\times10^7$, with $\tau_a=1.5$ and $\sigma_a \approx1.01$. (c) Scaling plot of $\langle s_a \rangle N^{-\gamma_a/{\bar{\nu}_a}}$ vs $\Delta z N^{1/{\bar{\nu}_a}}$ for different system sizes and $\gamma_a=0.5$. []{data-label="fig:aval"}](Fig6.pdf){width=".95\linewidth"}
![(Color online) (a) Plot of $\langle s_a(t) \rangle$ as a function of $t$ at $z_c$ for finite avalanches, showing $\langle s_a(t) \rangle \sim t^{2.0\pm 0.01}$. (b) The plot of $\langle t_{\rm finite} \rangle$ of finite avalanches vs. $N$ at $z_c$ on semi-logarithmic scale (left axis). Plot of $\langle t_{\infty} \rangle$ of infinite avalanches as a function of $N$ on double-logarithmic scale (right axis). The guide line has a slope of $1/3$. []{data-label="fig:hop"}](Fig7.pdf){width=".95\linewidth"}
We examine the mean avalanche size $\langle s_a\rangle\equiv \sum_{s_a=1}^{\prime} s_a{p_{s}}(z)\sim (\Delta z)^{-\gamma_a}$, where the prime indicates summation over finite avalanches. It follows that $\gamma_a=(2-\tau_a)/\sigma_a$ [@stauffer]. Thus, $\gamma_a=0.5$ is expected. Our simulation confirms this value in the large $\Delta z$ region (Fig. \[fig:aval\](c)). Data from different system sizes are well collapsed in the plot of $\langle s_a \rangle N^{-\gamma_a/{\bar{\nu}_a}}$ vs $\Delta z N^{1/{\bar{\nu}_a}}$ with $\gamma_a=0.5$ and ${\bar{\nu}}_a=1.85$. This means that there exists crossover points $z_a^*(N)$ such that $z_a^*(N)-z_c\sim N^{-1/{\bar{\nu}}_a}$ in finite systems. In the thermodynamic limit, $\langle s_a \rangle/N$ is equal to $0$ for $z < z_c$, $s_0$ for $z = z_c$ and $w(z-z_c)^{-\gamma_a}$ for $z > z_c$ where $s_0$ is constant and $w\sim O(N^{-1})$. This result shows that the avalanche statistics also exhibits HPT.
Statistics of the number of hops {#sec:hop-number}
--------------------------------
When investigating the avalanche dynamics we first focus on [*finite avalanches*]{}. Let $t_i(z)$ be the number of hopping steps between the two layers in avalanche processes, when the $i$th node is removed from the GMCC at $z$. $\langle s_a(t) \rangle_i$ is the avalanche size averaged over $i$, that is, the mean number of nodes removed, accumulated up to steps $t$. It is found in Fig. \[fig:hop\](a) that $\langle s_a(t) \rangle \sim t^2$ for finite avalanches, similarly to [@zhou]. Using the [avalanche size distribution]{} $p_s(z)$, we set up the duration time distribution $p_t(z)$ through the relations $p_s \mathrm{d}s =p_t \mathrm{d}t$ and $s_a \sim t^2$ as $p_{t}(z) \sim t^{-2\tau_a+1} f(t^2/(\Delta z)^{-1/\sigma_a})$. The mean number of hopping steps for finite avalanches is $\langle t_{\rm finite} \rangle \equiv \sum_{t=1}^{\prime} t p_t(z)$. Because of $\tau_a=3/2$, $\langle t \rangle \sim -\ln (\Delta z)$ for $z > z_c$ and $\langle t_{\rm finite} \rangle \sim \ln N$ at $z=z_c$ (Figs. \[fig:hop\](b) and \[fig:avhops\]). The number of hopping steps of [*infinite avalanches*]{} which can appear in the region $z < z_a^*(N)$ lead to $\langle t_{\infty} \rangle \sim N^{1/3}$, as shown in Fig. \[fig:hop\](b), in agreement with [@zhou].
The scaling plot $p_t(z)(z-t_c)^{(-2\tau_a+1)/2\sigma_a}$ vs. $t(z-z_c)^{1/2\sigma_a}$ displayed in \[fig:hopscaling\] proves our hypothesis of $p_t(z)\sim t^{(-2\tau_a+1)}f(-t^2/(\Delta z)^{-1\sigma_a})$. The exactly known special value $\tau_a=3/2$ yields $\langle t_{\mathrm{finite}}\rangle \sim \ln N$ at $z=z_c$ as observed in Fig. \[fig:hop\](b).
![(Color online) The scaling collapse of the distribution $p_t$ of the number of hops $t$ for finite avalanches. []{data-label="fig:hopscaling"}](Fig8.pdf){width=".95\linewidth"}
![(Color online) The mean number of hops of finite avalanches $\left< t_{\text{finite}} \right>$ as a function of the mean degree $z$. []{data-label="fig:avhops"}](Fig9.pdf){width=".95\linewidth"}
Simulation results on the 2D interdependent networks {#sec:num_result_2D}
====================================================
Let us describe the CF model on two layers of randomly interdependent 2D networks [@son_grassberger; @Li]. At the beginning the layers consist of topologically identical square lattices of size $N=L\times L$ sites with nearest-neighbor connectivity links within each layer. As it was the case for ER networks, the set of nodes in one layer has a random one-to-one correspondence via dependency links with the set of nodes in the other layer.
The control parameter is defined as the fraction $q$ of original nodes kept in a layer [@buldyrev], analogously to the site percolation problem. Each node shares its fate with its interdependent node on the other layer. The order parameter $m(q)$ is defined as the relative size of the GMCC.
We applied two boundary conditions (BC-s) to the system: periodic and semiperiodic. In the periodic BC the system is on a torus, while in the semiperiodic BC it is on a cylinder, i.e., open in one direction and periodic in the other one. The order of the characteristic parameter values ($q_c(\infty)$, $q_c(N)$ and $q_m^*(N)$) depends on the BC. For the periodic (semiperiodic) BC we have $q_c(N)<q_c(\infty)<q_m^*(N)$ ($q_c(\infty)<q_c(N)<q_m^*(N)$).
The average order parameter $m_c(N)$ before collapse is defined as the smallest nonzero values of the relative size of the giant component averaged over all runs with size $N$. For periodic (semiperiodic) BC we have $m_c(\infty)>m_c(N)$ ($m_c(\infty)<m_c(N)$). The figures are for semiperiodic BC if not indicated otherwise.
Critical behavior of GMCC {#ss:2d_crit_order_parameter}
-------------------------
![(Color online) $m-m_0$ vs $q-q_c$ is plotted for different system sizes on double logarithmic scales. The data seem to collapse into a single line of slope $\beta_m \approx 0.53$ in the large-$\Delta q$ region. Inset: Plot of the rescaled order parameter $(m-m_0)N^{\beta_m/{\bar{\nu}}_m}$ vs $\Delta q N^{1/{\bar{\nu}}_m}$. In order to achieve data collapse, we had to use ${\bar{\nu}}_m \approx 2.1$ and $\beta_m \approx 0.53$. []{data-label="fig:2dorder"}](Fig10.pdf){width=".95\linewidth"}
The method of \[sec:beta-proof\] can be used to numerically calculate the critical threshold $q_c$ and the jump size $m_0$. However, throughout this subsection, we will adopt the values $q_c=0.682892(5)$ and $m_0=0.603(2)$ which were recently obtained by Grassberger [@grassberger].
Theoretical consideration for the value of $\beta_m$ suggest $\beta_m=0.5$. \[fig:2dorder\] shows a plot of $m(q)-m_0$ vs. $\Delta q \equiv q-q_c$ for various system sizes. In the not too small $\Delta q$ region, the data collapses into a single line, which enables us to measure $\beta_m \approx 0.53$. Since the region of agreement is quite short, we suspect that this deviation from the theoretical value is due to the finite size corrections. The scaling plot $(m-m_0)N^{\beta_m/{\bar{\nu}}_m}$ vs $\Delta q N^{1/{\bar{\nu}}_m}$ suggests ${\bar{\nu}}_m =2.1 \pm 0.2$.
![(Color online) Plot of the susceptibility $\chi\equiv N(\langle m^2 \rangle-\langle m\rangle^2)$ vs $q-q_c(\infty)$ for different system sizes on double logarithmic scales using systems with periodic boundary conditions. Inset: Data collapsed plot of ($\langle m^2 \rangle$-$\langle m \rangle^2$)$N^{1-\gamma_m/{\bar{\nu}}_m}$ vs $w\equiv \Delta q N^{1/{\bar{\nu}}_m}$ for the different system sizes. The best collapse is achieved using $\gamma_m\approx 1.35$ with ${\bar{\nu}}_m\approx 2.4$. The boundary conditions have a strong effect on the corrections to scaling and on the measured effective values of the exponents. []{data-label="fig:2dsusc"}](Fig11.pdf){width=".95\linewidth"}
\[fig:2dsusc\] shows the raw plot of the susceptibility $\chi = N(\langle m^2\rangle-\langle m\rangle^2)$ against $q-q_c$ for different system sizes. Due to strong corrections to scaling the exponent $\gamma$ is less accurate than for the ER case. The inset of \[fig:2dsusc\] shows $(\langle m^2 \rangle-\langle m \rangle^2)N^{1-\gamma_m/{\bar{\nu}}_m}$ vs $\Delta q N^{1/{\bar{\nu}}_m}$ using $\gamma_m = 1.35 \pm 0.15$ and ${\bar{\nu}}_m = 2.4 \pm 0.2$, and one can observe that the deviation from power law leads to failure of collapse for large-$\Delta q$ region.
Our simulation data shows the following values of the exponents $\beta_m = 0.53 \pm 0.02$, $\gamma_m = 1.35 \pm 0.10$, and ${\bar{\nu}}_m = 2.2 \pm 0.2$. These exponents satisfy the scaling relation ${\bar{\nu}}_m = 2 \beta_m + \gamma_m$ within their error ranges.
Critical behavior of avalanche dynamics
---------------------------------------
We now examine the avalanche dynamics 2D lattices described above. Analogously to the case of double-layer ER networks, we denote the avalanche size at $q$ by $s_a(q)$, and the distribution of avalanche size by ${p_{s}}(q)$.
![(Color online) Plot of the [avalanche size distribution]{} ${p_{s}}(q_c)$ vs $s_a$ for different system sizes. The power-law regime is longer for larger system sizes. Inset: Plot of the [avalanche size distribution]{} in a scaling form ${p_{s}}(q_c) N^{\tau_a/\sigma_a {\bar{\nu}_a}}$ vs $s_aN^{-1/\sigma_a {\bar{\nu}_a}}$ for different system sizes. The data are well collapsed onto a single curve with $\sigma_a {\bar{\nu}_a}$. []{data-label="fig:2davalqc"}](Fig12.pdf){width=".95\linewidth"}
The avalanche size distribution follows a power law at $q_c$ as ${p_{s}}(q_c) \sim s_a^{-\tau_a}$. The exponent is measured to be $\tau_a = 1.59 \pm 0.02$, see \[fig:2davalqc\]. The avalanche size distribution follows this power-law up to a characteristic size $s_a^*$ that scales with the size of the system as $s_a^* \sim N^{1/\sigma_a{\bar{\nu}_a}}$, from which point it decays exponentially. The inset of \[fig:2davalqc\] plots ${p_{s}}N^{\tau_a/\sigma_a{\bar{\nu}_a}}$ against $s_a N^{-1/\sigma_a {\bar{\nu}_a}}$ using $\sigma_a {\bar{\nu}_a}\approx 1.47$, with which the data collapses into a single line.
\[fig:2dpsa\] shows plots the [avalanche size distribution]{} at various $q>q_c$ for a fixed system size $N=4096^2$. The distribution ${p_{s}}$ follows ${p_{s}}\sim s_a^{-\tau_a}f(-s_a/s_a^*)$, where $f$ is the so-called “master curve” (a scaling function) and we assume $s_a^* \sim (q-q_c)^{-1/\sigma_a}$. We obtain the exponent $\sigma_a$ by plotting ${p_{s}}(q)/(\Delta q)^{\tau_a/\sigma_a}$ versus $s_a(\Delta q)^{1/\sigma_a}$. The best data collapse is observed with $\sigma_a=0.70 \pm 0.05$, implying ${\bar{\nu}_a}= 2.1 \pm 0.2$, see \[fig:2dpsa\]. These values are confirmed by a somewhat more reliable method using the cumulative distribution function $P_s(q)$ which scales as $1-P_s(q)\sim s_a^{1-\tau_a}F(-s_a/s_a^*)$ where $F$ is another scaling function.
Notice in \[fig:2dpsa\] that the cutoff sizes $s_a^*$ are small, and to increase them one has to carry out the measurement of the cascade size distribution close to the critical point. For this, a trade-off is to be made. Going too close to the critical point of the infinite system the critical behavior of the finite system is lost.
![(Color online) Plot of ${p_{s}}(q)$ vs $s_a$ at different $\Delta q$ for $N=4096^2$. Inset: Plot of ${p_{s}}(q)/(\Delta q)^{-\tau_a/\sigma_a}$ vs $s_a(\Delta q)^{1/\sigma_a}$ at various $\Delta q$ for $N=4096^2$. With $\sigma_a \approx 0.70$, the data collapse into a single line. []{data-label="fig:2dpsa"}](Fig13.pdf){width=".95\linewidth"}
This observation supports the speculation that even systems as large as $N=4096^2$ are not enough to correctly assess power-law behaviors in the near-$q_c$ regions. As we shall see now, it is also related to the behavior of the first moment of the [avalanche size distribution]{}. The first moment of ${p_{s}}$ follows $\langle s_a\rangle \equiv \sum_{s_a=1}'s_a{p_{s}}(q) \sim (q-q_c)^{-\gamma_a}$. \[fig:2d\_avg\_s\] depicts our simulation results for the average size of finite avalanches for various $\Delta q$ and $N$, with a guideline of slope $\gamma_a = 0.5$ giving the best estimate for $\gamma_a$. The inset of \[fig:2d\_avg\_s\] is a plot of $\langle s_a\rangle N^{-\gamma_a/{\bar{\nu}_a}}$ versus $\Delta q N^{1/{\bar{\nu}_a}}$ for different system sizes. Collapse is achieved with $\gamma_a \approx 0.50 \pm 0.05$ and ${\bar{\nu}_a}\approx 2.1 \pm 0.2$. This value of $\gamma_a$ reasonably satisfies the scaling relation $\gamma_a=(2-\tau_a)/\sigma_a$ within error ranges. The quality of the collapse is still unsatisfactory, which makes the values of these exponents questionable.
![(Color online) The average size $\langle s_a \rangle$ of finite avalanches is plotted against $\Delta q$ for various system sizes. A line exhibiting the expected power-law with exponent $\gamma_m$ is drawn for comparison. The smaller systems seem to be too small to observe power-law behavior. Inset: Plot of $\langle s_a \rangle N^{-\gamma_a/{\bar{\nu}_a}}$ vs $\Delta q N^{1/{\bar{\nu}_a}}$ for various system sizes. Using $\gamma_a \approx 0.5$ and ${\bar{\nu}_a}\approx 2.1$, the data roughly collapse in the mid-$\Delta q$ region. However, collapse fails in the large-$\Delta q$ region. []{data-label="fig:2d_avg_s"}](Fig14.pdf){width=".95\linewidth"}
Statistics of the number of hops {#statistics-of-the-number-of-hops}
--------------------------------
We now turn our attention to the number of hops $t$, starting with the hops in finite avalanche processes. One can see in \[fig:2dh\](a) that the average size of avalanches $\langle s_a \rangle_t$ roughly scales with $t$ as $\langle s_a \rangle_t \sim t^{2.75}$, meaning that the fractal dimension of avalanche trees is $d_b=2.75$, which is different from that of the case of ER networks. This allows us to assume that the characteristic number of hops $t^*$ roughly scales as $t^* \sim (s_a^*)^{1/d_b} \sim (q-q_c)^{(-1/d_b\sigma_a)}$. Then, the distribution of the number of hops for finite avalanches would satisfy $p_t(q) \sim t^{-d_b \tau_a + d_b -1}f(t^{d_b}/(\Delta q)^{-1/d_b\sigma_a})$. This behavior is confirmed by \[fig:2dh\](b), which shows a scaling plot of this distribution.
![(Color online) (a) Plot of the mean avalanche sizes $\langle s_a \rangle$ vs the number of hops $t$ between the two layers. The overall slope is estimated to be about $2.75$. (b) The distribution $p_t(q)$ of the number of hops $t$ vs $\Delta q$ in scaling form. $p_t(q)(\Delta q)^{(-d_b\tau_a+d_b -1)/d_b\sigma_a}$ is plotted against $t(\Delta q)^{1/d_b\sigma_a}$, with $d_b \approx 2.75$. []{data-label="fig:2dh"}](Fig15.pdf){width=".95\linewidth"}
Recall that the value of $\tau_a$ was measured to be $\tau_a \approx 1.59$. This implies that, in contrast to the case of ER networks, the average number of hops of finite avalanches $\langle t \rangle \equiv \sum_{t=1}' t p_t(q)$ does not decrease logarithmically but rather follows a power-law with exponent $1-d_b\tau_a+d_b$. Also, the average number of finite hops $\langle t \rangle$ at $q=q_c$ approaches some value with a power-law, rather than increasing logarithmically \[fig:2d\_h\_vs\_Deltaq\] and \[fig:2dh\_at\_pc\](a) illustrate these points.
![ (Color online) The average number of finite hops $\langle t_\textrm{finite} \rangle$ is plotted against $\Delta q$ in double logarithmic scales. []{data-label="fig:2d_h_vs_Deltaq"}](Fig16.pdf){width=".95\linewidth"}
![ (Color online) (a) Plot of $\langle t_\textrm{finite}(\infty) \rangle - \langle t_\textrm{finite}(N) \rangle$ vs $N$ at $q_c$ in logarithmic scale, where $\langle t_\textrm{finite}(\infty) \rangle = 1.96$ was used. (b) Plot of $\langle t_\infty \rangle$ of infinite avalanches vs $N$ in logarithmic scale. The guideline has a slope $0.33$. []{data-label="fig:2dh_at_pc"}](Fig17.pdf){width=".95\linewidth"}
Lastly, we consider the number of hops that constitute the infinite avalanches. Our simulation results reveal that this number scales as $N^{0.33}$. \[fig:2dh\_at\_pc\](b) shows these behaviors by plotting $\langle t_\infty \rangle$ against system size $N$.
In short, analogously to the two-layered ER network, the two sets of exponents $\{\beta_m, \gamma_m, {\bar{\nu}}_m \}$ and $\{\gamma_a, {\bar{\nu}}_a, \sigma_a, \tau_a \}$ are measured to be distinct. In this model too, values of the critical exponents measured through simulation satisfy the scaling relation $1-\beta_m=\gamma_a$ that relates these sets. However, in all aspects the scaling behavior of 2D interdependent networks is much worse than that of the ER interdependent networks, indicating severe corrections to scaling.
Analytic results {#sec:analytic}
================
In the following we derive two rules that hold for general interdependent networks.
Proof of $\beta_m=1/2$ and $\gamma_a=1/2$ {#sec:beta-proof}
-----------------------------------------
For the exponent $\beta_m$ values close to $1/2$ were measured for very different network settings [@baxter]. We prove that $\beta_m=1/2$ holds for a wide range of mutual percolation processes. Let $P_\infty^s(q)$ denote the fraction of nodes belonging to the giant component of the classical (single layer) percolation problem where $q$ is the fraction of occupied nodes. Let $q_c^s$ denote the critical point of this single layer percolation. If an additional layer is added to the percolation process with dependency links the critical point for the mutual percolation is $q_c\geq q_c^s$ [@buldyrev; @bashan]. Now let’s consider a two-layered interdependent network with random infinite range interdependency links that represent a random one-to-one mapping between the layers. The control parameter $q$ denotes the fraction of the nodes kept. It has been shown that the size of the MCGC after the $i$th step is $P_\infty^s(q_i)$ where $q_i$ is an equivalent random attack given by the recursion [@buldyrev] $$\label{eq:rec} q_i=\frac{q}{q_{i-1}} P_\infty^s(q_{i-1}).$$ The recursion has a fixed point $x(q)$ corresponding to the steady state $m(q)\equiv P_\infty^s(x(q))$ of the system: $$\label{eq:fixed} x^2=q P_\infty^s(x).$$ As $q_c > p_c^s$ the $P_\infty^s$ curve of single layer percolation can be approximated by its series near $q_c$: $$\label{eq:lin} P_\infty^s(q)= a+b\cdot q + O(q^2)$$ with $a\equiv P_\infty^s(q_c)$.
For the critical behavior close to $q_c$ we need to solve $ x^2(q) = q\cdot\big(a+bx(q)\big)$ resulting $$\label{eq:roots} x=\frac{bq\pm\sqrt{b^2q^2+4aq}}{2}.$$ At the critical point $q_c$ the determinant $b^2(q_c)^2+4aq_c$ is zero. Introducing $q:=q_c+\Delta q$ and substituting into the valid (greater) result of \[eq:roots\] we have $$\label{eq:diff} x(q)-x(q_c)=\frac{b\Delta q+\sqrt{b^2(q_c+\Delta q)^2+4a(q_c+\Delta q)}}{2}.$$ By and $b^2q_c^2+4aq_c=0$, we get $$m(q)-m(q_c)\sim (q-q_c)^{1/2}+ O(q-q_c).$$ Thus, we conclude $\beta_m=1/2$. Due to the sum rule (see next subsection) this also implies $\gamma_a=1/2$.
Sum rule for interdependent networks and $\gamma_a=1-\beta_m$ {#sec:sum-rule}
-------------------------------------------------------------
For the avalanche dynamics we summarize over the whole history of the network: $$\label{eq:newsum}
1=m(q)+\int_q^1{\sum_s}^{\prime}{s p_s(\tilde{q})} \,\mathrm{d}\tilde{q}.$$ This formula expresses the fact that a site can either belong to the MCGC (first term on the r.h.s. of \[eq:newsum\]) or it is eliminated in one of the avalanches (the sum in \[eq:newsum\]). Here $p_s(q)$ is the number of avalanches of size $s$ occurring per site per attack $\mathrm{d}q$ at $q$. The summation is carried out over all finite avalanches and the integral takes into account any events that were triggered for $\tilde{q}\in[q,1]$. It is useful to write \[eq:newsum\] in differential form: $$\frac{\mathrm{d}m(q)}{\mathrm{d}q}={\sum_{s}}^{\prime}{sp_s(q)}.$$ Since $m(q)-m(q_c)\propto (q-q_c)^\beta_m$, it yields $\mathrm{d} m(q)/\mathrm{d}q\propto (q-q_c)^{\beta_m-1}$. The right hand side describes the average size of finite avalanches which scales as $\langle s_a(q)\rangle\sim(q-q_c)^{-\gamma_a}$. Comparing the two sides we find that the relation $\gamma_a=1-\beta_m$ between the two set of exponents holds universally.
{width=".9\linewidth"}
Summary {#sec:summary}
=======
Our aim has been in this paper to clarify the unusual features of the HPT as observed in the interdependent CF model. Due to the efficient algorithm [@hwang] we were able to carry out large scale simulations for the ER and 2D interdependent networks and determine numerically the exponents and the finite size scaling functions.
The specific challanges related to the HPT for the interdependent CF model come from the fact that, in contrast to ordinary percolation, we have here two divergent length scales as the system approaches the transition point and, correspondingly, two sets of exponents. The critical properties we obtained are schematically shown in Fig.17 for the Erdős–Rényi (ER) and for the 2D interdependent networks. One set of exponents, $\{\beta_m, \gamma_m, {\bar{\nu}}_m \}$ is associated with the order parameter and its related quantities, and the other set $\{\tau_a, \sigma_a, \gamma_a, {\bar{\nu}}_a \}$ is associated with the avalanche size distribution and its related ones. The subscripts $m$ and $a$ refer to the order parameter and avalanche dynamics respectively.
The numerically estimated values of the critical exponents for the ER and the 2D cases are listed in Table I. They reveal the unconventional character of the transition: the exponents $\bar{\nu}_m$ and $\bar{\nu}_a$ and $\gamma_m$ and $\gamma_a$ are different from each other, respectively. For the ER and 2D cases, the hyperscaling relation ${\bar{\nu}}_m=2\beta_m+\gamma_m$ holds even though data collapsing for the 2D case is not as satisfactory as for the ER case. The relation $\sigma_a {\bar{\nu}}_a=\tau_a$ does not hold (Sec. \[sec:num\_result\_ER\] and \[sec:num\_result\_2D\]).
We showed analytically that the two sets of critical exponents are not completely independent of each other; they are coupled through the relation $m(z)+\int_z^{z_0} \langle s_a(z) \rangle \mathrm{d}z =1$, where $z_0$ is the mean degree at the beginning of cascading processes. This relation leads to $\mathrm{d}m(z)/\mathrm{d}z= \langle s_a(z)\rangle$ and yields $1-\beta_m=\gamma_a$. We also showed that for random interdependence links $\beta_m =1/2$. Our numerical values support these relations.
We classified avalanches in the critical region as finite and infinite avalanches. When an infinite avalanche occurs, the GMCC completely collapses, and the system falls into an absorbing state. We found that the mean number of hopping steps denoted as $\langle t \rangle$ between the two layers in avalanche processes depends on the system size $N$ in different ways for the different types of avalanches: $\langle t \rangle \sim \ln N$ for finite avalanches, and $\sim N^{1/3}$ for infinite avalanches on the ER interdependent network. This difference in the scaling again underlines the peculiarities of the HPT: The infinite avalanche give rise to $m_0$, while the finite ones contribute to the critical avalanche statistics.
Our results present a unified picture of HPT, however, there are still open questions for further research. The strong corrections to scaling, especially for the 2D case should be understood. We have realized that the boundary conditions have a strong impact on the corrections and one should persue the investigation along this line. A real challange is to understand how the hybrid transition can be properly treated with the method of the renormalization group. Furthermore, it would be very interesting to see how other hybrid transitions fit into the presented framework.
This work was supported by National Research Foundation in Korea with the grant No. NRF-2014R1A3A2069005. JK acknowledges support from EU FP7 FET Open Grant No. 317532, Multiplex.
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abstract: 'Direct measurements of dark matter distributions in galaxies are currently only possible through the use of gravitational lensing observations. Combinations of lens modelling and stellar velocity dispersion measurements provide the best constraints on dark matter distributions in individual galaxies, however they can be quite complex. In this paper, we use observations and simulations of gravitational microlensing to measure the smooth (dark) matter mass fraction at the position of lensed images in three lens galaxies: , and . The first two systems consist of early-type lens galaxies, and both display a flux ratio anomaly in their close image pair. Anomalies such as these suggest a high smooth matter percentage is likely, and indeed we prefer $\sim50$ per cent smooth matter in MG 0414+0534, and $\sim80$ per cent in SDSS J0924+0219 at the projected locations of the lensed images. Q2237+0305 differs somewhat in that its lensed images lie in the central kiloparsec of the barred spiral lens galaxy, where we expect stars to dominate the mass distribution. In this system, we find a smooth matter percentage that is consistent with zero.'
author:
- 'N. F. Bate, D. J. E. Floyd, R. L. Webster and J. S. B. Wyithe'
title: A microlensing measurement of dark matter fractions in three lensing galaxies
---
Introduction
============
Gravitational lensing measurements provide the only direct method to probe the non-luminous matter component of lensing systems. Weak lensing measurements of cluster masses and mass distributions are becoming routine, however the measurement of the dark matter components of individual galaxies is a new and exciting prospect. By analysing gravitational microlensing signals in some strongly lensed quasars, we are able to infer the ratio of clumpy to continuously distributed matter along the line of sight to the background source. This gives us a probe of the dark matter content in a lensing galaxy at projected radii of $\sim2$–10kpc from the centre of the galaxy (@sw04; @pooley+09).
Recent years have seen considerable advancement in the mapping of dark and stellar mass in lensing galaxies (e.g. @keeton+98; @ferreras+05; @barnabe+09). Usually, the total mass within the Einstein Radius is constrained by modelling the lensing galaxy to fit the observed lensed image positions. Photometry of the lensing galaxy, in combination with stellar population synthesis models, provides the distribution of stellar mass. Finally, observations of stellar velocity dispersions can be used to break the mass-sheet and mass-anisotropy degeneracies, and thus constrain the overall mass density profile (see for example @kt03; @tk04; @ferreras+05; @ferreras+08; @barnabe+09; @auger+09, and references therein).
These analyses can provide a very detailed picture of the structure of lensing galaxies. However, they are quite complex, relying on often difficult observations and detailed modelling. A complementary method exists, in which observations of microlensing in the lensed quasar images are used to constrain the dark matter percentage along those lines of sight directly.
Cosmological microlensing occurs when the light path to a lensed quasar image intersects a starfield in a foreground lensing galaxy. The lensing galaxy as a whole magnifies the lensed image; microlensing by individual stars induces variations about this macro-magnification. This is most readily detected in lightcurves of quasar images, where relative motion between observer, lens and source causes uncorrelated fluctuations in brightness between lensed images. This effect was first observed in the lensed quasar Q2237+0305 [@irwin+89].
In some lensing systems we observe a close pair of quasar images. Basic lensing theory suggests that these two images should have approximately equal magnifications (@cr79; @bn86). On the contrary, in eight out of ten known cases we find that the brightness of the image located at the saddle point in the time delay surface is suppressed relative to the image at the minimum in the time delay surface [@pooley+07]. Microlensing is one possible explanation for such flux ratio anomalies. However, microlensing by a purely stellar component is not sufficient. @sw02 showed that this discrepancy could be accounted for by adding a significant smooth matter component to the lens at the image positions, since minimum and saddle point images are microlensed differently when a smooth matter component is added.
We have previously developed a technique for using single-epoch multi-wavelength observations of anomalous lensed quasars to constrain the radius and radial profile of the background quasar accretion discs (@bfww08; @fbw09). In those analyses, we marginalised over the smooth matter percentage in the lens as a nuisance parameter. Here, we turn the problem around and instead marginalise over the quasar parameters to obtain constraints on the smooth matter percentage in the lens at the image positions.
Rough microlensing measurements of smooth matter percentages have been reported previously. Spectroscopy of SDSS J0924+0219 undertaken by @keeton+06 suggested a smooth matter percentage of 80 to 85 per cent in that lens at the location of the $D$ and $A$ images. Using X-ray monitoring of HE 1104+1805, @chartas+09 reported a smooth matter percentage of $\sim80$ per cent is favoured. @pooley+09 measured the smooth matter percentage in PG 1115+080 to be $\sim90$ per cent, using X-ray observations. @metal08 found a weak trend supporting this result. Most recently, @dai+09 favoured a smooth matter fraction of $\sim70$ per cent using X-ray and optical monitoring of RXJ 1131-1231. Microlensing analyses consistently predict a significant smooth matter percentage in the lensing galaxy at the position of anomalous images.
In this paper, we present constraints on the dark matter percentages in three lensing galaxies: MG 0414+0534, SDSS J0924+0219 and Q2237+0305. MG 0414+0534 and SDSS J0924+0219 are both lensed by early-type galaxies, and consist of close image pairs displaying a flux ratio anomaly. MG 0414+0534 is moderately anomalous, whereas SDSS J0924+0219 is the most anomalous lensed quasar currently known. Q2237+0305 differs from the previous sources in two key ways: it is lensed by a barred spiral galaxy, and it does not contain a close image pair. Nevertheless, it is known to be affected by microlensing (e.g. @irwin+89).
This paper is laid out as follows: in Section \[sec:obs\] we discuss the observational data on the three systems of interest. The simulation technique is briefly described in Section \[sec:sims\]. We present our results and discussion in Section \[sec:results\], and conclude in Section \[sec:conclusions\]. Throughout this paper we use a cosmology with $H_0=70\rm{kms^{-1}Mpc^{-1}}$, $\Omega_m=0.3$ and $\Omega_{\Lambda}=0.7$.
Observational data {#sec:obs}
==================
MG 0414+0534
------------
MG 0414+0534 was discovered by @hewitt+92. It consists of a background quasar at $z_s=2.64$ [@lejt95] and a foreground early-type lensing galaxy at $z_l=0.96$ [@tk99]. Four images of the quasar are observed, with the close image pair (images $A_1$ and $A_2$) displaying a flux ratio anomaly. This anomaly is weak in both the mid-infrared ($A_2/A_1 = 0.90 \pm 0.04$ on 2005 October 10, @minezaki+09) and the radio ($A_2/A_1 = 0.90 \pm 0.02$ on 1990 April 2, @kh93), but somewhat stronger in the optical ($A_2/A_1 = 0.45\pm 0.06$ on 1991 November 2-4, @sm93).
In our analysis, we used three epochs of multi-wavelength MG 0414+0534 observations, presented in Table \[0414obs\]. The first two epochs were archival HST data, obtained from the CASTLES Survey webpage[^1] [@fls97]. The third epoch was obtained by us using the Magellan 6.5-metre Baade telescope. These data were first presented in @bfww08.
[lrll]{} $H$ & 16500 & $0.67\pm0.05$ & 2007 November 3\
$J$ & 12500 & $0.60\pm0.2$ & 2007 November 3\
$z^\prime$ & 9134 & $0.34\pm0.1$ & 2007 November 3\
$i^\prime$ & 7625 & $0.26\pm0.1$ & 2007 November 3\
$r^\prime$ & 6231 & $0.21\pm0.1$ & 2007 November 3\
F205W & 20650 & $0.83\pm0.03$ & 1997 August 14\
F110W & 11250 & $0.64\pm0.04$ & 1997 August 14\
F814W & 7940 & $0.47\pm0.01$ & 1994 November 8\
F675W & 6714 & $0.40\pm0.01$ & 1994 November 8\
SDSS J0924+0219
---------------
SDSS J0924+0219 is the most anomalous lensed quasar currently known. The minimum image $A$ has been observed to be a factor of $\sim20$ brighter than the saddle point image $D$ in the optical [@keeton+06]. The quasar was discovered by @inada03 in Sloan Digital Sky Survey (SDSS) imaging, and consists of an early-type lensing galaxy at $z_l = 0.394$ [@eigenbrod06a] and a background quasar at $z_s=1.524$ [@inada03].
Again, we use three epochs of observational data. These are presented in Table \[0924obs\]. The 2008 March 21 data were obtained by us using the Magellan 6.5-metre Baade telescope [@fbw09]. The 2003 November 18-23 data were taken using the HST/NICMOS and WFPC2 instruments as part of the CASTLES Survey [@keeton+06]. The 2001 December 15 were obtained by @inada03 using the MagIC instrument on the Baade telescope, and re-reduced by us (details can be found in @fbw09).
[lrll]{} $H$ & $16500\pm1450$ & $0.23\pm0.05$ & 2008 March 21\
$J$ & $12500\pm800$ & $0.15\pm0.05$ & 2008 March 21\
$Y$ & $10200\pm500$ & $0.14\pm0.05$ & 2008 March 21\
$z^\prime$ & $9134\pm800$ & $0.19\pm0.10$ & 2008 March 21\
$i^\prime$ & $7625\pm650$ & $0.16\pm0.10$ & 2008 March 21\
$r^\prime$ & $6231\pm650$ & $0.10\pm0.10$ & 2008 March 21\
$g^\prime$ & $4750\pm750$ & $0.08\pm0.08$ & 2008 March 21\
$i^\prime$ & $7625\pm650$ & $0.08\pm0.05$ & 2001 December 15\
$r^\prime$ & $6231\pm650$ & $0.07\pm0.05$ & 2001 December 15\
$g^\prime$ & $4750\pm750$ & $0.06\pm0.05$ & 2001 December 15\
$u^\prime$ & $3540\pm310$ & $<$0.09 & 2001 December 15\
F160W & $15950\pm2000$ & $0.08\pm0.01$ & 2003 November 18\
F814W & $8269\pm850$ & $0.05\pm0.005$ & 2003 November 18-19\
F555W & $5202\pm600$ & $0.05\pm0.01$ & 2003 November 23\
Q2237+0305
----------
Q2237+0305 is perhaps the most well-studied gravitationally lensed quasar. It was discovered by @huchra+85, and consists of a lensing galaxy at $z_l=0.0394$ and a background quasar at $z_s=1.695$. The two previous sources had early type lensing galaxies; the lens in Q2237+0305 is a barred spiral. Near-perfect alignment between observer, lens and quasar results in four virtually symmetric images of the background source, located in the bulge of the lensing galaxy. The optical depth to stars is therefore quite high, making the system an excellent target for microlensing analyses. Typically, this is taken to mean the smooth matter percentage in microlensing simulations can be set to zero. We will test this assumption here.
Q2237+0305 also differs from the two previous sources in that it does not contain a close image pair displaying a flux ratio anomaly. Nevertheless, its images are known to vary in brightness due to microlensing. We choose to examine images $A$ and $B$ both because they are well modelled, and they are roughly equi-distant from the centre of the lensing galaxy.
Our observational data were obtained from @eigenbrod+08b, in which 43 epochs of Q2237+0305 spectroscopic data from the FORS1 spectrograph on the Very Large Telescope (VLT) were analysed. Eigenbrod and collaborators deconvolved their Q2237+0305 spectra into a broad emission line component, a continuum emission component, and an iron pseudo-continuum. The continuum emission was fit with a power-law of the form $f_\nu \propto \nu^{\alpha_\nu}$. The power-law fit was then split into six wavelength bands, each with a width of 250Å in the quasar rest frame, and integrated in each band. The result is pseudo-broadband photometry in six wavebands, with contamination from broad emission lines and the iron continuum removed. We selected two epochs from these data for our analysis, separated by approximately a year. The flux ratios are presented in Table \[2237obs\]. Following the @eigenbrod+08b numbering, the 2005 November 11 dataset is epoch number 17, and the 2006 November 10 dataset is epoch number 28.
[lrrll]{} 1 & $1625\pm125$ & $4379\pm337$ & $0.52\pm0.02$ & 2005 November 11\
2 & $1875\pm125$ & $5053\pm337$ & $0.51\pm0.02$ & 2005 November 11\
3 & $2125\pm125$ & $5727\pm337$ & $0.50\pm0.01$ & 2005 November 11\
4 & $2375\pm125$ & $6401\pm337$ & $0.49\pm0.01$ & 2005 November 11\
5 & $2625\pm125$ & $7074\pm337$ & $0.48\pm0.01$ & 2005 November 11\
6 & $2875\pm125$ & $7748\pm337$ & $0.47\pm0.01$ & 2005 November 11\
1 & $1625\pm125$ & $4379\pm337$ & $0.33\pm0.02$ & 2006 November 10\
2 & $1875\pm125$ & $5053\pm337$ & $0.34\pm0.02$ & 2006 November 10\
3 & $2125\pm125$ & $5725\pm337$ & $0.35\pm0.01$ & 2006 November 10\
4 & $2375\pm125$ & $6401\pm337$ & $0.36\pm0.02$ & 2006 November 10\
5 & $2625\pm125$ & $7074\pm337$ & $0.37\pm0.02$ & 2006 November 10\
6 & $2875\pm125$ & $7748\pm337$ & $0.37\pm0.01$ & 2006 November 10\
Microlensing simulations {#sec:sims}
========================
The simulation technique used in this work has been presented previously in @bfww08 and @fbw09. In those papers, we marginalised over the smooth matter percentage $s$ as a nuisance parameter. Here, we instead marginalise over the accretion disc radius $\sigma_0$ and the power-law index $\zeta$ relating the radius of the accretion disc to the observed wavelength.
Our microlensing simulations were conducted using an inverse ray-shooting technique (@krs86; @wpk90). The key parameters in these simulations are the convergence $\kappa_{tot}$, which is a measure of the focussing power of the lens, and shear $\gamma$, which is a measure of the distortion introduced by the lens. The lensing parameters used in this analysis are presented in Table \[tab:lensparams\].
[lcccc]{} MG 0414+0534 & $A_1$ & 0.472 & 0.488 & 24.2\
MG 0414+0534 & $A_2$ & 0.485 & 0.550 & -26.8\
SDSS J0924+0219 & $A$ & 0.502 & 0.458 & 26.2\
SDSS J0924+0219 & $D$ & 0.476 & 0.565 & -22.4\
Q2237+0305 & $A$ & 0.413 & 0.382 & 5.03\
Q2237+0305 & $B$ & 0.410 & 0.384 & 4.98\
The convergence can be split into two components $\kappa_{tot} = \kappa_* + \kappa_s$, where $\kappa_*$ describes a clumpy stellar component and $\kappa_s$ a smoothly distributed component. We define the smooth matter percentage $s$ to be the ratio of the continuously distributed component to the total convergence:
$$s = \kappa_s / \kappa_{tot}$$
We allowed the smooth matter percentage to vary from 0 to 99 per cent, in 10 per cent increments. The smooth matter percentage is thus relatively coarsely sampled; our simulations were optimised to probe the accretion disc sources in each system.
The microlenses in our simulations were drawn from a Salpeter mass function $dN/dM \propto M^{-2.35}$ with a mass range $M_{max}/M_{min} = 50$. Physical sizes are therefore scaled by the average Einstein Radius projected on to the source plane $\eta_0$, which varies from system to system. Magnification maps were generated covering an area of $24\eta_0 \times 24\eta_0$, with a resolution of $2048\times2048$ pixels. Ten maps were generated for each image and smooth matter percentage.
As discussed in @bfww08, we randomly selected source positions in each combination of magnification maps, to build up a simulated library of flux ratio curves as a function of wavelength. Comparing these with the observations allows us to construct a three dimensional likelihood distribution for the observed flux ratio spectrum $F^{obs}$ given three model parameters: the radius of the quasar source in the bluest filter $\sigma_0$, the power-law index relating observed wavelength to radius of the source $\zeta$, and the smooth matter percentage $s$. We can convert these likelihoods to an *a posteriori* probability distribution for the three model parameters given the observations using Bayes theorem:
$$\frac{\rm{d}^3P}{\rm{d}\sigma_0\rm{d}\zeta\rm{d}s} \propto L(F^{obs}|\sigma_0,\zeta,s) \frac{\rm{d}P_{prior}}{\rm{d}\sigma_0} \frac{\rm{d}P_{prior}}{\rm{d}\zeta} \frac{\rm{d}P_{prior}}{\rm{d}s}$$
Uniform priors were used for the two dimensionless quantities, smooth matter percentage $s$ and power-law index $\zeta$. A logarithmic prior was used for the radius $\sigma_0$. We note that this differs slightly from the analyses in @bfww08 and @fbw09, where a uniform prior was also used for $\sigma_0$. We will briefly discuss prior dependence in Section \[sec:results\]. We marginalise over the accretion disc parameters $\sigma_0$ and $\zeta$ to obtain a probability distribution for the smooth matter percentage $s$:
$$\frac{\rm{d}P}{\rm{d}s} = \int \int \frac{\rm{d}^3P}{\rm{d}\sigma_0\rm{d}\zeta\rm{d}s} \rm{d}\sigma_0 \rm{d}\zeta$$
Our analysis focusses on two lensed images in each system. By dealing with flux ratios between images only, we remove the intrinsic quasar flux from the problem, provided the difference in light travel time between images is short. We assume that the smooth matter percentages in the two lensed images are identical. This is reasonable for the two anomalous systems, MG 0414+0534 and SDSS J0924+0219, as the anomalous images lie very close to each other. In Q2237+0305, where the images are widely separated, we choose to analyse image $A$ and $B$ only as they are essentially equi-distant from the centre of the lensing galaxy, and do not lie atop any obvious spiral features.
The probability distributions we obtain for smooth matter percentage are presented for MG 0414+0534 (Figure \[0414smooth\]), SDSS J0924+0219 (Figure \[0924smooth\]) and Q2237+0305 (Figure \[2237smooth\]). The dashed histograms show the differential probability distributions, and the solid lines show the cumulative probability distributions.
Results and discussion {#sec:results}
======================
We obtain the following formal constraints on smooth matter percentage at the image positions in each system: $50^{+30}_{-40}$ per cent in MG 0414+0534, $80^{+10}_{-10}$ per cent in SDSS J0924+0219, and $\leq50$ in Q2237+0305 (68 per cent confidence limits are quoted). Our simulations are not optimised to probe smooth matter percentage; we sample smooth matter parameter space only sparsely, in order to reduce simulation time. These results should therefore be considered estimates, rather than exact measurements. Nevertheless, they provide an interesting, and currently poorly explored, measurement of smooth matter content within only a few effective radii of early type galaxies.
In MG 0414+0534, the differential probability distribution (Figure \[0414smooth\], dashed line) does not particularly favour any single smooth matter percentage. This leads to a quite broad formal constraint on the smooth matter percentage.
Conversely, our measured smooth matter percentage in SDSS J0924+0219 is high, as we would expect for such an anomalous system. There is one other measurement of smooth matter percentage of this system in the literature. @keeton+06 estimated it to be 80 to 85 per cent, based on the estimated size of the broad emission line region in the system. Our analysis is focussed on the accretion disc, and finds essentially identical results. We do use the @keeton+06 observational data in obtaining our constraints, however excluding it and working only with the Magellan data does not significantly alter our results.
As has been discussed earlier, the lensed images in Q2237+0305 lie in the bulge of the lensing galaxy. Stars are therefore expected to dominate the microlensing signal, rather than a smooth matter component. Indeed, we find a smooth matter percentage that is consistent with zero in this system. There is a peak in the differential probability distribution (Figure \[2237smooth\], dashed line) at $\sim20$ per cent. Though we are reluctant to suggest that this peak is real, we do note that such a feature could be evidence of additional absorbing features along the line of sight (see for example @foltz+92, who reported MgII absorption features in the spectrum of Q2237+0305 at a redshift of 0.97).
To confirm that our results are not dominated by the choice of prior probability for the radius of the background quasar accretion disc $\sigma_0$, we repeated our analysis using a uniform prior rather than a logarithmic prior. Within their errors, we found no significant variation in our constraints on smooth matter percentage in any of our systems.
We can perform a simple calculation to obtain a rough theoretical prediction of the percentage of dark matter we expect to see at the image positions in each source. The method is briefly described by @sw02, but will be repeated here for clarity. It is only applicable to MG 0414+0534 and SDSS J0924+0219 as it assumes an early type lensing galaxy. We begin by working out a stellar surface mass density $\Sigma_s$ at the image positions. To do this, we take the observed effective radius of the lensing galaxy $R_e$, and compare it with Figure 10 in @bernardi03 to obtain a $g$-band surface brightness in magnitudes per square arcsecond (assuming no evolution in early type galaxies between redshift $z=0$ and the redshift of the lensing galaxy).
@kauffmann03 provide a relationship between mass-to-light ratio $M/L$ and $g$-band magnitude derived from $10^5$ Sloan galaxies in their Figure 14. Using the $g$-band surface brightness obtained from @bernardi03, we can get a rough mass-to-light ratio for our lensing galaxy. We convert our surface brightness from magnitudes per square arcsecond into solar luminosities per square parsec using the following relationship:
$$S[\rm{mag}/\rm{arcsec}^2] = M_{\odot g} + 21.572 -2.5\rm{log}_{10}S[\rm{L}_\odot/\rm{pc}^2]$$
where S is the surface brightness and $M_{\odot g}=5.45$ is the solar magnitude in the $g$-band. Now that we have the lens galaxy surface brightness in solar magnitudes, we can use the mass-to-light ratio to convert into a stellar surface mass density in solar masses per square parsec. This is the stellar surface mass density at the effective radius $R_e$ – we use the deVaucoulers profile to extrapolate this value out to the Einstein Radius, which is the approximate location of the lensed images.
The final piece of information we need is the surface mass density at the image positions. The critical surface mass density $\Sigma_{cr}$ for lensing is obtained from the following relationship:
$$\Sigma_{cr} = \frac{c^2}{4\pi G} \frac{D_s}{D_d D_{ds}}$$
where $D_d$ is the angular diameter distance to the lens, $D_s$ is the angular diameter distance to the source, and $D_{ds}$ is the angular diameter distance between lens and source. The other symbols have their usual meanings. For an isothermal sphere, the surface mass density at the Einstein Radius is half the critical surface mass density $\Sigma_{cr}$. The smooth matter percentage $s$ is simply obtained as follows:
$$s = 1 - \frac{\Sigma_s}{0.5\Sigma{cr}}$$
MG 0414+0534 has a deVaucoulers effective radius $R_e = 0\farcs78$ [@ketal00], which gives a $g$-band magnitude of $\sim21.75$ and a mass-to-light ratio $M/L \sim4$. This gives a stellar surface mass density at the effective radius of $514 M_\odot/pc^2$. For an Einstein Radius of $1\farcs15$ [@trotter00], the stellar surface mass density on the Einstein Ring is smaller by a factor of 0.46, giving $\Sigma_s=236 M_\odot/pc^2$. For a source at $z=2.64$ and a lens at $z=0.96$, the critical surface mass density for lensing is $\Sigma_{cr} = 2189 M_\odot/pc^2$. This gives a theoretically predicted smooth matter percentage of 78 per cent. Note that this number differs slightly from the figure quoted in @sw02 as they were using preprints of @bernardi03 and @kauffmann03. This prediction is on the high side of our measured smooth matter percentage of $50^{+30}_{-40}$ per cent, although it is formally consistent.
SDSS J0924+0219 has two measured effective radii in the literature: $R_e=0\farcs31\pm0\farcs02$ [@metal06] and $R_e=0\farcs50\pm0\farcs05$ [@eigenbrod06a]. Both results were obtained using [*HST*]{} data. @metal06 fit a deVaucoulers profile to the lensing galaxy, whereas @eigenbrod06a chose a two-dimensional exponential disk. Since an exponential profile is shallower than a deVaucoulers profile, it would tend to give a larger effective radius. This may allow the two effective radii to be reconciled, however we will deal with each case separately. The system has a lensing galaxy at $z=0.39$ and a source at $z=1.524$, giving a critical surface mass density of $\Sigma_{cr} = 2323 M_\odot/pc^2$.
For an effective radius $R_e=0\farcs50$ we expect a $g$-band surface brightness of $\sim20.5\rm{mag}/\rm{arcsec}^2$, and a mass-to-light ratio $M/L=5.6$. At the effective radius, the stellar surface mass density is $2285 M_\odot/pc^2$. For an Einstein Radius of $0\farcs85$ [@inada03] and a deVaucoulers profile, the stellar surface mass density at the Einstein Radius is lower by a factor of 0.35, giving $\Sigma_s = 791 M_\odot/pc^2$. Thus, we predict a smooth matter percentage of 32 per cent. This is significantly lower than our $1\sigma$ measured value of $80^{+10}_{-10}$. Even at the 95 per cent level, our measured smooth matter percentage is not consistent with the prediction of this rough calculation.
The situation is somewhat different for an effective radius $R_e = 0\farcs31$. Here, an early type galaxy with no evolution between $z=0$ and $z=0.39$ should have a $g$-band surface brightness of $\sim19.75\rm{mag}/\rm{arcsec}^2$ and a mass-to-light ratio $M/L=5$. This gives a stellar surface mass density of $4052 M_\odot/pc^2$ at the effective radius. Again assuming a deVaucoulers profile, the stellar surface mass density is reduced by a factor of 0.13 at the Einstein Radius of $0\farcs85$, giving $539 M_\odot/pc^2$. We therefore expect 46 per cent of the mass to be in stars, and 54 per cent in smooth matter. This is consistent with our measured smooth matter percentage at the 95 per cent level ($80^{+10}_{-30}$).
In Figure \[smooth\_compare\] we show a comparison between the results of this rough theoretical calculation and our microlensing measurements for MG 0414+0534 (circle) and SDSS J0924+0219 (square). No errors are provided for the rough theoretical calculations, although we emphasise that they should only be considered estimates. With only two data points, there does not seem to be any systematic difference between the microlensing measurements and the rough calculation.
It is, however, not correct to connect smooth matter percentage in our simulations directly with dark matter content in the lens. Intervening systems (which may not be detected otherwise) can contribute smooth matter to the surface mass density, as can gas and dust in the lens. As the lensing galaxies in both MG 0414+0534 and SDSS J0924+0219 are early type galaxies, we would expect this contribution to be very small. More importantly, @lg06 showed that small compact masses can mimic a smooth matter component. This was first demonstrated analytically in @dr81. For a source with a radius $0.1\eta_0$, @lg06 find that compact masses smaller than $\sim0.01\rm{M}_\odot$ can mimic a smoothly distributed mass component. This may help to explain why our measured microlensing smooth matter percentage in SDSS J0924+0219 appears to be slightly higher than we would expect from a simple deVaucoulers galaxy profile in an isothermal dark matter profile.
We note that the three lensing systems analysed in this paper do not represent a fair statistical sample of gravitationally lensed quasars. The single-epoch imaging technique used here requires short time delays between images, and so we preferentially select lenses with close image pairs (or high symmetry in the case of Q2237+0305). We have chosen close image pairs that exhibit strongly anomalous flux ratios for our analysis. Since a priori, anomalous flux ratios are driven by a smooth matter fraction [@sw02], this selection biases our estimates towards larger smooth dark matter fractions. Thus our results cannot be used to make global statements regarding smooth dark matter in galaxies. However eight out of ten close image systems display anomalous flux ratios [@pooley+07], and so we expect the smooth matter percentage at those image positions to be high. This indicates that any bias in our analysis of individual systems would be relatively minor. In the future our analysis could be extended to a statistical sample through the use of monitoring data and the light curve technique discussed in @k04.
Conclusions {#sec:conclusions}
===========
We have presented estimates of the smooth matter percentage in three lensing galaxies along the line of sight to the lensed images. We find a smooth matter percentage of $50^{+30}_{-40}$ in MG 0414+0534, $80^{+10}_{-10}$ in SDSS J0924+0219, and $\leq50$ in Q2237+0305, with 68 per cent confidence. In the two systems where the lensed images lie in the outer regions of the lensing galaxies (5 to 10 kpc from their centres), these measurements are inconsistent with zero smooth dark matter in the lensing galaxies. In Q2237+0305, where the lensed images lie in the central bulge of the lensing galaxy and so stars are expected to dominate the microlensing signal, our result is consistent with zero per cent smooth matter, as expected.
These measurements were obtained using a single-epoch imaging technique that is free from the need for long-term monitoring campaigns, which are required for detailed analysis of the lens mass profile. Our results also do not depend upon the typically unknown velocities of the stars in the lensing galaxy, and of the lensing galaxy and background source themselves, which enter into analyses of microlensing lightcurves. It is however only appropriate for systems with time delays between images of less than a day, so we can ensure we are observing the background source in the same state along each line of sight. It also does not provide us with any information on the slope of the dark matter profile; for that, stellar velocity dispersions in the lensing galaxy are required.
Nevertheless, our technique does provide an observationally inexpensive method for estimating the dark matter fraction in lensing galaxies at the location of lensed images. Gravitational lensing remains the only method for directly probing the dark matter content of galaxies.
NFB acknowledges the support of an Australian Postgraduate Award. DJEF acknowledges the support of a Magellan Fellowship from Astronomy Australia Limited. We are indebted to Joachim Wambsganss for the use of his inverse ray-shooting code. We thank the referee for comments which helped improve the final version of this manuscript.
[*Facilities:*]{} , , .
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[^1]: http://cfa-www.harvard.edu/castles/
|
---
author:
- |
Ivan Naumkin[^1] and Ricardo Weder[^2]\
Departamento de Física Matemática,\
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas.\
Universidad Nacional Autónoma de México.\
Apartado Postal 20-726, México DF 01000, México.
title: 'High-energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and applications[^3]'
---
[**Abstract**]{}
We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short-range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high-energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy $E.$ Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in $L^{2}\left( \Omega\right) ,$ where $\Omega$ is any connected bounded open set in $\mathbb{R}^{3}$ with smooth boundary, and we show that if we know an electric potential and a magnetic field for $\mathbb{R}^{3}\diagdown\Omega,$ then the scattering amplitude, given for some energy $E$, uniquely determines these electric potential and magnetic field everywhere in $\mathbb{R}^{3}$. Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some $E$, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge-conjugation and time-reversal transformations for the Dirac operator.
Introduction.
=============
The free Dirac operator $H_{0}$ is given by $$H_{0}=-i\alpha\cdot\nabla+m\alpha_{4}, \label{basicnotions1}$$ where $m$ is the mass of the particle, $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ and $\alpha_{j},$ $j=1,2,3,4,$ are $4\times4$ Hermitian matrices that satisfy the relation:$$\alpha_{j}\alpha_{k}+\alpha_{k}\alpha_{j}=2\delta_{jk},\text{ }j,k=1,2,3,4,
\label{basicnotions2}$$ where $\delta_{jk}$ denotes the Kronecker symbol. The standard choice of $\alpha_{j}$ is ([@16]): $$\alpha_{j}=\begin{pmatrix}
0 & \sigma_{j}\\
\sigma_{j} & 0
\end{pmatrix}
,\text{ \ }j\leq3,\text{ \ \ \ \ }\alpha_{4}=\begin{pmatrix}
I_{2} & 0\\
0 & -I_{2}\end{pmatrix}
=\beta,$$ where $I_{n}$ is $n\times n$ unit matrix and $$\sigma_{1}=\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}
,\sigma_{2}=\begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix}
,\sigma_{3}=\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}
\label{basicnotions22}$$ are the Pauli matrices.
The perturbed Dirac operator is defined by $$H=H_{0}+\mathbf{V}. \label{basicnotions6}$$ Here the potential $\mathbf{V}\left( x\right) $ is an Hermitian $4\times4$ matrix valued function defined for $x\in\mathbb{R}^{3}$.
In this paper we consider the scattering theory for the Dirac operator with electromagnetic potential. This problem has been extensively studied. Results on existence and completeness of the wave operators have been proved for a wide class of perturbations, including the long-range case, by the stationary method in ([@60],[@61],[@9],[@33], and the references therein), and by the time-dependent method in [@49] and [@16]. Also, in [@33], radiation estimates and asymptotics for large time of solutions of the time-dependent Dirac equation are obtained. The study of the point spectrum was made in ([@12],[@54],[@50],[@11], and the references quoted there). The limiting absorption principle was proved in [@51],[@14] and [@13]. Meromorphic extensions and resonances of the scattering matrix were treated in [@63] and [@14]. The high-energy behavior of the resolvent and the scattering amplitude was studied by the stationary method in [@15], and by the time-dependent method of [@62], in [@52] and [@57]. High-energy and low-energy behavior of the solutions of the Dirac equation, as well as Levinson theorem, were obtained in [@18] for spherically symmetric potentials. Holomorphy of the scattering matrix at fixed energy with respect to $c^{-2}$ for abstract Dirac operators was studied in [@59]. Finally, a detailed study of the Dirac equation is made in the monographs of Thaller [@16] and, Balinsky and Evans [@46].
The inverse scattering problem consists in establishing a relation between the scattering data (for example the scattering amplitude) and the potential. The most complete and difficult problem is to find a one-to-one correspondence between the scattering amplitude and the potential; that is to say, to give necessary and sufficient conditions on a scattering amplitude, such that it is associated to the scattering matrix $S\left( E\right) $ of a unique potential $V(x)$ in a given class. This is the characterization problem (see [@53],[@56] and [@55] for a discussion of this problem for the Schrödinger equation).
Another problem is uniqueness and reconstruction of the potential from the scattering data. We note that this problem is not well defined in the electromagnetic case, because the scattering amplitude is invariant under gauge transformation of the magnetic potential and thus, the problem cannot be solved in a unique way. Nevertheless, we can ask about uniqueness and reconstruction of the electric potential $V\left( x\right) $ and the magnetic field $B\left( x\right) =\operatorname*{rot}A\left( x\right) $, associated to a magnetic potential $A\left( x\right) $.
The uniqueness and reconstruction problem has different settings. One of them is uniqueness and reconstruction from the high-energy limit of the scattering amplitude. Using the stationary approach Ito [@15] solved this problem for electromagnetic potentials of the form$$\mathbf{V}\left( x\right) =\begin{pmatrix}
V & \sigma\cdot A\\
\sigma\cdot A & V
\end{pmatrix}
, \label{intro4}$$ decaying faster then $\left\vert x\right\vert ^{-3}$ at infinity. Using an adaptation to Dirac operators of the Green function of Faddeev ([@53]) and the Green function introduced by Eskin and Ralston ([@64]) in the inverse scattering theory for Schrödinger operators, the same problem was solved by Isozaki [@22], for electric potentials of the form$$\mathbf{V}\left( x\right) =\begin{pmatrix}
V_{+} & 0\\
0 & V_{-}\end{pmatrix}
, \label{intro1}$$ decaying faster than $\left\vert x\right\vert ^{-2}$. The time-dependent method ([@62]) was used by Jung [@52] to solve the uniqueness and reconstruction problem for continuous, Hermitian matrix valued potentials $\mathbf{V}\left( x\right) ,$ with integrable decay, which satisfy $$\lbrack\left( \mathbf{V}\left( x_{1}\right) +\left( \alpha\cdot
\omega\right) \mathbf{V}\left( x_{1}\right) \left( \alpha\cdot
\omega\right) \right) ,\left( \mathbf{V}\left( x_{2}\right) +\left(
\alpha\cdot\omega\right) \mathbf{V}\left( x_{2}\right) \left( \alpha
\cdot\omega\right) \right) ]=0, \label{intro7}$$ for all $x_{1},x_{2}\in\mathbb{R}^{n}.$ Also using the time-dependent method Ito [@57] gave conditions on a time-dependent potential of the form (\[intro4\]), so that it can be reconstructed from the scattering operator. In particular, if a short-range potential is time-independent, then the electric potential $V\left( x\right) $ and the magnetic field $B\left(
x\right) $ can be completely reconstructed from the scattering operator.
A similar problem is uniqueness and reconstruction of the electric potential and the magnetic field from the scattering amplitude at a fixed energy (ISPFE). Isozaki [@22] solved the ISPFE problem for potentials of the form (\[intro1\]) and decaying exponentially. A problem related to the ISPFE is the inverse boundary value problem (IBVP), where the Dirac operator at a fixed energy is considered in a bounded domain $\Omega$ in $\mathbb{R}^{3},$ and the uniqueness and reconstruction of the electric potential and the magnetic field from the Dirichlet-to-Dirichlet (D-D) map (see (\[completeness9\]), Section 8) is studied. Tsuchida [@20] solved the IBVP problem for potentials $\mathbf{V}$ of the form$$\mathbf{V}\left( x\right) =\begin{pmatrix}
V_{+} & \sigma\cdot A\\
\sigma\cdot A & V_{-}\end{pmatrix}
, \label{intro3}$$ and showed that the D-D map determines uniquely small potentials $V_{+},V_{-}$ and $B=\operatorname*{rot}A.$ Nakamura and Tsuchida [@19] solved the IBVP for potentials $\mathbf{V}\in C^{\infty}\left( \Omega\right) ,$ of the form (\[intro3\]). Moreover, establishing a relation between the D-D map and the scattering amplitude, they showed the uniqueness of the ISPFE problem for potentials $\mathbf{V}\in C^{\infty}\left( \mathbb{R}^{3}\right) $ of the form (\[intro3\]) with compact support. Goto [@23] solved the ISPFE problem for exponentially-decaying potentials of the form (\[intro4\]), and Li [@28] considered both the IBVP and the ISPFE problems for potentials that are general Hermitian matrices with compact support.
We note that in case of the Schrödinger equation with general short-range potentials the scattering matrix at a fixed positive energy does not uniquely determine the potential. Indeed, Chadan and Sabatier (see pag 207 of [@24]) give examples of non-trivial radial oscillating potentials decaying as $\left\vert x\right\vert ^{-3/2}$ at infinity, such that the corresponding scattering amplitude is identically zero at some positive energy. Thus, for the Schrödinger equation with general short-range potentials, it is necessary to impose extra conditions for the uniqueness of the ISPFE problem. Weder and Yafaev [@25] considered the ISPFE problem for the Schrödinger equation with general short-range potentials (see also [@65] for the case of long-range potentials) which are asymptotic sums of homogeneous terms (see (\[basicnotions16\]) and (\[basicnotions17\])). They used an explicit formula for the singularities of the scattering amplitude, obtained by Yafaev in [@27] and [@30], to show that the asymptotics of the electric potential and the magnetic field can be recovered from the diagonal singularities of the scattering amplitude (see also [@68]-[@67] and [@69]). Moreover, Weder in [@58] and [@26] proved that if two short-range electromagnetic potentials $\left( V_{1},A_{1}\right) $ and $\left( V_{2},A_{2}\right) $ in $\mathbb{R}^{n}$, $n\geq3$ have the same scattering matrix at a fixed positive energy and if the electric potentials $V_{j}$ and the magnetic fields $B_{j}=\operatorname*{rot}A_{j,}$ $j=1,2,$ coincide outside of some ball they necessarily coincide everywhere. The combination of this uniqueness result and the result of Weder and Yafaev [@25] implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field ([@26]). We proceed similarly in the Dirac case.
We follow the methods of the works of Weder [@58], [@26], Yafaev [@27],[@30], and Weder and Yafaev [@25], [@65] for the Schrödinger equation. For the Schrödinger equation with short-range potentials, the approximate solutions are given by $u_{N}\left( x,\xi\right)
=e^{i\left\langle x,\xi\right\rangle }+e^{i\left\langle x,\xi\right\rangle
}a_{N}\left( x,\xi\right) ,$ where $a$ solves the transport equation (see [@27]). These solutions are constructed in such a way that outside some conical neighborhood of the direction $x=\pm\xi,$ the remainder satisfies the estimate $$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}r_{N}\left(
x,\xi\right) \right\vert \leq C_{\alpha,\beta}\left\langle x\right\rangle
^{-p\left( N\right) }\left\langle \xi\right\rangle ^{-q\left( N\right)
},\text{ where }p\left( N\right) ,q\left( N\right) \rightarrow
+\infty,\text{ as }N\rightarrow+\infty. \label{intro5}$$ In the case of the Schrödinger equation with long-range potentials, the approximate solutions are of the form $u_{N}\left( x,\xi\right)
=e^{i\left\langle x,\xi\right\rangle +i\phi\left( x,\xi\right) }\left(
1+a_{N}\left( x,\xi\right) \right) ,$ where $\phi\left( x,\xi\right) $ solves the eikonal equation and $a\left(
x,\xi\right) $ is the solution of the transport equation ([@30]). Again, $u_{N}\left(
x,\xi\right) $ is constructed so that the remainder satisfies the estimate (\[intro5\]).
We observe that Gâtel and Yafaev [@33] constructed approximate solutions for the Dirac equation with long-range potentials of the form (\[intro4\]) satisfying for all $\alpha$ the estimate $$\left\vert \partial_{x}^{\alpha}V\left( x\right) \right\vert +\left\vert
\partial_{x}^{\alpha}A\left( x\right) \right\vert \leq C_{\alpha
}\left\langle x\right\rangle ^{-\rho-\left\vert \alpha\right\vert },\text{ for
some }\rho\in(0,1). \label{intro6}$$ These solutions are given by $e^{i\left\langle x,\xi\right\rangle
+i\Phi\left( x,\xi;E\right) }p\left( x,\xi;E\right) ,$ where the phase $\Phi\left( x,\xi;E\right) $ solves the eikonal equation and satisfies, for all $\alpha$ and $\beta,$ the estimate $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta
}\Phi\left( x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left\langle
x\right\rangle ^{1-\rho-\left\vert \alpha\right\vert }$. The function $p\left( x,\xi;E\right) $ satisfies a transport equation and it results to be explicit and exact. It satisfies $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}\left( p\left( x,\xi;E\right) -P_{\omega}\left(
E\right) \right) \right\vert \leq C_{\alpha,\beta}\left\langle
x\right\rangle ^{-\rho-\left\vert \alpha\right\vert },$ for all $\alpha$ and $\beta,$ where $P_{\omega}\left( E\right) $ is the amplitude in the plane wave solution $P_{\omega}\left( E\right) e^{i\sqrt{E^{2}-m^{2}}\left\langle
\omega,x\right\rangle }$ of energy $E$ and momentum in direction $\omega$ and it is defined by (\[basicnotions43\]) below. Then the remainder satisfies, outside a conical neighborhood of the direction $x=\pm\xi,$ the estimate $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}r_{N}\left( x,\xi;E\right) \right\vert \leq
C_{\alpha,\beta}\left\langle x\right\rangle ^{-1-\varepsilon},$ for some $\varepsilon>0,$ for all $\alpha$ and $\beta.$ Using these solutions, they construct special identities $J_{\pm},$ in order to prove the existence and completeness of the wave operators and to obtain the asymptotics for large time of the solutions of the time-dependent Dirac equation.
Even in the case of the Dirac equation with short-range electric potentials, it is not enough to consider only the transport equation, in order to obtain the estimate (\[intro5\]). Thus, we also need to consider the eikonal equation. It also turns out that we need to decompose the transport equation in two parts, one for the positive energies and another for the negative energies.
We consider potentials of the form (\[intro4\]) satisfying the estimate (\[intro6\]), with $\rho>1$. We take the approximate solutions of the form $e^{i\left\langle x,\xi\right\rangle +i\Phi\left( x,\xi;E\right) }w_{N}\left( x,\xi;E\right) ,$ where $\Phi\left( x,\xi;E\right) $ solves the eikonal equation and satisfies the estimate $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}\Phi\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\left( \rho-1\right) -\left\vert \alpha\right\vert
}\left\vert \xi\right\vert ^{-\left\vert \beta\right\vert }.$ The function $w_{N}$ decomposes in the sum $\left( w_{1}\right) _{N}+\left(
w_{2}\right) _{N},$ where the functions $\left( w_{1}\right) _{N}$ $\ $and $\left( w_{2}\right) _{N}$ satisfy two different transport equations and the estimate $\left\vert
\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\left( w_{j}\right) _{N}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert },$ for $j=1,2.$ This construction of the approximate solutions to the Dirac equation assures that the remainder $r_{N}\left(
x,\xi;E\right) $ satisfies the estimate (\[intro5\]). We note that we do not reduce the Dirac equation to the Schrödinger type equation. We deal directly with the Dirac equation to obtain the approximate solutions.
Following the idea of [@27] and [@30], and using our approximate solutions to the Dirac equation, we construct special identities $J_{\pm}\ $and use the stationary equation for the scattering matrix to find an explicit formula for the singularities of the scattering amplitude in a neighborhood of the diagonal for potentials satisfying the estimate (\[intro6\]) with $\rho>1$. Using this formula we express the leading singularity of the scattering amplitude in terms of the Fourier transform of the electric and the magnetic potentials. Furthermore, we obtain an error bound for the difference between the scattering amplitude and the leading singularity. We also show that this error bound is optimal in the case of electric and magnetic potentials that are homogeneous outside a sphere. We use the formula for the singularities of the scattering amplitude also to study the high-energy limit for the scattering matrix. Moreover, we uniquely reconstruct the electric potential and the magnetic field from the high-energy limit of the scattering amplitude. We recall that this result was proved in [@15], by studying the high-energy behavior of the resolvent, for smooth potentials which satisfy (\[intro6\]), for $\left\vert \alpha\right\vert
\leq d,$ $d\geq2,$ with $\rho>3.$ Also we show that for potentials, homogeneous and non-trivial outside of a sphere, satisfying the estimate (\[intro6\]) with $1<\rho\leq2,$ the total scattering cross-section is infinite.
If the scattering amplitude is given for some energy $E$, then, in particular we know all its singularities for this energy. As in [@25], assuming that the electric potential $V$ and the magnetic field $B=\operatorname*{rot}A$ satisfy estimates (\[eig32\]) and (\[basicnotions18\]), for all $d,$ respectively, and they are asymptotic to a sum of homogeneous terms we uniquely recover these asymptotics from the scattering amplitude singularities.
On the other hand, inspired by [@58],[@26], we consider a special set of solutions to the Dirac equation for fixed energy, called averaged scattering solutions and show that for potentials $\mathbf{V,}$ satisfying Condition \[basicnotions26\] and estimate (\[completeness2\]), for some connected bounded open set $\Omega,$ with smooth boundary, this set of solutions is strongly dense in the set of all solutions to the Dirac equation that are in $L^{2}\left( \Omega\right) $. This fact allows us to prove that if $\mathbf{V}_{j},$ $j=1,2,$ are of the form (\[intro3\]), where $V_{\pm}^{\left( j\right) }\in C^{\infty}\left(
\mathbb{R}^{3}\right) ,$ $j=1,2,$ satisfying (\[completeness2\]) and $B_{j}\in C^{\infty}\left( \mathbb{R}^{3}\right) ,$ $j=1,2,$ satisfying (\[basicnotions18\]), for $d=1,$ are such that $V_{\pm}^{\left( 1\right)
}=V_{\pm}^{\left( 2\right) },$ and $B_{1}=B_{2},$ for $x$ outside some connected bounded open set $\Omega^{\prime},$ with smooth boundary, and the scattering amplitudes for $\mathbf{V}_{1}$ and $\mathbf{V}_{2}$ coincide for some energy, then $V_{\pm}^{\left( 1\right) }=V_{\pm}^{\left( 2\right) }$ and $B_{1}=B_{2}$ for all $x$.
Finally, if the asymptotic decomposition of the electric potential $V\in C^{\infty}\left( \mathbb{R}^{3}\right) $ and the magnetic field $B\in C^{\infty}\left( \mathbb{R}^{3}\right) $ actually converge, respectively, to $V$ and $B,$ outside some bounded set, then combining both results, we show that the scattering matrix given for some fixed energy uniquely determines the electric potential $V$ and the magnetic field $B$. The paper is organized as follows. In Section 2 we give some known results about scattering theory for the Dirac operators. In Section 3, we define the scattering solutions, we calculate their asymptotics for large $x,$ and we give a relation between the coefficient of the term, decaying as $\frac
{1}{\left\vert x\right\vert },$ for $x\rightarrow\infty,$ in this asymptotic expansion and the kernel of the scattering matrix. In Section 4, we present symmetries of the kernel of the scattering matrix, that follow from the time-reversal, parity and charge conjugation transformations of the Dirac equation. These symmetries, interesting on their own, can be useful in a study of the characterization problem. In Section 5, we construct approximate generalized eigenfunctions for the Dirac equation that we use in Section 6 to obtain an explicit formula for the singularities of the scattering amplitude. Applications of the formula for the singularities of the kernel of the scattering matrix are presented in Section 7. The result on the completeness of the averaged scattering solutions is given in Section 8. In Section 9 we present the results on the uniqueness of ISPFE problem.
**Acknowledgement** *We thank Osanobu Yamada for informing us of reference [@45].*
Basic notions.
==============
The free Dirac operator $H_{0}$ (\[basicnotions1\]) is a self-adjoint operator on $L^{2}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) $ with domain $D\left( H_{0}\right) =\mathcal{H}^{1}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) ,$ the Sobolev space of order $1$ ([@16]). When there is no place of confusion we will write $L^{2}$ and$\ \mathcal{H}^{1}$ to simplify the notation. We can diagonalize $H_{0}$ by the Fourier transform $\mathcal{F}$ given by $\left( \mathcal{F}f\right) \left( x\right)
=\left( 2\pi\right) ^{-3/2}\int_{\mathbb{R}^{3}}e^{-i\left\langle
x,\xi\right\rangle }f\left( \xi\right) d\xi.$ Actually, $\mathcal{F}H_{0}\mathcal{F}^{\ast}$ acts as multiplication by the matrix $h_{0}\left(
\xi\right) =\alpha\cdot\xi+m\beta.$ This matrix has two eigenvalues $E=\pm\sqrt{\xi^{2}+m^{2}}$ and each eigenspace $X^{\pm}\left( \xi\right) $ is a two-dimensional subspace of $\mathbb{C}^{4}.$ The orthogonal projections onto these eigenspaces are given by (see [@16], page 9) $P^{\pm}\left(
\xi\right) :=\frac{1}{2}\left( I_{4}\pm\left( \xi^{2}+m^{2}\right)
^{-1/2}\left( \alpha\cdot\xi+m\beta\right) \right) .$ The spectrum of $H_{0}$ is purely absolutely continuous and it is given by $\sigma\left(
H_{0}\right) =\sigma_{ac}\left( H_{0}\right) =(-\infty,-m]\cup\lbrack
m,\infty).$
Let us introduce the weighted $L^{2}$ spaces for $s\in\mathbb{R},$ $L_{s}^{2}:=\{f:\left\langle x\right\rangle ^{s}f\left( x\right) \in L^{2}\},$ $\left\Vert f\right\Vert _{L_{s}^{2}}:=\left\Vert \left\langle x\right\rangle
^{s}f\left( x\right) \right\Vert _{L^{2}},$ where $\left\langle
x\right\rangle =\left( 1+\left\vert x\right\vert ^{2}\right) ^{1/2}.$ Moreover, for any $\alpha,s\in\mathbb{R}$ we define $\mathcal{H}^{\alpha
,s}:=\{f:\left\langle x\right\rangle ^{s}f\left( x\right) \in\mathcal{H}^{\alpha}\},$ $\left\Vert f\right\Vert _{\mathcal{H}^{\alpha,s}}:=\left\Vert
\left\langle x\right\rangle ^{s}f\left( x\right) \right\Vert _{\mathcal{H}^{\alpha}},$ where $\left\Vert f\left( x\right) \right\Vert _{\mathcal{H}^{\alpha}}=\left( \int_{\mathbb{R}^{3}}\left\langle \xi\right\rangle
^{2\alpha}\left\vert \hat{f}\left( \xi\right) \right\vert ^{2}d\xi\right)
^{1/2}.$
Let us now consider the perturbed Dirac operator $H$, given by (\[basicnotions6\]). We make the following assumption on the Hermitian $4\times4$ matrix valued potential $\mathbf{V,}$ defined for $x\in
\mathbb{R}^{3}$:
\[basicnotions26\]For some $s_{0}>1/2,$ $\left\langle x\right\rangle
^{2s_{0}}\mathbf{V}$ is a compact operator from $\mathcal{H}^{1}$ to $L^{2}.$
The assumptions on a potential $\mathbf{V,}$ assuring Condition \[basicnotions26\] are well known (see, for example, [@81]). In particular, Condition \[basicnotions26\] for $\mathbf{V}$ holds, if for some $\varepsilon>0,$ $\sup_{x\in\mathbb{R}^{3}}\int_{\left\vert x-y\right\vert
\leq1}\left\vert \left\langle y\right\rangle ^{2s_{0}}\mathbf{V}\left(
y\right) \right\vert ^{3+\varepsilon}dy<\infty$ and $\int_{\left\vert
x-y\right\vert \leq1}\left\vert \left\langle y\right\rangle ^{2s_{0}}\mathbf{V}\left( y\right) \right\vert ^{3+\varepsilon}dy\rightarrow0,$ as $\left\vert x\right\vert \rightarrow0$ (see Theorem 9.6, Chapter 6, of [@81]). Of course, the last two relations are true if the following estimate is valid $$\left\vert \mathbf{V}\left( x\right) \right\vert \leq C\left\langle
x\right\rangle ^{-\rho},\text{ for some }\rho>1. \label{basicnotions46}$$
Since $\mathbf{V}$ is an Hermitian $4\times4$ matrix valued potential $\mathbf{V,}$ Condition \[basicnotions26\] implies assumptions (A$_{1}$)-(A$_{3}$) of [@14]. Thus, under Condition \[basicnotions26\] $H$ is a self-adjoint operator on $D\left( H\right) =\mathcal{H}^{1}$ and the essential spectrum $\sigma_{ess}\left( H\right) =\sigma\left( H_{0}\right)
$. The wave operators (WO), defined as the following strong limit$$W_{\pm}\left( H,H_{0}\right) :=s-\lim_{t\rightarrow\pm\infty}e^{iHt}e^{-iH_{0}t}, \label{basicnotions9}$$ exist and are complete, i.e., $\operatorname*{Range}W_{\pm}=\mathcal{H}_{ac}$ (the subspace of absolutely continuity of $H$) and the singular continuous spectrum of $H$ is absent.
We recall that the study about the absence of eigenvalues embedded in the absolutely continuous spectrum was made in [@12],[@54],[@50],[@11], and the references quoted there. For example, it follows from Theorem 6 of [@12] that $\{(-\infty,-m)\cup(m,\infty)\}\cap\sigma
_{p}\left( H\right) =\varnothing$, if $\mathbf{V}\in L_{\operatorname*{loc}}^{5}\left( \mathbb{R}^{3}\right) \ $satisfies the following: for any $\varepsilon>0,$ there exists $R\left( \varepsilon\right) >0,$ such that for any $u\in\mathcal{H}_{\operatorname*{c}}^{1}\left( \left\vert x\right\vert
>R\left( \varepsilon\right) \right) $ (the set of functions from $\mathcal{H}^{1}\left( \left\vert x\right\vert >R\left( \varepsilon\right)
\right) $ with compact support in $\left\vert x\right\vert >R\left(
\varepsilon\right) $) $$\left\Vert \left\vert x\right\vert \mathbf{V}u\right\Vert _{L^{2}}\leq\varepsilon\left\Vert u\right\Vert _{\mathcal{H}^{1}},
\label{basicnotions47}$$ and, moreover,$$\left\Vert \left\vert x\right\vert ^{1/2}\mathbf{V}\right\Vert _{L^{\infty
}\left( \left\vert x\right\vert \geq R\right) }<\infty,\text{ for some }R>0.
\label{basicnotions48}$$ Note that $\mathbf{V}$ satisfies these relations, if the estimate (\[basicnotions46\]) holds.
From the existence of the WO it follows that $HW_{\pm}=W_{\pm}H_{0}$ (intertwining relations). The scattering operator, defined by$$\mathbf{S=S}\left( H,H_{0}\right) :=W_{+}^{\ast}W_{-},
\label{basicnotions10}$$ commutes with $H_{0}$ and it is unitary.
Let $H_{0S}:=-\triangle$ be the free Schrödinger operator in $L^{2}\left(
\mathbb{R}^{3};\mathbb{C}^{4}\right) .$ The limiting absorption principle (LAP) is the following statement. For $z$ in the resolvent set of $H_{0S}$ let $R_{0S}\left( z\right) :=\left( H_{0S}-z\right) ^{-1}$ be the resolvent. The limits $R_{0S}\left( \lambda\pm i0\right) =\lim_{\varepsilon
\rightarrow+0}R_{0S}\left( \lambda\pm i\varepsilon\right) ,$ ($\varepsilon
\rightarrow+0$ means $\varepsilon\rightarrow0$ with $\varepsilon>0$) exist in the uniform operator topology in $\mathcal{B}\left( L_{s}^{2},\mathcal{H}^{\alpha,-s}\right) ,$ $s>1/2,$ $\left\vert \alpha\right\vert \leq2$ ([@4],[@71],[@37],[@8]) and, moreover, $\left\Vert
R_{0S}\left( \lambda\pm i0\right) f\right\Vert _{\mathcal{H}^{\alpha,-s}}\leq C_{s,\delta}\lambda^{-\left( 1-\left\vert \alpha\right\vert \right)
/2}\left\Vert f\right\Vert _{L_{s}^{2}},$ for $\lambda\in\lbrack\delta
,\infty),$ $\delta>0$. Here for any pair of Banach spaces $X,Y,$ $\mathcal{B}\left( X,Y\right) $ denotes the Banach space of all bounded operators from $X$ into $Y.$ The functions $R_{0S}^{\pm}\left( \lambda
\right) ,$ given by $R_{0S}\left( \lambda\right) $ if $\operatorname{Im}\lambda\neq0,$ and $R_{0S}\left( \lambda\pm i0\right) ,$ if $\lambda
\in(0,\infty),$ are defined for $\lambda\in\mathbb{C}^{\pm}\cup\left(
0,\infty\right) $ ($\mathbb{C}^{\pm}$ denotes, respectively, the upper, lower, open complex half-plane) with values in $\mathcal{B}\left( L_{s}^{2},\mathcal{H}^{\alpha,-s}\right) $ and they are analytic for $\operatorname{Im}\lambda\neq0$ and locally Hölder continuous for $\lambda\in(0,\infty)$ with exponent $\vartheta$ satisfying the estimates $0<\vartheta\leq s-1/2$ and $\vartheta<1.$
For $z$ in the resolvent set of $H_{0}$ let $R_{0}\left( z\right) :=\left(
H_{0}-z\right) ^{-1}$ be the resolvent. From the LAP for $H_{0S}$ it follows that the limits (see Lemma 3.1 of [@14])$$\left. R_{0}\left( E\pm i0\right) =\lim_{\varepsilon\rightarrow+0}R_{0}\left( E\pm i\varepsilon\right) =\left\{
\begin{array}
[c]{c}\left( H_{0}+E\right) R_{0S}\left( \left( E^{2}-m^{2}\right) \pm
i0\right) \text{ for }E>m\\
\left( H_{0}+E\right) R_{0S}\left( \left( E^{2}-m^{2}\right) \mp
i0\right) \text{ for }E<-m,
\end{array}
\right. \right. \label{basicnotions27}$$ exist for $E\in(-\infty,-m)\cup(m,\infty)$ in the uniform operator topology in $\mathcal{B}\left( L_{s}^{2},\mathcal{H}^{\alpha,-s}\right) ,$ $s>1/2,$ $\alpha\leq1,$ and $\left\Vert R_{0}\left( E\pm i0\right) f\right\Vert
_{\mathcal{H}^{\alpha,-s}}$ $\leq C_{s,\delta}\left\vert E\right\vert
^{\left\vert \alpha\right\vert }\left\Vert f\right\Vert _{L_{s}^{2}},$ for $\left\vert E\right\vert \in\lbrack m+\delta,\infty),$ $\delta>0$. Furthermore, the functions, $R_{0}^{\pm}\left( E\right) ,$ given by $R_{0}\left( E\right) ,$ if $\operatorname{Im}E\neq0,$ and by $R_{0}\left(
E\pm i0\right) ,$ if $E\in(-\infty,-m)\cup(m,\infty),$ are defined for $E\in\mathbb{C}^{\pm}\cup\left( -\infty,-m\right) \cup\left( m,+\infty
\right) $ with values in $\mathcal{B}\left( L_{s}^{2},\mathcal{H}^{\alpha,-s}\right) ,$ and moreover, they are analytic for $\operatorname{Im}E\neq0$ and locally Hölder continuous for $E\in(-\infty,-m)\cup(m,\infty)$ with exponent $\vartheta$ such that $0<\vartheta\leq s-1/2$ and $\vartheta<1.$
Next we consider the resolvent $R\left( z\right) :=\left( H-z\right)
^{-1}$ for $z$ in the resolvent set of $H.$ The following limits exist for $E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right) $ in the uniform operator topology in $\mathcal{B}\left( L_{s}^{2},\mathcal{H}^{\alpha,-s}\right) ,$ $s\in\left( 1/2,s_{0}\right] ,$ $\left\vert
\alpha\right\vert \leq1,$ where $s_{0}$ is defined by Condition \[basicnotions26\] (see Theorem 3.9 of [@14])$$\left. R\left( E\pm i0\right) =\lim_{\varepsilon\rightarrow+0}R\left( E\pm
i\varepsilon\right) =R_{0}\left( E\pm i0\right) \left( 1+\mathbf{V}R_{0}\left( E\pm i0\right) \right) ^{-1}.\right. \label{basicnotions13}$$ From this relation and the properties of $R_{0}^{\pm}\left( E\right) $ it follows that the functions, $R^{\pm}\left( E\right) :=\{R\left( E\right) $ if $\operatorname{Im}E\neq0,$ and $R\left( E\pm i0\right) ,$ $E\in
\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right) \},$ defined for $E\in\mathbb{C}^{\pm}\cup\{(-\infty,-m)\cup(m,\infty
)\}\backslash\sigma_{p}\left( H\right) ,$ with values in $\mathcal{B}\left(
L_{s}^{2},\mathcal{H}^{\alpha,-s}\right) $ are analytic for $\operatorname{Im}E\neq0$ and locally Hölder continuous for $E\in
\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right) $ with exponent $\vartheta$ such that $0<\vartheta\leq s-1/2,$ $s<\min\{s_{0},3/2\}.$
The Foldy-Wouthuysen (F-W) transform [@48], that diagonalizes the free Dirac operator, is defined as follows: Let $\hat{G}\left( \xi\right) $ be the unitary $4\times4$ matrix defined by $\hat{G}\left( \xi\right)
=\exp\{\beta\left( \alpha\cdot\xi\right) \theta\left( \left\vert
\xi\right\vert \right) \},$ where $\theta\left( t\right) =\frac{1}{2t}\arctan\frac{t}{m}$ for $t>0.$ Note that $$\tilde{h}_{0}\left( \xi\right) :=\hat{G}\left( \xi\right) h_{0}\left(
\xi\right) \hat{G}\left( \xi\right) ^{-1}=\left( \xi^{2}+m^{2}\right)
^{1/2}\beta. \label{basicnotions30}$$
The F-W transform is the unitary operator $G$ on $L^{2}$ given by $G=\mathcal{F}^{-1}\hat{G}\left( \xi\right) \mathcal{F}$. $G$ transforms $H_{0}$ into $\tilde{H}_{0}:=GH_{0}G^{-1}=\left( H_{0S}+m^{2}\right)
^{1/2}\beta.$ We use the F-W transform to define the trace operator $T_{0}\left( E\right) $ for the free Dirac operator ([@14]). Let $P_{+}=\left(
\begin{array}
[c]{cc}I_{2} & 0\\
0 & 0
\end{array}
\right) $ and $P_{-}=\left(
\begin{array}
[c]{cc}0 & 0\\
0 & I_{2}\end{array}
\right) $. We define $T_{0}^{\pm}\left( E\right) \in\mathcal{B}\left(
L_{s}^{2};L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \right) $ by $\left( T_{0}^{\pm}\left( E\right) f\right) \left( \omega\right)
=\left( 2\pi\right) ^{-\frac{3}{2}}\upsilon\left( E\right)
{\displaystyle\int_{\mathbb{R}^{3}}}
e^{-i\nu\left( E\right) \left\langle \omega,x\right\rangle }$ $\ \times
P_{\pm}Gf\left( x\right) dx,$ where $\upsilon\left( E\right) =\left(
E^{2}\left( E^{2}-m^{2}\right) \right) ^{\frac{1}{4}}$ and $\nu\left(
E\right) =\sqrt{E^{2}-m^{2}}.$ The trace operator $T_{0}\left( E\right) $ for the free Dirac operator is defined by $T_{0}\left( E\right) =T_{0}^{+}\left( E\right) ,$ for $E>m$, and $T_{0}\left( E\right) =T_{0}^{-}\left( E\right) $, for $E<-m$. The operator valued function $T_{0}\left( E\right) :(-\infty,-m)\cup(m,\infty)\rightarrow\mathcal{B}\left( L_{s}^{2};L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \right)
$ is locally Hölder continuous with exponent $\vartheta$ satisfying $0<\vartheta\leq s-1/2$ and $\vartheta<1$ ([@70], [@80], [@8]). Moreover, the operator $\left( \mathcal{\tilde{F}}_{0}f\right) \left(
E,\omega\right) :=\left( T_{0}\left( E\right) f\right) \left(
\omega\right) $ extends to a unitary operator from $L^{2}$ onto $\mathcal{\hat{H}}^{\prime}:=L^{2}\left( \left( -\infty,-m\right)
;L^{2}\left( \mathbb{S}^{2};P_{-}\mathbb{C}^{4}\right) \right) \oplus
L^{2}\left( \left( m,+\infty\right) ;L^{2}\left( \mathbb{S}^{2};P_{+}\mathbb{C}^{4}\right) \right) =\left( \int_{\left( -\infty
,-m\right) }^{\oplus}L^{2}\left( \mathbb{S}^{2};P_{-}\mathbb{C}^{4}\right)
dE\right) $ $\oplus\left( \int_{\left( m,+\infty\right) }^{\oplus}L^{2}\left( \mathbb{S}^{2};P_{+}\mathbb{C}^{4}\right) dE\right) ,$ that gives a spectral representation of $H_{0},$ i.e., $\mathcal{\tilde{F}}_{0}H_{0}\mathcal{\tilde{F}}_{0}^{\ast}=E,$ the operator of multiplication by $E$ in $\mathcal{\hat{H}}^{\prime}$.
The perturbed trace operators are defined by, $T_{\pm}\left( E\right)
:=T_{0}\left( E\right) \left( I-\mathbf{V}R\left( E\pm i0\right) \right)
,$ for $E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left(
H\right) .$ They are bounded from $L_{s}^{2}$ into $L^{2}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ for $s\in\left( 1/2,s_{0}\right] $. Furthermore, the operator valued functions $E\rightarrow T_{\pm}\left(
E\right) $ from $\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left(
H\right) $ into $\mathcal{B}\left( L_{s}^{2};L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \right) $ are locally Hölder continuous for $E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right) $ with exponent $\vartheta$ satisfying the estimate $0<\vartheta\leq s-1/2,$ $s<\min\{s_{0},3/2\}$. The operators, $\left( \mathcal{\tilde{F}}_{\pm
}f\right) \left( E,\omega\right) :=\left( T_{\pm}\left( E\right)
f\right) \left( \omega\right) $ extend to unitary operators from $\mathcal{H}_{ac}$ onto $\mathcal{\hat{H}}^{\prime}$ and they give a spectral representations for the restriction of $H$ to $\mathcal{H}_{ac},$ $\mathcal{F}_{\pm}H\mathcal{F}_{\pm}^{\ast}=E,$ the operator of multiplication by $E$ in $\mathcal{\hat{H}}^{\prime}$.
Since the scattering operator $\mathbf{S}$ commutes with $H_{0},$ the operator $\mathcal{\tilde{F}}_{0}\mathbf{S}\mathcal{\tilde{F}}_{0}^{\ast}$ acts as a multiplication by the operator valued function $\tilde{S}\left( E\right)
:L^{2}\left( \mathbb{S}^{2};P_{\pm}\mathbb{C}^{4}\right) \rightarrow
L^{2}\left( \mathbb{S}^{2};P_{\pm}\mathbb{C}^{4}\right) ,$ $\pm E>m,$ called the scattering matrix. The scattering matrix satisfies the equality (see Theorem 4.2 of [@14] and also [@42],[@37],[@8])$$\tilde{S}\left( E\right) T_{-}\left( E\right) =T_{+}\left( E\right) ,
\label{basicnotions33}$$ and, moreover, it has the following stationary representation for $E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right) ,$$$\tilde{S}\left( E\right) =I-2\pi iT_{0}\left( E\right) \left(
\mathbf{V}-\mathbf{V}R\left( E+i0\right) \mathbf{V}\right) T_{0}\left(
E\right) ^{\ast}. \label{basicnotions31}$$
We consider now another spectral representation of $H_{0}$ that we find more convenient for our purposes. Let us define ([@15]) $$\left( \Gamma_{0}\left( E\right) f\right) \left( \omega\right) :=\left(
2\pi\right) ^{-\frac{3}{2}}\upsilon\left( E\right) P_{\omega}\left(
E\right) \int_{\mathbb{R}^{3}}e^{-i\nu\left( E\right) \left\langle
\omega,x\right\rangle }f\left( x\right) dx, \label{basicnotions11}$$ with$$P_{\omega}\left( E\right) :=\left\{
\begin{array}
[c]{c}P^{+}\left( \nu\left( E\right) \omega\right) ,\text{ \ }E>m,\\
P^{-}\left( \nu\left( E\right) \omega\right) ,\text{ \ }E<-m,
\end{array}
\right. \label{basicnotions43}$$ that is bounded from $L_{s}^{2},$ $s>1/2,$ into $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) .$ The adjoint operator $\Gamma_{0}^{\ast}\left(
E\right) :L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \rightarrow
L_{-s}^{2},$ $s>1/2,$ is given by $$\left( \Gamma_{0}^{\ast}\left( E\right) f\right) \left( \omega\right)
:=\left( 2\pi\right) ^{-\frac{3}{2}}\upsilon\left( E\right) \int_{\mathbb{S}^{2}}e^{i\nu\left( E\right) \left\langle x,\omega\right\rangle
}P_{\omega}\left( E\right) f\left( \omega\right) d\omega.
\label{basicnotions39}$$ Using (\[basicnotions30\]) we have $\hat{G}\left( \nu\left( E\right)
\omega\right) P_{\omega}\left( E\right) =\frac{1}{2}\left( I_{4}\pm
\beta\right) \hat{G}\left( \nu\left( E\right) \omega\right) =P_{\pm}\hat{G}\left( \nu\left( E\right) \omega\right) ,$ for $\pm E>m.$ This relation means that $\hat{G}\left( \nu\left( E\right) \omega\right) $ takes $X^{\pm}\left( \nu\left( E\right) \omega\right) $ onto $P_{\pm
}\mathbb{C}^{4},$ for $\pm E>m,$ and moreover, it implies that $$\Gamma_{0}\left( E\right) =\hat{G}\left( \nu\left( E\right)
\omega\right) ^{-1}T_{0}\left( E\right) . \label{basicnotions28}$$ As $\hat{G}\left( \nu\left( E\right) \omega\right) ^{-1}$ is differentiable on $E$, then from the last relation it follows that the operator valued function $\Gamma_{0}\left( E\right) $ is locally Hölder continuous on $(-\infty,-m)\cup(m,\infty)$ with the same exponent as $T_{0}\left( E\right) .$
Let us define the unitary operator $\mathcal{U}$ from $\mathcal{\hat{H}}^{\prime}$ onto $\mathcal{\hat{H}}:=\int_{\left( -\infty,-m\right)
\cup\left( m,+\infty\right) }^{\oplus}\mathcal{H}\left( E\right) dE,$ where$$\mathcal{H}\left( E\right) :=\int_{\mathbb{S}^{2}}^{\oplus}X^{\pm}\left(
\nu\left( E\right) \omega\right) d\omega,\text{ \ }\pm E>m,
\label{basicnotions34}$$ by $\left( \mathcal{U}f\right) \left( E,\omega\right) :=\hat{G}\left(
\nu\left( E\right) \omega\right) ^{-1}f\left( E,\omega\right) .$ Then the operator $\left( \mathcal{F}_{0}f\right) \left( E,\omega\right) :=\left(
\Gamma_{0}\left( E\right) f\right) \left( \omega\right) )=\left(
\mathcal{U\tilde{F}}_{0}f\right) \left( E,\omega\right) $ extends to unitary operator from $L^{2}$ onto $\mathcal{\hat{H}}$, that gives a spectral representation of $H_{0}$$$\mathcal{F}_{0}H_{0}\mathcal{F}_{0}^{\ast}=E, \label{basicnotions12}$$ the operator of multiplication by $E$ in $\mathcal{\hat{H}}$. Note that in $\mathcal{\hat{H}}^{\prime}$ the fibers $L^{2}\left( \mathbb{S}^{2};P_{\pm
}\mathbb{C}^{4}\right) $ can be written as $L^{2}\left( \mathbb{S}^{2};P_{\pm}\mathbb{C}^{4}\right) =\int_{\mathbb{S}^{2}}^{\oplus}P_{\pm
}\mathbb{C}^{4}d\omega,$ for $\pm E>m,$ where $P_{\pm}\mathbb{C}^{4}$ is independent of $\omega.$ However, in $\mathcal{\hat{H}}$ the fibers are given by (\[basicnotions34\]), where $X^{\pm}\left( \nu\left( E\right)
\omega\right) $ depend on $\omega.$
Let us define $$\Gamma_{\pm}\left( E\right) :=\Gamma_{0}\left( E\right) \left(
I-\mathbf{V}R\left( E\pm i0\right) \right) , \label{basicnotions25}$$ for $E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right)
.$ From relation (\[basicnotions28\]) it follows that$$\Gamma_{\pm}\left( E\right) =\hat{G}\left( \nu\left( E\right)
\omega\right) ^{-1}T_{\pm}\left( E\right) . \label{basicnotions32}$$ Then, the operator valued functions $\Gamma_{\pm}\left( E\right) $ are bounded from $L_{s}^{2}$ into $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ for $s\in\left( 1/2,s_{0}\right] ,$ and they are locally Hölder continuous on $(-\infty,-m)\cup(m,\infty)$ with the same exponent as $T_{\pm}\left( E\right) .$ The operators, $$\left( \mathcal{F}_{\pm}f\right) \left( E,\omega\right) :=\left(
\Gamma_{\pm}\left( E\right) f\right) \left( \omega\right) =\left(
\mathcal{U\tilde{F}}_{\pm}f\right) \left( E,\omega\right)
\label{basicnotions40}$$ extend to unitary operators from $\mathcal{H}_{ac}$ onto $\mathcal{\hat{H}}$ and they give a spectral representations for the restriction of $H$ to $\mathcal{H}_{ac},$ $\mathcal{F}_{\pm}H\mathcal{F}_{\pm}^{\ast}=E,$ the operator of multiplication by $E$ in $\mathcal{\hat{H}}$.
In the spectral representation (\[basicnotions12\]) the scattering matrix acts as a multiplication by the operator valued function $S\left( E\right)
:\mathcal{H}\left( E\right) \rightarrow\mathcal{H}\left( E\right) .$ Note that relation (\[basicnotions28\]) implies$$S\left( E\right) =\hat{G}\left( \nu\left( E\right) \omega\right)
^{-1}\tilde{S}\left( E\right) \hat{G}\left( \nu\left( E\right)
\omega\right) . \label{basicnotions35}$$ Thus, the scattering matrices $S\left( E\right) $ and $\tilde{S}\left(
E\right) $ are unitary equivalent. Moreover, from relation (\[basicnotions35\]), (\[basicnotions28\]) and representation (\[basicnotions31\]) we obtain the following stationary formula for $S\left( E\right) ,$ $$S\left( E\right) =I-2\pi i\Gamma_{0}\left( E\right) \left( \mathbf{V}-\mathbf{V}R\left( E+i0\right) \mathbf{V}\right) \Gamma_{0}\left(
E\right) ^{\ast}, \label{basicnotions14}$$ for $E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right)
.$ Here $I$ is the identity operator on $\mathcal{H}\left( E\right) .$
By the Schwartz Theorem (see, for example, Theorem 5.2.1 of [@77]) for every continuous and linear operator $T$ from $C^{\infty}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) $ to $D^{\prime}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $ (the set of distributions in $\mathbb{S}^{2}$) there is one, and only one distribution $t\left( \omega,\theta\right) $ on $\mathbb{S}^{2}\times\mathbb{S}^{2}$ such that for all $f\in C^{\infty}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ $\left( Tf\right) \left(
\omega\right) =\int_{\mathbb{S}^{2}}t\left( \omega,\theta\right) f\left(
\theta\right) d\theta.$ The integral in the R.H.S. of the last equation represents the duality parenthesis between the test functions and distributions on the variable $\theta.$ The distribution $t\left( \omega,\theta\right) $ is named the kernel of the operator $T.$
We say that the operator $T$ is an integral operator, if its kernel $t\left(
\omega,\theta\right) $ is actually a continuous function for $\omega
\neq\theta$ which satisfies the following estimate $\left\vert t\left(
\omega,\theta\right) \right\vert \leq\frac{C}{\left\vert \omega
-\theta\right\vert ^{2-\varepsilon}},$ for some $\varepsilon>0.$ Note that in this case $t\left( \omega,\theta\right) \in L^{1}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $ and moreover, using the Young’s inequality we have $\left\Vert Tf\right\Vert _{L^{2}\left( \mathbb{S}^{2}\right) }^{2}\leq\int_{\mathbb{S}^{2}}\left\vert \int_{\mathbb{S}^{2}}\frac
{C}{\left\vert \omega-\theta\right\vert ^{2-\varepsilon}}f\left(
\theta\right) d\theta\right\vert ^{2}d\omega\leq C\left\Vert f\right\Vert
_{L^{2}\left( \mathbb{S}^{2}\right) }^{2}.$ Thus, if $T$ is an integral operator, then it is bounded in $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) .$ Below we show (Theorem \[representation25\]) that if a magnetic potential $A$ and an electric potential $V$ satisfy estimates (\[eig31\]) and (\[eig32\]) respectively, then $S\left( E\right) -I$ is an integral operator. We call scattering amplitude to the integral kernel $s^{\operatorname{int}}\left( \omega,\theta;E\right) $ of $S\left( E\right) -I.$
From the unitary of $S\left( E\right) $ it follows $\left( S\left(
E\right) -I\right) ^{\ast}\left( S\left( E\right) -I\right) =-\left(
S\left( E\right) -I\right) -\left( S\left( E\right) -I\right) ^{\ast}.$ Then,$${\displaystyle\int_{\mathbb{S}^{2}}}
s^{\operatorname{int}}\left( \eta,\omega;E\right) ^{\ast}s^{\operatorname{int}}\left( \eta,\theta;E\right) d\eta
=-s^{\operatorname{int}}\left( \omega,\theta;E\right) -s^{\operatorname{int}}\left( \theta,\omega;E\right) ^{\ast}. \label{representation103}$$ This equality is known in the physics literature as Optical Theorem (see for example [@24]).
We had already mentioned in the introduction that the scattering operator $\mathbf{S}$ and the scattering matrix $S\left( E\right) $ are invariant under the gauge transformation $A\rightarrow A+\nabla\psi,$ for $\psi\in
C^{\infty}\left( \mathbb{R}^{3}\right) $ such that $\partial^{\alpha}\psi=O\left( \left\vert x\right\vert ^{-\rho-\left\vert \alpha\right\vert
}\right) $ for $0\leq\left\vert \alpha\right\vert \leq1$ and some $\rho>0$ as $\left\vert x\right\vert \rightarrow\infty$. Indeed, note that $\tilde
{H}:=e^{-i\psi}He^{i\psi}$ satisfies $\tilde{H}=H+\left( \alpha\cdot
\nabla\psi\right) .$ Then we get $W_{\pm}\left( \tilde{H},H_{0}\right)
=s-\lim_{t\rightarrow\pm\infty}e^{i\tilde{H}t}e^{-itH_{0}}=s-\lim
_{t\rightarrow\pm\infty}e^{-i\psi}e^{iHt}e^{i\psi}e^{-itH_{0}}.$ Under the assumptions on $\psi$, the operator of multiplication by the function $e^{i\psi}-1$ is a compact operator from $\mathcal{H}^{1}$ to $L^{2}.$ Since $e^{-iH_{0}t}$ converges weakly to $0,$ as $t\rightarrow\pm\infty,$ then $s-\lim_{t\rightarrow\pm\infty}\left( e^{-i\psi}e^{iHt}e^{i\psi}e^{-itH_{0}}\right) =s-\lim_{t\rightarrow\pm\infty}\left( e^{-i\psi}e^{iHt}\left(
e^{i\psi}-1\right) e^{-itH_{0}}\right) +s-\lim_{t\rightarrow\pm\infty
}\left( e^{-i\psi}e^{iHt}e^{-itH_{0}}\right) =s-\lim_{t\rightarrow\pm\infty
}\left( e^{-i\psi}e^{iHt}e^{-itH_{0}}\right) ,$ and then $W_{\pm}\left(
\tilde{H},H_{0}\right) =e^{i\psi}W_{\pm}\left( H,H_{0}\right) .$ Thus, we conclude that $\mathbf{S}\left( \tilde{H},H_{0}\right) =\mathbf{S}\left(
H,H_{0}\right) .$
It results convenient for us to associate the scattering matrix $S\left(
E\right) $ directly with the magnetic field $B\left( x\right)
=\operatorname*{rot}A\left( x\right) .$ However, as $S\left( E\right) $ was defined in terms of the magnetic potential $A\left( x\right) ,$ we recall the procedure given in [@25] and [@31] for the construction of a short-range magnetic potential from an arbitrary magnetic field satisfying the condition for some $d\geq1$ $$\left\vert \partial^{\alpha}B\left( x\right) \right\vert \leq C_{\alpha
}\left( 1+\left\vert x\right\vert \right) ^{-r-\left\vert \alpha\right\vert
},\text{ }r>2,\text{ }0\leq\left\vert \alpha\right\vert \leq d.
\label{basicnotions18}$$ We note that the magnetic potential can be reconstructed from the magnetic field $B\left( x\right) $ such that $\operatorname*{div}B(x)=0$ only up to arbitrary gauge transformations. It is not convenient to work in the standard transversal gauge $\left\langle x,A_{tr}\left( x\right) \right\rangle =0$ since even for magnetic fields of compact support the potential $A_{tr}\left(
x\right) $ decays only as $\left\vert x\right\vert ^{-1}$ at infinity.
Let $B(x)=(B_{1}(x),B_{2}(x),B_{3}(x))$ be a magnetic field satisfying estimates (\[basicnotions18\]) such that $\operatorname*{div}B(x)=0$. Let us define the following matrix $$F=\left(
\begin{array}
[c]{ccc}0 & B_{3} & -B_{2}\\
-B_{3} & 0 & B_{1}\\
B_{2} & -B_{1} & 0
\end{array}
\right)$$ and introduce the auxiliary potentials $$A_{i}^{\left( reg\right) }\left( x\right) ={\displaystyle\int_{1}^{\infty}}
s\sum_{j=1}^{3}F^{\left( ij\right) }\left( sx\right) x_{j}ds,\text{
\ }A_{i}^{\left( \infty\right) }\left( x\right) =-{\displaystyle\int_{0}^{\infty}}
s\sum_{j=1}^{3}F^{\left( ij\right) }\left( sx\right) x_{j}ds,
\label{basicnotions19}$$ where $F^{\left( ij\right) }$ is the $\left( i,j\right) $-th element of $F$. Note that $A^{\left( \infty\right) }\left( x\right) $ is a homogeneous function of degree $-1,$ $A^{\left( reg\right) }\left(
x\right) =O\left( \left\vert x\right\vert ^{-\rho}\right) $ with $\rho=r-1$ as $\left\vert x\right\vert \rightarrow\infty,$ $\operatorname*{rot}A^{\left(
\infty\right) }\left( x\right) =0$ for $x\neq0$ and $A_{tr}=A^{\left(
reg\right) }\left( x\right) +A^{\left( \infty\right) }\left( x\right)
.$ We define the function $U\left( x\right) $ for $x\neq0$ as a curvilinear integral $$U\left( x\right) =\int_{\Gamma_{x_{0},x}}\left\langle A^{\left(
\infty\right) }\left( y\right) ,dy\right\rangle \label{basicnotions20}$$ taken between some fixed point $x_{0}\neq0$ and a variable point $x.$ If $0\notin\Gamma_{x_{0},x},$ it follows from the Stokes theorem that the function $U\left( x\right) $ does not depend on the choice of a contour $\Gamma_{x_{0},x}$ and $\operatorname*{grad}U\left( x\right) =A_{\infty
}\left( x\right) .$ Now we define the magnetic potential as $$A\left( x\right) =A_{tr}\left( x\right) -\operatorname*{grad}\left(
\eta\left( x\right) U\left( x\right) \right) =A_{reg}\left( x\right)
+\left( 1-\eta\left( x\right) \right) A_{\infty}\left( x\right)
-U\left( x\right) \operatorname*{grad}\eta\left( x\right) ,
\label{basicnotions21}$$ where $\eta\left( x\right) \in C^{\infty}\left( \mathbb{R}^{3}\right) ,$ $\eta\left( x\right) =0$ in a neighborhood of zero and $\eta\left(
x\right) =1$ for $\left\vert x\right\vert \geq1.$ Note that $\operatorname*{rot}A\left( x\right) =B\left( x\right) ,$ $A\in C^{\infty
}$ if $B\in C^{\infty}$ and $A\left( x\right) =A_{reg}\left( x\right) $ for $\left\vert x\right\vert \geq1.$ Moreover, it follows from the assumption (\[basicnotions18\]) that $A\left( x\right) $ satisfies the estimates $\left\vert \partial^{\alpha}A\left( x\right) \right\vert \leq C_{\alpha
}\left( 1+\left\vert x\right\vert \right) ^{-\rho-\left\vert \alpha
\right\vert },$ $\rho=r-1,$ for all $0\leq\left\vert \alpha\right\vert \leq
d.$
For a given magnetic field $B\left( x\right) $ we associate the magnetic potential $A\left( x\right) $ by formulae (\[basicnotions19\])–(\[basicnotions21\]) and then construct the scattering matrix $S\left(
E\right) $ in terms of the Dirac operator (\[basicnotions6\]). As we showed above, if another magnetic potential $\tilde{A}\left( x\right) $ satisfies $\operatorname*{rot}\tilde{A}\left( x\right) $ $=B(x)$, then the scattering matrices corresponding to potentials $A$ and $\tilde{A}$ coincide. This allows us to speak about the scattering matrix $S(E)$ corresponding to the magnetic field $B$.
Now we introduce some notation. Let $\mathit{\dot{S}}^{-\rho}=\mathit{\dot{S}}^{-\rho}\left( \mathbb{R}^{3}\right) $ be the set of $C^{\infty}\left(
\mathbb{R}^{3}\backslash\{0\}\right) $ functions $f\left( x\right) $ such that $\partial^{\alpha}f\left( x\right) =O\left( \left\vert x\right\vert
^{-\rho-\left\vert \alpha\right\vert }\right) ,$ as $\left\vert x\right\vert
\rightarrow\infty,$ for all $\alpha.$ Define also a subspace of $\mathit{\dot
{S}}^{-\rho}$ by $\mathit{S}^{-\rho}:=\mathit{\dot{S}}^{-\rho}\cap C^{\infty
}\left( \mathbb{R}^{3}\right) .$ An example of functions from the class $\mathit{\dot{S}}^{-\rho}$ are the homogeneous functions $f\in C^{\infty
}\left( \mathbb{R}^{3}\backslash\{0\}\right) $ of order $-\rho$, i.e. such that $f\left( tx\right) =t^{-\rho}f\left( x\right) $ for all $x\neq0$ and $t>0.$
Let functions $f_{j}\in\mathit{\dot{S}}^{-\rho_{j}}$ with $\rho_{j}\rightarrow\infty.$ The notation $$f\left( x\right) \simeq\sum_{j=1}^{\infty}f_{j}\left( x\right)
\label{basicnotions16}$$ means that, for any $N,$ the remainder$$f\left( x\right) -\sum_{j=1}^{N}f_{j}\left( x\right) \in\mathit{\dot{S}}^{-\rho},\text{ where }\rho=\min_{j\geq N+1}\rho_{j}. \label{basicnotions17}$$ Note that the function $f\in C^{\infty}$ is determined by its expansion (\[basicnotions16\]) only up to a term from the Schwartz class $\mathit{S=S}^{-\infty}\mathit{.}$
Finally, let us recall some facts about pseudodifferential operators. We define a pseudodifferential operator (PDO) by ([@32],[@36]) $$\left( Af\right) \left( x\right) =\left( 2\pi\right) ^{-d}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{i\left\langle \xi,x-\xi^{\prime
}\right\rangle }a\left( x,\xi\right) f\left( \xi^{\prime}\right)
d\xi^{\prime}d\xi, \label{basicnotions42}$$ where $f\in\mathcal{S(}\mathbb{R}^{d};\mathbb{C}^{4}\mathcal{)}$ and $a\left(
x,\xi\right) $ is a $\left( 4\times4\right) -$matrix. Here $d$ is the dimension (equals $2$ or $3$). We denote by $\mathcal{S}^{n,m}$ the class of PDO, which symbols are of $C^{\infty}\left( \mathbb{R}^{d}\times
\mathbb{R}^{d}\right) $ class and for all $x,\xi$ and for all multi-indices $\alpha,\beta$,$$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}a\left( x,\xi\right)
\right\vert \leq C_{\alpha,\beta}\left\langle x\right\rangle ^{n-\left\vert
\alpha\right\vert }\left\langle \xi\right\rangle ^{m-\left\vert \beta
\right\vert }. \label{representation197}$$ Note that $\mathcal{S}^{n,m}\subset\Gamma_{1}^{2m_{1}},$ for $m_{1}=\max\{n,m\}$, where the clases $\Gamma_{\rho}^{m_{1}}$ are defined in [@36]. Moreover we need a more special class $\mathcal{S}_{\pm}^{n,m}\subset$ $\mathcal{S}^{n,m}$ satisfying the additional property $a\left( x,\xi\right) =0$ if $\mp\left\langle \hat{x},\hat{\xi}\right\rangle
\leq\varepsilon,$ $\varepsilon>0,$ and $a\left( x,\xi\right) =0$ if $\left\vert x\right\vert \leq\varepsilon_{1}$ or $\left\vert \xi\right\vert
\leq\varepsilon_{1},$ $\varepsilon_{1}>0.$ (Here $\hat{x}=x/\left\vert
x\right\vert $ and $\hat{\xi}=\xi/\left\vert \xi\right\vert $).
It is convenient for us to consider a more general formula for the action of the PDO’s, determined by their amplitude. We define a PDO $\mathbf{A}$ by $$\left( \mathbf{A}f\right) \left( x\right) =\left( 2\pi\right) ^{-d}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{i\left\langle \xi,x-\xi^{\prime
}\right\rangle }a\left( x,\xi,\xi^{\prime}\right) f\left( \xi^{\prime
}\right) d\xi^{\prime}d\xi, \label{basicnotions41}$$ where $a\left( x,\xi,\xi^{\prime}\right) $ is called the amplitude of $\mathbf{A}.$ We say that $a\left( x,\xi,\xi^{\prime}\right) $ belongs to the class $\mathcal{S}^{n}$ if for all indices $\alpha,\beta\,,\gamma$ the following estimate holds $$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial_{\xi^{\prime}}^{\gamma}a\left( x,\xi,\xi^{\prime}\right) \right\vert \leq C_{\alpha
,\beta,\gamma}\left\langle \left( x,\xi,\xi^{\prime}\right) \right\rangle
^{n-\left\vert \alpha+\beta+\gamma\right\vert },\text{ }\left( x,\xi
,\xi^{\prime}\right) \in\mathbb{R}^{3d}. \label{representation198}$$ We note that $\mathcal{S}^{n}$ is contained in $\Pi_{1}^{n}$ ($\Pi_{\rho}^{n}$ are defined in [@36])). We can make the passage from the amplitude to the correspondent (left) symbol by the relation $$a_{\operatorname*{left}}\left( x,\xi\right) \simeq\sum_{\alpha}\frac
{1}{\alpha!}\left. \partial_{\xi}^{\alpha}\left( -i\partial_{\xi^{\prime}}^{\alpha}\right) a\left( x,\xi,\xi^{\prime}\right) \right\vert
_{\xi^{\prime}=x}. \label{basicnotions45}$$
For arbitrary $n,$ the integrals in the R.H.S. of relations (\[basicnotions42\]) and (\[basicnotions41\]) are understood as oscillating integrals. Furthermore, we recall the following results from the PDO calculus (see [@32] or [@36])
\[basicnotions24\]If $a\left( x,\xi\right) \in\mathcal{S}^{n,m}$ with $n\leq0$ and $m\leq0,$ then the PDO $A$ can be extended to a bounded operator in $L^{2}.$ The $L^{2}$- norm of $A\left\langle x\right\rangle ^{-n}$ is estimated by some constant $C,$ that depends only on the constants $C_{\alpha,\beta}$, given by (\[representation197\])$.$ Moreover, if $a\left( x,\xi\right) \in\mathcal{S}^{n,m}$ with $n<0$ and $m<0,$ then $A$ can be extended to a compact operator in $L^{2}.$
\[representation196\]Let $A_{j}$ be PDO with symbols $a_{j}\in
\mathcal{S}^{n_{j},m_{j}},$ for $j=1,2.$ Then the symbol $a$ of the product $A_{1}A_{2}$ belongs to the class $\mathcal{S}^{n_{1}+n_{2},m_{1}+m_{2}}$ and it admits the following asymptotic expansion $a\left( x,\xi\right)
=\sum_{\left\vert \alpha\right\vert <N}\frac{\left( -i\right) ^{\left\vert
\alpha\right\vert }}{\alpha!}\partial_{\xi}^{\alpha}a_{1}\left( x,\xi\right)
\partial_{x}^{\alpha}a_{2}\left( x,\xi\right) +r^{\left( N\right) }\left(
x,\xi\right) ,$ where $r^{\left( N\right) }\in\mathcal{S}^{n_{1}+n_{2}-N,m_{1}+m_{2}-N}.$
Note that, in particular, Propositions \[basicnotions24\] and \[representation196\] imply the following result
\[representation131\]Let $A_{j}$ be PDO with symbols $a_{j}\in
\mathcal{S}^{0,0},$ $j=1,2,$ and let $A$ be the PDO with symbol $a_{1}\left(
x,\xi\right) a_{2}\left( x,\xi\right) .$ Then, $A_{1}A_{2}-A$ can be extended to a compact operator.
For a PDO $A$ defined by its amplitude $a\left( x,\xi,\xi^{\prime}\right) $ we have
\[representation76\]If $a\left( x,\xi,\xi^{\prime}\right) \in
\mathcal{S}^{n}$, then the PDO $A$ can be extended to a bounded operator in $L^{2}$ for $n=0,$ and it operator norm is bounded by a constant $C$ depending only on $C_{\alpha,\beta,\gamma}$ given by (\[representation198\]). Moreover, $A$ can be extended to a compact operator in $L^{2}$ for $n<0.$
Scattering solutions.
=====================
In the stationary approach to the scattering theory it is useful to consider special solutions to the Dirac equation $$\left( H_{0}+\mathbf{V}\right) u=Eu,\text{ for }x\in\mathbb{R}^{3},
\label{re7}$$ called scattering solutions, or generalized eigenfunctions of continuous spectrum. Suppose that $\mathbf{V}$ satisfies Condition \[basicnotions26\] and $\mathbf{V}\in L_{s}^{2},$ for some $s>1/2.$ Then, for all $E\in
\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right) ,$ we define the scattering solutions to equation (\[re7\]) by$$u_{\pm}\left( x,\theta;E\right) =P_{\theta}\left( E\right) e^{i\nu\left(
E\right) \left\langle x,\theta\right\rangle }-\left( R\left( E\pm
i0\right) \mathbf{V}\left( \cdot\right) P_{\theta}\left( E\right)
e^{i\nu\left( E\right) \left\langle \cdot,\theta\right\rangle }\right)
\left( x\right) . \label{representation29}$$ We observe that under suitable assumptions on the solutions $u$ to (\[re7\]), known as “radiation conditions”, and on the potential $\mathbf{V,}$ $u_{\pm}$ is characterized as the unique solution to (\[re7\]), satisfying these “radiation conditions”. The problem of existence and unicity of solutions to (\[re7\]) was treated in [@41], by studying the formula (\[basicnotions27\]), for $f\in L_{s}^{2},$ $s>1/2,$ with radiation conditions $v_{\pm}\in L_{-s}^{2}\ $and$\ (\frac{\partial}{\partial
x_{j}}v_{\pm}\left( x\right) \mp iv_{\pm}\left( E\right) \frac{x_{j}}{\left\vert x\right\vert }u\left( x\right) )\in L_{s-1}^{2},$ $1/2<s\leq1,$ for $j=1,2,3.$ The radiation estimates, in the sense that the operators $\left\langle x\right\rangle ^{-1/2}\left( \frac{\partial}{\partial x_{j}}-\left\vert x\right\vert ^{-2}x_{j}\sum_{k=1}^{3}x_{k}\frac{\partial
}{\partial x_{k}}\right) $, for $j=1,2,3,$ are $H-$smooth, for the Dirac equation with long-range potentials, were obtained in [@33].
We want to give an asymptotic formula for $u_{\pm}$, as $\left\vert
x\right\vert \rightarrow\infty,$ where the asymptotic is understood in an appropriate sense. We denote by $\tilde{o}\left( \left\vert x\right\vert
^{-1}\right) $ a function $g\left( x\right) $ such that $\lim
_{r\rightarrow\infty}\left( \frac{1}{r}\int_{\left\vert x\right\vert \leq
r}\left\vert g\left( x\right) \right\vert ^{2}dx\right) =0.$ Of course a $o\left( \left\vert x\right\vert ^{-1}\right) $ function is also a $\tilde{o}\left( \left\vert x\right\vert ^{-1}\right) $ function. We prove the following
\[re17\]Suppose that $\mathbf{V}$ satisfies Condition \[basicnotions26\] and $\mathbf{V}\in L_{s}^{2},$ for some $s>1/2.$ Then, the scattering solutions admit the asymptotic expansion$$u_{\pm}\left( x,\theta;E\right) =P_{\theta}\left( E\right) e^{i\nu\left(
E\right) \left\langle x,\theta\right\rangle }+a_{\pm}\left( \hat{x},\theta;E\right) \frac{e^{\pm i\left( \operatorname*{sgn}E\right)
\nu\left( E\right) \left\vert x\right\vert }}{\left\vert x\right\vert
}+\tilde{o}\left( \left\vert x\right\vert ^{-1}\right) , \label{re16}$$ where the functions $a_{\pm}\left( \hat{x},\theta;E\right) :=-\left(
\operatorname*{sgn}E\right) \left( \frac{2\pi\left\vert E\right\vert }{\nu\left( E\right) }\right) ^{1/2}\left( \Gamma_{\pm}\left( E\right)
\mathbf{V}\left( \cdot\right) P_{\theta}\left( E\right) e^{i\nu\left(
E\right) \left\langle \cdot,\theta\right\rangle }\right) \left( \pm\left(
\operatorname*{sgn}E\right) \hat{x}\right) \ $can be recovered from $u_{\pm
}\left( x,\theta;E\right) $ by the formula $$a_{\pm}\left( \hat{x},\theta;E\right) =-\left( \operatorname*{sgn}E\right)
\left( \frac{2\pi\left\vert E\right\vert }{\nu\left( E\right) }\right)
^{1/2}\left( \Gamma_{0}\left( E\right) \mathbf{V}u_{\pm}\right) \left(
\pm\left( \operatorname*{sgn}E\right) \hat{x}\right) . \label{re18}$$ Moreover $a_{+}\left( x,\theta;E\right) $ is related to the scattering amplitude $s^{\operatorname{int}}\left( \hat{x},\theta;E\right) $ by the formula $a_{+}\left( \hat{x},\theta;E\right) =-i\left( \operatorname*{sgn}E\right) \left( 2\pi\right) \nu\left( E\right) ^{-1}$ $\times
s^{\operatorname{int}}\left( \left( \operatorname*{sgn}E\right) \hat
{x},\theta;E\right) .$
Note that the coefficient of the leading term in the asymptotics (\[re16\]) is explicit. Similar asymptotic was obtained in [@41], but the expression for the asymptotics (\[re16\]) is not explicit there.
In order to prove Theorem \[re17\] we need some results.
\[re13\]For all $\left\vert E\right\vert >m$ and all functions $f\in
L_{2}$ with compact support we have $$R_{0}\left( E\pm i0\right) f\left( x\right) =\left( \operatorname*{sgn}E\right) \left( \frac{2\pi\left\vert E\right\vert }{\nu\left( E\right)
}\right) ^{1/2}\left( \Gamma_{0}\left( E\right) f\right) \left(
\pm\left( \operatorname*{sgn}E\right) \hat{x}\right) \frac{e^{\pm i\left(
\operatorname*{sgn}E\right) \nu\left( E\right) \left\vert x\right\vert }}{\left\vert x\right\vert }+O\left( \left\vert x\right\vert ^{-2}\right) ,
\label{re1}$$ and $$\left( \partial_{\left\vert x\right\vert }R_{0}\left( E\pm i0\right)
\right) f\left( x\right) =\pm i\left( 2\pi\right) ^{1/2}\upsilon\left(
E\right) \left( \Gamma_{0}\left( E\right) f\right) \left( \pm\left(
\operatorname*{sgn}E\right) \hat{x}\right) \frac{e^{\pm i\left(
\operatorname*{sgn}E\right) \nu\left( E\right) \left\vert x\right\vert }}{\left\vert x\right\vert }+O\left( \left\vert x\right\vert ^{-2}\right) .
\label{re2}$$
From the relation $H_{0}^{2}=-\Delta+m^{2}$ we get, for all $z\in\mathbb{C},$ $\left( H_{0}-z\right) \left( H_{0}+z\right) =-\Delta-\left( z^{2}-m^{2}\right) ,$ and hence $R_{0}\left( z\right) =\left( H_{0}+z\right)
R_{0S}\left( z^{2}-m^{2}\right) .$ Recall that $R_{0S}\left( z^{2}-m^{2}\right) $ is an integral operator. Its kernel (Green function) is given by $\left( 4\pi\left\vert x-y\right\vert \right) ^{-1}e^{i\sqrt{z^{2}-m^{2}}\left\vert x-y\right\vert }.$ Therefore, $R_{0}\left( z\right) $ is an integral operator with kernel $R_{0}\left( x,y;z\right) :=\left(
H_{0}+z\right) (\left( 4\pi\left\vert x-y\right\vert \right) ^{-1}$ $\times
e^{i\sqrt{z^{2}-m^{2}}\left\vert x-y\right\vert }).$ Proceeding similarly to the proof of Lemma 2.6 of [@8] (page 79), we get (\[re1\]) and (\[re2\]).
From the LAP for the perturbed Dirac operator and Lemma \[re13\], we get
\[re10\]For all $f\in L_{s}^{2},$ $s>1/2,$ asymptotics (\[re1\]) and (\[re2\]) hold, with $O\left( \left\vert x\right\vert ^{-2}\right) $ replaced by $\tilde{o}\left( \left\vert x\right\vert ^{-1}\right) .$ Suppose that $\mathbf{V}$ satisfies Condition \[basicnotions26\]. Then, for all $f\in L_{s}^{2},$ $s>1/2,$ and $E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash
\sigma_{p}\left( H\right) \}$ we have$$R\left( E\pm i0\right) f\left( x\right) =\left( \operatorname*{sgn}E\right) \left( \frac{2\pi\left\vert E\right\vert }{\nu\left( E\right)
}\right) ^{1/2}\left( \Gamma_{\pm}\left( E\right) f\right) \left(
\pm\left( \operatorname*{sgn}E\right) \hat{x}\right) \frac{e^{\pm i\left(
\operatorname*{sgn}E\right) \nu\left( E\right) \left\vert x\right\vert }}{\left\vert x\right\vert }+\tilde{o}\left( \left\vert x\right\vert
^{-1}\right) ,$$ and $$\left( \partial_{\left\vert x\right\vert }R\left( E\pm i0\right) \right)
f\left( x\right) =\pm i\left( 2\pi\right) ^{1/2}\upsilon\left( E\right)
\left( \Gamma_{\pm}\left( E\right) f\right) \left( \pm\left(
\operatorname*{sgn}E\right) \hat{x}\right) \frac{e^{\pm i\left(
\operatorname*{sgn}E\right) \nu\left( E\right) \left\vert x\right\vert }}{\left\vert x\right\vert }+\tilde{o}\left( \left\vert x\right\vert
^{-1}\right) .$$
The proof is the same as in the case of the Schrödinger equation (see Proposition 4.3 and Theorem 4.4 of [@8], page 240).
We now are able to prove Theorem \[re17\]**.**
**Proof of Theorem \[re17\]. **Note that $\mathbf{V}\left( x\right)
P_{\theta}\left( E\right) e^{i\nu\left( E\right) \left\langle
x,\theta\right\rangle }\in L_{s}^{2}\mathbf{,}$ for some $s>1/2.$ The asymptotic expansion (\[re16\]) is consequence of (\[re18\]). Multiplying (\[representation29\]) from the left side by $I+R_{0}\left( E\pm i0\right)
\mathbf{V}$ and using the resolvent identity $R=R_{0}-R_{0}\mathbf{V}R,$ we get that $u_{\pm}\left( x,\theta;E\right) =P_{\theta}\left( E\right)
e^{i\nu\left( E\right) \left\langle x,\theta\right\rangle }-R_{0}\left(
E\pm i0\right) \mathbf{V}u_{\pm}.$ Then relation (\[re18\]) follows from Lemma \[re10\]. Using the representation (\[basicnotions14\]) of the scattering matrix $S\left( E\right) ,$ the definition (\[basicnotions25\]) of $\Gamma_{+}\left( E\right) $ and the relation (\[basicnotions39\]) for $\Gamma_{0}^{\ast}\left( E\right) $ we get $s^{\operatorname{int}}\left(
\omega,\theta;E\right) =-i\left( 2\pi\right) ^{-\frac{1}{2}}\upsilon\left(
E\right) (\Gamma_{+}\left( E\right) \mathbf{V}\left( \cdot\right)
P_{\theta}\left( E\right) e^{i\nu\left( E\right) \left\langle \cdot
,\theta\right\rangle })\left( \omega\right) .$ Then, the relation between $a_{+}$ and $s^{\operatorname{int}}$, follows from the definition of $a_{+}.$
Symmetries of the kernel of the scattering matrix.
==================================================
In this Section we discuss the parity, charge-conjugation and time-reversal transformations for the Dirac operator (see, for example [@48]). In particular, we want to study the symmetries that these transformations imply for the kernel $s\left( \omega,\theta;E\right) $ of the scattering matrix $S\left( E\right) $. These symmetries can give necessary conditions when one studies the characterization problem. We consider the Dirac operator with potential $\mathbf{V}$ of the form (\[intro4\]). Suppose that $\mathbf{V}$ satisfies Condition \[basicnotions26\]. The latter assumption is made only in order to guarantee the existence of the wave operators $W_{\pm}$ and the scattering operator $\mathbf{S},$ and may be relaxed.
The parity transformation is defined as $\mathcal{P=}e^{i\phi}\beta\varkappa,$ where $\left( \varkappa f\right) \left( x\right) =f\left( -x\right) $ is the space reflection operator and $\phi$ is a fixed phase. The transformation $\mathcal{P}$ commutes with $H_{0}$. For the perturbed operator $H$ we have $\mathcal{P}\left( -i\alpha\nabla+m\beta+\alpha A\left( x\right) +V\left(
x\right) \right) =\left( -i\alpha\nabla+m\beta-\alpha A\left( -x\right)
+V\left( -x\right) \right) \mathcal{P}.$ Therefore, for the operator $H$ of the form (\[intro4\]) with general electromagnetic potential $\mathbf{V}$ the parity transformation $\mathcal{P}$ is not a symmetry. If we consider an even electric potential, $V\left( x\right) =V\left( -x\right) ,$ and an odd magnetic potential, $A\left( -x\right) =-A\left( x\right) $, then $\mathcal{P}H=H\mathcal{P}$. In this case it follows that the parity transformation $\mathcal{P}$ commutes with the wave operators $W_{\pm}$ and the scattering operator $\mathbf{S.}$
Noting that$$\beta\hat{\varkappa}\Gamma_{0}\left( E\right) =\Gamma_{0}\left( E\right)
e^{-i\phi}\mathcal{P}, \label{sym1}$$ and$$e^{-i\phi}\mathcal{P}\Gamma_{0}^{\ast}\left( E\right) =\Gamma_{0}^{\ast
}\left( E\right) \beta\hat{\varkappa}, \label{sym7}$$ where $\left( \hat{\varkappa}b\right) \left( \omega\right) =b\left(
-\omega\right) ,$ is the reflection operator on the unit sphere, we obtain the equality $\beta\hat{\varkappa}S\left( E\right) =\beta\hat{\varkappa
}\Gamma_{0}\left( E\right) \mathbf{S}\Gamma_{0}^{\ast}\left( E\right)
=\Gamma_{0}\left( E\right) \mathbf{S}\Gamma_{0}^{\ast}\left( E\right)
\hat{\varkappa}\beta$ $=S\left( E\right) \hat{\varkappa}\beta.$ This means that the kernel $s\left( \omega,\theta;E\right) $ of the scattering matrix $S\left( E\right) $ satisfy the relation$$s\left( \omega,\theta;E\right) =\beta s\left( -\omega,-\theta;E\right)
\beta. \label{sym4}$$
For the Dirac operator the charge-conjugation transformation is defined as $\mathcal{C}=i\left( \beta\alpha_{2}\right) \mathbf{C,}$ where $\mathbf{C}$ is the complex conjugation. Note that $\mathcal{C}\left( -i\alpha
\nabla+m\beta+\alpha A+V\right) =i\left( \beta\alpha_{2}\right) \left(
i\overline{\alpha}\nabla+m\beta+\overline{\alpha}A+V\right) \mathbf{C}=-\left( -i\alpha\nabla+m\beta-\alpha A-V\right) \mathcal{C}\mathbf{.}$ We consider now an odd electric potential $V\left( x\right) $ and an even magnetic potential $A\left( x\right) .$ Using the charge-conjugation transformation $\mathcal{C}$ we see that $\mathcal{CP}H=-H\mathcal{CP}$, $\mathcal{CP}W_{\pm}=W_{\pm}\mathcal{CP}$ and $\mathcal{CP}\mathbf{S=S}\mathcal{CP}$. Moreover, as$$\mathcal{C}\hat{\varkappa}\Gamma_{0}\left( E\right) =\Gamma_{0}\left(
-E\right) \mathcal{C}, \label{representation199}$$ and$$\mathcal{C}\Gamma_{0}^{\ast}\left( E\right) =\Gamma_{0}^{\ast}\left(
-E\right) \mathcal{C}\hat{\varkappa}, \label{representation200}$$ then using (\[sym1\]) and (\[sym7\]) we have $\mathcal{C}\beta\Gamma
_{0}\left( E\right) =\mathcal{C}\hat{\varkappa}\Gamma_{0}\left( E\right)
e^{-i\phi}\mathcal{P}=\Gamma_{0}\left( -E\right) \mathcal{C}e^{-i\phi
}\mathcal{P}$, and $\mathcal{C}e^{-i\phi}\mathcal{P}\Gamma_{0}^{\ast}\left(
E\right) =\mathcal{C}\Gamma_{0}^{\ast}\left( E\right) \beta\hat{\varkappa
}=\Gamma_{0}^{\ast}\left( -E\right) \mathcal{C}\beta$. The last two equalities imply $\mathcal{C}\beta S\left( E\right) =\mathcal{C}\beta
\Gamma_{0}\left( E\right) \mathbf{S}\Gamma_{0}^{\ast}\left( E\right)
=\Gamma_{0}\left( -E\right) \mathbf{S}\Gamma_{0}^{\ast}\left( -E\right)
\mathcal{C}\beta=S\left( -E\right) \mathcal{C}\beta.$ Thus, we obtain the following relation for the kernel $s\left( \omega,\theta;E\right) $ of the scattering matrix $S\left( E\right) :$$$\overline{s\left( \omega,\theta;E\right) }=\alpha_{2}s\left( \omega
,\theta;-E\right) \alpha_{2}. \label{sym5}$$
Another symmetry of the free Dirac operator is the time-reversal transformation $\mathcal{T}=-i\left( \alpha_{1}\alpha_{3}\right) \mathbf{C}$. Note that $$\left. \mathcal{T}\left( -i\alpha\nabla+m\beta+\alpha A+V\right) =-i\left(
\alpha_{1}\alpha_{3}\right) \left( i\overline{\alpha}\nabla+m\beta
+\overline{\alpha}A+V\right) \mathbf{C}=\left( -i\alpha\nabla+m\beta-\alpha
A+V\right) \mathcal{T}\mathbf{.}\right. \label{representation104}$$ If $A=0,$ then relation (\[representation104\]) implies that $\mathcal{T}H=H\mathcal{T}$ and $\mathcal{T}e^{itH}=e^{-itH}\mathcal{T}$. Thus, we have the following relations $$\mathcal{T}W_{\pm}=W_{\mp}\mathcal{T}\text{ and }\mathcal{T}\mathbf{S=S}^{\ast}\mathcal{T}. \label{representation105}$$ Noting that$$\mathcal{T}\hat{\varkappa}\Gamma_{0}\left( E\right) =\Gamma_{0}\left(
E\right) \mathcal{T}, \label{sym2}$$ and$$\mathcal{T}\Gamma_{0}^{\ast}\left( E\right) =\Gamma_{0}^{\ast}\left(
E\right) \mathcal{T}\hat{\varkappa} \label{sym9}$$ we obtain $\mathcal{T}\hat{\varkappa}S\left( E\right) =\mathcal{T}\hat{\varkappa}\Gamma_{0}\left( E\right) \mathbf{S}\Gamma_{0}^{\ast}\left(
E\right) =\Gamma_{0}\left( E\right) \mathbf{S}^{\ast}\Gamma_{0}^{\ast
}\left( E\right) \mathcal{T}\hat{\varkappa}=S\left( E\right) ^{\ast
}\mathcal{T}\hat{\varkappa}.$ The last equality for the scattering matrix $S\left( E\right) $ leads to the following symmetry relation for the kernel $s\left( \omega,\theta;E\right) :$$$\left( \alpha_{1}\alpha_{3}\right) \overline{s\left( \omega,\theta
;E\right) }=\left( s\left( -\theta,-\omega;E\right) \right) ^{\ast
}\left( \alpha_{1}\alpha_{3}\right) . \label{representation106}$$
If $A\neq0$, then relation (\[representation105\]) is not satisfied. In this case, in addition to $\mathcal{T}$, we need to apply some other transformation to $H$ to get a relation similar to (\[representation105\]). Note that the parity transformation $\mathcal{P}$ changes the sign of the magnetic potential $A.$ Therefore, in case of even potentials $\mathbf{V}$ we have $\mathcal{TP}H=H\mathcal{TP}$, and hence, $\mathcal{TP}W_{\pm}=W_{\mp}\mathcal{TP}$ and $\mathcal{TP}\mathbf{S=S}^{\ast}\mathcal{TP}.$ Using relations (\[sym1\]) and (\[sym2\]) we get $\mathcal{T}\beta\Gamma_{0}\left( E\right)
=\mathcal{T}\hat{\varkappa}\Gamma_{0}\left( E\right) e^{-i\phi}\mathcal{P}=\Gamma_{0}\left( E\right) \mathcal{T}e^{-i\phi}\mathcal{P},$ and, by using (\[sym7\]) and (\[sym9\]) we obtain $\mathcal{T}e^{-i\phi
}\mathcal{P}\Gamma_{0}^{\ast}\left( E\right) =\mathcal{T}\Gamma_{0}^{\ast
}\left( E\right) \beta\hat{\varkappa}=\Gamma_{0}^{\ast}\left( E\right)
\mathcal{T}\beta.$ From the above equalities we have $\mathcal{T}\beta
S\left( E\right) =\mathcal{T}\beta\Gamma_{0}\left( E\right) \mathbf{S}\Gamma_{0}^{\ast}\left( E\right) =\Gamma_{0}\left( E\right) \mathbf{S}^{\ast}\Gamma_{0}^{\ast}\left( E\right) \mathcal{T}\beta=S\left( E\right)
^{\ast}\mathcal{T}\beta,$ and, thus,$$\left( \alpha_{1}\alpha_{3}\beta\right) \overline{s\left( \omega
,\theta;E\right) }=\left( s\left( \theta,\omega;E\right) \right) ^{\ast
}\left( \alpha_{1}\alpha_{3}\beta\right) . \label{sym6}$$
Let us consider the case when the electric potential $V=0$ and the magnetic potential $A$ is a general function of $x$. As the charge-conjugation transformation changes the sign of the magnetic potential $A$, we get a relation, similar to (\[representation105\]) for the following transformation $\Lambda=\mathcal{CT}.$ As $\Lambda\left( iH\right)
=-iH\Lambda,$ then $\Lambda e^{itH}=e^{-itH}\Lambda,$ which implies that $\Lambda W_{\pm}=W_{\mp}\Lambda$ and $\Lambda\mathbf{S=S}^{\ast}\Lambda.$ From relations (\[representation199\]) and (\[sym2\]) it follows that $\Lambda\Gamma_{0}\left( E\right) =\Gamma_{0}\left( -E\right) \Lambda
.\ $Moreover, using equalities (\[representation200\]) and (\[sym9\]) we get $\Lambda\Gamma_{0}^{\ast}\left( E\right) =\Gamma_{0}^{\ast}\left(
-E\right) \Lambda.$ Therefore, we obtain $\Lambda S\left( E\right)
=\Lambda\Gamma_{0}\left( E\right) \mathbf{S}\Gamma_{0}^{\ast}\left(
E\right) =\Gamma_{0}\left( -E\right) \mathbf{S}^{\ast}\Gamma_{0}^{\ast
}\left( -E\right) \Lambda=S\left( -E\right) ^{\ast}\Lambda.$ Therefore we obtain the following symmetry relation$$s\left( \omega,\theta;E\right) =\gamma\left( s\left( \theta,\omega
;-E\right) \right) ^{\ast}\gamma, \label{representation107}$$ where $\gamma=\alpha_{1}\alpha_{2}\alpha_{3}\beta.$
Finally suppose that $\mathbf{V}\left( x\right) $ is an odd function. Then the transformation $\Pi=\mathcal{CTP}$ satisfies the equality $\Pi H=-H\Pi$ and $\Pi e^{itH}=e^{-itH}\Pi.$ This implies that $\Pi W_{\pm}=W_{\mp}\Pi$ and $\Pi\mathbf{S=S}^{\ast}\Pi.$ Moreover, as $\Lambda\beta\hat{\varkappa}\Gamma_{0}\left( E\right) =\Gamma_{0}\left( -E\right) e^{-i\phi}\Pi,$ and $e^{-i\phi}\Pi\Gamma_{0}^{\ast}\left( E\right) =\Gamma_{0}^{\ast}\left(
-E\right) \Lambda\beta\hat{\varkappa},$ then we have $\Lambda\beta
\hat{\varkappa}S\left( E\right) =\Lambda\beta\hat{\varkappa}\Gamma
_{0}\left( E\right) \mathbf{S}\Gamma_{0}^{\ast}\left( E\right) =\Gamma
_{0}\left( -E\right) \mathbf{S}^{\ast}\Gamma_{0}^{\ast}\left( -E\right)
\Lambda\beta\hat{\varkappa}=S\left( -E\right) ^{\ast}\Lambda\beta
\hat{\varkappa},$ and$$s\left( \omega,\theta;E\right) =\beta\gamma\left( s\left( -\theta
,-\omega;-E\right) \right) ^{\ast}\gamma\beta. \label{sym10}$$
Approximate solutions.
======================
In this Section we construct approximate generalized eigenfunctions for the Dirac equation. For the Schrödinger equation with short-range potentials, the approximate solutions are given by $u\left( x,\xi\right)
=e^{i\left\langle x,\xi\right\rangle }+e^{i\left\langle x,\xi\right\rangle
}a\left( x,\xi\right) ,$ where $a$ solves the transport equation (see [@27]). In the case of the Schrödinger equation with long-range potentials, the approximate solutions are of the form $u\left( x,\xi\right) =e^{i\left\langle x,\xi\right\rangle
+i\phi}\left( 1+a\left( x,\xi\right) \right) ,$ where $\phi$ solves the eikonal equation and $a\left(
x,\xi\right) $ is the solution of the transport equation ([@30]).
For the Dirac equation with short-range potentials it is not enough to consider only the transport equation, in order to obtain the desired estimates. Thus, we need to consider the eikonal equation too. It also turns out that we need to decompose the transport equation in two equations, one for the positive energies and another for the negative energies, to obtain a smoothness and high-energy expansion of the generalized eigenfunctions for the Dirac equation.
For an arbitrary $\xi\in\mathbb{R}^{3}$ let us consider the Dirac equation $$Hu=\left( \alpha\left( -i\triangledown+A\right) +m\beta+V\right)
u=Eu,\text{ }E=\pm\sqrt{\xi^{2}+m^{2}}, \label{eig1}$$ where $A=\left( A_{1},A_{2},A_{3}\right) $ is a magnetic potential satisfying the estimate$$\left\vert \partial_{x}^{\alpha}A\left( x\right) \right\vert \leq
C_{\alpha,\beta}\left( 1+\left\vert x\right\vert \right) ^{-\rho
_{m}-\left\vert \alpha\right\vert },\text{ }\rho_{m}>1, \label{eig31}$$ for all $\alpha,$ and $V$ is a scalar electric potential, which satisfies, for all $\alpha,$ the estimate$$\left\vert \partial_{x}^{\alpha}V\left( x\right) \right\vert \leq
C_{\alpha,\beta}\left( 1+\left\vert x\right\vert \right) ^{-\rho
_{e}-\left\vert \alpha\right\vert },\text{ }\rho_{e}>1. \label{eig32}$$
\[eig40\]Let $\omega=\xi/\left\vert \xi\right\vert ,$ $\hat{x}=x/\left\vert x\right\vert $ and $\Xi^{\pm}\left( E\right) :=\Xi^{\pm
}\left( \varepsilon_{0},R;E\right) \subset\mathbb{R}^{3}\times\mathbb{R}^{3}$ be the domain $\Xi^{\pm}\left( E\right) =[\left( x,\xi\right)
\in\mathbb{R}^{3}\times\mathbb{R}^{3}\mid$ $\pm\left( \operatorname*{sgn}E\right) \left\langle \hat{x},\omega\right\rangle \geq-1+\varepsilon_{0}$ for $\left\vert x\right\vert \geq R\},$ for some $0<\varepsilon_{0}<1$ and $0<R<\infty.$
We aim to construct $4\times4$ matrices $u_{N}^{\pm}\left( x,\xi;E\right) $ whose columns are approximate solutions to equation (\[eig1\]) in such way that the remainders $$r_{N}^{\pm}\left( x,\xi;E\right) \left. :=\right. e^{-i\left\langle
x,\xi\right\rangle }\left( H-E\right) u_{N}^{\pm}\left( x,\xi;E\right) ,
\label{eig30}$$ satisfy the following estimates $$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}r_{N}^{\pm}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\rho-N-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-N-\left\vert \beta\right\vert },\text{ }N\geq0,
\label{eig28}$$ for $\rho=\min\{\rho_{e},\rho_{m}\},$ $\left( x,\xi\right) \in\Xi^{\pm
}\left( E\right) $ and all multi-indices $\alpha$ and $\beta.$ It is natural for us to seek the matrices $u_{N}^{\pm}\left( x,\xi;E\right)
$ as$$u_{N}^{\pm}\left( x,\xi;E\right) =e^{i\phi^{\pm}\left( x,\xi;E\right)
}w_{N}^{\pm}\left( x,\xi;E\right) , \label{eig2}$$ where $\phi^{\pm}\left( x,\xi;E\right) $ is a real-valued function and$\ w_{N}^{\pm}\left( x,\xi;E\right) $ are $4\times4$ matrix-valued functions. Introducing (\[eig2\]) into equation (\[eig1\]) and using (\[eig30\]) we get$$\left( \alpha\left( -i\triangledown+\triangledown\phi^{\pm}+A\right)
+m\beta+V-E\right) w_{N}^{\pm}=e^{i\left\langle x,\xi\right\rangle
-i\phi^{\pm}}r_{N}^{\pm}. \label{eig3}$$ Let us write $\phi^{\pm}$ as$$\phi^{\pm}\left( x,\xi;E\right) =\left\langle x,\xi\right\rangle +\Phi^{\pm
}\left( x,\xi;E\right) , \label{eig25}$$ where $\Phi^{\pm}$ tends to $0$ as $\left\vert x\right\vert \rightarrow\infty$ for $\left( x,\xi\right) \in\Xi^{\pm}\left( E\right) .$ Then, (\[eig3\]) takes the form$$\left( \alpha\cdot\xi+m\beta-E+\alpha\left( -i\triangledown+\triangledown
\Phi^{\pm}+A\right) +V\right) w_{N}^{\pm}=e^{-i\Phi^{\pm}}r_{N}^{\pm}.
\label{eig4}$$ Now let us decompose $w_{N}^{\pm}$ as $$w_{N}^{\pm}=\left( w_{1}\right) _{N}^{\pm}+P_{\omega}\left( E\right)
\left( w_{2}\right) _{N}^{\pm}. \label{eig42}$$ Then, we get $\left( -2EP_{\omega}\left( -E\right) +\alpha\left(
-i\triangledown+\triangledown\Phi^{\pm}+A\right) +V\right) \left(
w_{1}\right) _{N}^{\pm}+\left( \alpha\left( -i\triangledown+\triangledown
\Phi^{\pm}+A\right) +V\right) P_{\omega}\left( E\right) \left(
w_{2}\right) _{N}^{\pm}=e^{-i\Phi^{\pm}}r_{N}^{\pm},$ where we used that $\alpha\cdot\xi+m\beta-E=-2EP_{\omega}\left( -E\right) .$ Using the algebra of the matrices $\alpha_{j}$ we get the equality $\alpha\left(
-i\triangledown+\triangledown\Phi^{\pm}+A\right) \left( \alpha\cdot
\xi\right) $ $=2\left\langle \xi,\left( -i\triangledown+\triangledown
\Phi^{\pm}+A\right) \right\rangle -\left( \alpha\cdot\xi\right)
\alpha\left( -i\triangledown+\triangledown\Phi^{\pm}+A\right) .$ This relation and equality $P_{\omega}^{2}\left( E\right) =P_{\omega}\left(
E\right) $ imply $$\left.
\begin{array}
[c]{c}\left( -2EP_{\omega}\left( -E\right) +\alpha\left( -i\triangledown
+\triangledown\Phi^{\pm}+A\right) +V\right) \left( w_{1}\right) _{N}^{\pm
}+P_{\omega}\left( -E\right) \alpha\left( -i\triangledown+\triangledown
\Phi^{\pm}+A\right) P_{\omega}\left( E\right) \left( w_{2}\right)
_{N}^{\pm}\\
+\frac{1}{E}\left\langle \xi,\left( -i\triangledown+\triangledown\Phi^{\pm
}+A\right) \right\rangle P_{\omega}\left( E\right) \left( w_{2}\right)
_{N}^{\pm}+VP_{\omega}\left( E\right) \left( w_{2}\right) _{N}^{\pm
}=e^{-i\Phi^{\pm}}r_{N}^{\pm}.
\end{array}
\right. \label{eig5}$$
Let the functions $\Phi^{\pm}$ satisfy the eikonal equation$$\left\langle \omega,\triangledown\Phi^{\pm}+A\right\rangle +\frac
{E}{\left\vert \xi\right\vert }V=0. \label{eig6}$$ Then, we need that the functions $\left( w_{1}\right) _{N}^{\pm}$ and $\left( w_{2}\right) _{N}^{\pm}$ are approximate solutions for the transport equation$$\left.
\begin{array}
[c]{c}\left( -2EP_{\omega}\left( -E\right) +\alpha\left( -i\triangledown
+\triangledown\Phi^{\pm}+A\right) +V\right) \left( w_{1}\right) _{N}^{\pm
}\\
+P_{\omega}\left( -E\right) \alpha\left( -i\triangledown+\triangledown
\Phi^{\pm}+A\right) P_{\omega}\left( E\right) \left( w_{2}\right)
_{N}^{\pm}+\frac{1}{E}\left\langle \xi,\left( -i\triangledown\right)
\right\rangle P_{\omega}\left( E\right) \left( w_{2}\right) _{N}^{\pm
}=e^{-i\Phi^{\pm}}r_{N}^{\pm}.
\end{array}
\right. \label{eig7}$$ We will search the functions $\left( w_{1}\right) _{N}^{\pm}$ and $\left(
w_{2}\right) _{N}^{\pm}~$as $$\left( w_{1}\right) _{N}^{\pm}=\sum_{j=1}^{N}\frac{1}{\left\vert
\xi\right\vert ^{j}}b_{j}^{\pm}\left( x,\xi;E\right) \label{eig8}$$ and$$\left( w_{2}\right) _{N}^{\pm}=\sum_{j=0}^{N}\frac{1}{\left\vert
\xi\right\vert ^{j}}c_{j}^{\pm}\left( x,\xi;E\right) . \label{eig9}$$ We want that the functions $w_{N}^{\pm}$ in relation (\[eig2\]) tend to $P_{\omega}\left( E\right) ,$ as $\left\vert x\right\vert \rightarrow
\infty.$ Therefore we set $b_{0}^{\pm}=0$ and $c_{0}^{\pm}=I$. Plugging (\[eig8\]) and (\[eig9\]) in (\[eig7\]) and multiplying the resulting equation on the left-hand side by $P_{\omega}\left( -E\right) $ we get$$\left.
\begin{array}
[c]{c}\sum_{j=1}^{N}\frac{1}{\left\vert \xi\right\vert ^{j}}\left( -2EP_{\omega
}\left( -E\right) +P_{\omega}\left( -E\right) \left( \alpha\left(
-i\triangledown+\triangledown\Phi^{\pm}+A\right) +V\right) \right)
b_{j}^{\pm}\\
+\sum_{j=0}^{N}\frac{1}{\left\vert \xi\right\vert ^{j}}P_{\omega}\left(
-E\right) \alpha\left( -i\triangledown+\triangledown\Phi^{\pm}+A\right)
P_{\omega}\left( E\right) c_{j}^{\pm}=e^{-i\Phi^{\pm}}P_{\omega}\left(
-E\right) r_{N}^{\pm},
\end{array}
\right. \label{eig11}$$ and by multiplying by $P_{\omega}\left( E\right) $ we obtain$$\left. \sum_{j=1}^{N}\frac{1}{\left\vert \xi\right\vert ^{j}}\left( \left(
P_{\omega}\left( E\right) \left( \alpha\left( -i\triangledown
+\triangledown\Phi^{\pm}+A\right) +V\right) \right) b_{j}^{\pm}+\frac{\left\vert \xi\right\vert }{E}\left\langle \omega,\left(
-i\triangledown\right) \right\rangle P_{\omega}\left( E\right) c_{j}^{\pm
}\right) =e^{-i\Phi^{\pm}}P_{\omega}\left( E\right) r_{N}^{\pm}.\right.
\label{eig10}$$ In order to get the desired estimates for $r_{N}^{\pm}$ we need that the terms in (\[eig11\]) and (\[eig10\]), which contain powers of $\frac
{1}{\left\vert \xi\right\vert }$ smaller than $N,$ are equal to $0.$ Then, comparing the terms of the same power of $\frac{1}{\left\vert \xi\right\vert
}$ in (\[eig11\]) and (\[eig10\]) we obtain the following equations ($E$ behaves like $\left( \operatorname*{sgn}E\right) \left\vert \xi\right\vert $ for large $\left\vert \xi\right\vert $) $$\left. b_{j+1}^{\pm}\left( x,\xi;E\right) =\frac{\left\vert \xi\right\vert
}{2E}P_{\omega}\left( -E\right) \left( \alpha\left( -i\triangledown
+\triangledown\Phi^{\pm}+A\right) +V\right) b_{j}^{\pm}+\frac{\left\vert
\xi\right\vert }{2E}P_{\omega}\left( -E\right) \alpha\left( -i\triangledown
+\triangledown\Phi^{\pm}+A\right) P_{\omega}\left( E\right) c_{j}^{\pm
},\right. \label{eig12}$$ for $0\leq j\leq N-1$ and$$\left\langle \omega,\triangledown c_{j}^{\pm}\right\rangle =-i\frac
{E}{\left\vert \xi\right\vert }P_{\omega}\left( E\right) \left(
\alpha\left( -i\triangledown+\triangledown\Phi^{\pm}+A\right) +V\right)
b_{j}^{\pm}, \label{eig37}$$ for $0\leq j\leq N.$ It follows from (\[eig12\]) that $b_{j}^{\pm}=P_{\omega}\left( -E\right) b_{j}^{\pm}.$ Then, the term $P_{\omega}\left(
E\right) Vb_{j}^{\pm}$ in equation (\[eig37\]) is equals to zero. Thus, $c_{j}^{\pm}$ satisfies equation$$\left\langle \omega,\triangledown c_{j}^{\pm}\right\rangle =-i\frac
{E}{\left\vert \xi\right\vert }P_{\omega}\left( E\right) \left(
\alpha\left( -i\triangledown+\triangledown\Phi^{\pm}+A\right) \right)
b_{j}^{\pm}. \label{eig13}$$
Our problem is reduced now to solve equations (\[eig6\]) and (\[eig13\]). Both of the equations are of the form$$\left\langle \omega,\triangledown d\right\rangle =F\left( x,\xi;E\right) ,
\label{eig39}$$ where $d$ and $F$ are either scalars or matrices. A simple substitution shows that the functions $$\Phi^{\pm}\left( x,\xi;E\right) =\left\{ \left.
\begin{array}
[c]{c}\pm\int\limits_{0}^{\infty}\left( \frac{\left\vert E\right\vert }{\left\vert
\xi\right\vert }V\left( x\pm t\omega\right) +\left\langle \omega,A\left(
x\pm t\omega\right) \right\rangle \right) dt,\text{ }E>m,\\
\pm\int\limits_{0}^{\infty}\left( \frac{\left\vert E\right\vert }{\left\vert
\xi\right\vert }V\left( x\mp t\omega\right) -\left\langle \omega,A\left(
x\mp t\omega\right) \right\rangle \right) dt,\text{ }-E>m,
\end{array}
\right. \right. \label{eig15}$$ formally satisfy equation (\[eig6\]) and the matrices$$c_{j}^{\pm}\left( x,\xi;E\right) =\left\{ \left.
\begin{array}
[c]{c}\pm\int\limits_{0}^{\infty}F_{j}^{\pm}\left( x\pm t\omega,\xi;E\right)
dt,\text{ }E>m,\\
\pm\int\limits_{0}^{\infty}F_{j}^{\pm}\left( x\mp t\omega,\xi;E\right)
dt,\text{ }-E>m,
\end{array}
\right. \right. \label{eig16}$$ for $j\geq1,$ where$$F_{j}^{\pm}\left( x,\xi;E\right) =i\frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }P_{\omega}\left( E\right) \left( \alpha\left(
-i\triangledown+\triangledown\Phi^{\pm}\left( x,\xi;E\right) +A\right)
\right) b_{j}^{\pm}\left( x,\xi;E\right) , \label{eig41}$$ solve, at least formally, equation (\[eig13\]).
Note that relations (\[eig12\]) and (\[eig16\]) imply, by induction that $$b_{j}^{\pm}P_{\omega}\left( E\right) =b_{j}^{\pm}\text{ and }c_{j}^{\pm
}P_{\omega}\left( E\right) =c_{j}^{\pm},\text{ }j\geq1. \label{eig35}$$
We need the following result to give a precise sense to expressions (\[eig15\]) and (\[eig16\]), and to get the desired estimates for the functions $\Phi^{\pm},$ $b_{j}^{\pm}$ and $c_{j}^{\pm}$ (see Lemma 2.1, [@30])
\[eig17\]Suppose that the function (or matrix) $F$ satisfies the estimate $$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}F\left( x,\xi;E\right)
\right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert x\right\vert \right)
^{-\rho-\left\vert \alpha\right\vert }\left\vert \xi\right\vert ^{-\left\vert
\beta\right\vert }\text{,} \label{eig34}$$ for $\left( x,\xi\right) \in\Xi^{\pm}\left( E\right) $ and some $\rho>1.$ Then the scalar (or a matrix-valued) functions $d^{\pm}\left( x,\xi;E\right)
=\pm\int\limits_{0}^{\infty}F\left( x\pm t\omega,\xi;E\right) dt$ satisfy equation (\[eig39\]) and the estimates $\left\vert \partial_{x}^{\alpha
}\partial_{\xi}^{\beta}d^{\pm}\left( x,\xi;E\right) \right\vert \leq
C_{\alpha,\beta}\left( 1+\left\vert x\right\vert \right) ^{-\left(
\rho-1\right) -\left\vert \alpha\right\vert }\left\vert \xi\right\vert
^{-\left\vert \beta\right\vert },$ on $\Xi^{\pm}\left( E\right) $ for all $\alpha$ and $\beta.$
It follows from Lemma \[eig17\] that under assumptions (\[eig31\]) and (\[eig32\]) the phase functions $\Phi^{\pm},$ defined by relation (\[eig15\]) are solutions to equation (\[eig6\]) and satisfy on $\Xi^{\pm
}\left( E\right) $ the estimates$$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}\Phi^{\pm}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\left( \rho-1\right) -\left\vert \alpha\right\vert
}\left\vert \xi\right\vert ^{-\left\vert \beta\right\vert },\text{ \ }\rho=\min\{\rho_{e},\rho_{m}\}. \label{eig22}$$
Moreover, we obtain from Lemma \[eig17\] the following
\[eig18\]Suppose that the magnetic potential $A$ and the electric potential $V$ satisfy the estimates (\[eig31\]) and (\[eig32\]), respectively. Then, $c_{j}^{\pm}$ defined by (\[eig16\]) solve equation (\[eig13\]) and the following estimates hold on $\Xi^{\pm}\left( E\right)
$ $$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}b_{j}^{\pm}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\rho-j+1-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert },\text{ }j\geq1,\text{ }
\label{eig19}$$ and$$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}c_{j}^{\pm}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\rho-j+1-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert },\text{ }j\geq1.\text{ }
\label{eig20}$$
We argue by induction in $j.$ Set $j=1.$ First note that $$\left\vert \partial_{\xi}^{\beta}P_{\omega}\left( \pm E\right) \right\vert
\leq C\left\vert \xi\right\vert ^{-\left\vert \beta\right\vert }\text{ \ and
}\left\vert \partial_{\xi}^{\beta}\left( \frac{\left\vert \xi\right\vert }{E}P_{\omega}\left( \pm E\right) \right) \right\vert \leq C\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert }. \label{eig23}$$ Differentiating the relation (\[eig12\]) we see that $\partial_{x}^{\alpha
}\partial_{\xi}^{\beta}b_{1}^{\pm}$ is a sum of terms of the form $\partial_{\xi}^{\beta_{1}}\left( \frac{\left\vert \xi\right\vert }{2E}P_{\omega}\left( -E\right) \right) \left( \partial_{x}^{\alpha
}\partial_{\xi}^{\beta_{2}}\left( \alpha\left( \triangledown\Phi^{\pm
}+A\right) \right) \right) $ $\times\left( \partial_{\xi}^{\beta_{3}}P_{\omega}\left( E\right) \right) ,$ with $\sum_{j=1}^{3}\beta_{j}=\beta.$ Then, from inequalities (\[eig22\]) and (\[eig23\]) it follows the estimate (\[eig19\]) for $b_{1}^{\pm}.$ Using (\[eig22\]),(\[eig23\]) and (\[eig19\]) with $j=1$ we see that $F_{1}^{\pm}$ in equality (\[eig16\]) satisfies the estimate $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}F_{1}^{\pm}\left( x,\xi;E\right) \right\vert \leq
C_{\alpha,\beta}\left( 1+\left\vert x\right\vert \right) ^{-\rho
-1-\left\vert \alpha\right\vert }\left\vert \xi\right\vert ^{-\left\vert
\beta\right\vert }.$ Then, using Lemma \[eig17\] it follow that $c_{1}^{\pm
}\left( x,\xi;E\right) $ solve equation (\[eig13\]) and satisfy estimates (\[eig20\])$.$
By induction assume that (\[eig19\]) and (\[eig20\]) are true for $j=n-1.$ Differentiating (\[eig12\]) it follows that $\partial_{x}^{\alpha}\partial_{\xi}^{\beta}b_{n}^{\pm}$ is a sum of terms of the form $$\left.
\begin{array}
[c]{c}\partial_{\xi}^{\beta_{1}^{\prime}}\left( \frac{\left\vert \xi\right\vert
}{2E}P_{\omega}\left( -E\right) \right) \left( \partial_{x}^{\alpha
_{1}^{\prime}}\partial_{\xi}^{\beta_{2}^{\prime}}\left( \alpha\left(
-i\triangledown+\triangledown\Phi^{\pm}+A\right) +V\right) \right)
\partial_{x}^{\alpha_{2}^{\prime}}\partial_{\xi}^{\beta_{3}^{\prime}}b_{n-1}\\
+\partial_{\xi}^{\beta_{1}}\left( \frac{\left\vert \xi\right\vert }{2E}P_{\omega}\left( -E\right) \right) \left( \partial_{x}^{\alpha_{1}}\partial_{\xi}^{\beta_{2}}\left( \alpha\left( -i\triangledown
+\triangledown\Phi^{\pm}+A\right) \right) \right) \left( \partial_{\xi
}^{\beta_{3}}P_{\omega}\left( E\right) \right) \partial_{x}^{\alpha_{2}}\partial_{\xi}^{\beta_{4}}c_{n-1}\end{array}
\right. \label{eig29}$$ with $\alpha_{1}+\alpha_{2}=\alpha,$ $\alpha_{1}^{\prime}+\alpha_{2}^{\prime
}=\alpha,$ $\sum_{j=1}^{4}\beta_{j}=\beta$ and $\sum_{j=1}^{3}\beta
_{j}^{\prime}=\beta.$ Therefore, from the hypothesis of induction, and inequalities (\[eig22\]) and (\[eig23\]) we get estimates (\[eig19\]) for $b_{n}^{\pm}.$ Similarly we see that $F_{n}$ satisfy the estimate $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}F_{n}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\rho-n-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert }.$ Using Lemma \[eig17\] we conclude that $c_{n}^{\pm}\left( x,\xi;E\right) $ are solutions to equation (\[eig13\]) satisfying estimates (\[eig20\]).
Proposition \[eig18\] implies that the solutions to equation (\[eig1\]) are given by (\[eig2\]). Let us define the functions$$a_{N}^{\pm}\left( x,\xi;E\right) :=e^{i\Phi^{\pm}\left( x,\xi;E\right)
}w_{N}^{\pm}\left( x,\xi;E\right) . \label{eig33}$$ Note that $$u_{N}^{\pm}\left( x,\xi;E\right) =e^{i\left\langle x,\xi\right\rangle }a_{N}^{\pm}\left( x,\xi;E\right) . \label{eig43}$$ Relation (\[eig35\]) implies that $$u_{N}^{\pm}P_{\omega}(E)=u_{N}^{\pm}\text{, }a_{N}^{\pm}P_{\omega}\left(
E\right) =a_{N}^{\pm}\text{ and }r_{N}^{\pm}P_{\omega}\left( E\right)
=r_{N}^{\pm}. \label{eig36}$$
We conclude this Section with the following result
Suppose that the magnetic potential $A$ and the electric potential $V$ satisfy the estimates (\[eig31\]) and (\[eig32\]), respectively. Then, for every $\left( x,\xi\right) \in\Xi^{\pm}\left( E\right) $ the following estimates hold$$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}w_{N}^{\pm}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert }, \label{eig26}$$$$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}a_{N}^{\pm}\left(
x,\xi;E\right) \right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert
x\right\vert \right) ^{-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert }. \label{eig27}$$ Moreover the remainder $r_{N}^{\pm}\left( x,\xi;E\right) $ satisfy estimate (\[eig28\]).
Estimate (\[eig26\]) is a consequence of Proposition \[eig18\].
Note that $\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a_{N}^{\pm}\left(
x,\xi;E\right) $ is a sum of terms of the form $\left( \partial_{x}^{\alpha_{1}}\partial_{\xi}^{\beta_{1}}e^{i\Phi^{\pm}\left( x,\xi;E\right)
}\right) \left( \partial_{x}^{\alpha_{2}}\partial_{\xi}^{\beta_{2}}w_{N}^{\pm}\left( x,\xi;E\right) \right) ,$ with $\alpha_{1}+\alpha
_{2}=\alpha$ and $\beta_{1}+\beta_{2}=\beta.$ Thus, relation (\[eig27\]) follows from the estimates (\[eig22\]) and (\[eig26\]).
To prove the estimate (\[eig28\]) we observe first that relations (\[eig11\]) and (\[eig10\]) imply $r_{N}^{\pm}\left( x,\xi;E\right)
=\left\vert \xi\right\vert ^{-N}e^{i\Phi^{\pm}\left( x,\xi;E\right)
}\{P_{\omega}\left( -E\right) $ $\times\left( \alpha\left( -i\triangledown
+\triangledown\Phi^{\pm}+A\right) +V\right) b_{N}^{\pm}+\left( P_{\omega
}\left( -E\right) \alpha\left( -i\triangledown+\triangledown\Phi^{\pm
}+A\right) \right) P_{\omega}\left( E\right) c_{N}^{\pm}\}.$ Differentiating $r_{N}^{\pm}$ we see that $\partial_{x}^{\alpha}\partial_{\xi
}^{\beta}r_{N}^{\pm}$ is a sum of terms similar to (\[eig29\]). Thus, using estimates (\[eig22\]), (\[eig23\]), (\[eig19\]) and (\[eig20\]) we obtain (\[eig28\]).
Estimates for the scattering matrix kernel.
===========================================
Statement of the results.
-------------------------
In this Section we study the diagonal singularities and the high-energy behavior of the scattering amplitude for potentials of the form (\[intro4\]) satisfying estimates (\[eig31\]) and (\[eig32\]). We follow the method of Yafaev [@27] and [@30] for the Schrödinger operator for this problem, that consist in defining special identifications $J_{\pm}$ and in studying the perturbed stationary formula for the scattering matrix. In other words, we will use the approximate solutions (\[eig2\]) to construct explicit functions $s_{N}\left( \omega,\theta;E\right) ,$ such that the difference $s-s_{N}$ is increasingly smoother as $N\rightarrow\infty.$ Moreover, as $N\rightarrow\infty,$ the difference $s-s_{N}$ decays increasingly faster when $E\rightarrow\infty.$
Let us announce the main result of this Section. First we prepare some results.
For an arbitrary point $\omega_{0}\in\mathbb{S}^{2}$ let $\Pi_{\omega_{0}}$ be the plane orthogonal to $\omega_{0}$ and $$\Omega_{\pm}\left( \omega_{0},\delta\right) :=\{\omega\in\mathbb{S}^{2}|\pm\left\langle \omega,\omega_{0}\right\rangle >\delta\},
\label{representation64}$$ for some $0<\delta<1$. For any $\omega_{j}\in\mathbb{S}^{2}$ let us define $O_{j}^{\pm}=\Omega_{\pm}\left( \omega_{j},\sqrt{\left( 1+\delta\right)
/2}\right) $ and set $O_{j}:=O_{j}^{+}\cup O_{j}^{-}.$ Let us prove the following result
\[representation221\]Let $j$ and $k$ be such that $O_{j}\cap O_{k}\neq\varnothing.$ Then, if $\omega_{jk}\in O_{j}^{\pm}\cap O_{k}^{\pm},$ we get $O_{j}^{+}\cup O_{k}^{+}\subseteq\Omega_{\pm}\left( \omega_{jk},\delta\right) $ and $O_{j}^{-}\cup O_{k}^{-}\subseteq\Omega_{\mp}\left(
\omega_{jk},\delta\right) .$ Moreover, if $\omega_{jk}\in O_{j}^{\pm}\cap
O_{k}^{\mp},$ we have $O_{j}^{+}\cup O_{k}^{-}\subseteq\Omega_{\pm}\left(
\omega_{jk},\delta\right) $ and $O_{j}^{-}\cup O_{k}^{+}\subseteq\Omega_{\mp
}\left( \omega_{jk},\delta\right) .$
Let $\omega_{jk}\in O_{j}^{+}$ and $\omega$ be in $O_{j}^{+}.$ If $\omega
_{j}=\omega_{jk}$ or $\omega=\omega_{j}$, then $\omega$ belongs to $\Omega
_{+}\left( \omega_{jk},\delta\right) $. Thus, we can suppose that $\omega_{j}\neq\omega_{jk}$ and $\omega\neq\omega_{j}.$ Let $\theta_{\omega}$ be a unit vector in the plane generated by $\omega$ and $\omega_{j},$ that is orthogonal to $\omega_{j}:\left\langle \omega_{j},\theta_{\omega}\right\rangle
=0.$ We decompose $\omega$ as $\omega=\left\langle \omega,\omega
_{j}\right\rangle \omega_{j}+\left\langle \omega,\theta_{\omega}\right\rangle
\theta_{\omega}.$ Similarly, for $\omega_{j}\neq\omega_{jk}$ we take a unit vector $\theta_{\omega_{jk}}$ such that $\omega_{jk}=\left\langle \omega
_{jk},\omega_{j}\right\rangle \omega_{j}+\left\langle \omega_{jk},\theta_{\omega_{jk}}\right\rangle \theta_{\omega_{jk}},$ $\left\langle
\omega_{j},\theta_{\omega_{jk}}\right\rangle =0.$ Then, we have $\left\langle
\omega,\omega_{jk}\right\rangle =\left\langle \omega,\omega_{j}\right\rangle
\left\langle \omega_{jk},\omega_{j}\right\rangle +\left\langle \omega
,\theta_{\omega}\right\rangle \left\langle \omega_{jk},\theta_{\omega_{jk}}\right\rangle \left\langle \theta_{\omega},\theta_{\omega_{jk}}\right\rangle
.$ As $\left\vert \left\langle \omega,\omega_{j}\right\rangle \right\vert
>\sqrt{\left( 1+\delta\right) /2},$ we get $\left\vert \left\langle
\omega,\theta_{\omega}\right\rangle \right\vert <\sqrt{1-\left(
1+\delta\right) /2}$ and, similarly $\left\vert \left\langle \omega
_{jk},\theta_{\omega_{jk}}\right\rangle \right\vert <\sqrt{1-\left(
1+\delta\right) /2}.$ Using this inequalities and the estimate $\left\langle
\theta_{\omega},\theta_{\omega_{jk}}\right\rangle \geq-1$, we obtain $\left\langle \omega,\omega_{jk}\right\rangle >\left( 1+\delta\right)
/2-\left( 1-\left( 1+\delta\right) /2\right) =\delta,$ and then, $\omega\in\Omega_{+}\left( \omega_{jk},\delta\right) .$ If $\omega$ belongs to $O_{j}^{-},$ $\left( -\omega\right) \in O_{j}^{+}.$ Thus, $-\omega$ belongs to $\Omega_{+}\left( \omega_{jk},\delta\right) $ and hence, $\omega\in\Omega_{-}\left( \omega_{jk},\delta\right) .$ If $\omega_{jk}\in
O_{j}^{-}$, then $-\omega_{jk}\in O_{j}^{+}$, that implies $O_{j}^{\pm
}\subseteq\Omega_{\pm}\left( -\omega_{jk},\delta\right) =\Omega_{\mp}\left(
\omega_{jk},\delta\right) .$ Proceeding similarly for $\omega_{jk},\omega\in
O_{k},$ we obtain the result of Lemma \[representation221\]$.$
Let us take $\{O_{j}\}_{j=1,2,...,n},$ such that for some $n,$ they are an open cover of $\mathbb{S}^{2}$ with the following property: if $O_{j}\cap
O_{k}=\varnothing,$ then $\operatorname*{dist}\left( O_{j},O_{k}\right) >0.$ For every $j$ we take $\chi_{j}\left( \omega\right) \in C^{\infty}\left(
\mathbb{S}^{2}\right) ,$ $\ \chi_{j}\left( \omega\right) =\chi_{j}\left(
-\omega\right) ,$ such that $\sum_{j=1}^{n}\chi_{j}\left( \omega\right)
=1,$ for any $\omega\in\mathbb{S}^{2}.$
We decompose $S\left( E\right) $ as the sum $$S\left( E\right) =\sum_{j,k=1}^{n}\chi_{j}S\left( E\right) \chi_{k}.
\label{representation170}$$ Then the kernel $s\left( \omega,\theta;E\right) ~$of the scattering matrix $S\left( E\right) $ decomposes as the sum $$s\left( \omega,\theta;E\right) =\sum_{j,k=1}^{n}\chi_{j}\left(
\omega\right) s\left( \omega,\theta;E\right) \chi_{k}\left( \theta\right)
. \label{representation187}$$
Suppose that $O_{j}\cap O_{k}\neq\varnothing$ and let $\omega_{jk}\in
O_{j}\cap O_{k}$ be fixed. We take $\omega_{kj}=\omega_{jk}.$ Let us define $$\chi_{jk}\left( \omega,\theta\right) :=\chi_{jk}^{+}\left( \omega\right)
\chi_{jk}^{+}\left( \theta\right) -\chi_{jk}^{-}\left( \omega\right)
\chi_{jk}^{-}\left( \theta\right) , \label{representation225}$$ where $\chi_{jk}^{\pm}\left( \omega\right) \in C^{\infty}\left(
\mathbb{S}^{2}\right) $ are such that $\chi_{jk}^{\pm}\left( \omega\right)
=\chi_{kj}^{\pm}\left( \omega\right) ,$ $\chi_{jk}^{\pm}\left(
\omega\right) =1$ for $\omega\in\Omega_{\pm}\left( \omega_{jk},\delta\right) $ and $\chi_{jk}^{\pm}\left( \omega\right) =0$ for $\pm\left\langle \omega,\omega_{jk}\right\rangle <0.$ Note that $\chi
_{jk}\left( \omega,\theta\right) =\chi_{jk}\left( \theta,\omega\right) $ and $\chi_{jk}\left( \omega,\theta\right) =-\chi_{jk}\left( -\omega
,-\theta\right) $ Moreover, Lemma \[representation221\] implies the following properties of the function $\chi_{jk}\left( \omega,\theta\right)
:$ if $\omega_{jk}\in O_{j}^{+}\cap O_{k}^{+},$ $\chi_{jk}\left(
\omega,\theta\right) =\pm1$ for $\left( \omega,\theta\right) \in O_{j}^{\pm}\times O_{k}^{\pm}\subseteq\Omega_{\pm}\left( \omega_{jk},\delta\right) \times\Omega_{\pm}\left( \omega_{jk},\delta\right) ,$ and $\chi_{jk}\left( \omega,\theta\right) =0$ for $\left( \omega,\theta\right)
\in O_{j}^{\pm}\times O_{k}^{\mp}\subseteq\Omega_{\pm}\left( \omega
_{jk},\delta\right) \times\Omega_{\mp}\left( \omega_{jk},\delta\right) $; if $\omega_{jk}\in O_{j}^{-}\cap O_{k}^{-},$ $\chi_{jk}\left( \omega
,\theta\right) =\mp1$ for $\left( \omega,\theta\right) \in O_{j}^{\pm
}\times O_{k}^{\pm}\subseteq\Omega_{\mp}\left( \omega_{jk},\delta\right)
\times\Omega_{\mp}\left( \omega_{jk},\delta\right) ,$ and $\chi_{jk}\left(
\omega,\theta\right) =0$ for $\left( \omega,\theta\right) \in O_{j}^{\pm
}\times O_{k}^{\mp}\subseteq\Omega_{\mp}\left( \omega_{jk},\delta\right)
\times\Omega_{\pm}\left( \omega_{jk},\delta\right) ;$ if $\omega_{jk}\in
O_{j}^{+}\cap O_{k}^{-},$ $\chi_{jk}\left( \omega,\theta\right) =\pm1$ for $\left( \omega,\theta\right) \in O_{j}^{\pm}\times O_{k}^{\mp}\subseteq\Omega_{\pm}\left( \omega_{jk},\delta\right) \times\Omega_{\pm
}\left( \omega_{jk},\delta\right) ,$ and $\chi_{jk}\left( \omega
,\theta\right) =0$ for $\left( \omega,\theta\right) \in O_{j}^{\pm}\times
O_{k}^{\pm}\subseteq\Omega_{\pm}\left( \omega_{jk},\delta\right)
\times\Omega_{\mp}\left( \omega_{jk},\delta\right) ;$ if $\omega_{jk}\in
O_{j}^{-}\cap O_{k}^{+},$ $\chi_{jk}\left( \omega,\theta\right) =\pm1$ for $\left( \omega,\theta\right) \in O_{j}^{\mp}\times O_{k}^{\pm}\subseteq\Omega_{\pm}\left( \omega_{jk},\delta\right) \times\Omega_{\pm
}\left( \omega_{jk},\delta\right) ,$ and $\chi_{jk}\left( \omega
,\theta\right) =0$ for $\left( \omega,\theta\right) \in O_{j}^{\pm}\times
O_{k}^{\pm}\subseteq\Omega_{\mp}\left( \omega_{jk},\delta\right)
\times\Omega_{\pm}\left( \omega_{jk},\delta\right) .$ We set $$s_{N,jk}\left( \omega,\theta;E\right) :=\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\chi_{jk}\left( \omega,\theta\right) \chi
_{j}\left( \omega\right) \chi_{k}\left( \theta\right) \int\limits_{\Pi
_{\omega_{jk}}}e^{i\nu\left( E\right) \left\langle y,\theta-\omega
\right\rangle }\mathbf{h}_{N,jk}\left( y,\omega,\theta;E\right) dy,
\label{representation26}$$ where $$\mathbf{h}_{N,jk}\left( y,\omega,\theta;E\right) :=\left(
\operatorname*{sgn}E\right) \left( a_{N}^{+}\left( y,\nu\left( E\right)
\omega;E\right) \right) ^{\ast}\left( \alpha\cdot\omega_{jk}\right)
\left( a_{N}^{-}\left( y,\nu\left( E\right) \theta;E\right) \right) ,
\label{representation27}$$ where $a_{N}^{\pm}\left( x,\xi;E\right) $ are the functions (\[eig33\]). The integral in (\[representation26\]) is understood as an oscillatory integral.
\[representation75\][ Note that the operator $S_{\operatorname{pr}}\left( E\right) $ with kernel $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing
}s_{N,jk}\left( \omega,\theta;E\right) $ is a PDO on the sphere $S^{2},$ with amplitude of the class $S^{0}.$ Indeed, let us denote by $\zeta$ the orthogonal projection of $\omega\in\Omega_{+}\left( \omega_{jk},\delta\right) $ on the two-dimensional plane $\Pi_{\omega_{jk}},$ and let $\Sigma$ be the projection of $\Omega_{+}\left( \omega_{jk},\delta\right) $ on $\Pi_{\omega_{jk}}.$ We identify below the points $\omega\in\Omega
_{+}\left( \omega_{jk},\delta\right) $ and $\zeta\in\Sigma$ and for any function $f\left( \omega\right) $, $\omega\in\Omega_{+}\left( \omega
_{jk},\delta\right) ,$ we define $\tilde{f}\left( \zeta\right) :=f\left(
\omega\right) .$ If $f\left( \omega,\theta\right) $ is a function of two variables $\omega,\theta\in\Omega_{+}\left( \omega_{jk},\delta\right) ,$ then we put $\tilde{f}\left( \zeta,\zeta^{\prime}\right) :=f\left(
\omega,\theta\right) .$ We have,$$\left.
{\displaystyle\int}
s_{N,jk}\left( \omega,\theta;E\right) f\left( \theta\right) d\theta
=\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}{\displaystyle\int_{\Pi_{\omega_{jk}}}}
{\displaystyle\int_{\Pi_{\omega_{jk}}}}
e^{i\left\langle y,\zeta^{\prime}-\zeta\right\rangle }\mathbf{\tilde{h}}_{N,jk}^{\prime}\left( y,\zeta,\zeta^{\prime}\right) \tilde{f}\left(
\zeta^{\prime}\right) d\zeta^{\prime}dy,\right. \label{representation74}$$ with $\tilde{h}_{N,jk}^{\prime}\left( y,\zeta,\zeta^{\prime}\right)
:=\frac{\tilde{\chi}_{jk}\left( \zeta,\zeta^{\prime}\right) \tilde{\chi}_{j}\left( \zeta\right) \tilde{\chi}_{k}\left( \zeta^{\prime}\right)
}{\left( 1-\left\vert \zeta^{\prime}\right\vert ^{2}\right) ^{1/2}}\tilde
{h}_{N,jk}\left( y,\zeta,\zeta^{\prime};E\right) .$ Note that for $\omega,\theta\in$ $\Omega_{\pm}\left( \omega_{jk},\delta\right) ,$ the functions $a_{N}^{\pm}$ satisfy (\[eig27\]), for all $y\in\Pi_{\omega_{jk}}.$ Therefore, the amplitude $\tilde{h}_{N,jk}^{\prime}\left( y,\zeta
,\zeta^{\prime};E\right) $ of $S_{\operatorname{pr}}\left( E\right) $ belongs to the class $S^{0}.$]{}
We define also the function $g_{N,jk}^{\operatorname{int}}\left(
\omega,\theta;E\right) $ as $$\mathbf{g}_{N,jk}^{\operatorname{int}}\left( \omega,\theta;E\right)
:=s_{N,jk}\left( \omega,\theta;E\right) -s_{00}^{(jk)}\left( \omega
,\theta;E\right) , \label{representation181}$$ where$$\left. s_{00}^{(jk)}\left( \omega,\theta;E\right) :=\left(
\operatorname*{sgn}E\right) \left( 2\pi\right) ^{-2}\upsilon\left(
E\right) ^{2}\chi_{jk}\left( \omega,\theta\right) \chi_{j}\left(
\omega\right) \chi_{k}\left( \theta\right) \int\limits_{\Pi_{\omega_{jk}}}e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }P_{\omega}\left( E\right) \left( \alpha\cdot\omega_{jk}\right) P_{\theta
}\left( E\right) dy.\right. \label{representation182}$$ Proposition \[representation89\] shows that $\sum\limits_{O_{j}\cap
O_{k}\neq\varnothing}s_{00}^{(jk)}\left( \omega,\theta;E\right) $ is a Dirac-function on $\mathcal{H}\left( E\right) .$
We now formulate the results that we will prove in this Section. For $\omega_{0}\in\mathbb{S}^{2}$ we introduce cut-off function $\Psi_{\pm}\left(
\omega,\theta;\omega_{0}\right) \in C^{\infty}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) ,$ supported on $\Omega_{\pm}\left( \omega
_{0},\delta\right) \times\Omega_{\pm}\left( \omega_{0},\delta\right) .$ Moreover, let $\Psi_{1}\left( \omega,\theta\right) \in C^{\infty}\left(
\mathbb{S}^{2}\times\mathbb{S}^{2}\right) $ be supported on $O\times
O^{\prime},$ where $O,O^{\prime}\subseteq\mathbb{S}^{2}$ are open sets such that $\overline{O}\cap\overline{O}^{\prime}=\varnothing.$ We define $$s_{\operatorname{sing}}^{(N)}\left( \omega,\theta;E;\omega_{0}\right)
:=\pm\Psi_{\pm}\left( \omega,\theta;\omega_{0}\right) \left( 2\pi\right)
^{-2}\upsilon\left( E\right) ^{2}\int\limits_{\Pi_{\omega_{0}}}e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }\mathbf{h}_{N}\left( y,\omega,\theta;E;\omega_{0}\right) dy,
\label{representation248}$$ as an oscillatory integral, where $$\mathbf{h}_{N}\left( y,\omega,\theta;E;\omega_{0}\right) :=\left(
\operatorname*{sgn}E\right) \left( a_{N}^{+}\left( y,\nu\left( E\right)
\omega;E\right) \right) ^{\ast}\left( \alpha\cdot\omega_{0}\right) \left(
a_{N}^{-}\left( y,\nu\left( E\right) \theta;E\right) \right)
\label{representation252}$$ and $s_{\operatorname*{reg}}\left( \omega,\theta;E\right) :=\Psi_{1}\left(
\omega,\theta\right) s\left( \omega,\theta;E\right) .$
\[representation179\]Let the magnetic potential $A\left( x\right) $ and the electric potential $V\left( x\right) $ satisfy the estimates (\[eig31\]) and (\[eig32\]), respectively. For any $p$ and $q,$ $s_{\operatorname*{reg}}\left( \omega,\theta;E\right) \ $belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $ and its $C^{p}$-norm is a $O\left( E^{-q}\right) $ function. Moreover, for any $p$ and $q$ there exists $N,$ sufficiently large, such that, $\Psi_{\pm}\left(
\omega,\theta;\omega_{0}\right) s\left( \omega,\theta;E\right)
-s_{\operatorname{sing}}^{(N)}\left( \omega,\theta;E\right) $ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and moreover, its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert E\right\vert \rightarrow\infty$. These estimates are uniform in $\omega_{0},$ in the case when the $C^{p}$-norms of the function $\Psi_{\pm
}\left( \omega,\theta;\omega_{0}\right) $ are uniformly bounded on $\omega_{0}\in\mathbb{S}^{2}.$
Let us decompose the scattering matrix $S\left( E\right) $ in the sum (\[representation170\]). Then, taking $\Psi_{\pm}\left( \omega
,\theta;\omega_{jk}\right) =\chi_{jk}^{\pm}\left( \omega\right) \chi
_{jk}^{\pm}\left( \theta\right) \chi_{j}\left( \omega\right) \chi
_{k}\left( \theta\right) ,$ $\omega_{0}=\omega_{jk},$ in the definition of $s_{\operatorname{sing}}^{(N)},$ and noting that in this case $s_{\operatorname{sing}}^{(N)}=s_{N,jk},$ we obtain
If $O_{j}\cap O_{k}=\varnothing,$ then for any $p$ and $q,$ $\chi_{j}\left(
\omega\right) s\left( \omega,\theta;E\right) \chi_{k}\left( \theta\right)
$ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $ and its $C^{p}$-norm is a $O\left( E^{-q}\right) $ function. If $O_{j}\cap O_{k}\neq\varnothing,$ then for any $p$ and $q$ there exists $N,$ sufficiently large, such that, $\chi_{j}\left( \omega\right) s\chi
_{k}\left( \theta\right) -s_{N,jk}$ belongs to the class $C^{p}\left(
\mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and moreover, its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert E\right\vert
\rightarrow\infty$.
\[representation25\]Let the magnetic potential $A\left( x\right) $ and the electric potential $V\left( x\right) $ satisfy the estimates (\[eig31\]) and (\[eig32\]), respectively. Then, the scattering matrix $S\left( E\right) $ admits the following decomposition$$S\left( E\right) =I+\mathcal{G+R}\text{,} \label{representation180}$$ where $I$ is the identity in $\mathcal{H}\left( E\right) ,$ $\mathcal{G}$ is an integral operator with kernel $\mathbf{g}_{N}\left( \omega,\theta
;E\right) \mathbf{:=}\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}\chi
_{j}\left( \omega\right) \mathbf{g}_{N,jk}^{\operatorname{int}}\left(
\omega,\theta;E\right) \chi_{k}\left( \theta\right) ,$ which satisfies the estimate $$\left\vert \mathbf{g}_{N}\left( \omega,\theta;E\right) \right\vert \leq
C\left\vert \omega-\theta\right\vert ^{-\left( 3-\rho\right) },\text{
}\omega\neq\theta,\text{ for }\rho=\min\{\rho_{e},\rho_{m}\}<3,
\label{representation83}$$ and it is a continuous function of $\omega$ and $\theta,$ for $\rho>3;$ and $\mathcal{R}$ is an integral operator with kernel $r_{N}\left( \omega
,\theta;E\right) $. For any $p$ and $q$ there exists $N,$ sufficiently large, such that $r_{N}\left( \omega,\theta;E\right) $ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and moreover, its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert
E\right\vert \rightarrow\infty$.
Symmetries of the approximate kernel of the scattering matrix.
--------------------------------------------------------------
We note that the approximate kernels $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}s_{N,jk}$ satisfy the symmetry relations (\[sym4\]), (\[sym5\]), (\[representation106\]), (\[sym6\]), (\[representation107\]) and (\[sym10\]). Below we suppose that $E>m.$ The case $E<-m$ is analogous.
Let us first show that the approximate kernels $\sum\limits_{O_{j}\cap
O_{k}\neq\varnothing}s_{N,jk}$ are invariant under the gauge transformation $A\rightarrow A+\nabla\psi,$ for $\psi\in C^{\infty}\left( \mathbb{R}^{3}\right) $ such that $\partial^{\alpha}\psi=O\left( \left\vert
x\right\vert ^{-\rho-\left\vert \alpha\right\vert }\right) $ for $0\leq\left\vert \alpha\right\vert \leq1$ and some $\rho>0$ as $\left\vert
x\right\vert \rightarrow\infty$. We emphasize the dependence of different functions on $A.$ We get from (\[eig15\]) that$$\left. \Phi^{\pm}\left( x,\xi;E;A+\nabla\psi\right) =\pm\left(
\int\limits_{0}^{\infty}\left( \frac{\left\vert E\right\vert }{\left\vert
\xi\right\vert }V\left( x\pm t\omega\right) +\left\langle \omega,A\left(
x\pm t\omega\right) \right\rangle +\left\langle \omega,\nabla\psi\left( x\pm
t\omega\right) \right\rangle \right) dt\right) =\Phi^{\pm}\left(
x,\xi;E;A\right) -\psi\left( x\right) .\right. \label{sym40}$$ Next we show that$$b_{j}^{\pm}\left( x,\xi;E;A+\nabla\psi\right) =b_{j}^{\pm}\left(
x,\xi;E;A\right) \text{,} \label{sym41}$$ and$$c_{j}^{\pm}\left( x,\xi;E;A+\nabla\psi\right) =c_{j}^{\pm}\left(
x,\xi;E;A\right) \text{,} \label{sym42}$$ for $j\geq1.$ First we prove relations (\[sym41\]) and (\[sym42\]) for $j=1.$ From (\[eig12\]) and using (\[sym40\]) we get $b_{1}^{\pm}\left(
x,\xi;E;A+\nabla\psi\right) =\frac{\left\vert \xi\right\vert }{2E}P_{\omega
}\left( -E\right) \alpha\left( \triangledown\Phi^{\pm}\left(
x,\xi;E;A\right) -\triangledown\psi\left( x\right) +A+\nabla\psi\right)
P_{\omega}\left( E\right) =b_{1}^{\pm}\left( x,\xi;E;A\right) .$ Then, by (\[eig41\]), we have $F_{1}^{\pm}\left( x,\xi;E;A+\nabla\psi\right) $ $=F_{1}^{\pm}\left( x,\xi;E;A\right) $. Moreover, using the relation (\[eig16\]), we get $c_{1}^{\pm}\left( x,\xi;E;A+\nabla\psi\right)
=\pm\int\limits_{0}^{\infty}F_{1}^{\pm}\left( x\pm\omega t,\xi;E;A+\nabla
\psi\right) dt=\pm\int\limits_{0}^{\infty}F_{1}^{\pm}\left( x\pm\omega
t,\xi;E;A\right) dt=c_{1}^{\pm}\left( x,\xi;E;A\right) .$ By an argument similar to the case $j=1$ we prove relations (\[sym41\]) and (\[sym42\]) by induction for any $j.$ The definition (\[representation27\]) of $\mathbf{h}_{N,jk}$ and relations (\[eig33\]), (\[eig42\]), (\[eig8\]), (\[eig9\]), (\[sym40\])-(\[sym42\]) imply $\mathbf{h}_{N,jk}\left(
y,\omega,\theta;E;A+\nabla\psi\right) =\mathbf{h}_{N,jk}\left(
y,\omega,\theta;E;A\right) ,$ and hence, $s_{N,jk}\left( y,\omega
,\theta;E;A+\nabla\psi\right) =s_{N,jk}\left( y,\omega,\theta;E;A\right) .$
Now we show that relation (\[sym4\]) for $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}s_{N,jk}$ with an even electric potential $V$ and an odd magnetic potential $A$ holds. From (\[representation26\]) and the relation $\chi_{jk}\left( \omega,\theta\right) \chi_{j}\left( \omega\right)
\chi_{k}\left( \theta\right) =-\chi_{jk}\left( -\omega,-\theta\right)
\chi_{j}\left( -\omega\right) \chi_{k}\left( -\theta\right) $ we get $\beta s_{N,jk}\left( -\omega,-\theta;E\right) \beta=-\left( 2\pi\right)
^{-2}\upsilon\left( E\right) ^{2}\chi_{jk}\left( \omega,\theta\right)
\chi_{j}\left( \omega\right) \chi_{k}\left( \theta\right) \int_{\Pi_{\omega_{jk}}}e^{i\nu\left( E\right) \left\langle y,\theta
-\omega\right\rangle }\beta\mathbf{h}_{N,jk}\left( -y,-\omega,-\theta
;E\right) \beta dy.$ Thus, we need to show that $$\beta\mathbf{h}_{N,jk}\left( y,\omega,\theta;E\right) =-\mathbf{h}_{N,jk}\left( -y,-\omega,-\theta;E\right) \beta. \label{representation120}$$ By relation (\[representation27\]), equation (\[representation120\]) is equivalent to$$\left. \beta\left( a_{N}^{+}\left( y,\nu\left( E\right) \omega;E\right)
\right) ^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{-}\left( y,\nu\left( E\right) \theta;E\right) \right) =-\left(
a_{N}^{+}\left( -y,-\nu\left( E\right) \omega;E\right) \right) ^{\ast
}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{-}\left(
-y,-\nu\left( E\right) \theta;E\right) \right) \beta.\right.
\label{sym43}$$ Under the assumptions on $V$ and $A$ we get from (\[eig15\]) that$$\left.
\begin{array}
[c]{c}\Phi^{\pm}\left( x,\xi;E\right) =\pm\left( \int\limits_{0}^{\infty}\left(
\frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left( x\pm
t\omega\right) +\left\langle \omega,A\left( x\pm t\omega\right)
\right\rangle \right) dt\right) \\
=\pm\left( \int\limits_{0}^{\infty}\left( \frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }V\left( -x\pm t\left( -\omega\right) \right)
+\left\langle \left( -\omega\right) ,A\left( -x\pm t\left( -\omega\right)
\right) \right\rangle \right) dt\right) =\Phi^{\pm}\left( -x,-\xi
;E\right) .
\end{array}
\right. \label{sym32}$$ Let us prove that$$\beta b_{j}^{\pm}\left( x,\xi;E\right) =b_{j}^{\pm}\left( -x,-\xi;E\right)
\beta\text{ and }\beta c_{j}^{\pm}\left( x,\xi;E\right) =c_{j}^{\pm}\left(
-x,-\xi,E\right) \beta, \label{sym34}$$ for $j\geq1.$ By (\[eig12\]), for $j=1,$ we have $\beta b_{1}^{\pm}\left(
x,\xi;E\right) =\frac{\left\vert \xi\right\vert }{2E}\beta P_{\omega}\left(
-E\right) \alpha\left( \triangledown\Phi^{\pm}\left( x,\xi;E\right)
+A\right) P_{\omega}\left( E\right) .$ Using that $\beta\alpha=-\alpha
\beta$, $-A\left( x\right) =A\left( -x\right) ,$ relation (\[sym32\]) and $-\triangledown\Phi^{\pm}\left( x,\xi;E\right) =\left( \triangledown
\Phi^{\pm}\right) \left( -x,-\xi;E\right) $ we obtain$$\left.
\begin{array}
[c]{c}\beta b_{1}^{\pm}\left( x,\xi;E\right) =-\frac{\left\vert \xi\right\vert
}{2E}P_{-\omega}\left( -E\right) \alpha\left( \triangledown\Phi^{\pm
}\left( x,\xi;E\right) +A\left( x\right) \right) P_{-\omega}\left(
E\right) \beta\\
=\frac{\left\vert \xi\right\vert }{2E}P_{-\omega}\left( -E\right)
\alpha\left( \left( \triangledown\Phi^{\pm}\right) \left( -x,-\xi
;E\right) +A\left( -x\right) \right) P_{-\omega}\left( E\right)
\beta=b_{1}^{\pm}\left( -x,-\xi;E\right) \beta.
\end{array}
\right. \label{sym33}$$ Using (\[eig41\]), (\[sym33\]) and equality $i\beta\triangledown
b_{1}^{\pm}\left( x,\xi;E\right) =-\left( i\triangledown b_{1}^{\pm
}\right) \left( -x,-\xi;E\right) \beta,$ we obtain $\beta F_{1}^{\pm
}\left( x,\xi;E\right) =-i\frac{\left\vert E\right\vert }{\left\vert
\xi\right\vert }P_{-\omega}\left( E\right) \alpha(-i\triangledown
+\triangledown\Phi^{\pm}\left( x,\xi;E\right) $ $+A\left( x\right) )\beta
b_{1}^{\pm}\left( x,\xi;E\right) =F_{1}^{\pm}\left( -x,-\xi;E\right)
\beta$ and therefore we get $\beta c_{1}^{\pm}\left( x,\xi;E\right) =\pm
\int\limits_{0}^{\infty}\beta F_{1}^{\pm}\left( x\pm\omega t,\xi;E\right)
dt=\pm\int\limits_{0}^{\infty}F_{1}^{\pm}\left( -x\pm\left( -\omega\right)
t,-\xi;E\right) \beta dt=c_{1}^{\pm}\left( -x,-\xi;E\right) \beta.$ By an argument similar to the case $j=1$ we prove relation (\[sym34\]) by induction$.$ Using (\[sym32\]), (\[sym34\]) and equality $\beta
\alpha=-\alpha\beta$ we obtain (\[sym43\]) and then, we get relation (\[representation120\]).
Let us consider now an odd electric potential $V\left( x\right) $ and an even magnetic potential $A\left( x\right) $ and prove equality (\[sym5\]). From (\[representation26\]) we get $\overline{s_{N,jk}\left( \omega
,\theta;E\right) }=\left( 2\pi\right) ^{-2}\upsilon\left( E\right)
^{2}\chi_{jk}\left( \omega,\theta\right) \chi_{j}\left( \omega\right)
\chi_{k}\left( \theta\right) \int\limits_{\Pi_{\omega_{jk}}}e^{i\nu\left(
E\right) \left\langle y,\theta-\omega\right\rangle }\overline{\mathbf{h}_{N,jk}\left( -y,\omega,\theta;E\right) }dy.$ Thus, we have to show that $\alpha_{2}\left( \overline{\mathbf{h}_{N,jk}\left( -y,\omega,\theta
;E\right) }\right) =\mathbf{h}_{N,jk}\left( y,\omega,\theta;-E\right)
\alpha_{2}.$ That is$$\left.
\begin{array}
[c]{c}\alpha_{2}\overline{\left( a_{N}^{+}\left( -y,\nu\left( E\right)
\omega;E\right) \right) ^{\ast}\left( \alpha\cdot\omega_{jk}\right)
\left( a_{N}^{-}\left( -y,\nu\left( E\right) \theta;E\right) \right) }\\
=-\left( a_{N}^{+}\left( y,\nu\left( E\right) \omega;-E\right) \right)
^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{-}\left(
y,\nu\left( E\right) \theta;-E\right) \right) \alpha_{2}.
\end{array}
\right. \label{sym36}$$ Let us show that $$\alpha_{2}\left( \overline{a_{N}^{\pm}\left( -y,\nu\left( E\right)
\omega;E\right) }\right) =a_{N}^{\pm}\left( y,\nu\left( E\right)
\omega;-E\right) \alpha_{2}. \label{sym15}$$ For the phase functions $\Phi^{\pm}$ we have the following equality$$\left.
\begin{array}
[c]{c}-\Phi^{\pm}\left( x,\xi;E\right) =\pm\left( -\int\limits_{0}^{\infty
}\left( \frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left(
x\pm t\omega\right) +\left\langle \omega,A\left( x\pm t\omega\right)
\right\rangle \right) dt\right) \\
=\pm\left( \int\limits_{0}^{\infty}\left( \frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }V\left( -x\mp t\omega\right) -\left\langle
\omega,A\left( -x\mp t\omega\right) \right\rangle \right) dt\right)
=\Phi^{\pm}\left( -x,\xi;-E\right) .
\end{array}
\right. \label{sym11}$$ Let us prove that$$\alpha_{2}\left( \overline{b_{j}^{\pm}\left( x,\xi;E\right) }\right)
=b_{j}^{\pm}\left( -x,\xi;-E\right) \alpha_{2} \label{sym12}$$ and$$\alpha_{2}\left( \overline{c_{j}^{\pm}\left( x,\xi;E\right) }\right)
=c_{j}^{\pm}\left( -x,\xi;-E\right) \alpha_{2}, \label{sym35}$$ for any $j\geq1.$ For $j=1,$ using that $\alpha_{2}\overline{\alpha}=-\alpha\alpha_{2}$, $\alpha_{2}\overline{P_{\omega}\left( E\right)
}=P_{\omega}\left( -E\right) \alpha_{2},$ $A\left( x\right) =A\left(
-x\right) $ and (\[sym11\]) we obtain$$\left.
\begin{array}
[c]{c}\alpha_{2}\overline{b_{1}^{\pm}\left( x,\xi;E\right) }=-\frac{\left\vert
\xi\right\vert }{2E}P_{\omega}\left( E\right) \alpha\left( \triangledown
\Phi^{\pm}\left( x,\xi;E\right) +A\left( x\right) \right) P_{\omega
}\left( -E\right) \alpha_{2}\\
=-\frac{\left\vert \xi\right\vert }{2E}P_{\omega}\left( E\right)
\alpha\left( \left( \triangledown\Phi^{\pm}\right) \left( -x,\xi
;-E\right) +A\left( -x\right) \right) P_{\omega}\left( -E\right)
\alpha_{2}=b_{1}^{\pm}\left( -x,\xi;-E\right) \alpha_{2}.
\end{array}
\right. \label{sym14}$$ From (\[sym11\]) and (\[sym14\]) we have the equality $\alpha_{2}\overline{F_{1}^{\pm}\left( x,\xi;E\right) }=i\frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }P_{\omega}\left( -E\right) \left(
\alpha\left( i\triangledown+\triangledown\Phi^{\pm}\left( x,\xi;E\right)
+A\left( x\right) \right) \right) \alpha_{2}\overline{b_{1}^{\pm}\left(
x,\xi;E\right) }=F_{1}^{\pm}\left( -x,\xi;-E\right) \alpha_{2}$ and $\alpha_{2}\overline{c_{1}^{\pm}\left( x,\xi;E\right) }=\pm\int\limits_{0}^{\infty}\alpha_{2}\overline{F_{1}^{\pm}\left( x\pm\omega
t,\xi;E\right) }dt=\pm\int\limits_{0}^{\infty}F_{1}^{\pm}\left( -x\mp\omega
t,\xi;-E\right) \alpha_{2}dt=c_{1}^{\pm}\left( -x,\xi;-E\right) \alpha
_{2}.$ Relation (\[sym12\]) and (\[sym35\]) for any $j$ can be proved similarly by induction$.$ From (\[sym11\]), (\[sym12\]) and (\[sym35\]) using the definition of $a_{N}^{\pm}$ (see (\[eig33\])) and recalling that $\alpha_{2}\overline{P_{\omega}\left( E\right) }=P_{\omega}\left(
-E\right) \alpha_{2},$ we get (\[sym15\]). Multiplying equation (\[sym15\]) on the left and on the right by $\alpha_{2}$ and taking adjoint, we prove that $$\alpha_{2}\left( \overline{a_{N}^{\pm}\left( -y,\nu\left( E\right)
\omega;E\right) }\right) ^{\ast}=\left( a_{N}^{\pm}\left( y,\nu\left(
E\right) \omega;-E\right) \right) ^{\ast}\alpha_{2}. \label{sym37}$$ By (\[sym15\]), (\[sym37\]) and $\alpha_{2}\overline{\alpha}=-\alpha
\alpha_{2}$ we obtain (\[sym36\]), what proves (\[sym5\]).
We suppose now that $A=0$ and prove the equality (\[representation106\]) for $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}s_{N,jk}$. Note that $$\chi_{jk}\left( \omega,\theta\right) \chi_{j}\left( \omega\right) \chi
_{k}\left( \theta\right) +\chi_{kj}\left( \omega,\theta\right) \chi
_{k}\left( \omega\right) \chi_{j}\left( \theta\right) =-(\chi_{jk}\left(
-\theta,-\omega\right) \chi_{j}\left( -\theta\right) \chi_{k}\left(
-\omega\right) +\chi_{kj}\left( -\theta,-\omega\right) \chi_{k}\left(
-\theta\right) \chi_{j}\left( -\omega\right) ). \label{sym45}$$ Thus, it is enough to prove that $\left( \alpha_{1}\alpha_{3}\right)
\overline{\mathbf{h}_{N,jk}\left( y,\omega,\theta;E\right) }=-\left(
\mathbf{h}_{N,jk}\left( y,-\theta,-\omega;E\right) \right) ^{\ast}\left(
\alpha_{1}\alpha_{3}\right) ,$ or$$\left.
\begin{array}
[c]{c}\left( \alpha_{1}\alpha_{3}\right) \overline{\left( a_{N}^{+}\left(
y,\nu\left( E\right) \omega;E\right) \right) ^{\ast}\left( \alpha
\cdot\omega_{jk}\right) \left( a_{N}^{-}\left( y,\nu\left( E\right)
\theta;E\right) \right) }\\
=-\left( a_{N}^{-}\left( y,-\nu\left( E\right) \omega;E\right) \right)
^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{+}\left(
y,-\nu\left( E\right) \theta;E\right) \right) \left( \alpha_{1}\alpha
_{3}\right) .
\end{array}
\right. \label{sym20}$$ First of all note that$$\left. -\Phi^{\pm}\left( x,\xi;E\right) =\pm\left( -\int\limits_{0}^{\infty}\frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left(
x\pm t\omega\right) dt\right) =\mp\left( \int\limits_{0}^{\infty}\frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left( x\mp
t\left( -\omega\right) \right) dt\right) =\Phi^{\mp}\left( x,-\xi
;E\right) .\right. \label{sym16}$$ Let us prove that$$\left( \alpha_{1}\alpha_{3}\right) \overline{b_{j}^{\pm}\left(
x,\xi;E\right) }=b_{j}^{\mp}\left( x,-\xi;E\right) \left( \alpha_{1}\alpha_{3}\right) \label{sym18}$$ and that$$\left( \alpha_{1}\alpha_{3}\right) \overline{c_{j}^{\pm}\left(
x,\xi;E\right) }=c_{j}^{\mp}\left( x,-\xi;E\right) \left( \alpha_{1}\alpha_{3}\right) , \label{sym19}$$ for $j\geq1.$ Consider the case $j=1.$ We have $\left( \alpha_{1}\alpha
_{3}\right) \overline{b_{1}^{\pm}\left( x,\xi;E\right) }=\frac{\left\vert
\xi\right\vert }{2E}\left( \alpha_{1}\alpha_{3}\right) \overline{P_{\omega
}\left( -E\right) \left( \alpha\cdot\triangledown\Phi^{\pm}\left(
x,\xi;E\right) \right) P_{\omega}\left( E\right) }.$ Using that $\left(
\alpha_{1}\alpha_{3}\right) \overline{\alpha}=-\alpha\left( \alpha_{1}\alpha_{3}\right) $, $\left( \alpha_{1}\alpha_{3}\right) \beta=\beta\left(
\alpha_{1}\alpha_{3}\right) $ and relation (\[sym16\]) we have$$\left. \left( \alpha_{1}\alpha_{3}\right) \overline{b_{1}^{\pm}\left(
x,\xi;E\right) }=\frac{\left\vert \xi\right\vert }{2E}P_{-\omega}\left(
-E\right) \alpha\left( \triangledown\left( \Phi^{\mp}\left( x,-\xi
;E\right) \right) \right) P_{-\omega}\left( E\right) \left( \alpha
_{1}\alpha_{3}\right) =b_{1}^{\mp}\left( x,-\xi;E\right) \left( \alpha
_{1}\alpha_{3}\right) .\right. \label{sym17}$$ From (\[sym17\]) we get $\left( \alpha_{1}\alpha_{3}\right) \overline
{F_{1}^{\pm}\left( x,\xi;E\right) }=i\frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }P_{-\omega}\left( E\right) \left(
\alpha\left( i\triangledown+\triangledown\Phi^{\pm}\left( x,\xi;E\right)
\right) \right) \left( \alpha_{1}\alpha_{3}\right) \overline{b_{1}^{\pm
}\left( x,\xi;E\right) }=-F_{1}^{\mp}\left( x,-\xi;E\right) \left(
\alpha_{1}\alpha_{3}\right) $ and $\left( \alpha_{1}\alpha_{3}\right)
\overline{c_{1}^{\pm}\left( x,\xi;E\right) }=\pm\int\limits_{0}^{\infty
}\left( \alpha_{1}\alpha_{3}\right) \overline{F_{1}^{\pm}\left( x\pm\omega
t,\xi;E\right) }dt=\mp\int\limits_{0}^{\infty}F_{1}^{\mp}\left( x\mp\left(
-\omega\right) t,-\xi;E\right) \left( \alpha_{1}\alpha_{3}\right)
dt=c_{1}^{\mp}\left( x,-\xi;E\right) \left( \alpha_{1}\alpha_{3}\right) .$ Similarly we prove relations (\[sym18\]) and (\[sym19\]) for any $j.$ From (\[sym16\])-(\[sym19\]), using the identity $\left( \alpha_{1}\alpha
_{3}\right) \overline{\alpha}=-\alpha\left( \alpha_{1}\alpha_{3}\right) $ we obtain (\[sym20\]).
Let us prove (\[sym6\]). Suppose that $\mathbf{V}$ is even. From (\[representation26\]) we get $\overline{s_{N,jk}\left( \omega
,\theta;E\right) }=\left( 2\pi\right) ^{-2}\upsilon\left( E\right)
^{2}\chi_{jk}\left( \omega,\theta\right) \chi_{j}\left( \omega\right)
\chi_{k}\left( \theta\right) \int\limits_{\Pi_{\omega_{jk}}}$ $\times
e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }\overline{\mathbf{h}_{N,jk}\left( -y,\omega,\theta;E\right) }dy.$ Moreover, note that $$\chi_{jk}\left( \omega,\theta\right) \chi_{j}\left( \omega\right) \chi
_{k}\left( \theta\right) +\chi_{kj}\left( \omega,\theta\right) \chi
_{k}\left( \omega\right) \chi_{j}\left( \theta\right) =\chi_{jk}\left(
\theta,\omega\right) \chi_{j}\left( \theta\right) \chi_{k}\left(
\omega\right) +\chi_{kj}\left( \theta,\omega\right) \chi_{k}\left(
\theta\right) \chi_{j}\left( \omega\right) ). \label{sym44}$$ Thus, in order to prove relation (\[sym6\]) we need to show that$$\left( \alpha_{1}\alpha_{3}\beta\right) \overline{\mathbf{h}_{N,jk}\left(
-y,\omega,\theta;E\right) }=\left( \mathbf{h}_{N,jk}\left( y,\theta
,\omega;E\right) \right) ^{\ast}\left( \alpha_{1}\alpha_{3}\beta\right) ,
\label{sym24}$$ which follows from$$\left.
\begin{array}
[c]{c}\left( \alpha_{1}\alpha_{3}\beta\right) \overline{\left( a_{N}^{+}\left(
-y,\nu\left( E\right) \omega;E\right) \right) ^{\ast}\left( \alpha
\cdot\omega_{jk}\right) \left( a_{N}^{-}\left( -y,\nu\left( E\right)
\theta;E\right) \right) }\\
=\left( a_{N}^{-}\left( y,\nu\left( E\right) \omega;E\right) \right)
^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{+}\left(
y,\nu\left( E\right) \theta;E\right) \right) \left( \alpha_{1}\alpha
_{3}\beta\right) .
\end{array}
\right. \label{sym25}$$ Note that$$\left.
\begin{array}
[c]{c}-\Phi^{\pm}\left( x,\xi;E\right) =\pm\left( -\int\limits_{0}^{\infty
}\left( \frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left(
x\pm t\omega\right) +\left\langle \omega,A\left( x\pm t\omega\right)
\right\rangle \right) dt\right) \\
=\mp\left( \int\limits_{0}^{\infty}\left( \frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }V\left( -x\mp t\omega\right) +\left\langle
\omega,A\left( -x\mp t\omega\right) \right\rangle \right) dt\right)
=\Phi^{\mp}\left( -x,\xi;E\right) .
\end{array}
\right. \label{sym21}$$ For $j=1,$ using that $\left( \alpha_{1}\alpha_{3}\beta\right)
\overline{\alpha}=\alpha\left( \alpha_{1}\alpha_{3}\beta\right) $ and (\[sym21\]) we have $\left( \alpha_{1}\alpha_{3}\beta\right)
\overline{b_{1}^{\pm}\left( x,\xi;E\right) }=\frac{\left\vert \xi\right\vert
}{2E}P_{\omega}\left( -E\right) \alpha(\left( \left( \triangledown
\Phi^{\mp}\right) \left( -x,\xi;E\right) \right) $ $+A\left( -x\right)
)P_{\omega}\left( E\right) \left( \alpha_{1}\alpha_{3}\beta\right)
=b_{1}^{\mp}\left( -x,\xi;E\right) \left( \alpha_{1}\alpha_{3}\beta\right)
,$ and $\left( \alpha_{1}\alpha_{3}\beta\right) \overline{F_{1}^{\pm}\left(
x,\xi;E\right) }=-i\frac{\left\vert E\right\vert }{\left\vert \xi\right\vert
}P_{\omega}\left( E\right) \left( \alpha\left( i\triangledown
+\triangledown\Phi^{\pm}\left( x,\xi;E\right) \right) +A\right) $ $\times\left( \alpha_{1}\alpha_{3}\beta\right) \overline{b_{1}^{\pm}\left(
x,\xi;E\right) }=-F_{1}^{\mp}\left( -x,\xi;E\right) \left( \alpha
_{1}\alpha_{3}\beta\right) .$ Therefore, we get $\left( \alpha_{1}\alpha
_{3}\beta\right) \overline{c_{1}^{\pm}\left( x,\xi;E\right) }=\pm\int_{0}^{\infty}\left( \alpha_{1}\alpha_{3}\beta\right) \overline{F_{1}^{\pm
}\left( x\pm\omega t,\xi;E\right) }$ $=\mp\int_{0}^{\infty}F_{1}^{\mp
}\left( -x\mp\omega t,\xi;E\right) \left( \alpha_{1}\alpha_{3}\beta\right)
dt=c_{1}^{\mp}\left( -x,\xi;E\right) \left( \alpha_{1}\alpha_{3}\beta\right) .$ Then, by induction in $j$ we obtain$$\left( \alpha_{1}\alpha_{3}\beta\right) \overline{b_{j}^{\pm}\left(
x,\xi;E\right) }=b_{j}^{\mp}\left( -x,\xi;E\right) \left( \alpha_{1}\alpha_{3}\beta\right) , \label{sym22}$$ and$$\left( \alpha_{1}\alpha_{3}\beta\right) \overline{c_{j}^{\pm}\left(
x,\xi;E\right) }=c_{j}^{\mp}\left( -x,\xi;E\right) \left( \alpha_{1}\alpha_{3}\beta\right) , \label{sym23}$$ for any $j\geq1.$ As before, relations (\[sym21\]), (\[sym22\]) and (\[sym23\]) imply (\[sym25\]).
Now suppose that $V$ is equal to zero and prove relation (\[representation107\]). As relation (\[sym44\]) holds, we have to show that $\gamma\mathbf{h}_{N,jk}\left( y,\omega,\theta;E\right) =\left(
\mathbf{h}_{N,jk}\left( y,\theta,\omega;-E\right) \right) ^{\ast}\gamma,$ or, which is the same$$\left.
\begin{array}
[c]{c}\gamma\left( a_{N}^{+}\left( y,\nu\left( E\right) \omega;E\right)
\right) ^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{-}\left( y,\nu\left( E\right) \theta;E\right) \right) \\
=-\left( a_{N}^{-}\left( y,\nu\left( E\right) \omega;-E\right) \right)
^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{+}\left(
y,\nu\left( E\right) \theta;-E\right) \right) \gamma.
\end{array}
\right. \label{sym27}$$ Noting that $\gamma\alpha=-\alpha\gamma,$ $\gamma\beta=-\beta\gamma$ and$$\left. \Phi^{\pm}\left( x,\xi;E\right) =\pm\left( \int\limits_{0}^{\infty
}\left\langle \omega,A\left( x\pm t\omega\right) \right\rangle dt\right)
=\mp\left( -\int\limits_{0}^{\infty}\left\langle \omega,A\left( x\pm
t\omega\right) \right\rangle dt\right) =\Phi^{\mp}\left( x,\xi;-E\right)
,\right. \label{sym26}$$ we have $\gamma b_{1}^{\pm}\left( x,\xi;E\right) =-\frac{\left\vert
\xi\right\vert }{2E}P_{\omega}\left( E\right) \alpha\left( \left(
\triangledown\Phi^{\mp}\left( x,\xi;-E\right) \right) +A\right) P_{\omega
}\left( -E\right) \gamma=b_{1}^{\mp}\left( x,\xi;-E\right) \gamma.$ It follows that $\gamma F_{1}^{\pm}\left( x,\xi;E\right) =-i\frac{\left\vert
E\right\vert }{\left\vert \xi\right\vert }P_{\omega}\left( -E\right) \left(
\alpha\left( -i\triangledown+\triangledown\Phi^{\pm}\left( x,\xi;E\right)
\right) +A\right) \gamma b_{1}^{\pm}\left( x,\xi;E\right) =-F_{1}^{\mp
}\left( x,\xi;-E\right) \gamma\ $and $\gamma c_{1}^{\pm}\left(
x,\xi;E\right) =\pm\int\limits_{0}^{\infty}\gamma F_{1}^{\pm}\left(
x\pm\omega t,\xi;E\right) dt$ $=\mp\int\limits_{0}^{\infty}F_{1}^{\mp}\left(
x\pm\omega t,\xi;-E\right) \gamma dt=c_{1}^{\mp}\left( x,\xi;-E\right)
\gamma.$ By induction in $j$ we get the equalities $\gamma b_{j}^{\pm}\left(
x,\xi;E\right) =b_{j}^{\mp}\left( x,\xi;-E\right) \gamma,$ and $\gamma
c_{j}^{\pm}\left( x,\xi;E\right) =c_{j}^{\mp}\left( x,\xi;-E\right)
\gamma,$ with $j\geq1.$ These two relations, together with (\[sym26\]) and identities $\gamma\alpha=-\alpha\gamma,$ $\gamma\beta=-\beta\gamma$ imply equality (\[sym27\]).
Finally, we consider an odd function $\mathbf{V.}$ From (\[representation26\]) we get $\left( s_{N,jk}\left( -\theta
,-\omega;-E\right) \right) ^{\ast}=\left( 2\pi\right) ^{-2}\chi
_{jk}\left( -\theta,-\omega\right) \chi_{j}\left( -\theta\right) \chi
_{k}\left( -\omega\right) $ $\times\int_{\Pi_{\omega_{jk}}}e^{i\nu\left(
E\right) \left\langle y,\theta-\omega\right\rangle }\left( \mathbf{h}_{N,jk}\left( -y,-\theta,-\omega;-E\right) \right) ^{\ast}dy.$ Then, by (\[sym45\]), relation (\[sym10\]) for $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}s_{N,jk}$ follows from $$\left.
\begin{array}
[c]{c}\gamma\beta\left( a_{N}^{+}\left( y,\nu\left( E\right) \omega;E\right)
\right) ^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{-}\left( y,\nu\left( E\right) \theta;E\right) \right) \\
=\left( a_{N}^{-}\left( -y,-\nu\left( E\right) \omega;-E\right) \right)
^{\ast}\left( \alpha\cdot\omega_{jk}\right) \left( a_{N}^{+}\left(
-y,-\nu\left( E\right) \theta;-E\right) \right) \gamma\beta.
\end{array}
\right. \label{sym31}$$ Note that$$\left.
\begin{array}
[c]{c}\Phi^{\pm}\left( x,\xi;E\right) =\pm\left( \int\limits_{0}^{\infty}\left(
\frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left( x\pm
t\omega\right) +\left\langle \omega,A\left( x\pm t\omega\right)
\right\rangle \right) dt\right) \\
=\mp\left( \int\limits_{0}^{\infty}\left( \frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }V\left( -x\pm t\left( -\omega\right) \right)
-\left\langle -\omega,A\left( -x\pm t\left( -\omega\right) \right)
\right\rangle \right) dt\right) =\Phi^{\mp}\left( -x,-\xi;-E\right) .
\end{array}
\right. \label{sym28}$$ Then, we have$$\left. \gamma\beta b_{1}^{\pm}\left( x,\xi;E\right) =-\frac{\left\vert
\xi\right\vert }{2E}P_{-\omega}\left( E\right) \alpha\left( \left( \left(
\triangledown\Phi^{\mp}\right) \left( -x,-\xi;-E\right) \right) +A\left(
-x\right) \right) P_{-\omega}\left( -E\right) \gamma\beta=b_{1}^{\mp
}\left( -x,-\xi;-E\right) \gamma\beta.\right. \label{sym29}$$ Using relations (\[sym28\]) and (\[sym29\]) we get $\gamma\beta F_{1}^{\pm}\left( x,\xi;E\right) =\left( i\frac{\left\vert E\right\vert
}{\left\vert \xi\right\vert }P_{-\omega}\left( -E\right) \left(
\alpha\left( -i\triangledown+\triangledown\Phi^{\pm}\left( x,\xi;E\right)
\right) +A\right) \right) \gamma\beta b_{1}^{\pm}\left( x,\xi;E\right)
=-F_{1}^{\mp}\left( -x,-\xi;-E\right) \gamma\beta.$ Thus, we have $\gamma\beta c_{1}^{\pm}\left( x,\xi;E\right) =\pm\int\limits_{0}^{\infty
}\gamma\beta F_{1}^{\pm}\left( x\pm\omega t,\xi;E\right) dt$ $=\mp
\int\limits_{0}^{\infty}F_{1}^{\mp}\left( -x\pm\left( -\omega\right)
t,-\xi;-E\right) \gamma\beta dt=c_{1}^{\mp}\left( -x,-\xi;-E\right)
\gamma\beta.$ By induction in $j$ we obtain$$\gamma\beta b_{j}^{\pm}\left( x,\xi;E\right) =b_{j}^{\mp}\left(
-x,-\xi;-E\right) \gamma\beta\text{,} \label{sym30}$$ and$$\gamma\beta c_{j}^{\pm}\left( x,\xi;E\right) =c_{j}^{\mp}\left(
-x,-\xi;-E\right) \gamma\beta, \label{sym39}$$ with $j\geq1.$ Using (\[sym28\]), (\[sym30\]) and (\[sym39\]) we get equality (\[sym31\]).
The identification operators.
-----------------------------
The proofs of Theorems \[representation179\] and \[representation25\] are based in a stationary formula for the scattering matrix $S\left( E\right) $.
In the general case, where $H_{0}$ and $H$ are self-adjoint operators in different Hilbert spaces $\mathcal{H}_{0}$ and $\mathcal{H}$ respectively, for $\Lambda\subset\sigma_{ac}\left( H_{0}\right) ,$ the wave operators are defined by the relation $$W_{\pm}\left( H,H_{0};J;\Lambda\right) :=s-\lim_{t\rightarrow\pm\infty
}e^{itH}Je^{-itH_{0}}E_{0}\left( \Lambda\right) , \label{representation35}$$ where $J$ is a bounded identification operator between the spaces $\mathcal{H}_{0}$ and $\mathcal{H},$ and $E_{0}\left( \Lambda\right) $ is the resolution of the identity for $H_{0}.$ When $\Lambda=\sigma_{ac}\left(
H_{0}\right) $ we write $W_{\pm}\left( H,H_{0};J\right) $ instead of $W_{\pm}\left( H,H_{0};J;\Lambda\right) .$ In our case, $\sigma_{ac}\left(
H_{0}\right) =\sigma\left( H_{0}\right) ,$ the spaces $\mathcal{H}_{0}$ and $\mathcal{H}$ coincide and the wave operators $W_{\pm}\left( H,H_{0}\right)
=W_{\pm}\left( H,H_{0};I\right) $ ($I$ is the identity operator) exist and are complete ([@9], [@14], [@33]). Thus, there is no need to consider an identification operator $J.$ However, it is convenient for us to introduce special identifications $J_{\pm}$ and to study the scattering matrix $\tilde{S}\left( E\right) $ associated to the wave operators $W_{\pm}\left(
H,H_{0};J;\Lambda\right) .$ We have to construct $J_{\pm}$ in such way that for a given $E,$ $\tilde{S}\left( E\right) =S\left( E\right) .$
We begin by defining the identifications $J_{\pm}=J_{\pm}^{\left( N\right)
}.$ We take $\varepsilon_{0}$ and $R$ as in Definition \[eig40\]. Let $\varepsilon>0$ be such that $$\sqrt{1-\delta^{2}}<\varepsilon<1-\varepsilon_{0}, \label{representation63}$$ where $\delta$ is given in the definition of the sets $\Omega_{\pm}\left(
\omega_{0},\delta\right) $ (see (\[representation64\])). Let $\sigma_{+}\in
C^{\infty}\left[ -1,1\right] ,$ be such that $\sigma_{+}\left( \tau\right)
=1$ if $\tau\in(-\varepsilon,1]$ and $\sigma_{+}\left( \tau\right) =0$ if $\tau\in\lbrack-1,-1+\varepsilon_{0}].$ We take $\sigma_{-}\left(
\tau\right) =\sigma_{+}\left( -\tau\right) .$ Now let $\eta\in C^{\infty
}\left( \mathbb{R}^{3}\right) ,$ $0\leq\eta\leq1,$ be such that $\eta\left(
x\right) =0$ in a neighborhood of zero and $\eta\left( x\right) =1$ for $\left\vert x\right\vert \geq R$. Let the function $\theta\left( t\right)
\in C^{\infty}\left( \mathbb{R}_{+}\right) $ be equal to zero if $t\leq c$ and equal to $1$ for $t\geq c_{1},$ with some $0<c<c_{1}<\nu\left( E\right)
$. Finally, we define $\zeta_{\pm}^{+}\left( x,\xi\right) \left.
:=\right. \sigma_{\pm}\left( \eta\left( x\right) \left\langle \hat{x},\hat{\xi}\right\rangle \right) \theta\left( \left\vert \xi\right\vert
\right) ,$ and $\zeta_{\pm}^{-}\left( x,\xi\right) \left. :=\right.
\sigma_{\mp}\left( \eta\left( x\right) \left\langle \hat{x},\hat{\xi
}\right\rangle \right) \theta\left( \left\vert \xi\right\vert \right) .$ Note that $\zeta_{\pm}^{+}$ is supported on $\Xi^{\pm}\left( E\right) ,$ for $E>m$ and $\zeta_{\pm}^{-}$ is supported on $\Xi^{\pm}\left( E\right) ,$ for $E<-m.$
We define the identifications $J_{\pm}=J_{\pm}^{\left( N\right) }$ as the PDO’s $$\left( J_{\pm}f\right) \left( x\right) :=\left( 2\pi\right) ^{-3/2}\int_{\mathbb{R}^{3}}e^{i\left\langle x,\xi\right\rangle }j_{N}^{\pm}\left(
x,\xi\right) \hat{f}\left( \xi\right) d\xi, \label{representation15}$$ where $j_{N}^{\pm}\left( x,\xi\right) :=a_{N}^{\pm}\left( x,\xi;\left\vert
\lambda\left( \xi\right) \right\vert \right) \zeta_{\pm}^{+}\left(
x,\xi\right) +a_{N}^{\pm}\left( x,\xi;-\left\vert \lambda\left( \xi\right)
\right\vert \right) \zeta_{\pm}^{-}\left( x,\xi\right) ,$ $\lambda\left(
\xi\right) =\lambda\left( \xi;E\right) :=\left( \operatorname*{sgn}E\right) \sqrt{\left\vert \xi\right\vert ^{2}+m^{2}}$ and the functions $a_{N}^{\pm}\left( x,\xi;E\right) $ are given by (\[eig33\]). As $a_{N}^{\pm}\left( x,\xi;E\right) $ satisfies the estimate (\[eig27\]) on $\Xi^{\pm}\left( E\right) $, then $j_{N}^{\pm}\left( x,\xi\right)
\in\mathit{S}^{0,0}$. Thus, using Proposition \[basicnotions24\] we see that $J_{\pm}$ are bounded. It follows from relation (\[eig36\]) that $$j_{N}^{\pm}\left( x,\xi\right) =a_{N}^{\pm}\left( x,\xi;\left\vert
\lambda\left( \xi\right) \right\vert \right) P^{+}\left( \xi\right)
\zeta_{\pm}^{+}\left( x,\xi\right) +a_{N}^{\pm}\left( x,\xi;-\left\vert
\lambda\left( \xi\right) \right\vert \right) P^{-}\left( \xi\right)
\zeta_{\pm}^{-}\left( x,\xi\right) . \label{representation133}$$
[ We take $\zeta_{\pm}^{+}$ for the projector $P^{+}\left(
\xi\right) $ on the positive energies $E$ and $\zeta_{\pm}^{-}$ for the projector $P^{-}\left( \xi\right) $ on the negative energies $E$, in order to assure that $J_{\pm}$ correspond to $W_{\pm}.$ This also explains the different definitions of $\Phi^{\pm}$ and $c_{j}^{\pm}$ for $E>m$ and $-E>m.$]{}
The following result is analogous to Lemma 1.1 of [@72] for the Schrödinger operator, that can be proved by using a stationary phase argument
\[representation56\]Let $A_{\pm}^{+}$ and $A_{\pm}^{-}$ be PDO operators with symbols $a_{\pm}^{+}\left( x,\xi\right) ,a_{\pm}^{-}\left(
x,\xi\right) \in\mathit{S}^{0,0},$ respectively, satisfying $$a_{\pm}^{+}\left( x,\xi\right) =0\text{ if }\pm\left\langle \hat{x},\hat
{\xi}\right\rangle \leq-1+\varepsilon_{0},\text{ and }a_{\pm}^{-}\left(
x,\xi\right) =0\text{ if }\mp\left\langle \hat{x},\hat{\xi}\right\rangle
\leq-1+\varepsilon_{0},\text{ }\varepsilon_{0}>0, \label{representation58}$$ for $\left\vert x\right\vert \geq R>0,$ and$$\text{ }a_{\pm}^{+}\left( x,\xi\right) =a_{\pm}^{-}\left( x,\xi\right)
=0\text{ for }\left\vert \xi\right\vert \leq c, \label{representation122}$$ for some $c>0.$ Moreover, suppose that $$a_{\pm}^{+}\left( x,\xi\right) P^{+}\left( \xi\right) =a_{\pm}^{+}\left(
x,\xi\right) \text{ and }a_{\pm}^{-}\left( x,\xi\right) P^{-}\left(
\xi\right) =a_{\pm}^{-}\left( x,\xi\right) . \label{representation59}$$ Then, for any $f\in\mathcal{S}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) $ and any $N$ there is a constant $C_{N,f}$ such that $$\left. \left\Vert A_{\pm}e^{-itH_{0}}f\right\Vert \leq C_{N,f}\left(
1+\left\vert t\right\vert \right) ^{-N},\text{ }\mp t>0,\right.
\label{representation57}$$ where $A_{\pm}$ is either $A_{\pm}^{+}$ or $A_{\pm}^{-}.$
Note that the symbols $a_{N}^{\pm}\left( x,\xi;\left\vert \lambda\left(
\xi\right) \right\vert \right) \zeta_{\pm}^{+}\left( x,\xi\right) $ and $a_{N}^{\pm}\left( x,\xi;-\left\vert \lambda\left( \xi\right) \right\vert
\right) \zeta_{\pm}^{-}\left( x,\xi\right) $ satisfy the assumptions of Lemma \[representation56\].
Let us define $\left( \mathbf{J}_{\pm}f\right) \left( x\right) :=\left(
2\pi\right) ^{-3/2}\int_{\mathbb{R}^{3}}e^{i\left\langle x,\xi\right\rangle
}\mathbf{j}_{\pm}\left( x,\xi\right) \hat{f}\left( \xi\right) d\xi,$ with $\mathbf{j}_{\pm}\left( x,\xi\right) :=P^{+}\left( \xi\right) \zeta_{\pm
}^{+}\left( x,\xi\right) +P^{-}\left( \xi\right) \zeta_{\pm}^{-}\left(
x,\xi\right) .$ Then, we have $\left( \left( J_{\pm}-\mathbf{J}_{\pm
}\right) f\right) \left( x\right) =\left( 2\pi\right) ^{-3/2}\int_{\mathbb{R}^{3}}e^{i\left\langle x,\xi\right\rangle }\tilde{j}_{\pm
}\left( x,\xi\right) \hat{f}\left( \xi\right) d\xi,$ where $\tilde{j}_{\pm}\left( x,\xi\right) \in\mathit{S}^{-\left( \rho-1\right) ,0}.$ Note that $$\lim_{\left\vert t\right\vert \rightarrow\infty}\left\Vert \left( J_{\pm
}-\mathbf{J}_{\pm}\right) e^{-iH_{0}t}f\right\Vert =0.
\label{representation82}$$ The last equality is consequence of the following Proposition, that also can be proved by using a stationary phase argument
\[representation118\]Let $A$ be a PDO with symbol $a\left( x,\xi\right)
\in\mathit{S}^{-\sigma,0},$ for some $\sigma>0,$ such that $a\left(
x,\xi\right) =0$ for $\left\vert \xi\right\vert \leq c,$ $c>0$. Then, for $f\in\mathcal{S}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) $ the following estimate holds$$\left\Vert Ae^{-iH_{0}t}f\right\Vert \leq C\left\langle t\right\rangle
^{-\sigma}. \label{representation203}$$ In particular, we get $$\lim_{t\rightarrow\pm\infty}\left\Vert Ae^{-iH_{0}t}f\right\Vert =0,\text{ for
any }f\in L^{2}. \label{representation117}$$
Using Lemma \[representation56\] and equality (\[representation82\]) we obtain, by a stationary phase argument, the following result
The following equalities hold $$s-\lim_{t\rightarrow\pm\infty}\left( J_{\pm}-\theta\left( \sqrt{H_{0}^{2}-m^{2}}\right) \right) e^{-iH_{0}t}=0, \label{representation17}$$$$s-\lim_{\left\vert t\right\vert \rightarrow\infty}J_{+}^{\ast}J_{-}e^{-iH_{0}t}=0. \label{representation18}$$
As the wave operators $W_{\pm}\left( H,H_{0}\right) $ exist, relation (\[representation17\]) implies that $W_{\pm}\left( H,H_{0};J_{\pm}\right)
$ exist and the following equality holds$$W_{\pm}\left( H,H_{0}\right) \theta\left( \sqrt{H_{0}^{2}-m^{2}}\right)
=W_{\pm}\left( J_{\pm}\right) . \label{representation91}$$ Note that the existence of the wave operators $W_{\pm}\left( H,H_{0};J_{\pm
}\right) $ can be proved in the same way as in [@33], where similar identification operators $J_{\pm}$ were defined.
We define the scattering operator $\mathbf{S}\left( J_{+},J_{-}\right) ,$ associated to the wave operators $W_{\pm}\left( H,H_{0};J_{\pm}\right) ,$ by the relation $\mathbf{S}\left( J_{+},J_{-}\,\right) $ $:=W_{+}^{\ast}\left(
H,H_{0};J_{+}\right) W_{-}\left( H,H_{0};J_{-}\right) .$ To simplify the notation we denote $\mathbf{\tilde{S}=S}\left( J_{+},J_{-}\right) .$
Identity (\[representation91\]) implies that the scattering operators $\mathbf{S}$ and $\mathbf{\tilde{S}}$ are related by the equality $$\theta\left( \sqrt{H_{0}^{2}-m^{2}}\right) \mathbf{S}\theta\left(
\sqrt{H_{0}^{2}-m^{2}}\right) =\mathbf{\tilde{S}}. \label{representation206}$$
The perturbation and the Mourre estimate.
-----------------------------------------
We define the perturbations $T_{\pm}$ as $$T_{\pm}=HJ_{\pm}-J_{\pm}H_{0}. \label{representation34}$$ Note that $\left( J_{\pm}H_{0}f\right) \left( x\right) =\left(
2\pi\right) ^{-3/2}\int_{\mathbb{R}^{3}}e^{i\left\langle x,\xi\right\rangle
}\left\vert \lambda\left( \xi\right) \right\vert \left( a_{N}^{\pm}\left(
x,\xi;\left\vert \lambda\left( \xi\right) \right\vert \right) \zeta_{\pm
}^{+}\left( x,\xi\right) -a_{N}^{\pm}\left( x,\xi;-\left\vert
\lambda\left( \xi\right) \right\vert \right) \zeta_{\pm}^{-}\left(
x,\xi\right) \right) \hat{f}\left( \xi\right) d\xi.$ Using relation (\[eig30\]) we have $$\left.
\begin{array}
[c]{c}g_{\pm}\left( x,\xi\right) :=\left( H-\left\vert \lambda\left( \xi\right)
\right\vert \right) \left( u_{N}^{\pm}\left( x,\xi;\left\vert
\lambda\left( \xi\right) \right\vert \right) \zeta_{\pm}^{+}\left(
x,\xi\right) \right) +\left( H+\left\vert \lambda\left( \xi\right)
\right\vert \right) \left( u_{N}^{\pm}\left( x,\xi;-\left\vert
\lambda\left( \xi\right) \right\vert \right) \zeta_{\pm}^{-}\left(
x,\xi\right) \right) \\
=e^{i\left\langle x,\xi\right\rangle }\left( r_{N}^{\pm}\left(
x,\xi;\left\vert \lambda\left( \xi\right) \right\vert \right) \zeta_{\pm
}^{+}\left( x,\xi\right) +r_{N}^{\pm}\left( x,\xi;-\left\vert
\lambda\left( \xi\right) \right\vert \right) \zeta_{\pm}^{-}\left(
x,\xi\right) \right) \\
-i\sum_{j=1}^{3}\left( \partial_{x_{j}}\zeta_{\pm}^{+}\left( x,\xi\right)
\right) \alpha_{j}u_{N}^{\pm}\left( x,\xi;\left\vert \lambda\left(
\xi\right) \right\vert \right) -i\sum_{j=1}^{3}\left( \partial_{x_{j}}\zeta_{\pm}^{-}\left( x,\xi\right) \right) \alpha_{j}u_{N}^{\pm}\left(
x,\xi;-\left\vert \lambda\left( \xi\right) \right\vert \right) .
\end{array}
\right. \label{representation16}$$ Then, taking $t_{\pm}\left( x,\xi\right) =e^{-i\left\langle x,\xi
\right\rangle }g_{\pm}\left( x,\xi\right) $ and using relation (\[eig43\]) we obtain the following representation for $T_{\pm}:$$$\left( T_{\pm}f\right) \left( x\right) =\int e^{i\left\langle
x,\xi\right\rangle }t_{\pm}\left( x,\xi\right) \hat{f}\left( \xi\right)
d\xi=\left( T_{\pm}^{1}f\right) \left( x\right) +\left( T_{\pm}^{2}f\right) \left( x\right) , \label{representation36}$$ where the parts $T_{\pm}^{1}$ and $T_{\pm}^{2}$ have the symbols$$t_{\pm}^{1}=r_{N}^{\pm}\left( x,\xi;\left\vert \lambda\left( \xi\right)
\right\vert \right) \zeta_{\pm}^{+}\left( x,\xi\right) +r_{N}^{\pm}\left(
x,\xi;-\left\vert \lambda\left( \xi\right) \right\vert \right) \zeta_{\pm
}^{-}\left( x,\xi\right) \label{representation37}$$ and $$t_{\pm}^{2}=-i\sum_{j=1}^{3}\left( \partial_{x_{j}}\zeta_{\pm}^{+}\left(
x,\xi\right) \right) \alpha_{j}a_{N}^{\pm}\left( x,\xi;\left\vert
\lambda\left( \xi\right) \right\vert \right) -i\sum_{j=1}^{3}\left(
\partial_{x_{j}}\zeta_{\pm}^{-}\left( x,\xi\right) \right) \alpha_{j}a_{N}^{\pm}\left( x,\xi;-\left\vert \lambda\left( \xi\right) \right\vert
\right) , \label{representation38}$$ respectively. Using (\[eig28\]) we get $$t_{\pm}^{1}\in\mathcal{S}^{-\rho-N,-N},\text{ }N\geq0,
\label{representation23}$$ and, using (\[eig27\]) we obtain $$t_{\pm}^{2}\in\mathcal{S}_{\pm}^{-1,0}. \label{representation24}$$
Let us introduce the operator $\mathbf{A}$, known as the generator of dilation, $\mathbf{A=}\frac{1}{2i}\sum
_{j=1}^{3}\left( x\cdot\nabla+\nabla\cdot x\right) .$ Note that $$i[H_{0},\mathbf{A}]=H_{0}-m\beta, \label{representation22}$$ and$$i[H,\mathbf{A}]=H-m\beta-\mathbf{V}+i[\mathbf{V,A}]=H-m\beta-\mathbf{V}-\left\langle x,\left( \nabla\mathbf{V}\right) \left( x\right)
\right\rangle . \label{representation121}$$ We recall that if$\ \mathbf{V}\ $satisfies the estimate (\[basicnotions46\]), there are no eigenvalues embedded in the absolutely continuous spectrum of $H$. Thus, the Mourre estimate$$\pm E_{H}\left( I\right) i[H,\mathbf{A}]E_{H}\left( I\right) \geq
cE_{H}\left( I\right) ,\text{ }c>0,\text{ }I=\left( E-\eta_{E},E+\eta
_{E}\right) ,\text{ }\pm E>m, \label{representation20}$$ is satisfied for some $\eta_{E}>0$ (Theorem 2.5 of [@33]).
The results we need below were proved in [@5] (see also [@34]) by introducing the so-called conjugate operator. Condition $\left( c_{n}\right)
$ of Definition 3.1 of [@5] for an operator $\mathbf{B}$ to be conjugate to $H,$ asks the following: The form $i[H,\mathbf{B}]$, defined on $D(\mathbf{B})\cap D(H)$, is bounded from below and closable. The self-adjoint operator associated with its closure is denoted $iB_{1}$. Assume $D(B_{1})\supset D(H)$. If $n>1$, assume for $j=2,3,...,n$ that the form $i[iB_{j-1},\mathbf{B}],$ defined on $D(\mathbf{B})\cap D(H)$, is bounded from below and closable. The associated self-adjoint operator is denoted $iB_{j}$, and $D(B_{j})\supset D(H)$ is assumed. If we know that the forms $i[iB_{j-1},\mathbf{B}]$ extend to self-adjoint operators $iB_{j}$, for all $j,$ and $D(B_{j})\supset D(H),$ then there is no need to ask the boundeness from below of $i[iB_{j-1},\mathbf{B}],$ $j\geq1,$ in order to obtain the results of [@5].
For $\pm E>m,$ let us consider the operator $\pm\mathbf{A.}$ Note that $i[iB_{j-1},\pm\mathbf{A}],$ $j\geq1,$ ($iB_{0}=H$) are not bounded from below. Nevertheless, using equality (\[representation121\]) we see that $i[iB_{j-1},\mathbf{A}]=H-m\beta-\mathbf{V+V}_{j},$ $j\geq1,$ where $\mathbf{V}_{0}=\mathbf{V}$ and $\mathbf{V}_{j}$ is defined recursively by $\mathbf{V}_{j}=-\left\langle x,\left( \nabla\mathbf{V}_{j-1}\right) \left(
x\right) \right\rangle ,$ $j\geq1.$ Then, the forms $i[iB_{j-1},\mathbf{A}],$ $j\geq1,$ extend to self-adjoint operators $iB_{j}$, and $D(B_{j})=D(H).$ Moreover $\pm\mathbf{A}$ satisfies relation (\[representation20\]). Thus, we can apply the results of [@5], taking, for $\pm E>m,$ the operator $\pm\mathbf{A}$ as conjugate to $H.$ Furthermore, we use the dilatation transformation argument (see [@39],[@30]) in order to prove that these results hold uniformly for $\left\vert E\right\vert \geq E_{0}>m,$ for any $E_{0}.$ We get the following
\[representation9\]Let estimates (\[eig31\]) and (\[eig32\]) hold. Define $\mathbf{P}_{+}:=E_{\mathbf{A}}\left( 0,\infty\right) ~$and $\mathbf{P}_{-}:=E_{\mathbf{A}}\left( -\infty,0\right) $ as the spectral projections of the operator $\mathbf{A}$ ($E_{\mathbf{A}}$ is the resolution of the identity for $\mathbf{A}$). For $\pm\operatorname{Re}z>m$ and $\operatorname{Im}z\geq0,$ the operators$$\left\langle \mathbf{A}\right\rangle ^{-p}R\left( z\right) \left\langle
\mathbf{A}\right\rangle ^{-p},p>\frac{1}{2}, \label{representation1}$$$$\left\langle \mathbf{A}\right\rangle ^{-1+p}\mathbf{P}_{\pm}R\left( z\right)
\left\langle \mathbf{A}\right\rangle ^{-q},\text{ \ }\left\langle
\mathbf{A}\right\rangle ^{-q}R\left( z\right) \mathbf{P}_{\pm}\left\langle
\mathbf{A}\right\rangle ^{-1+p} \label{representation2}$$ with $q>\frac{1}{2},$ $p<q$ and$$\left\langle \mathbf{A}\right\rangle ^{p}\mathbf{P}_{\mp}R\left( z\right)
\mathbf{P}_{\pm}\left\langle \mathbf{A}\right\rangle ^{p},\text{ \ }\forall p
\label{representation3}$$ are continuous in norm with respect to $z.$ Moreover, the norms of operators (\[representation1\])-(\[representation3\]) at $z=E+i0$ are bounded by $C\left\vert E\right\vert ^{-1}$ as $\left\vert E\right\vert \rightarrow
\infty.$
We now present the following two assertions ([@27],[@30]). We denote by $T$ the PDO with symbol $t.$ Recall that the clases $\mathcal{S}_{\pm
}^{m,n}$ where defined below (\[representation197\]).
\[representation7\]Let $t\in\mathcal{S}_{\pm}^{0,0}$ for one of the signs and let $p>0$ be an entire number. Then, the operator $\left\langle
x\right\rangle ^{p}\left\langle \nabla\right\rangle ^{p}T\left\langle
\mathbf{A}\right\rangle ^{-q}$ is bounded, for $q\geq p$.
Let $t\in\mathcal{S}_{\pm}^{n,m}$ for some $n$ and $m.$ Then the operator $\left\langle x\right\rangle ^{q}\left\langle \nabla\right\rangle
^{s}T\mathbf{P}_{\pm}\left\langle \mathbf{A}\right\rangle ^{p}$ is bounded for all real numbers $p,q,s.$
Since for $\mathbf{V}\ $satisfying the estimate (\[basicnotions46\]), there are no eigenvalues embedded in the absolutely continuous spectrum of $H,$ the resolvent $R\left( E\pm i0\right) $ is locally Hölder continuous on $(-\infty,-m)\cup(m,\infty).$ Thus, from Proposition 4.1 of [@15] we obtain
\[representation8\]Let $\mathbf{V}=\{V_{jk}\}_{j,k=1,2,3,4,}$ be an Hermitian $4\times4$-matrix valued function, such that $\left\vert \left(
x\cdot\nabla\right) ^{l}V_{jk}\left( x\right) \right\vert ,$ $j,k=1,2,3,4,$ are bounded for all $x\in\mathbb{R}^{3}$ and $l=0,1,2.$ Then, for any $E_{0}>m,$ the following estimate holds$$\sup_{\substack{0<\varepsilon<1\\\left\vert E\right\vert \geq E_{0}}}\left\Vert R\left( E\pm i\varepsilon\right) f\right\Vert _{L_{-s}^{2}}\leq
C_{s,E_{0}}\left\Vert f\right\Vert _{L_{s}^{2}},\text{ }1/2<s\leq1.$$
Using Propositions \[representation9\]-\[representation8\] we get, similarly to Proposition 3.5 of [@27] or Proposition 4.1 of [@30], the following result
\[representation33\]For any $p$, $q,$ and $N$ such that $N>p-\rho+1/2,$ $N\geq q,$ for $\left\vert \operatorname{Re}z\right\vert >m$ and $\operatorname{Im}z\geq0,$ the operator $$\left\langle x\right\rangle ^{p}\left\langle \nabla\right\rangle ^{q}T_{+}^{\ast}R\left( z\right) T_{-}\left\langle \nabla\right\rangle
^{q}\left\langle x\right\rangle ^{p}, \label{representation21}$$ is continuous in norm with respect to $z$ and, moreover, the operator $\left\langle x\right\rangle ^{p}\left\langle \nabla\right\rangle ^{q}T_{+}^{\ast}R_{+}\left( E\right) T_{-}\left\langle \nabla\right\rangle
^{q}\left\langle x\right\rangle ^{p},$ is uniformly bounded for $\left\vert
E\right\vert \geq E_{0}>m$, for all $E_{0}.$
The regular and singular parts of the scattering matrix.
--------------------------------------------------------
For $\left\vert E\right\vert >m,$ let us define the following operators$$S_{1}\left( E\right) =2\pi i\Gamma_{0}\left( E\right) T_{+}^{\ast}R_{+}\left( E\right) T_{-}\Gamma_{0}^{\ast}\left( E\right) ,
\label{representation14}$$ and for $f_{j}\in\mathcal{H}\left( E\right) $ such that $f_{1},f_{2}\in
C^{\infty}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ $j=1,2,$ $S_{2}\left( E\right) $ is defined as the following form$$\left( S_{2}\left( E\right) f_{1},f_{2}\right) :=-2\pi i\lim
_{\mu\downarrow0}\left( J_{+}^{\ast}T_{-}\delta_{\mu}\left( H_{0}-E\right)
g_{1},\Gamma_{0}^{\ast}\left( E\right) \Gamma_{0}\left( E\right)
g_{2}\right) , \label{representation134}$$ where $\delta_{\mu}\left( H_{0}-E\right) :=\left( 2\pi i\right)
^{-1}\left( R_{0}\left( E+i\mu\right) -R_{0}\left( E-i\mu\right) \right)
$ and $g_{j}$ are such that $\hat{g}_{j}\left( \xi\right) =\upsilon\left(
E\right) ^{-1}f_{j}\left( \hat{\xi}\right) \gamma\left( \left\vert
\xi\right\vert \right) $, $j=1,2,$ with $\hat{\xi}=\xi/\left\vert
\xi\right\vert ,$ $\gamma\in C_{0}^{\infty}\left( \mathbb{R}^{+}\setminus\{0\}\right) $ and $\gamma\left( \nu\left( E\right) \right) =1.$ Below we compute the limit in the R.H.S. of (\[representation134\]) and we prove that it is a bounded operator, that is independent of $\gamma.$
The kernel of the operator $S_{1}\left( E\right) $ is smooth:
\[representation32\]Let the magnetic potential $A\left( x\right) $ and the electric potential $V\left( x\right) $ satisfy the estimates (\[eig31\]) and (\[eig32\]) respectively. For any $p$ and $q,$ there is $N$ such that the kernel $s_{1}\left( \omega,\theta;E\right) $ of the operator $S_{1}\left( E\right) $ belongs to the class of $C^{p}\left(
\mathbb{S}^{2}\times\mathbb{S}^{2}\right) $-functions$,$ and furthermore its $C^{p}-$norm is $O\left( \left\vert E\right\vert ^{-q}\right) $ when $\left\vert E\right\vert \rightarrow\infty.$
Using the definition (\[basicnotions11\]) of $\Gamma_{0}\left( E\right) $, the relation (\[basicnotions39\]) for $\Gamma_{0}^{\ast}\left( E\right) ,$ and $\Gamma_{0}\left( E\right) \left\langle \nabla\right\rangle ^{-q_{0}}=\left( 1+\nu\left( E\right) ^{2}\right) ^{-\frac{q_{0}}{2}}\Gamma
_{0}\left( E\right) ,$ we get $\left( S_{1}\left( E\right) f\right)
\left( \omega\right) =i\left( 2\pi\right) ^{-2}\upsilon\left( E\right)
^{2}\left( 1+\nu\left( E\right) ^{2}\right) ^{-q_{0}}\int\int\limits_{\mathbb{S}^{2}}e^{-i\nu\left( E\right) \left\langle \omega
,x\right\rangle }\left[ P_{\omega}\left( E\right) \left\langle
\nabla\right\rangle ^{q_{0}}T_{+}^{\ast}R_{+}\left( E\right) T_{-}\left\langle \nabla\right\rangle ^{q_{0}}P_{\theta}\left( E\right) \right]
e^{i\nu\left( E\right) \left\langle \theta,x\right\rangle }$ $\times
f\left( \theta\right) d\theta dx.$ Note that for $s>3/2,$ $e^{i\nu\left(
E\right) \left\langle \theta,x\right\rangle }\left\langle x\right\rangle
^{-s}\in L^{2}.$ Using Lemma \[representation33\] with $N\geq3/2-\rho$ we see that $$\left.
\begin{array}
[c]{c}s_{1}\left( \omega,\theta;E\right) \left. :=\right. i\left( 2\pi\right)
^{-2}\upsilon\left( E\right) ^{2}\left( 1+\nu\left( E\right) ^{2}\right)
^{-q_{0}}{\displaystyle\int}
\left( e^{-i\nu\left( E\right) \left\langle \omega,x\right\rangle
}\left\langle x\right\rangle ^{-s}\right) \\
\times\left[ P_{\omega}\left( E\right) \left\langle x\right\rangle
^{s}\left\langle \nabla\right\rangle ^{q_{0}}T_{+}^{\ast}R_{+}\left(
E\right) T_{-}\left\langle \nabla\right\rangle ^{q_{0}}\left\langle
x\right\rangle ^{s}P_{\theta}\left( E\right) \right] \left( e^{i\nu\left(
E\right) \left\langle \theta,x\right\rangle }\left\langle x\right\rangle
^{-s}\right) dx,
\end{array}
\right. \label{representation177}$$ is a continuous function of $\omega$ and$\ \theta.$ Differentiating (\[representation177\]) $p$ times with respect to $\omega$ or $\theta$ we see that $\partial_{\omega}^{p}\partial_{\theta}^{p}\left( s_{1}\left(
\omega,\theta;E\right) \right) $ is continuous in $\omega$ and $\theta,$ and bounded by $C\left\vert E\right\vert ^{-q},$ if the operator $$\left\langle x\right\rangle ^{p_{0}}\left\langle \nabla\right\rangle ^{q_{0}}T_{+}^{\ast}R_{+}\left( E\right) T_{-}\left\langle \nabla\right\rangle
^{q_{0}}\left\langle x\right\rangle ^{p_{0}} \label{representation19}$$ with $p_{0}>p+3/2$ and $q_{0}\geq1+q/2+p,$ is bounded uniformly for $\left\vert E\right\vert \geq E_{0}>m$. Taking $N\geq p_{0}-\rho$ and $N\geq
q_{0}$ in Lemma \[representation33\] we get the desired result.
Let us study now the limit (\[representation134\]). It follows from (\[representation134\]) that$$\left. \left( S_{2}\left( E\right) f_{1},f_{2}\right) =-2\pi i\lim
_{\mu\downarrow0}\left( T_{-}\mathcal{F}^{\ast}\tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1},J_{+}\Gamma_{0}^{\ast}\left(
E\right) f_{2}\right) ,\right. \label{representation146}$$ where $\tilde{\delta}_{\mu}^{\left( \operatorname*{sgn}E\right) }=\frac{\mu
}{\pi}\left( \left( \lambda\left( \xi\right) -E\right) ^{2}+\mu
^{2}\right) ^{-1}P^{\left( \operatorname*{sgn}E\right) }\left( \xi\right)
,$ where $\lambda\left( \xi\right) $ are defined below (\[representation15\]). Note that the equality $\lambda\left( \xi\right)
=E$ is valid only if $\left\vert \xi\right\vert =\nu\left( E\right) .$
Using the relation (\[representation133\]) we obtain the following equation $J_{+}\Gamma_{0}^{\ast}\left( E\right) f_{2}=\left( 2\pi\right)
^{-\frac{3}{2}}\upsilon\left( E\right) \int e^{i\left\langle x,\nu\left(
E\right) \omega\right\rangle }j_{N}^{+}\left( x,\nu\left( E\right)
\omega;E\right) $ $\times f_{2}\left( \omega\right) d\omega,$ where $j_{N}^{+}\left( x,\xi;E\right) :=a_{N}^{+}\left( x,\xi;\lambda\left(
\xi\right) \right) \zeta_{+}^{\left( \operatorname*{sgn}E\right) }\left(
x,\xi\right) $ and moreover $$\left. \left( T_{-}f,J_{+}\Gamma_{0}^{\ast}\left( E\right) f_{2}\right)
=\left( 2\pi\right) ^{-3}\upsilon\left( E\right)
{\displaystyle\int}
\left(
{\displaystyle\int}
{\displaystyle\int}
e^{i\left\langle x,\xi^{\prime}-\nu\left( E\right) \omega\right\rangle
}\left( \left( j_{N}^{+}\left( x,\nu\left( E\right) \omega;E\right)
\right) ^{\ast}t_{-}\left( x,\xi^{\prime}\right) \hat{f}\left( \xi
^{\prime}\right) ,f_{2}\left( \omega\right) \right) d\omega d\xi^{\prime
}\right) dx,\right. \label{representation51}$$ for $f\in\mathcal{S}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) .$
Let us define $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{3}\right) $ such that $\varphi\left( 0\right) =1.$ Then, taking $f=\mathcal{F}^{\ast}\tilde{\delta}_{\mu}^{\left( \operatorname*{sgn}E\right) }\hat{g}_{1}$ in (\[representation51\]) and using relations (\[representation36\]), (\[eig36\]) and (\[representation146\]) we get$$\left. \left( S_{2}\left( E\right) f_{1},f_{2}\right) =-i\left(
2\pi\right) ^{-2}\upsilon\left( E\right) \lim_{\mu\downarrow0}\lim_{\varepsilon\rightarrow0}{\displaystyle\int}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( G^{\left( \varepsilon\right) }\left( \nu\left( E\right)
\omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) \right) d\omega d\xi^{\prime}.\right.
\label{representation188}$$ where, for some $\varsigma\in C_{0}^{\infty}\left( \mathbb{R}^{+}\right) ,$ such that $\varsigma\left( t\right) =1,$ in some neighborhood of $\nu\left(
E\right) $ and $\varsigma\left( t\right) =0,$ for $t<c_{1},$ $$G^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
:=\varsigma\left( \left\vert \xi\right\vert \right) \varsigma\left(
\left\vert \xi^{\prime}\right\vert \right) \int e^{i\left\langle
x,\xi^{\prime}-\xi\right\rangle }\left( j_{N}^{+}\left( x,\xi;E\right)
\right) ^{\ast}t_{-}\left( x,\xi^{\prime};E\right) \varphi\left(
\varepsilon x\right) dx, \label{representation40}$$ and $$\left. t_{-}\left( x,\xi^{\prime};E\right) :=t_{-}^{1}\left( x,\xi
^{\prime};E\right) +t_{-}^{2}\left( x,\xi^{\prime};E\right) ,\right.
\label{representation223}$$ with $t_{-}^{1}\left( x,\xi^{\prime};E\right) :=r_{N}^{-}\left(
x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) \zeta_{-}^{\left(
\operatorname*{sgn}E\right) }\left( x,\xi^{\prime}\right) $ and $t_{-}^{2}\left( x,\xi^{\prime};E\right) :=-i\sum_{j=1}^{3}\left( \partial
_{x_{j}}\zeta_{-}^{\left( \operatorname*{sgn}E\right) }\left( x,\xi
^{\prime}\right) \right) \alpha_{j}a_{N}^{-}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) .$ Below we study the limit of $G^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right) $, as $\varepsilon\rightarrow0$. Then, calculating the limit (\[representation188\]), we recover information about the smoothness and behavior for $\left\vert E\right\vert \rightarrow\infty$ of the kernel $s_{2}\left( \omega,\theta;E\right) $ of $S_{2}\left( E\right) .$
Let us denote $\hat{\xi}=\xi/\left\vert \xi\right\vert $ and $\hat{\xi
}^{\prime}=\xi^{\prime}/\left\vert \xi^{\prime}\right\vert .$ We use the following result
\[representation54\]Let $G\left( \xi,\xi^{\prime};E\right) ,$ defined on $\mathbb{R}^{3}\times\mathbb{R}^{3},$ be such that for each $E,$ $\left\vert
E\right\vert >m,$ the function $G\left( \nu\left( E\right) \hat{\xi},\xi^{\prime};E\right) $ is Hölder-continuous on $\xi^{\prime},$ uniformly for $\omega\in\mathbb{S}^{2}.$ Let $g_{1},$ $f_{1}$ and $f_{2}$ be as in (\[representation134\])$.$ Then, the following identity holds$$\left. \lim_{\mu\downarrow0}{\displaystyle\int}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( G\left( \nu\left( E\right) \omega,\xi^{\prime};E\right)
\tilde{\delta}_{\mu}^{\left( \operatorname*{sgn}E\right) }\hat{g}_{1}\left(
\xi^{\prime}\right) ,f_{2}\left( \omega\right) \right) d\omega
d\xi^{\prime}=\upsilon\left( E\right)
{\displaystyle\int_{\mathbb{S}^{2}}}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( G\left( \nu\left( E\right) \omega,\nu\left( E\right)
\theta;E\right) f_{1}\left( \theta\right) ,f_{2}\left( \omega\right)
\right) d\omega d\theta.\right. \label{representation65}$$
Note that for $f\in L^{1}\left( 0,\infty\right) ,$ Hölder-continuous in the point $\nu\left( E\right) ,$ the following relation holds$$\lim_{\mu\downarrow0}\frac{\mu}{\pi}\int_{0}^{\infty}f\left( r\right)
\left( \left( \sqrt{r^{2}+m^{2}}-\left\vert E\right\vert \right) ^{2}+\mu^{2}\right) ^{-1}dr=\frac{f\left( \nu\left( E\right) \right)
\left\vert E\right\vert }{\nu\left( E\right) }. \label{representation66}$$ Then, passing to the polar coordinate system in (\[representation65\]) and using that $\hat{g}_{1}\left( \xi^{\prime}\right) =\upsilon\left( E\right)
^{-1}f_{1}\left( \hat{\xi}^{\prime}\right) \gamma\left( \left\vert
\xi^{\prime}\right\vert \right) $ we get$$\left. \lim_{\mu\downarrow0}{\displaystyle\int}
{\displaystyle\int}
\left( G\left( \nu\left( E\right) \omega,\xi^{\prime};E\right)
\tilde{\delta}_{\mu}^{\left( \operatorname*{sgn}E\right) }\hat{g}_{1}\left(
\xi^{\prime}\right) ,f_{2}\left( \omega\right) \right) d\omega
d\xi^{\prime}=\lim_{\mu\downarrow0}\frac{\mu}{\pi}{\displaystyle\int_{0}^{\infty}}
\left( \left( \sqrt{r^{2}+m^{2}}-\left\vert E\right\vert \right) ^{2}+\mu^{2}\right) ^{-1}W\left( r\right) r^{2}dr,\right.
\label{representation153}$$ where $W\left( r\right) =\upsilon\left( E\right) ^{-1}\gamma\left(
r\right) \int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}\left( G\left(
\nu\left( E\right) \omega,r\theta;E\right) f_{1}\left( \theta\right)
,f_{2}\left( \omega\right) \right) d\omega d\theta.$ As $G\left(
\nu\left( E\right) \hat{\xi},\xi^{\prime};E\right) $ is a Hölder-continuous function of the variable $\xi^{\prime},$ uniformly in $\hat{\xi},$ we conclude that $W\left( r\right) $ is a Hölder-continuous function of $r.$ Therefore, applying the equality (\[representation66\]) to the R.H.S. of (\[representation153\]) and recalling that $\upsilon\left(
E\right) =\left( \left\vert E\right\vert \nu\left( E\right) \right)
^{1/2},$ we get the desired result.
Using (\[representation223\]) we decompose $G^{\left( \varepsilon\right)
}\left( \xi,\xi^{\prime};E\right) $ as $$G^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
=G_{1}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
+G_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right) ,
\label{representation166}$$ where $$\left. G_{1}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
=\varsigma\left( \left\vert \xi\right\vert \right) \varsigma\left(
\left\vert \xi^{\prime}\right\vert \right)
{\displaystyle\int}
e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }\left( j_{N}^{+}\left(
x,\xi;E\right) \right) ^{\ast}t_{-}^{1}\left( x,\xi^{\prime};E\right)
\varphi\left( \varepsilon x\right) dx,\right. \label{representation61}$$ and$$\left. G_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
=\varsigma\left( \left\vert \xi\right\vert \right) \varsigma\left(
\left\vert \xi^{\prime}\right\vert \right)
{\displaystyle\int}
e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }\left( j_{N}^{+}\left(
x,\xi;E\right) \right) ^{\ast}t_{-}^{2}\left( x,\xi^{\prime};E\right)
\varphi\left( \varepsilon x\right) dx.\right. \label{representation62}$$ Let us prove now the following result
\[representation183\]The limit $G_{1}^{\left( 0\right) }\left(
\nu\left( E\right) \hat{\xi},\nu\left( E\right) \hat{\xi}^{\prime
};E\right) :=\lim\limits_{\varepsilon\rightarrow\infty}G_{1}^{\left(
\varepsilon\right) }\left( \nu\left( E\right) \hat{\xi},\nu\left(
E\right) \hat{\xi}^{\prime};E\right) $ exists. For any $p$ and $q,$ there is $N,$ such that $G_{1}^{\left( 0\right) }\left( \nu\left( E\right)
\hat{\xi},\nu\left( E\right) \hat{\xi}^{\prime};E\right) \in C^{p}\left(
\mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q},$ as $\left\vert E\right\vert
\rightarrow\infty.$ Moreover, the limit in the R.H.S. of (\[representation188\]) with $G_{1}^{\left( \varepsilon\right) },$ instead of $G^{\left( \varepsilon\right) },$ exists, and $-i\left( 2\pi\right)
^{-2}\upsilon\left( E\right) \lim\limits_{\mu\downarrow0}\lim
\limits_{\varepsilon\rightarrow0}{\displaystyle\int}
{\displaystyle\int_{\mathbb{S}^{2}}}
(G_{1}^{\left( \varepsilon\right) }\left( \nu\left( E\right) \omega
,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left( \operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right) ,$ $f_{2}\left(
\omega\right) )d\omega d\xi^{\prime}$ $=-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}{\displaystyle\int_{\mathbb{S}^{2}}}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( G_{1}^{\left( 0\right) }\left( \nu\left( E\right) \omega
,\nu\left( E\right) \theta;E\right) f_{1}\left( \theta\right)
,f_{2}\left( \omega\right) \right) d\theta d\omega.$
As $j_{N}^{\pm}\left( x,\xi;E\right) \in\mathit{S}^{0,0}$, using (\[representation23\]) we see that $\left\vert \left( j_{N}^{+}\left(
x,\xi;E\right) \right) ^{\ast}t_{-}^{1}\left( x,\xi^{\prime};E\right)
\right\vert \leq C_{\alpha,\beta}\left( 1+\left\vert x\right\vert \right)
^{-\rho-N+\left\vert \alpha\right\vert +\left\vert \beta\right\vert
}\left\vert \xi^{\prime}\right\vert ^{-N},$ for $N\geq0.$ Thus, for $N$ big enough, the limit in (\[representation61\]), as $\varepsilon\rightarrow0,$ exists, and, moreover, $G_{1}^{\left( 0\right) }\left( \xi,\xi^{\prime
};E\right) $ is a function in $C^{p\left( N\right) }\left( \mathbb{R}^{3}\times\mathbb{R}^{3}\right) ,$ that decreases as $C\left\vert \xi
^{\prime}\right\vert ^{-q\left( N\right) },$ when $\left\vert \xi^{\prime
}\right\vert \rightarrow\infty$. Replacing $G^{\left( \varepsilon\right) }$ with $G_{1}^{\left( 0\right) }$ in (\[representation188\]) and using Lemma \[representation54\] to calculate the resulting limit we complete the proof.
We now study the term $G_{2}^{\left( \varepsilon\right) }\left( \xi
,\xi^{\prime};E\right) .$ Let us consider first the function $G_{2}^{\left(
\varepsilon\right) }\left( \xi,\xi^{\prime};E\right) $ for $\xi\neq
\xi^{\prime}.$ We prove the following result
\[representation145\]Let $O,O^{\prime}\subseteq\mathbb{S}^{2}$ be open sets such that $\overline{O}\cap\overline{O}^{\prime}=\varnothing.$ Then, there exists a function $G_{2,O,O^{\prime}}^{\left( 0\right) }\left(
\xi,\xi^{\prime};E\right) ,$ such that for any $p$ and $q$, $G_{2,O,O^{\prime
}}^{\left( 0\right) }\left( \xi,\xi^{\prime};E\right) $ is of $C^{p}\left( \mathbb{R}^{3}\times\mathbb{R}^{3}\right) $-class, and its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$ as $\left\vert
E\right\vert \rightarrow\infty,$ and moreover, for $g_{1},$ $f_{1}$ and $f_{2}$ as in (\[representation134\]), with the additional property $f_{1}\in C_{0}^{\infty}\left( O^{\prime}\right) $ and $f_{2}\in
C_{0}^{\infty}\left( O\right) ,$ $-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) \lim\limits_{\mu\downarrow0}\lim\limits_{\varepsilon
\rightarrow0}{\displaystyle\int}
{\displaystyle\int\limits_{\mathbb{S}^{2}}}
$ $\times(G_{2}^{\left( \varepsilon\right) }\left( \nu\left( E\right)
\omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) )d\omega d\xi^{\prime}=-i\left( 2\pi\right)
^{-2}\upsilon\left( E\right) ^{2}{\displaystyle\int\limits_{\mathbb{S}^{2}}}
{\displaystyle\int\limits_{\mathbb{S}^{2}}}
(G_{2,O,O^{\prime}}^{\left( 0\right) }\left( \nu\left( E\right)
\omega,\nu\left( E\right) \theta;E\right) f_{1}\left( \theta\right)
,f_{2}\left( \omega\right) )d\theta d\omega.$ In particular, the function $G_{2,O,O^{\prime}}^{\left( 0\right) }$ satisfies the estimate$$\left\Vert G_{2,O,O^{\prime}}^{\left( 0\right) }\left( \nu\left( E\right)
\hat{\xi},\nu\left( E\right) \hat{\xi}^{\prime};E\right) \right\Vert
_{C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) }\leq C_{p}\left(
O,O^{\prime}\right) \left\vert E\right\vert ^{-q}, \label{representation224}$$ for any $p$ and $q.$
Choosing $l$ such that $\sqrt{3}\left\vert \xi_{l}-\xi_{l}^{\prime}\right\vert
\geq\left\vert \xi-\xi^{\prime}\right\vert >0$ and integrating (\[representation62\]) by parts $n$ times we get $$G_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
=G_{2,O,O^{\prime}}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime
};E\right) +R_{jk}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime
};E\right) , \label{representation245}$$ where $G_{2,O,O^{\prime}}^{\left( \varepsilon\right) }\left( \xi
,\xi^{\prime};E\right) :=\left( \xi_{l}^{\prime}-\xi_{l}\right)
^{-n}\varsigma\left( \left\vert \xi\right\vert \right) \varsigma\left(
\left\vert \xi^{\prime}\right\vert \right) \int e^{i\left\langle
x,\xi^{\prime}-\xi\right\rangle }\varphi\left( \varepsilon x\right) \left(
i\partial_{x_{l}}\right) ^{n}\left( \left( j_{N}^{+}\left( x,\xi;E\right)
\right) ^{\ast}t_{-}^{2}\left( x,\xi^{\prime};E\right) \right) dx$ and $R_{jk}^{\left( \varepsilon\right) }$ is given by $R_{jk}^{\left(
\varepsilon\right) }\left( \xi,\xi^{\prime};E\right) :=\sum_{m=1}^{n}\varepsilon^{m}\left( \xi_{l}^{\prime}-\xi_{l}\right) ^{-n}\int
e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }g_{m}\left( x,\xi
,\xi^{\prime};E\right) \varphi^{\left( m\right) }\left( \varepsilon
x\right) dx,$ with $g_{m}\in\mathcal{S}^{m-n},$ for $m\leq n.$ Note that $$\lim_{\varepsilon\rightarrow0}{\displaystyle\int}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( R_{jk}^{\left( \varepsilon\right) }\left( \nu\left( E\right)
\omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) \right) d\omega d\xi^{\prime}=0.
\label{representation243}$$ Indeed, substituting the definition of $R_{jk}^{\left( \varepsilon\right) }$ in (\[representation243\]) and integrating several times by parts in the variable $\xi^{\prime}$ the resulting expression, we obtain a product of an absolutely convergent integral, uniformly bounded on $\varepsilon,$ and $\varepsilon^{m}.$
For $n>2$ the integral in the relation for $G_{2,O,O^{\prime}}^{\left(
\varepsilon\right) }$ is absolutely convergent. Hence, the limit of $G_{2,O,O^{\prime}}^{\left( \varepsilon\right) },$ as $\varepsilon
\rightarrow0,$ exists and it is equal to the absolutely convergent integral$$G_{2,O,O^{\prime}}^{\left( 0\right) }\left( \xi,\xi^{\prime};E\right)
=\left( \xi_{j}-\xi_{j}^{\prime}\right) ^{-n}\varsigma\left( \left\vert
\xi\right\vert \right) \varsigma\left( \left\vert \xi^{\prime}\right\vert
\right) \int e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }\left(
i\partial_{x_{j}}\right) ^{n}\left( \left( j_{N}^{+}\left( x,\xi;E\right)
\right) ^{\ast}t_{-}\left( x,\xi^{\prime};E\right) \right) dx.
\label{representation41}$$ Moreover, for any $n$ we have the following estimate $$\left\vert \left( \partial_{\xi}^{\beta}\partial_{\xi^{\prime}}^{\beta^{\prime}}G_{2,O,O^{\prime}}^{\left( 0\right) }\right) \left(
\xi,\xi^{\prime};E\right) \right\vert \leq C_{p,jk}\left\vert \xi-\xi
^{\prime}\right\vert ^{-n},\text{ }\left\vert \beta\right\vert +\left\vert
\beta^{\prime}\right\vert =p, \label{representation137}$$ for $p<n-2.$ Introducing decomposition (\[representation245\]) in the R.H.S. of (\[representation188\]) and using relation (\[representation243\]) for the part corresponding to $R_{jk}^{\left( \varepsilon\right) }$ and Lemma \[representation54\] to calculate the limit, as $\mu\rightarrow0,$ of the part $G_{2,O,O^{\prime}}^{\left( 0\right) },$ we conclude the proof.
Now let us study the singularities of $G_{2}^{\left( \varepsilon\right)
}\left( \xi,\xi^{\prime};E\right) $ for $\xi^{\prime}=\xi.$ For an arbitrary $\omega_{0}\in\mathbb{S}^{2},$ we introduce cut-off functions $\Psi_{\pm
}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) \in C^{\infty}\left(
\mathbb{S}^{2}\times\mathbb{S}^{2}\right) ,$ supported on $\{\left( \hat
{\xi},\hat{\xi}^{\prime}\right) \in\mathbb{S}^{2}\times\mathbb{S}^{2}|\hat{\xi},\hat{\xi}^{\prime}\in\Omega_{\pm}\left( \omega_{0},\delta\right)
\}$ and consider $\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega
_{0}\right) G_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime
};E\right) .$ We need the following result
Let us take $\omega,\theta\in\Omega_{+}\left( \omega_{0},\delta\right) $ or $\omega,\theta\in\Omega_{-}\left( \omega_{0},\delta\right) $. Suppose that $\left\langle \theta,\hat{x}\right\rangle >\varepsilon,$ where $\varepsilon
>\sqrt{1-\delta^{2}}.$ Then, $\left\langle \omega,\hat{x}\right\rangle
>-\varepsilon.$
The case $\theta=\omega_{0}$ is immediate. Suppose that $\theta\neq\omega
_{0}.$ We prove the case $\omega,\theta\in\Omega_{+}\left( \omega_{0},\delta\right) .$ The proof for the case $\omega,\theta\in\Omega_{-}\left(
\omega_{0},\delta\right) $ is analogous. Let $\hat{y}$ be a unit vector in the plane generated by $\theta$ and $\omega_{0},$ that is orthogonal to $\omega_{0}:\left\langle \omega_{0},\hat{y}\right\rangle =0.$ We write $\theta$ as $\theta=\left\langle \theta,\omega_{0}\right\rangle \omega
_{0}+\left\langle \theta,\hat{y}\right\rangle \hat{y}.$ Let $x$ be such that $\left\langle \theta,\hat{x}\right\rangle >\varepsilon.$ Then, from the relation $\left\langle \theta,\hat{x}\right\rangle =\left\langle \theta
,\omega_{0}\right\rangle \left\langle \omega_{0},\hat{x}\right\rangle
+\left\langle \theta,\hat{y}\right\rangle \left\langle \hat{y},\hat
{x}\right\rangle ,$ and $\left\vert \left\langle \theta,\hat{y}\right\rangle
\right\vert <\sqrt{1-\delta^{2}},$ for $\theta\in\Omega_{+}\left( \omega
_{0},\delta\right) ,$ it follows that $\left\langle \omega_{0},\hat
{x}\right\rangle >0.$ For $\omega\in\Omega_{+}\left( \omega_{0},\delta\right) $ and for $\hat{z}$ such that $\left\langle \hat{y},\hat
{z}\right\rangle =\left\langle \omega_{0},\hat{z}\right\rangle =0,$ we have $\omega=\left\langle \omega,\omega_{0}\right\rangle \omega_{0}+\left\langle
\omega,\hat{y}\right\rangle \hat{y}+\left\langle \omega,\hat{z}\right\rangle
\hat{z}.$ Then, it follows$$\left. \left\langle \omega,\hat{x}\right\rangle =\left\langle \omega
,\omega_{0}\right\rangle \left\langle \omega_{0},\hat{x}\right\rangle
+\left\langle \omega,\hat{y}\right\rangle \left\langle \hat{y},\hat
{x}\right\rangle +\left\langle \omega,\hat{z}\right\rangle \left\langle
\hat{z},\hat{x}\right\rangle >\left\langle \omega,\hat{y}\right\rangle
\left\langle \hat{y},\hat{x}\right\rangle +\left\langle \omega,\hat
{z}\right\rangle \left\langle \hat{z},\hat{x}\right\rangle .\text{ }\right.
\label{representation173}$$ If $\left\langle \omega,\hat{y}\right\rangle \hat{y}+\left\langle \omega
,\hat{z}\right\rangle \hat{z}=0\ $or $\left\langle \hat{x},\hat{y}\right\rangle \hat{y}+\left\langle \hat{x},\hat{z}\right\rangle \hat{z}=0,$ then from (\[representation173\]) we get $\left\langle \omega,\hat
{x}\right\rangle >0>-\delta.$ Suppose that $\left\langle \omega,\hat
{y}\right\rangle \hat{y}+\left\langle \omega,\hat{z}\right\rangle \hat{z}\neq0$ and $\left\langle \hat{y},\hat{x}\right\rangle \hat{y}+\left\langle
\hat{z},\hat{x}\right\rangle \hat{z}\neq0.$ Let us define $\omega_{\hat
{y},\hat{z}}=\frac{1}{\sqrt{\left\langle \omega,\hat{y}\right\rangle
^{2}+\left\langle \omega,\hat{z}\right\rangle ^{2}}}\left( \left\langle
\omega,\hat{y}\right\rangle \hat{y}+\left\langle \omega,\hat{z}\right\rangle
\hat{z}\right) $ and $\hat{x}_{\hat{y},\hat{z}}=\frac{1}{\sqrt{\left\langle
\hat{x},\hat{y}\right\rangle ^{2}+\left\langle \hat{x},\hat{z}\right\rangle
^{2}}}\left( \left\langle \hat{x},\hat{y}\right\rangle \hat{y}+\left\langle
\hat{x},\hat{z}\right\rangle \hat{z}\right) .$ Then, using that $\varepsilon>\sqrt{1-\delta^{2}}$, it follows from (\[representation173\]) that $\left\langle \omega,\hat{x}\right\rangle >\left( \sqrt{\left\langle
\omega,\hat{y}\right\rangle ^{2}+\left\langle \omega,\hat{z}\right\rangle
^{2}}\right) \left( \sqrt{\left\langle \hat{x},\hat{y}\right\rangle
^{2}+\left\langle \hat{x},\hat{z}\right\rangle ^{2}}\right) \left\langle
\omega_{\hat{y},\hat{z}},\hat{x}_{\hat{y},\hat{z}}\right\rangle >-\sqrt
{1-\delta^{2}}>-\varepsilon.$
Note that, $\partial_{x_{j}}\zeta_{-}^{\pm}\left( x,\xi^{\prime}\right) $ is equal to $0$ for $\pm\left\langle \hat{\xi}^{\prime},\hat{x}\right\rangle
<\varepsilon$ and $\zeta_{+}^{\pm}\left( x,\xi\right) =1$ for $\pm
\left\langle \hat{\xi},\hat{x}\right\rangle >-\varepsilon$ and $\left\vert
\xi\right\vert \geq c_{1}.$ Then, Lemma above implies $\zeta_{+}^{\pm}\left(
x,\xi\right) \partial_{x_{j}}\zeta_{-}^{\pm}\left( x,\xi^{\prime}\right)
=\partial_{x_{j}}\zeta_{-}^{\pm}\left( x,\xi^{\prime}\right) ,$ for $\left\vert \xi\right\vert \geq c_{1}$ and $\hat{\xi},\hat{\xi}^{\prime}\in\Omega_{+}\left( \omega_{0},\delta\right) $ or $\hat{\xi},\hat{\xi
}^{\prime}\in\Omega_{-}\left( \omega_{0},\delta\right) .$ Thus, from (\[representation62\]) we obtain the following equality, for $\hat{\xi},\hat{\xi}^{\prime}\in\Omega_{+}\left( \omega_{0},\delta\right) $ or $\hat{\xi},\hat{\xi}^{\prime}\in\Omega_{-}\left( \omega_{0},\delta\right)
,$ $$\left. G_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
=-i\varsigma\left( \left\vert \xi\right\vert \right) \varsigma\left(
\left\vert \xi^{\prime}\right\vert \right)
{\displaystyle\sum\limits_{j=1}^{3}}
{\displaystyle\int}
\left( u_{N}^{+}\left( x,\xi;\lambda\left( \xi\right) \right) \right)
^{\ast}\alpha_{j}u_{N}^{-}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime
}\right) \right) \left( \partial_{x_{j}}\zeta_{-}^{\left(
\operatorname*{sgn}E\right) }\left( x,\xi^{\prime}\right) \right)
\varphi\left( \varepsilon x\right) dx.\right. \label{representation156}$$
Recall the notation of Remark \[representation75\]. We need the two following results (see Proposition 5.4 and 5.5 of [@30])
\[representation147\]Let us consider $\mathcal{A}_{\pm}^{\left(
\varepsilon\right) }\left( \xi,\xi^{\prime};E\right) =\int\limits_{\Pi
_{\omega_{0}}}e^{i\left\langle y,\xi^{\prime}-\xi\right\rangle }g_{\pm}\left(
y,\xi,\xi^{\prime};E\right) \varphi\left( \varepsilon y\right) dy,$ where $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{3}\right) $ is such that $\varphi\left( 0\right) =1,$ $\Pi_{\omega_{0}}:=\left\{ y\in\mathbb{R}^{3}|\left\langle y,\omega_{0}\right\rangle =0,\omega_{0}\in\mathbb{S}^{2}\right\} $ and $g_{\pm}\in\mathcal{S}^{p}$ for some real $p,$ satisfies $$\operatorname*{supp}g_{\pm}\subset\{\left( \hat{\xi},\hat{\xi}^{\prime
}\right) \in\mathbb{R}^{3}\times\mathbb{R}^{3}\mid\hat{\xi},\hat{\xi}^{\prime}\in\Omega_{\pm}\left( \omega_{0},\delta\right) ,\text{ }\delta>0,\text{ }\left\vert \xi^{\prime}\right\vert \geq c\}.
\label{representation228}$$ Let $g_{1},$ $f_{1}$ and $f_{2}$ be as in (\[representation134\])$.$ Then, for $\left\vert E\right\vert >m$ and even $n$ we have $$\left.
\begin{array}
[c]{c}\lim_{\mu\downarrow0}\lim_{\varepsilon\rightarrow0}{\displaystyle\int_{\mathbb{S}^{2}}}
{\displaystyle\int}
\left( \mathcal{A}_{\pm}^{\left( \varepsilon\right) }\left( \nu\left(
E\right) \omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) \right) d\xi^{\prime}d\omega\\
=\upsilon\left( E\right)
{\displaystyle\int_{\Pi_{\omega_{0}}}}
{\displaystyle\int_{\Pi_{\omega_{0}}}}
{\displaystyle\int_{\Pi_{\omega_{0}}}}
e^{i\nu\left( E\right) \left\langle y,\zeta^{\prime}-\zeta\right\rangle
}\left\langle \nu\left( E\right) y\right\rangle ^{-n}\left\langle
D_{\zeta^{\prime}}\right\rangle ^{n}\left( \tilde{g}_{\pm}^{\prime}\left(
y,\nu\left( E\right) \zeta,\nu\left( E\right) \zeta^{\prime};E\right)
\tilde{f}_{1}\left( \zeta^{\prime}\right) ,\tilde{f}_{2}\left(
\zeta\right) \right) dyd\zeta^{\prime}d\zeta,
\end{array}
\right. \label{representation151}$$ where $\left\langle \nu\left( E\right) y\right\rangle =\sqrt{1+\left(
\nu\left( E\right) y\right) ^{2}},$ $\left\langle D_{\zeta^{\prime}}\right\rangle ^{2}=1-\partial_{\zeta^{\prime}}^{2}$ and $\tilde{g}_{\pm
}^{\prime}\left( y,\nu\left( E\right) \zeta,\nu\left( E\right)
\zeta^{\prime};E\right) :=\frac{\tilde{g}_{\pm}\left( y,\nu\left( E\right)
\zeta,\nu\left( E\right) \zeta^{\prime};E\right) }{\left( \left(
1-\left\vert \zeta\right\vert ^{2}\right) \left( 1-\left\vert \zeta^{\prime
}\right\vert ^{2}\right) \right) ^{1/2}}.$
We define $\Pi_{\omega_{0}}^{\pm}\left( E\right) :=\{x\in\mathbb{R}^{3}|x=z\omega_{0}+y,$ $y\in\Pi_{\omega_{0}}$ and $\pm\left(
\operatorname*{sgn}E\right) z\geq0\}.$
\[representation176\]Let us consider $\mathcal{A}_{\pm}^{\left(
\varepsilon\right) }\left( \xi,\xi^{\prime};E\right) =\left( \left\vert
\lambda\left( \xi\right) \right\vert -\left\vert \lambda\left( \xi^{\prime
}\right) \right\vert \right)
{\displaystyle\int\limits_{\Pi_{\omega_{0}}^{\pm}\left( E\right) }}
e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }g_{\pm}\left( x,\xi
,\xi^{\prime};E\right) \varphi\left( \varepsilon x\right) dx,$ where $\varphi\in C_{0}^{\infty}\left( \mathbb{R}^{3}\right) $ is such that $\varphi\left( 0\right) =1,$ and $g_{\pm}$ satisfies the assumption (\[representation228\]) and the estimate $$\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial_{\xi^{\prime}}^{\beta^{\prime}}g_{\pm}\left( x,\xi,\xi^{\prime};E\right) \right\vert \leq
C_{\alpha,\beta,\beta^{\prime}}\left( 1+\left\vert x\right\vert \right)
^{p-\left\vert \alpha\right\vert }, \label{representation227}$$ for some real $p,$ and all $x\in\Pi_{\omega_{0}}^{\pm}\left( E\right) $. Moreover, the following relation holds $$g_{\pm}\left( x,\xi,\xi^{\prime};E\right) =0\text{ if\ }\left(
\operatorname*{sgn}E\right) \left\langle \eta,x\right\rangle \geq
c_{0}\left\vert \eta\right\vert \left\vert x\right\vert ,\text{ for }\eta
=\xi+\xi^{\prime},\text{ }c_{0}\in\left( 0,1\right) ,
\label{representation226}$$ for all $x\in\Pi_{\omega_{0}}^{\pm}\left( E\right) $, $\left\vert
x\right\vert \geq R.$ Then, we have $$\lim_{\mu\downarrow0}\lim_{\varepsilon\rightarrow0}{\displaystyle\int_{\mathbb{S}^{2}}}
{\displaystyle\int}
\left( \mathcal{A}_{\pm}^{\left( \varepsilon\right) }\left( \nu\left(
E\right) \omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) \right) d\xi^{\prime}d\omega=0.
\label{representation175}$$
We want to find an expression for $G_{2}^{\left( \varepsilon\right) }\left(
\xi,\xi^{\prime};E\right) $ independent on the cut-off functions $\zeta
_{-}^{\left( \operatorname*{sgn}E\right) }$. If $\hat{\xi}^{\prime}\in
\Omega_{\pm}\left( \omega_{0},\delta\right) ,$ the function $\partial
_{x_{j}}\zeta_{-}^{\left( \operatorname*{sgn}E\right) }\left( x,\xi
^{\prime}\right) $ is equal to zero for $\pm\left( \operatorname*{sgn}E\right) z<0$, so we can consider the integral in (\[representation156\]) only in the region $\Pi_{\omega_{0}}^{\pm}\left( E\right) .$ Integrating by parts in $G_{2}^{\left( \varepsilon\right) }$ and noting that $\zeta
_{-}^{\left( \operatorname*{sgn}E\right) }\left( y,\xi^{\prime}\right)
=1,$ for $\left\vert \xi^{\prime}\right\vert \geq c_{1},$ we get $$\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right)
G_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
=\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) \left(
\pm\breve{G}_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime
};E\right) +R_{\pm}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime
};E\right) \right) , \label{representation159}$$ where$$\left. \breve{G}_{2}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime
};E\right) :=i\left( \operatorname*{sgn}E\right) \varsigma\left(
\left\vert \xi\right\vert \right) \varsigma\left( \left\vert \xi^{\prime
}\right\vert \right)
{\displaystyle\int\limits_{\Pi_{\omega_{0}}}}
\left( u_{N}^{+}\left( y,\xi;\lambda\left( \xi\right) \right) \right)
^{\ast}\left( \alpha\cdot\omega_{0}\right) u_{N}^{-}\left( y,\xi^{\prime
};\lambda\left( \xi^{\prime}\right) \right) \varphi\left( \varepsilon
y\right) dy,\right. \label{representation150}$$ and $R_{\pm}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
:=i\left( \operatorname*{sgn}E\right) \varsigma\left( \left\vert
\xi\right\vert \right) \varsigma\left( \left\vert \xi^{\prime}\right\vert
\right)
{\displaystyle\sum\limits_{i=1}^{3}}
$ $\
{\displaystyle\int\limits_{\Pi_{\omega_{0}}^{\pm}\left( E\right) }}
\partial_{x_{i}}\left( \left( u_{N}^{+}\left( x,\xi;\lambda\left(
\xi\right) \right) ^{\ast}\alpha_{i}u_{N}^{-}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) \right) \varphi\left(
\varepsilon x\right) \right) \zeta_{-}^{\left( \operatorname*{sgn}E\right)
}\left( x,\xi^{\prime}\right) dx.$ Using the definition (\[eig30\]) of the functions $r_{N}^{_{\pm}}\left( x,\xi;E\right) $ we obtain $i\sum_{j=1}^{3}\partial_{x_{j}}\left( \left( u_{N}^{_{+}}\left( x,\xi;\lambda\left(
\xi\right) \right) \right) ^{\ast}\alpha_{j}u_{N}^{_{-}}\left(
x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) \right)
=e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }$ $\times\lbrack\left(
r_{N}^{_{+}}\left( x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast
}a_{N}^{_{-}}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right)
\right) $ $-\left( a_{N}^{_{+}}\left( x,\xi;\lambda\left( \xi\right)
\right) \right) ^{\ast}r_{N}^{_{-}}\left( x,\xi^{\prime};\lambda\left(
\xi^{\prime}\right) \right) +\left( \lambda\left( \xi\right)
-\lambda\left( \xi^{\prime}\right) \right) \left( a_{N}^{_{+}}\left(
x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast}a_{N}^{_{-}}\left(
x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) ].$
Let us decompose $\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega
_{0}\right) R_{\pm}^{\left( \varepsilon\right) }$ in the sum $$\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) R_{\pm
}^{\left( \varepsilon\right) }=\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) \left( \left( R_{1}^{\left( \varepsilon
\right) }\right) _{\pm}+\left( R_{2}^{\left( \varepsilon\right) }\right)
_{\pm}+\left( R_{3}^{\left( \varepsilon\right) }\right) _{\pm}\right) ,
\label{representation249}$$ where$$\left.
\begin{array}
[c]{c}\left( R_{1}^{\left( \varepsilon\right) }\right) _{\pm}\left( \xi
,\xi^{\prime};E\right) :=i\varepsilon\left( \operatorname*{sgn}E\right)
\varsigma\left( \left\vert \xi\right\vert \right) \varsigma\left(
\left\vert \xi^{\prime}\right\vert \right) \\
\times{\displaystyle\sum\limits_{j=1}^{3}}
\text{ }\left.
{\displaystyle\int\limits_{\Pi_{\omega_{0}}^{\pm}\left( E\right) }}
e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }\left( \left( a_{N}^{_{+}}\left( x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast}\alpha
_{j}a_{N}^{_{-}}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right)
\right) \right) \zeta_{-}^{\left( \operatorname*{sgn}E\right) }\left(
x,\xi^{\prime}\right) \left( \partial_{x_{j}}\varphi\right) \left(
\varepsilon x\right) dx\right. ,
\end{array}
\right.$$ $\ $$$\left.
\begin{array}
[c]{c}\ \left( R_{2}^{\left( \varepsilon\right) }\right) _{\pm}\left( \xi
,\xi^{\prime};E\right) :=\left( \operatorname*{sgn}E\right) \varsigma
\left( \left\vert \xi\right\vert \right) \varsigma\left( \left\vert
\xi^{\prime}\right\vert \right) \\
\times{\displaystyle\int\limits_{\Pi_{\omega_{0}}^{\pm}\left( E\right) }}
e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }\left( \left( r_{N}^{+}\left( x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast}a_{N}^{-}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right)
-\left( a_{N}^{+}\left( x,\xi;\lambda\left( \xi\right) \right) \right)
^{\ast}r_{N}^{-}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right)
\right) \right) \zeta_{-}^{\left( \operatorname*{sgn}E\right) }\left(
x,\xi^{\prime}\right) \varphi\left( \varepsilon x\right) dx,
\end{array}
\right.$$ and $$\left.
\begin{array}
[c]{c}\left( R_{3}^{\left( \varepsilon\right) }\right) _{\pm}\left( \xi
,\xi^{\prime};E\right) :=\left( \left\vert \lambda\left( \xi\right)
\right\vert -\left\vert \lambda\left( \xi^{\prime}\right) \right\vert
\right) \varsigma\left( \left\vert \xi\right\vert \right) \varsigma\left(
\left\vert \xi^{\prime}\right\vert \right) \\
\times{\displaystyle\int\limits_{\Pi_{\omega_{0}}^{\pm}\left( E\right) }}
e^{i\left\langle x,\xi^{\prime}-\xi\right\rangle }\left( a_{N}^{+}\left(
x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast}a_{N}^{-}\left(
x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) \zeta_{-}^{\left(
\operatorname*{sgn}E\right) }\left( x,\xi^{\prime}\right) \varphi\left(
\varepsilon x\right) dx.
\end{array}
\right.$$ We first prove the following
\[representation184\]The limit $\Psi_{\pm}\left( \hat{\xi},\hat{\xi
}^{\prime};\omega_{0}\right) \left( R_{2}^{\left( 0\right) }\right)
_{\pm}\left( \xi,\xi^{\prime};E\right) :=\lim\limits_{\varepsilon
\rightarrow0}\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right)
\left( R_{2}^{\left( \varepsilon\right) }\right) _{\pm}\left( \xi
,\xi^{\prime};E\right) $ exists. For any $p$ and $q,$ there is $N,$ such that the function $\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega
_{0}\right) \left( R_{2}^{\left( 0\right) }\right) _{\pm}\left(
\nu\left( E\right) \hat{\xi},\nu\left( E\right) \hat{\xi}^{\prime
};E\right) $ is of the $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $ class, and its $C^{p}-$norm is bounded by $C\left\vert
E\right\vert ^{-q},$ as $\left\vert E\right\vert \rightarrow\infty.$ Moreover, the following relation holds $$\left.
\begin{array}
[c]{c}-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) \lim_{\mu\downarrow
0}\lim_{\varepsilon\rightarrow0}{\displaystyle\int}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( \Psi_{\pm}\left( \omega,\hat{\xi}^{\prime}\,;\omega_{0}\right)
R_{\pm}^{\left( \varepsilon\right) }\left( \nu\left( E\right) \omega
,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left( \operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right) ,f_{2}\left(
\omega\right) \right) d\omega d\xi^{\prime}\\
=-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}{\displaystyle\int_{\mathbb{S}^{2}}}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( \Psi_{\pm}\left( \omega,\theta;\omega_{0}\right) \left(
R_{2}^{\left( 0\right) }\right) _{\pm}\left( \nu\left( E\right)
\omega,\nu\left( E\right) \theta;E\right) f_{1}\left( \theta\right)
,f_{2}\left( \omega\right) \right) d\theta d\omega.
\end{array}
\right. \label{representation161}$$
Since $\zeta_{-}^{\left( \operatorname*{sgn}E\right) }$ is supported on $\Xi^{-}\left( E\right) ,$ $a_{N}^{_{-}}\zeta_{-}^{\left(
\operatorname*{sgn}E\right) }$ satisfies the estimate (\[eig27\]). If $x\in\Pi_{\omega_{0}}^{\pm}\left( E\right) $ and $\hat{\xi}\in\Omega_{\pm
}\left( \omega_{0},\delta\right) ,$ using (\[representation63\]), we get $\left( x,\xi\right) \in$ $\Xi^{+}\left( E\right) .$ Then, $\left(
a_{N}^{_{+}}\left( x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast
}$ also satisfies (\[eig27\]). Thus, for $x\in\Pi_{\omega_{0}}^{\pm}\left(
E\right) $ and $\hat{\xi}\in\Omega_{\pm}\left( \omega_{0},\delta\right) $, we obtain $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial
_{\xi^{\prime}}^{\beta^{\prime}}\left( \left( a_{N}^{_{+}}\left(
x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast}\alpha_{j}a_{N}^{_{-}}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right)
\right) \right) \zeta_{-}^{\left( \operatorname*{sgn}E\right) }\left(
x,\xi^{\prime}\right) \right\vert \leq C_{\alpha,\beta,\beta^{\prime}}\left(
1+\left\vert x\right\vert \right) ^{-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert }\left\vert \xi^{\prime
}\right\vert ^{-\left\vert \beta^{\prime}\right\vert },$ for all indices $\alpha$ and $\beta$ . This estimate implies that for all $f,g\in
\mathcal{S}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) ,$ (see relation (\[representation243\]) $$\lim_{\varepsilon\rightarrow0}\int\int\left( \Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) \left( R_{1}^{\left( \varepsilon
\right) }\right) _{\pm}\left( \xi,\xi^{\prime};E\right) f\left(
\xi^{\prime}\right) ,g\left( \xi\right) \right) d\xi^{\prime}d\xi=0.
\label{representation157}$$ The proof of this relation is analogous to that of relation (\[representation243\]).
Now observe that the functions $r_{N}^{-}\left( x,\xi^{\prime};\lambda\left(
\xi^{\prime}\right) \right) \zeta_{-}^{\left( \operatorname*{sgn}E\right)
}\left( x,\xi^{\prime}\right) $ and $a_{N}^{-}\left( x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) \zeta_{-}^{\left(
\operatorname*{sgn}E\right) }\left( x,\xi^{\prime}\right) $ satisfy the estimates (\[eig28\]) and (\[eig27\]), respectively, for all $x,\xi^{\prime}\in\mathbb{R}^{3}$. Moreover, the estimate (\[eig28\]) for the function $\left( r_{N}^{+}\left( x,\xi;\lambda\left( \xi\right)
\right) \right) ^{\ast}$ and the estimate (\[eig27\]) for $\left(
a_{N}^{+}\left( x,\xi;\lambda\left( \xi\right) \right) \right) ^{\ast}$ hold for $x\in\Pi_{\omega_{0}}^{\pm}\left( E\right) $ and $\hat{\xi}\in\Omega_{\pm}\left( \omega_{0},\delta\right) $. Hence, for $N$ big enough, we conclude that the limit, as $\varepsilon\rightarrow0,$ of $\Psi_{\pm
}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) \left(
R_{2}^{\left( \varepsilon\right) }\right) _{\pm}\left( \xi,\xi^{\prime
};E\right) $ exists and it is equal to $\Psi_{\pm}\left( \hat{\xi},\hat{\xi
}^{\prime};\omega_{0}\right) \left( R_{2}^{\left( 0\right) }\right)
_{\pm}\left( \xi,\xi^{\prime};E\right) .$ Moreover, for any $p$ and $q$ there exist $N,$ such that $\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime
};\omega_{0}\right) \left( R_{2}^{\left( 0\right) }\right) _{\pm}\left(
\xi,\xi^{\prime};E\right) $ is a $C^{p}-$function of variables $\xi$ and $\xi^{\prime},$ and its $C^{p}-$norm is bounded by $C\left\vert E\right\vert
^{-q},$ as $\left\vert E\right\vert \rightarrow\infty$.
Note that $\left\vert \partial_{x}^{\alpha}\partial_{\xi}^{\beta}\partial
_{\xi^{\prime}}^{\beta^{\prime}}\left( \left( a_{N}^{+}\left( x,\xi
;\lambda\left( \xi\right) \right) \right) ^{\ast}a_{N}^{-}\left(
x,\xi^{\prime};\lambda\left( \xi^{\prime}\right) \right) \zeta_{-}^{\left(
\operatorname*{sgn}E\right) }\left( x,\xi^{\prime}\right) \right)
\right\vert \leq C_{\alpha,\beta,\beta^{\prime}}\left( 1+\left\vert
x\right\vert \right) ^{-\left\vert \alpha\right\vert }\left\vert
\xi\right\vert ^{-\left\vert \beta\right\vert }\left\vert \xi^{\prime
}\right\vert ^{-\left\vert \beta^{\prime}\right\vert },$ for all $x\in
\Pi_{\omega_{0}}^{\pm}\left( E\right) \ $and $\hat{\xi}\in\Omega_{\pm
}\left( \omega_{0},\delta\right) ,$ and all indices $\alpha,\beta
,\beta^{\prime}$. For some $0<\kappa_{0}<\kappa_{1},$ let $\chi\in
C_{0}^{\infty}\left( \mathbb{R}^{+}\right) $ be such that $\chi\left(
\kappa\right) =1$ for $0\leq\kappa\leq\kappa_{0}$ and $\chi\left(
\kappa\right) =0$ for $\kappa\geq\kappa_{1}.$ We split $\left(
R_{3}^{\left( \varepsilon\right) }\right) _{\pm}$ in two parts, $\chi\left( \left\vert \xi-\xi^{\prime}\right\vert \right) \left(
R_{3}^{\left( \varepsilon\right) }\right) _{\pm}$ and $\left(
1-\chi\left( \left\vert \xi-\xi^{\prime}\right\vert \right) \right) \left(
R_{3}^{\left( \varepsilon\right) }\right) _{\pm}.$ Taking $\kappa_{1}$ small enough, we see that the cut-off function $\chi\left( \left\vert \xi
-\xi^{\prime}\right\vert \right) \zeta_{-}^{\left( \operatorname*{sgn}E\right) }\left( x,\xi^{\prime}\right) $ satisfies relation (\[representation226\]). Then, applying Lemma \[representation176\] to the term $\chi\left( \left\vert \xi-\xi^{\prime}\right\vert \right) \Psi_{\pm
}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) \left(
R_{3}^{\left( \varepsilon\right) }\right) _{\pm}$ and Lemma \[representation145\] to $\left( 1-\chi\left( \left\vert \xi-\xi^{\prime
}\right\vert \right) \right) \Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime
};\omega_{0}\right) \left( R_{3}^{\left( \varepsilon\right) }\right)
_{\pm}$ we have$$\lim_{\mu\downarrow0}\lim_{\varepsilon\rightarrow0}{\displaystyle\int_{\mathbb{S}^{2}}}
{\displaystyle\int}
\left( \Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right)
\left( R_{3}^{\left( \varepsilon\right) }\right) _{\pm}\left( \xi
,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left( \operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right) ,f_{2}\left(
\omega\right) \right) d\xi^{\prime}d\omega=0. \label{representation158}$$ Introducing decomposition (\[representation249\]) in the L.H.S. of (\[representation161\]) and using relations (\[representation157\]), (\[representation158\]) and Lemma \[representation54\] to calculate the resulting limit, we conclude that relation (\[representation161\]) holds.
Let us now prove the following result
\[representation185\]Let $g_{1},$ $f_{1}$ and $f_{2}$ be as in (\[representation134\]). For an arbitrary $\omega_{0}\in\mathbb{S}^{2},$ the equality $$\left.
\begin{array}
[c]{c}-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) \lim_{\mu\downarrow
0}\lim_{\varepsilon\downarrow0}{\displaystyle\int}
{\displaystyle\int}
\left( \pm\Psi_{\pm}\left( \omega,\hat{\xi}^{\prime}\,;\omega_{0}\right)
\breve{G}_{2}^{\left( \varepsilon\right) }\left( \nu\left( E\right)
\omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) \right) d\xi^{\prime}d\omega\\
={\displaystyle\int_{\mathbb{S}^{2}}}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( s_{\operatorname{sing}}^{(N)}\left( \omega,\theta;E;\omega
_{0}\right) f_{1}\left( \theta\right) ,f_{2}\left( \omega\right) \right)
d\theta d\omega,
\end{array}
\right. \label{representation189}$$ holds, where $s_{\operatorname{sing}}^{(N)}\left( \omega,\theta;E;\omega
_{0}\right) $ is given by (\[representation248\]).
Applying Lemma \[representation147\] to $\pm\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) \breve{G}_{2}^{\left( \varepsilon
\right) }\left( \xi,\xi^{\prime};E\right) $ we get $$\left.
\begin{array}
[c]{c}-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) \lim_{\mu\downarrow
0}\lim_{\varepsilon\downarrow0}{\displaystyle\int}
{\displaystyle\int}
\left( \pm\Psi_{\pm}\left( \omega,\hat{\xi}^{\prime}\,;\omega_{0}\right)
\breve{G}_{2}^{\left( \varepsilon\right) }\left( \nu\left( E\right)
\omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) \right) d\xi^{\prime}d\omega\\
=\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}{\displaystyle\int_{\Pi_{\omega_{0}}}}
{\displaystyle\int_{\Pi_{\omega_{0}}}}
{\displaystyle\int_{\Pi_{\omega_{0}}}}
e^{i\nu\left( E\right) \left\langle y,\zeta^{\prime}-\zeta\right\rangle
}\left\langle \nu\left( E\right) y\right\rangle ^{-n}\left\langle
D_{\zeta^{\prime}}\right\rangle ^{n}\left( \mathbf{\tilde{h}}_{N}^{\prime
}\left( y,\zeta,\zeta^{\prime};E\right) \tilde{f}_{1}\left( \zeta^{\prime
}\right) ,\tilde{f}_{2}\left( \zeta\right) \right) d\zeta^{\prime}d\zeta,
\end{array}
\right. \label{representation236}$$ for even $n,$ where $\mathbf{\tilde{h}}_{N,jk}^{\prime}\left( y,\zeta
,\zeta^{\prime};E\right) :=\pm\tilde{\Psi}_{\pm}\left( \zeta,\zeta^{\prime
};\omega_{0}\right) \frac{1}{\left( 1-\left\vert \zeta\right\vert
^{2}\right) ^{1/2}}\frac{1}{\left( 1-\left\vert \zeta^{\prime}\right\vert
^{2}\right) ^{1/2}}\mathbf{\tilde{h}}_{N}\left( y,\zeta,\zeta^{\prime
};E\right) .$ Integrating back by parts in the R.H.S of (\[representation236\]) and understanding the resulting expression as an oscillatory integral, we obtain the expression (\[representation189\]).
Recall the function $\Psi_{1}\left( \hat{\xi},\hat{\xi}^{\prime}\right) ,$ defined above (\[representation248\]). We are able to prove the following result for $S_{2}\left( E\right) .$
\[representation241\]Let $s_{2}\left( \omega,\theta;E\right) $ be the kernel of the operator $S_{2}\left( E\right) ,$ defined as the limit (\[representation134\]). For any $p$ and $q,$ $\Psi_{1}\left( \omega
,\theta\right) $ $\times s_{2}\left( \omega,\theta;E\right) $ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $ and its $C^{p}$-norm is a $O\left( E^{-q}\right) $ function. Moreover, for any $p$ and $q$ there exists $N,$ sufficiently large, such that, $\Psi_{\pm}\left(
\omega,\theta;\omega_{0}\right) s_{2}\left( \omega,\theta;E\right)
-s_{\operatorname{sing}}^{\left( N\right) }\left( \omega,\theta
;E;\omega_{0}\right) $ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and moreover, its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert E\right\vert
\rightarrow\infty$. These estimates are uniform in $\omega_{0}\in
\mathbb{S}^{2}.$
From the relations (\[representation188\]) and (\[representation166\]) we get $$\left.
\begin{array}
[c]{c}\left( \left( S_{2}\left( E\right) \Psi\left( \omega,\cdot;\omega
_{0}\right) f_{1}\right) \left( \omega\right) ,f_{2}\left( \omega\right)
\right) =-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) \\
\times\lim_{\mu\downarrow0}\lim_{\varepsilon\rightarrow0}{\displaystyle\int}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( \Psi\left( \omega,\hat{\xi}^{\prime}\right) \left( G_{1}^{\left(
\varepsilon\right) }+G_{2}^{\left( \varepsilon\right) }\right) \left(
\nu\left( E\right) \omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu
}^{\left( \operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime
}\right) ,f_{2}\left( \omega\right) \right) d\omega d\xi^{\prime},
\end{array}
\right. \label{representation239}$$ where $\Psi\left( \hat{\xi},\hat{\xi}^{\prime}\right) $ is either $\Psi
_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) $ or $\Psi
_{1}\left( \hat{\xi},\hat{\xi}^{\prime}\right) .$ Suppose first that $\Psi\left( \hat{\xi},\hat{\xi}^{\prime}\right) =\Psi_{1}\left( \hat{\xi
},\hat{\xi}^{\prime}\right) .$ By Lemma \[representation183\] and Lemma \[representation145\] the limit (\[representation239\]) exists and, for any $p$ and $q$ there exists $N,$ such that $\Psi_{1}\left( \hat{\xi},\hat{\xi}^{\prime}\right) \left( G_{1}^{\left( 0\right) }+G_{2}^{\left(
\varepsilon\right) }\right) \left( \nu\left( E\right) \hat{\xi},\nu\left( E\right) \hat{\xi}^{\prime};E\right) $ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert
E\right\vert \rightarrow\infty$.
Now let us consider the case $\Psi\left( \hat{\xi},\hat{\xi}^{\prime}\right)
=\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) .$ Again, Lemma \[representation183\] implies that the part in the limit (\[representation239\]) corresponding to the term $\Psi_{\pm}\left(
\hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) G_{1}^{\left( \varepsilon
\right) }\left( \nu\left( E\right) \hat{\xi},\xi^{\prime};E\right) $ exists and the function $\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{0}\right) G_{1}^{\left( \varepsilon\right) }\left( \nu\left(
E\right) \hat{\xi},\xi^{\prime};E\right) $ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert
E\right\vert \rightarrow\infty$. Using relation (\[representation159\]) and applying Lemma \[representation184\] to the part $\Psi_{\pm}R_{\pm}^{\left(
\varepsilon\right) }$ and Lemma \[representation185\] to $\Psi_{\pm}\breve{G}_{2}^{\left( \varepsilon\right) },$ in order to calculate the limit in the R.H.S of (\[representation239\]) corresponding to the term $\Psi
_{\pm}\left( \omega,\theta;\omega_{0}\right) G_{2}^{\left( \varepsilon
\right) }\left( \nu\left( E\right) \hat{\xi},\xi^{\prime};E\right) ,$ and noting that all the estimates are uniform on $\omega_{0}$ if the $C^{p}$-norms of the function $\Psi_{\pm}\left( \omega,\theta;\omega_{0}\right) $ are uniformly bounded on $\omega_{0}\in\mathbb{S}^{2},$ we complete the proof.
Let us present a result that we use below (see Lemma 4.1 of [@47])
\[representation201\]Let $f\left( x,\xi\right) \in C^{\infty}\left(
\mathbb{R}^{N}\times\mathbb{R}^{N}\right) $ satisfy $\left\vert \partial
_{x}^{\alpha}f\left( x,\xi\right) \right\vert \leq C_{\alpha}\left\langle
x\right\rangle ^{-\rho-\left\vert \alpha\right\vert }$ for some $0<\rho<N.$ Then we have $\left\vert \int_{\mathbb{R}^{N}}(e^{-i\left\langle
x,\xi\right\rangle }f\left( x,\xi\right) )dx\right\vert \leq C\left\vert
\xi\right\vert ^{-\left( N-\rho\right) }$ as $\left\vert \xi\right\vert
\rightarrow0.$
Let us define the function $\mathbf{h}_{N}^{\operatorname{int}}$ by the relation $$\left.
\begin{array}
[c]{c}\mathbf{h}_{N}^{\operatorname{int}}\left( y,\omega,\theta;E;\omega
_{0}\right) :=\left( \operatorname*{sgn}E\right) (\left( a^{+}\left(
y,\nu\left( E\right) \omega;E\right) -P_{\omega}\left( E\right) \right)
^{\ast}\left( \alpha\cdot\omega_{0}\right) \left( a^{_{-}}\left(
y,\nu\left( E\right) \theta;E\right) -P_{\theta}\left( E\right) \right)
\\
+P_{\omega}\left( E\right) \left( \alpha\cdot\omega_{0}\right) (a^{_{-}}\left( y,\nu\left( E\right) \theta;E\right) -P_{\theta}\left( E\right)
)+\left( a^{+}\left( y,\nu\left( E\right) \omega;E\right) -P_{\omega
}\left( E\right) \right) \left( \alpha\cdot\omega_{0}\right) P_{\theta
}\left( E\right) ).
\end{array}
\right. \label{representation253}$$ We prove now the following
\[representation186\]The function $\mathbf{b}_{N},$ given by $\mathbf{b}_{N}\left( \omega,\theta;E;\omega_{0}\right) :=\pm\left(
2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\Psi_{\pm}\left(
\omega,\theta;\omega_{0}\right) \int\limits_{\Pi_{\omega_{0}}}e^{i\nu\left(
E\right) \left\langle y,\theta-\omega\right\rangle }(\mathbf{h}_{N}\left(
y,\omega,\theta;E;\omega_{0}\right) $ $-\left( \operatorname*{sgn}E\right)
P_{\omega}\left( E\right) \left( \alpha\cdot\omega_{0}\right) P_{\theta
}\left( E\right) )dy,$ satisfies the estimate (\[representation83\]).
We first note that $\mathbf{b}_{N}\left( \omega,\theta;E;\omega_{0}\right)
=\pm\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\Psi_{\pm}\left(
\omega,\theta;\omega_{0}\right) \int\limits_{\Pi_{\omega_{0}}}e^{i\nu\left(
E\right) \left\langle y,\theta-\omega\right\rangle }\mathbf{h}_{N}^{\operatorname{int}}\left( y,\omega,\theta;E;\omega_{0}\right) dy,$ where the function $\mathbf{h}_{N}^{\operatorname{int}}$ is defined by $\mathbf{h}_{N}^{\operatorname{int}}\left( y,\omega,\theta;E;\omega
_{0}\right) :=\left( \operatorname*{sgn}E\right) (\left( a^{+}\left(
y,\nu\left( E\right) \omega;E\right) -P_{\omega}\left( E\right) \right)
^{\ast}\left( \alpha\cdot\omega_{0}\right) \left( a^{_{-}}\left(
y,\nu\left( E\right) \theta;E\right) -P_{\theta}\left( E\right) \right)
+P_{\omega}\left( E\right) \left( \alpha\cdot\omega_{0}\right) (a^{_{-}}\left( y,\nu\left( E\right) \theta;E\right) -P_{\theta}\left( E\right)
)+\left( a^{+}\left( y,\nu\left( E\right) \omega;E\right) -P_{\omega
}\left( E\right) \right) \left( \alpha\cdot\omega_{0}\right) P_{\theta
}\left( E\right) ).$ Using the notation of Remark \[representation75\] and relations (\[eig22\]), (\[eig42\]), (\[eig8\]), (\[eig9\]), (\[eig19\]) and (\[eig20\]) we get $\mathbf{b}_{N}\left( y,\omega
,\theta;E;\omega_{0}\right) =\pm\left( 2\pi\right) ^{-2}\upsilon\left(
E\right) ^{2}\int_{\Pi_{\omega_{0}}}e^{i\nu\left( E\right) \left(
y,\zeta^{\prime}-\zeta\right) }$ $\times\left( \mathbf{\tilde{h}}_{N}^{\operatorname{int}}\right) ^{\prime}\left( y,\zeta,\zeta^{\prime
};E;\omega_{0}\right) dy,$ where $\left( \mathbf{\tilde{h}}_{N}^{\operatorname{int}}\right) ^{\prime}\in\mathcal{S}^{-\left( \rho-1\right)
}.$ From Lemma \[representation201\] and the inequality $\left\vert
\omega-\theta\right\vert $ $\leq\frac{1}{\delta}\left\vert \zeta^{\prime
}-\zeta\right\vert $ we obtain the estimate (\[representation83\]) for $\mathbf{b}_{N}.$
We now use the partition of the unity that we introduce in (\[representation170\]). Recall the definition (\[representation182\]) of $s_{00}^{(jk)}.$ For $f,g\in\mathcal{H}\left( E\right) ,$ let us define $I_{jk}:=\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}\left( s_{00}^{(jk)}\left( \omega,\theta;E\right) f\left( \theta\right) ,g\left(
\omega\right) \right) _{\mathcal{H}\left( E\right) }d\theta d\omega.$ We prove the following
\[representation89\]The function $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}s_{00}^{(jk)}\left( \omega,\theta;E\right) $ is a Dirac-function over $\mathcal{H}\left( E\right) $. That is $$\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}I_{jk}=\left( f,g\right)
_{\mathcal{H}\left( E\right) }. \label{representation88}$$
Observe that $I_{jk}=\left( \operatorname*{sgn}E\right) \left( 2\pi\right)
^{-2}\upsilon\left( E\right) ^{2}\int\limits_{\Pi_{\omega_{jk}}}\int\limits_{\Pi_{\omega_{jk}}}\int\limits_{\Pi_{\omega_{jk}}}e^{i\nu\left(
E\right) \left\langle y,\zeta^{\prime}-\zeta\right\rangle }\left(
\mathbf{\tilde{h}}_{00,jk}^{\prime}\left( \zeta,\zeta^{\prime};E\right)
\tilde{f}\left( \zeta^{\prime}\right) ,\tilde{g}\left( \zeta\right)
\right) d\zeta^{\prime}dyd\zeta,$ where $\mathbf{\tilde{h}}_{00,jk}^{\prime}=\frac{\tilde{\chi}_{j}\left( \zeta\right) }{\left( 1-\left\vert
\zeta\right\vert ^{2}\right) ^{1/2}}\frac{\tilde{\chi}_{k}\left(
\zeta^{\prime}\right) }{\left( 1-\left\vert \zeta^{\prime}\right\vert
^{2}\right) ^{1/2}}\tilde{\chi}_{jk}\left( \zeta,\zeta^{\prime}\right)
\tilde{P}_{\zeta}\left( E\right) \left( \alpha\cdot\omega_{jk}\right)
\tilde{P}_{\zeta^{\prime}}\left( E\right) .$ (Here we used the notation of Remark \[representation75\]). Calculating the integrals over $\zeta^{\prime
}$ and $y$ we get $I_{jk}=\left( \operatorname*{sgn}E\right) \frac
{\upsilon\left( E\right) ^{2}}{\nu\left( E\right) ^{2}}\int_{\Pi
_{\omega_{jk}}}\frac{1}{1-\left\vert \zeta\right\vert ^{2}}\tilde{\chi}_{j}\left( \zeta\right) \tilde{\chi}_{k}\left( \zeta^{\prime}\right)
\tilde{\chi}_{jk}\left( \zeta,\zeta^{\prime}\right) \left( \tilde{P}_{\zeta}\left( E\right) \left( \alpha\cdot\omega_{jk}\right) \tilde
{P}_{\zeta^{\prime}}\left( E\right) \tilde{f}\left( \zeta\right)
,\tilde{g}\left( \zeta\right) \right) d\zeta.$ Since$$P_{\omega}\left( E\right) \left( \alpha\cdot\omega_{jk}\right) =\left(
\alpha\cdot\omega_{jk}\right) P_{\omega}\left( -E\right) +\frac{\nu\left(
E\right) }{E}\left\langle \omega,\omega_{jk}\right\rangle ,
\label{representation113}$$ we have $P_{\omega}\left( E\right) \left( \alpha\cdot\omega_{jk}\right)
P_{\omega}\left( E\right) =\frac{\nu\left( E\right) }{E}\left\langle
\omega,\omega_{jk}\right\rangle P_{\omega}\left( E\right) .$ As $\pm\left\langle \omega,\omega_{jk}\right\rangle =\sqrt{1-\left\vert
\zeta\right\vert ^{2}}$ for $\omega\in\Omega_{\pm}\left( \omega_{jk},\delta\right) ,$ then $\chi_{j}\left( \omega\right) \chi_{k}\left(
\omega\right) $ $\times\chi_{jk}\left( \omega,\omega\right) \left\langle
\omega,\omega_{jk}\right\rangle =\chi_{j}\left( \omega\right) \chi
_{k}\left( \omega\right) \chi_{jk}^{\prime}\left( \omega,\omega\right)
\sqrt{1-\left\vert \zeta\right\vert ^{2}},$ where $\chi_{jk}^{\prime}\left(
\omega,\theta\right) :=\chi_{jk}^{+}\left( \omega\right) \chi_{jk}^{+}\left( \theta\right) +\chi_{jk}^{-}\left( \omega\right) \chi_{jk}^{-}\left( \theta\right) .$ Thus, using the relation $\chi_{j}\left(
\omega\right) \chi_{k}\left( \omega\right) \chi_{jk}^{\prime}\left(
\omega,\omega\right) =\chi_{j}\left( \omega\right) \chi_{k}\left(
\omega\right) ,$ we get $I_{jk}=\int_{\Pi_{\omega_{jk}}}\frac{1}{\sqrt{1-\left\vert \zeta\right\vert ^{2}}}\tilde{\chi}_{j}\left(
\zeta\right) \tilde{\chi}_{k}\left( \zeta\right) \left( \tilde{P}_{\zeta
}\left( E\right) \tilde{f}\left( \zeta\right) ,\tilde{g}\left(
\zeta\right) \right) d\zeta.$ Returning back to variable $\omega$ in the last expression we obtain$$I_{jk}=\int_{\mathbb{S}^{2}}\chi_{j}\left( \omega\right) \chi_{k}\left(
\omega\right) \left( f_{1}\left( \omega\right) ,f_{2}\left(
\omega\right) \right) d\omega. \label{representation86}$$ Noting that $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}\chi_{j}\left(
\omega\right) \chi_{k}\left( \omega\right) =\sum\limits_{j,k=1}^{n}\chi
_{j}\left( \omega\right) \chi_{k}\left( \omega\right) =1$ we obtain (\[representation88\]).
We obtain the following
\[representation168\]The limit (\[representation134\]) exists and the operator $S_{2}\left( E\right) $ is decomposed as follows$$S_{2}\left( E\right) =I+\mathcal{G+R}_{1}\text{,} \label{representation171}$$ where $I$ is the identity in $\mathcal{H}\left( E\right) ,$ $\mathcal{G}$ is an integral operator with kernel $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing
}\mathbf{g}_{N,jk}^{\operatorname{int}}\left( \omega,\theta;E\right) $ satisfying (\[representation83\]) and $\mathcal{R}_{1}$ is an integral operator with kernel $r_{1}\left( \omega,\theta;E\right) :=-i\left(
2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\left( \sum\limits_{O_{j}\cap
O_{k}=\varnothing}\chi_{j}\left( \omega\right) \left( G_{1}^{\left(
0\right) }+G_{2,O_{j},O_{k}}^{\left( 0\right) }\right) \left( \nu\left(
E\right) \omega,\nu\left( E\right) \theta;E\right) \chi_{k}\left(
\theta\right) \right. $ $\left. +\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}r_{jk}\left( \omega,\theta;E\right) \right) ,$ where the function $r_{jk}$ is defined by $r_{jk}\left( \omega,\theta;E\right)
:=\chi_{j}\left( \omega\right) \left( G_{1}^{\left( 0\right) }+\left(
1-\chi_{jk}^{\prime}\left( \omega,\theta\right) \right) G_{2}^{\left(
0\right) }+\chi_{jk}^{+}\left( \omega\right) \right. $ $\left. \times
\chi_{jk}^{+}\left( \theta\right) \left( R_{2}^{\left( 0\right) }\right)
_{+}+\chi_{jk}^{-}\left( \omega\right) \chi_{jk}^{-}\left( \theta\right)
\left( R_{2}^{\left( 0\right) }\right) _{-}\right) \chi_{k}\left(
\theta\right) ,$ with $\chi_{jk}^{\prime}\left( \omega,\theta\right)
=\chi_{jk}^{+}\left( \omega\right) \chi_{jk}^{+}\left( \theta\right)
+\chi_{jk}^{-}\left( \omega\right) \chi_{jk}^{-}\left( \theta\right) .$ For any $p$ and $q$ there exists $N,$ sufficiently large, such that, $r_{1}\left( \omega,\theta;E\right) $ belongs to the class $C^{p}\left(
\mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and moreover, its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert E\right\vert
\rightarrow\infty$. Furthermore, the operator $S_{2}\left( E\right) -I$ is a compact operator on $\mathcal{H}\left( E\right) .$
Let us write $S_{2}\left( E\right) $ as in relation (\[representation170\]). First we consider the case $O_{j}\cap O_{k}=\varnothing.$ The relation (\[representation166\]), Lemma \[representation183\] and Lemma \[representation145\] imply that $\chi_{j}\left( \omega\right)
s_{2}\left( \omega,\theta;E\right) \chi_{k}\left( \theta\right) =-i\left(
2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\chi_{j}\left( \omega\right)
\left( G_{1}^{\left( 0\right) }+G_{2,O_{j},O_{k}}^{\left( 0\right)
}\right) \left( \nu\left( E\right) \omega,\nu\left( E\right)
\theta;E\right) \chi_{k}\left( \theta\right) $, where the function $\chi_{j}\left( \omega\right) \left( G_{1}^{\left( 0\right) }+G_{2,O_{j},O_{k}}^{\left( 0\right) }\right) \left( \nu\left( E\right)
\omega,\nu\left( E\right) \theta;E\right) \chi_{k}\left( \theta\right) $ satisfies the estimate (\[representation224\])$.$ Suppose now that $O_{j}\cap O_{k}\neq\varnothing.$ We take $\Psi_{\pm}\left( \hat{\xi},\hat{\xi}^{\prime};\omega_{jk}\right) =\chi_{j}\left( \hat{\xi}\right)
\chi_{k}\left( \hat{\xi}^{\prime}\right) \chi_{jk}^{\pm}\left( \hat{\xi
}\right) \chi_{jk}^{\pm}\left( \hat{\xi}^{\prime}\right) $ (here we use Lemma \[representation221\]). Let us decompose $G_{2}^{\left(
\varepsilon\right) }=\left( 1-\chi_{jk}^{\prime}\left( \hat{\xi},\hat{\xi
}^{\prime}\right) \right) G_{2}^{\left( \varepsilon\right) }+\chi
_{jk}^{\prime}\left( \hat{\xi},\hat{\xi}^{\prime}\right) G_{2}^{\left(
\varepsilon\right) }.$ Then using the relations (\[representation188\]), (\[representation166\]) and (\[representation159\]) we get $$\left. \left( \chi_{j}\left( S_{2}\left( E\right) \chi_{k}f_{1}\right)
,f_{2}\right) =-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right)
\lim_{\mu\downarrow0}\lim_{\varepsilon\rightarrow0}{\displaystyle\int}
{\displaystyle\int_{\mathbb{S}^{2}}}
\left( G_{jk}^{\left( \varepsilon\right) }\left( \nu\left( E\right)
\omega,\xi^{\prime};E\right) \tilde{\delta}_{\mu}^{\left(
\operatorname*{sgn}E\right) }\hat{g}_{1}\left( \xi^{\prime}\right)
,f_{2}\left( \omega\right) \right) d\omega d\xi^{\prime},\right.
\label{representation250}$$ with $G_{jk}^{\left( \varepsilon\right) }\left( \xi,\xi^{\prime};E\right)
:=\chi_{j}\left( \hat{\xi}\right) \left( G_{1}^{\left( \varepsilon\right)
}+\left( 1-\chi_{jk}^{\prime}\left( \hat{\xi},\hat{\xi}^{\prime}\right)
\right) G_{2}^{\left( \varepsilon\right) }+\chi_{jk}\left( \hat{\xi},\hat{\xi}^{\prime}\right) \tilde{G}_{2}^{\left( \varepsilon\right) }+\sum_{\tau=-1}^{1}\chi_{jk}^{\operatorname*{sgn}\tau}\left( \hat{\xi
}\right) \chi_{jk}^{\operatorname*{sgn}\tau}\left( \hat{\xi}^{\prime
}\right) R_{\operatorname*{sgn}\tau}^{\left( \varepsilon\right) }\right)
\chi_{k}\left( \hat{\xi}^{\prime}\right) .$ Using Lemma \[representation183\] for the part in the limit (\[representation250\]) corresponding to the term $\chi_{j}\left( \hat{\xi}\right) G_{1}^{\left(
\varepsilon\right) }\chi_{k}\left( \hat{\xi}^{\prime}\right) ,$ Lemma \[representation145\] for $\chi_{j}\left( \hat{\xi}\right) $ $\times\left( 1-\chi_{jk}^{\prime}\left( \hat{\xi},\hat{\xi}^{\prime
}\right) \right) G_{2}^{\left( \varepsilon\right) }\chi_{k}\left(
\hat{\xi}\right) ,$ Lemma \[representation184\] for $\chi_{j}\left(
\hat{\xi}\right) \chi_{k}\left( \hat{\xi}^{\prime}\right) \chi_{jk}^{\pm
}\left( \hat{\xi}\right) \chi_{jk}^{\pm}\left( \hat{\xi}^{\prime}\right)
R_{\pm}^{\left( \varepsilon\right) }$ and Lemma \[representation185\] for $\chi_{j}\left( \hat{\xi}\right) \chi_{k}\left( \hat{\xi}^{\prime}\right)
$ $\times\chi_{jk}^{\pm}\left( \hat{\xi}\right) \chi_{jk}^{\pm}\left(
\hat{\xi}^{\prime}\right) \tilde{G}_{2}^{\left( \varepsilon\right) },$ in order to calculate the limit (\[representation250\]), we get $\chi
_{j}\left( \omega\right) s_{2}\left( \omega,\theta;E\right) \chi
_{k}\left( \theta\right) =-i\left( 2\pi\right) ^{-2}\upsilon\left(
E\right) ^{2}r_{jk}\left( \omega,\theta;E\right) $ $+s_{N,jk}\left(
\omega,\theta;E\right) .$ For any $p$ and $q$ there exists $N,$ sufficiently large, such that, $r_{jk}\left( \omega,\theta;E\right) $ belongs to the class $C^{p}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $, and moreover, its $C^{p}-$norm is bounded by $C\left\vert E\right\vert ^{-q}$, as $\left\vert E\right\vert \rightarrow\infty$.
Taking the sum over $j$ and $k$, such that $O_{j}\cap O_{k}\neq\varnothing,$ of $s_{N,jk}\left( \omega,\theta;E\right) ,$ and using Proposition \[representation89\] and Lemma \[representation186\], with $\Psi_{\pm
}\left( \omega,\theta;\omega_{jk}\right) =\chi_{j}\left( \omega\right)
\chi_{k}\left( \theta\right) \chi_{jk}^{\pm}\left( \omega\right) \chi
_{jk}^{\pm}\left( \theta\right) ,$ we obtain the term $I+\mathcal{G}$, where $\mathcal{G}$ satisfies the assertions of Lemma \[representation168\]. Moreover, taking the sum over $j$ and $k$, such that $O_{j}\cap O_{k}\neq\varnothing,$ of the parts $\chi_{j}\left( \omega\right) s_{2}\left(
\omega,\theta;E\right) \chi_{k}\left( \theta\right) $ we see that the kernel of $\mathcal{R}_{1}$ is given by $r_{1}\left( \omega,\theta;E\right)
.$
Noting that the amplitude of $S_{2}\left( E\right) -I$ belongs to the class $\mathcal{S}^{-\left( \rho-1\right) },$ it follows from Proposition \[representation76\] that the operator $S_{2}\left( E\right) -I$ is compact.
Stationary representation for the scattering matrix and proofs of Theorems \[representation179\] and \[representation25\].
--------------------------------------------------------------------------------------------------------------------------
In order to prove Theorem \[representation25\] we need a stationary formula for the scattering matrix $S\left( E\right) $ in terms of the identifications $J_{\pm}$ ([@47], [@42], [@43], [@37], [@8]). The scattering operators $\mathbf{S}$ and $\mathbf{\tilde{S}}$ are related by equation (\[representation206\]) and hence, $\mathbf{\tilde{S}}$ commutes with $H_{0}.$ This implies that $\mathcal{F}_{0}\mathbf{\tilde{S}}\mathcal{F}_{0}^{\ast}$ acts as a multiplication by the operator valued function $\tilde{S}\left( E\right) .$ Let us first give a stationary formula for $\tilde{S}\left( E\right) .$
\[representation135\]For $\left\vert E\right\vert >m,$ the scattering matrix $\tilde{S}\left( E\right) $ can be represented as $$\tilde{S}\left( E\right) =S_{1}\left( E\right) +S_{2}\left( E\right) ,
\label{representation30}$$ where $S_{1}\left( E\right) $ is given by relation (\[representation14\]) and $S_{2}\left( E\right) $ is defined by (\[representation171\]).
We follow the proof of Theorem 3.3 of [@47] for the case of the Schrödinger equation. Let $\Lambda\subset(-\infty,-m)\cup(m,+\infty)$ be bounded. We first note that for $g_{j}\in\mathcal{S}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) \cap E_{0}\left( \Lambda\right) L^{2},$ $j=1,2,$ we have $$\int\limits_{\Lambda}\left( \tilde{S}\left( E\right) \Gamma_{0}\left(
E\right) g_{1},\Gamma_{0}\left( E\right) g_{2}\right) _{\mathcal{H}\left(
E\right) }dE=\left( \mathbf{\tilde{S}}g_{1},g_{2}\right) .
\label{representation142}$$ We will work with the form $\left( \mathbf{\tilde{S}}g_{1},g_{2}\right) ,$ and then use equality (\[representation142\]) to get the desired result for $\tilde{S}\left( E\right) .$ From (\[representation18\]) we get $W_{+}\left( H,H_{0};J_{+}\right) ^{\ast}W_{+}\left( H,H_{0};J_{-}\right)
=0,$ and thus, $\mathbf{\tilde{S}}=W_{+}^{\ast}\left( H,H_{0};J_{+}\right)
\left( W_{-}\left( H,H_{0};J_{-}\right) -W_{+}\left( H,H_{0};J_{-}\right)
\right) .$ Noting that $$W_{+}\left( H,H_{0};J_{-}\right) -W_{-}\left( H,H_{0};J_{-}\right)
=\left( i\int\limits_{-\infty}^{\infty}e^{itH}T_{-}e^{-itH_{0}}dt\right) ,
\label{representation43}$$ and using the intertwining property of $W_{+}\left( H,H_{0};J_{+}\right) $ we have $$\left.
\begin{array}
[c]{c}\left( \mathbf{\tilde{S}}g_{1},g_{2}\right) =\left( \left( W_{-}\left(
H,H_{0};J_{-}\right) -W_{+}\left( H,H_{0};J_{-}\right) \right) g_{1},W_{+}\left( H,H_{0};J_{+}\right) g_{2}\right) \\
=-i{\displaystyle\int\limits_{-\infty}^{\infty}}
\left( T_{-}e^{-itH_{0}}g_{1},W_{+}\left( H,H_{0};J_{+}\right) e^{-itH_{0}}g_{2}\right) .
\end{array}
\right. \label{representation60}$$ Let us split $T_{-}=T_{-}^{1}+T_{-}^{2}$ in the R.H.S. of (\[representation60\]). By relation (\[representation23\]), using Proposition \[representation118\] we see that the integral in the R.H.S. of (\[representation60\]) for $T_{-}^{1}$ is well defined. Moreover, noting that $t_{-}^{2}$ is equal to $0$ in some neighborhood of the directions $\left\langle \hat{x},\hat{\xi}\right\rangle =\pm1,$ $\left\vert x\right\vert
\geq R,$ and applying Lemma \[representation56\] to the operator $T_{-}^{2}$ we conclude that the integral in the R.H.S. of (\[representation60\]) is well defined$.$ Using the equality $W_{+}\left( H,H_{0};J_{+}\right)
-J_{+}=i\int\limits_{0}^{\infty}e^{i\tau H}T_{+}e^{-i\tau H_{0}}d\tau,$ we obtain$$\left. \left( \mathbf{\tilde{S}}g_{1},g_{2}\right) =i\int\limits_{0}^{\infty}i\left( \int\limits_{-\infty}^{\infty}\left( T_{-}e^{-itH_{0}}g_{1},e^{i\tau H}T_{+}e^{-i\left( \tau+t\right) H_{0}}g_{2}\right)
dt\right) d\tau-i{\displaystyle\int\limits_{-\infty}^{\infty}}
\left( T_{-}e^{-itH_{0}}g_{1},J_{+}e^{-itH_{0}}g_{2}\right) dt.\right.
\label{representation44}$$ By the same argument as that we used in (\[representation60\]) we show the convergence of the integrals in (\[representation44\]).
Let us consider the first term of the R.H.S. of (\[representation44\]). We represent this term as the following double limit$$\left.
\begin{array}
[c]{c}i\int\limits_{0}^{\infty}i\left( \int\limits_{-\infty}^{\infty}\left(
T_{-}e^{-itH_{0}}g_{1},e^{i\tau H}T_{+}e^{-i\left( \tau+t\right) H_{0}}g_{2}\right) dt\right) d\tau\\
=\lim\limits_{\mu,\mu^{\prime}\downarrow0}i{\displaystyle\int\limits_{0}^{\infty}}
e^{-\mu\tau}i\left(
{\displaystyle\int\limits_{-\infty}^{\infty}}
e^{-\mu^{\prime}\left\vert t\right\vert }\left( e^{i\left( \tau+t\right)
H_{0}}T_{+}^{\ast}e^{-i\tau H}T_{-}e^{-itH_{0}}g_{1},g_{2}\right) dt\right)
d\tau.
\end{array}
\right. \label{representation45}$$
As $\mathcal{F}_{0}$ gives a spectral representation of $H_{0}$ (see \[basicnotions12\]), then $$\left( g_{1},g_{2}\right) =\int\limits_{\Lambda}\left( \Gamma_{0}\left(
E\right) g_{1}\left( \cdot\right) ,\Gamma_{0}\left( E\right) g_{2}\left(
\cdot\right) \right) _{\mathcal{H}\left( E\right) }dE,
\label{representation46}$$ for $g_{1},g_{2}\in L_{s}^{2}\cap E_{0}\left( \Lambda\right) L^{2},$ $s>1/2,$ where $\Gamma_{0}\left( E\right) $ is given by the relation (\[basicnotions11\]). Applying (\[representation46\]) to the R.H.S. of (\[representation45\]) and using the identities $i\int\limits_{0}^{\infty
}e^{-\mu\tau}e^{-i\tau\left( H-E\right) }d\tau=R\left( E+i\mu\right) $ and $i\int\limits_{-\infty}^{\infty}e^{-\mu^{\prime}\left\vert t\right\vert
}e^{-it\left( H_{0}-E\right) }dt=2\pi i\delta_{\mu^{\prime}}\left(
H_{0}-E\right) ,$ to calculate the integrals on $t$ and $\tau$ of the resulting expression we get $$\left.
\begin{array}
[c]{c}i\int\limits_{0}^{\infty}i\left( \int\limits_{-\infty}^{\infty}\left(
T_{-}e^{-itH_{0}}g_{1},e^{i\tau H}T_{+}e^{-i\left( \tau+t\right) H_{0}}g_{2}\right) dt\right) d\tau\\
=\lim\limits_{\mu,\mu^{\prime}\downarrow0}2\pi i\int\limits_{\Lambda}\left(
\Gamma_{0}\left( E\right) T_{+}^{\ast}R\left( E+i\mu\right) T_{-}\delta_{\mu^{\prime}}\left( H_{0}-E\right) g_{1},\Gamma_{0}\left( E\right)
g_{2}\right) dE.
\end{array}
\right. \label{representation247}$$ Let $f_{1},f_{2}\in\mathcal{H}\left( E\right) $ be $C^{\infty}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) $ functions and take $g_{j}$ as in (\[representation134\])$.$ Note that $g_{j}\in\mathcal{S}\left(
\mathbb{R}^{3};\mathbb{C}^{4}\right) $ and $\Gamma_{0}\left( E\right)
g_{j}=f_{j}.$ Moreover, observe that the operator $\Gamma_{0}\left( E\right)
\left\langle x\right\rangle ^{-s},$ for $s>1/2$, is bounded$.$ Then, relation $\lim_{\mu\downarrow0}\left\Vert (\delta_{\mu}\left( H_{0}-E\right) \right.
$ $\left. -\Gamma_{0}^{\ast}\left( E\right) \Gamma_{0}\left( E\right)
)f\right\Vert _{L_{-s}^{2}}=0$ and Lemma \[representation33\] imply that the limit in the R.H.S. of (\[representation247\]) exists. Hence, we obtain$$\left. i\int\limits_{0}^{\infty}i\left( \int\limits_{-\infty}^{\infty
}\left( T_{-}e^{-itH_{0}}g_{1},e^{i\tau H}T_{+}e^{-i\left( \tau+t\right)
H_{0}}g_{2}\right) dt\right) d\tau=2\pi i{\displaystyle\int\limits_{\Lambda}}
\left( \Gamma_{0}\left( E\right) T_{+}^{\ast}R_{+}\left( E\right)
T_{-}\Gamma_{0}\left( E\right) ^{\ast}f_{1},f_{2}\right) _{\mathcal{H}\left( E\right) }dE.\right. \label{representation143}$$
Applying (\[representation46\]) to the second term of the R.H.S. of (\[representation44\]) we have $$\left. i{\displaystyle\int\limits_{-\infty}^{\infty}}
\left( T_{-}e^{-itH_{0}}g_{1},J_{+}e^{-itH_{0}}g_{2}\right) dt={\displaystyle\int\limits_{\Lambda}}
\left( i{\displaystyle\int\limits_{-\infty}^{\infty}}
\left( \Gamma_{0}\left( E\right) J_{+}^{\ast}T_{-}e^{-it\left(
H_{0}-E\right) }g_{1},\Gamma_{0}\left( E\right) g_{2}\right)
_{\mathcal{H}\left( E\right) }dt\right) dE.\right.
\label{representation204}$$ As the operators $\Gamma_{0}\left( E\right) \left\langle x\right\rangle
^{-s}$ and $\left\langle x\right\rangle ^{s}J_{+}^{\ast}\left\langle
x\right\rangle ^{-s},$ for $s>1/2$, are bounded, splitting $\left\langle
x\right\rangle ^{s}T_{-}=\left\langle x\right\rangle ^{s}T_{-}^{1}+\left\langle x\right\rangle ^{s}T_{-}^{2}$ and using Proposition \[representation118\] for $\left\langle x\right\rangle ^{s}T_{-}^{1}$ (with $N\geq1$) and Lemma \[representation56\] for $\left\langle x\right\rangle
^{s}T_{-}^{2}$ we conclude that the integral in the variable $t$ in the R.H.S. of (\[representation204\]) is absolutely convergent. Then we get $i\int\limits_{-\infty}^{\infty}\left( T_{-}e^{-itH_{0}}g_{1},J_{+}e^{-itH_{0}}g_{2}\right) dt=\int\limits_{\Lambda}\lim\limits_{\mu\downarrow
0}i\int\limits_{-\infty}^{\infty}e^{-\mu\left\vert t\right\vert }(J_{+}^{\ast
}T_{-}e^{-it\left( H_{0}-E\right) }g_{1},$ $\Gamma_{0}^{\ast}\left(
E\right) \Gamma_{0}\left( E\right) g_{2})dtdE.$ Calculating the integral on the variable $t$ we finally obtain$$\left. i{\displaystyle\int\limits_{-\infty}^{\infty}}
\left( T_{-}e^{-itH_{0}}g_{1},J_{+}e^{-itH_{0}}g_{2}\right) dt=2\pi i{\displaystyle\int\limits_{\Lambda}}
\lim_{\mu\downarrow0}\left( J_{+}^{\ast}T_{-}\delta_{\mu}\left(
H_{0}-E\right) g_{1},\Gamma_{0}^{\ast}\left( E\right) \Gamma_{0}\left(
E\right) g_{2}\right) dE.\right. \label{representation144}$$
Using equalities (\[representation142\]), (\[representation44\]), (\[representation143\]) and (\[representation144\]) we obtain$$\left.
\begin{array}
[c]{c}{\displaystyle\int\limits_{\Lambda}}
\left( \tilde{S}\left( E\right) f_{1},f_{2}\right) _{\mathcal{H}\left(
E\right) }dE\\
=2\pi i{\displaystyle\int\limits_{\Lambda}}
\left( \Gamma_{0}\left( E\right) T_{+}^{\ast}R_{+}\left( E\right)
T_{-}\Gamma_{0}\left( E\right) ^{\ast}f_{1},f_{2}\right) _{\mathcal{H}\left( E\right) }dE-2\pi i{\displaystyle\int\limits_{\Lambda}}
\lim_{\mu\downarrow0}\left( J_{+}^{\ast}T_{-}\delta_{\mu}\left(
H_{0}-E\right) g_{1},\Gamma_{0}^{\ast}\left( E\right) \Gamma_{0}\left(
E\right) g_{2}\right) dE.
\end{array}
\right. \label{representation190}$$ Note that the limit in the second term in the R.H.S. of (\[representation190\]) is equals to the limit in relation (\[representation134\]). Then applying Lemma \[representation168\] to calculate this limit we get$${\displaystyle\int\limits_{\Lambda}}
\left( \tilde{S}\left( E\right) f_{1},f_{2}\right) _{\mathcal{H}\left(
E\right) }dE={\displaystyle\int\limits_{\Lambda}}
\left( \left( S_{1}\left( E\right) +S_{2}\left( E\right) \right)
f_{1},f_{2}\right) _{\mathcal{H}\left( E\right) }dE,
\label{representation205}$$ where $S_{1}\left( E\right) $ is given by relation (\[representation14\]) and $S_{2}\left( E\right) $ is defined by (\[representation171\]). Since relation (\[representation205\]) holds for all bounded $\Lambda
\subset(-\infty,-m)\cup(m,+\infty)$ and all $f_{1},f_{2}\in\mathcal{H}\left(
E\right) \cap C^{\infty}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $, we obtain relation (\[representation30\]).
\[representation191\]For any $\left\vert E\right\vert >m$ the scattering matrix $S\left( E\right) $ satisfy the relation $$S\left( E\right) =S_{1}\left( E\right) +S_{2}\left( E\right) .$$
Recall that the scattering operators $\mathbf{S}$ and $\mathbf{\tilde{S}}$ are related by equation (\[representation206\]). As $\tilde{S}\left( E\right)
$ satisfies equation (\[representation30\]) and since $c_{1}$ in the definition of $\theta\left( t\right) $ is arbitrary, we conclude that for every $\left\vert E\right\vert >m$ there is $c_{1}$ such that the scattering matrix $S\left( E\right) $ is equal to the scattering matrix $\tilde
{S}\left( E\right) .$ Thus, we get the relation (\[representation30\]) also for $S\left( E\right) .$
Theorem \[representation25\] is now consequence of Corollary \[representation191\], Theorem \[representation32\] and Lemma \[representation168\]. Theorem \[representation179\] follows from Corollary \[representation191\], Theorem \[representation32\] and Theorem \[representation241\].
Applications of the formula for the singularities of the kernel of the scattering matrix.
=========================================================================================
High energy limit of the scattering matrix.
-------------------------------------------
Let us consider the principal part $S_{0}\left( E\right) ,$ of $S\left(
E\right) .$ We take $N=0$ in the relation (\[representation26\]). Then, the kernel $s_{0}\left( \omega,\theta;E\right) $ of $S_{0}\left( E\right) $ is given by $s_{0}\left( \omega,\theta;E\right) =\sum\limits_{O_{j}\cap
O_{k}\neq\varnothing}s_{0,jk}\left( \omega,\theta;E\right) $
Let us define the operator $\mathbf{P}\left( E\right) :L^{2}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) \rightarrow L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $ by the relation $\left( \mathbf{P}\left(
E\right) f\right) \left( \omega\right) :=P_{\omega}\left( E\right)
f\left( \omega\right) .$ Moreover we denote $\mathbf{P}\left( \pm
\infty\right) :L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \rightarrow
L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $ by the relation $\left(
\mathbf{P}\left( \pm\infty\right) f\right) \left( \omega\right)
:=P_{\omega}\left( \pm\infty\right) f\left( \omega\right) ,$ where $P_{\omega}\left( \pm\infty\right) =\frac{1}{2}\left( 1\pm\left(
\alpha\cdot\omega\right) \right) .$ Note that $\mathbf{P}\left( E\right) $ converges in $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $ norm to $\mathbf{P}\left( \pm\infty\right) ,$ as $\pm E\rightarrow\infty.$ We prove the following result.
The operator $S_{0}\left( E\right) $ is uniformly bounded in $L^{2}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ for all $\left\vert E\right\vert \geq
E_{0}$, $E_{0}>m,$ and the following estimate holds$$\left\Vert S\left( E\right) -S_{0}\left( E\right) \right\Vert =O\left(
\left\vert E\right\vert ^{-1}\right) ,\text{ }\left\vert E\right\vert
\rightarrow\infty. \label{representation77}$$ Moreover, $S\left( E\right) \mathbf{P}\left( E\right) $ converges strongly$,$ as $\pm E\rightarrow\infty,$ in $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $ to the operator $S\left( \pm\infty\right)
\mathbf{P}\left( \pm\infty\right) $ where $S\left( \pm\infty\right) $ is the operator of multiplication by the function $\sum\limits_{O_{j}\cap
O_{k}\neq\varnothing}\chi_{j}\left( \omega\right) \chi_{k}\left(
\omega\right) e^{-i\int_{-\infty}^{\infty}\left( V\left( t\omega\right)
\pm\left\langle \omega,A\left( t\omega\right) \right\rangle \right) dt}.$
Using the notation of Remark \[representation75\] and making the change $z=\nu\left( E\right) y$ in the relation for $\tilde{s}_{0}\left(
\zeta,\zeta^{\prime};E\right) ,$ we obtain $\tilde{s}_{0}\left( \zeta
,\zeta^{\prime};E\right) =\left( 2\pi\right) ^{-2}\left( \frac
{\upsilon\left( E\right) }{\nu\left( E\right) }\right) ^{2}{\displaystyle\int\limits_{\Pi_{\omega_{jk}}}}
e^{i\left\langle z,\zeta^{\prime}-\zeta\right\rangle }\mathbf{\tilde{h}}_{N,jk}^{\prime}\left( \frac{z}{\nu\left( E\right) },\zeta,\zeta^{\prime
};E\right) dz.$ Since $\left\vert \partial_{z}^{\alpha}\partial_{\zeta
}^{\beta}\partial_{\zeta^{\prime}}^{\gamma}\mathbf{\tilde{h}}_{N,jk}^{\prime
}\left( \frac{z}{\nu\left( E\right) },\zeta,\zeta^{\prime};E\right)
\right\vert \leq C_{\alpha,\beta,\,\gamma}\left\langle z\right\rangle
^{-\left\vert \alpha\right\vert },$ where $C_{\alpha,\beta,\,\gamma}$ is independent on $E,$ for $\left\vert E\right\vert \geq E_{0}$, and $\mathbf{\tilde{h}}_{N,jk}^{\prime}$ is a compact-supported function of $\zeta$ and $\zeta^{\prime},$ it follows from Proposition \[representation76\] that $S_{0}\left( E\right) $ is uniformly bounded in $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ for $\left\vert
E\right\vert \geq E_{0}.$
Using (\[eig42\]), (\[eig8\]), (\[eig9\]), (\[eig22\]), (\[eig19\]), (\[eig20\]) and (\[eig33\]) we see that $$\left\vert \partial_{z}^{\alpha}\partial_{\zeta}^{\beta}\partial
_{\zeta^{\prime}}^{\gamma}\left( \mathbf{\tilde{h}}_{N,jk}^{\prime
}-\mathbf{\tilde{h}}_{0,jk}^{\prime}\right) \left( \frac{z}{\nu\left(
E\right) },\zeta,\zeta^{\prime};E\right) \right\vert \leq C_{\alpha
,\beta,\,\gamma}\nu\left( E\right) ^{-1}\left\langle z\right\rangle
^{-\rho-\left\vert \alpha\right\vert }, \label{representation193}$$ where $C_{\alpha,\beta,\,\gamma}$ is independent on $E,$ for $\left\vert
E\right\vert \geq E_{0}.$ Then, equality (\[representation77\]) follows from Proposition \[representation76\].
Now we calculate the limit of $S\left( E\right) $ as $\pm E\rightarrow
\infty.$ Note that integrating by parts on the variable $\zeta^{\prime}$ we have ${\displaystyle\int_{\mathbb{S}^{2}}}
s_{0,jk}\left( \omega,\theta;E\right) $ $\times f\left( \theta\right)
d\theta=\left( 2\pi\right) ^{-2}{\displaystyle\int_{\Pi_{\omega_{jk}}}}
{\displaystyle\int_{\Pi_{\omega_{jk}}}}
e^{i\left\langle z,\zeta^{\prime}-\zeta\right\rangle }\left\langle
z\right\rangle ^{-n}\left\langle D_{\zeta^{\prime}}\right\rangle ^{n}\left(
\mathbf{\tilde{h}}_{0,jk}^{\prime}\left( \frac{z}{\nu\left( E\right)
},\zeta,\zeta^{\prime};E\right) \tilde{f}\left( \zeta^{\prime}\right)
\right) d\zeta^{\prime}dz$, for even $n$ and any $f\in C^{\infty}.$ Thus, taking the limit, as $\pm E\rightarrow\infty,$ in the R.H.S. of the last relation and then integrating back by parts we get $$\left. \lim_{\pm E\rightarrow\infty}{\displaystyle\int_{\mathbb{S}^{2}}}
s_{0}\left( \omega,\theta;E\right) f\left( \theta\right) d\theta=\left(
2\pi\right) ^{-2}{\displaystyle\int_{\Pi_{\omega_{jk}}}}
{\displaystyle\int_{\Pi_{\omega_{jk}}}}
e^{i\left\langle z,\zeta^{\prime}-\zeta\right\rangle }\mathbf{\tilde{h}}_{0,jk}^{\prime}\left( 0,\zeta,\zeta^{\prime};\infty\right) \tilde
{f}\left( \zeta^{\prime}\right) d\zeta^{\prime}dz,\right.
\label{representation207}$$ where $\mathbf{\tilde{h}}_{0,jk}^{\prime}\left( 0,\zeta,\zeta^{\prime};\pm\infty\right) :=\pm\frac{\tilde{\chi}_{jk}\left( \zeta,\zeta^{\prime
}\right) \tilde{\chi}_{j}\left( \zeta\right) \tilde{\chi}_{k}\left(
\zeta^{\prime}\right) }{\left( 1-\left\vert \zeta^{\prime}\right\vert
^{2}\right) ^{1/2}}\tilde{P}_{\zeta}\left( \pm\infty\right) \left(
\alpha\cdot\omega_{jk}\right) \tilde{P}_{\zeta^{\prime}}\left( \pm
\infty\right) .$ The integral in the variable $\zeta^{\prime}$ in (\[representation207\]) is the inverse of the Fourier transform of the function $\mathbf{\tilde{h}}_{0,jk}^{\prime}\left( 0,\zeta,\zeta^{\prime
};\infty\right) \tilde{f}\left( \zeta^{\prime}\right) .$ As $\mathbf{\tilde
{h}}_{0,jk}^{\prime}\left( 0,\zeta,\zeta^{\prime};\infty\right) \tilde
{f}\left( \zeta^{\prime}\right) $ has a compact support in $\zeta^{\prime},$ then calculating the integral with respect to $\zeta^{\prime}$ in (\[representation207\]) we obtain a function of the variable $z$ that belongs to $\mathcal{S}.$ Calculating the integral in $z$ we get back the function $\mathbf{\tilde{h}}_{0,jk}^{\prime}\left( 0,\zeta,\zeta
;\infty\right) \tilde{f}\left( \zeta\right) .$ From relation (\[representation113\]) we get $$P_{\omega}\left( \pm\infty\right) \left( \alpha\cdot\omega_{jk}\right)
P_{\omega}\left( \pm\infty\right) =\pm\left\langle \omega,\omega
_{jk}\right\rangle P_{\omega}\left( \pm\infty\right)
.\ \label{representation251}$$ As $\pm\left\langle \omega,\omega_{jk}\right\rangle =\left( 1-\left\vert
\zeta\right\vert ^{2}\right) ^{1/2},$ for $\omega\in\Omega_{\pm}\left(
\omega_{jk},\delta\right) ,$ then $\left\langle \omega,\omega_{jk}\right\rangle \chi_{jk}\left( \omega,\omega\right) \chi_{j}\left(
\omega\right) \chi_{k}\left( \omega\right) =\left( 1-\left\vert
\zeta\right\vert ^{2}\right) ^{1/2}\chi_{j}\left( \omega\right) \chi
_{k}\left( \omega\right) .$ Therefore, using these relations in the expression for $\mathbf{\tilde{h}}_{0,jk}^{\prime}\left( 0,\zeta,\zeta
;\pm\infty\right) ,$ substituting the result in (\[representation207\]), and taking in account that $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}\chi_{j}\left( \omega\right) \chi_{k}\left( \omega\right) =1,$ we complete the proof.
Leading diagonal singularity of the kernel of the scattering matrix.
--------------------------------------------------------------------
Recall that for $\rho=\min\{\rho_{e},\rho_{m}\}>3,$ $s^{\operatorname{int}}\left( \omega,\theta;E\right) \in C^{0}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) ,$ where $s^{\operatorname{int}}$ is the kernel of the operator $S\left( E\right) -I$ (see Theorem \[representation25\]). Let us now calculate the leading term on the diagonal of $s^{\operatorname{int}}\left( \omega,\theta;E\right) ,$ for $1<\rho<3,$ as $\omega-\theta\rightarrow0,$ with $E$ fixed$.$ For a fixed $\omega_{0}\in\mathbb{S}^{2},$ we take a cut-off function $\Psi_{+}\left( \omega
,\theta;\omega_{0}\right) ,$ supported on $\Omega_{+}\left( \omega
_{0},\delta\right) \times\Omega_{+}\left( \omega_{0},\delta\right) ,$ such that it is equal to $1$ in $\Omega_{+}\left( \omega_{0},\delta^{\prime
}\right) $, for some $\delta^{\prime}>\delta.$ We define $\mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( x\right) :=\frac{\left\vert
E\right\vert }{\nu\left( E\right) }V\left( x\right) +\left(
\operatorname*{sgn}E\right) \left\langle \omega,A\left( x\right)
\right\rangle .$ The following result is similar to Theorem 1.2 of [@47] for the Schrödinger equation:
\[representation231\]Let the magnetic potential $A\left( x\right) $ and the electric potential $V\left( x\right) $ satisfy the estimates (\[eig31\]) and (\[eig32\]), with $1<\rho<3,$ respectively. Then, for all fixed $\omega\in\mathbb{S}^{2}$ and $\theta\in\Omega_{+}\left( \omega
,\delta^{\prime}\right) ,$ $\omega\neq\theta,$ we have$$\left. \left\vert \left( s^{\operatorname{int}}\left( \omega,\theta
;E\right) -\frac{1}{i}\left( 2\pi\right) ^{-1/2}\upsilon\left( E\right)
^{2}\frac{\nu\left( E\right) }{\left\vert E\right\vert }\left(
\mathcal{FV}_{V,A;\omega}^{\left( E\right) }\right) \left( -\nu\left(
E\right) \tilde{\theta}\right) P_{\omega}\left( E\right) \right)
\right\vert \leq C\left\vert \omega-\theta\right\vert ^{-2+\rho_{1}},\right.
\label{representation102}$$ where $\tilde{\theta}=\theta-\left\langle \theta,\omega\right\rangle \omega,$ $\rho_{1}=2\left( \rho-1\right) ,$ if $\rho<2$ and $\rho_{1}=2-\varepsilon,$ with $\varepsilon>0,$ for $\rho=2.$ Here the constant $C$ is independent on $\omega.$ If $\rho>2,$ then the difference in the L.H.S. of (\[representation102\]) is continuous$.$
Note that $$\left\vert \omega-\theta\right\vert ^{2}=2\left( 1-\left\langle \theta
,\omega\right\rangle \right) =2\frac{\left\vert \tilde{\theta}\right\vert
^{2}}{1+\sqrt{1-\left\vert \tilde{\theta}\right\vert ^{2}}}.
\label{representation99}$$ Let us define $h\left( y,\omega,\theta;E\right) :=-i\left(
\operatorname*{sgn}E\right) \left( \Phi^{+}\left( y,\nu\left( E\right)
\omega;E\right) -\Phi^{-}\left( y,\nu\left( E\right) \theta;E\right)
\right) P_{\omega}\left( E\right) \left( \alpha\cdot\omega\right)
P_{\theta}\left( E\right) .$ Putting $\omega=\omega_{0}$ in (\[representation252\]) and using estimates (\[eig22\]), (\[eig19\]) and (\[eig20\]) we have $\left( \mathbf{h}_{N}-\left( \operatorname*{sgn}E\right) P_{\omega}\left( E\right) \left( \alpha\cdot\omega\right)
P_{\theta}\left( E\right) -h\right) \in\mathcal{S}^{-\rho_{1}}.$ Then, decomposing $\theta\neq\omega$ as $\theta=\left\langle \theta,\omega
\right\rangle \omega+\tilde{\theta},$ $\tilde{\theta}\in\Pi_{\omega}\ $and using Lemma \[representation201\] and (\[representation99\]) we get $$\left.
\begin{array}
[c]{c}\left\vert \left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\int\limits_{\Pi_{\omega}}e^{i\nu\left( E\right) \left\langle y,\tilde
{\theta}\right\rangle }\left( \mathbf{h}_{N}\left( y,\omega,\theta
;E;\omega\right) -\left( \operatorname*{sgn}E\right) P_{\omega}\left(
E\right) \left( \alpha\cdot\omega\right) P_{\theta}\left( E\right)
-h\left( y,\omega,\theta;E\right) \right) dy\right\vert \\
\leq\left\{
\begin{array}
[c]{c}C\left\vert \omega-\theta\right\vert ^{-2+\rho_{1}},\text{ }\rho<2,\\
C\left\vert \omega-\theta\right\vert ^{-\varepsilon},\text{ }\varepsilon
>0,\text{ }\rho=2.
\end{array}
\right.
\end{array}
\right. \label{representation96}$$ Moreover, for $\rho>2,$ as the integral in the L.H.S. of (\[representation96\]) is absolutely convergent, it is a continuous function of $\omega$ and $\theta.$
Let us show that for all $\alpha,$ $$\left\vert \int\limits_{0}^{\infty}\partial_{y}^{\alpha}\left( \mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( y\pm t\omega\right) -\mathcal{V}_{V,A;\theta}^{\left( E\right) }\left( y\pm t\theta\right) \right)
dt\right\vert \leq C_{\alpha}\left\vert \omega-\theta\right\vert \left\langle
y\right\rangle ^{-\left( \rho-1\right) -\left\vert \alpha\right\vert },
\label{representation93}$$ Since $\mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( y\pm
t\theta\right) -\mathcal{V}_{V,A;\theta}^{\left( E\right) }\left( y\pm
t\theta\right) =\left( \operatorname*{sgn}E\right) \left\langle
\omega-\theta,A\left( y\pm t\theta\right) \right\rangle ,$ and $A$ satisfies the estimate (\[eig31\]), then it is enough to prove the following relation $$\left\vert \int\limits_{0}^{\infty}\partial_{y}^{\alpha}\left( \mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( y\pm t\omega\right) -\mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( y\pm t\theta\right) \right)
dt\right\vert \leq C_{\alpha}\left\vert \omega-\theta\right\vert \left\langle
y\right\rangle ^{-\left( \rho-1\right) -\left\vert \alpha\right\vert }.
\label{representation242}$$ First take $\alpha=0.$ Using the mean value theorem we have$$\left. \mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( y\pm
t\omega\right) -\mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( y\pm
t\theta\right) =\pm t\left\langle \left( \left( \triangledown
\mathcal{V}_{V,A;\omega}^{\left( E\right) }\right) \left( \pm ct\left(
\theta-\omega\right) +\left( y\pm t\omega\right) \right) \right)
,\omega-\theta\right\rangle ,\right. \label{representation94}$$ for some $0\leq c\leq1.$ Estimates (\[eig31\]) and (\[eig32\]) for $A$ and $V$ imply $$\left. \left\vert \left( \triangledown\mathcal{V}_{V,A;\omega}^{\left(
E\right) }\right) \left( \pm ct\left( \theta-\omega\right) +\left( y\pm
t\omega\right) \right) \right\vert \leq C\left( 1+\left\vert ct\left(
\theta-\omega\right) \pm\left( y\pm t\omega\right) \right\vert \right)
^{-\rho-1}.\right. \label{representation95}$$ Let us take $\left\vert \tilde{\theta}\right\vert \leq\sqrt{1-\delta^{2}}$. Then, for $\delta$ close enough to $1,$ we get $\left\vert ct\left(
\theta-\omega\right) \pm\left( y\pm t\omega\right) \right\vert ^{2}=c^{2}t^{2}\left\vert \theta-\omega\right\vert ^{2}\pm2ct\left\vert
y\right\vert \left\langle \hat{y},\theta\right\rangle -2ct^{2}+2ct^{2}\left\langle \omega,\theta\right\rangle +\left\vert y\right\vert ^{2}+t^{2}~$ $\geq\left( 1-\sqrt{1-\delta^{2}}\right) \left\vert y\right\vert
^{2}+t^{2}\left( 1-\eta+2\left( \delta-1\right) \right) \geq c_{1}\left(
\left\vert y\right\vert ^{2}+t^{2}\right) ,$ for some $c_{1}>0.$ Using this estimate in (\[representation95\]) and substituting the resulting inequality in (\[representation94\]) we obtain estimate (\[representation242\]), and hence, relation (\[representation93\]), for $\alpha=0$. The proof of (\[representation93\]) for $\alpha>0$ is analogous.
From (\[representation113\]) we have that $\frac{\nu\left( E\right)
}{\left\vert E\right\vert }P_{\omega}\left( E\right) =\left(
\operatorname*{sgn}E\right) P_{\omega}\left( E\right) \left( \alpha
\cdot\omega\right) P_{\omega}\left( E\right) .$ Then, using $\left\vert
\partial_{y}^{\alpha}(-i\left( \operatorname*{sgn}E\right) (\Phi^{+}\left(
y,\nu\left( E\right) \omega;E\right) \right. $ $\left. -\Phi^{-}\left(
y,\nu\left( E\right) \theta;E\right) ))P_{\omega}\left( E\right) \left(
\alpha\cdot\omega\right) (P_{\theta}\left( E\right) -P_{\omega}\left(
E\right) )\right\vert \leq C_{\alpha}\left\vert \omega-\theta\right\vert
\left\langle y\right\rangle ^{-\left( \rho-1\right) -\left\vert
\alpha\right\vert }$ and (\[representation93\]) we get, for all $\alpha,$$$\left\vert \partial_{y}^{\alpha}\left( h\left( y,\omega,\theta;E\right)
-i\frac{\nu\left( E\right) }{\left\vert E\right\vert }R\left(
y,\omega;E\right) P_{\omega}\left( E\right) \right) \right\vert \leq
C_{\alpha}\left\vert \omega-\theta\right\vert \left\langle y\right\rangle
^{-\left( \rho-1\right) -\left\vert \alpha\right\vert },
\label{representation92}$$ where$$R\left( y,\omega;E\right) :=\int\limits_{-\infty}^{\infty}\left(
\mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( y+t\omega\right)
\right) dt. \label{representation219}$$
Using (\[representation92\]), Lemma \[representation201\] and (\[representation99\]) we have$$\left. \left\vert \int\nolimits_{\Pi_{\omega}}e^{i\nu\left( E\right)
\left\langle y,\tilde{\theta}\right\rangle }\left( h\left( y,\omega
,\theta;E\right) +i\frac{\nu\left( E\right) }{\left\vert E\right\vert
}R\left( y,\omega;E\right) P_{\omega}\left( E\right) \right)
dy\right\vert \leq C\left\vert \omega-\theta\right\vert ^{-2+\rho}\right. .
\label{representation97}$$ Then, relation (\[representation102\]) follows from Theorem \[representation179\], estimates (\[representation96\]), (\[representation97\]) and equation $-i\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\int\limits_{\Pi_{\omega}}e^{i\nu\left(
E\right) \left\langle y,\tilde{\theta}\right\rangle }R\left( y,\omega
;E\right) dy$ $=-i\left( 2\pi\right) ^{-1/2}\upsilon\left( E\right)
^{2}\left( \mathcal{FV}_{V,A;\omega}^{\left( E\right) }\right) \left(
-\nu\left( E\right) \tilde{\theta}\right) .$
[Suppose that $V,A\in C^{\infty}$ are such that $V=V_{0}\left\vert
x\right\vert ^{-\rho}\ $and $A=A_{0}\left\vert x\right\vert ^{-\rho},$ $1<\rho<3,$ for $\left\vert x\right\vert \geq R,$ for some $R>0,$ and $V_{0}$ is a real constant and $A_{0}$ is a constant, real vector, satisfying $V_{0}+\left\langle \omega,A_{0}\right\rangle $ $\neq0,$ for all $\omega\in
S^{2}.$ Since $FV_{V,A;\omega}^{\left( E\right) }=F\left( \mathcal{V}_{V,A;\omega}^{\left( E\right) }\mathbf{-}\left( V_{0}+\left\langle
\omega,A_{0}\right\rangle \right) \left\vert x\right\vert ^{-\rho}\right)
+\left( V_{0}+\left\langle \omega,A_{0}\right\rangle \right) F\left(
\left\vert x\right\vert ^{-\rho}\right) $ and $F\left( \left\vert
x\right\vert ^{-\rho}\right) =2^{3-\rho}\pi^{\frac{3}{2}}\frac{\Gamma\left(
\frac{3-\rho}{2}\right) }{\Gamma\left( \frac{\rho}{2}\right) }\left\vert
\xi\right\vert ^{-\left( 3-\rho\right) }=4\pi\rho\left( \rho-1\right)
\Gamma\left( -\rho\right) \left( \sin\frac{\pi\rho}{2}\right) \left\vert
\xi\right\vert ^{-\left( 3-\rho\right) },$ where $\Gamma$ is the Gamma function (see [@76]), then as in the non-relativistic case [@47], relations (\[representation102\]) and (\[representation99\]) imply that the estimate (\[representation83\]) is optimal. This implies that the relation $\left\vert s^{\operatorname{int}}\left( \omega,\theta;E\right)
\right\vert \leq C\left\vert \omega-\theta\right\vert ^{-3+\rho}$ is the best possible.]{}
The scattering cross-section.
-----------------------------
As before let $s^{\operatorname{int}}$ be the kernel of the operator $S\left(
E\right) -I.$ Let us consider the principal part $\mathcal{G}_{0}\left(
E\right) ,$ of $S\left( E\right) -I.$ We take $N=0$ in the relation (\[representation181\]). Then, using the definition (\[eig33\]) and relations (\[eig42\]), (\[eig8\]) and (\[eig9\]) we get that the kernel $\mathbf{g}_{0}^{\operatorname{int}}\left( \omega,\theta;E\right) $ of $\mathcal{G}_{0}\left( E\right) $ is given by$$\left. \mathbf{g}_{0}^{\operatorname{int}}\left( \omega,\theta;E\right)
=\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}\mathbf{g}_{0,jk}^{\operatorname{int}}\left( \omega,\theta;E\right) ,\right.
\label{representation80}$$ with $\mathbf{g}_{0,jk}^{\operatorname{int}}\left( \omega,\theta;E\right)
=\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\chi_{jk}\left(
\omega,\theta\right) \chi_{j}\left( \omega\right) \chi_{k}\left(
\theta\right)
{\displaystyle\int_{\Pi_{\omega_{jk}}}}
e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }\mathbf{h}_{pr}\left( y,\omega,\theta;E\right) dy,$ and $\mathbf{h}_{pr}\left( y,\omega,\theta;E\right) :=\left( \operatorname*{sgn}E\right)
$ $\times\left( e^{-i\Phi^{+}\left( y,\nu\left( E\right) \omega;E\right)
+i\Phi^{-}\left( y,\nu\left( E\right) \theta;E\right) }-1\right)
P_{\omega}\left( E\right) \left( \alpha\cdot\omega_{jk}\right) P_{\theta
}\left( E\right) .$
If $\rho>3,~$then Theorem \[representation25\] assures that $\mathbf{g}_{N,jk}^{\operatorname{int}}\left( \omega,\theta;E\right) $ is a continuous function of $\omega$ and$\ \theta$. Thus, we can consider the limit of $\mathbf{g}_{0,jk}^{\operatorname{int}}\left( \omega,\theta;E\right) $ as $\left\vert E\right\vert \rightarrow\infty$ on the diagonal $\omega
=\theta=\omega_{jk}.$ Taking in account (\[representation193\]) and relation (\[representation80\]), and using (\[representation251\]), we have $$\lim_{\pm E\rightarrow\infty}\left\Vert \frac{s^{\operatorname{int}}\left(
\omega,\omega;E\right) }{\upsilon\left( E\right) ^{2}}-\left( 2\pi\right)
^{-2}\int\limits_{\Pi_{\omega}}m_{\pm}\left( y,\omega\right) dy\right\Vert
_{\mathcal{B}\left( X^{\pm}\left( \nu\left( E\right) \omega\right)
\right) }=0, \label{representation79}$$ where $m_{\pm}\left( y,\omega\right) =\left( e^{-i\int_{-\infty}^{\infty
}\left( V\left( y+t\omega\right) \pm\left\langle \omega,A\left(
y+t\omega\right) \right\rangle \right) dt}-1\right) .$ Equality (\[representation79\]) was proved in [@15] by studying the high-energy limit of the resolvent.
Now let us prove the following result
\[representation210\]Suppose that $\rho>2.$ Then, the function $s^{\operatorname{int}}\left( \omega,\theta;E\right) +s^{\operatorname{int}}\left( \theta,\omega;E\right) ^{\ast}$ is continuous on $\mathbb{S}^{2}\times\mathbb{S}^{2}.$
It follows from estimates (\[eig22\]), (\[eig19\]) and (\[eig20\]), and definition (\[eig33\]) that $\mathbf{h}_{N,jk}-\mathbf{h}_{0,jk}\in\mathcal{S}^{-\rho}.$ Then, if $\rho>2,$ we have $\chi_{j}\left(
\omega\right) s\left( \omega,\theta;E\right) \chi_{k}\left( \theta\right)
-\mathbf{g}_{0,jk}^{\operatorname{int}}\left( \omega,\theta;E\right) \in
C^{0}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) .$ Thus, we only need to show that the sum $\mathbf{g}_{0,jk}^{\operatorname{int}}\left(
\omega,\theta;E\right) +\mathbf{g}_{0,jk}^{\operatorname{int}}\left(
\theta,\omega;E\right) ^{\ast}$ is continuous on $\mathbb{S}^{2}\times\mathbb{S}^{2}.$ From the definition of $\mathbf{g}_{0,jk}^{\operatorname{int}}$ we have$$\left.
\begin{array}
[c]{c}\mathbf{g}_{0,jk}^{\operatorname{int}}\left( \omega,\theta;E\right)
+\mathbf{g}_{0,jk}^{\operatorname{int}}\left( \theta,\omega;E\right) ^{\ast
}=\left( \operatorname*{sgn}E\right) \left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\chi_{jk}\left( \omega,\theta\right) \chi
_{j}\left( \omega\right) \chi_{k}\left( \theta\right) \\
\times{\displaystyle\int\limits_{\Pi_{\omega_{jk}}}}
e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }\left(
e^{-i\Phi^{+}\left( y,\nu\left( E\right) \omega;E\right) +i\Phi^{-}\left(
y,\nu\left( E\right) \theta;E\right) }+e^{i\Phi^{+}\left( y,\nu\left(
E\right) \omega;E\right) -i\Phi^{-}\left( y,\nu\left( E\right)
\theta;E\right) }-2\right. \\
\left. +e^{i\Phi^{+}\left( y,\nu\left( E\right) \theta;E\right)
-i\Phi^{-}\left( y,\nu\left( E\right) \omega;E\right) }-e^{i\Phi
^{+}\left( y,\nu\left( E\right) \omega;E\right) -i\Phi^{-}\left(
y,\nu\left( E\right) \theta;E\right) }\right) P_{\omega}\left( E\right)
\left( \alpha\cdot\omega_{jk}\right) P_{\theta}\left( E\right) dy.
\end{array}
\right. \label{representation211}$$ Note that$$\left.
\begin{array}
[c]{c}{\displaystyle\int\limits_{\Pi_{\omega_{jk}}}}
e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }\left(
e^{-i\Phi^{+}\left( y,\nu\left( E\right) \omega;E\right) +i\Phi^{-}\left(
y,\nu\left( E\right) \theta;E\right) }+e^{i\Phi^{+}\left( y,\nu\left(
E\right) \omega;E\right) -i\Phi^{-}\left( y,\nu\left( E\right)
\theta;E\right) }-2\right) dy\\
=2{\displaystyle\int\limits_{\Pi_{\omega_{jk}}}}
e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }\left(
\cos\left( \int\limits_{0}^{\infty}\mathcal{V}_{V,A;\omega}^{\left(
E\right) }\left( y+t\omega\right) +\mathcal{V}_{V,A;\theta}^{\left(
E\right) }\left( y-t\theta\right) dt\right) -1\right) dy.
\end{array}
\right. \label{representation111}$$ The R.H.S. of relation (\[representation111\]) is absolutely convergent if $\rho>2.$ Thus, to complete the proof, it is enough to show that ${\displaystyle\int\limits_{\Pi_{\omega_{jk}}}}
e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }(e^{i\Phi^{+}\left( y,\nu\left( E\right) \theta;E\right) -i\Phi^{-}\left(
y,\nu\left( E\right) \omega;E\right) }-e^{i\Phi^{+}\left( y,\nu\left(
E\right) \omega;E\right) -i\Phi^{-}\left( y,\nu\left( E\right)
\theta;E\right) })dy$ is continuous. Since $\left\vert \left( \Phi^{\pm
}\left( y,\nu\left( E\right) \omega;E\right) \right) ^{n}\right\vert $ $\leq C_{n}\left\langle y\right\rangle ^{-\left( \rho-1\right) n},$ we only have to prove the continuity of the following integral$$i{\displaystyle\int\limits_{\Pi_{\omega_{jk}}}}
e^{i\nu\left( E\right) \left\langle y,\theta-\omega\right\rangle }\left(
\left( \Phi^{+}\left( y,\nu\left( E\right) \theta;E\right) -\Phi
^{-}\left( y,\nu\left( E\right) \omega;E\right) \right) -\left( \Phi
^{+}\left( y,\nu\left( E\right) \omega;E\right) -\Phi^{-}\left(
y,\nu\left( E\right) \theta;E\right) \right) \right) dy.
\label{representation208}$$ Note that estimate (\[representation93\]) implies $\left\vert \partial
_{y}^{\alpha}\left( \left( \Phi^{+}\left( y,\nu\left( E\right)
\theta;E\right) -\Phi^{+}\left( y,\nu\left( E\right) \omega;E\right)
\right) +\left( \Phi^{-}\left( y,\nu\left( E\right) \theta;E\right)
-\Phi^{-}\left( y,\nu\left( E\right) \omega;E\right) \right) \right)
\right\vert $ $\leq C_{\alpha}\left\vert \omega-\theta\right\vert \left\langle
y\right\rangle ^{-\left( \rho-1\right) -\left\vert \alpha\right\vert }.$ Then, it follows from Lemma \[representation201\] that integral (\[representation208\]) is estimated by $C\left\vert \omega-\theta
\right\vert ^{\rho-2},$ and thus, it is continuous.
We define the scattering cross-section for a fixed incoming direction $\theta$ and all outgoing directions $\omega$ by the following relation $\sigma\left(
\theta;E;u\right) =\int_{\mathbb{S}^{2}}\left\vert s^{\operatorname{int}}\left( \omega,\theta;E\right) u\right\vert ^{2}d\omega,$ for a normalized initial state $u\in X^{\pm}\left( \nu\left( E\right) \theta\right) ,$ $\left\vert u\right\vert _{\mathbb{C}^{4}}=1.$ Using (\[representation103\]) we have $$\sigma\left( \theta;E;u\right) =-\left( \left( s^{\operatorname{int}}\left( \theta,\theta;E\right) +s^{\operatorname{int}}\left( \theta
,\theta;E\right) ^{\ast}\right) u,u\right) . \label{representation109}$$
The following Lemma is consequence of the relation (\[representation109\]) and Proposition \[representation210\].
\[representation234\]The scattering cross-section $\sigma\left(
\theta;E;u\right) $ is a continuous function of $\theta,$ for $\rho>2.$ Furthermore, the total scattering cross-section, given by the relation $\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}\left\vert s^{\operatorname{int}}\left( \omega,\theta;E\right) u\right\vert ^{2}d\theta d\omega,$ for a normalized initial state $u\in X^{\pm}\left( \nu\left( E\right)
\theta\right) ,$ $\left\vert u\right\vert _{\mathbb{C}^{4}}=1,$ is finite if $\rho>2.$
The estimate$$\left\vert \partial_{y}^{\alpha}\partial_{\zeta}^{\beta}\partial
_{\zeta^{\prime}}^{\gamma}\left( \mathbf{\tilde{h}}_{N}^{\operatorname{int}}\left( y,\zeta,\zeta^{\prime};E;\omega_{jk}\right) -\mathbf{\tilde{h}}_{pr}\left( y,\zeta,\zeta^{\prime};E\right) \right) \right\vert \leq
C_{\alpha,\beta,\,\gamma}\nu\left( E\right) ^{-1}\left\langle y\right\rangle
^{-\rho-\left\vert \alpha\right\vert }, \label{representation213}$$ where $\mathbf{h}_{N}^{int}$ is defined (\[representation253\]), and Theorem \[representation179\] imply $$\upsilon\left( E\right) ^{-2}\left( \chi_{j}\left( \omega\right)
s^{\operatorname{int}}\left( \omega,\theta;E\right) \chi_{k}\left(
\theta\right) -\mathbf{g}_{0,jk}^{\operatorname{int}}\left( \omega
,\theta;E\right) \right) =O\left( \left\vert E\right\vert ^{-1}\right)
,\text{ as }\left\vert E\right\vert \rightarrow\infty.
\label{representation112}$$ Then, for any $u\in X^{\pm}\left( \nu\left( E\right) \theta\right) ,$ $\left\vert u\right\vert _{\mathbb{C}^{4}}=1,$ taking $\omega=\omega
_{jk}=\theta$ and using relations (\[representation109\]), (\[representation112\]), (\[representation211\]), (\[representation111\]), (\[representation113\]) and equalities $P_{\theta}\left( E\right) u=u,$ $\chi_{jk}\left( \theta,\theta\right) \chi_{j}\left( \theta\right)
\chi_{k}\left( \theta\right) =\chi_{j}\left( \theta\right) \chi_{k}\left(
\theta\right) $ and $\sum\limits_{O_{j}\cap O_{k}\neq\varnothing}$ $\chi
_{j}\left( \theta\right) \chi_{k}\left( \theta\right) =1,$ we get $\left(
2\pi\right) ^{2}\upsilon\left( E\right) ^{-2}\sigma\left( \theta
;E;u\right) $ $=2\int\limits_{\Pi_{\theta}}\left( 1-\cos\int\limits_{-\infty
}^{\infty}\mathcal{V}_{V,A;\theta}^{\left( E\right) }\left( y+t\theta
\right) dt\right) dy+O\left( \left\vert E\right\vert ^{-1}\right) ,$ as $\left\vert E\right\vert \rightarrow\infty.$ A similar result was obtained in [@15] by studying the high-energy limit of the resolvent.
The following result is a consequence of Theorem \[representation25\] and Proposition \[representation231\]
Let the electric potential $V$ satisfy estimate (\[eig32\]) with some $\rho_{e}>1$ and the magnetic field $B$ satisfy the estimate (\[basicnotions18\]) with $r=\rho_{m}+1,$ $\rho_{m}>1$ and all $d.$ Let $V$ and $B$ be homogeneous functions of order $-\rho_{e}$ and $-\rho_{m}-1$, respectively, for $\left\vert x\right\vert \geq R,$ for some $R>0,$ and at least one of them is non-trivial for $\left\vert x\right\vert \geq R.$ Then the total scattering cross-section is infinite if and only if $\rho\leq2,$ where if both $V$ and $B$ are non-trivial for $\left\vert x\right\vert \geq
R,$ then $\rho=\min\{\rho_{e},\rho_{m}\},$ if $V$ is trivial, $\rho=\rho_{m}$ and if $B$ is trivial, $\rho=\rho_{e}.$
Note that Lemma \[representation234\] shows that the total scattering cross-section is finite if $\rho>2.$ Let the magnetic potential $A$ be defined by the equalities (\[basicnotions19\])-(\[basicnotions21\]). Since $B$ is homogeneous of order $-\rho_{m}-1$, $A$ is homogeneous of order $-\rho_{m}$. Thus we get $\mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( x\right)
=\left\vert x\right\vert ^{-\rho}\left( V_{\operatorname*{ang}}\left(
\hat{x}\right) +\left( \operatorname*{sgn}E\right) \left\langle
\omega,A_{\operatorname*{ang}}\left( \hat{x}\right) \right\rangle \right)
+W\left( x\right) $ for $\left\vert x\right\vert \geq R$, for some $V_{\operatorname*{ang}}\in C^{\infty}\left( \mathbb{S}^{2}\right) $, $A_{\operatorname*{ang}}\left( \hat{x}\right) \in C^{\infty}\left(
\mathbb{S}^{2};\mathbb{R}^{3}\right) $ and some $W\left( x\right) $ homogeneous of order $\rho_{1}=\max\{\rho_{e},\rho_{m}\}.$ Note that if $\rho_{e}=\rho_{m},$ $W\left( x\right) \equiv0.$ Then we have $$\left.
\begin{array}
[c]{c}\left\vert \left( \mathcal{FV}_{V,A;\omega}^{\left( E\right) }\right)
\left( -\nu\left( E\right) \tilde{\theta}\right) P_{\omega}\left(
E\right) u\right\vert =\frac{1}{\left( 2\pi\right) ^{3/2}}\left\vert
{\displaystyle\int}
\left( e^{i\nu\left( E\right) \left\langle \tilde{\theta},x\right\rangle
}\mathbf{V}_{h}^{\omega}\left( x\right) u\right) dx-{\displaystyle\int\limits_{\left\vert x\right\vert \leq R}}
\left( e^{i\nu\left( E\right) \left\langle \tilde{\theta},x\right\rangle
}\mathbf{V}_{h}^{\omega}\left( x\right) u\right) dx\right. \\
\left. +{\displaystyle\int\limits_{\left\vert x\right\vert \geq R}}
\left( e^{i\nu\left( E\right) \left\langle \tilde{\theta},x\right\rangle
}W\left( x\right) P_{\omega}\left( E\right) u\right) dx+{\displaystyle\int\limits_{\left\vert x\right\vert \leq R}}
\left( e^{i\nu\left( E\right) \left\langle \tilde{\theta},x\right\rangle
}\mathcal{V}_{V,A;\omega}^{\left( E\right) }\left( x\right) P_{\omega
}\left( E\right) u\right) dx\right\vert ,
\end{array}
\right. \label{representation232}$$ where $\mathbf{V}_{h}^{\omega}=\left( V_{h}^{\left( 1\right) }+\left(
\operatorname*{sgn}E\right) \left\langle \omega,V_{h}^{\left( 2\right)
}\right\rangle \right) P_{\omega}\left( E\right) $ and $V_{h}^{\left(
1\right) }\left( x\right) =\left\vert x\right\vert ^{-\rho}V_{\operatorname*{ang}}\left( \hat{x}\right) ,$ $V_{h}^{\left( 2\right)
}\left( x\right) =\left\vert x\right\vert ^{-\rho}A_{\operatorname*{ang}}\left( \hat{x}\right) $ for $x\in\mathbb{R}^{3}.$
Note that if a function $f\left( x\right) :=\left\vert x\right\vert ^{-\rho
}f_{\operatorname*{ang}}\left( \hat{x}\right) ,$ where $f_{\operatorname*{ang}}\left( \hat{x}\right) $ belongs to $C^{\infty
}\left( \mathbb{S}^{2}\right) $ and it is non-trivial, then its Fourier transform is given by $\hat{f}\left( \xi\right) =\left\vert \xi\right\vert
^{-3+\rho}\hat{f}\left( \hat{\xi}\right) $ and $\hat{f}\left( \hat{\xi
}\right) $ is also a non-trivial, $C^{\infty}\left( \mathbb{S}^{2}\right) $ function. This means that $\hat{V}_{h}^{\left( j\right) }\left( \xi\right)
\in C^{\infty}\left( \mathbb{R}^{3}\backslash\{0\}\right) $ and $\hat{V}_{h}^{\left( j\right) }\left( \xi\right) =\left\vert \xi\right\vert
^{-3+\rho}\hat{V}_{h}^{\left( j\right) }\left( \hat{\xi}\right) $ for $\xi\neq0,$ $j=1,2$, and hence, $\mathbf{\hat{V}}_{h}^{\omega}\left(
\xi\right) =\left\vert \xi\right\vert ^{-3+\rho}\mathbf{\hat{V}}_{h}^{\omega
}\left( \hat{\xi}\right) $.
Suppose that $\rho=\rho_{e}$, what implies that $\hat{V}_{h}^{\left(
1\right) }\left( \hat{\xi}\right) $ is non-trivial. We take $\hat{\xi}$ such that $\hat{V}_{h}^{\left( 1\right) }\left( \hat{\xi}\right) \neq0.$ Let $\omega_{1}$ be orthogonal to $\hat{\xi}$ and suppose that $\mathbf{\hat
{V}}_{h}^{\omega_{1}}\left( \hat{\xi}\right) \neq0.$ By continuity we get $\left\vert \mathbf{\hat{V}}_{h}^{\omega}\left( \hat{\zeta}\right)
\right\vert >c,$ for some constant $c>0,$ $\left\vert \omega-\omega
_{1}\right\vert <\delta_{1},$ for some $\delta_{1}>0,$ as in Proposition \[representation231\], and $\hat{\zeta}$ such that $\left\langle \hat{\zeta
},\hat{\xi}\right\rangle >1-\varepsilon$ for some $\varepsilon>0.$ Hence, $\left\vert \mathbf{\hat{V}}_{h}^{\omega}\left( \zeta\right) \right\vert
>c\left\vert \zeta\right\vert ^{-3+\rho},$ for $\left\vert \omega-\omega
_{1}\right\vert <\delta_{1},$ $\delta_{1}>0,$ and $\zeta$ such that $\left\langle \zeta,\xi\right\rangle >\left( 1-\varepsilon\right) \left\vert
\zeta\right\vert \left\vert \xi\right\vert \mathbf{.}$ Note that there are $W_{1}$ and $W_{2}$ such that $\mathcal{V}_{V,A;\omega}^{\left( E\right)
}\left( x\right) F\left( \left\vert x\right\vert \leq R\right) +W\left(
x\right) F\left( \left\vert x\right\vert \geq R\right) =W_{1}\left(
x\right) +W_{2}\left( x\right) ,$ where $W_{1}$ satisfies (\[eig32\]) with $\rho_{1},$ and $W_{2}$ is a $C^{\infty}$ function for $\left\vert
x\right\vert \leq R,$ and $W_{2}\equiv0,$ for $\left\vert x\right\vert >R$ ($F\left( \cdot\right) $ is the characteristic function of the correspondent set). Observe that, if $\rho_{e}=\rho_{m}$, then we have $W_{1}\left(
x\right) \equiv0$ and $W_{2}\left( x\right) =\mathcal{V}_{V,A;\omega
}^{\left( E\right) }\left( x\right) $ for $\left\vert x\right\vert \leq
R.$ It follows from Lemma \[representation201\] that $\left\vert
{\displaystyle\int}
\left( e^{i\nu\left( E\right) \left\langle \tilde{\theta},x\right\rangle
}W_{1}\left( x\right) u\right) dx\right\vert \leq C\left( \rho_{1}-\rho\right) \left\vert \omega-\theta\right\vert ^{-3+\rho_{2}},$ where $\rho_{2}=\rho_{1},$ for $\ \rho_{1}<3$, and $\rho_{2}=3-\varepsilon,$ $\varepsilon>0$ for $\rho_{1}\geq3.$ We note that $\left\vert -{\displaystyle\int\limits_{\left\vert x\right\vert \leq R}}
\left( e^{-i\left\langle y,x\right\rangle }\mathbf{V}_{h}^{\omega}\left(
x\right) u\right) dx+{\displaystyle\int\limits_{\left\vert x\right\vert \leq R}}
\left( e^{-i\left\langle y,x\right\rangle }W_{2}\left( x\right) P_{\omega
}\left( E\right) u\right) dx\right\vert \leq C,$ uniformly in $y$ and $\omega.$ Moreover, if $\left\vert \omega-\omega_{1}\right\vert <\delta_{1}$ and $-\left\langle \tilde{\theta}/\left\vert \tilde{\theta}\right\vert
,\hat{\xi}\right\rangle >1-\varepsilon$ we have $\frac{1}{\left( \nu\left(
E\right) \left\vert \tilde{\theta}\right\vert \right) ^{3-\rho}}\left\vert
\left\vert \mathbf{\hat{V}}_{h}^{\omega_{1}}\left( -\tilde{\theta}/\left\vert
\tilde{\theta}\right\vert \right) \right\vert -C(\left\vert \rho_{e}-\rho
_{m}\right\vert \right. $ $\left. \times\left( \nu\left( E\right)
\left\vert \tilde{\theta}\right\vert \right) ^{-3+\rho_{2}}+1)\left(
\nu\left( E\right) \left\vert \tilde{\theta}\right\vert \right) ^{3-\rho
}\right\vert $ $\geq\frac{c}{2\left( \nu\left( E\right) \left\vert
\tilde{\theta}\right\vert \right) ^{3-\rho}},$ for $\left\vert \tilde{\theta
}\right\vert <\varepsilon_{1}$ and some $\varepsilon_{1}>0$. Then, from relations (\[representation232\]) and (\[representation102\]) we get$$\left.
\begin{array}
[c]{c}\left\vert s^{\operatorname{int}}\left( \omega,\theta;E\right) \right\vert
\geq\left\vert \left( 2\pi\right) ^{-1/2}\upsilon\left( E\right) ^{2}\frac{\nu\left( E\right) }{\left\vert E\right\vert }\left( \mathcal{FV}_{V,A;\omega}^{\left( E\right) }\right) \left( -\nu\left( E\right)
\tilde{\theta}\right) P_{\omega}\left( E\right) \right\vert \\
-\left\vert \left( s^{\operatorname{int}}\left( \omega,\theta;E\right)
-\frac{1}{i}\left( 2\pi\right) ^{-1/2}\upsilon\left( E\right) ^{2}\frac{\nu\left( E\right) }{\left\vert E\right\vert }\left( \mathcal{FV}_{V,A;\omega}^{\left( E\right) }\right) \left( -\nu\left( E\right)
\tilde{\theta}\right) P_{\omega}\left( E\right) \right) \right\vert
\geq\frac{c}{\left( \nu\left( E\right) \left\vert \tilde{\theta}\right\vert
\right) ^{3-\rho}},
\end{array}
\right. \label{representation235}$$ for $\left\vert \tilde{\theta}\right\vert <\varepsilon_{2}\leq\varepsilon_{1}$ and some $\varepsilon_{2}>0$. Let $\delta_{1}$ be such that the set $\Theta_{\omega}:=\{\theta\in\Omega_{+}\left( \omega,\delta\right)
|-\left\langle \tilde{\theta}/\left\vert \tilde{\theta}\right\vert ,\hat{\xi
}\right\rangle >1-\varepsilon$ and $\left\vert \tilde{\theta}\right\vert
<\varepsilon_{2}\}$ is of positive measure. Then, using (\[representation235\]) we obtain $\int\limits_{\mathbb{S}^{2}}\int\limits_{\mathbb{S}^{2}}\left\vert s^{\operatorname{int}}\left(
\omega,\theta;E\right) \right\vert ^{2}d\theta d\omega\geq\frac{c}{\nu\left(
E\right) ^{6-2\rho}}\int\limits_{\left\vert \omega-\omega_{1}\right\vert
\leq\delta_{1}}\int\limits_{\Theta_{\omega}}\frac{1}{\left\vert \tilde{\theta
}\right\vert ^{6-2\rho}}d\theta d\omega.$ As $\int_{\Theta_{\omega}}\frac
{1}{\left\vert \omega-\theta\right\vert ^{6-2\rho}}d\theta$ is infinite, for $\rho\leq2,$ then using relation (\[representation99\]) we conclude that $\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}\left\vert s^{\operatorname{int}}\left( \omega,\theta;E\right) u\right\vert ^{2}d\theta d\omega=\infty$.
Suppose again that $\rho=\rho_{e}$ and $\hat{\xi}$ is such that $\hat{V}_{h}^{\left( 1\right) }\left( \hat{\xi}\right) \neq0,$ but $\mathbf{\hat
{V}}_{h}^{\omega_{1}}\left( \hat{\xi}\right) =0.$ Noting that $\mathbf{\hat
{V}}_{h}^{-\omega_{1}}\left( \hat{\xi}\right) \neq0$ and proceeding similarly as above we get $\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}\left\vert s^{\operatorname{int}}\left( \omega,\theta;E\right) u\right\vert
^{2}d\theta d\omega=\infty$ if $\rho=\rho_{e}$.
Now suppose that $\rho=\rho_{m}$. Let us take $\hat{\xi}_{1}$ such that $\hat{V}_{h}^{\left( 2\right) }\left( \hat{\xi}_{1}\right) \neq0.$ By continuity $\left\vert \hat{V}_{h}^{\left( 2\right) }\left( \hat{\xi
}\right) \right\vert >c>0$ for all $\hat{\xi}$ close enough to $\hat{\xi}_{1}$. Consider the set $\Psi\subset\mathbb{S}^{2}$ of all $\hat{\xi}$ such that $\hat{V}_{h}^{\left( 2\right) }\left( \hat{\xi}\right) \neq0.$ We claim that there is $\omega_{1}$ orthogonal to some $\hat{\xi}\in\Psi$ such that $\left\langle \omega_{1},\hat{V}_{h}^{\left( 2\right) }\left( \hat
{\xi}\right) \right\rangle \neq0.$ Suppose that this is not true. That is, for every $\hat{\xi}\in\Psi$, $\left\langle \omega,\hat{V}_{h}^{\left(
2\right) }\left( \hat{\xi}\right) \right\rangle =0,$ for all $\omega$ orthogonal to $\hat{\xi}.$ This implies that $\left\langle \hat{\xi},\hat
{V}_{h}^{\left( 2\right) }\left( \hat{\xi}\right) \right\rangle
=\pm\left\vert \hat{V}_{h}^{\left( 2\right) }\left( \hat{\xi}\right)
\right\vert .$ Hence, taking in account that $\hat{V}_{h}^{\left( 2\right)
}\left( \hat{\xi}\right) =0,$ for $\hat{\xi}\in\mathbb{S}^{2}\diagdown\Psi,$ we have $\xi\times\hat{V}_{h}^{\left( 2\right) }\left( \xi\right) =0,$ for all $\xi,$ what implies that $\operatorname{curl}V_{h}^{\left( 2\right)
}=B=0,$ for $\left\vert x\right\vert \geq R.$ This is a contradiction. Then there is $\omega_{1}$ orthogonal to some $\hat{\xi}$ such that $\left\langle
\omega_{1},\hat{V}_{h}^{\left( 2\right) }\left( \hat{\xi}\right)
\right\rangle \neq0.$ Similarly to the case when $\rho=\rho_{e}$ we obtain $\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}\left\vert s^{\operatorname{int}}\left( \omega,\theta;E\right) u\right\vert ^{2}d\theta d\omega=\infty$.
Reconstruction of the electric potential and the magnetic field from the high energy limit.
-------------------------------------------------------------------------------------------
Now we consider a special limit when $\left\vert E\right\vert \rightarrow
\infty$ and $\omega\left( E\right) ,\theta\left( E\right) \rightarrow
\omega,$ for an arbitrary $\omega\in\mathbb{S}^{2},$ in such way that $\eta:=\nu\left( E\right) \left( \omega\left( E\right) -\theta\left(
E\right) \right) \neq0$ is fixed (see [@75]). Let us take two families of vectors $\omega\left( E\right) ,\theta\left( E\right) \in\mathbb{S}^{2}$ with these properties. We obtain the following result
Let the magnetic potential $A\left( x\right) $ and the electric potential $V\left( x\right) $ satisfy the estimates (\[eig31\]) and (\[eig32\]) respectively. For $\eta\in\mathbb{R}^{3}\backslash\{0\},$ let $\omega\left(
E\right) ,\theta\left( E\right) \in\mathbb{S}^{2}$ be as above. Then, we have$$\left. \lim_{\substack{\pm E\rightarrow\infty}}\upsilon\left( E\right)
^{-2}s\left( \omega\left( E\right) ,\theta\left( E\right) ;E\right)
=\left( 2\pi\right) ^{-1}\mathcal{F}\left( e^{-iR\left( y,\omega;\pm
\infty\right) }P_{\omega}^{\pm}\left( \infty\right) \right) \left(
\eta\right) \right. , \label{representation114}$$ where $R\left( y,\omega;\pm\infty\right) :=\int\limits_{-\infty}^{\infty
}\left( V\left( y+t\omega\right) \pm\left\langle \omega,A\left(
y+t\omega\right) \right\rangle \right) dt$ and $P_{\omega}\left( \pm
\infty\right) =\frac{1}{2}\left( I\pm\left( a\cdot\omega\right) \right)
.$
We follow the proof of Proposition 6.7 of [@30] for the Schrödinger equation. For a fixed $\omega\in\mathbb{S}^{2},$ we take a cut-off function $\Psi_{+}\left( \omega,\theta;\omega\right) ,$ supported, as function of $\theta,$ on $\Omega_{+}\left( \omega,\delta\right) ,$ such that it is equal to $1$ in $\Omega_{+}\left( \omega,\delta^{\prime}\right) $, for some $\delta^{\prime}>\delta.$ Let the first coordinate axis in $\Pi_{\omega}$ be directed along $\eta.$ Then, integrating by parts in the R.H.S of relation (\[representation248\]) (understood as an oscillatory integral), with respect to $y_{1},$ we get$$\left. s_{\operatorname{sing}}^{(N)}\left( \omega\left( E\right)
,\theta\left( E\right) ;E;\omega\right) =\left( 2\pi\right) ^{-2}\upsilon\left( E\right) ^{2}\left( i\left\vert \eta\right\vert \right)
^{-n}\Psi_{+}\left( \omega\left( E\right) ,\theta\left( E\right)
;\omega\right)
{\displaystyle\int\limits_{\Pi_{\omega}}}
e^{-i\left\langle y,\eta\right\rangle }\partial_{y_{1}}^{n}\mathbf{h}_{N}\left( y,\omega\left( E\right) ,\theta\left( E\right) ;E;\omega
\right) dy.\right. \label{representation194}$$ For $n\geq2$ the integral in the last relation is absolutely convergent, as $$\left\vert \partial_{y_{1}}^{n}\mathbf{h}_{N}\left( y,\omega\left( E\right)
,\theta\left( E\right) ;E;\omega\right) \right\vert \leq C_{n}\left\langle
y\right\rangle ^{-\left( \rho-1\right) -n}. \label{representation214}$$ We have $\left\vert \partial_{y}^{\alpha}\partial_{\zeta}^{\beta}\partial_{\zeta^{\prime}}^{\gamma}\left( \tilde{\Psi}_{+}\left( \zeta\left(
E\right) ,\zeta^{\prime}\left( E\right) ;\omega\right) \left(
\mathbf{\tilde{h}}_{N}-\mathbf{\tilde{h}}_{0}\right) \right) \left(
y,\zeta,\zeta^{\prime};E\right) \right\vert \leq$ $C_{\alpha,\beta,\,\gamma
}\nu\left( E\right) ^{-1}\left\langle y\right\rangle ^{-\rho-\left\vert
\alpha\right\vert },$ for some constants $C_{\alpha,\beta,\,\gamma},$ independent of $\zeta$ and $\zeta^{\prime}.$ Then, Theorem \[representation179\] implies that $$\lim_{\substack{\left\vert E\right\vert \rightarrow\infty}}\upsilon\left(
E\right) ^{-2}s\left( \omega\left( E\right) ,\theta\left( E\right)
;E\right) =\lim\limits_{_{\substack{\left\vert E\right\vert \rightarrow
\infty}}}\upsilon\left( E\right) ^{-2}s_{\operatorname{sing}}^{(0)}\left(
\omega\left( E\right) ,\theta\left( E\right) ;E;\omega\right) .
\label{representation115}$$ Using equality (\[representation251\]) we see that$$\lim_{\pm E\rightarrow\infty}\left( \Psi_{+}\left( \omega\left( E\right)
,\theta\left( E\right) ;\omega\right) \partial_{y_{1}}^{n}\mathbf{h}_{0}\left( y,\omega\left( E\right) ,\theta\left( E\right) ;E;\omega
\right) \right) =\left( \partial_{y_{1}}^{n}e^{-iR\left( y,\omega
;\pm\infty\right) }\right) P_{\omega}\left( \pm\infty\right) .
\label{representation116}$$ Estimate (\[representation214\]) allows us to take the limit in (\[representation194\]), as $\pm E\rightarrow\infty$. Therefore, equalities (\[representation115\]) and (\[representation116\]) imply$$\left. \lim\limits_{\pm E\rightarrow\infty}\upsilon\left( E\right)
^{-2}s^{\operatorname{int}}\left( \omega\left( E\right) ,\theta\left(
E\right) ;E;\omega\right) =\left( 2\pi\right) ^{-2}\left( i\left\vert
\eta\right\vert \right) ^{-n}{\displaystyle\int_{\Pi_{\omega}}}
e^{-i\left\langle y,\eta\right\rangle }\left( \partial_{y_{1}}^{n}e^{-iR\left( y,\omega;\pm\infty\right) }\right) P_{\omega}\left( \pm
\infty\right) dy,\right. \label{representation195}$$ Integrating back by parts in the R.H.S. of \[representation195\], we obtain (\[representation114\]).
Let us prove that we can uniquely reconstruct the electric potential $V$ and the magnetic field $B$ from the limit (\[representation195\]). The integral in the R.H.S. of (\[representation195\]) is, up to a coefficient, the two-dimensional Fourier transform of $\left( \partial_{y_{1}}^{n}e^{-iR\left( y,\omega;\pm\infty\right) }\right) P_{\omega}^{\pm}\left(
\infty\right) .$ The matrices $P_{\omega}^{\pm}\left( \infty\right) $ are of the form $\frac{1}{2}\begin{pmatrix}
I & \pm\sum_{j=1}^{3}\sigma_{j}\omega_{j}\\
\pm\sum_{j=1}^{3}\sigma_{j}\omega_{j} & I
\end{pmatrix}
.$ Taking, for example, the first component of the matrix $\left(
\partial_{y_{1}}^{n}e^{-iR\left( y,\omega;\pm\infty\right) }\right)
P_{\omega}^{\pm}\left( \infty\right) $ we recover the function $\left(
\partial_{y_{1}}^{n}e^{-iR\left( y,\omega;\pm\infty\right) }\right) $. Let us take $y:=\left( y_{1},y_{2},y_{3}\right) \in\Pi_{\omega}.$ Since $\partial_{y_{1}}^{n-1}e^{-iR\left( y,\omega;\pm\infty\right) }$ tends to $0,$ as $\left\vert y\right\vert \rightarrow\infty,$ then $\left(
\partial_{y_{1}}^{n-1}e^{-iR\left( y,\omega;\pm\infty\right) }\right)
=-\left( \int_{0}^{\infty}\partial_{y_{1}}^{n}e^{-iR\left( y\left(
t\right) ,\omega;\pm\infty\right) }dt\right) ,$ where $y\left( t\right)
=\left( y_{1}+t,y_{2},y_{3}\right) \in\Pi_{\omega}.$ Applying this argument $n-1$ times we get the function $\partial_{y_{1}}e^{-iR\left( y,\omega
;\pm\infty\right) }.$ Since $R\left( y,\omega;\pm\infty\right) $ tends to $0,$ as $\left\vert y\right\vert \rightarrow\infty,$ we have $e^{-iR\left(
y,\omega;\pm\infty\right) }-1=-\left( \int_{0}^{\infty}\partial_{y_{1}}e^{-iR\left( y\left( t\right) ,\omega;\pm\infty\right) }dt\right) .$ Thus, we recover the function $e^{-iR\left( y,\omega;\pm\infty\right) }\ $from the limit (\[representation195\]). Since $R\left( y,\omega
;\pm\infty\right) $ is a continuous function of $y\in\Pi_{\omega}$ and tends to $0,$ as $\left\vert y\right\vert \rightarrow\infty,$ we can determine the function $R\left( y,\omega;\pm\infty\right) $ from the function $e^{-iR\left( y,\omega;\pm\infty\right) }.$ Similarly to [@79] or [@25], noting that $R_{e}\left( \omega,y;V\right) :=\int\limits_{-\infty
}^{\infty}V\left( y+t\omega\right) dt$ is even in $\omega$ and $R_{m}\left(
\omega,y;A\right) :=\int\limits_{-\infty}^{\infty}\left\langle \omega
,A\left( y+t\omega\right) \right\rangle dt$ is odd, we get $R_{e}$ and $R_{m}$ from the limit (\[representation195\]), and moreover, inverting the Radon transform we recover the electric potential $V$ and the magnetic field $B,$ from the scattering amplitude. We can formulate the obtained results as the following
\[representation218\]Suppose that the electric potential $V\left(
x\right) $ and the magnetic field $B\left( x\right) $ satisfy the estimates (\[eig32\]) and (\[basicnotions18\]), for all $\alpha$ and $d,$ respectively. Then the scattering amplitude $s\left( \omega,\theta;E\right)
,$ known in some neighborhood of the diagonal $\omega=\theta,$ for every $E\geq E_{0}$ or $-E\geq E_{0},$ for some $E_{0}>m,$ uniquely determines the electric potential $V\left( x\right) $ and the magnetic field $B\left(
x\right) .$ Moreover, one can reconstruct $V\left( x\right) $ and $B\left(
x\right) $ from the high-energy limit (\[representation195\]).
Using the high-energy asymptotics of the resolvent, Ito [@15] gave the relation (\[representation114\]) and he proved Theorem \[representation218\] for smooth electromagnetic potentials with $\rho
>3.$ Jung [@52], calculating the high-velocity limit for the scattering operator, by the time-dependent method of [@62], for continuous, Hermitian matrix valued potentials $\mathbf{V}\left( x\right) ,$ satisfying the conditions (\[intro7\]) and $\left\Vert VF\left( \left\vert x\right\vert
\geq R\right) \right\Vert \in L^{1}\left( \left[ 0;\infty\right)
;dR\right) ,$ where $F\left( \left\vert x\right\vert \geq R\right) $ is the characteristic function of the set $\left\vert x\right\vert \geq R,$ presents a reconstruction formula, that allows to uniquely recover the electric potential and magnetic field from the scattering operator. Also using the time-dependent method, Ito [@57] showed that for potentials $\mathbf{V}$ of the form (\[intro4\]), satisfying $\left\vert \partial_{x}^{\alpha
}V\left( x\right) \right\vert +\left\vert \partial_{x}^{\alpha}A\left(
x\right) \right\vert \leq C_{\alpha}\left\langle x\right\rangle ^{-\rho},$ $\rho>1,$ $\left\vert \alpha\right\vert \leq1,$ for all $x\in\mathbb{R}^{3},$ the electric potential $V\left( x\right) $ and the magnetic field $B\left(
x\right) $ can be completely reconstructed from the scattering operator. Ito also considered time-dependent potentials in [@57].
Inverse problem at fixed energy for homogeneous potentials.
-----------------------------------------------------------
We follow the approach of [@25]. For a fixed $\omega\in\mathbb{S}^{2},$ we take a cut-off function $\Psi_{+}\left( \omega,\theta;\omega\right) ,$ supported, as function of $\theta,$ on $\Omega_{+}\left( \omega
,\delta\right) ,$ such that it is equal to $1$ in $\Omega_{+}\left(
\omega,\delta^{\prime}\right) $, for some $\delta^{\prime}>\delta.$ It is convenient for us to reformulate Theorem \[representation179\] in terms of asymptotic series. Let us rewrite formula (\[representation252\]) in terms of powers of the potential $W\left( x\right) :=$ $\left( V\left( x\right)
,A\left( x\right) \right) .$ Note first that for $\left\vert E\right\vert
>m,$ $e^{i\Phi^{\pm}\left( x,\xi;E\right) }=\sum_{j=0}^{\infty}\frac{1}{j!}\left( \pm i{\displaystyle\int\limits_{0}^{\infty}}
\left( \frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left(
x\pm\left( \operatorname*{sgn}E\right) t\omega\right) +\left(
\operatorname*{sgn}E\right) \left\langle \omega,A\left( x\pm\left(
\operatorname*{sgn}E\right) t\omega\right) \right\rangle \right) dt\right)
^{j}.$ Introducing this expression in (\[eig33\]) we can write $a_{N}^{\pm
}\left( x,\xi;E\right) $ as an asymptotic expansion $a_{N}^{\pm}\left(
x,\xi;E\right) \simeq\sum_{m=0}^{\infty}a_{N,m}^{\pm}\left( x,\xi\right) ,$ where $a_{N,m}^{\pm}\left( x,\xi\right) $ is of order $m$ with respect to $W\left( x\right) $. Plugging this expansion in the representation (\[representation252\]) for $\mathbf{h}_{N}\left( y,\theta,\omega
;E\,;\omega\right) $ and collecting together the terms of the same power with respect to $W\left( x\right) $ we see that $\mathbf{h}_{N}\left(
y,\theta,\omega;E\,;\omega\right) $ admits the expansion into the asymptotic series $$\mathbf{h}_{N}\left( y,\theta,\omega;E\,;\omega\right) \simeq\sum
\limits_{n=0}^{\infty}\mathbf{a}_{n}\left( y,\omega,\theta;E\right) ,
\label{direct17}$$ where $\mathbf{a}_{n}\left( y,\omega,\theta;E\right) \ $is of order $n$ with respect to $W\left( x\right) .$ Note that $\mathbf{a}_{0}\left(
y,\omega,\theta;E\right) =\left( \operatorname*{sgn}E\right) P_{\omega
}\left( E\right) \left( \alpha\cdot\omega\right) P_{\theta}\left(
E\right) $ and that the lineal term with respect to $W\left( x\right) $ is given by the relation $$\left.
\begin{array}
[c]{c}\mathbf{a}_{1}\left( y,\omega,\theta;E\right) =\\
-i\left( \operatorname*{sgn}E\right) \left(
{\displaystyle\int\limits_{0}^{\infty}}
\left( \frac{\left\vert E\right\vert }{\left\vert \xi\right\vert }V\left(
y\pm t\omega\right) \pm\left\langle \omega,A\left( y\pm t\omega\right)
\right\rangle +\frac{\left\vert E\right\vert }{\left\vert \xi\right\vert
}V\left( y\mp t\theta\right) \pm\left\langle \theta,A\left( y\mp
t\theta\right) \right\rangle \right) dt\right) P_{\omega}\left( E\right)
\left( \alpha\cdot\omega\right) P_{\theta}\left( E\right) ,
\end{array}
\right. \label{direct1}$$ for $\pm E>m,$ up to a term from the class $\mathit{S}^{-\rho}$ (on variable $y$). Moreover, it follows that $\mathbf{a}_{n}\in\mathit{S}^{-\left(
\rho-1\right) n}.$
We will recover the asymptotics of the potential $W\left( x\right) $ from the linear part, with respect to $W\left( x\right) $, of the symbol of the operator $S_{\operatorname{sing}}\left( E\right) ,$ with kernel $s_{\operatorname{sing}}^{(N)}.$ We compute the symbol $a\left(
y,\omega;E\right) $ of $S_{\operatorname{sing}}\left( E\right) $ from its amplitude $\upsilon\left( E\right) ^{2}\Psi_{+}\left( \omega,\theta
;\omega\right) $ $\times\mathbf{h}_{N}\left( y,\omega,\theta;E;\omega
\right) ,$ by making the change of variables $z=-\nu\left( E\right) y$ in the definition (\[representation248\]) of $\mathbf{h}_{N}$ and applying (\[basicnotions45\]). Recall the notation of Remark \[representation75\]. We get, $$\tilde{a}\left( y,\zeta;E\right) \simeq\sum_{\beta}\frac{\upsilon\left(
E\right) ^{2}}{\beta!}\left( -i\nu\left( E\right) \right) ^{-\left\vert
\beta\right\vert }\left. \partial_{y}^{\beta}\partial_{\zeta^{\prime}}^{\beta}\mathbf{\tilde{h}}_{N}\left( y,\zeta,\zeta^{\prime};E;\omega\right)
\right\vert _{\zeta^{\prime}=\zeta}, \label{direct16}$$ for $y\in\Pi_{\omega}$ (here we used that $\Psi_{+}\left( \omega
,\theta;\omega\right) =1,$ for $\theta\in\Omega_{+}\left( \omega
,\delta^{\prime}\right) $)$.$ The explicit formula for the symbol $a\left(
y,\omega;E\right) $ can be obtained by plugging the expansion (\[direct17\]) in the relation (\[direct16\]). Note that $s_{\operatorname{sing}}^{(N)}$ is related with $a\left( y,\omega;E\right) $ by the expression $s_{\operatorname{sing}}^{(N)}\left( \omega,\theta;E\,;\omega\right)
=\left( 2\pi\right) ^{-2}\int\limits_{\Pi_{\omega}}e^{i\nu\left( E\right)
\left\langle y,\theta\right\rangle }a\left( y,\omega;E\right) dy.$ This relation together with Theorem \[representation179\] show that we can recover the function $a\left( y,\omega;E\right) $ from the scattering amplitude $s\left( \omega,\theta;E\,\right) $, up to a function from the class $\mathit{S}^{-p\left( N\right) },$ that is, $a\left( y,\omega
;E\right) =\int\limits_{\Pi_{\omega}}e^{i\nu\left( E\right) \left\langle
y,\theta\right\rangle }s\left( \omega,\theta;E\,\right) \Psi_{+}\left(
\omega,\theta;\omega\right) d\theta+a_{\operatorname*{reg}}^{\left(
N\right) }\left( y,\omega;E\right) ,$ where $a_{\operatorname*{reg}}^{\left( N\right) }\in\mathit{S}^{-p\left( N\right) }\ $and $p\left(
N\right) \rightarrow\infty,$ as $N\rightarrow\infty.$
Using the relations (\[representation113\]) and (\[direct1\]), we get the following result for the function $a\left( y,\omega;E\right) $:
\[direct19\]Let the magnetic potential $A\left( x\right) $ and the electric potential $V\left( x\right) $ satisfy the estimates (\[eig31\]) and (\[eig32\]) respectively. Then, the function $a\left( y,\omega
;E\right) $ admits the expansion $$a\left( y,\omega;E\right) \simeq\frac{\nu\left( E\right) }{\left\vert
E\right\vert }P_{\omega}\left( E\right) +\sum_{n=1}^{\infty}h_{n}\left(
y,\omega;E\right) , \label{direct9}$$ where $h_{n}\left( y,\omega;E\right) $ is of order $n$ with respect to $W\left( x\right) $ and $h_{n}\in\mathit{S}^{-\left( \rho-1\right) n}.$ Moreover, $(h_{1}+i\frac{\nu\left( E\right) }{\left\vert E\right\vert
}R\left( y,\omega;E\right) P_{\omega}\left( E\right) )\in\mathit{S}^{-\rho},$ where $R$ is defined by (\[representation219\]), and $(a-\frac{\nu\left( E\right) }{\left\vert E\right\vert }P_{\omega}\left(
E\right) +i\frac{\nu\left( E\right) }{\left\vert E\right\vert }R\left(
y,\omega;E\right) P_{\omega}\left( E\right) )\in\mathit{S}^{-\rho
+1-\varepsilon},$ $\varepsilon=\min\{\rho-1,1\}.$
Let the electric potential $V\left( x\right) \in C^{\infty}\left(
\mathbb{R}^{3}\right) $ and the magnetic field $B\left( x\right) \in
C^{\infty}\left( \mathbb{R}^{3}\right) ,$ with $\operatorname{div}B=0,$ admit the asymptotic expansions$$V\left( x\right) \simeq\sum_{j=1}^{\infty}V_{j}\left( x\right) ,
\label{direct2}$$ and $$B\left( x\right) \simeq\sum_{j=1}^{\infty}B_{j}\left( x\right) ,
\label{direct3}$$ respectively, where the functions $V_{j}\left( x\right) $ are homogeneous of order $-\rho_{j}^{\left( e\right) },$ with $1<\rho_{j}^{\left( e\right)
}<\rho_{k}^{\left( e\right) },$ and the functions $B_{j}\left( x\right) $ are homogeneous of order $-r_{j}^{\left( m\right) },$ with $2<r_{j}^{\left(
m\right) }<r_{k}^{\left( m\right) }$ for $k>j.$
It follow from relations (\[basicnotions19\])-(\[basicnotions21\]) that the magnetic field $B(x)$ is homogeneous of order $-r^{\left( m\right) }$ $<-2$ if and only if the magnetic potential $A(x)$ is homogeneous of order $-\rho^{\left( m\right) }=-r^{\left( m\right) }+1<-1$. Therefore, if $B$ satisfies relation (\[direct3\]), $A(x),$ defined by (\[basicnotions19\])-(\[basicnotions21\]), is an asymptotic sum $$A\left( x\right) \simeq\sum_{j=1}^{\infty}A_{j}\left( x\right) ,
\label{direct20}$$ where the functions $A_{j}\left( x\right) $ are homogeneous of order $-\rho_{j}^{\left( m\right) }$ with $1<\rho_{j}^{\left( m\right) }<\rho_{k}^{\left( m\right) }$ for $k>j.$
Let $V\left( x\right) $ and $B\left( x\right) $ be as above. We define the magnetic potential $A\left( x\right) $ by (\[basicnotions19\])-(\[basicnotions21\]). Thus, we obtain a potential $\mathbf{V}\left(
x\right) $ of the form (\[intro4\]), where $A$ and $V$ satisfy the estimates (\[eig31\]) and (\[eig32\]), respectively. Moreover, $V$ admits the expansion (\[direct2\]) and $A$ satisfies the relation (\[direct20\]).
Actually, adding terms which are equal to zero in (\[direct2\]) and (\[direct3\]) we can suppose that $r_{j}^{\left( m\right) }=\rho
_{j}^{\left( e\right) }+1.$ Then, expansions (\[direct2\]) and (\[direct20\]) are equivalent to the expansion $$W\left( x\right) \simeq\sum_{j=1}^{\infty}W_{j}\left( x\right)
\label{direct4}$$ where $W_{j}\left( x\right) =\left( V_{j}\left( x\right) ,A_{j}\left(
x\right) \right) $ is homogeneous of order $-\rho_{j}=-\rho_{j}^{\left(
m\right) }=-\rho_{j}^{\left( e\right) }.$
Plugging (\[direct4\]) in the expansion (\[direct9\]) we get the following result for the function $a\left( y,\omega;E\right) $, analogous to Theorem 3.4 of [@25] for the Schrödinger equation:
\[direct7\]Suppose that an electric potential $V\left( x\right) $ and a magnetic field $B\left( x\right) ,$ with $\operatorname{div}B=0,$ are $C^{\infty}\left( \mathbb{R}^{3}\right) $-functions and that they admit the asymptotic expansions (\[direct2\]) and (\[direct3\]), where $V_{j}$ and $B_{j}$ are homogeneous functions of orders $-\rho_{j}$ and $-r_{j}=-\rho
_{j}-1,$ respectively, where $1<\rho_{1}<\rho_{2}<\cdots.$ Let the magnetic potential $A(x)$ be defined by the equalities (\[basicnotions19\])-(\[basicnotions21\]). Then, the function $a\left( y,\omega;E\right) $ admits the asymptotic expansion $$a\left( y,\omega;E\right) \simeq\frac{\nu\left( E\right) }{\left\vert
E\right\vert }P_{\omega}\left( E\right) +\sum_{n=1}^{\infty}\sum
_{m=0}^{\infty}\sum_{j_{1},...,j_{n}}h_{n,m;j_{1},...,j_{n}}\left(
y,\omega;E\right) , \label{direct11}$$ where, for each $k=1,2,...,n,$ $j_{k}\ $takes values from $1$ to $\infty.$ The functions $h_{n,m;j_{1},...,j_{n}}\left( y,\omega;E\right) $ are of order $n$ with respect to the potential $W\left( x\right) ,$ they only depend on $W_{j_{1}},W_{j_{2}}...,W_{j_{k}},$ and they are homogeneous functions of order $n-m-\sum_{k=1}^{n}\rho_{j_{k}}$ with respect to the variable $y.$ We note that $h_{1,0;j}\left( y,\omega;E\right) =-i\frac{\nu\left( E\right)
}{\left\vert E\right\vert }R\left( y,\omega;E;W_{j}\right) P_{\omega}\left(
E\right) $ (here we emphasize the dependence of the function $R\left(
y,\omega;E\right) $ on $W_{j}$) is homogeneous function of order $1-\rho_{j}$ with respect to $y$.
Suppose that we know the matrix $-i\frac{\nu\left( E\right) }{\left\vert
E\right\vert }R\left( y,\omega;E;W\right) P_{\omega}\left( E\right) $. Note that the first diagonal component of the matrix $P_{\omega}\left(
E\right) $ is equal to $\left( \frac{1}{2}+\frac{m}{2E}\right) .$ As $\left( \frac{1}{2}+\frac{m}{2E}\right) \neq0,$ for $\left\vert E\right\vert
>m,$ we recover the function $\int\limits_{-\infty}^{\infty}\mathcal{V}_{V,A,\omega}^{\left( E\right) }\left( t\omega+y\right) dt$ from the first diagonal component of the matrix $-i\frac{\nu\left( E\right) }{\left\vert
E\right\vert }R\left( y,\omega;E;W\right) P_{\omega}\left( E\right) $. Therefore, similarly to [@79] or [@25], we determine the electric potential $V$ and the magnetic field $B$ from $R_{e}$ and $R_{m},$ respectively, by using the Radon transform.
Let us define a mapping that sends a potential $W\left( x\right) =\left(
V\left( x\right) ,A\left( x\right) \right) $ into $a^{\left( N\right)
}\left( y,\omega;E\right) +a_{\operatorname*{reg}}^{\left( N\right)
}\left( y,\omega;E\right) -\frac{\nu\left( E\right) }{\left\vert
E\right\vert }P_{\omega}\left( E\right) \ $by $T\left( y,\omega;E;W\right)
=a^{(N)}\left( y,\omega;E\right) +a_{\operatorname*{reg}}^{\left( N\right)
}\left( y,\omega;E\right) -\frac{\nu\left( E\right) }{\left\vert
E\right\vert }P_{\omega}\left( E\right) $ (here we emphasize the dependence of $a$ on $N$). As $a^{(N)}\left( y,\omega;E\right) $ is defined in terms of the asymptotic expansion (\[direct16\]), then $T$ is defined only up to a function from the class $\mathit{S}^{-\infty}.$ We separate the linear part $-i\frac{\nu\left( E\right) }{\left\vert E\right\vert }R\left(
y,\omega;E;W\right) P_{\omega}\left( E\right) $ of $T$, with respect to $W\left( x\right) ,$ and define $Q\left( y,\omega;E;W\right) =T\left(
y,\omega;E;W\right) +i\frac{\nu\left( E\right) }{\left\vert E\right\vert
}R\left( y,\omega;E;W\right) P_{\omega}\left( E\right) .$ We now can formulate the reconstruction result.
\[18\]Suppose that an electric potential $V\left( x\right) $ and a magnetic field $B\left( x\right) ,$ with $\operatorname{div}B=0,$ are $C^{\infty}\left( \mathbb{R}^{3}\right) $-functions and that they admit the asymptotic expansions (\[direct2\]) and (\[direct3\]), where $V_{j}$ and $B_{j}$ are homogeneous functions of orders $-\rho_{j}$ and $-r_{j}=-\rho
_{j}-1,$ respectively, where $1<\rho_{1}<\rho_{2}<\cdots.$ Let the magnetic potential $A(x)$ be defined by the equalities (\[basicnotions19\])-(\[basicnotions21\]). Then, the kernel $s\left( \omega,\theta;E\right) $ of the scattering matrix $S\left( E\right) $ for fixed $E\in(-\infty
,m)\cup(m,+\infty)$ in a neighborhood of the diagonal $\omega=\theta$ uniquely determines each one of $V_{j}\left( x\right) $ and $B_{j}\left( x\right)
$. Moreover, $V_{1}\left( x\right) $ and $B_{1}\left( x\right) $ can be reconstructed from the formula $-i\frac{\nu\left( E\right) }{\left\vert
E\right\vert }R\left( y,\omega;E;W_{1}\right) P_{\omega}\left( E\right)
=h_{1}\left( y,\omega;E\right) ,$ where $h_{1}$ is the term of highest homogeneous order with respect to $y$ in the expansion (\[direct11\]). The functions $V_{j}\left( x\right) $ and $B_{j}\left( x\right) ,$ for $j\geq2,$ can be recursively reconstructed from the formula $-i\frac
{\nu\left( E\right) }{\left\vert E\right\vert }R\left( y,\omega
;E;W_{j}\right) P_{\omega}\left( E\right) =\left( a\left( y,\omega
;E\right) -\frac{\nu\left( E\right) }{\left\vert E\right\vert }P_{\omega
}\left( E\right) -T\left( y,\omega;E;\sum_{i=1}^{j-1}W_{i}\right) \right)
^{{{}^\circ}}.$ Here we denote by $f^{{{}^\circ}}$, the highest order homogeneous term $f_{k}$ in (\[basicnotions16\]) that is not identically zero.
The proof is analogous to Theorem 4.2 of [@25], for the case of the Schrödinger equation.
Let $W^{\left( j\right) }$ satisfy the assumptions of Theorem \[18\] for $j=1,2.$ If $s_{1}\left( \omega,\theta;E\right) -s_{2}\left( \omega
,\theta;E\right) \in C^{\infty}\left( \mathbb{S}^{2}\times\mathbb{S}^{2}\right) $ for some $\left\vert E\right\vert >m,$ then $V_{1}-V_{2}$ and $B_{1}-B_{2}$ belong to the Schwartz class $\mathit{S.}$
Completeness of averaged scattering solutions.
==============================================
If the potential $\mathbf{V}$ satisfies Condition \[basicnotions26\] and decreases as $\left\vert x\right\vert ^{-\rho},$ when $\left\vert x\right\vert
\rightarrow\infty,$ with $\rho>2,$ then, for all $E\in\{(-\infty
,-m)\cup(m,\infty)\}\backslash\sigma_{p}\left( H\right) ,$ the scattering solution $u_{\pm}\left( x,\omega;E\right) $ are defined by (\[representation29\]) and they satisfy the asymptotic expansion (\[re16\]).
If $\rho\leq2,$ then relation (\[representation29\]) makes no sense. However, similarly to the Schrödinger operator case ([@37],[@38]), we can generalize definition (\[representation29\]).
We define the unperturbed averaged scattering solutions by $\psi_{0,f}\left( x;E\right) :=\int_{\mathbb{S}^{2}}e^{i\nu\left( E\right) \left\langle \omega,x\right\rangle
}P_{\omega}\left( E\right) f\left( \omega\right) d\omega,$ for any $f\in
L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) .$ Note that up to a coefficient $\psi_{0,f}$ is equal to $\Gamma_{0}^{\ast}\left( E\right) f,$ defined by (\[basicnotions39\]). Then, it follows that $\psi_{0,f}\in\mathcal{H}^{1,-s}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) ,$ $s>1/2,$ and $H_{0}\psi_{0,f}=E\psi_{0,f}.$
Let the potential $\mathbf{V}$ satisfy Condition \[basicnotions26\]. The perturbed averaged scattering solutions are defined by $$\psi_{+,f}\left( x;E\right) :=[I-R_{+}\left( E\right) \mathbf{V}]\psi_{0,f},\text{ }E\in\{(-\infty,-m)\cup(m,\infty)\}\backslash\sigma
_{p}\left( H\right) ,\text{ }f\in L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) . \label{completeness6}$$ Note that $\psi_{+,f}\in\mathcal{H}^{1,-s}\left( \mathbb{R}^{3};\mathbb{C}^{4}\right) ,$ $1/2<s\leq s_{0},$ and $H\psi_{+,f}=E\psi_{+,f}.$ If $\rho>2,$ the formula (\[representation29\]) holds and we have $\psi_{+,f}\left( x;E\right) =\int_{\mathbb{S}^{2}}\psi^{+}\left(
x,\omega;E\right) f\left( \omega\right) d\omega.$ The last equality justifies the name averaged scattering solutions.
Using the stationary representation (\[basicnotions14\]) we can write the scattering matrix $S\left( E\right) $ in terms of the averaged solutions. For $f,g\in L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $ we have $$\left. \left( S\left( E\right) f,g\right) _{\mathcal{H}\left( E\right)
}=\left( f,g\right) _{\mathcal{H}\left( E\right) }-i\left( 2\pi\right)
^{-2}\upsilon\left( E\right) ^{2}\left( \mathbf{V}\psi_{+,f},\psi
_{0,g}\right) _{L^{2}}.\right. \label{completeness5}$$
For us, the important property of the averaged scattering solutions is that the set (\[completeness6\]) is dense on the set of all solutions to the Dirac equation (\[eig1\]) in $L^{2}\left( \Omega;\mathbb{C}^{4}\right) ,$ where $\Omega$ is a connected open bounded set with smooth boundary $\partial\Omega$. We present this assertion as:
\[completeness12\]Let $\mathbf{V}\ $satisfies Condition \[basicnotions26\] and the following estimate $$\left\vert \partial_{x}^{\alpha}\mathbf{V}\left( x\right) \right\vert \leq
C_{\alpha}\left( 1+\left\vert x\right\vert \right) ^{-\rho},\text{ }\rho>1,\text{ }\left\vert \alpha\right\vert \leq1,\text{ for }x\in
\mathbb{R}^{3}\backslash\Omega, \label{completeness2}$$ where $\Omega$ is a connected open bounded set with smooth boundary $\partial\Omega$. Then, the set of averaged scattering solutions $\left\{
\psi_{+,f},\right. $ $\left. f\in\mathcal{H}\left( E\right) \right\} $ is strongly dense on the set of all solutions to (\[eig1\]) in $L^{2}\left(
\Omega;\mathbb{C}^{4}\right) $ for all fixed $E\in\{(-\infty,-m)\cup
(m,\infty)\}\backslash\sigma_{p}\left( H\right) .$
Let us note that the result of Theorem \[completeness12\] holds for all $\left\vert E\right\vert >m,$ if some result on absence of eigenvalues on $(-\infty,-m)\cup(m,\infty)$ is applied. For example, if $\mathbf{V}\in
L_{\operatorname*{loc}}^{5}\left( \mathbb{R}^{3}\right) $ satisfies relation (\[completeness2\]), then the result of Theorem \[completeness12\] remains true for all $\left\vert E\right\vert >m$ (see Section 2)$.$
We proceed as in the proof of Theorem 3.1 of [@26] (see also [@58]) for the Schrödinger case. Let us take a solution $\chi\in L^{2}\left(
\Omega;\mathbb{C}^{4}\right) $ that is orthogonal to $\psi_{+,f}$ for all $f\in\mathcal{H}\left( E\right) $. Then, as $\psi_{+,f}=\psi_{+,P_{\omega
}\left( E\right) f},$ for all $f\in L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ it follows that $$\left( \chi,\psi_{+,f}\right) _{L^{2}\left( \Omega;\mathbb{C}^{4}\right)
}=0,\text{ for }f\in L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) .
\label{completeness1}$$ We extend $\chi$ by zero to $\mathbb{R}^{3}\backslash\Omega$ and then, $\chi\in L_{s}^{2}$ for all $s.$ Let us define $\psi:=R_{+}\left( E\right)
\chi.$ Note that $\psi$ satisfy the equation $$\left( H-E\right) \psi=\chi. \label{completeness10}$$ Suppose that $\psi\in L_{-\sigma}^{2}$ for some $\sigma<1/2.$ Then, as $\chi=0$ on $\mathbb{R}^{3}\backslash\Omega,$ multiplying (\[completeness10\]) from the left side by $H_{0}+E$ we get that $\psi$ satisfies the following equation$$-\Delta\psi-\left( E^{2}-m^{2}\right) \psi+\left( H_{0}+E\right) \left(
\mathbf{V}\psi\right) =0, \label{completeness11}$$ for $x\in$ $\mathbb{R}^{3}\backslash\overline{\Omega}$. As the principal part $\Delta\psi$ of (\[completeness11\]) is diagonal, then, under assumption (\[completeness2\]), the proofs of [@45] in the case of a scalar Schrödinger equation apply for a system of Schrödinger equations (\[completeness11\]) and thus, we get that $\psi$ vanishes identically on $\mathbb{R}^{3}\backslash\overline{\Omega}.$ In particular, $\psi=0$ on $\partial\Omega$, in the trace sense. Then, as the boundary $\partial\Omega$ is smooth, we can approximate $\psi$ in the norm of $\mathcal{H}^{1}$ by functions $\psi_{n}\in C_{0}^{\infty}\left( \Omega\right) ,$ $n\in
\mathbb{N}.$ Noting that $\left( \left( H-E\right) \psi_{n},\chi\right)
_{L^{2}\left( \Omega;\mathbb{C}^{4}\right) }=\left( \psi_{n},\left(
H-E\right) \chi\right) _{L^{2}\left( \Omega;\mathbb{C}^{4}\right) }$ for all $n,$ we prove that $\left\Vert \chi\right\Vert _{L^{2}\left(
\Omega;\mathbb{C}^{4}\right) }^{2}=\left( \left( H-E\right) \psi
,\chi\right) _{L^{2}\left( \Omega;\mathbb{C}^{4}\right) }=\left(
\psi,\left( H-E\right) \chi\right) _{L^{2}\left( \Omega;\mathbb{C}^{4}\right) }=0,$ and hence, $\chi=0.$
Thus, to complete the proof we need to show that $\psi\in L_{-\sigma}^{2},$ for some $\sigma<1/2.$ Note that relation (\[completeness1\]) implies that $\Gamma_{-}\left( E\right) \chi=0$. Moreover, as the operator $\mathcal{F}_{-},$ defined by relation (\[basicnotions40\]), gives a spectral representation of $H$ and $\Gamma_{-}\left( \lambda\right) $ is locally Hölder continuous, it follows from the Privalov’s theorem that $\psi
=\psi_{1}+\psi_{2},$ where $$\psi_{1}=\int_{I}\Gamma_{-}^{\ast}\left( \lambda\right) \left( \frac
{1}{\lambda-E}\left( \Gamma_{-}\left( \lambda\right) -\Gamma_{-}\left(
E\right) \right) \chi\right) d\lambda, \label{completeness14}$$ and $\psi_{2}=R\left( E\right) E_{H}\left( \mathbb{R}\backslash I\right)
\chi,$ for some neighborhood $I$ of the point $E.$ Here $E_{H}$ is the resolution of the identity for $H.$ Note that $\psi_{2}$ is already from $L^{2}.$
Let us define the operator $J$ by $$Jg:=\int_{I}\Gamma_{-}^{\ast}\left( \lambda\right) g\left( \lambda\right)
d\lambda. \label{completeness15}$$ Since $\mathcal{F}_{-}$ is unitary from $\mathcal{H}_{ac}$ onto $\mathcal{\hat
{H}}$ and $Jg=E_{H}\left( I\right) \mathcal{F}_{-}^{\ast}g,$ the operator $J$ is bounded from $L^{2}\left( I;L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \right) $ into $L^{2}.$ Moreover, as the operator $\Gamma
_{-}^{\ast}\left( \lambda\right) $ is bounded from $L^{2}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) $ into $L_{-s}^{2},$ for $1/2<s\leq
s_{0}$, then $J$ is bounded from $L^{1}\left( I;L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \right) $ into $L_{-s}^{2}.$ Thus, by interpolation (see, for example, [@2]), $J$ is bounded from $L^{p}\left(
I;L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) \right) $ into $L_{-\sigma}^{2},$ with $\sigma=\left( 2/p-1\right) s$ and $1\leq p\leq2.$
Let us take $s_{1}=\frac{1}{2}+\vartheta,$ $\vartheta<\min\{s_{0}-1/2,1/2\}.$ Note that $\Gamma_{-}\left( \lambda\right) $ is locally Hölder continuous from $L_{s_{1}}^{2}$ to $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) $ with exponent $\vartheta.$ Then, as $\chi\in L_{s_{1}}^{2}\ $we get $\frac{1}{\lambda-E}\left( \Gamma_{-}\left( \lambda\right) -\Gamma
_{-}\left( E\right) \right) \chi\in L^{p}\left( I;L^{2}\left(
\mathbb{S}^{2};\mathbb{C}^{4}\right) \right) ,$ where $p<\frac
{1}{1-\vartheta}.$ Taking $p=\frac{1}{1-\vartheta/2}$ and $s=1/2+\vartheta/2$ we get that $\sigma\ <\frac{1}{2}.$ Using relations (\[completeness14\]) and (\[completeness15\]) we get $\psi_{1}=J\left( \frac{1}{\lambda-E}\left(
\Gamma_{-}\left( \lambda\right) -\Gamma_{-}\left( E\right) \right)
\chi\right) .$ Therefore, we conclude that $\psi\in L_{-\sigma}^{2}$. Theorem is proved.
Uniqueness of the electric potential and the magnetic field at fixed energy.
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In this Section we aim to show that the scattering matrix $S\left( E\right)
,$ given for some energy $E,$ determines uniquely the electric potential $\begin{pmatrix}
V_{+} & 0\\
0 & V_{-}\end{pmatrix}
$ and the magnetic field $B\left( x\right) =\operatorname*{rot}A\left(
x\right) .$ The averaged scattering solutions (\[completeness6\]) result useful here. With the help of these solutions we can prove that $S\left(
E\right) $ determines uniquely the Dirichlet to Dirichlet map (Definition (\[completeness9\])). Then, supposing that the electric potential $V_{\pm
}\left( x\right) $ and the magnetic field $B\left( x\right) $ are known outside some connected open bounded set $\Omega_{E}$ with smooth boundary $\partial\Omega_{E},$ we show that $V_{\pm}\left( x\right) $ and $B\left(
x\right) $ are uniquely determined everywhere in $\mathbb{R}^{3}.$
Let us consider the free Dirac operator $L_{0,\Omega_{E}}=\left(
\begin{array}
[c]{cc}0 & -i\sigma\cdot\nabla\\
-i\sigma\cdot\nabla & 0
\end{array}
\right) $ on $L^{2}\left( \Omega_{E};\mathbb{C}^{2}\right) \times
L^{2}\left( \Omega_{E};\mathbb{C}^{2}\right) ,$ where $\Omega_{E}$ is a connected open bounded set with smooth boundary $\partial\Omega_{E}$ and $\sigma=\left( \sigma_{1},\sigma_{2},\sigma_{3}\right) $ are the Pauli matrices (\[basicnotions22\]). $L_{0}$ is a self-adjoint operator on (see [@19]) $D_{L_{0,\Omega_{E}}}:=\{u=\left( u_{+},u_{-}\right) \mid
u_{+}\in\mathcal{H}_{0}^{1}\left( \Omega_{E};\mathbb{C}^{2}\right) ,u_{-}\in\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right) \},$ where $\mathcal{H}_{0}^{1}\left( \Omega_{E};\mathbb{C}^{2}\right) $ is the closure of $C_{0}^{\infty}\left( \Omega_{E}\right) $ in the space $\mathcal{H}^{1}\left( \Omega_{E};\mathbb{C}^{2}\right) $ and $\mathcal{H}\left(
\Omega_{E};\mathbb{C}^{2}\right) $ is the closure of $\mathcal{H}^{1}\left(
\Omega_{E};\mathbb{C}^{2}\right) $ in the norm $\left\Vert \cdot\right\Vert
_{\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right) }:=\left\Vert \left(
\sigma\cdot\nabla\right) \cdot\right\Vert _{L^{2}\left( \Omega
_{E};\mathbb{C}^{2}\right) }+\left\Vert \cdot\right\Vert _{L^{2}\left(
\Omega_{E};\mathbb{C}^{2}\right) }.$
Let $\mathbf{V}$ be an Hermitian $4\times4$-matrix-valued function whose entries belong to $L^{\infty}\left( \Omega_{E}\right) .$ Then, $L_{\mathbf{V,}\Omega_{E}}:=L_{0,\Omega_{E}}+\mathbf{V}$ is self-adjoint on $D_{L_{0,\Omega_{E}}}$. Consider the following Dirichlet problem$$\left\{
\begin{array}
[c]{c}\left( L_{\mathbf{V,}\Omega_{E}}-E\right) \left( u_{+},u_{-}\right)
=0,\text{ in }\Omega_{E},\\
\left. u_{+}\right\vert _{\partial\Omega_{E}}=g\in h\left( \partial
\Omega_{E}\right) ,\text{ on }\partial\Omega_{E}.
\end{array}
\right. \label{completeness4}$$ Here $h\left( \partial\Omega_{E}\right) $ is defined as the trace on $\partial\Omega_{E}$ of $\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right)
.$ Suppose that $E$ belongs to the resolvent set of $L_{\mathbf{V,\Omega_{E}}}$. Then, from Proposition 4.11 of [@19] we get that for every $f\in
h\left( \partial\Omega_{E}\right) $ there exist an unique solution $\left(
u_{+},u_{-}\right) \in\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right)
\times\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right) $ of the equation (\[completeness4\]).
For any $g\in h\left( \partial\Omega_{E}\right) ,$ we define the Dirichlet to Dirichlet (up-spinor to down-spinor) map by$$\Lambda_{\mathbf{V}}g=\left. u_{-}\right\vert _{\partial\Omega_{E}}\in
h\left( \partial\Omega_{E}\right) , \label{completeness9}$$ where $\left( u_{+},u_{-}\right) $ is the unique solution of (\[completeness4\]).
The uniqueness result of the potential at fixed energy is the following
\[completeness8\]Let the potentials $\mathbf{V}_{j}\left( x\right) ,$ $j=1,2,$ be given by$$\mathbf{V}_{j}\left( x\right) =\begin{pmatrix}
V_{+}^{\left( j\right) } & \sigma\cdot A^{\left( j\right) }\\
\sigma\cdot A^{\left( j\right) } & V_{-}^{\left( j\right) }\end{pmatrix}
\label{completeness17}$$ with real functions $V_{\pm}^{\left( j\right) },A_{k}^{\left( j\right)
}\in C^{\infty}\left( \mathbb{R}^{3}\right) ,$ $k=1,2,3,$ such that $V_{\pm
}^{\left( j\right) }$ and $A_{k}^{\left( j\right) },$ $j=1,2,$ satisfy (\[completeness2\]). Let $S_{j}\left( E\right) $ be the scattering matrices corresponding to $\mathbf{V}_{j},$ $j=1,2.$ Suppose that for some $E\in\left( -\infty,-m\right) \cup\left( m,+\infty\right) ,$ $S_{1}\left(
E\right) =S_{2}\left( E\right) ,$ and there is a connected open bounded set $\Omega_{E}$ with smooth boundary $\partial\Omega_{E}$ $,$ such that $E$ belongs to the resolvent set of $L_{\mathbf{V}_{j},\Omega_{E}}$ for both $j=1,2.$ Let $\mathbf{V}_{1}\left( x\right) $ be equal to $\mathbf{V}_{2}\left( x\right) $ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E}.$ Then we have that $V_{\pm}^{\left( 1\right) }\left( x\right) =V_{\pm}^{\left(
2\right) }\left( x\right) $ and $\operatorname*{rot}A^{\left( 1\right)
}\left( x\right) =\operatorname*{rot}A^{\left( 2\right) }\left( x\right)
$ for all $x\in\mathbb{R}^{3}.$
We follow the proof of [@26] for the Schrödinger case. Let us first show that $\psi_{+,f}^{\left( 1\right) }\left( x;E\right) =\psi
_{+,f}^{\left( 2\right) }\left( x;E\right) $ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E},$ $f\in\mathcal{H}\left( E\right) ,$ where $\psi_{+,f}^{\left( j\right) }$ are the averaged scattering solutions corresponding to $\mathbf{V}_{j}$ for $j=1,2.$ Denote$\ \psi:=\psi
_{+,f}^{\left( 2\right) }-\psi_{+,f}^{\left( 1\right) }$ and $\eta:=\mathbf{V}_{1}\psi_{+,f}^{\left( 1\right) }-\mathbf{V}_{2}\psi
_{+,f}^{\left( 2\right) }.$ Note that $\psi\in\mathcal{H}^{1,-s},$ $\eta\in
L_{s}^{2},$ $\frac{1}{2}<s\leq s_{0},$ and moreover, $$\left( H_{0}-E\right) \psi=\eta. \label{completeness3}$$ Since $S_{1}\left( E\right) =S_{2}\left( E\right) ,$ it follows from (\[completeness5\]) that $\Gamma_{0}\left( E\right) \eta=0.$ This implies that $\Gamma_{0}\left( E\right) \mathcal{F}^{\ast}\hat{\eta}=0.$ Then, as $\hat{\eta}\in\mathcal{H}^{s}$ and $\Gamma_{0}\left( E\right) \mathcal{F}^{\ast}$ is bounded from $\mathcal{H}^{s}$ into $L^{2}\left( \mathbb{S}^{2};\mathbb{C}^{4}\right) ,$ we conclude that $\hat{\eta}\left( \xi\right)
=0$ in the trace sense on the sphere of radius $\left\vert \xi\right\vert
=\nu\left( E\right) .$
Note that $\hat{\psi}=P^{+}\left( \xi\right) \hat{\psi}+P^{-}\left(
\xi\right) \hat{\psi}.$ From (\[completeness3\]) we get $$\left( \pm\sqrt{\xi^{2}+m^{2}}-E\right) P^{\pm}\left( \xi\right) \hat
{\psi}=P^{\pm}\left( \xi\right) \hat{\eta}. \label{representation220}$$ For $\pm E>m,$ the function $\frac{1}{\mp\sqrt{\xi^{2}+m^{2}}-E}$ is bounded. Then, as $P^{\pm}\left( \xi\right) \hat{\eta}\in\mathcal{H}^{s},$ we get, for $\pm E>m,$ $P^{\mp}\left( \xi\right) \hat{\psi}\in\mathcal{H}^{s}.$ Moreover, it follows from (\[representation220\]) that, $P^{\pm}\left(
\xi\right) \hat{\psi}=\frac{\pm\sqrt{\xi^{2}+m^{2}}+E}{\left\vert
\xi\right\vert ^{2}-\nu\left( E\right) ^{2}}P^{\pm}\left( \xi\right)
\hat{\eta},$ for $\pm E>m.$ As $P^{\pm}\left( \xi\right) \hat{\eta}\in\mathcal{H}^{s}$ and $P^{\pm}\left( \xi\right) \hat{\eta}=0$ in the trace sense on the sphere of radius $\left\vert \xi\right\vert =\nu\left( E\right)
,$ then Theorem 3.2 of [@4] implies that for $\pm E>m,$ $P^{\pm}\left(
\xi\right) \hat{\psi}\in\mathcal{H}^{s-1},$ $s>\frac{1}{2}.$ Therefore, we obtain that $\hat{\psi}\in\mathcal{H}^{s-1}$ and $\psi\in L_{s-1}^{2}.$ Note that $s-1>-1/2.$ As $\mathbf{V}_{1}\left( x\right) =\mathbf{V}_{2}\left(
x\right) ,$ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E}$ we see that $\left(
H_{0}+\mathbf{V}_{1}\right) \psi=E\psi,$ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E}.$ Thus, similarly to the proof of Theorem \[completeness12\], as $\psi$ satisfies (\[completeness11\]), for $x\in\mathbb{R}^{3}\diagdown\Omega_{E},$ we conclude that $\psi$ is identically zero for $x\in\mathbb{R}^{3}\diagdown\overline{\Omega_{E}}.$ In particular we obtain $\psi_{+,f}^{\left( 1\right) }\left( x;E\right)
=\psi_{+,f}^{\left( 2\right) }\left( x;E\right) $ in the trace sense on $\partial\Omega_{E}$
Let $\tau$ be the trace map $\tau:\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right) \rightarrow h\left( \partial\Omega_{E}\right) .$ Note that the scattering solution $\psi_{+,f}^{\left( j\right) }\left( x;E\right)
\in\mathcal{H}^{1}\left( \Omega_{E};\mathbb{C}^{4}\right) ,$ $j=1,2,$ solves (\[completeness4\]) with $g=\tau\left( \left( \psi_{+,f}^{\left(
j\right) }\right) _{+}\right) $ (here $(u)_{\pm}$ denotes the first or the last two components of a $\mathbb{C}^{4}$ vector respectively). As $\psi
_{+,f}^{\left( 1\right) }\left( x;E\right) =\psi_{+,f}^{\left( 2\right)
}\left( x;E\right) $ in the trace sense on $\partial\Omega_{E},$ we get $$\left. \Lambda_{\mathbf{V}_{2}}\left( \tau\left( \left( \psi
_{+,f}^{\left( 1\right) }\right) _{+}\right) \right) =\Lambda
_{\mathbf{V}_{2}}\left( \tau\left( \left( \psi_{+,f}^{\left( 2\right)
}\right) _{+}\right) \right) =\tau\left( \left( \psi_{+,f}^{\left(
2\right) }\right) _{-}\right) =\tau\left( \left( \psi_{+,f}^{\left(
1\right) }\right) _{-}\right) =\Lambda_{\mathbf{V}_{1}}\left( \tau\left(
\left( \psi_{+,f}^{\left( 1\right) }\right) _{+}\right) \right)
.\right. \label{completeness13}$$
For any solution $u^{\left( j\right) }=\left( u_{+}^{\left( j\right)
},u_{-}^{\left( j\right) }\right) \in\mathcal{H}\left( \Omega
_{E};\mathbb{C}^{2}\right) \times\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right) $ of $\left( L_{\mathbf{V}_{j}}-E\right) u^{\left( j\right)
}=0,$ $j=1,2,$ we have (see Lemma 2.1 of [@19])$$_{h\left( \partial\Omega_{E}\right) }\left\langle \overline{u_{+}^{\left(
2\right) }},\left( i\sigma\cdot N\right) \left( \Lambda_{\mathbf{V}_{1}}-\Lambda_{\mathbf{V}_{2}}\right) u_{+}^{\left( 1\right) }\right\rangle
\text{ }_{h\left( \partial\Omega_{E}\right) ^{\ast}}=\int_{\Omega_{E}}\left( u^{\left( 2\right) },\left( \mathbf{V}_{1}-\mathbf{V}_{2}\right)
u^{\left( 1\right) }\right) dx, \label{completeness16}$$ where $h\left( \partial\Omega_{E}\right) ^{\ast}$ is the dual space to $h\left( \partial\Omega_{E}\right) $ with respect to the duality $_{h\left(
\partial\Omega_{E}\right) }\left\langle u,\overline{v}\right\rangle
_{h\left( \partial\Omega_{E}\right) ^{\ast}}=\int_{\partial\Omega_{E}}u\cdot\overline{v}dS$ and $N$ is the unit outer normal vector to $\partial\Omega_{E}.$ Taking $u^{\left( 1\right) }=\psi_{+,f}^{\left(
1\right) }$ in (\[completeness16\]) and using (\[completeness13\]) we get $\int_{\Omega_{E}}\left( u^{\left( 2\right) },\left( \mathbf{V}_{1}-\mathbf{V}_{2}\right) \psi_{+,f}^{\left( 1\right) }\right) dx=0$ for all averaged scattering solutions $\psi_{+,f}^{\left( 1\right) }.$ Since these solutions are dense on the set of all solutions to (\[completeness4\]) (here we used Theorem \[completeness12\], observing that $(-\infty
,-m)\cup(m,\infty)\}\cap\sigma_{p}\left( H\right) =\varnothing$ if $\mathbf{V}$ satisfies relation (\[basicnotions46\])), $\int_{\Omega_{E}}(u^{\left( 2\right) },\left( \mathbf{V}_{1}-\mathbf{V}_{2}\right)
u^{\left( 1\right) })dx=0,$ for any solution $u^{\left( j\right) }=\left(
u_{+}^{\left( j\right) },u_{-}^{\left( j\right) }\right) \in
\mathcal{H}\left( \Omega_{E};\mathbb{C}^{2}\right) \times\mathcal{H}\left(
\Omega_{E};\mathbb{C}^{2}\right) $ of $\left( L_{\mathbf{V}_{j}}-E\right)
u^{\left( j\right) }=0,$ $j=1,2.$ Thus, it follows from relation (\[completeness16\]) that $_{h\left( \Gamma\right) }\left\langle
\overline{u_{+}^{\left( 2\right) }},\left( i\sigma\cdot N\right) \left(
\Lambda_{\mathbf{V}_{1}}-\Lambda_{\mathbf{V}_{2}}\right) u_{+}^{\left(
1\right) }\right\rangle _{h\left( \Gamma\right) ^{\ast}}=0$ for any solution $u^{\left( j\right) }=\left( u_{+}^{\left( j\right) },u_{-}^{\left( j\right) }\right) $ of $\left( L_{\mathbf{V}_{j}}-E\right)
u^{\left( j\right) }=0,$ $j=1,2.$ As for any $f_{j}\in h\left(
\Gamma\right) ,$ $j=1,2,$ there exist an unique solution $u^{\left(
j\right) }$ of $\left( L_{\mathbf{V}_{j}}-E\right) u^{\left( j\right)
}=0,$ such that $\left. u_{+}^{\left( j\right) }\right\vert _{h\left(
\Gamma\right) }=f_{j},$ we conclude that $\left( i\sigma\cdot N\right)
\left( \Lambda_{\mathbf{V}_{1}}-\Lambda_{\mathbf{V}_{2}}\right) f_{1}\in
h\left( \Gamma\right) ^{\ast}$ is the functional $0,\ $for all $f_{1}\in
h\left( \Gamma\right) ,$ and hence, $\Lambda_{\mathbf{V}_{1}}=\Lambda
_{\mathbf{V}_{2}}.$ Since $\mathbf{V}_{1}\left( x\right) =\mathbf{V}_{2}\left( x\right) $ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E}$ and $\mathbf{V}_{j}\in C^{\infty}\left( \mathbb{R}^{3}\right) ,$ $j=1,2,$ we get that $A_{k}^{\left( 1\right) }=A_{k}^{\left( 2\right) },$ for $k=1,2,3,$ to infinite order on $\partial\Omega_{E}.$ Thus, it follows from Theorem 1 of [@19] that $V_{1}\left( x\right) =V_{2}\left( x\right) ,$ $\operatorname*{rot}A_{1}\left( x\right) =\operatorname*{rot}A_{2}\left(
x\right) $ on $\Omega_{E}.$ Theorem is proved.
\[22\]Let the potentials $\mathbf{V}_{j}\left( x\right) ,$ $j=1,2,$ be given by (\[completeness17\]), with real functions $V_{\pm}^{\left(
j\right) },A_{k}^{\left( j\right) }\in C^{\infty}\left( \mathbb{R}^{3}\right) ,$ $k=1,2,3,$ such that $V_{\pm}^{\left( j\right) }$ satisfy (\[completeness2\]) and the magnetic fields $B_{j},$ with $\operatorname{div}B_{j}=0,$ satisfy the estimate (\[basicnotions18\]) with $d=1$, $j=1,2$. Suppose that for some $E\in\left( -\infty,-m\right)
\cup\left( m,+\infty\right) ,$ $S_{1}\left( E\right) =S_{2}\left(
E\right) ,$ and there is a connected open bounded set $\Omega_{E}$ with smooth boundary $\partial\Omega_{E}$ $,$ such that $E$ belongs to the resolvent set of $L_{\mathbf{V}_{j},\Omega_{E}}$ for both $j=1,2.$ Let $V_{\pm}^{\left( 1\right) }=V_{\pm}^{\left( 2\right) }$ and $B_{1}=B_{2}$ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E}.$ Then we have that $V_{\pm
}^{\left( 1\right) }=V_{\pm}^{\left( 2\right) }$ and $B^{\left( 1\right)
}=B^{\left( 2\right) }$ for all $x\in\mathbb{R}^{3}.$
Let us define the magnetic potentials $\tilde{A}_{j},$ $j=1,2,$ from the magnetic fields $B_{j},$ $j=1,2,$ by the equalities (\[basicnotions19\])-(\[basicnotions21\]). Let $\mathbf{\tilde{V}}_{j}\left( x\right) ,$ $j=1,2,$ be the correspondent potentials. Observe that $\tilde{S}_{j}=S_{j}$, where $\tilde{S}_{j}$ is associated to potential $\mathbf{\tilde{V}}_{j},$ $j=1,2.$ Moreover, as $L_{\mathbf{V}_{j},\Omega_{E}}$ and $L_{\mathbf{\tilde
{V}}_{j},\Omega_{E}}$ are unitary equivalent, $E$ belongs to the resolvent set of $L_{\mathbf{\tilde{V}}_{j},\Omega_{E}}.$ Since $B_{1}=B_{2}$ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E},$ then by construction $\tilde{A}_{1}=\tilde{A}_{2}$ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E},$ and hence $\mathbf{\tilde{V}}_{1}\left( x\right) =\mathbf{\tilde{V}}_{2}\left(
x\right) ,$ for $x\in\mathbb{R}^{3}\diagdown\Omega_{E}.$ Applying Theorem \[completeness8\] we conclude that $V_{\pm}^{\left( 1\right) }=V_{\pm
}^{\left( 2\right) }$ and $B^{\left( 1\right) }=B^{\left( 2\right) }$ for all $x\in\mathbb{R}^{3}.$
If the asymptotic expansions (\[direct2\]) and (\[direct3\]) actually converge, in pointwise sense, for $\left\vert x\right\vert $ large enough, respectively to $V\left( x\right) $ and $B\left( x\right) ,$ then collecting the result of Corollary \[22\] and the reconstruction result of Theorem \[18\] we are able to formulate the following uniqueness result for the inverse scattering problem:
Let the expansion (\[direct2\]) for the electric potentials $V_{j}\in
C^{\infty}\left( \mathbb{R}^{3}\right) $ and the expansion (\[direct3\]) for the magnetic fields $B_{j}\in C^{\infty}\left( \mathbb{R}^{3}\right) ,$ with $\operatorname{div}B_{j}=0,$ $j=1,2,$ hold. Let the magnetic potentials $A_{j},$ $j=1,2,$ be defined by the equalities (\[basicnotions19\])-(\[basicnotions21\]). Suppose that for some $E\in\left( -\infty
,-m\right) \cup\left( m,+\infty\right) ,$ $S_{1}\left( E\right)
=S_{2}\left( E\right) ,$ and there is a connected open bounded set $\Omega_{E}$ with smooth boundary $\partial\Omega_{E}$ $,$ such that $E$ belongs to the resolvent set of $L_{\mathbf{V}_{j},\Omega_{E}}$ for both $j=1,2.$ Moreover, suppose that the asymptotic expansions (\[direct2\]) and (\[direct3\]), for $V$ and $B$, respectively, actually converge in pointwise sense for $x\in\mathbb{R}^{3}\diagdown\Omega_{E}.$ Then, we have that $V_{1}\left( x\right) =$ $V_{2}\left( x\right) $ and $B_{1}\left(
x\right) =B_{2}\left( x\right) $ for all $x\in\mathbb{R}^{3}.$
As $S_{1}\left( E\right) =S_{2}\left( E\right) ,$ it follows from the reconstruction result of Theorem \[18\] that the asymptotic terms $V_{k}^{\left( 1\right) }=V_{k}^{\left( 2\right) }$ and $B_{k}^{\left(
1\right) }=B_{k}^{\left( 2\right) }$ coincide for all $k.$ Moreover, as the asymptotic expansions (\[direct2\]) and (\[direct3\]) converge, $V_{1}\left( x\right) =$ $V_{2}\left( x\right) $ and $B_{1}\left(
x\right) =B_{2}\left( x\right) $ for $x\in\mathbb{R}^{3}\diagdown\Omega
_{E}$. Using Corollary \[22\] we conclude that $V_{1}\left( x\right) =$ $V_{2}\left( x\right) $ and $B_{1}\left( x\right) =B_{2}\left( x\right)
$ for all $x\in\mathbb{R}^{3}.$
[99]{}
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[^1]: Electronic Mail: [email protected]
[^2]: Fellow, Sistema Nacional de Investigadores. Electronic mail: [email protected]
[^3]: Research partially supported by CONACYT under Project CB-2008-01-99100.
|
---
abstract: 'We study the competition of magneto-dipole, anisotropy and exchange interactions in composite three dimensional multiferroics. Using Monte Carlo simulations we show that magneto-dipole interaction does not suppress the ferromagnetic state caused by the interaction of the ferroelectric matrix and magnetic subsystem. However, the presence of magneto-dipole interaction influences the order-disorder transition: depending on the strength of magneto-dipole interaction the transition from the ferromagnetic to the superparamagnetic state is accompanied either by creation of vortices or domains of opposite magnetization. We show that the temperature hysteresis loop occurs due to non-monotonic behavior of exchange interaction versus temperature. The origin of this hysteresis is related to the presence of stable magnetic domains which are robust against thermal fluctuations.'
author:
- 'A. M. Belemuk'
- 'O. G. Udalov'
- 'N. M. Chtchelkatchev'
- 'I. S. Beloborodov'
bibliography:
- 'NumericalMod.bib'
title: 'Competition of magneto-dipole, anisotropy and exchange interactions in composite multiferroics'
---
Introduction\[sec:intro\]
=========================
Multiferroic materials are materials with coupled magnetic and electric degrees of freedom. [@Cheong07; @Khomskii06; @Katsuara05; @Mostovoy06; @Kimura07] One example of this coupling is due to spin-orbit interaction in certain crystals. However, this coupling is relatively weak. Currently there is an active search for multiferroic materials with strong coupling. [@Ederer05; @Kornev07]
One way to strongly coupled magnetic and electric degrees of freedom is to develop hybrid ferroelectric-ferromagnetic layered materials where mechanical stress produces strong correlations between the layers. [@Fiebig05; @Scott2006; @Spal2007; @Sone12] Recently another promising possibility has been suggested based on granular materials where small metallic ferromagnetic (FM) grains were embedded into ferroelectric (FE) matrix or these grains were located in close proximity to the FE substrate. [@Bel2014ME] The presence of small metallic grains increases the strength of Coulomb interaction providing the necessary coupling between the FE and FM degrees of freedom.
An important question is to understand the nature of multiferroic state in granular multiferroic materials. On the mean-field level the properties of these materials have been understood. [@Bel2014ME] It was shown that the exchange coupling $J$ depends on the properties of the FE matrix/substrate, in particular on the dielectric permittivity $\epsilon$ of the surrounding medium. [@Bel2014ME; @Bel2014ME1] Due to temperature dependence of dielectric permittivity, $\epsilon(T)$, the exchange interaction depends non-monotonically on temperature leading to the inverse phase transition with paramagnetic phase appearing at lower temperatures compared to the ferromagnetic phase. The qualitative behavior of $J$ is different for small and large (compared to the grain size $r_{\rm gr}$) inter-grain distances $a$: for large inter-grain distances, $a > r_{\rm gr}$, the $J$-value is increased in the vicinity of the FE Curie point due to suppression of the Coulomb blockade effects leading to a different magnetic state at these temperatures.
However, it is still an open question whether the magneto-electric coupling obtained in the mean-field theory is robust against the magneto-dipole and anisotropy interactions neglected in the mean-field approach. We use the numerical modelling to address this question.
We investigate the non-equilibrium meta-stable states in granular system and study the nature of new meta-stable phases appearing in the system with temperature dependent exchange interaction. In addition, we answer the question if temperature dependent exchange interaction can lead to unusual blocking effects.
To be more specific, we study the magnetic behavior of composite multiferroics with $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}<\operatorname{\mathit T_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FM}}}$, where $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$ is the FE Curie temperature of paraelectric-ferroelectric transition of the FE matrix and $\operatorname{\mathit T_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FM}}}$ is the Curie temperature of FM grains. We focus on the temperature range $T \ll \operatorname{\mathit T_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FM}}}$ where all grains are in the FM state and study the phase diagram of granular multiferroics beyond the mean-field approximation using Monte-Carlo simulations. In particular, we study the combined effect of magnetic anisotropy, the long-range magneto-dipole (MD) interaction and the exchange coupling. We show that there is an inverse magnetic transition in the system which is robust with respect to the MD interaction. This inverse transition disappears only for strong MD interaction, stronger than the exchange coupling. In addition, we show that non-monotonic temperature behavior of intergrain exchange interaction leads to a new type of hysteresis in composite multiferroics.
Three-dimensional (3D) nanostructures composed of single-domain ferromagnetic particles has been intensively studied both experimentally and theoretically. [@Twardowski2013; @Bertram2001; @Chen1991; @Parkin1989; @Chien1988; @Bedanta09; @Maylin2000; @Trohidou1998; @Kleemann2004] The interplay of magnetic anisotropy, long-range magneto-dipole interaction and short range exchange interaction defines the magnetic state of the system. Depending on the ratio of these interactions different magnetic states are possible in granular ferromagnets. [@Freitas2001; @Freitas2006] Among them are superparamagnetic (SPM), super spin-glass (SSG) and superferromagnetic (SFM) states.
The most studied situation is related to the case of large intergrain distances ($ \geq 2$nm) and small exchange interaction where magnetic state is defined by the competition of MD interaction and anisotropy. [@Berkowitz2000; @Trohidou1998; @Grady1998] The magnetic anisotropy is responsible for “blocking” phenomena and defines the blocking temperature $T_{\mathrm b}$. [@Chien1988; @Chien1991] The weak MD interaction modifies the blocking temperature, while the strong MD interaction leads to the SSG state. [@Ayton95; @Ravichandran96; @Mamiya99; @Djurberg97; @Sahoo03]
For small distances between grains ($\sim 1$ nm) the exchange interaction is crucial. It leads to the formation of the SFM state. [@Sobolev2012; @Freitas2001; @Hembree1996; @Lutz1998; @Freitas2007] In such systems the SPM-SFM transition occurs. Even a weak exchange interaction can influence the magnetic state of the system. [@Hembree1996] In particular, the FM ordering with long range ferromagnetic correlations appears.
Different theoretical methods have been used to study granular magnets. The mean-field approach allows to study granular systems with finite short range exchange interaction and zero (or weak) MD interaction neglecting fluctuation effects. [@Beattie1990; @Munakata2009] Modelling based on Landau-Lifshitz equation allows to consider the MD, magnetic anisotropy and exchange interactions. [@Ziemann2004] However, this approach needs to be generalized in the presence of thermal fluctuations by introducing the Langevin forces. [@Lazaro1998] These fluctuations are important for granular magnetic systems since the granular magnetic moment is relatively small and fluctuation effects are pronounced especially near the phase transitions. However, the inclusion of Langevin forces in the nonlinear spin dynamics is not numerically efficient. Therefore we use Monte-Carlo (MC) simulations which allow to study phase transitions in composite multiferroics with strong long-range MD interaction and arbitrary thermal fluctuations. [@Levanyuk1993; @Chantrell1994; @Trohidou1998; @Grady1998; @Freitas2006; @Bertram2001; @Asselin2010PhysRevB; @Chantrell2010PhysRevB; @Cheng2010; @Twardowski2013]
MC modelling strongly depends on the degree of anisotropy. At strong anisotropy the problem reduces to the Ising model with magnetic moment of each particle having only two directions defined by the anisotropy axis. In this case the MC modelling is very efficient and is based on the trial spin flips. Another type of MC modelling is based on the Heisenberg model with arbitrary magnetization direction. This model is more general but it requires more computational time due to spin rotations over the sphere rather than spin flips. [@Matsubara1993; @Hinzke1999; @Asselin2010PhysRevB; @Chantrell2010PhysRevB] We use our own MC code with random spin-flips and random spin-rotations that is valid for any anisotropy.
The paper is organized as follows: In Sec. \[results\] we formulate our main results. In Sec. \[Sec:Model\] we discuss the model of composite multiferroics. In Sec. \[quantities\] we introduce important physical quantities which we calculate. We discuss our results in Sec. \[Sec:Results\]. The details of our numerical calculations are presented in Appendix \[App:CalcProc\].
Main results {#results}
============
Here we summarize our main results. The non-monotonic temperature dependence of exchange interaction in composite multiferroics leads to the unusual evolution of the magnetic state with temperature. The intergrain exchange interaction has either peak or deep in the vicinity of the FE phase transition due to coupling of electric and magnetic degrees of freedom. \cite{} In the mean field approximation the peak in the exchange interaction leads to the onset of FM state in the vicinity of FE phase transition. The deep in the exchange interaction suppresses the FM state in the vicinity of the FE Curie point. We use Monte-Carlo simulations to investigate the influence of long-range MD interaction and magnetic anisotropy on the magnetic phase diagram of composite multiferroics. We show that MD interaction and anisotropy do not suppress the magneto-electric coupling in these materials, however their interplay produces a new type of hysteresis. Our results are the following:
1\) The Monte-Carlo simulations reproduce the mean field results in the absence of MD interaction and magnetic anisotropy. Similar to the mean field approach, the FM state exists in the vicinity of the FE Curie point and the disordered state appears away from this region.
2\) The finite MD interaction does not suppress the FM ordering in the vicinity of the FE phase transition even if the MD interaction is twice stronger than the exchange interaction. The presence of MD interaction leads to the appearance of the domain structure and to splitting of uniform FM state. This result is similar to Ref. , where a weak FM interaction leads to the formation of FM domains, while strong MD interaction produces a vortex structure.
3\) The magnetic state depends on the strength of MD interaction outside the FM region: the system is in the SPM state for weak MD interaction and in the antiferromagnetic stripe phase for strong MD interaction.
4\) The magnetic anisotropy does not influence the FM state. However, it prevents the formation of vortices in the transition region and leads to a widening of the FM domain.
5\) The “blocking phenomenon” does not appear at finite magnetic anisotropy and zero MD interaction at considered temperatures meaning that the system has enough time to reach the ground state such that the non-equilibrium state does not appear.
6\) The “blocking phenomenon” appears at finite magnetic anisotropy and finite MD interaction. The temperature hysteresis loop occurs due to non-monotonic behavior of exchange interaction vs. temperature. The origin of this hysteresis is related to the presence of stable magnetic domains which are robust against thermal fluctuations, MD, exchange, and anisotropy interactions.
7\) The AFM stripes appear in the case of deep in the exchange interaction.
Below we discuss these results in details.
The model {#Sec:Model}
=========
Magnetic subsystem
------------------
In this subsection we consider the magnetic subsystem. We model a composite multiferroic as an ensemble of FM grains embedded into FE matrix. All grains are homogeneously magnetized single domain FM particles of the same volume $V$ and saturation magnetization $M_s$. For temperatures $T \ll \operatorname{\mathit T_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FM}}}$, the saturation magnetization $M_s$ is constant. Each grain with volume $V$ has a magnetic moment $\mu= M_s V$ and is treated as a point dipole located at the centre of the grain. The grains are pinned to the sites of the regular cubic lattice with lattice spacing $a$ and can freely rotate their adjusting magnetic moments.
The whole system is modelled as a $3D$ lattice of classical spins, with magnetic moment of the $i$th grain being ${\bm \mu}_i= \mu {\bf S}_i$, where the unit vector ${\bf S}_i= (S_i^x, S_i^y, S_i^z)$ is the spin of $i$th particle representing the direction of the magnetic moment.
We assume that each grain has a uniaxial anisotropy. Spatial distributions of anisotropy axes varies in different experiments and depends on the preparation condition. The anisotropy axes can be homogeneously distributed over the solid angle, or uniformly distributed in a certain plane. We assume that the easy axes of all grains are oriented in the z-direction. This situation is realized in experiment with magnetic field applied during the sample preparation. [@Freitas2007]
The Hamiltonian of the system has the form $$\label{ham}
{\cal H}= {\cal H}_{\rm exc}+ {\cal H}_{\rm dip}+ {\cal H}_{\rm an}.$$ The first term, ${\cal H}_{\rm exc}$, describes the exchange coupling between grains $i$ and $j$ $$\label{exc}
{\cal H}_{\rm exc}= - J \sum \limits_{\langle i, j \rangle} {\bf S}_i \cdot {\bf S}_j,$$ where the sum is over the nearest neighbour pairs of grains.
The second term, ${\cal H}_{\rm dip}$, in Eq. describes the long-range magneto-dipole (MD) interaction between magnetic moments ${\bm \mu}_i$ and ${\bm \mu}_j$ of individual grains $$\label{dip}
{\cal H}_{\rm dip}=g \sum \limits_{i < j} {\,} \frac{{\bf S}_i \cdot {\bf S}_j {\,} r^2_{ij}- 3({\bf S}_i \cdot {\bf r}_{ij}) ({\bf S}_j \cdot {\bf r}_{ij})}{r^5_{ij}},$$ where ${\bf r}_{ij}$ is the distance between magnetic moments at sites $i$ and $j$ measured in units of lattice spacing $a$ and $g$ is the MD interaction constant.
The third term, ${\cal H}_{\rm an}$, in Eq. describes uniaxial anisotropy energy $$\label{haman}
{\cal H}_{\rm an}= -K \sum_i ({\bf e}_z \cdot {\bf S}_i)^2,$$ where $K$ is the temperature independent magnetic anisotropy energy of a single grain. The unit vector ${\bf e}_z$ defines the direction of the anisotropy easy axis.
We consider the energy parameters $(J, g, K, T)$ in arbitrary units. Parameters, $g$ and $K$ depend on a grain volume and can be controlled by varying the grain size. The dipole coupling $g$ is additionally depends on the lattice spacing $a$. This allows one to vary parameters $g$ and $K$ in a wide range. The exchange interaction is proportional to the grain surface and scales with the volume as $V^{2/3}$. Moreover, the ratio of MD and exchange interactions can be controlled by varying the interparticle distance $a$. The MD interaction decays as $1/a^3$ with distance, while exchange interaction decays exponentially $e^{-\kappa a}$, where $\kappa$ is the inverse length which depend on the band structure of the surrounding FE matrix and the Fermi energy of electrons inside grains.
Ferroelectric subsystem
-----------------------
The FE matrix is characterized by the Curie temperature $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$. We study the temperature region in the vicinity of $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$. The most important characteristic of the FE matrix in our consideration is the FE dielectric permittivity $\epsilon$ which has a peculiarity in the vicinity of the phase transition point, $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$. [@Levan1983] This peculiar behavior of $\epsilon$ provides a strong coupling between electric and magnetic degrees of freedom. [@Bel2014ME; @Bel2014ME1]
Interaction of magnetic and ferroelectric subsystems
----------------------------------------------------
Recently the intergrain exchange interaction, $J$, in composite multiferroics was studied and its temperature behavior was predicted. [@Bel2014ME; @Bel2014ME1] In the vicinity of the FE phase transition the exchange interaction has a peculiarity. Depending on the system parameters it has either peak or deep. Such peculiarity appears due to combine influence of Coulomb blockade effects and the temperature dependence of dielectric permittivity of the FE matrix on the exchange interaction. The peculiarity of the exchange interaction in the vicinity of the FE Curie point $T=\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$ is related to the peculiarity in the dielectric permittivity $\epsilon$ of the FE matrix. The exchange interaction as a function of $\epsilon$ has the form [@Bel2014ME] $$\label{Eq_J_peak}
J(T)=J_{0}\epsilon^{\gamma a/r_{\rm gr}- 1},$$ where parameter $J_0>0$ is the $\epsilon$-independent part of exchange interaction, $J_0$ decays exponentially with intergrain distance $a$; $\gamma$ is the numerical coefficient of order one. The dielectric permittivity $\epsilon$ has a peak at temperature $T=\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$. For $\gamma < r_{\rm gr}/a$ (small intergrain distances) the exchange interaction has a deep, while for $\gamma > r_{\rm gr}/a$ (large intergrain distances) it has a peak.
The dependence of the intergrain exchange interaction $J$ on the FE permittivity $\epsilon$ is the signature of magneto-electric coupling emerging in composite multiferroics. The peak of exchange interaction leads to the unusual magnetic phase diagram: the FM state appears in the vicinity of the FE Curie temperature $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$, while away from $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$ the system is in the SPM state. Thus, the FM state in the system exists in a finite region around FE Curie point, $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$. In case of deep in the temperature dependence of exchange constant $J$ the opposite situation occurs: the FM state is suppressed in the vicinity of $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$ due to interaction of magnetic and FE subsystems.
Above effects have been studied in Ref. using the mean field approximation without taking into account the MD interaction and anisotropy. Here we take into account both interactions and study the influence of MD interaction and magnetic anisotropy on the magneto-electric coupling in composite multiferroics.
For large inter-grain distances, $\gamma > r_{\rm gr}/a$, the exchange interaction $J(T)$ has the peak. In this work we model this peak as follows $$\label{Jmfrr}
J(T)= J_0 e^{-(T- \operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}})^2/w^2},$$ where $w$ is the width of the exchange peak and $J_0$ is the amplitude of intergrain exchange interaction. The peak in $J(T)$ occurs at temperatures $T = \operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$, when the Coulomb blockade is suppressed and the electron wave functions become weakly localized leading to a strong overlap and strong exchange interaction. [@Bel2014ME] The Coulomb blockade is suppressed when permittivity $\epsilon(T)$ reaches its maximum value at $T=\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$.
For small inter-grain distances, $\gamma< r_{\rm gr}/a$, the exchange interaction $J(T)$ has a deep which we model as follows $$\label{Eq_J_pit}
J(T)= J_0 \left( 1- e^{-(T- \operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}})^2/w^2} \right).$$
Calculated quantities {#quantities}
=====================
In this section we introduce physical quantities which we calculate. The calculation procedure is described in Appendix \[App:CalcProc\]. The first quantity we calculate is the average magnetization [@Lau89; @Holm93; @Peczak91; @Chen93] $$\label{magn_part}
M(T)=\left| \frac1N \sum \limits_{i= 1}^N {\bf S}_i \right|,$$ where $N$ is the number of lattice sites.
In the presence of MD interaction and zero external magnetic field, the average magnetization $M(T)$ is not an efficient quantity: the lattice spins form complex magnetic patterns, either domains or vortices, and the mean magnetization vanishes, $1/N \sum_{i} {\bf S}_i= 0$, even if locally the system is in the FM state. To account for local FM correlations we introduce the cell averaged magnetization, $m(T)$, over a cell with linear size $L_c$. We find that the optimal size for such cell average is $L_c=5$ $$\label{mcell}
m(T)= \left \langle\left|\frac{1}{N_c} \sum \limits_{i} {\bf S}_{i}\right| \right \rangle,$$ where summation is over the all grains in a given cell $N_c=L_c^3$ and the averaging is defined as summation over the all possible positions of the cell centre, $\langle\rangle=N^{-1}\sum_{i}$.
Next, we introduce the spin-spin correlation function $G$ as an averaged correlation function for nearest-neighbour pairs $$\label{Cg}
G= \frac1N \sum \limits_{\bf R} \frac16 \sum \limits_{\bf g} \langle {\bf S}_{\bf R} \cdot {\bf S}_{\bf R+g} \rangle=
\frac{1}{3N} \sum \limits_{\langle i,j \rangle} \langle {\bf S}_i \cdot {\bf S}_j \rangle,$$ where ${\bf g}$ are the six vectors of nearest neighbours and the last summation ${\langle i,j \rangle}$ is over the all nearest-neighbour pairs in the lattice. The correlation function $G$ is important for understanding magneto-transport in granular magnets. The magneto-resistance (MR) of granular magnetic film is proportional to this correlation function, MR(T)$\sim G$. The MR measurements can be considered as the probes of the magnetic state of the system.
Discussion of results {#Sec:Results}
=====================
Influence of magneto-dipole (MD) interaction on the magnetic phase diagram of composite multiferroics
-----------------------------------------------------------------------------------------------------
In this subsection we discuss the influence of MD interaction on the magnetic phase diagram of composite multiferroics for the case of large inter-grain distances, where exchange interaction $J$ has a peak around $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$, see Eq. (\[Jmfrr\]).
![(Color online) Magnetic phase diagram of composite multiferroic vs. temperature for zero magneto-dipole interaction ($g=0$) and magnetic anisotropy ($K=0$). Solid (red) line shows the temperature dependence of intergrain exchange interaction, $J(T)$. Straight dash dotted (orange) line stands for temperature $T$. Gray dash dot-dotted and green dotted lines show the average magnetization $M(T)$ and cell averaged magnetization $m(T)$, respectively. Blue dashed line shows the nearest neighbour correlation function $G$. Transition temperatures $T_{\mathrm{\scriptscriptstyle C}}^{\pm}$ are defined using the mean-field equation, $T=2J(T)$.[]{data-label="Fig_PU_0_0"}](Fig1_g00_K00.pdf){width="1\columnwidth"}
![(Color online) Snapshots of single magnetic layer of composite multiferroic. Panels (a) and (b) show the disordered (SPM) magnetic state and the ordered FM state, respectively. Position of these panels is shown in Fig. 1 by black arrows.[]{data-label="Fig_DisOr"}](Fig_g00K00DisOr.pdf){width="1\columnwidth"}
### Zero magneto-dipole (MD) interaction
In the absence of long-range MD interaction and anisotropy the magnetic phase diagram obtained using the Monte-Carlo simulations coincides with the mean-field phase diagram (see Fig. \[Fig\_PU\_0\_0\]). We used the following parameters: $J_0=2.5$, $\operatorname{\Delta \mathit{T}^{\mathrm{\scriptscriptstyle FE}}}=0.1$, $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}=0.5$. These parameters correspond to Fe grains embedded into organic TTF-CA FE matrix. The grain size is $5-10$ nm and the intergrain distance $ \geq 1$ nm. For these parameters the intergrain exchange interaction is about $J \sim 300$K. The Curie temperature of bulk ferroelectric TTF-CA is $80$ K. However, for granular array it can be smaller; the Curie temperature of TTF-CA in composite granular metal/FE system is $50$ K. [@Keller2014]
The FM state appears around $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$, where $J(T)$ has a maximum (see snapshot in Fig. \[Fig\_DisOr\]b) and the SPM state exists outside this region (see Fig. \[Fig\_DisOr\]a). The finite FM region appears with two magnetic phase transitions due to peak in the exchange interaction. In the mean-field approximation both transition temperatures $T_{\mathrm{\scriptscriptstyle C}}^{\pm}$ are defined as $T=2J(T)$. Due to zero MD interaction domains are not formed, the ground state of the system is the homogeneous FM state and the magnetization $M(T)$ is coincide with the cell averaged magnetization $m(T)$. In the SPM state the saturation magnetization and the correlator $G$ tends to zero since the system is in the disordered state due to thermal fluctuations.
![(Color online) Magnetic phase diagram of composite multiferroic vs. temperature $T$ for weak magneto-dipole interaction ($g=0.5$) and zero magnetic anisotropy ($K=0$). All notations are defined in Fig. 1.[]{data-label="Fig_PU_05_0"}](Fig2_g05_K00.pdf "fig:"){width="1\columnwidth"}\
![(Color online) Snapshots of single magnetic layer of composite multiferroic. Panel (a) shows the FM state divided into stripe domains appearing due to the interplay of exchange and weak MD interactions. Panel (b) shows the vortex state with magnetic vortices appearing due to strong MD interaction. Panel (c) shows FM chains forming antiferromagnetic pattern appearing at low temperatures and MD interaction being stronger than exchange interaction. Position of panels (a) and (c) is shown in Fig. 3 by black arrows.[]{data-label="Fig_DomVor"}](Fig_DomVor.pdf){width="1\columnwidth"}
### Weak and moderate magneto-dipole (MD) interaction
The long-range MD interaction competes with intergrain exchange interaction suppressing the FM state in the system. Figure \[Fig\_PU\_05\_0\] shows the case of weak MD interaction with the dipole constant, $g=0.5$. This value of $g$ is typical for Fe grains with size $a=4$ nm, where $g=(2.2 \mu_{\mathrm B}V/\lambda^3_{\mathrm{Fe}})^2/a^3\approx50$ K, $\lambda_{\mathrm{Fe}}=0.28$ nm is the Fe lattice parameter. [@Chelikowsky2006] For these parameters the FM region exists. However, the MD interaction reduces the size of FM state and leads to the formation of domains in the system (see the left panel in Fig. \[Fig\_DomVor\]). Above the FM region the SPM state appears, similar to the case of zero MD interaction, meaning that the thermal fluctuations exceed the exchange and MD interactions. Below the FM region the thermal fluctuations exceed the exchange interaction but not the MD interaction. As a result the antiferromagnetic stripes appear at low temperatures, $T < 0.35$ (see panel (c) in Fig. \[Fig\_DomVor\]). The transition to stripe structure occurs via formation of large antiferromagnetic domains with temperature dependent sizes.
![(Color online) Magnetic phase diagram of composite multiferroic vs. temperature $T$ for intermediate magneto-dipole interaction $g=2.0$ and zero magnetic anisotropy ($K=0$). All notations are defined in Fig. 1.[]{data-label="Fig_PU_2_0"}](Fig5_g20_K00.pdf "fig:"){width="1\columnwidth"}\
### Strong magneto-dipole (MD) interaction
For strong MD interaction, $g\geq 2.0$, the uniform FM state does not appear in the system (see Fig. \[Fig\_PU\_2\_0\]). However, the long-range magnetic correlations are still exist in the system close to $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$. For such a strong MD interaction the vortex-like structure appears in the system (see the central snapshots in Fig. \[Fig\_DomVor\](b)). The vortex structure transforms into the stripe structure outside the $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$ region. For strong MD interaction the SPM state does not appear for temperatures above and below $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$, instead the stripe structure appears in these regions. At higher temperatures the stripe structure transforms into SPM state due to thermal fluctuations.
The MD interaction grows with particle volume as $V^2$. For Fe grains of size $6$nm and interparticle distance $1$nm the dipole constant is $g\approx 230$ K. This value of MD interaction equals to the peak value of exchange interaction. The exchange interaction grows with grain surface as $V^{2/3}$ and it is intergrain distance dependent.
![(Color online) Magnetic phase diagram of composite multiferroic vs. temperature $T$ for strong magneto-dipole interaction, $g=5.0$, and zero magnetic anisotropy, $K=0$. All notations are defined in Fig. 1.[]{data-label="Fig_PU_5_0"}](Fig7_g50_K00.pdf "fig:"){width="1\columnwidth"}\
Figure \[Fig\_PU\_5\_0\] shows the case of strong MD interaction with dipole constant $g=5$ being twice larger than the peak value of the exchange interaction. This case is typical for $10$nm $Fe$ grains with MD constant $g$ exceeding the room temperature. Even in this case the FM domains exist in the vicinity of the FE Curie point. The FM state in the vicinity of the FE phase transition is robust against the MD interaction leading to the unusual magnetic phase diagram of composite multiferroics due to ME coupling of ferroelectric and magnetic subsystems.
Influence of magnetic anisotropy on the magnetic phase diagram of composite multiferroics
-----------------------------------------------------------------------------------------
In this subsection we discuss the influence of magnetic anisotropy on the magnetic phase diagram of composite multiferroics. The magnetic anisotropy in granular materials is much stronger than in bulk magnets due to surface and shape anisotropy. [@Chien1988; @Chen1991] It plays an important role in formation of magnetic state of granular magnets. The magnetic relaxation time in the system of non-interacting particles exponentially depends on the ratio of anisotropy energy and temperature, $\tau_{\mathrm{r}}\sim \exp(K/(k_{\mathrm B}T))$. At low temperatures the relaxation time becomes larger than the characteristic measurement time. At these temperatures the measured magnetic properties are the properties of non-equilibrium or “blocked” state. The temperature hysteresis of magnetic properties is the signature of “blocking” phenomenon. The Monte-Carlo (MC) calculations in some way are similar to real experiment: simulations start with a certain non-equilibrium state and the system “relaxes” to the equilibrium state via discrete steps during the simulations. If number of MC steps $N_{\mathrm{MC}}$ (which can be associated with measurement time if the attempt frequency of the system is known) exceeds a certain value $N_{\mathrm r}$ (which can be associated with relaxation time $\tau_{\mathrm{r}}$) then the system relaxes to the equilibrium state during simulations. In the opposite case, the system is locked into some non-equilibrium magnetic state.
![(Color online) Magnetic phase diagram of composite multiferroic vs. temperature $T$ for zero magneto-dipole interaction and strong magnetic anisotropy ($K=6.0$). Insert: Snapshot of a single magnetic layer of composite multiferroic with disordered magnetic state due to strong anisotropy. Black arrow indicates the position of this snapshot on phase diagram. All notations are defined in Fig. 1.[]{data-label="Fig_PU_6_00"}](MsY_K60_g00updt2.pdf "fig:"){width="1\columnwidth"}\
Figure \[Fig\_PU\_6\_00\] shows the magnetic behavior of granular multiferroics with strong anisotropy, $K=6.0$. This anisotropy is twice larger than the peak value of the exchange interaction. This situation is realized for $6$ nm Fe grains with anisotropy constant $K=0.8\cdot10^{-6}$ erg/cm$^3$. [@Chen1991] Figure \[Fig\_PU\_6\_00\] coincides with mean-field theory, where FM state exists in the vicinity of the FE phase transition and the disordered state appears away from $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}$. The only difference with mean field theory is related to the fact that magnetic moments in the disordered state have only two possible directions along the anisotropy axis (see inset in Fig. \[Fig\_PU\_6\_00\]).
For zero MD interaction the MC results do not depend on the initial state of the system. We use two different approaches for MC calculations, see the details in the Appendix: 1) The starting configuration is the FM alignment along the z-direction at the initial temperature point. After a certain number of MC steps the resulting spin configuration is used for the next temperature point. 2) The starting configuration is the disordered state at each temperature point. In addition, we change the direction of temperature evolution and the number of MC steps. Both approaches lead to the same results without hysteresis behavior. Thus, we conclude that in our modelling $N_{\mathrm{MC}}\gg N_{\mathrm r}$. And the magnetic anisotropy alone does not qualitatively change the magnetic phase diagram of composite multiferroics and it does not lead to the suppression of ME effects in the system.
Hysteresis behavior of composite multiferroics with strong magnetic anisotropy and magneto-dipole interaction
-------------------------------------------------------------------------------------------------------------
In this subsection we discuss the influence of strong magnetic anisotropy and MD interaction on the phase diagram of composite multiferroics and show that these interactions lead to new features. We use the following approach: 1) We move from high to low temperatures starting from the FM state at initial temperature $T=0.72$. At each next temperature point we begin with the final state of the previous temperature point. 2) We move from low $T=0.28$ to high $T=0.72$ temperatures with the initial state corresponding to the FM state. Results are shown in Fig. \[Fig\_Hyst1\].
![(Color online) Temperature hysteresis of composite multiferroic. Left and right panels correspond to the case of increasing and decreasing temperature, respectively.[]{data-label="Fig_Hyst1"}](MsY_K30_g05_upd4_wf02_nMC500.pdf){width="1\columnwidth"}
The presence of MD interaction increases the blocking temperature leading to larger magnetic relaxation time, $\tau_{\mathrm{r}}$. [@Bertram2001; @Maylin2000] However, this is not the case for our system. To understand the nature of hysteresis in composite multiferroics consider the starting point $T=0.28$ in the “warming” case, left panel, with the uniform FM state as the initial state. The final state at this temperature obtained after MC simulations is the disordered (SPM) state. The system relaxes from FM to SPM state in the process of calculations. Therefore neither magnetic anisotropy nor the MD interaction produce “blocking” in the system at these temperatures.
However, if we move from high to low temperatures the relaxation to the SPM state does not occur at temperature $T=0.28$. The reason for the cooling process is related to the fact that one needs to pass the peak in the exchange interaction at temperature $T=0.5$ where multiple inhomogeneous FM state with domains of different orientations is formed. Around $T=0.5$ the average magnetization (the red curve) is zero. But the cell averaged magnetization is finite meaning that the system is divided into domains of opposite magnetization. These domains occur due to the interplay of exchange and MD interactions. This multidomain state is robust against thermal fluctuations for $T<0.5$ even for zero exchange interaction. This is in contrast to the uniform FM state, which can be destroyed in the absence of exchange interaction even at low temperatures. As a result the magnetization vs. temperature has a hysteresis behavior.
At higher temperature, $T\approx 0.7$, the multidomain state is unstable against thermal fluctuations at these temperatures and the hysteresis behavior is absent.
![(Color online) Evolution of magnetic phase diagram of composite multiferroic as a function of exchange interaction at fixed temperature.[]{data-label="Fig_Hys2"}](MsJ_K30_T03.pdf){width="\columnwidth"}
Figure \[Fig\_Hys2\] shows the magnetic phase diagram as a function of exchange constant $J$ at fixed temperature $T=0.3$. On the left panel we increase the exchange constant $J$ starting from zero to a certain high value and decrease it back to zero. The initial spin configuration at $J=0$ is the uniform FM state. On the right panel in Fig. \[Fig\_Hys2\] we start with finite value of exchange interaction $J$, decrease it to zero and return back to the same value. Obviously, such a nonmonotonic behavior of exchange interaction occurs with changing the temperature in composite multiferroic. Figure \[Fig\_Hys2\] shows the hysteresis behavior caused by the non monotonic change of the intergrain exchange interaction. Similar hysteresis occurs as a function of temperature.
To summarize, the above hysteresis is specific to composite multiferroics - materials with non-monotonic behavior of exchange interaction. The hysteresis is absent for systems with temperature independent exchange interaction. The peculiar feature of this hysteresis is related to the fact that it appears in the vicinity of the FE Curie point.
Deep in the exchange interaction in the vicinity of FE Curie point
------------------------------------------------------------------
{width="2\columnwidth"}\
The intergrain exchange interaction has either peak or deep in the vicinity of FE Curie point depending on the system parameters. [@Bel2014ME] Here we study the situation with deep in the exchange interaction in the vicinity of the FE Curie point. Figure \[Fig\_Pd\_0\_0\]a shows the case of zero MD interaction and zero anisotropy with the following parameters: the deep is $J_0 = 0.75$, $\operatorname{\Delta \mathit{T}^{\mathrm{\scriptscriptstyle FE}}}=0.07$, and $\operatorname{\mathit{T}_{\mathrm{\scriptscriptstyle C}}^{\mathrm{\scriptscriptstyle FE}}}=0.5$. In this case the Monte-Carlo simulations and the mean field approximation coincide. For chosen parameters the exchange interaction exceeds the temperature in the whole range except the close vicinity of FE transition, where exchange interaction is small and the system is in the SPM state. Outside this region the system is in the FM state. A single domain state with two magnetic phase transitions is realized for zero MD interaction.
A moderate MD interaction leads to the domains formation in the FM phase and to the formation of magnetic vortices in the transition regions. Figure \[Fig\_Pd\_0\_0\]b shows this behavior for dipole constant $g = 0.5$.
Strong MD interaction ($g=5.0$) leads to the suppression of the FM state at high temperatures, see Fig. \[Fig\_Pd\_0\_0\]c. Here the AFM stripe structure appears instead of FM ordering. At low temperatures the FM state exists. Therefore in the case of deep in the exchange interaction the MD interaction leads to the suppression of high temperature magnetic phase transition in contrast to the case of peak in the exchange interaction. At higher temperatures, $T > 1$, the AFM state is suppressed due to thermal fluctuations.
Applicability of results
------------------------
Here we discuss the applicability of our approach. First, we study the case of regular magnetic array with fixed intergrain distances. In real materials this distance fluctuates leading to the dispersion of MD and exchange interactions.
Second, we consider 3D multiferroic materials which produced via bottom-up method. A different top-down fabrication, based on layer by layer growth, is used to produce a single layer of magnetic grains. In 2D systems the influence of MD interaction on the magnetic phase diagram is different from 3D case. This situation requires further investigation.
Conclusion
==========
We studied the competition of magneto-dipole, anisotropy and exchange interactions in composite three dimensional multiferroics – materials with magnetic grains embedded into FE matrix. The peculiarity of composite (or granular) multiferroics is related to the fact that interparticle interaction is affected by the FE matrix. Granular multiferroics show the magneto-electric coupling effect. Using Monte Carlo simulations we showed that magneto-dipole interaction does not suppress the ferromagnetic state caused by the interaction of the ferroelectric matrix and magnetic subsystem. Thus, MD interaction does not suppress the ME effect in granular multiferroics. However, the presence of magneto-dipole interaction influences the order-disorder transition: depending on the strength of magneto-dipole interaction the transition from the FM to the SPM state is accompanied either by creation of vortices or domains of opposite magnetization.
We showed that “blocking phenomenon” appears at finite magnetic anisotropy and finite MD interaction. The temperature hysteresis loop occurs due to non-monotonic behavior of exchange interaction vs. temperature. The origin of this hysteresis is related to the presence of stable magnetic domains which are robust against thermal fluctuations.
Acknowledgements
================
We thank Shauna Robbennolt and Sarah Tolbert for useful discussions. I. B. was supported by NSF under Cooperative Agreement Award EEC-1160504, NSF Award DMR-1158666 and the U.S. Civilian Research and Development Foundation (CRDF Global). A. B. and N. C. are grateful to Russian Academy of Sciences for the access to JSCC and “Uran” clusters and Kurchatov center for access to HCP supercomputer cluster. A. B. acknowledges support of the Russian foundation for Basic Research (grant No. 13-02-00579). N. C. acknowledges Laboratoire de Physique Théorique, Toulouse and CNRS for hospitality and Russian Fundamental Research foundation (grant No. 14-12-01185) for support of supercomputer simulations.
Calculation procedure {#App:CalcProc}
=====================
We use classical Monte Carlo (MC) simulations and the standard Metropolis algorithm to model magnetic properties of the system. [@Landau2000; @Kretschmer79; @Binder76; @Binder69; @Landau81; @Freitas2006] We consider $L \times L \times L$ ($L= 20$) cubic lattice with periodic boundary conditions. To efficiently evaluate the long-range MD interaction in systems with relatively small number of particles (as, for example, $L= 5, 6, 7$ considered in Ref. [@Kretschmer79; @Kechrakos98]) one has to implement Ewald summation technique. [@Allen87; @Wang01] We account the MD interaction by direct summation in the real space applying the minimum image convention. [@Allen87] In terms of the range of the interaction that have been taken into account, this scheme is equivalent to the fast Fourier transform method used for micromagnetic simulations. [@Hinzke2000; @Yuan92]
We use the FM state ordered along the $z$-direction, ${\bf S}_i= {\bf e}_z$, as an initial spin configuration for simulating at the first temperature point. The resulting spin state is used as an initial state for the next temperature point and so on. To study hysteresis effects we make two passages: first, we start with low temperature and increase the temperature during our calculations; second, we do the opposite.
One MC step consists of $L \times L \times L$ consecutive changes in the lattice spin orientations. We calculated the change in the energy of the system $\Delta {\cal H}$: if change is negative, $\Delta {\cal H} \le 0$, a new state is accepted; if change is positive, $\Delta {\cal H} > 0$, the new state is accepted with probability $e^{- \Delta {\cal H}/T}$. In our simulations we use $N_{MC}= 12000$ MC steps per spin and study 60 samples in every 200 MC steps to calculate thermal properties. To check the stability of final configuration on the number of MC steps we increased the number of MC steps five times (up to $N_{MC}= 6 \cdot 10^4$) and find no difference in the resulting state.
We generate the update in spin directions using two ways. First, the spin orientations were distributed uniformly over the unit sphere’s surface [@Wang01] $$\label{update2}
\cos \theta_i= \xi, \quad \varphi_i= \pi \xi',$$ where $\xi, \xi'$ are some random numbers from the interval $(-1, 1)$. This algorithm becomes inefficient at low temperatures or strong anisotropy. In this case the majority of randomly chosen spin directions has to be rejected due to large energy change $\Delta {\cal H}$. We use such an update to check the results of the second algorithm with tuned step of change in the spin direction.
Our main algorithm for spin change was an algorithm where a new spin direction is chosen within a small angle near a given spin ${\bf S}_i$ [@Serena93]. First, a random unit vector ${\bf w}$ perpendicular to the chosen spin ${\bf S}_i$ is generated. Then new trial configuration is chosen as $$\label{cone}
{\bf S}'_i= \cos \theta_i {\,} {\bf S}_i+ \sin \theta_i {\,} {\bf w},$$ where $\theta_i$, the rotation angle from ${\bf S}_i$ to ${\bf S}'_i$, is chosen according to $$\cos \theta_i= 1+ \xi (\cos \theta_{max}- 1),$$ $\xi$ is a random number varying in the interval $(0, 1)$, the angle $\theta_{max}$ is a maximum allowed amplitude for the change of the polar angle $\theta_i$ of the initial spin ${\bf S}_i$, $0 < \theta_{max} \leqslant \pi$. The new spin direction ${\bf S}'_i$ lies within a cone around the initial direction with aperture angle $2\theta_{max}$ and all the directions inside this cone can be reached with the same probability [@Serena93]. The value of $\theta_{max}$ is adjusted after one full MC sweep over the lattice to keep, when possible, the number of accepted spin changes around $\sim 50\%$. Also we kept the lower bound for $\theta_{max} \gtrsim \pi/6$ to prevent too small MC moves which are inefficient to thermalize the system.
This algorithm is not valid at low temperatures [@Freitas2006] or strong anisotropy, $K \gg 1$, when the system tends to the Ising limit which does not allow the spin flips. We improve the situation allowing spins to flip with a certain small probability ($\lesssim 0.1 - 0.2$). With this modification we reproduce the correct values of the critical temperature $T_c$ for the Heisenberg, $T_c \simeq 1.44$, [@Lau89; @Holm93; @Peczak91; @Chen93; @Watson69; @Landau81; @Binder69; @Binder76; @Watson69] and the Ising, $T_c \simeq 4.51$, models. [@Binder01]
|
---
author:
- 'P. Englmaier'
- 'M. Pohl'
- 'N. Bissantz'
subtitle: 3D Distribution of Molecular Gas
title: The Milky Way Spiral Arm Pattern
---
Introduction
============
Half a century ago, [@Oort58] used the Leiden/Sydney 21 cm line survey to construct a map of the neutral atomic hydrogen gas distribution for the Milky Way (Fig. \[oort\]). This map, the first large scale map of the Milky Way’s gas distribution - mostly located in spiral arms, was disturbed by distance errors and excluded the inner part of the Galaxy as well as the region beyond. The apparent expansion of the Galactic centre region was speculated to be due to a bar, and this view has been generally accepted in the last decade. The method used by Oort, however, assumed circular rotation of the gas, since distance information was not known, and therefore was not applicable in the innermost $\sim5\,\kpc$’s of the Galaxy. The map showed spiral arms, but also many fingers pointing towards earth (the “Finger-of-God” effect), and the pattern was incomplete in direction of the centre and anti-centre. There was also an apparent difference between spiral arms on the left and right side of the Galaxy; not a single spiral pattern could fit the observations.
Since then, many studies attempted to chart the spiral arm pattern of the Milky Way in several tracers, but often assuming circular rotation laws for translating radial velocity into distance. Some tracers follow a 4-armed spiral pattern, while others only follow 2 arms (see [@Vallee95] for a review). The perhaps most successful attempt to chart spiral arms, was achieved by [@GG76], who used HII regions and also a circular rotation law for distance estimation, or more direct methods for nearby objects.
More recently, [@NakanishiSofue] used the $^{12}$CO ($J=1-0$) survey data of [@Dame01] to recover the 3D distribution of the molecular gas in the Milky Way. Again, a circular rotation law was assumed, and the area beyond the Galactic centre was excluded. The face-on view is compatible with a 4-armed spiral pattern.
In the outer disk, spiral arms have been traced by analyzing the HI layer thickness [@Levine06], again finding at least four spiral arms.
Method
======
[@Pohl08] used the velocity field from the standard model of [@Bissantz03] to recover the gas distribution in the Milky Way using a probabilistic method to match the observed CO gas distribution from [@Dame01] to the model prediction along the line-of-sight. The underlying kinematic model is not a simple circular rotation law, but has been calculated from a realistic mass model including a tri-axial model of the bulge/bar which has been determined using the observed COBE/DIRBE near-IR light distribution [@BissantzGerhard02]. At radii larger than $7\,\kpc$ we use a circular rotation law (after a smooth transition).
When multiple distances are permitted by the model for a given measured signal, the signal is distributed over the allowed distance bins according to certain weights. These weights have been chosen to avoid placing gas at unrealistic large distances or above or below the warping and flaring plane. Our approach is based on the ideas of regularization methods which are used commonly e.g. for non-parametric reconstruction problems. Comparison with a mock density model allows us to identify artefacts caused by the inversion. The resulting map for the gas distribution is shown in Fig. \[gas\] (blue-green inner part) together with the HI layer thickness (red-gray outer part) from [@Levine06]. Major inversion artefacts in this map are: the circle between Sun and galactic centre, the linear structure behind the galactic centre on along the line-of-sight, and the structure seen beyond the solar radius in the far side of the disk.
For further details of the method we refer the reader to [@Pohl08].
Interpretation
==============
Two or four spiral arms?
------------------------
When we trace by eye the spiral arms in Fig. \[gas\] starting at the bar ends, the situation becomes complicated when we reach $\sim7\,\kpc$ in radius. Spiral arms seem to end or branch and any picture drawn is highly subjective. However, we can trace the arms with confidence at small and large radii. On the other hand, we can make a sensible connection between the spiral arms, since arms cannot cross, only branch. When we interpret the spiral pattern this way, we can draw the pattern shown in Fig. \[arms\], a 2-armed spiral pattern in the inner Galaxy, which branches in two more arms at about the solar radius. Similarly, there seems to be some indication of short branches starting of the minor axis of the bar when the spiral arms pass by near the Langrangian points of the bar. Those short branches might be due to the assumed kinematical gas flow model. Unfortunately, the 3-kpc-arms are only hinted at in our map. This is a consequence of the poor fit of the gas dynamics near the 3-kpc-arm, which causes the gas from the near 3-kpc-arm to be broken into two pieces. Nevertheless it can be identified in the map and we also see a weak signal for the counter arm.
Since the spiral arms here are matched to structure seen in the deprojected gas distribution map, there is no reason to expect symmetry in the derived spiral pattern. However, surprisingly we find an almost perfect 180-degree rotational symmetry in the inner Galaxy. Since artefacts, real spurs, and gaps in the map are not expected to be symmetrically distributed, we conclude that the observed symmetry and the 2-armed spiral pattern must be real.
In the transition region, at 7..8 kpc galactic radius, we observe spiral arm branching which seems to occur at two locations which are not 180 degree apart in azimuth. When we overlay the spiral pattern with the pattern rotated by 180 degrees, we observe that the spiral arms from both patterns alternately cross and interleave each other. This seems to indicate, that the outer galaxy spiral pattern is a superposition of even and odd spiral modes. The Galaxy nevertheless appears rather symmetric 4-armed, but this might be an illusion. It remains to be seen, if the situation for the Milky Way is similar to the hidden 3-armed spiral mode found in many late type spirals as observed by [@Elmegreen92]. If true, three arms should be closer together, while one stronger arm, resulting from two superimposed arms, should be more isolated.
The 3-kpc-arms
--------------
The 3-kpc-arm is a well known and studied feature of the (l,v)-diagram. Its main characteristics are, that it passes in front of the galactic centre with a large radial velocity of $53\,\kms$ which indicates that it is driven by the bar into non-circular motion. Alternatively, the 3-kpc-arm’s peculiar non-circular motion and apparent lack of a counter-arm, led [@Fux99] to the conclusion, that the arm is pushed around by a $m=1$ mode in the inner disk, caused by an off-centre bar tumbling around the centre with a low pattern speed of $\sim20..30\,\kmskpc$. The counter arm is pushed to much larger non-circular velocities explaining the $+135\,\kms$ feature.
Aligned with the 3-kpc arm is a group of OH/IR stars, which has lead to the interpretation of a material arm [@Sevenster99] in the context of the [@Fux99] model, but can also be understood in terms of the OH/IR progenitors being formed near the Lagrange points at corotation and perpendicular to the bar [@Englmaier00].
Very recently [@Dame08] have found that a possible complementary far 3-kpc-arm has been overlooked in the $(l,v)$-diagram, which is sursprisingly symmetric to the near 3-kpc-arm. While [@Dame08] have so far only uncovered part of the structure, it seems like a long searched for piece of the galactic puzzle has fallen in place. The vertical thickness of the new arm is about half the value for the near arm, which places it at the same distance from the galactic centre as the near arm. Moreover, the two arms, if symmetric, allow estimation of the position angle of the bar. By assuming that the arms are bisymmetric and start on the major axis of the bar, we can estimate that the bar’s position angle is in the range of 20 to 40 degrees. The value depends critically on the longitude extent of the far arm. [@Dame08] find the arm extends to $l=-7..-8\deg$ (corresponding to $25..20\deg$ for the bar angle), or $l=-12\deg$ if two isolated clouds which lie in the continuation of the observed part of the far 3-kpc-arm also belong to it (Dame, priv. comm.). This, however, is unlikely, because the far arm would then not appear as a coherent structure, very unlike the near arm.
Inner Galaxy
------------
In the central kpc of the reconstructed map, we observe a ring with radius $\sim200\,\pc$, which is off-centre, density peaks before and behind the galactic centre, and gas along the leading edges of the nuclear bar (see Fig. \[centre\]; the dashed line indicates the position of the bar). We can make a direct comparison with the results from [@Sawada06], which used a non-kinematical method to map the molecular gas in the same region. [@Sawada06] found a slightly different distribution. Our reconstruction of the inner Galaxy is resolution limited, smoothing out the innermost few $100\,\pc$. Both methods have distance errors, causing “Finger-of-God” structure pointing to earth.
In [@Sawada06], the clump named Bania’s clump 2 [@Bania77] is stretched along the line-of-sight. A similar structure exists in our map, but stretched out over a smaller distance (at (x,y)=(0.4,-0.4) kpc). Another clump on the far side, but somewhat closer to the centre, is seen in our map (at (x,y)=(-0.3,0.1), and might be the counter object to Bania’s clump 2. Or, it might be misplaced in distance and it should truly sit at (x,y)=(0,0.1).
Outer Galaxy
------------
In the outer galaxy we can compare and continue our map with the map produced by [@Levine06]. They used the thickness of the HI layer to trace spiral arms in the outer galaxy out to more than 20 kpc in radius. The maps overlap at $r\sim8\,\kpc$ allowing us to compare and continue the spiral arms between the two studies. We find both maps to agree very well, all four arms can be continued into the map provided by [@Levine06].
Comparison with other studies
=============================
In the previous section, we already compared to [@Sawada06] in the inner galaxy and [@Levine06] in the outer galaxy. Another study, which used the same data and created a face-on map of the Milky Way was recently done by [@NakanishiSofue]. We find excellent agreement in the outer part of the model and can cross-identify features in both studies.
The main difference seems to be the interpretation in terms of spiral arms.
Conclusions
===========
The Milky Way has four symmetric spiral arms in the inner part, two of which, the near and far 3-kpc-arm, end inside corotation, two other arms start at $\sim4\,\kpc$ on the major axis of the bar, continue through corotation, and branch at $\sim7\,\kpc$ into four spiral arms which continue to $\sim20\,\kpc$.
The outer pattern very likely is a superposition of $m=2$ and $m=3$ spiral density waves, as has been observed in external galaxies similar to the Milky Way.
The observed symmetries might be useful in future attempts to invert the gas distribution. One of the spiral arm branches occurs near to us and might leave an imprint in the local stellar velocity distribution. Models of the gas dynamics should take the symmetry of the two 3-kpc-arms into account.
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|
---
address: '[Department of Mathematics, University of Bayreuth, Germany, [email protected]]{.nodecor}'
author:
- Sascha Kurz
title: Lifted codes and the multilevel construction for constant dimension codes
---
**Abstract:** Constant dimension codes are e.g. used for error correction and detection in random linear network coding, so that constructions for these codes have achieved wide attention. Here, we improve over $150$ lower bounds by describing better constructions for subspace distance $4$.
**Keywords:** constant dimension codes, multilevel construction, Echelon–Ferrers construction, linkage, network coding
Introduction {#sec_intro}
============
Let $q$ be a prime power and ${\mathbb{F}}_q$ be the finite field with $q$ elements. For two integers $0\le k\le n$ we denote by ${\mathcal{G}}_q(n,k)$ the set of all $k$-dimensional subspaces in ${\mathbb{F}}_q^n$. The so-called subspace distance ${d_S}(U,W):=\dim(U)+\dim(W)-2\dim(U\cap W)=2k-2\dim(U\cap W)$ defines a metric on ${\mathcal{G}}_q(n,k)$. A subset ${\mathcal{C}}\subseteq{\mathcal{G}}_q(n,k)$ is called a *constant dimension code* ([`CDC`]{}), its elements are also called codewords, and ${d_S}({\mathcal{C}})=\min\!\left\{{d_S}(U,W)\,:\, U,W\in {\mathcal{C}},U\neq W\right\}$ is the corresponding *minimum (subspace) distance*. We also call ${\mathcal{C}}$ an $(n,M,d,k)_q$ [[`CDC`]{}]{} if ${\mathcal{C}}$ has cardinality $M$ and ${d_S}({\mathcal{C}})\ge d$. The maximum possible cardinality of an $(n,M,d,k)_q$ [[`CDC`]{}]{} is denoted by $A_q(n,d;k)$. Constant dimensions codes are e.g. applied in random linear network coding, see e.g. [@koetter-kschischang08], and the determination of bounds for $A_q(n,d;k)$ is one of the main problems. Here we improve more than $150$ of the previously best known constructions for [[`CDC`]{}]{}s. An online table for bounds for $A_q(n,d;k)$ can be found at [subspacecodes.uni-bayreuth.de](subspacecodes.uni-bayreuth.de), see the corresponding technical manual [@TableSubspacecodes]. The remaining part of this paper is structured as follows. In Section \[sec\_preliminaries\] we introduce the necessary preliminaries and describe constructions for [[`CDC`]{}]{}s from the literature. Our theoretical and algorithmical results are the topic of Section \[sec\_results\]. The resulting numerical improvements for lower bounds for $A_q(n,d;k)$ are listed in Appendix \[sec\_improved\_lower\_bounds\]. Extensive computational data about the details of the underlying constructions are given in Appendix \[sec\_skeleton\_codes\].
Preliminaries {#sec_preliminaries}
=============
Given a [[`CDC`]{}]{} ${\mathcal{C}}$ we first consider the question how to represent its codewords, i.e., $k$-dimensional subspaces $U\in{\mathcal{G}}_q(n,k)$. Starting from a generator matrix whose $k$ rows form a basis of $U$ the application of the Gaussian elimination algorithm gives a unique generator matrix in *reduced row echelon form* denoted by $E(U)$. In the other direction we write $U=\langle E(U)\rangle$. By $v(U)\in{\mathbb{F}}_2^n$ we denote the characteristic vector of the pivot columns in $E(U)$, which is also called *identifying vector*. The *Ferrers tableaux* $T(U)$ of $U$ arises from $E(U)$ by removing the zeroes from each row of $E(U)$ left to the pivots and afterwards removing all pivot columns. If we then replace all remaining entries by dots we obtain the *Ferrers diagram* ${\mathcal{F}}(U)$ of $U$ which only depends on the identifying vector $v(U)$. As an example we consider $$U=\left\langle\begin{pmatrix}
1&0&1&1&0&1&0&1&0&1\\
1&0&0&1&1&1&1&1&1&1\\
0&0&0&1&1&0&0&0&1&0\\
0&0&0&0&0&0&1&1&0&1
\end{pmatrix}\right\rangle\in{\mathcal{G}}_2(10,4),$$ where we have $$E(U)=
\begin{pmatrix}
1&0&0&0&0&1&0&0&0&0\\
0&0&1&0&1&0&0&1&1&1\\
0&0&0&1&1&0&0&0&1&0\\
0&0&0&0&0&0&1&1&0&1
\end{pmatrix},$$ $v(U)=1 0 1 1 0 0 1 0 0 0\in{\mathbb{F}}_2^{10}$, and $${\mathcal{F}}(U)=
\begin{array}{llllll}
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\
& \bullet & \bullet & \bullet & \bullet & \bullet \\
& \bullet & \bullet & \bullet & \bullet & \bullet \\
& & & \bullet & \bullet & \bullet
\end{array}.$$
The *Hamming distance* ${d_H}(u,w)=\#\left\{1\le i \le n\,:\, u_i\neq w_i\right\}$, for $u,w\in{\mathbb{F}}_q^n$, can be used to lower bound the subspace distance between two codewords $U,W\in{\mathcal{G}}_q(n,k)$:
([@etzion2009error Lemma 2])\
\[lemma\_dist\_subspace\_hamming\]
For $U,W\in{\mathcal{G}}_q(n,k)$ we have ${d_S}(U,W)\ge {d_H}(v(U),v(W))$.
If the identifying vectors of two codewords coincide, then we can utilize the *rank distance* ${d_R}(A,B):=\operatorname{rank}(A-B)$ for matrices $A,B\in{\mathbb{F}}_q^{m\times l}$:
([@silberstein2011large Corollary 3])\
\[lemma\_dist\_subspace\_rank\]
For $U,W\in{\mathcal{G}}_q(n,k)$ with $v(U)=v(W)$ we have ${d_S}(U,W)=2{d_R}(E(U),E(W))$.
Since ${d_R}$ is a metric, we call a subset $C\subseteq{\mathbb{F}}_q^{m\times l}$ of matrices a *rank-metric code*. If $C$ is a linear subspace of ${\mathbb{F}}_q^{m\times l}$ we call the code *linear*. Given a Ferrers diagram ${\mathcal{F}}$ with $m$ dots in the rightmost column and $l$ dots in the top row, we call a rank-metric code $C_{{\mathcal{F}}}$ a *Ferrers diagram rank-metric* ([`FDRM`]{}) code if for any codeword $M\in {\mathbb{F}}_q^{m\times l}$ of $C_{{\mathcal{F}}}$ all entries not in ${\mathcal{F}}$ are zero. By ${d_R}(C_{{\mathcal{F}}})$ we denote the minimum rank distance, i.e., the minimum of the rank distance between pairs of different codewords.
\[definition\_lifted\] ([@silberstein2015error])\
Let ${\mathcal{F}}$ be a Ferrers diagram and $C_{{\mathcal{F}}}\subseteq {\mathbb{F}}_q^{k\times(n-k)}$ be an [[`FDRM`]{}]{} code. The corresponding *lifted [[`FDRM`]{}]{} code* ${\mathcal{C}}_{{\mathcal{F}}}$ is given by $${\mathcal{C}}_{{\mathcal{F}}}=\left\{U\in{\mathcal{G}}_q(n,k)\,:\, {\mathcal{F}}(U)={\mathcal{F}}, T(U)\in C_{{\mathcal{F}}}\right\}.$$
We remark that the bijection between the codewords of the [[`CDC`]{}]{} ${\mathcal{C}}_{{\mathcal{F}}}$ and the [[`FDRM`]{}]{} code $C_{{\mathcal{F}}}$ generalizes the construction of lifted *maximum rank distance* ([`MRD`]{}) codes. An [[`MRD`]{}]{} code corresponds to the case of a Ferrers diagram ${\mathcal{F}}$ with $k$ dots in each column and $n-k$ dots in each row (more details below).
Directly from Lemma \[lemma\_dist\_subspace\_rank\] and Definition \[definition\_lifted\] we can conclude:
([@etzion2009error Lemma 4])\
\[lemma\_FDRM\_CDC\_equivalence\] Let $C_{{\mathcal{F}}}\subseteq {\mathbb{F}}_q^{k\times(n-k}$ be an [[`FDRM`]{}]{} code with minimum rank distance $\delta$, then the lifted [[`FDRM`]{}]{} code ${\mathcal{C}}_{{\mathcal{F}}}\subseteq {\mathcal{G}}_q(n,k)$ is an $(n,\#C_{{\mathcal{F}}},2\delta,k)_q$ [[`CDC`]{}]{}.
Let $v({\mathcal{F}})$ be the identifying vector of a given Ferrers diagram ${\mathcal{F}}$. In general, we denote by $A_q(n,d;k;v)$ the maximum cardinality $M$ of an $(n,M,d,k)_q$ [[`CDC`]{}]{} where all codewords have $v\in{\mathbb{F}}_2^n$ as identifying vector. We also speak of an $(n,M,d,k,v)_q$ [[`CDC`]{}]{}. With this (and Lemma \[lemma\_FDRM\_CDC\_equivalence\]) the upper bound for the cardinality of $C_{{\mathcal{F}}}$ from [@etzion2009error Theorem 1][^1] can be rewritten to:
\[thm\_upper\_bound\_ef\] $$A_q(n,d;k;v({\mathcal{F}}))\le q^{\min\!\left\{\nu_i\,:\,0\le i\le d/2-1\right\}},$$ where $\nu_i$ is the number of dots in ${\mathcal{F}}$, which are neither contained in the first $i$ rows nor contained in the last $\tfrac{d}{2}-1-i$ columns.
If we choose a minimum subspace distance of $d=6$, then we obtain $$A_2(10,6;4;1011001000)\le 2^8$$ due to $$\begin{array}{llllll}
{\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}\\
& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}\\
& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}\\
& & & {\color{blue}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}\end{array}
\quad
\begin{array}{llllll}
{\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}\\
& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{red}{\bullet}}\\
& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{red}{\bullet}}\\
& & & {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{red}{\bullet}}\end{array}
\quad
\begin{array}{llllll}
{\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}\\
& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}& {\color{red}{\bullet}}\\
& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}\\
& & & {\color{blue}{\bullet}}& {\color{blue}{\bullet}}& {\color{blue}{\bullet}}\end{array}
.$$
The Hamming weight of a vector $v\in{\mathbb{F}}_2^n$ is the Hamming distance ${d_H}(v,\mathbf{0})$ of $v$ to the zero vector.
([@etzion2009error Theorem 3])\
\[thm\_EF\] If ${\mathcal{S}}\subseteq{\mathbb{F}}_2^n$ is a set of binary vectors with Hamming weight $k$ that has minimum Hamming distance $d$ and ${\mathcal{C}}_v\subseteq{\mathcal{G}}_q(n,k)$ is an $(n,\star,d,k,v)_q$ [[`CDC`]{}]{} for each $v\in {\mathcal{S}}$, then ${\mathcal{C}}=\cup_{v\in{\mathcal{S}}} {\mathcal{C}}_v$ is an $(n,\star,d,k)_q$ [[`CDC`]{}]{} with cardinality $\sum_{v\in {\mathcal{S}}} \#{\mathcal{C}}_v$.
Choosing the ${\mathcal{C}}_v$ as lifted [[`FDRM`]{}]{} codes, the underlying construction is called *multilevel construction* in [@etzion2009error] and *Echelon-Ferrers construction* in some other papers. The binary code ${\mathcal{S}}\subseteq{\mathbb{F}}_2^n$ is called *skeleton code*. Using our notation, a given skeleton code ${\mathcal{S}}$ with minimum Hamming distance $\tfrac{d}{2}$ gives $$A_q(n,d;k)\ge \sum_{v\in {\mathcal{S}}} A_q(n,d;k;v).$$ The upper bound of Theorem \[thm\_upper\_bound\_ef\] is attained in many cases including $d\le 4$ and rectangular Ferrers diagrams. For other cases we refer e.g.. to [@antrobus2019maximal; @liu2019constructions] and the references mentioned therein. Indeed, for finite fields no strict improvement of Theorem \[thm\_upper\_bound\_ef\] is known so that it is conjectured that the upper bound can always be attained. If $2\le 2k\le n$ and ${\mathcal{F}}$ is the rectangular Ferrers diagrams with $k$ dots in each column and $n-k$ dots in each row, then a rank-metric code $C_{{\mathcal{F}}}\subseteq{\mathbb{F}}_q^{k\times(n-k)}$ attaining the maximum possible cardinality $q^{(n-k)(k-d/2+1)}$ for a given minimum subspace distance $d\le 2k$ is called *maximum rank distance* ([`MRD`]{}) code. Even linear [[`MRD`]{}]{} codes exist for all parameters, so that lifting gives the well-known lower bound $$A_q(n,d;k)\ge q^{(n-k)(k-d/2+1)}$$ (assuming $2k\le n$), which is at least half the optimal value for $d\ge 4$, see e.g. [@heinlein2017asymptotic Proposition 8].
Instead of starting with an [[`FDRM`]{}]{} code $C_{{\mathcal{F}}}$ and lift it to a [[`CDC`]{}]{} ${\mathcal{C}}_{{\mathcal{F}}}$ one can also start from an $(m,N,d,k)_q$ [[`CDC`]{}]{} ${\mathcal{C}}$ and an [[`MRD`]{}]{} code ${\mathcal{M}}\subseteq {\mathbb{F}}_q^{k\times(n-m)}$ with minimum rank distance $d/2$. With this we can construct a [[`CDC`]{}]{} $${\mathcal{C}}'=\left\{\langle E(U)|M\rangle\,:\,U\in{\mathcal{C}}, M\in{\mathcal{M}}\right\}\subseteq {\mathcal{G}}_q(n,k)$$ with ${d_S}({\mathcal{C}}')=d$ and $\#{\mathcal{C}}'=\#{\mathcal{C}}\cdot\#{\mathcal{M}}$, where $A|B$ denotes the concatenation of two matrices $A$ and $B$ with the same number of rows. This lifting variant was called *Construction D* in [@silberstein2015error Theorem 37], cf. [@gluesing9cyclic Theorem 5.1]. By construction, the identifying vectors of the codewords of ${\mathcal{C}}'$ contain their $k$ ones in the first $m$ positions. More generally, we denote by $${n_1 \choose k_1}\dots {n_l \choose k_l}$$ the set of binary vectors which contain exactly $k_i$ ones in positions $1+\sum_{j=1}^{i-1} n_j$ to $\sum_{j=1}^{i} n_j$ for all $1\le i\le l$. With this, we write $A_q\!\left(n,d;k;{n_1 \choose k_1}\dots {n_l \choose k_l}\right)$ for the maximum cardinality of an $(n,\star,d,k)_q$ [[`CDC`]{}]{} whose codewords all have identifying vectors in this set and state:
\[thm\_construction\_d\] For each $0\le \Delta<n$ we have $$A_q(n,d;k)\ge A_q\!\left(n,d;k;{{n-\Delta}\choose k},{\Delta\choose 0}\right)\ge q^{\Delta(k-d/2+1)}A_q(n-\Delta,d;k).$$
The special structure of the identifying vectors can be used to add further codewords. In our notation the *linkage construction* from [@gluesing10construction Theorem 2.3], [@silberstein2015error Corollary 39] can be written as $$A_q(n,d;k)\ge A_q\!\left(n,d;k;{{n-\Delta}\choose k},{\Delta\choose 0}\right)+A_q\!\left(n,d;k;{{n-\Delta}\choose 0},{\Delta\choose k}\right),$$ which was improved to $$\begin{aligned}
A_q(n,d;k)&\ge& A_q\!\left(n,d;k;{{n-\Delta}\choose k},{\Delta\choose 0}\right)\\
&&+A_q\!\left(n,d;k;{{n-\Delta-k+d/2}\choose 0},{\Delta+k+d/2\choose k}\right)\end{aligned}$$ in [@heinlein2017asymptotic Theorem 18, Corollary 4], taking Theorem \[thm\_construction\_d\] and $A_q(n,d;k;{n-m \choose 0},{m\choose k})=A_q(m,d;k)$ into account. By using the notation ${{n-n'}\choose {\le k-k'}},{{n'}\choose {\ge k'}}$ for the set of vectors in ${\mathbb{F}}_2^n$ with at most $k-k'$ ones in the first $n-n'$ positions and at least $k'$ ones in the last $n'$ positions we can denote by $A_q(n,d;k;{{n-n'}\choose {\le k-k'}},{{n'}\choose {\ge k'}})$ the maximum cardinality of an $(n,\star,d,k)_q$ [[`CDC`]{}]{} whose codewords have identifying vectors in this set, so that Lemma \[lemma\_dist\_subspace\_hamming\] gives:
\[lemma\_ef\_comb\_special\] For each $0\le \Delta<n$ we have $$\begin{aligned}
A_q(n,d;k)&\ge& A_q\!\left(n,d;k;{{n-\Delta}\choose k},{\Delta\choose 0}\right)\\
&& +A_q\!\left(n,d;k;{{n-\Delta}\choose {\le k-d/2}},{\Delta\choose {\ge d/2}}\right).
\end{aligned}$$
We remark that Lemma \[lemma\_ef\_comb\_special\] was implicitly contained in the proofs of many papers improving lower bounds for $A_q(n,d;k)$, see e.g. [@cossidente2019combining; @xu2018new] and the references cited therein. In [@kurz2019note] the quantity $A_q\!\left(n,d;k;{{n-\Delta}\choose {\le k-d/2}},{\Delta\choose {\ge d/2}}\right)$ was introduced as $B_q(n,\Delta,d;k)$. For the special case $\Delta=k$ a lower bound for $A_q\!\left(n,d;k;{{n-\Delta}\choose {\le k-d/2}},{\Delta\choose {\ge d/2}}\right)$ was constructed in [@xu2018new] via $$\left\{\langle M|I_k\rangle\,:\, M\in {\mathcal{M}},\operatorname{rank}(M)\le k-d/2\right\},$$ where $I_k$ denotes the $k\times k$ unit matrix and $\mathcal{M}\subseteq {\mathbb{F}}_q^{k\times(n-k)}$ is a rank metric code with ${d_R}({\mathcal{M}})\ge d/2$. By replacing $I_k$ by $E(U)$ for all codewords of a $(\Delta,\star,d,k)_q$ [[`CDC`]{}]{} we obtain yet another variant of the lifting idea. One of the most general versions can be found in [@cossidente2019combining Lemma 4.1].
Of course we can also utilize the multilevel/Echelon-Ferrers construction from Theorem \[thm\_EF\] to obtain lower bounds for $A_q\!\left(n,d;k;{{n-\Delta}\choose {\le k-d/2}},{\Delta\choose {\ge d/2}}\right)$ by restricting the skeleton code ${\mathcal{S}}$ to subsets of the set ${{n-\Delta}\choose {\le k-d/2}},{\Delta\choose {\ge d/2}}$ of identifying vectors. More generally, we can define the Hamming distance ${d_H}({\mathcal{S}},{\mathcal{S}}')$ between two sets ${\mathcal{S}},{\mathcal{S}}'\subseteq {\mathbb{F}}_2^n$ as $${d_H}({\mathcal{S}},{\mathcal{S}}')=\min\left\{{d_H}(v,v')\,:\,v\in{\mathcal{S}}, v'\in{\mathcal{S}}'\right\}$$ and slightly generalize Theorem \[thm\_EF\] to:
\[thm\_EF\_generalized\] Let ${\mathcal{S}}_i\subseteq{\mathbb{F}}_2^n$ and ${\mathcal{C}}_i$ be $(n,\star,d,k,{\mathcal{S}}_i)_q$ [[`CDC`]{}s]{} for all $1\le i\le n$ such that ${d_H}({\mathcal{S}}_i,{\mathcal{S}}_j)\ge d$ for all $1\le i<j\le l$. Then ${\mathcal{C}}=\cup_{1\le i\le l} {\mathcal{C}}_i$ is an $(n,\star,d,k)_q$ [[`CDC`]{}]{} with cardinality $\#{\mathcal{C}}=\sum_{i=1}^l \#{\mathcal{C}}_i$.
If we choose $n=n_1+n_2$, ${\mathcal{S}}_1=\left({n_1 \choose k},{n_2\choose 0}\right)\subseteq {\mathbb{F}}_2^n$, ${\mathcal{S}}_2=\left({n_1 \choose 0},{n_2\choose k}\right)\subseteq {\mathbb{F}}_2^n$, and ${\mathcal{S}}_3,\dots,{\mathcal{S}}_l\subseteq {n\choose k} \cap \left({n_1\choose \ge d/2},{n_2\choose \ge d/2}\right)\subseteq {\mathbb{F}}_2^n$ of cardinality $\#{\mathcal{S}}_j=1$ (for $3\le j\le l$) with ${d_H}({\mathcal{S}}_3,\dots,{\mathcal{S}}_l)\ge d$, then the conditions of Theorem \[thm\_EF\_generalized\] are satisfied. This is [@li2019construction Theorem 3.1].[^2] More examples where constant dimension codes with different sets of identifying vectors are combined can be found in [@heinlein2017coset].
Results {#sec_results}
=======
We want to apply Theorem \[thm\_EF\_generalized\] in order to obtain improved lower bounds for $A_q(n,d;k)$. In order to avoid to explicitly deal with the existence and construction of [[`FDRM`]{}]{} codes we restrict ourselves to subspace distance $d=4$, where the upper bound of Theorem \[thm\_upper\_bound\_ef\] can be always attained. As an introductory example we state:
\[prop\_13\_4\_5\] We have $A_2(13,4;5)\ge 4\,796\,825\,069$ and $A_q(13,4;5) \ge q^{32}+q^{28}+q^{26}+8q^{24}+q^{23}+3q^{22}+q^{21}+4q^{20}+4q^{19}
+5q^{18}+q^{17}+9q^{16}+8q^{15}+9q^{14}+6q^{13}+7q^{12}+5q^{11}+q^{10}+5q^9
+3q^8+q^7+3q^6+4q^5+3q^4+q^3+3q^2$.
We apply Theorem \[thm\_EF\] with skeleton codes $\mathcal{S}_{13,4,5}^1$ and $\mathcal{S}_{13,4,5}^2$, see Appendix \[sec\_skeleton\_codes\].
For $q\in\{2,3\}$ the previously best lower bounds were $A_2(13,4;5)\ge 4\,796\,417\,559$ and $A_3(13,4;5)\ge 1\,880\,918\,023\,783\,990$ [@he2019hierarchical], while our parametric lower bound yields $A_3(13,4;5)\ge 1\,880\,918\,252\,176\,932$. The only algorithmical challenge is to find good skeleton codes. Note that $\mathcal{S}_{13,4,5}^1$ gives $A_q(13,4;5)\ge q^{32}+q^{28}+q^{26}+8q^{24}+q^{23}+2q^{22}+3q^{21}+5q^{20}+3q^{19}+3q^{18}+3q^{17}+8q^{16}+8q^{15}+9q^{14}
+5q^{13}+8q^{12}+9q^{11}+q^{10}+7q^9+2q^8+2q^7+2q^6+q^5+3q^4+2q^3+3q^2+q^0$, i.e., the coefficient for $q^{22}$ is only $2$ instead of $3$, which pays off for $q=2$ since the coefficient for $q^{21}$ is $3$ instead of $1$ and the coefficient for $q^{20}$ is $5$ instead of $4$.
If the maximum sizes of the [[`FDRM`]{}]{} codes are known, as it is the case for subspace distance $d=4$, for given parameters $n$, $d$, $k$, and $q$ the problem of determining the best lower bound for $A_q(n,d;k)$ based on Theorem \[thm\_EF\] can be easily formulated as maximum weighted clique problem. To this end we denote by ${\mathcal{G}}_{n,d,k,q}=\left({\mathcal{V}}_{n,d,k,q},{\mathcal{E}}_{n,d,k,q}\right)$ the graph consisting of vertices corresponding to the binary vectors in ${\mathbb{F}}_2^n$ with Hamming weight $k$. W.l.o.g. we label the elements of ${\mathcal{V}}_{n,d,k,q}$ from $1$ to ${n\choose k}$ such that $w(1)\ge w(2)\ge \dots$, where $w(v)$ denotes the weight of $v$ that is given by the maximum possible cardinality of the corresponding [[`FDRM`]{}]{} code. For two different vertices $v,v'\in {\mathcal{V}}_{n,d,k,q}$ the edge $\{v,v'\}$ is in ${\mathcal{E}}_{n,d,k,q}$ iff the Hamming distance between $v$ and $v'$ is at least $d$. With this, the feasible skeleton codes are in bijection to the cliques of ${\mathcal{G}}_{n,d,k,q}$ and we are searching for the maximum weight cliques. Looping over all cliques or all inclusion maximal cliques of ${\mathcal{G}}_{n,d,k,q}$ becomes computationally intractable even for moderate sized parameters due to the quickly increasing number of vertices. Given an upper bound $ub$ for the maximum clique size in ${\mathcal{G}}_{n,d,k,q}$, see e.g. [@brouwer1990new] and <https://www.win.tue.nl/~aeb/codes/Andw.html>, we can compute an upper bound on the weight $w({\mathcal{S}}'):=\sum_{v\in{\mathcal{S}}'}w(v)$ of every clique that contains a subclique:
\[lemma\_ub\_clique\] Let ${\mathcal{S}},{\mathcal{S}}'$ be cliques in ${\mathcal{G}}_{n,d,k,q}$ with ${\mathcal{S}}\subseteq{\mathcal{S}}'$ and $\max\left\{w(v)\,:\, v\in {\mathcal{S}}\right\}<\min\left\{w(v)\,:\,w\in{\mathcal{S}}'\backslash {\mathcal{S}}\right\}$, where $w(i)\ge w(j)\ge 0$ for all $i\le j$. Then, we have $w({\mathcal{S}}')\le \Omega$, where $\Omega$ is the value computed by Algorithm \[alg:clique\_ub\] applied to ${\mathcal{G}}_{n,d,k,q}$, $w$, and ${\mathcal{S}}$.
By $cand$ we denote the set of vertices $m+1\le v\le \#{\mathcal{V}}$ with $\Big\{\left\{x,v\right\}\,:\,x\in {\mathcal{S}}\Big\}\subseteq {\mathcal{E}}$, where $m$, ${\mathcal{V}}$, and ${\mathcal{E}}$ are as in Algorithm \[alg:clique\_ub\]. Since ${\mathcal{S}}\subseteq{\mathcal{S}}'$ and ${\mathcal{S}}'$ is a clique, we have $\Big\{\left\{x,v\right\}\,:\,x\in {\mathcal{S}}\Big\}\subseteq {\mathcal{E}}$ for all vertices $v\in{\mathcal{S}}'\backslash{\mathcal{S}}$. Due to our assumption $\max\left\{w(v)\,:\, v\in {\mathcal{S}}\right\}<\min\left\{w(v)\,:\,w\in{\mathcal{S}}'\backslash {\mathcal{S}}\right\}$ we also have $m+1\le v\le \#{\mathcal{V}}$ for all $v\in{\mathcal{S}}'\backslash{\mathcal{S}}$. Thus, we have ${\mathcal{S}}'\subseteq{\mathcal{S}}\cup cand$. If $\#{\mathcal{S}}+\# cand\le ub$, then $\hat{{\mathcal{S}}}={\mathcal{S}}\cup cand$ and $w({\mathcal{S}}')\le w({\mathcal{S}}\cup cand)=w(\hat{{\mathcal{S}}})=\Omega$. Otherwise we have $\#{\mathcal{S}}'\le ub=\#\hat{{\mathcal{S}}}$, set $s=\# \left({\mathcal{S}}'\backslash{\mathcal{S}}\right)$, , $\hat{s}=
\# \left(\hat{{\mathcal{S}}}\backslash{\mathcal{S}}\right)$, and write ${\mathcal{S}}'={\mathcal{S}}\cup\left\{a_1,\dots,a_s\right\}$, $\hat{{\mathcal{S}}}={\mathcal{S}}\cup\left\{b_1,\dots,b_{\hat{s}}\right\}$, where we assume that the sequences $a_i$ and $b_i$ are increasing. Due to our assumption on the weight function $w$ and the choice of $\hat{{\mathcal{S}}}\backslash{\mathcal{S}}$ as those vertices with the smallest elements in $cand$ we have $w(a_i)\le w(b_i)$ for all $1\le i\le s$, so that $$w({\mathcal{S}}')=w({\mathcal{S}})+\sum_{i=1}^s w(a_i)\le w({\mathcal{S}})+\sum_{i=1}^s w(b_i)\le w({\mathcal{S}})+\sum_{i=1}^{\hat{s}} w(b_i)=w(\hat{{\mathcal{S}}})=\Omega.$$
$\Omega\longleftarrow w({\mathcal{S}})$ $\hat{{\mathcal{S}}}\longleftarrow {\mathcal{S}}$ $m\longleftarrow 0$
We can use Algorithm \[alg:clique\_ub\] to determine a maximum weight clique of ${\mathcal{G}}_{n,d,k,q}$ without explicitly traversing all cliques:
\[prop\_max\_weight\_clique\_exhaustive\] Algorithm \[alg:maximum\_weight\_clique\] determines a maximum weight clique ${\mathcal{U}}$ in ${\mathcal{G}}_{n,d,k,q}$.
Let ${\mathcal{U}}'$ be a maximum weight clique in ${\mathcal{G}}_{n,d,k,q}$ and ${\mathcal{U}}'_i$ be the subset of the smallest $i$ elements of ${\mathcal{U}}$ for $1\le i\le \#{\mathcal{U}}'$. Now let $m$ be the largest index such that `Dive` is called with ${\mathcal{S}}={\mathcal{U}}'_m$. Since `Dive` is initially called with ${\mathcal{S}}=\emptyset={\mathcal{U}}'_0$, $0\le m\le \#{\mathcal{U}}'$ is well defined. If $m=\#{\mathcal{U}}'$ then either ${\mathcal{U}}$ is set to ${\mathcal{U}}'$ or we already have $w({\mathcal{U}})\ge w({\mathcal{U}}')$. Note that every replacement of ${\mathcal{U}}$ strictly increases the value of $w({\mathcal{U}})$. In the remaining cases we assume $m<\#{\mathcal{U}}$ and set $v'={\mathcal{U}}'_{m+1}\backslash{\mathcal{U}}'_m$. By construction we have $l\le v'$, so that the for loop of Algorithm \[alg:dive\] attains $v=v'$. Since ${\mathcal{U}}'_{m+1}$ is a clique we have $\Big\{\left\{x,v\right\}\,:\,x\in {\mathcal{S}}\Big\}\subseteq {\mathcal{E}}$ and since $\texttt{Dive}$ is not called with ${\mathcal{U}}'_{m+1}={\mathcal{S}}\cup\{v\}$ we have `UB`$({\mathcal{G}},{\mathcal{S}}\cup\{v\},ub)\le w({\mathcal{U}})$, so that Lemma \[lemma\_ub\_clique\] gives $w({\mathcal{U}})\ge w({\mathcal{U}}')$. Since every replacement of ${\mathcal{U}}$ strictly increases the value of $w({\mathcal{U}})$ the proposed statement follows.
${\mathcal{U}}\longleftarrow \emptyset$ ${\mathcal{S}}\longleftarrow \emptyset$ (${\mathcal{G}}$, $w$, ${\mathcal{S}}$, $ub$)
Let $1\le l\le \# V$ be the smallest index such that the elements in ${\mathcal{S}}$ have strictly smaller indices; return if no such index exists
As an application of Proposition \[prop\_max\_weight\_clique\_exhaustive\] we remark that for $(n,d,k)=(13,4,5)$ and $q\in\{2,\dots,9\}$ the lower bound stated in Proposition \[prop\_13\_4\_5\] is indeed the optimal multilevel/Echelon-Ferrers construction. So, based on Lemma \[lemma\_ub\_clique\], an exhaustive search is indeed possible, while in [@he2019hierarchical] only a heuristic was used. However, we also have to use heuristics for larger parameters and replace Algorithm \[alg:maximum\_weight\_clique\] by Algorithm \[alg:maximum\_weight\_clique\_heuristic\]. (The enlargement of ${\mathcal{L}}'$ has to be implemented in a modified version of `Dive` to be technically correct.)
${\mathcal{U}}\longleftarrow \emptyset$ ${\mathcal{L}}' \longleftarrow \emptyset$
We can iteratively apply Algorithm \[alg:maximum\_weight\_clique\_heuristic\] in order to heuristically find a clique of large weight in ${\mathcal{G}}_{n,d,k,q}$. Starting from ${\mathcal{L}}=\emptyset$ we can use the determined list ${\mathcal{L}}'$ in the next round increasing $ub$ incrementally, say by $10$ in each iteration, until no further improvement is found. The advantage of that approach is that using the parameters $\Delta_1$ and $\Delta_2$ we can control the extend to which we want to perform an exhaustive search. The iterative process also partially prevents from stalling in local neighborhoods of partial cliques that are very diverse to a global optimum. As an example for a result obtained by the application of Algorithm \[alg:maximum\_weight\_clique\_heuristic\] we state:
\[prop\_17\_4\_6\] $A_q(17,4;6) \ge q^{55}+q^{51}+q^{49}+8q^{47}+3q^{45}+3q^{44}+5q^{43}+q^{42}+5q^{41}+9q^{40}
+22q^{39}+7q^{38}+11q^{37}+13q^{36}+19q^{35}+3q^{34}+17q^{33}+15q^{32}+69q^{31}+20q^{30}
+49q^{29}+22q^{28}+33q^{27}+15q^{26}+23q^{25}+20q^{24}+38q^{23}+17q^{22}+29q^{21}+24q^{20}
+40q^{19}+19q^{18}+20q^{17}+15q^{16}+28q^{15}+15q^{14}+13q^{13}+8q^{12}+7q^{11}+5q^{10}
+3q^9+10q^8+q^7+2q^6+q^5+2q^4+q^3+q^2+q^1+q^0$.
We apply Theorem \[thm\_EF\] with skeleton codes $\mathcal{S}_{17,4,6}$, see Appendix \[sec\_skeleton\_codes\].
In Table \[table\_improved\_ef\_parameters\] we list the cases where Algorithm \[alg:maximum\_weight\_clique\] or Algorithm \[alg:maximum\_weight\_clique\_heuristic\] yields a skeleton code such that Theorem \[thm\_EF\] gives a strictly better constructive lower bound for $A_q(n,4;k)$. In all cases we obtain improvements for all $q\in\{2,\dots,9\}$, see Table \[table\_improved\_ef\_numerical\] in Appendix \[sec\_improved\_lower\_bounds\]. The utilized skeleton codes are listed in Appendix \[sec\_skeleton\_codes\]. Some of them consist of over 1000 identifying vectors. Whenever we state two skeleton codes, for given parameters $n$ and $k$, the first code gives the lower bound for $A_2(n,4;k)$ and the second code gives the lower bound for general field sizes $q$, which is larger for $q\ge 3$, see Proposition \[prop\_13\_4\_5\] for an example. Whenever we only list one skeleton code, it yields the utilized lower bound for all field sizes $q\ge 2$, see Proposition \[prop\_17\_4\_6\] for an example. We remark that for $d=4$ all explicitly stated numerical results of [@he2019hierarchical; @liu2019parallel] have been strictly improved. (The other results of [@he2019hierarchical] depend on the unproven assumption, nevertheless the author states otherwise, that the upper bound of Theorem \[thm\_upper\_bound\_ef\] is attained for $d=6$.)
$k$ $n$
----- ----------------
5 13, 14
6 14, 15, 16, 17
7 16, 17, 18, 19
8 18, 19
9 19
: Parameters where improved codes for $A_q(n,4;k)$ have been found using Algorithm \[alg:maximum\_weight\_clique\] or Algorithm \[alg:maximum\_weight\_clique\_heuristic\].[]{data-label="table_improved_ef_parameters"}
Another approach, to obtain improved lower bounds, is based on Theorem \[thm\_construction\_d\] and Lemma \[lemma\_ef\_comb\_special\]:
\[prop\_11\_4\_3\] $A_q(11,4;3) \ge q^8\cdot A_q(7,4;3)+q^4 + q^3 + 2q^2 + q + q^0$.
We apply Theorem \[thm\_construction\_d\] with $\Delta=4$ to deduce $A_q\!\left(11,4;3;{7\choose 3},{4\choose 0}\right)\ge q^{8}\cdot A_q(7,4;3)$ and Theorem \[thm\_EF\] with the skeleton code ${\mathcal{S}}_{11,7,4,3}$, see Appendix \[sec\_skeleton\_codes\], to deduce\
$A_q\!\left(11,4;3;{7\choose \le 1},{4\choose \ge 2}\right)\ge
q^4 + q^3 + 2q^2 + q + q^0$. With this, the stated lower bound follows from Lemma \[lemma\_ef\_comb\_special\].
For $q\ge 3$ Proposition \[prop\_11\_4\_3\] yields a strictly larger lower bound than the previous record from [@heinlein2019generalized]. Using $\Delta=3$ and skeleton code $${\mathcal{S}}_{10,7,4,3}=\left\{ 0000100110, 0000010101, 0000001011 \right\}\,\hat{=}\,\{38, 21, 11\},$$ where the integers give the binary identifying vectors reading them in base $2$ representation, gives $A_q(10,4;3) \ge q^6\cdot A_q(7,4;3)+q^2+q+1$, which equals the lower bound from [@heinlein2019generalized]. For the naming of the skeleton codes ${\mathcal{S}}_{n,m,d,k}$ for $A_q(n,d;k)$ we use $m=n-\Delta$ as an abbreviation.
$k$ $n$
----- ----------------------------------------------------
3 11(7), 12(7), 13(7), 14(11,7), 15(11,7), 16(13,7),
17(13,7), 18(13,7), 19(13,7)
4 13(8), 14(8), 15(8), 17(12), 18(12), 19(12)
: Parameters where improved codes for $A_q(n,4;k)$ have been found using Theorem \[thm\_construction\_d\] and Lemma \[lemma\_ef\_comb\_special\].[]{data-label="table_improved_ef_linkage"}
In Table \[table\_improved\_ef\_linkage\] we list the parameters $n$ and $k$ where the approach based on Theorem \[thm\_construction\_d\] and Lemma \[lemma\_ef\_comb\_special\] yields strict improvements. The corresponding value of $m$ is stated in brackets, where the last is for general field sizes $q$ and the last but one for $q=2$. In some cases, when $n$ is rather small, better lower bounds for $A_2(n,4;k)$ have been found using integer linear programming and prescribed automorphisms, see e.g. [@kohnert2008construction] for an introductory paper and [@TableSubspacecodes] for the precise reference per specific instance. The numerical improvements are listed in Table \[table\_improved\_ef\_linkage\_numerical\] in Appendix \[sec\_improved\_lower\_bounds\] and the corresponding skeleton codes in Appendix \[sec\_skeleton\_codes\]. We remark that for $k=4$ and $n\in\{12,16\}$ the obtained codes are inferior compared to those from [@cossidente2019combining]. We remark that for $d=4$ all numerical results of [@li2019construction] have been strictly improved.
In order to find the skeleton codes Algorithm \[alg:maximum\_weight\_clique\] and Algorithm \[alg:maximum\_weight\_clique\_heuristic\] are applied by modifying the input graphs ${\mathcal{G}}_{n,m,d,k,q}$, where $m=n-\Delta$, accordingly. Since Lemma \[lemma\_ub\_clique\] and Algorithm \[alg:clique\_ub\] significantly speed up the solution process by cutting parts of the search tree as early as possible, we also want to use good upper bounds on the maximum clique size in ${\mathcal{G}}_{n,m,d,k,q}$. While of cause the maximum clique size in ${\mathcal{G}}_{n,d,k,q}$ is an upper bound this can usually be improved.
\[prop\_ILP\_ub\] Let $1\le k\le m\le n$ and $2\le d/2\le k$ be integers. Then, the maximum cardinality of a set ${\mathcal{S}}\subseteq{\mathbb{F}}_2^n$ of binary vectors with Hamming weight $k$ that have at most $k-d/2$ ones in their first $m$ coordinates with minimum Hamming distance $d$ is upper bounded by $\sum_{j\in J} c_j$, where $J=\left\{j\in\mathbb{N}\,:\, j\ge k-n+m,j\le k-d/2\right\}$ and the $c_j$ are integers satisfying the constraints $$\sum_{j\in J} {j\choose i}\cdot {{k-j}\choose{k-d/2+1-i}}c_j\le {m\choose i}{{n-m}\choose{k-d/2+1-i}}$$ for all $0\le i\le k-d/2+1$. Moreover, we have $c_0\le A_1(n-m,d;k)$, which denotes the maximum cardinality of a subset of binary vectors in ${\mathbb{F}}_2^{n-m}$ with Hamming weight $k$ and minimum Hamming distance $d$.
Given ${\mathcal{S}}$ we let $c_j$ be the number of elements in ${\mathcal{S}}$ that have exactly $j$ ones in their first $m$ coordinates. It can be easily checked that $j\in J$ implies $0\le j\le m$, $0\le k-j\le n-m$, and $j\le k-d/2$, i.e., the counts $c_j$ are well defined. Since ${\mathcal{S}}$ has minimum Hamming distance $d$ every subset $I\subseteq{\mathbb{F}}_2^n$ of cardinality $k-d/2+1$ is contained in the support of at most one element in ${\mathcal{S}}$. For a given integer $0\le i\le k-d/2+1$ we consider all those sets $I$ containing exactly $i$ elements in $\{1,\dots,m\}$. There are exactly ${m\choose i}{{n-m}\choose{k-d/2+1-i}}$ of those sets and every element of ${\mathcal{S}}$ with $j$ ones in the first $m$ coordinates has a support that contains exactly ${j\choose i}\cdot {{k-j}\choose{k-d/2+1-i}}$ of those sets. Thus, the stated inequalities are valid. The additional upper bound for $c_0$ follows directly from our choice of the count $c_0$.
Given fixed parameters the integer linear program of Proposition \[prop\_ILP\_ub\] can be solved easily. As an example we will show that also parametric upper bounds can be concluded. To this end, let $k=4$, $d=4$, and $n-m\ge 4$, so that $J=\{0,1,2\}$. For $i\in\{0,1,2\}$ the constraints of Proposition \[prop\_ILP\_ub\] read $$\begin{aligned}
4c_0+c_1 &\le& {m\choose 0}{{n-m}\choose 3}={{n-m}\choose 3},\\
3c_1+2c_2&\le& {m\choose 1}{{n-m}\choose 2}=\frac{m(n-m)(n-m-1)}{2},\\
2c_2&\le& {m\choose 2}{{n-m}\choose 1}=\frac{m(m-1)(n-m)}{2},\end{aligned}$$ noting that that the constraint for $i=3=k-d/2+1$ is trivially satisfied. Since $c_1$ and $c_2$ are non-negative integers, dividing the second constraint by two gives $$c_1+c_2\le \left\lfloor\frac{m(n-m)(n-m-1)}{4}\right\rfloor,$$ so that $c_0\le A_1(n-m,4;4)$ gives $$c_0+c_1+c_2\le \left\lfloor\frac{m(n-m)(n-m-1)}{4}\right\rfloor +A_1(n-m,4;4).$$ For $(n,m)=(13,8)$ we obtain an upper bound of $40+1=41$, which indeed is attained by ${\mathcal{S}}_{13,8,4,4}$, see Appendix \[sec\_skeleton\_codes\], while $A_1(13,4;4)=65$. Similarly, for $(n,m)=(14,8)$ we obtain an upper bound of $60+3=63$, which indeed is attained by ${\mathcal{S}}_{14,8,4,4}$, while $A_1(14,4;4)=91$ is much larger.
We remark that Proposition \[prop\_ILP\_ub\] mimics [@kurz2019note Lemma 4.1], which proves an upper bound for $A_q\!\left(n,d;k;{{n-\Delta}\choose {\le k-d/2}},{\Delta\choose {\ge d/2}}\right)$. Possibly the quantity $A_1(n,m,d;k)$ upper bounded in Proposition \[prop\_ILP\_ub\], generalizing $A_1(n,d;k)$, is interesting on its own and similar techniques as for the (partial) determination of $A_1(n,d;k)$ can be applied. To this end, we remark that the upper bound of Proposition \[prop\_ILP\_ub\] can surely be improved in most cases but was sufficiently good for our purpose.
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Improved lower bounds {#sec_improved_lower_bounds}
=====================
In Table \[table\_improved\_ef\_numerical\] we compare the numerical improvements for $A_q(n,4;k)$ based on Theorem \[thm\_EF\]. The previously best known lower bounds are mostly from [@he2019hierarchical]. However, the lower bounds for $A_q(19,4;7)$ from [@he2019hierarchical] are flawed, i.e., some magnitudes too small.
$A_q(n,4;k)$ New Old
--------------- ------------------------------------------- -----------------------------------------------------------------
$A_2(13,4;5)$ 4796825069 4796417559 [@he2019hierarchical]
$A_3(13,4;5)$ 1880918252176932 1880918023783990 [@he2019hierarchical]
$A_4(13,4;5)$ 18525690519076963184 18525690479132333173 [@he2019hierarchical]
$A_5(13,4;5)$ 23322304251254415373950 23322304248923865096456 [@he2019hierarchical]
$A_7(13,4;5)$ 1104898620940893642387898640 1104898620939789578683671514 [@he2019hierarchical]
$A_8(13,4;5)$ 79247846163928378274466378432 79247846163915655208442806985 [@he2019hierarchical]
$A_9(13,4;5)$ 3434214279120463268840947142394 3434214279120353599762054717228 [@he2019hierarchical]
$A_2(14,4;5)$ 76749681496 76745404672 [@he2019hierarchical]
$A_3(14,4;5)$ 152354381482802889 152354354408240436 [@he2019hierarchical]
$A_4(14,4;5)$ 4742576773448941977744 4742576757171205745408 [@he2019hierarchical]
$A_5(14,4;5)$ 14576440157067948253027775 14576440154794852120820500 [@he2019hierarchical]
$A_7(14,4;5)$ 2652861588879102858398037208053 2652861588875282767080909163052 [@he2019hierarchical]
$A_8(14,4;5)$ 324599177887450844354761706030144 324599177887378338095324360943616 [@he2019hierarchical]
$A_9(14,4;5)$ 22531879885309361371329789150871659 22531879885308389971852673089028988 [@he2019hierarchical]
$A_2(13,4;6)$ 38327432465 38325131657 [@he2019hierarchical]
$A_3(13,4;6)$ 50782269101589019 50782269101569336 [@he2019hierarchical]
$A_4(13,4;6)$ 1185639430145591548865 1185639430145591024577 [@he2019hierarchical]
$A_5(13,4;6)$ 2915286427121720403833501 2915286427121720397974126 [@he2019hierarchical]
$A_7(13,4;6)$ 378980216844611802379905892367 378980216844611802379704124332 [@he2019hierarchical]
$A_8(13,4;6)$ 40574896910456482568428114359809 40574896910456482568427309053441 [@he2019hierarchical]
$A_9(13,4;6)$ 2503542202545027727509322118254705 2503542202545027727509319406311282 [@he2019hierarchical]
$A_2(14,4;6)$ 1227267234053 1227203232293 [@he2019hierarchical]
$A_3(14,4;6)$ 12340234566815274820 12340234566810426241 [@he2019hierarchical]
$A_4(14,4;6)$ 1214095649227435851637265 1214095649227435312865809 [@he2019hierarchical]
$A_5(14,4;6)$ 9110270832108553596766282526 9110270832108553578429954401 [@he2019hierarchical]
$A_7(14,4;6)$ 6369520523839151727825310046433656 6369520523839151727821917674342793 [@he2019hierarchical]
$A_8(14,4;6)$ 1329558223045414729174436187566776385 1329558223045414729174409793499570241 [@he2019hierarchical]
$A_9(14,4;6)$ 147831663555770209444761899682581639650 147831663555770209444761739522908440329 [@he2019hierarchical]
$A_2(15,4;6)$ 39273527139056 39267675031563 [@he2019hierarchical]
$A_3(15,4;6)$ 2998677038028861083854 2998676636295383433055 [@he2019hierarchical]
$A_4(15,4;6)$ 1243233944887303700963929029 1243233943362040432057180581 [@he2019hierarchical]
$A_5(15,4;6)$ 28469596350369158409165204552256 28469596349440811610995019309681 [@he2019hierarchical]
$A_7(15,4;6)$ 107052531444164864809019626364096185842 107052531444149851093995761837649623043 [@he2019hierarchical]
$A_8(15,4;6)$ 43566963852752158504511827652740340200969 43566963852751451455533741586485379191945 [@he2019hierarchical]
$A_9(15,4;6)$ 87293119013046753015079265888857783791854 87293119013046541123889721019479429050233
44 81 [@he2019hierarchical]
$A_2(16,4;6)$ 1256765678235469 1256703351587805 [@he2019hierarchical]
$A_3(16,4;6)$ 728678523485248028210242 728678523483522880513165 [@he2019hierarchical]
$A_4(16,4;6)$ 1273071559584675915665964748625 1273071559584674249524907514705 [@he2019hierarchical]
$A_5(16,4;6)$ 88967488594921579438014390756229526 88967488594921579094529973785226401 [@he2019hierarchical]
$A_7(16,4;6)$ 17992318959820794052190487867406638758853 17992318959820794052179837126816751848892
38 05 [@he2019hierarchical]
$A_8(16,4;6)$ 14276022715269827610082997266445121659360 14276022715269827610082737902847157431165
67137 87585 [@he2019hierarchical]
: Improvements based on Theorem \[thm\_EF\].[]{data-label="table_improved_ef_numerical"}
$A_q(n,4;k)$ New Old
--------------- ------------------------------------------- -----------------------------------------------------------------
$A_9(16,4;6)$ 51545713846013977302875956389540675158560 51545713846013977302875913003333046529357
7978830 7720745 [@he2019hierarchical]
$A_2(17,4;6)$ 40214593296350543 40210734642430233 [@he2019hierarchical]
$A_3(17,4;6)$ 177068881245303116902339546 177068857538981556600415147 [@he2019hierarchical]
$A_4(17,4;6)$ 1303625277015154268047530588005973 1303625275416014562978042328889121 [@he2019hierarchical]
$A_5(17,4;6)$ 278023401859130594858143611346347770156 278023401850065051300001033963426380051 [@he2019hierarchical]
$A_7(17,4;6)$ 30239690475770808604751881256801280488268 30239690475766567624866522189453324986367
311358 024243 [@he2019hierarchical]
$A_8(17,4;6)$ 46779671233396171116050537713696416375261 46779671233395411929881098933346105696388
813580361 708610177 [@he2019hierarchical]
$A_9(17,4;6)$ 30437228568932793457735943799142491471855 30437228568932719575938776421624397010040
909627410848 042569507531 [@he2019hierarchical]
$A_2(15,4;7)$ 313939996903443 313923840120169 [@he2019hierarchical]
$A_3(15,4;7)$ 80962390735680572668348 80962387333738514962426 [@he2019hierarchical]
$A_4(15,4;7)$ 79566863776146089092873059065 79566863724904828874349525569 [@he2019hierarchical]
$A_5(15,4;7)$ 3558699030838267109468908231148586 3558699030750375431966367668488876 [@he2019hierarchical]
$A_7(15,4;7)$ 36719018111675892766467253005230234472800 36719018111669485366326051726236511270134 [@he2019hierarchical]
$A_8(15,4;7)$ 22306285465490553868002054600166649426159 22306285465490013824377105083555006005322
313 241 [@he2019hierarchical]
$A_9(15,4;7)$ 63636683737167279191968272796410112452114 63636683737167010339661770016731801211973
39310 68344 [@he2019hierarchical]
$A_2(16,4;7)$ 20093092605969267 20090530823175168 [@TableSubspacecodes]
$A_3(16,4;7)$ 59021599901810630842384564 59021591907098096238648717 [@he2019hierarchical]
$A_4(16,4;7)$ 325905875863710195123556939265157 325905875503966895183219736444928 [@he2019hierarchical]
$A_5(16,4;7)$ 55604672371466910516074031340231456506 55604672369921077974140644073486328125 [@he2019hierarchical]
$A_7(16,4;7)$ 43199557618314831473921116433704053011403 43199557618309916202868392749119713436798
04392 04101 [@he2019hierarchical]
$A_8(16,4;7)$ 58474588970678832897891612620850698965996 58474588970678073098225984456969906775715
13506569 97238272 [@he2019hierarchical]
$A_9(16,4;7)$ 33819142841966544686367447197905513991858 33819142841966479587063098904956143968989
06891914222 57540722053 [@he2019hierarchical]
$A_2(17,4;7)$ 1285973764408635208 1285780755925958656 [@he2019hierarchical]
$A_3(17,4;7)$ 43026746493711590523056074275 43026740586030433477849264332 [@he2019hierarchical]
$A_4(17,4;7)$ 1334910467554985714047715561596194368 1334910466075153545917778842397704192 [@he2019hierarchical]
$A_5(17,4;7)$ 86882300580430813669650616907760767647637 86882300578011697800830006599426269531250
5 0 [@he2019hierarchical]
$A_7(17,4;7)$ 50823847542371226941060627330811586531967 50823847542365442865880653737635543921067
5271592683 3571016516 [@he2019hierarchical]
$A_8(17,4;7)$ 15328762651129632211002306082390007789441 15328762651129433020259724863226574216922
94589626442240 82703427141632 [@he2019hierarchical]
$A_9(17,4;7)$ 17972879091077542502798423597933679533502 17972879091077507906314355839720905589861
14940243153381903 92090028734124332 [@he2019hierarchical]
$A_2(18,4;7)$ 82302571633443282819 82291970549255555624 [@TableSubspacecodes]
$A_3(18,4;7)$ 31366498204356725852831211333673 31366481390109307710095048570592 [@he2019hierarchical]
$A_4(18,4;7)$ 5467793275108157387543615361694729718789 5467793218060404819993105682513015078912 [@he2019hierarchical]
$A_5(18,4;7)$ 13575359465692365684898552467453540627598 13575359458996180137997725978493690490722
430631 656250 [@he2019hierarchical]
$A_7(18,4;7)$ 59793748395124324914384236296038210886457 59793748394835040265326746482178302037689
090991018946109 516829976915564 [@he2019hierarchical]
$A_8(18,4;7)$ 40183431564177263067797197215986973010693 40183431564146475997690442413946479412809
5844085497257345033 5421048885275000832 [@he2019hierarchical]
$A_9(18,4;7)$ 95515248370413402653106548360544977074290 95515248370399043860359211584572109956785
8421375268890568749019 5543645243756034671222 [@he2019hierarchical]
: Improvements based on Theorem \[thm\_EF\] cont.
$A_q(n,4;k)$ New Old
--------------- ------------------------------------------- --------------------------------------------------------------
$A_2(19,4;7)$ 5267367924445148864092 5058097205000347197549 [@silberstein2015error]
$A_3(19,4;7)$ 22866177190621892757679222318915925 22813524065165704375162588681706875 [@silberstein2015error]
$A_4(19,4;7)$ 22396081254840251014738769405257047429902 22388542052518188728938813857333763516548
630 177 [@silberstein2015error]
$A_5(19,4;7)$ 21211499165144290480179096294676880646602 21209813680090451272236350632742053588039
6993016307 4984750151 [@silberstein2015error]
$A_7(19,4;7)$ 70346747049379816915369395320383414712645 70346075648600428027267630601996366936175
47397843085217471817 78982540335678343419 [@silberstein2015error]
$A_8(19,4;7)$ 10533845483959684448674419012790949807456 10533801523606741064564183358274781734572
1619935428942997712801930 1344118067187280665018945 [@silberstein2015error]
$A_9(19,4;7)$ 50760719109220869118965528769591000765261 50760616504319115174434295841792410847448
3420811027477177039803795175 9256852919544839333945104727 [@silberstein2015error]
$A_2(18,4;8)$ 1316667538397101428149 1264601087568682942805 [@silberstein2015error]
$A_3(18,4;8)$ 2540681546251523549771895302399518 2534836537121138399731153735006570 [@silberstein2015error]
$A_4(18,4;8)$ 13997550036505113139933121791886674834127 13992838834950406313750769467311428427492
53 01 [@silberstein2015error]
$A_5(18,4;8)$ 84845996396175179770599196313863292814169 84839254734261831517960672499611973762512
28756276 21110026 [@silberstein2015error]
$A_7(18,4;8)$ 14356478989082867395087999913040607219209 14356341969123295746286010290803218889077
1727223436678152394 1110979148690800890 [@silberstein2015error]
$A_8(18,4;8)$ 16459133568567622026207934780783177598554 16459064880639275271231203508694685723605
08541359722889311735873 96441738049974310277185 [@silberstein2015error]
$A_9(18,4;8)$ 62667554455729492107715380469110602507032 62667427783112758078838097157679059423642
42571284112409913460035888 52612630126994566337005626 [@silberstein2015error]
$A_2(19,4;8)$ 168534060204346081643054 161868939208791416732918[@silberstein2015error]
$A_3(19,4;8)$ 5556470543990117093920796733024932097 5543687506683929680212033218489009029[@silberstein2015error]
$A_4(19,4;8)$ 22933585979825739912419408751656551857477 22925867147182745704449260695243044338837
750740 225537[@silberstein2015error]
$A_5(19,4;8)$ 66285934684512179124945660759025164232422 66280667761142055873406775390321854501962
7507808667905 6771492187626[@silberstein2015error]
$A_7(19,4;8)$ 11823177776106271896222089221602557589481 11823064934277706348783619772918955293567
1105121482204632859666129 2313049101078765671318222[@silberstein2015error]
$A_8(19,4;8)$ 34517304881588725672548617783010034714431 34517160832562417413613060900666077554631
24750791181363571603492943432 69548983834979971097956450817[@silberstein2015error]
$A_9(19,4;8)$ 29973697026756603314019362623281023278094 29973636439636704539558217472416705317244
694159378234061636310178009770385 006954378905881075605724274378842[@silberstein2015error]
$A_2(19,4;9)$ 1348002146261417447406857 1289520797394170812563456 [@etzion2016optimal]
$A_3(19,4;9)$ 150024595081884393007012600338236544880 149656439781495352647490043484854812672 [@etzion2016optimal]
$A_4(19,4;9)$ 14677494853604982256349239711358766939534 14672330163008228150280295382725315655876
33631137 80198656 [@etzion2016optimal]
$A_5(19,4;9)$ 82857418316609426935359109104035114463034 82850622250904212594169271527415901448337
035070083878426 330052257546240 [@etzion2016optimal]
$A_7(19,4;9)$ 40553499771898134480982178003920834382400 40553105686976525598537903185027802877404
027150097709434199076454452 377818341526587313754210304 [@etzion2016optimal]
$A_8(19,4;9)$ 17672860099364199843916651708091969078559 17672785292630676941870781019917737428555
21985408931899768042741180322945 76219388246358086384262766919680 [@etzion2016optimal]
$A_9(19,4;9)$ 21850825132503488679494454968297144272741 21850780456810680205404210364952225732272
734751304289430201060161991357689262 903279105184451471542312700846538752 [@etzion2016optimal]
: Improvements based on Theorem \[thm\_EF\] cont.
In Table \[table\_improved\_ef\_numerical\] we compare the numerical improvements for $A_q(n,4;k)$ based on Theorem \[thm\_EF\], Theorem \[thm\_construction\_d\], and Lemma \[lemma\_ef\_comb\_special\]. We also list a few cases which result in the same lower bound that was previously known. For consistency reasons we also list the obtained lower bounds for $A_q(12,4;4)$ and $A_q(16,4;4)$ which are inferior to the lower bounds obtained in [@cossidente2019combining].
$A_q(n,4;k)$ New Old
--------------- --------------------------- ------------------------------------------------------
$A_3(10,4;3)$ 5086975 5086975 [@heinlein2019generalized]
$A_4(10,4;3)$ 273727509 273727509 [@heinlein2019generalized]
$A_5(10,4;3)$ 6162421906 6162421906 [@heinlein2019generalized]
$A_7(10,4;3)$ 680487816156 680487816156 [@heinlein2019generalized]
$A_8(10,4;3)$ 4407724867657 4407724867657 [@heinlein2019generalized]
$A_9(10,4;3)$ 22911698562814 22911698562814 [@heinlein2019generalized]
$A_3(11,4;3)$ 45782788 45782686 [@heinlein2019generalized]
$A_4(11,4;3)$ 4379640165 4379639873 [@heinlein2019generalized]
$A_5(11,4;3)$ 154060547681 154060547001 [@heinlein2019generalized]
$A_7(11,4;3)$ 33343902991701 33343902989195 [@heinlein2019generalized]
$A_8(11,4;3)$ 282094391530121 282094391525889 [@heinlein2019generalized]
$A_9(11,4;3)$ 1855847583588025 1855847583581293 [@heinlein2019generalized]
$A_3(12,4;3)$ 412045132 412044676 [@heinlein2019generalized]
$A_4(12,4;3)$ 70074242725 70074241065 [@heinlein2019generalized]
$A_5(12,4;3)$ 3851513692181 3851513687561 [@heinlein2019generalized]
$A_7(12,4;3)$ 1633851246593749 1633851246571461 [@heinlein2019generalized]
$A_8(12,4;3)$ 18054041057928329 18054041057886353 [@heinlein2019generalized]
$A_9(12,4;3)$ 150323654270630845 150323654270557225 [@heinlein2019generalized]
$A_3(13,4;3)$ 3708406309 3708405100 [@heinlein2019generalized]
$A_4(13,4;3)$ 1121187883941 1121187877725 [@heinlein2019generalized]
$A_5(13,4;3)$ 96287842305306 96287842283141 [@heinlein2019generalized]
$A_7(13,4;3)$ 80058711083096502 80058711082943685 [@heinlein2019generalized]
$A_8(13,4;3)$ 1155458627707417737 1155458627707087193 [@heinlein2019generalized]
$A_9(13,4;3)$ 12176215995921105826 12176215995920451445 [@heinlein2019generalized]
$A_2(14,4;3)$ 6241671 6241671 [@heinlein2019generalized]
$A_3(14,4;3)$ 33375657145 33375648396 [@heinlein2019generalized]
$A_4(14,4;3)$ 17939006144421 17939006056956 [@heinlein2019generalized]
$A_5(14,4;3)$ 2407196057636556 2407196057148120 [@heinlein2019generalized]
$A_7(14,4;3)$ 3922876843071748206 3922876843065022206 [@heinlein2019generalized]
$A_8(14,4;3)$ 73949352173274772617 73949352173255598072 [@heinlein2019generalized]
$A_9(14,4;3)$ 986273495669609638336 986273495669561209956 [@heinlein2019generalized]
$A_3(15,4;3)$ 300380915398 300380802505 [@heinlein2019generalized]
$A_4(15,4;3)$ 287024098316197 287024096598921 [@heinlein2019generalized]
$A_5(15,4;3)$ 60179901440933431 60179901426410736 [@heinlein2019generalized]
$A_7(15,4;3)$ 192220965310515799351 192220965310140305694 [@heinlein2019generalized]
$A_8(15,4;3)$ 4732758539089585747081 4732758539088207997713 [@heinlein2019generalized]
$A_9(15,4;3)$ 79888153149238381303087 79888153149234028941436 [@heinlein2019generalized]
$A_2(16,4;3)$ 102223687 102223687 [@heinlein2019generalized]
$A_3(16,4;3)$ 2703428238582 2703427322125 [@heinlein2019generalized]
$A_4(16,4;3)$ 4592385573059152 4592385547188501 [@heinlein2019generalized]
$A_5(16,4;3)$ 1504497536023335775 1504497535674194406 [@heinlein2019generalized]
$A_7(16,4;3)$ 9418827300215274168199 9418827300197242691478 [@heinlein2019generalized]
$A_8(16,4;3)$ 302896546501733487813184 302896546501646667812937 [@heinlein2019generalized]
$A_9(16,4;3)$ 6470940405088308885550047 6470940405087960642352846 [@heinlein2019generalized]
: Improvements based on Theorem \[thm\_EF\], Theorem \[thm\_construction\_d\], and Lemma \[lemma\_ef\_comb\_special\].[]{data-label="table_improved_ef_linkage_numerical"}
$A_q(n,4;k)$ New Old
--------------- ------------------------------------- ----------------------------------------------------------------
$A_2(17,4;3)$ 408894755 408894729 [@heinlein2019generalized]
$A_3(17,4;3)$ 24330854147239 24330847680853 [@heinlein2019generalized]
$A_4(17,4;3)$ 73478169168946433 73478168809292217 [@heinlein2019generalized]
$A_5(17,4;3)$ 37612438400583394376 37612438392939375961 [@heinlein2019generalized]
$A_7(17,4;3)$ 461522537710548434241752 461522537709756797516943 [@heinlein2019generalized]
$A_8(17,4;3)$ 19385378976110943220043777 19385378976105915500991697 [@heinlein2019generalized]
$A_9(17,4;3)$ 524146172812153019729553808 524146172812127278980234877 [@heinlein2019generalized]
$A_2(18,4;3)$ 1635579035 1635578957 [@heinlein2019generalized]
$A_3(18,4;3)$ 218977687325155 218977629126520 [@heinlein2019generalized]
$A_4(18,4;3)$ 1175650706703142933 1175650700948669781 [@heinlein2019generalized]
$A_5(18,4;3)$ 940310960014584859406 940310959823484378906 [@heinlein2019generalized]
$A_7(18,4;3)$ 22614604347816873277845856 22614604347778083078190614 [@heinlein2019generalized]
$A_8(18,4;3)$ 1240664254471100366082801737 1240664254470778592063165001 [@heinlein2019generalized]
$A_9(18,4;3)$ 42455839997784394598093858458 42455839997782309597398420706 [@heinlein2019generalized]
$A_2(19,4;3)$ 6542316171 6542316059 [@heinlein2019generalized]
$A_3(19,4;3)$ 1970799185926408 1970798662145206 [@heinlein2019generalized]
$A_4(19,4;3)$ 18810411307250286949 18810411215178781533 [@heinlein2019generalized]
$A_5(19,4;3)$ 23507774000364621485181 23507773995587109861266 [@heinlein2019generalized]
$A_7(19,4;3)$ 1108115613043026790614447001 1108115613041126070837091728 [@heinlein2019generalized]
$A_8(19,4;3)$ 79402512286150423429299311241 79402512286129829892059310425 [@heinlein2019generalized]
$A_9(19,4;3)$ 3438923039820535962445602535189 3438923039820367077389315073958 [@heinlein2019generalized]
$A_2(12,4;4)$ 19674269 19676797 [@cossidente2019combining]
$A_3(12,4;4)$ 288648673507 288648887023 [@cossidente2019combining]
$A_4(12,4;4)$ 283104148286289 283104153226065 [@cossidente2019combining]
$A_5(12,4;4)$ 59732550564570151 59732550620930151 [@cossidente2019combining]
$A_7(12,4;4)$ 191677878196845899475 191677878199060649103 [@cossidente2019combining]
$A_8(12,4;4)$ 4723722950504908124737 4723722950514423444033 [@cossidente2019combining]
$A_9(12,4;4)$ 79780441020720237308359 79780441020754680563815 [@cossidente2019combining]
$A_2(13,4;4)$ 157396313 157332190 [@cossidente2019combining]
$A_3(13,4;4)$ 7793514240823 7793495430036 [@cossidente2019combining]
$A_4(13,4;4)$ 18118665490931521 18118664249474716 [@cossidente2019combining]
$A_5(13,4;4)$ 7466568820575245751 7466568787180077320 [@cossidente2019combining]
$A_7(13,4;4)$ 65745512221518213208951 65745512216555289614188 [@cossidente2019combining]
$A_8(13,4;4)$ 2418546150658513179095553 2418546150622126921477496 [@cossidente2019combining]
$A_9(13,4;4)$ 58159941504105053602711351 58159941503893673245551936 [@cossidente2019combining]
$A_2(14,4;4)$ 1259181253 1258757174 [@cossidente2019combining]
$A_3(14,4;4)$ 210424885316173 210424421624298 [@cossidente2019combining]
$A_4(14,4;4)$ 1159594591440766481 1159594516050838620 [@cossidente2019combining]
$A_5(14,4;4)$ 933321102572187566901 933321098538702991570 [@cossidente2019combining]
$A_7(14,4;4)$ 22550710691980761977054117 22550710690309028764671498 [@cossidente2019combining]
$A_8(14,4;4)$ 1238295629137158820145963073 1238295629118788686643907448 [@cossidente2019combining]
$A_9(14,4;4)$ 42398597356492584370698238141 42398597356340204444957848530 [@cossidente2019combining]
$A_2(15,4;4)$ 10073483841 10071464646 [@cossidente2019combining]
$A_3(15,4;4)$ 5681471907358915 5681463153275925 [@cossidente2019combining]
$A_4(15,4;4)$ 74214053852334056793 74214050169101548368 [@cossidente2019combining]
$A_5(15,4;4)$ 116665137821525417847661 116665137415279661027650 [@cossidente2019combining]
$A_7(15,4;4)$ 7734893767349401492485075543 7734893766857015258769289566 [@cossidente2019combining]
$A_8(15,4;4)$ 634007362118225316643319878225 634007362109986775858834010688 [@cossidente2019combining]
$A_9(15,4;4)$ 30908577472883094009493870125553 30908577472784286989399940957138 [@cossidente2019combining]
$A_2(16,4;4)$ 80596312221 80596320222 [@cossidente2019combining]
$A_3(16,4;4)$ 153399853228893616 153399853246113244 [@cossidente2019combining]
$A_4(16,4;4)$ 4749699529175914867537 4749699529177178098513 [@cossidente2019combining]
$A_5(16,4;4)$ 14583142241438194910273276 14583142241438230122883276 [@cossidente2019combining]
$A_7(16,4;4)$ 2653068562231495127604331943392 2653068562231495132921600978204 [@cossidente2019combining]
$A_8(16,4;4)$ 324611769405185201386954976457281 324611769405185201425928435085889 [@cossidente2019combining]
$A_9(16,4;4)$ 22532352977741502934357344934823692 22532352977741502934583323007263948 [@cossidente2019combining]
: Improvements based on Theorem \[thm\_EF\], Theorem \[thm\_construction\_d\], and Lemma \[lemma\_ef\_comb\_special\] cont.
$A_q(n,4;k)$ New Old
--------------- ------------------------------------------- ----------------------------------------------------------------------
$A_2(17,4;4)$ 644770526929 644769492958 [@cossidente2019combining]
$A_3(17,4;4)$ 4141796037183639766 4141796035658667783 [@cossidente2019combining]
$A_4(17,4;4)$ 303980769867258670043393 303980769866940801875612 [@cossidente2019combining]
$A_5(17,4;4)$ 1822892780179774365686344376 1822892780179753492675780445 [@cossidente2019combining]
$A_7(17,4;4)$ 910002516845402828768417638272482 910002516845402816852352967620731 [@cossidente2019combining]
$A_8(17,4;4)$ 166201225935454823110121665687982081 166201225935454822961083064550157688 [@cossidente2019combining]
$A_9(17,4;4)$ 16426085320773555639146507673669565108 16426085320773555637759638858654591537 [@cossidente2019combining]
$A_2(18,4;4)$ 5158164354661 5158157544758 [@cossidente2019combining]
$A_3(18,4;4)$ 111828493003995506044 111828492966430770027 [@cossidente2019combining]
$A_4(18,4;4)$ 19454769271504557075730961 19454769271485256959951964 [@cossidente2019combining]
$A_5(18,4;4)$ 227861597522471795764877751276 227861597522469274830565882195 [@cossidente2019combining]
$A_7(18,4;4)$ 312130863277973170267574410726205300 312130863277973166253742271546610147 [@cossidente2019combining]
$A_8(18,4;4)$ 85095027678952869432382343292909789249 85095027678952869357138271983639907192 [@cossidente2019combining]
$A_9(18,4;4)$ 11974616198843922060937804378020488874436 11974616198843922059938039662012799420059 [@cossidente2019combining]
$A_2(19,4;4)$ 41265315376833 41265282958278 [@cossidente2019combining]
$A_3(19,4;4)$ 3019369311108187930600 3019369310399000457648 [@cossidente2019combining]
$A_4(19,4;4)$ 1245105233376291684834977113 1245105233375348762973895504 [@cossidente2019combining]
$A_5(19,4;4)$ 28482699690308974471842016207036 28482699690308720567594897355775 [@cossidente2019combining]
$A_7(19,4;4)$ 107060886104344797401778345454852597360 107060886104344796219558793302396711773 [@cossidente2019combining]
$A_8(19,4;4)$ 43568654171623869149379762750201287709265 43568654171623869115634697686019566694976 [@cossidente2019combining]
$A_9(19,4;4)$ 87294952089572191824236594129317862395535 87294952089572191817753865390378237377288
08 19 [@cossidente2019combining]
: Improvements based on Theorem \[thm\_EF\], Theorem \[thm\_construction\_d\], and Lemma \[lemma\_ef\_comb\_special\] cont.
Skeleton codes for the multilevel construction {#sec_skeleton_codes}
==============================================
In this appendix we list the used skeleton codes from the results of Section \[sec\_results\]. For the ease of a more compact representation we replace each vector $v\in{\mathbb{F}}_2^n$ by the integer $\sum_{i=1}^n v_i\cdot 2^{n-i}$. As an example, the integer $6168$ corresponds to the vector $1100000011000\in{\mathbb{F}}_2^{13}$. Starting from an integer, the value of $n$ needs to be clear from the context.
${\mathcal{S}}_{13,4,5}^1=\{$7936, 7360, 6816, 1984, 3488, 3680, 5776, 6496, 6544, 6736, 7216, 3720, 5456, 5704, 3400, 5000, 5508, 2948, 4912, 5416, 5668, 7180, 2856, 3604, 4932, 1816, 2882, 6666, 1826, 3346, 4802, 2753, 4336, 5218, 5281, 6406, 6409, 6661, 7171, 1256, 1380, 2706, 2833, 3217, 4514, 4545, 5258, 740, 929, 1618, 2264, 2388, 2636, 3206, 3333, 4705, 696, 1236, 2274, 2442, 3114, 1417, 1669, 2228, 4300, 4426, 4636, 4867, 5189, 376, 466, 841, 1202, 1329, 1577, 2598, 5142, 5145, 428, 618, 790, 1347, 1550, 2161, 2217, 4389, 1084, 2339, 405, 4154, 597, 4243, 651, 563, 2078, 2123, 118, 199, 109, 283, 1063, 4111$\}$
${\mathcal{S}}_{13,4,5}^2=\{$7936, 7360, 6816, 3008, 3488, 3680, 5776, 6496, 6544, 6736, 7216,3720, 5456, 5512, 5704, 3400, 1840, 1924, 5668, 7180, 3604, 4904, 4932, 4994, 1858, 2840, 2852, 5410, 6666, 3346, 1264, 1698, 4516, 4801, 5281, 6406, 6409, 6661, 7171, 1380, 1473, 2706, 3217, 4328, 4706, 4748, 4881, 740, 929, 1801, 2264, 2388, 2444, 2636, 3333, 5254, 696, 1617, 2274, 3114, 3142, 4308, 1228, 2228, 2609, 2819, 4426, 5146, 5189, 376, 466, 714, 1308, 4274, 4630, 426, 1550, 2217, 2245, 4209, 409, 617, 4156, 661, 355, 1078, 1081, 309, 333, 2131, 4235, 391, 1099,2078, 583, 110, 539, 2087$\}$
${\mathcal{S}}_{14,4,5}^1=\{$15872, 14720, 13632, 3968, 7456, 11072, 11456, 12992, 13088, 13472, 14432, 11536, 6816, 7312, 5904, 6920, 10896, 3680, 6736, 7240, 9992, 14360, 5768, 9808, 11304, 9860, 13332, 5508, 5700, 6468, 10788, 992, 2864, 3608, 9602, 12812, 12818, 13322, 14342, 1488, 1712, 2760, 3394, 4994, 5232, 5666, 6338, 9508, 10530, 3236, 4528, 4552, 6418, 8872, 9089, 9368, 9761, 10508, 10762, 12561, 1448, 1730, 1857, 4712, 5313, 6284, 7173, 8560, 8644, 8980, 10324, 10401, 920, 2288, 2408, 3590, 4756, 6196, 9314, 11267, 12425, 2452, 2497, 3210, 3337, 6659, 9292, 3122, 3153, 4324, 4450, 5164, 6186, 10313, 12357, 844, 850, 2821, 5379, 4657, 8372, 8402, 8522, 8774, 12323, 628, 721, 810, 1308, 1329, 2601, 5145, 4393, 8729, 678, 1557, 4188, 4250, 421, 1114, 1129, 1174, 2150, 2201, 4366, 234, 345, 1187, 1547, 8250, 8334, 403, 611, 653, 310, 4179, 8455, 542, 2183, 8237, 1095$\}$
${\mathcal{S}}_{14,4,5}^2=\{$15872, 14720, 13632, 6016, 6976, 7360, 11552, 12992, 13088, 13472, 14432, 7440, 10912, 11024, 11408, 6800, 3680, 3848, 11336, 14360, 7208, 9808, 9864, 9988, 3716, 5680, 5704, 10820, 13332, 6692, 992, 3396, 9032, 9602, 10562, 12812, 12818, 13322, 14342, 2760, 2946, 3457, 5412, 6434, 8656, 9412, 9496, 9762, 1480, 1858, 3602, 4528, 4776, 4888, 5272, 6412, 9089, 10433, 10762, 12561, 1392, 3234, 4548, 6228, 6305, 6665, 8616, 10380, 11269, 1729, 2288, 2408, 2456, 2468, 2849, 5218, 5638, 6282, 7171, 8852, 10292, 10505, 12425, 1428, 1809, 2616, 5385, 8548, 9313, 852, 908, 4705, 5201, 9260, 12357, 2641, 3100, 3121, 4434, 4869, 5253, 8418, 1234, 8312, 8498, 8753, 12323, 690, 1322, 1577, 4739, 710, 806, 4204, 8771, 10259, 618, 2326, 4262, 8346, 451, 665, 677, 2138, 2150, 8462, 220, 233, 316, 345, 1166, 2197, 4154, 8278, 779, 1078, 2567, 589, 1287, 4149, 542, 2093, 4171, 8327, 115, 8221, 1051$\}$
${\mathcal{S}}_{13,4,6}^1=\{$8064, 7776, 7504, 6624, 6864, 6960, 6984, 7344, 7368, 7464, 7704, 3908, 2016, 3752, 5828, 5924, 5954, 6820, 3778, 3874, 6552, 7686, 3492, 3732, 5032, 5794, 3521, 3857, 5524, 6794, 7429, 1944, 3474, 5057, 5537, 5777, 5897, 2956, 2977, 4948, 5010, 5452, 5514, 6264, 6534, 6915, 7299, 2898, 3402, 3657, 5330, 5426, 6697, 6725, 2516, 3188, 3356, 5292, 1656, 1925, 2738, 3621, 2668, 4722, 4764, 5226, 5233, 6246, 948, 2474, 2673, 3225, 4340, 6293, 970, 1478, 1814, 3130, 3214, 4878, 504, 1385, 1638, 3171, 4412, 4442, 4697, 6227, 1244, 1621, 2236, 2266, 2281, 2393, 2405, 6174, 867, 1253, 1675, 2358, 4282, 4493, 4661, 825, 1333, 1363, 1587, 2695, 4302, 4323, 5149, 5191, 723, 845, 1206, 1326, 2589, 3095, 4395, 435, 685, 4205, 4375, 414, 606, 2319, 63$\}$
${\mathcal{S}}_{13,4,6}^2=\{$8064, 7776, 7504, 6624, 6864, 6960, 6984, 7344, 7368, 7464, 7704, 3908, 6824, 2016, 5828, 5924, 5954, 3748, 3778, 3874, 6552, 7686, 5794, 3521, 3857, 5524, 7429, 1944, 2964, 3468, 3474, 5057, 5537, 5777, 5897, 2977, 3284, 3380, 3721, 4948, 5004, 5010, 5452, 5514, 6264, 6534, 6789, 6915, 7299, 2898, 2954, 3402, 5330, 5426, 2764, 2860, 4788, 5236, 5292, 1656, 1925, 2676, 2738, 3180, 3186, 3242, 4810, 4906, 4716, 4722, 5226, 6246, 2666, 1478, 1814, 2396, 2417, 3161, 6229, 504, 1265, 1638, 4316, 4412, 4442, 4457, 4697, 5177, 934, 1381, 1621, 1678, 2236, 2266, 2281, 2362, 2617, 6174, 6189, 6195, 6219, 867, 4282, 4325, 1363, 1581, 1587, 1611, 3143, 723, 821, 845, 1206, 1229, 1326, 4679, 5159, 469, 1309, 2599, 435, 459, 683, 795, 374, 429, 669, 1118, 1179, 2327, 414, 4247, 4367, 238, 574, 2191, 123$\}$
${\mathcal{S}}_{14,4,6}^1=\{$16128, 15552, 15008, 13248, 13728, 13920, 13968, 14688, 14736, 14928, 15408, 4032, 7816, 11600, 11656, 11848, 11908, 13640, 7556, 7748, 13104, 15372, 5968, 6984, 7464, 3888, 7042, 7490, 7714, 11048, 11076, 11556, 13588, 14858, 6948, 10114, 11137, 11794, 11809, 6017, 7256, 7697, 9896, 10008, 10904, 10946, 11426, 12528, 13068, 13830, 13833, 14598, 14601, 14853, 15363, 5796, 5826, 6804, 6849, 7314, 7329, 9668, 10049, 13394, 13450, 3312, 3852, 5032, 5528, 6376, 6712, 11409, 5476, 5898, 6484, 9124, 9812, 10468, 10804, 13445, 1896, 9432, 9570, 11334, 12492, 12625, 13059, 1940, 2914, 3425, 4824, 5332, 6450, 8680, 10360, 10570, 12586, 1008, 2520, 3276, 3843, 4961, 5660, 9042, 9396, 9441, 10524, 10545, 2897, 4578, 5425, 6342, 10441, 12454, 12457, 12581, 1745, 2484, 2732, 2738, 3354, 4472, 4724, 4934, 5321, 6257, 9324, 9513, 12348, 12483, 1490, 3178, 4785, 6297, 8564, 8626, 10390, 972, 1452, 1650, 1829, 2412, 2474, 2652, 2838, 3132, 3267, 3349, 4714, 4889, 8817, 8901, 9486, 1690, 2501, 2665, 5178, 5389, 8806, 8981, 9491, 10325, 10339, 963, 1372, 1699, 4502, 4750, 5198, 5219, 6190, 6221, 8601, 8762, 12339, 1593, 1677, 8851, 10285, 828, 1254, 2451, 2699, 3123, 4325, 4691, 9291, 1334, 1419, 1582, 4653, 8525, 9245, 10267, 726, 819, 1607, 2166, 4427, 6167, 8583, 9255, 12303, 1141, 2599, 252, 3087, 8286, 243, 359, 783, 1175, 207, 63$\}$
${\mathcal{S}}_{14,4,6}^2=\{$16128, 15552, 15008, 13248, 13728, 13920, 13968, 14688, 14736, 14928, 15408, 4032, 7816, 13648, 11656, 11848, 11908, 7496, 7556, 7748, 13104, 15372, 11588, 3888, 7042, 7714, 11048, 14858, 5928, 6936, 6948, 10114, 11074, 11137, 11554, 11794, 11809, 5954, 6017, 6568, 6760, 6977, 7442, 7457, 7697, 9896, 10008, 10020, 10904, 11028, 12528, 13068, 13578, 13830, 13833, 14598, 14601, 14853, 15363, 5796, 5908, 6804, 10049, 10660, 10852, 11537, 3312, 3852, 5528, 5720, 9576, 10472, 10584, 13573, 5352, 5476, 6360, 6372, 6484, 9620, 9812, 9432, 9444, 10452, 12492, 13059, 5332, 1008, 2786, 3276, 3843, 4792, 6322, 12458, 4578, 4818, 4833, 8632, 8824, 8884, 9394, 10354, 10417, 2732, 2769, 3242, 4472, 4532, 4724, 5234, 5297, 6257, 8658, 8673, 8906, 12348, 12378, 12390, 12393, 12438, 12441, 12453, 12483, 1489, 1737, 2506, 8564, 9329, 972, 1452, 1644, 1692, 1734, 2412, 2460, 2652, 3132, 3162, 3174, 3177, 3222, 3225, 3237, 3267, 4553, 6286, 8901, 12373, 2501, 9358, 10318, 10381, 937, 963, 1372, 1699, 2618, 3157, 5198, 5261, 6221, 12339, 874, 922, 934, 1593, 2467, 8803, 8851, 9293, 828, 857, 869, 917, 1338, 1379, 1590, 1619, 2358, 2361, 2613, 2699, 3123, 4499, 4654, 6187, 854, 1333, 1419, 2387, 4683, 4743, 8494, 8734, 8749, 9259, 10267, 10279, 819, 2631, 4382, 4397, 4637, 5147, 5159, 6167, 8523, 8583, 12303, 4423, 8477, 9239, 252, 3087, 243, 783, 207, 63$\}$
${\mathcal{S}}_{15,4,6}^1=\{$32256, 31104, 30016, 26496, 27456, 27840, 27936, 29376, 29472, 29856, 30816, 8064, 15632, 23200, 23312, 23696, 23816, 27280, 15112, 15496, 26208, 30744, 11936, 13968, 14928, 7776, 14084, 14980, 15428, 22096, 22152, 23112, 27176, 29716, 13896, 20228, 22274, 23588, 23618, 12034, 14512, 15394, 19792, 20016, 21808, 21892, 22852, 25056, 26136, 27660, 27666, 29196, 29202, 29706, 30726, 11592, 11652, 13608, 13698, 14628, 14658, 19336, 20098, 26788, 26900, 6624, 7704, 10064, 11056, 12752, 13424, 14881, 22818, 28945, 3792, 10952, 11796, 11841, 12968, 13185, 14529, 18248, 19624, 19841, 20936, 21608, 22049, 26890, 3880, 7489, 18864, 19140, 21313, 21697, 22668, 24984, 25250, 26118, 27141, 28809, 2016, 5828, 6850, 9648, 10664, 12900, 17360, 20720, 21140, 25172, 25745, 25865, 26705, 5040, 6552, 7329, 7686, 9922, 11320, 18084, 18792, 18882, 19233, 19977, 21048, 21090, 23043, 28741, 5794, 9156, 10017, 10850, 12684, 13827, 19553, 20882, 20897, 22577, 24908, 24914, 25162, 26755, 2980, 3428, 3490, 3724, 3857, 4968, 5464, 6708, 6801, 8944, 9448, 9665, 10642, 12514, 12641, 13445, 18648, 19026, 24696, 24966, 25137, 25347, 28707, 1944, 2904, 3009, 5474, 5897, 6356, 6917, 12594, 17128, 17252, 20780, 25637, 25667, 3658, 4824, 4948, 5304, 5521, 5676, 6264, 6534, 6698, 9428, 9778, 9865, 13337, 14357, 17634, 17802, 18972, 2744, 3252, 5002, 5330, 10356, 10778, 17612, 17962, 18982, 20650, 20678, 1844, 3356, 4833, 6441, 9004, 9542, 10396, 10438, 10565, 12380, 12442, 14347, 17202, 17524, 17989, 19477, 21517, 21523, 24678, 1656, 1925, 2508, 2886, 3186, 3273, 3621, 9033, 9498, 10521, 17702, 20570, 1713, 2668, 3226, 4902, 5398, 5413, 5653, 6246, 8650, 9382, 10345, 10531, 18582, 1452, 1859, 2673, 6419, 8982, 9493, 11271, 17050, 17177, 18490, 18701, 20534, 20747, 1393, 4332, 4549, 5262, 8869, 12334, 12371, 12551, 17091, 17166, 17497, 18510, 24606, 504, 937, 1638, 1686, 2282, 2723, 3339, 6221, 8846, 2453, 3118, 4686, 4749, 6174, 8851, 9293, 16572, 1253, 2358, 5195, 8409, 16725, 16739, 17543, 4659, 9259, 16941, 486, 725, 2259, 4277, 8733, 8775, 1563, 2221, 4381, 16563, 1181, 2583, 619, 1118, 365, 411, 574, 207, 1079$\}$
${\mathcal{S}}_{15,4,6}^2=\{$32256, 31104, 30016, 26496, 27456, 27840, 27936, 29376, 29472, 29856, 30816, 8064, 15632, 27296, 23312, 23696, 23816, 14992, 15112, 15496, 26208, 30744, 23176, 7776, 14084, 22096, 23620, 29716, 11856, 13872, 13896, 14916, 15396, 15426, 20228, 22148, 22274, 11908, 12034, 13136, 13520, 13954, 19792, 20016, 20040, 21808, 22056, 23076, 23106, 23586, 25056, 26136, 27156, 27660, 27666, 29196, 29202, 29706, 30726, 11592, 11816, 13608, 14882, 20098, 21320, 21704, 6624, 7704, 11056, 11440, 14657, 19152, 20944, 21168, 21889, 27146, 28945, 10704, 10952, 11649, 12720, 12744, 12968, 13185, 19240, 19624, 22721, 22817, 14497, 18864, 18888, 19329, 20904, 24984, 26118, 26769, 26889, 28809, 2016, 10664, 3524, 6552, 7686, 9584, 12644, 13409, 18241, 24916, 25681, 25861, 28741, 9921, 10017, 17264, 17348, 17648, 17768, 17828, 17858, 18788, 19553, 20708, 20834, 21089, 2980, 3010, 3490, 3745, 3857, 5464, 5524, 6484, 7249, 7429, 8944, 9064, 9448, 10468, 10594, 10849, 12514, 13445, 13571, 21253, 24696, 24756, 24780, 24786, 24876, 24882, 24906, 24966, 25137, 25161, 25221, 25347, 25641, 25731, 26661, 26691, 28707, 1944, 2904, 3288, 3384, 5026, 5777, 5897, 9108, 9612, 9618, 11013, 17128, 18658, 19589, 19715, 4824, 4920, 5004, 5304, 5514, 6264, 6324, 6348, 6354, 6444, 6450, 6474, 6534, 6705, 6729, 6789, 6915, 7209, 7299, 11333, 12572, 13337, 14357, 17298, 18057, 21123, 21541, 21571, 24746, 1876, 2744, 9098, 10883, 12837, 12867, 18716, 19013, 19481, 20636, 20762, 21017, 22541, 22547, 3188, 5446, 5701, 6314, 10396, 10522, 10777, 11299, 12442, 14347, 24678, 1656, 1716, 1740, 1746, 1836, 1842, 1866, 1925, 3651, 4724, 5228, 5234, 9030, 9414, 9510, 18586, 18979, 1489, 2668, 2674, 3178, 3350, 4806, 5286, 5667, 6246, 9308, 9749, 12374, 17190, 1706, 3597, 4714, 4886, 8870, 16988, 17468, 17498, 17558, 17678, 17939, 18518, 20534, 20558, 945, 969, 1449, 2417, 2710, 2830, 3214, 4549, 8764, 8794, 9274, 9739, 10294, 10318, 12334, 24606, 504, 1638, 4337, 4457, 8613, 8643, 16954, 17038, 18478, 1381, 2281, 2453, 6174, 8537, 16803, 17485, 867, 4499, 16601, 16697, 16781, 741, 1251, 2381, 2443, 4405, 8377, 8405, 16723, 16941, 486, 1566, 1309, 2227, 4269, 4299, 5143, 8491, 16501, 795, 2247, 2343, 8301, 8307, 669, 1179, 4189, 4623, 8471, 16491, 414, 2109, 2139, 599, 4155, 8335, 1071, 126$\}$
${\mathcal{S}}_{16,4,6}^1=\{$64512, 62208, 60032, 52992, 54912, 55680, 55872, 58752, 58944, 59712, 61632, 16128, 31264, 46400, 46624, 47392, 47632, 54560, 30224, 30992, 52416, 61488, 23872, 27936, 29856, 15552, 28168, 29960, 30856, 44192, 44304, 46224, 54352, 59432, 27792, 40456, 44548, 47176, 47236, 24068, 29024, 30788, 39584, 40032, 43616, 43784, 45704, 50112, 52272, 55320, 55332, 58392, 58404, 59412, 61452, 23184, 23304, 27216, 27396, 29256, 29316, 38672, 40196, 53576, 53800, 13248, 15408, 20128, 22112, 25504, 26848, 29762, 45636, 57890, 7584, 21904, 23592, 23682, 25936, 26370, 29058, 36496, 39248, 39682, 41872, 43216, 44098, 53780, 4032, 7760, 14978, 27713, 37728, 38280, 40065, 42626, 42753, 43394, 45336, 45441, 49968, 50500, 52236, 54282, 57618, 57633, 57873, 61443, 11656, 13700, 19296, 21328, 22273, 25800, 26241, 27009, 34720, 41440, 42280, 50344, 51490, 51730, 51745, 53410, 10080, 13104, 14658, 14913, 15372, 19844, 22640, 36168, 37584, 37764, 38466, 39954, 42096, 42180, 46086, 46089, 53521, 54277, 57482, 11588, 18312, 20034, 21700, 23569, 25368, 27654, 39106, 41764, 41794, 45154, 49816, 49828, 50324, 53345, 53510, 3888, 5960, 6856, 6980, 7448, 7714, 9936, 10928, 13416, 13602, 17888, 18896, 19330, 21284, 22721, 22817, 25028, 25282, 25409, 26890, 28817, 36417, 37296, 38052, 49392, 49932, 50274, 50694, 50697, 51345, 51465, 52227, 53385, 57414, 57417, 57477, 5808, 6018, 7041, 10948, 11794, 11809, 12712, 13834, 14497, 25188, 28933, 34256, 34504, 41560, 41665, 45137, 51274, 51334, 7316, 9648, 9896, 10608, 11042, 11352, 12528, 13068, 13396, 13585, 14601, 15363, 18856, 19556, 19730, 26674, 27139, 28714, 35268, 35604, 36129, 37944, 39429, 43145, 43269, 3688, 5488, 6504, 10004, 10660, 13829, 20712, 21153, 21556, 35224, 35924, 37964, 41300, 41356, 43057, 51269, 3312, 3852, 5825, 6712, 9666, 11025, 12882, 18008, 19084, 20792, 20876, 21130, 24760, 24884, 25649, 28694, 34404, 35048, 35978, 37641, 38954, 43034, 43046, 45093, 49356, 49923, 5016, 5772, 6372, 6546, 7242, 18066, 18996, 19593, 21042, 21577, 26649, 26661, 34570, 35404, 38147, 41140, 2904, 3426, 5336, 6452, 9804, 10796, 10826, 11306, 11397, 12492, 13059, 17300, 18764, 20690, 21062, 25129, 37164, 37425, 3473, 3718, 5346, 7237, 12838, 14357, 17964, 18986, 19017, 22542, 34100, 34354, 36980, 37402, 37507, 41068, 41494, 42005, 42019, 1008, 2786, 3276, 3843, 8664, 9098, 9441, 10524, 12825, 17738, 21539, 24668, 24742, 25102, 25613, 34182, 34332, 34437, 34994, 35865, 37020, 37189, 49212, 49347, 1873, 4564, 5446, 5673, 6678, 9545, 12442, 17617, 17692, 17989, 20803, 41485, 2529, 2769, 4906, 6236, 9093, 9372, 9498, 9795, 12348, 12483, 17702, 17705, 17795, 18586, 21005, 33144, 33969, 35139, 2506, 2857, 3605, 4716, 5413, 6297, 10345, 10390, 16818, 18709, 24675, 33450, 33478, 34981, 35086, 41107, 41227, 972, 1452, 3132, 3267, 4553, 5651, 6667, 8817, 9318, 17573, 18518, 18595, 20565, 33882, 33897, 49203, 1477, 1689, 4518, 4885, 6243, 8554, 16753, 17466, 17550, 33193, 34061, 963, 1699, 2661, 4442, 6407, 8762, 10323, 12339, 16793, 20615, 33126, 36939, 1622, 2362, 5166, 5259, 6189, 8549, 8611, 9479, 33173, 33365, 33379, 745, 828, 2445, 3123, 8505, 16979, 20507, 32985, 49167, 730, 5149, 8405, 33863, 36887, 819, 1206, 2222, 4217, 4277, 12303, 1141, 8365, 16494, 16935, 17431, 252, 3087, 243, 414, 663, 783, 343, 207, 63$\}$
${\mathcal{S}}_{16,4,6}^2=\{$64512, 62208, 60032, 52992, 54912, 55680, 55872, 58752, 58944, 59712, 61632, 16128, 31264, 54592, 46624, 47392, 47632, 29984, 30224, 30992, 52416, 61488, 46352, 15552, 28168, 44192, 47240, 59432, 23712, 27744, 27792, 29832, 30792, 30852, 40456, 44296, 44548, 23816, 24068, 26272, 27040, 27908, 39584, 40032, 40080, 43616, 44112, 46152, 46212, 47172, 50112, 52272, 54312, 55320, 55332, 58392, 58404, 59412, 61452, 23184, 23632, 27216, 29764, 40196, 42640, 43408, 13248, 15408, 22112, 22880, 29314, 38304, 41888, 42336, 43778, 54292, 57890, 21408, 21904, 23298, 25440, 25488, 25936, 26370, 27393, 38480, 39248, 45442, 45634, 45697, 4032, 28994, 29057, 29249, 37728, 37776, 38658, 39681, 41808, 42753, 49968, 52236, 53538, 53778, 53793, 57618, 57633, 57873, 61443, 21328, 22273, 45377, 7048, 13104, 15372, 19168, 25288, 26818, 36482, 49832, 51362, 51722, 53521, 57482, 7554, 7746, 7809, 10120, 11080, 11140, 34528, 35296, 35536, 37576, 39106, 41416, 41668, 42178, 43201, 3888, 5960, 6020, 6980, 10928, 11586, 11649, 11810, 11841, 12968, 14498, 14858, 17888, 18128, 18896, 19240, 20936, 21188, 21698, 22721, 25028, 25793, 26890, 27142, 42506, 43529, 49392, 49512, 49560, 49572, 49752, 49764, 49812, 49932, 50274, 50322, 50337, 50442, 50694, 50697, 51282, 51297, 51345, 51462, 51465, 51717, 52227, 53322, 53382, 53385, 57414, 57417, 57477, 5808, 6576, 6768, 7489, 10052, 19746, 19986, 20001, 22026, 23049, 34256, 34600, 35608, 35620, 37316, 38081, 39178, 39430, 3752, 9648, 9840, 10608, 12528, 12648, 12696, 12708, 12888, 12900, 12948, 13068, 13410, 13458, 13473, 13578, 13830, 13833, 14418, 14433, 14481, 14598, 14601, 14853, 15363, 18200, 18212, 19220, 22666, 25144, 25865, 26117, 26674, 26761, 28714, 36114, 36129, 36369, 42246, 43082, 43142, 43269, 49492, 50257, 50437, 53317, 5488, 19729, 21766, 22789, 25674, 25734, 34580, 37432, 38026, 38153, 38405, 38962, 41272, 41524, 42034, 42121, 43057, 45082, 45094, 45097, 3312, 3432, 3480, 3492, 3672, 3684, 3732, 3852, 6376, 11402, 12628, 13393, 13573, 19084, 20792, 21044, 21554, 22577, 22598, 24884, 25649, 26693, 28694, 28697, 28709, 38985, 39045, 49356, 49923, 9448, 10456, 10468, 21577, 21637, 34444, 35212, 35404, 37172, 37937, 37958, 42053, 45077, 2978, 3412, 5336, 5348, 6356, 6700, 7210, 12492, 13059, 17804, 17996, 18616, 18764, 24748, 1953, 9428, 9772, 10540, 10780, 11305, 33976, 34124, 34936, 34996, 35878, 37036, 41068, 41116, 1008, 1890, 1938, 2898, 2913, 2961, 3276, 3843, 4834, 5420, 5660, 6428, 7190, 7193, 7205, 10883, 17290, 17528, 17588, 18548, 20588, 20636, 24668, 49212, 49347, 1873, 8674, 8914, 8929, 9500, 11285, 18051, 18819, 19011, 33610, 33670, 33673, 33908, 36956, 2762, 4562, 4577, 4817, 4906, 6691, 12348, 12483, 17074, 17222, 17225, 17285, 17578, 18538, 24739, 25099, 34179, 34371, 34970, 34985, 35139, 1737, 8657, 8986, 8998, 9001, 10531, 17731, 18582, 18597, 33202, 33394, 33457, 33605, 37027, 37387, 41059, 41107, 41227, 41479, 972, 1482, 1734, 2502, 2505, 2757, 3132, 3267, 4762, 4886, 4889, 4901, 5411, 5651, 6419, 8810, 16754, 16817, 17009, 17561, 18521, 20579, 20627, 20747, 20999, 24659, 24839, 33882, 33897, 49203, 1477, 4713, 8857, 8981, 9491, 17494, 17509, 33137, 33941, 34901, 36947, 37127, 963, 2618, 4454, 8598, 8846, 12339, 33002, 1593, 4494, 4501, 4686, 4749, 8549, 16602, 16614, 16617, 828, 1338, 1590, 2358, 2361, 2613, 3123, 8378, 8526, 8589, 8781, 16942, 32982, 32985, 32997, 49167, 1333, 4218, 4278, 4281, 4429, 10267, 16597, 33070, 33310, 33325, 819, 2222, 8310, 8313, 8373, 9255, 12303, 16670, 16685, 16925, 1197, 4213, 33053, 252, 683, 1134, 1182, 2142, 2157, 2205, 3087, 423, 1117, 243, 363, 411, 603, 615, 663, 783, 343, 207, 63$\}$
${\mathcal{S}}_{17,4,6}=\{$129024, 124416, 120064, 105984, 109824, 111360, 111744, 117504, 117888, 119424, 123264, 32256, 62528, 109184, 93248, 94784, 95264, 59968, 60448, 61984, 104832, 122976, 92704, 31104, 56336, 88384, 94480, 118864, 47424, 55488, 55584, 59664, 61584, 61704, 80912, 88592, 89096, 47632, 48136, 52544, 54080, 55816, 79168, 80064, 80160, 87232, 88224, 92304, 92424, 94344, 100224, 104544, 108624, 110640, 110664, 116784, 116808, 118824, 122904, 46368, 47264, 54432, 59528, 80392, 85280, 86816, 26496, 30816, 44224, 45760, 58628, 76608, 83776, 84672, 87556, 108584, 115780, 42816, 43808, 46596, 50880, 50976, 51872, 52740, 54786, 76960, 78496, 90884, 91268, 91394, 8064, 57988, 58114, 58498, 75456, 75552, 77316, 79362, 83616, 85506, 99936, 104472, 107076, 107556, 107586, 115236, 115266, 115746, 122886, 42656, 44546, 90754, 14096, 26208, 29953, 30744, 38336, 50576, 53636, 72964, 99664, 102724, 103444, 103489, 107042, 114964, 115009, 115729, 118789, 23684, 23810, 36624, 38544, 38664, 39684, 69056, 70592, 71072, 75152, 78212, 82832, 83336, 84356, 86402, 7776, 11920, 12040, 13960, 15106, 15428, 15490, 21856, 22096, 25936, 27393, 27777, 28996, 29313, 29716, 35776, 36256, 37792, 40193, 41872, 42376, 43396, 45442, 50056, 51586, 53780, 54284, 72324, 85012, 87058, 98784, 99024, 99120, 99144, 99504, 99528, 99624, 99864, 100548, 100644, 100674, 100884, 100929, 101388, 101394, 101409, 102564, 102594, 102690, 102924, 102930, 102945, 103434, 104454, 106644, 106689, 106764, 106770, 106785, 107025, 107529, 108549, 110595, 114828, 114834, 114849, 114954, 115209, 116739, 11616, 13152, 13536, 20104, 23108, 23170, 39972, 40002, 44052, 46098, 68512, 69200, 71216, 71240, 72449, 72833, 74632, 76162, 78356, 78860, 7504, 14884, 19296, 19680, 20016, 21216, 22056, 23617, 25056, 25296, 25392, 25416, 25776, 25800, 25896, 26136, 26820, 26916, 26946, 27156, 27660, 27666, 28836, 28866, 28962, 29196, 29202, 29706, 30726, 36424, 39553, 45332, 50288, 51730, 52234, 53348, 53522, 57428, 72258, 72738, 84492, 86164, 86284, 86538, 98984, 100514, 100874, 102673, 106634, 10976, 14913, 15393, 39458, 43532, 45578, 51348, 51468, 69160, 74864, 76052, 76306, 76810, 77924, 82544, 83048, 84068, 84242, 86114, 90164, 90188, 90194, 6624, 6864, 6960, 6984, 7344, 7368, 7464, 7704, 12752, 21784, 22804, 23073, 25256, 26786, 27146, 41584, 42088, 43108, 45154, 45196, 49768, 51298, 53386, 57388, 57394, 57418, 77970, 78090, 98712, 99846, 100497, 100617, 102537, 14484, 14604, 18896, 20912, 20936, 43154, 43274, 68888, 70424, 70808, 74344, 75874, 75916, 84106, 90154, 3908, 6824, 10672, 10696, 12712, 13400, 14609, 19224, 21144, 24984, 26118, 28753, 35992, 37232, 38996, 42257, 49496, 49937, 50441, 51281, 53297, 53321, 99589, 2016, 5828, 5924, 5954, 14418, 18856, 19544, 22673, 22793, 37464, 37944, 67952, 68248, 69872, 69992, 71732, 71756, 74072, 82136, 82232, 3748, 3778, 3874, 6552, 7686, 9668, 10840, 11320, 12856, 13457, 13574, 14473, 18196, 19729, 22572, 22578, 22602, 26673, 26697, 28713, 35056, 35176, 37096, 41176, 41272, 41737, 45125, 49336, 50309, 51241, 70921, 71761, 86041, 98424, 98694, 98949, 99075, 99459, 5794, 14378, 17348, 17828, 17858, 21254, 21641, 35384, 36102, 38022, 45081, 67220, 67340, 67346, 67816, 70289, 73912, 3476, 9124, 9154, 9634, 9812, 11345, 11525, 12934, 13061, 18066, 18186, 21574, 24696, 24966, 25669, 34148, 34444, 43075, 49478, 49733, 49795, 50198, 50243, 68358, 68742, 69970, 71721, 75801, 82981, 1944, 5012, 5516, 5522, 17314, 17996, 34356, 34386, 35462, 37164, 37194, 37446, 66404, 66788, 66914, 67210, 68681, 68867, 74054, 74774, 82118, 82214, 82454, 82958, 2956, 2962, 3466, 6264, 6534, 9524, 9772, 9778, 9802, 10822, 10885, 11302, 12617, 13379, 17620, 25125, 25155, 25635, 33508, 33620, 33634, 34018, 35122, 35909, 37042, 37073, 41137, 41158, 41254, 41494, 41998, 49318, 49445, 49678, 67794, 68145, 70182, 70213, 82467, 98406, 5002, 9426, 9987, 10793, 17234, 17714, 18982, 34346, 34988, 35018, 37923, 38925, 38931, 66274, 67882, 68227, 69802, 73894, 74254, 1873, 3188, 5425, 6469, 7189, 8908, 12835, 17076, 17196, 20572, 20677, 20803, 24678, 24853, 34076, 35365, 66004, 66769, 69925, 98389, 1656, 1925, 4724, 5228, 5234, 8988, 9002, 9372, 9498, 17098, 17578, 33204, 33228, 33234, 67139, 1713, 1737, 1833, 2668, 2674, 3178, 3241, 6246, 6309, 6339, 6435, 7179, 16756, 20643, 24717, 24723, 24843, 33562, 35093, 65964, 65970, 65994, 66652, 70663, 98334, 98349, 98355, 98379, 2883, 3597, 3603, 4714, 8436, 8556, 8562, 8858, 9381, 12342, 12366, 17468, 18518, 19463, 33194, 2396, 16620, 16626, 16746, 24606, 33433, 33881, 37127, 49181, 66108, 66138, 66325, 66618, 504, 1637, 4883, 8426, 16954, 16985, 41003, 66701, 2645, 4284, 4314, 4410, 4749, 6174, 9355, 1366, 2234, 2265, 2361, 67655, 1357, 18459, 20503, 483, 723, 822, 846, 1206, 1230, 1326, 1566, 10263, 65717, 4381, 683, 65819, 8285, 407, 2319, 32911, 16463, 4143, 119$\}$
${\mathcal{S}}_{15,4,7}^1=\{$32512, 31936, 31392, 29632, 30112, 30304, 30352, 31072, 31120, 31312, 31792, 28296, 30032, 12224, 15752, 15944, 16004, 31756, 20384, 23432, 23880, 23940, 24104, 24132, 24194, 27440, 27464, 27524, 27944, 27972, 28034, 28196, 28226, 29480, 31242, 8032, 8080, 14128, 15172, 15234, 15652, 15682, 15906, 20304, 29460, 15128, 22216, 23332, 23362, 23704, 23842, 26056, 26308, 26392, 26497, 27736, 27796, 28177, 28912, 29962, 30214, 30217, 30982, 30985, 31237, 31747, 11952, 13992, 26402, 27329, 27809, 29324, 13764, 14018, 14145, 15137, 15444, 15506, 15633, 21424, 22180, 23188, 26856, 27282, 29957, 11632, 12052, 13672, 14092, 14785, 15457, 21954, 22290, 22305, 23313, 23634, 29004, 29234, 7873, 11176, 13220, 14552, 14564, 14900, 14993, 19696, 20236, 21860, 22840, 22868, 22945, 23137, 25444, 26962, 29443, 12042, 19905, 21336, 22754, 25304, 25784, 25826, 25908, 25953, 27164, 28874, 28977, 6896, 7400, 7945, 13154, 14642, 18280, 20968, 21716, 21905, 22097, 22732, 25482, 26740, 27177, 7942, 13012, 13492, 18324, 19172, 20018, 21624, 21729, 22044, 22737, 23589, 25044, 25268, 25425, 25809, 26822, 28838, 28841, 28869, 5616, 6050, 7729, 9200, 10130, 10978, 11468, 12728, 12920, 13025, 13193, 13850, 14506, 14915, 18904, 19128, 20041, 20101, 20227, 21202, 21382, 21682, 24952, 25708, 26917, 28732, 5972, 11593, 11653, 11845, 11907, 12754, 13426, 13513, 13699, 13861, 19628, 19658, 19761, 21108, 25004, 25010, 26777, 28758, 28761, 3800, 3896, 7045, 7452, 7493, 10676, 12660, 13590, 19242, 19738, 21162, 21317, 25194, 26682, 28771, 28819, 3556, 5962, 6828, 6858, 7366, 9844, 10054, 10698, 10860, 11046, 11370, 11430, 17378, 20850, 20892, 25742, 25875, 9164, 9884, 10025, 10865, 11324, 11555, 14393, 14414, 14477, 18097, 18796, 19222, 19558, 21145, 22582, 25027, 25251, 25357, 26701, 3028, 3041, 3538, 5073, 5548, 5740, 6513, 6748, 7257, 7317, 7331, 9642, 9649, 10588, 10842, 10917, 18074, 18854, 21582, 21645, 22667, 3753, 3877, 6506, 6553, 6758, 7226, 10894, 11027, 12686, 12693, 13358, 17353, 17756, 17833, 18115, 19541, 19603, 25675, 2994, 5530, 5782, 6550, 9562, 9817, 11347, 12877, 18006, 18021, 18837, 19022, 22557, 22599, 25629, 3683, 4969, 9020, 12935, 13383, 14375, 17317, 18787, 18997, 19027, 20782, 21037, 21067, 21547, 24862, 25159, 25639, 26647, 28687, 1905, 1989, 5433, 5475, 5771, 9430, 9445, 9557, 10453, 12589, 12619, 12843, 16881, 17966, 20661, 24875, 25115, 1778, 1945, 2905, 3467, 4922, 8678, 8681, 8890, 9779, 17209, 17299, 18553, 21527, 3382, 3406, 4581, 4917, 5331, 6683, 10419, 10567, 17109, 17799, 18590, 20763, 1934, 3613, 5237, 6253, 7183, 8444, 8915, 16846, 2765, 4787, 16634, 17523, 986, 4345, 9371, 9743, 18703, 1703, 1815, 2477, 4342, 4523, 8563, 2711, 4445, 4879, 16621, 886, 1389, 2283, 2622, 4695, 17039, 1262, 2363, 16727, 1245, 4303, 701, 2167, 8367, 415, 623, 8287, 1087$\}$
${\mathcal{S}}_{15,4,7}^2=\{$32512, 31936, 31392, 29632, 30112, 30304, 30352, 31072, 31120, 31312, 31792, 28296, 30032, 12224, 15944, 23944, 24196, 27976, 28036, 28228, 29488, 31756, 15240, 16002, 20384, 24104, 15656, 15908, 23368, 23876, 24130, 27432, 27522, 28194, 31242, 8032, 8080, 14212, 15172, 15682, 15745, 20304, 12080, 15512, 15636, 22216, 22402, 23332, 23425, 23842, 26392, 26404, 26434, 26497, 27412, 27457, 27922, 27937, 28177, 28912, 29452, 29962, 30214, 30217, 30982, 30985, 31237, 31747, 13768, 13992, 15138, 23192, 23640, 14145, 14756, 14996, 15041, 22292, 22978, 23314, 23698, 23713, 23825, 26856, 29957, 14098, 14680, 14904, 15121, 15457, 21416, 21864, 22305, 11504, 12044, 13160, 13668, 13908, 19396, 20161, 21953, 22756, 22840, 23137, 25816, 25828, 26836, 28876, 29443, 11682, 11937, 14562, 19184, 20234, 7400, 7945, 11858, 13016, 13028, 13217, 20020, 21346, 21688, 21716, 21730, 22066, 22097, 25272, 25314, 26802, 28842, 5872, 6640, 7942, 13432, 13492, 13522, 13873, 14452, 17904, 20229, 20952, 22737, 5976, 9200, 11601, 12035, 12728, 12756, 14513, 18316, 19660, 19756, 21112, 21172, 21202, 22642, 24952, 25012, 25042, 25057, 25204, 25297, 25714, 25777, 26737, 28732, 28762, 28774, 28777, 28822, 28825, 28837, 28867, 7708, 7749, 10122, 10956, 11082, 11436, 11466, 12978, 3800, 3812, 11141, 13509, 13593, 19116, 19146, 19249, 19626, 19843, 20852, 20914, 21317, 26766, 28757, 3048, 5932, 6604, 7366, 7450, 7461, 7715, 10922, 11356, 12658, 13603, 18249, 21273, 3512, 3540, 3553, 6764, 6825, 6979, 9836, 9884, 9926, 9929, 10025, 10604, 10652, 10694, 10697, 11366, 11414, 13454, 17866, 19036, 19561, 19609, 21605, 22606, 22669, 25678, 25741, 26701, 28723, 3762, 5073, 5553, 6570, 6822, 7317, 9132, 10857, 10905, 11322, 14382, 14419, 14475, 18026, 18074, 18086, 18089, 18794, 18842, 18854, 18857, 19046, 19094, 19542, 21134, 21155, 22581, 2018, 2996, 3026, 3697, 5066, 5532, 5737, 5785, 5797, 5827, 6505, 6553, 6745, 7225, 9578, 10005, 12878, 12899, 12941, 17260, 17756, 19002, 20885, 21550, 21643, 25134, 25227, 26667, 3442, 5542, 5734, 5782, 6502, 6550, 6742, 6803, 7222, 9052, 9532, 9637, 9667, 10806, 10837, 11317, 12691, 13357, 13387, 14365, 17766, 17814, 18069, 18195, 18777, 18789, 1908, 6332, 6362, 6485, 9062, 9110, 9558, 9875, 10469, 10553, 10595, 12597, 12621, 17242, 17317, 17347, 17722, 19027, 19507, 20819, 21037, 21067, 21533, 22555, 24862, 24877, 24907, 24967, 25117, 25159, 25627, 25639, 26647, 28687, 1989, 2979, 3723, 9018, 18659, 3469, 3662, 5454, 6451, 12827, 13335, 17206, 17717, 20779, 1516, 2958, 3283, 3403, 3629, 4581, 9043, 21015, 2893, 4999, 7183, 8444, 8885, 17101, 1497, 3607, 4725, 4894, 5235, 9387, 10767, 16634, 17607, 1831, 2759, 2843, 3239, 4345, 17935, 18703, 953, 4342, 9487, 16629, 8435, 1627, 2286, 2398, 875, 1598, 2269, 1246, 2235, 1213, 4303, 471, 687, 2167, 446, 1135, 8351, 381, 16479, 4159$\}$
${\mathcal{S}}_{16,4,7}=\{$65024, 63872, 62784, 59264, 60224, 60608, 60704, 62144, 62240, 62624, 63584, 56592, 60064, 31504, 32008, 40832, 48272, 55952, 56072, 56456, 58976, 63512, 16192, 31824, 46864, 48388, 24256, 24352, 30352, 31368, 31876, 47696, 48200, 54864, 55044, 56388, 62484, 16032, 44808, 46728, 47748, 47874, 28048, 28420, 30280, 30466, 31300, 40544, 47408, 47656, 52784, 52808, 52868, 52994, 54824, 54914, 55844, 55874, 56354, 57824, 58904, 59924, 60428, 60434, 61964, 61970, 62474, 63494, 30000, 30896, 43920, 44368, 46660, 28200, 29572, 30244, 31012, 31042, 31266, 31777, 44674, 45896, 46376, 46466, 53712, 54657, 55617, 59914, 61713, 26448, 27344, 28226, 28289, 44580, 45744, 46192, 46626, 47298, 15896, 22408, 23008, 23938, 27522, 29128, 29296, 29890, 30913, 42704, 43720, 44200, 44417, 44609, 45953, 51632, 51656, 52097, 53672, 55457, 57752, 58886, 59537, 59657, 61577, 13792, 14800, 24084, 27969, 29505, 39748, 40258, 45508, 46273, 12080, 15650, 23656, 26308, 26992, 27048, 27076, 27748, 27810, 36320, 37856, 40100, 40460, 40466, 42416, 42440, 42818, 48131, 50544, 50628, 51009, 53604, 54369, 57684, 58449, 58629, 61509, 11232, 24074, 25520, 27425, 29090, 29345, 40225, 43248, 43368, 43428, 44130, 45288, 15116, 15633, 15878, 18400, 20248, 22936, 23128, 25840, 25960, 26020, 28900, 39586, 39617, 39992, 41840, 41896, 42785, 45410, 45665, 49904, 50024, 50084, 50114, 50408, 50594, 50849, 51428, 51554, 51809, 53474, 57464, 57524, 57548, 57554, 57644, 57650, 57674, 57734, 57905, 57929, 57989, 58115, 58409, 58499, 59429, 59459, 61475, 7600, 7624, 13720, 14994, 15412, 15498, 23697, 36504, 36628, 38548, 39256, 39316, 39697, 42340, 6096, 11714, 11924, 13250, 14753, 21848, 21908, 22114, 22289, 22868, 23302, 23813, 25320, 25444, 26250, 26377, 37784, 38666, 40017, 43402, 43570, 43781, 44081, 53532, 54297, 55317, 57514, 7024, 7080, 13652, 13906, 14922, 20114, 26162, 38104, 38284, 41700, 43590, 46117, 7524, 7945, 11096, 11480, 11660, 11852, 13016, 13112, 14085, 14456, 14540, 14726, 14897, 15429, 19348, 19873, 21394, 21688, 22738, 22834, 23173, 26826, 26908, 27673, 28828, 28954, 28969, 29209, 30733, 30739, 36242, 38196, 39212, 39305, 51356, 51482, 51737, 53402, 55307, 57446, 4036, 5992, 6052, 10146, 10177, 13642, 14665, 19668, 19764, 19788, 19794, 20049, 21204, 21300, 21324, 21708, 21804, 22700, 23081, 23619, 25926, 28778, 34648, 35640, 35724, 37714, 38449, 38473, 39028, 42268, 45148, 45222, 45334, 45589, 3824, 6884, 7393, 10936, 12035, 13484, 13490, 13865, 13955, 18136, 19170, 21620, 22085, 25386, 25801, 35540, 36044, 36394, 37496, 37580, 37766, 42182, 43205, 50268, 50454, 50709, 53334, 54279, 11145, 12724, 19148, 19244, 19274, 19754, 20006, 21194, 22641, 25041, 26729, 26774, 34530, 34705, 36169, 37996, 38058, 38213, 39018, 41628, 41754, 41766, 42138, 43068, 43098, 43299, 45114, 45325, 45579, 3937, 7452, 7717, 9848, 10028, 10868, 11546, 11561, 12908, 12970, 14438, 14489, 18100, 18310, 21670, 25180, 25254, 25366, 25660, 25690, 25891, 26682, 27655, 28726, 29191, 34610, 36134, 37298, 37322, 37673, 41578, 42509, 42515, 45203, 49724, 49754, 49814, 49934, 50234, 50318, 50699, 51254, 51278, 51719, 53294, 57374, 3978, 7043, 7282, 7366, 9172, 10674, 12997, 18680, 19121, 19269, 20913, 20934, 25283, 25749, 26709, 34420, 34476, 35281, 35436, 39054, 5958, 6812, 6938, 9905, 11429, 12657, 13091, 13401, 17866, 18028, 19981, 35270, 35497, 35651, 36035, 37478, 37541, 38163, 41411, 41635, 42038, 42251, 49497, 49557, 3448, 3785, 6597, 9586, 11350, 13411, 18213, 18854, 21050, 21771, 34501, 35941, 35989, 37105, 38989, 39175, 1976, 3026, 5065, 5545, 5724, 5785, 7309, 8696, 9830, 10894, 11411, 14366, 17266, 18883, 19030, 19219, 20713, 24973, 34154, 37052, 38963, 41177, 41273, 49337, 49547, 3747, 5571, 6741, 6755, 8946, 9450, 9622, 10597, 10645, 11310, 12502, 14379, 16884, 17321, 17817, 19595, 20835, 21075, 21534, 21555, 24885, 25133, 34213, 35475, 3644, 3674, 5774, 6457, 6710, 9065, 9125, 9813, 10574, 10835, 11339, 12590, 12683, 17977, 22567, 33260, 34382, 35002, 35381, 35870, 37406, 43031, 49269, 49479, 4965, 5013, 5434, 6483, 10467, 17637, 17749, 24755, 34567, 41133, 2918, 4572, 5333, 6325, 7195, 8678, 9107, 9549, 9758, 10781, 12615, 13335, 33621, 34937, 41075, 49259, 49319, 1521, 2537, 3257, 3379, 3463, 3655, 6347, 10791, 16826, 17619, 35101, 37063, 1875, 1933, 2492, 3254, 20573, 20759, 33971, 1002, 1750, 2859, 4787, 5415, 6671, 12349, 17181, 33142, 33483, 36955, 4346, 16750, 18493, 18523, 18703, 20623, 24655, 33182, 892, 1276, 3165, 8437, 17051, 33853, 950, 967, 1454, 1643, 2459, 8975, 9275, 9359, 16606, 16999, 33879, 761, 2286, 5199, 8382, 8539, 33339, 34863, 1374, 1591, 17455, 2263, 687, 4215, 607, 319$\}$
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142607, 168251, 283703, 458815, 8037, 8085, 12003, 12117, 12171, 13797, 14157, 15051, 23211, 23662, 26283, 27051, 27243, 27291, 36533, 36653, 38573, 39341, 39533, 39581, 42413, 42605, 42653, 43253, 43373, 43421, 43613, 45293, 51773, 57533, 69294, 71030, 76014, 100542, 134835, 134955, 136875, 137643, 137835, 137883, 140715, 140907, 140955, 140967, 141555, 141675, 141723, 141735, 141915, 143595, 150075, 155835, 166959, 265466, 265901, 267637, 272621, 273551, 297149, 7910, 8078, 11750, 11990, 12110, 14023, 14791, 20150, 20263, 21941, 22183, 22301, 22901, 22951, 23198, 25973, 26215, 26270, 26870, 26983, 27038, 27230, 28903, 29021, 36278, 36470, 36523, 36638, 38259, 38318, 38510, 38558, 39158, 39278, 39326, 39518, 42230, 42350, 42398, 42590, 43243, 43358, 45278, 50750, 51518, 53438, 57470, 68857, 71447, 76431, 78167, 86327, 100911, 106671, 166075, 281647, 298015, 7651, 7891, 8011, 11731, 13771, 19891, 20083, 20251, 21931, 22123, 22171, 22771, 22891, 22939, 23127, 25518, 25843, 25963, 26007, 26203, 26971, 28891, 36213, 38133, 38253, 38301, 38493, 39261, 42333, 50493, 53373, 68854, 68974, 69022, 69214, 70894, 71902, 74974, 83134, 84094, 99454, 134515, 134935, 136435, 136555, 136603, 136795, 136807, 136855, 137563, 137575, 137623, 140635, 140887, 141655, 143575, 148795, 149047, 149815, 150559, 151675, 151735, 155767, 265461, 265581, 265629, 265821, 267501, 268509, 269391, 271581, 279741, 280701, 296061, 7638, 19830, 20125, 21750, 21863, 21918, 22110, 22765, 22878, 25837, 25950, 26845, 36083, 36203, 36251, 36443, 38123, 38238, 39131, 42203, 50363, 51323, 69031, 69223, 69271, 69391, 71311, 71911, 72079, 72271, 74983, 75151, 75343, 75991, 76111, 78031, 83503, 84151, 84271, 84511, 86191, 90223, 90271, 99511, 99631, 99871, 100471, 100639, 102511, 102559, 106591, 114751, 132092, 164987, 14903, 21358, 21851, 36077, 134379, 164535, 164655, 263162, 265127, 271079, 271247, 13871, 14639, 19687, 19805, 21725, 36062, 66553, 68951, 70871, 70991, 83063, 83231, 86111, 137423, 140495, 148655, 149615, 279223, 279343, 295351, 295543, 295711, 11835, 13623, 14523, 19182, 19675, 66550, 134359, 148255, 164215, 263157, 264039, 264087, 265047, 266727, 266967, 267087, 270807, 7742, 11581, 13501, 14462, 33779, 68815, 77887, 133007, 133967, 278903, 7455, 11447, 13435, 17902, 33773, 132075, 164079, 264911, 7343, 11375, 17383, 18877, 25663, 33758, 133751, 137279, 278767, 295135, 7287, 17371, 18045, 132055, 132535, 147679, 263631, 9181, 66511, 68671, 139583, 9143, 264767, 270527, 5023, 9071, 3003, 4983, 8955, 263487, 266367, 1981, 2941, 4847, 8702, 66367, 1887, 4603, 1783, 2527, 65791$\}$
${\mathcal{S}}_{10,7,4,3}=\{$38, 21, 11$\}$
${\mathcal{S}}_{11,7,4,3}=\{$44, 74, 25, 134, 69, 35$\}$
${\mathcal{S}}_{12,7,4,3}=\{$56, 84, 140, 146, 74, 273, 38, 521, 1029, 2051$\}$
${\mathcal{S}}_{13,7,4,3}=\{$112, 168, 280, 292, 148, 546, 76, 1042, 1057, 529, 2058, 4102, 4105, 2053, 67$\}$
${\mathcal{S}}_{14,11,4,3}=\{$38, 21, 11$\}$
${\mathcal{S}}_{14,7,4,3}=\{$224, 336, 560, 584, 296, 1092, 152, 2084, 2114, 1058, 4116, 4161, 8204, 8210, 8225, 1041, 4106, 134, 2057, 261, 515$\}$
${\mathcal{S}}_{15,11,4,3}=\{$44, 74, 25, 134, 69, 35$\}$
${\mathcal{S}}_{15,7,4,3}=\{$448, 672, 1120, 1168, 592, 2184, 304, 4168, 4228, 2116, 8232, 8322, 16408, 16420, 16450, 16513, 2082, 8212, 8257, 268, 4114, 4129, 522, 2065, 1030, 1033, 517, 259$\}$
${\mathcal{S}}_{16,13,4,3}=\{$38, 21, 11$\}$
${\mathcal{S}}_{16,7,4,3}=\{$896, 1344, 2240, 2336, 1184, 4368, 608, 8336, 8456, 4232, 16464, 16644, 32816, 32840, 32900, 33026, 4164, 16424, 16514, 536, 8228, 8258, 1044, 4130, 2060, 2066, 1034, 518$\}$
${\mathcal{S}}_{17,13,4,3}=\{$44, 74, 25, 134, 69, 35$\}$
${\mathcal{S}}_{17,7,4,3}=\{$1792, 2688, 4480, 4672, 2368, 8736, 1216, 16672, 16912, 8464, 32928, 33288, 65632, 65680, 65800, 66052, 8328, 32848, 33028, 1072, 16456, 16516, 2088, 8260, 4120, 4132, 2068, 1036, 7$\}$
${\mathcal{S}}_{18,13,4,3}=\{$56, 84, 140, 146, 74, 273, 38, 521, 1029, 2051$\}$
${\mathcal{S}}_{18,7,4,3}=\{$3584, 5376, 8960, 9344, 4736, 17472, 2432, 33344, 33824, 16928, 65856, 66576, 131264, 131360, 131600, 132104, 16656, 65696, 66056, 2144, 32912, 33032, 4176, 16520, 8240, 8264, 4136, 2072, 38, 21, 11$\}$
${\mathcal{S}}_{19,13,4,3}=\{$112, 168, 280, 292, 148, 546, 76, 1042, 1057, 529, 2058, 4102, 4105, 2053, 67$\}$
${\mathcal{S}}_{19,7,4,3}=\{$7168, 10752, 17920, 18688, 9472, 34944, 4864, 66688, 67648, 33856, 131712, 133152, 262528, 262720, 263200, 264208, 33312, 131392, 132112, 4288, 65824, 66064, 8352, 33040, 16480, 16528, 8272, 4144, 44, 74, 25, 134, 69, 35$\}$
${\mathcal{S}}_{12,8,4,4}=\{$3084, 780, 1546, 2314, 2566, 2569, 204, 1286, 1289, 1541, 3075, 2309, 170, 771, 105, 60, 90, 102, 150, 153, 165, 195, 85, 51, 15$\}$
${\mathcal{S}}_{13,8,4,4}=\{$6168, 1560, 3092, 4628, 5132, 5138, 408, 2572, 2578, 3082, 6150, 4618, 212, 785, 1542, 120, 308, 332, 338, 1169, 1289, 172, 178, 202, 390, 649, 2129, 2309, 298, 4145, 4169, 4229, 4355, 581, 2089, 2179, 102, 1061, 1091, 547, 15$\}$
${\mathcal{S}}_{14,8,4,4}=\{$12336, 3120, 6184, 9256, 10264, 10276, 816, 5144, 5156, 6164, 12300, 9236, 240, 424, 1570, 3084, 616, 664, 676, 2338, 2578, 2593, 344, 356, 404, 780, 1298, 1313, 1553, 4258, 4618, 12291, 596, 2321, 8290, 8338, 8353, 8458, 8710, 8713, 204, 1162, 3075, 4178, 4193, 4241, 4358, 4361, 4613, 2122, 2182, 2185, 8273, 8453, 771, 1094, 1097, 1157, 2117, 60, 195, 51, 15$\}$
${\mathcal{S}}_{15,8,4,4}=\{$24672, 6240, 12368, 18512, 20528, 20552, 1632, 10288, 10312, 12328, 24600, 18472, 480, 848, 3140, 6168, 1232, 1328, 1352, 4676, 5156, 5186, 688, 712, 808, 1560, 2596, 2626, 3106, 8516, 9236, 9281, 24582, 1192, 4642, 16580, 16676, 16706, 16916, 16961, 17420, 17426, 17441, 408, 2324, 2369, 3089, 6150, 8356, 8386, 8482, 8716, 8722, 8737, 9226, 12293, 4244, 4289, 4364, 4370, 4385, 4625, 5129, 16546, 16906, 18437, 20483, 1542, 2188, 2194, 2209, 2314, 2569, 8465, 10243, 120, 4234, 16529, 16649, 390, 773, 8329, 1157, 1283, 643, 101, 86, 51, 46, 75, 29$\}$
${\mathcal{S}}_{16,12,4,4}=\{$49164, 12300, 24586, 36874, 40966, 40969, 3084, 20486, 20489, 24581, 49155, 36869, 780, 1546, 12291, 2314, 2566, 2569, 204, 1286, 1289, 1541, 3075, 2309, 170, 771, 105, 60, 90, 102, 150, 153, 165, 195, 85, 51, 15$\}$
${\mathcal{S}}_{17,12,4,4}=\{$98328, 24600, 49172, 73748, 81932, 81938, 6168, 40972, 40978, 49162, 98310, 73738, 1560, 3092, 12305, 24582, 4628, 5132, 5138, 18449, 20489, 408, 2572, 2578, 3082, 6150, 10249, 33809, 36869, 4618, 66065, 66569, 67589, 69635, 212, 1542, 9221, 33289, 34819, 120, 308, 332, 338, 16901, 17411, 172, 178, 202, 390, 8707, 298, 101, 15$\}$
${\mathcal{S}}_{18,12,4,4}=\{$196656, 49200, 98344, 147496, 163864, 163876, 12336, 81944, 81956, 98324, 196620, 147476, 3120, 6184, 24610, 49164, 9256, 10264, 10276, 36898, 40978, 40993, 816, 5144, 5156, 6164, 12300, 20498, 20513, 24593, 67618, 73738, 196611, 9236, 36881, 132130, 133138, 133153, 135178, 139270, 139273, 240, 424, 3084, 18442, 49155, 66578, 66593, 67601, 69638, 69641, 73733, 616, 664, 676, 33802, 34822, 34825, 132113, 135173, 344, 356, 404, 780, 12291, 17414, 17417, 18437, 596, 33797, 204, 3075, 771, 60, 195, 51, 15$\}$
${\mathcal{S}}_{19,12,4,4}=\{$393312, 98400, 196688, 294992, 327728, 327752, 24672, 163888, 163912, 196648, 393240, 294952, 6240, 12368, 49220, 98328, 18512, 20528, 20552, 73796, 81956, 81986, 1632, 10288, 10312, 12328, 24600, 40996, 41026, 49186, 135236, 147476, 147521, 393222, 18472, 73762, 264260, 266276, 266306, 270356, 270401, 278540, 278546, 278561, 480, 848, 6168, 36884, 36929, 49169, 98310, 133156, 133186, 135202, 139276, 139282, 139297, 147466, 196613, 1232, 1328, 1352, 67604, 67649, 69644, 69650, 69665, 73745, 81929, 264226, 270346, 294917, 327683, 688, 712, 808, 1560, 24582, 34828, 34834, 34849, 36874, 40969, 135185, 163843, 1192, 67594, 264209, 266249, 408, 6150, 12293, 133129, 18437, 20483, 1542, 10243, 120, 390, 773, 1157, 1283, 643, 101, 86, 51, 46, 75, 29$\}$
[^1]: In the cited paper the upper bound was stated for linear [[`FDRM`]{}]{} codes only. However, the statement is also true without this assumption, as observed by e.g. the same authors later.
[^2]: In [@he2020note] it was claimed that [@li2019construction Theorem 3.1] is incorrect. However, the stated counterexample is flawed since the example for $C_2$ is not of the form specified in [@li2019construction Theorem 3.1] since there are e.g. non-zero entries in the first four columns of $a_1$.
|
---
abstract: 'The appearance of debris disks around distant stars depends upon the scattering/phase function (SPF) of the material in the disk. However, characterizing the SPFs of these extrasolar debris disks is challenging because only a limited range of scattering angles are visible to Earth-based observers. By contrast, Saturn’s tenuous rings can be observed over a much broader range of geometries, so their SPFs can be much better constrained. Since these rings are composed of small particles released from the surfaces of larger bodies, they are reasonable analogs to debris disks and so their SPFs can provide insights into the plausible scattering properties of debris disks. This work examines two of Saturn’s dusty rings: the G ring (at 167,500 km from Saturn’s center) and the D68 ringlet (at 67,600 km). Using data from the cameras onboard the Cassini spacecraft, we are able to estimate the rings’ brightnesses at scattering angles ranging from 170$^\circ$ to 0.5$^\circ$. We find that both of the rings exhibit extremely strong forward-scattering peaks, but for scattering angles above 60$^\circ$ their brightnesses are nearly constant. These SPFs can be well approximated by a linear combination of three Henyey-Greenstein functions, and are roughly consistent with the SPFs of irregular particles from laboratory measurements. Comparing these data to Fraunhofer and Mie models highlights several challenges involved in extracting information about particle compositions and size distributions from SPFs alone. The SPFs of these rings also indicate that the degree of forward scattering in debris disks may be greatly underestimated.'
author:
- 'Matthew M. Hedman, Christopher C. Stark'
title: |
Saturn’s G and D rings provide nearly complete\
measured scattering/phase functions of nearby debris disks
---
Introduction
============
The material in extrasolar debris disks consists of fine particles lofted from the surfaces of planetesimals by collisions and other processes. This debris can exhibit a wide variety of structures, including rings [e.g., @kalas2005; @schneider2006], warps [e.g., @golimowski2006; @krist2005], and brightness asymmetries [e.g., @hines2007; @kalas2007]. One of the most common and pronounced asymmetries in moderately-inclined debris disks is a brightness inequality along the projected minor axis [e.g., @schneider2014]. This asymmetry is likely the result of anisotropic “forward" scattering of starlight by dust grains, with the near side of the disk appearing brighter than the far side.
Brightness variations due to the anisotropic scattering properties of the circumstellar debris can be quantified by the material’s scattering/phase function (SPF). This function specifies the relative brightness of the material as a function of either the phase angle $\alpha$ (i.e. the angle between the rays followed by the incident and scattered starlight) or its supplement, the scattering angle $\theta$. If the SPF of the disk material is sufficiently well known, then the brightness variations due to the changing lighting geometry could be removed, revealing the density variations that could be generated by unseen planets [@stark2014]. The SPF can also be used to constrain the apparent albedo, and thus the mass, of a disk. Finally, if debris disks are strongly forward-scattering as recent observations suggest [@stark2014; @perrin2014], the forward-scattered starlight from regions more than 5 AU from the star may create additional pseudo-zodiacal light in edge-on systems with which future exoEarth-imaging missions must contend [@stark2015]. The SPF of debris dust therefore plays a critical role in determining the dynamical state of a planetary system, constraining its composition, and designing future missions.
Unfortunately, the SPFs of the material in exoplanetary systems are poorly constrained. To determine the SPF, one must observe the disk over a large range of scattering angles, especially those near the forward-scattering peak. For debris disks not observed edge-on, the range of scattering angles that nature provides are limited to a range of values around $90^\circ$ that depends upon the inclination of the system. Currently-available observations of debris disks reveal that their SPFs vary slowly at scattering angles around $90^{\degree}$, with typical Henyey-Greenstein fits adopting values of $0.0 < g \lesssim 0.3$ . These results are somewhat surprising, given that classic Mie theory predicts highly forward-scattering dust, with the first moment of the SPF, $\langle\cos{\theta}\rangle \sim 0.9$. While there are SPFs that exhibit both a strong forward-scattering peak and a flat phase function near $\theta\sim90\degree$, these findings still highlight the difficulties involved in extrapolating the measured SPF of exoplanetary disks beyond the narrow observed range of phase angles. Larger ranges of phase angles are potentially observable in edge-on disks, but in these situations there are still limitations imposed by the inner working angle of the observations. Furthermore, in these systems the SPF becomes degenerate with the radial variations in the dust density and size distribution. Estimates of our local zodiacal cloud’s scattering phase function suffer from these same degeneracies [e.g., @hong1985].
In lieu of being able to measure the complete SPF of debris disks directly, it is useful to consider dusty tenuous rings that surround the giant planets. Unlike the famous dense rings of Saturn, which are composed mostly of pebble-to-boulder-sized chunks of ice [see @cuzzi2009 and references therein], these much fainter and more tenuous rings appear to contain much smaller particles. Indeed, these tenuous rings appear much brighter when viewed at high phase angles, which indicates that most of the visible particles are less than 100 microns across. Such small particles will be destroyed or ejected from the planetary system on timescales less than a few thousand years [@burns2001; @horanyi2009], so they need to be continuously supplied to the ring from larger objects. Indeed, many tenuous rings contain small moons that are likely the largest of those source bodies, and visible dusty ring particles probably consist of material released from the surfaces of these objects by collisions with meteoroids, much like how the material in debris disks is thought to arise from the collisions among planetesimals in orbit around the star. Furthermore, these dusty rings have been observed over a broad range of viewing geometries by spacecraft, yielding a much more complete picture of their SPFs. Analyses of Voyager and ground-based data yielded sparse phase curves of both Saturn’s E and G rings [@showalter1991; @showalter1993; @throop1998], but even these were enough to clearly show the forward-scattering peak. More recently, the combined data from multiple missions have provided well-sampled phase curves of both Jupiter’s main ring [@throop2004] and Saturn’s F ring [@french2012], both of which show a clear forward-scattering peak and a relatively flat SPF for phase angles less than 120$^\circ$. However, both rings show significant longitudinal and/or temporal variations in their brightness, which complicates efforts to interpret the details of these SPFs.
In this paper we will use data from the cameras onboard Cassini spacecraft to constrain the SPF of two of Saturn’s tenuous dusty rings over a range of scattering angles from 170$^\circ$ to 0.5$^\circ$ (i.e. phase angles from 10$^\circ$ to $179.5^\circ$), thereby probing further up the forward-scattering peak than ever before. Specifically, we will examine here the inner edge of the G ring and a narrow ringlet called D68 in the D ring. The G ring is located at approximately 167,500 km from Saturn’s center, outside Saturn’s main ring system (see Figure \[dgcontext\]). This ring probably consists of micrometeorite-ejected debris from larger objects (including a small moon called Aegaeon) confined near the inner edge of the ring by a co-rotation resonance with Mimas [@showalter1993; @lissauer2000; @hedman2007; @hedman2010]. D68, on the other hand, is found only 67,600 km from Saturn’s center (see Figure \[dgcontext\]) and is the innermost discrete ringlet in the ring system [@showalter1996; @hedman2007d; @hedman2014]. Unlike the G ring, there is no obvious local source of dust in this region. However, D68 is embedded in a broad sheet of dust extending interior to Saturn’s main rings, and so it probably represents material that has become trapped as it drifted inwards towards the planet. While Saturn possesses many other dusty rings, we have chosen to focus of these two because they are comparatively narrow, which means the relevant signals can be more easily extracted from instrumental and astronomical backgrounds. Furthermore, while both these rings exhibit some brightness variations, neither one is as obviously clumpy and time-variable as the F ring [@french2012], or as strongly asymmetric as the E ring [@hedman2012]. These two rings therefore promise to provide reasonably coherent phase functions that could be useful for modeling exoplanetary debris disks.
In Section \[observations\] we describe the Cassini observations and reduction techniques, as well as our methods for measuring the SPF. In Section \[results\] we present the measured scattering phase functions for D68 and the G ring, while in Section \[fits\] we attempt to fit these SPFs to Henyey-Greenstein functions, laboratory-measured phase functions, and models based on Faunhofer diffraction and Mie theory. We discuss some implications of the measured SPFs for exoplanetary disks in Section \[discussion\] and summarize our findings in Section \[conclusions\].
Observations & Data Reduction {#observations}
=============================
Data sources
------------
The raw data for this investigation consists of images obtained by the Imaging Science Subsystem (ISS) onboard the Cassini Spacecraft. We performed a comprehensive search for images of both the G ring and D68, and then selected images with sufficient signal to noise to detect these ring features. This search included images obtained by both the Narrow Angle Camera (NAC) and the Wide Angle Camera (WAC) components of the Imaging Science Subsystem [@porco2004; @west2010]. Both of these cameras have multiple filters, but for the purposes of this analysis we only considered images obtained through the clear filters or the RED filter on the WAC. Note that the effective wavelength of the RED filter (647 nm) is close to the effective wavelengths of the NAC and WAC clear filters (651 nm and 634 nm, respectively, Porco [*et al.*]{} 2004). Thus the measured brightness through these different filters should be insensitive to the rings’ spectral slopes. Each image was calibrated using the standard CISSCAL calibration routines [@porco2004; @west2010], which remove instrumental backgrounds, flat-field the image and convert the raw data numbers into $I/F$, a measure of surface reflectance that is unity for a Lambertian surface viewed and illuminated at normal incidence (see [http://pds-rings.seti.org/cassini/iss/calibration.html]{}). Each image was also geometrically navigated using the appropriate SPICE kernels [@acton1996], and the detailed geometry of the image was refined based on the locations of stars and/or sharp ring edges in the field of view.
From images to radial profiles
------------------------------
The data from these images were reduced to profiles of the ring’s observed brightness as a function of ring radius (distance from the planet’s spin axis). For images obtained at scattering angles greater than 15$^\circ$ (i.e. phase angles less than 165$^\circ$) and ring opening angles greater than 0.5$^\circ$, these profiles were computed by simply averaging the brightness in the image over a range of longitudes, avoiding any part of the ring that was in shadow or had low radial resolution due to projection effects. During this process, we also compute the scattering angle at the ring $\theta$ and the opening angle of the ring to the spacecraft $B$. The ring opening angle is relevant to this analysis because the total path length through the ring along the line of sight is proportional to $1/\sin|B|$. Hence, for any tenuous ring the total amount of material along the line of sight and the apparent brightness of the ring are proportional to $1/\sin|B|$. We therefore multiply the observed ring brightness by $\sin|B|$ to convert the observed $I/F$ into a “normal $I/F$" (also denoted $\mu I/F$). For low optical depth rings like the G and D rings, multiplying through by this factor removes the dependance on $B$ (indeed, the normal $I/F$ can be regarded as the $I/F$ that would be measured if the ring was viewed face-on, with $B=90^\circ$) and so this quantity should be independent of viewing geometry for a given scattering angle.
A small number of G-ring images were obtained when the spacecraft was less than 0.5$^\circ$ from the ringplane (see for example Figure \[dgcontext\]d). In these nearly edge-on images, the finite vertical extent of the ring can influence the detailed shape of a brightness profile derived using the above methods, and these changes can interfere with efforts to quantify the ring’s brightness. Hence we use a different set of procedures for these images. First, we sum the $I/F$ values along pixel columns orthogonal to the radial direction to produce a profile of the vertically integrated $I/F$ versus radius. In such a profile, the signal at a given radius $r_0$ includes contributions from material at all radii $r>r_0$. However, these projection effects can be removed using an onion-peeling algorithm, which iteratively estimates the signal at each radius and removes that material’s contribution from the rest of the data [@showalter1985; @showalter1987; @depater2004; @hedman2012]. When properly normalized, this algorithm yields a profile of the ring’s normal $I/F$ that can be directly compared with the data obtained at higher elevation angles. However, the repeated differencing of the vertically-integrated $I/F$ profile also amplifies small noise fluctuations in the final brightness data, so we only use this method on the lowest-elevation images where other methods of generating the brightness profile clearly distort the rings’ shape.
We also used different reduction procedures on images where the rings were viewed at scattering angles less than 15$^\circ$ (phase angles greater than 165$^\circ$). In these cases, the SPF of the relevant rings is very steep and so the brightness of the ring could vary significantly over the range of scattering angles visible in a single image. Hence, instead of reducing each image to a single profile, we extracted multiple brightness profiles from each image. This was done in slightly different ways for the two ring features.
For the narrow D68 ringlet, the low-scattering-angle data consist primarily of individual images that captured an entire ring ansa (like that shown in Figure \[dgcontext\]b). Thus multiple profiles were obtained from each image by simply averaging together different ranges of ring longitudes within each image.
For the more diffuse G ring, individual images were less likely to capture a range of radii and longitudes needed to yield suitable ring profiles. Thus we instead considered four observation sequences that were particularly informative about the rings’ scattering/phase function. Three of these sequences (Rev 028[^1] HIPHWAC, Rev 173 HIPHWAC and Rev 195 HIPHASEC) were mosaics of many images that together captured nearly the entire ring, while the fourth observation (Rev 028 HIPHASE002) was a movie where the camera observed the brightness of one part of the G ring change as the Sun moved relative to the observation point. Since the relevant data could be spread across multiple images, for these observations it made more sense to compute profiles by averaging together data from several images covering a restricted range of phase angles rather than just consider a limited range of longitudes in each image. While this procedure was straightforward for the movie sequence, for the three large-scale mosaics there was an added complication; between two and four disjoint portions of the ring could have the same phase angle (see Figure \[gphaselab\]). We therefore separated the relevant datasets into four “quadrants” prior to computing the averaged profiles.
From Profiles to Brightness Estimates
-------------------------------------
We used automated procedures the extract estimates of the relevant ring’s brightness from each of the profiles derived above. Given the differences in the shapes and environments of D68 and the G ring, different algorithms were employed to quantify the brightness of the two ring structures.
D68 is a relatively isolated ringlet embedded in a rather homogeneous sheet of material (see Figure \[d68prof\]), and the ringlet itself is unresolved in almost all Cassini images [@hedman2007d; @hedman2014]. Quantities like the ringlet’s peak brightness therefore depend on the image resolution and are not ideal for this investigation. Hence we instead quantify the brightness of this feature in terms of its normal equivalent width (NEW), which is the ringlet’s radially-integrated normal $I/F$ above any smoothly varying background level. This parameter is independent of image resolution and elevation angle for any ring with sufficiently low optical depth. In practice, we use two different methods to compute the normal equivalent width of D68 from each profile:
- [**Int Method:**]{} The brightness of a selected part of the brightness profile is directly integrated after removing a quadratic background fit to two 1000-km-wide zones on either side of the selected region (see Figure \[d68prof\]). For scattering angles greater than 15$^\circ$, the selected region ran from 67,500 km to 67,750 km, and so the background was based on a fit to the zones between 66,500-67,500 km and 67,750-68,750 km. Due to differences in the image resolution and the appearance of the ringlet, different ranges were used for profiles obtained at low scattering angles. For scattering angles between 3$^\circ$ and 15$^\circ$, the integral was compute for the region between 67,000-68,000 km, and for scattering angles below 3$^\circ$, the region was 66,000-68,500 km.
- [**Fit Method:**]{} The profile of D68 is fit to a Lorentzian peak plus linear background. The integrated brightness under the peak is then determined from the product of the peak amplitude and the peak width. This fit was performed on the data within the same radial range that was integrated over with the Int Method.
As discussed in more detail below, these two methods generally yield similar estimates of the ringlet’s normal equivalent width.
The G ring, on the other hand, is an extended asymmetric feature (see Figure \[gprof\]), making it more difficult to quantify in terms of an equivalent width. In particular, we cannot fit the ring’s asymmetric profile to a simple shape like a Gaussian or a Lorentzian peak. Also, the outer edge is very diffuse, gradually fading out against the background E ring over several tens of thousands of kilometers, so it is difficult to isolate the ring signal from background trends in this region. In order to avoid these difficulties, we quantify the brightness of the G ring in terms of the [*brightness contrast*]{} across the ring’s relatively sharp inner edge. This contrast is computed by fitting linear trends to the brightness-versus-radius curves in two regions on either side of the edge (164,000-166,000 km and 169,000-171,000 km) and extrapolating both those trends to 167,500 km. The difference between these two numbers provides an estimate of the brightness contrast across the edge, and the uncertainty in this value was estimated from the errors on the two linear fits.
Selection of high-quality brightness estimates
----------------------------------------------
The above procedures will only provide sensible estimates of the relevant ring’s brightness if they cover a sufficient range of radii, so we visually examined the relevant profiles to ensure that they contained sufficient data to fit the background models for D68 or determine the linear trends on either side of the G ring. These algorithms would also only yield suitable brightness estimates if the data had adequate signal-to-noise, and so additional criteria were used to exclude less reliable measurements.
For the observations at scattering angles greater than 15$^\circ$, where the relevant ring features were subtle, we visually screened the relevant profiles and removed data where spurious peaks from cosmic rays corrupted the data in the vicinity of the relevant ring structure. For the G ring, we also excluded profiles where the inner slope was negative or the outer slope was positive, because such results indicate that the images had complex backgrounds that might contaminate the final calculations. Furthermore, we excluded G-ring images where the fractional error on the contrast was greater than 20% if the scattering angle was greater than 70$^\circ$, or more than 5% the contrast if the scattering angle was smaller than 70$^\circ$. Finally, we excluded the G-ring images that captured the bright arc near the ring’s inner edge [@hedman2007] because this analysis is concerned with the background G ring.
For observations made at scattering angles less than $15^\circ$, the rings are considerably brighter and so signal-to-noise was less of a concern. Still, we needed to exclude observations where instrumental backgrounds or signals from Saturn’s limb contaminated the profiles. For D68, we simply excluded any observation where the background brightness profile did not increase with distance from the planet (indicating a significant background from Saturn’s limb) or where there was no peak in the profile $<$ 500 km wide at the expected location for D68. For the G ring, we again required that the inner slope to be positive or the outer slope to be negative, and we excluded G-ring images where the error on the contrast was greater than 5% the estimated contrast. In this case, we did not deliberately exclude regions containing the G-ring arc because at low scattering angles the arc is a small perturbation on the overall profile.
Results
=======
------------- ------------------ ---------- ------- --------- --------- --
Image Name Ephemeris Time $\theta$ B NEW-Int NEW-Fit
(deg) (deg) (m) (m)
N1721658014 396235731.719373 10.67 -9.57 6.59 5.42
N1547167883 221746779.842845 17.32 25.93 3.03 2.75
N1547168273 221747169.840369 17.32 25.90 2.58 2.72
N1547167493 221746389.845322 17.32 25.97 2.95 2.77
N1547167103 221745999.847799 17.33 26.01 3.43 2.85
N1547166713 221745609.850276 17.33 26.04 2.62 2.88
N1547166323 221745219.852752 17.33 26.08 3.05 2.95
N1547165933 221744829.855229 17.33 26.12 3.11 2.90
N1547165543 221744439.857706 17.33 26.15 3.02 2.92
------------- ------------------ ---------- ------- --------- --------- --
: Normal Equivalent Width estimates for D68 derived from entire images, almost all of which were obtained at scattering angles above 15$^\circ$ (Full Table available in online supplement).[]{data-label="dtab1"}
------------- ------------------ ---------- ------- --------------- ---------- ----------
Image Name Ephemeris Time $\theta$ B Longitude NEW-Int NEW-Fit
(deg) (deg) Range (deg) (m) (m)
W1537005177 256801306.743551 0.56 15.42 293.41-299.88 10048.16 15297.32
W1537006505 256801339.747224 0.61 15.40 293.37-299.97 9060.68 15362.94
W1537005177 336715092.863261 0.63 15.37 286.71-293.29 8709.91 12524.48
W1537006505 336719142.834170 0.68 15.35 286.66-293.25 7779.63 12043.77
W1537005177 414063384.998397 0.69 15.31 280.00-286.59 7818.31 10072.35
W1537005177 414063701.031543 0.73 15.25 273.29-279.88 7119.29 9211.29
W1537006505 211584138.991393 0.74 15.29 280.06-286.54 7002.43 9404.19
W1537005177 211585466.986816 0.76 15.20 266.71-273.17 6876.74 9798.16
W1537006505 211607287.795406 0.78 15.23 273.34-279.94 6584.09 8547.55
------------- ------------------ ---------- ------- --------------- ---------- ----------
: Normal Equivalent Width estimates for D68 derived from observations at scattering angles below 15$^\circ$ (Full Table available in online supplement).[]{data-label="dtab2"}
------------- ------------------ ---------- ------- ----------- -----------
Image Name Ephemeris Time $\theta$ B Contrast Error
(deg) (deg) (10$^-6$) (10$^-6$)
N1537372644 211951594.636115 17.74 9.91 2.90561 0.06985
N1537373650 211952600.629690 17.78 9.90 2.85942 0.06722
N1537374656 211953606.623264 17.83 9.88 2.78850 0.07595
N1537375662 211954612.616838 17.87 9.86 2.81788 0.06576
N1537378680 211957630.597561 18.00 9.81 2.81277 0.06664
N1537379686 211958636.591135 18.05 9.80 2.64955 0.06149
N1537380692 211959642.584710 18.09 9.78 2.74368 0.07522
N1537381698 211960648.578284 18.14 9.76 2.87448 0.08296
N1537382704 211961654.571858 18.18 9.75 2.80607 0.07743
N1537383710 211962660.565433 18.23 9.73 2.79316 0.08281
------------- ------------------ ---------- ------- ----------- -----------
: Normal $I/F$ contrast across the G-ring’s inner edge derived from observations at scattering angles above 15$^\circ$ (Full Table available in online supplement).[]{data-label="gtab1"}
--------------------- ---------- ---------- ----------- -----------
Observation Quadrant $\theta$ Contrast Error
Sequence (deg) (10$^-6$) (10$^-6$)
Rev 028 HIPHWAC001 2 0.80 2046.49 46.83
Rev 028 HIPHASE002 3 1.00 1265.64 35.21
Rev 028 HIPHWAC001 4 1.20 821.38 10.79
Rev 028 HIPHASE002 1 1.20 819.20 32.94
Rev 195 HIPHASEC001 2 1.20 1124.98 39.04
Rev 028 HIPHWAC001 4 1.40 614.81 5.45
Rev 028 HIPHWAC001 1 1.60 466.21 12.89
Rev 028 HIPHWAC001 4 1.60 443.47 5.28
Rev 195 HIPHASEC001 4 1.60 549.82 2.04
--------------------- ---------- ---------- ----------- -----------
: Normal $I/F$ contrast across the G-ring’s inner edge derived from observations at scattering angles below 15$^\circ$ (Full Table available in online supplement).[]{data-label="gtab2"}
---------------- --------------- ---- ----------- ----------- ----------- -----------
$\theta$ Range Ave. $\theta$ N NEW-Int Error$^a$ NEW-Fit Error$^a$
(deg) (deg) (m) (m) (m) (m)
0.55 0.60 0.56 1 10048.161 5024.081 15297.324 7648.662
0.60 0.65 0.62 2 8885.293 3141.426 13943.709 4929.846
0.65 0.70 0.68 2 7798.970 2757.352 11058.062 3909.615
0.70 0.75 0.73 2 7060.861 2496.392 9307.736 3290.782
0.75 0.80 0.78 3 6733.740 1943.863 9697.604 2799.457
0.80 0.85 0.82 2 6281.713 2220.921 9759.794 3450.608
1.45 1.75 1.65 12 909.391 131.259 955.829 137.962
1.75 2.00 1.85 17 652.560 79.134 639.408 77.540
2.00 2.50 2.31 7 255.434 48.273 296.533 56.040
3.00 4.00 3.63 22 64.945 6.923 64.655 6.892
4.00 5.00 4.54 8 46.305 8.186 45.396 8.025
5.50 6.50 6.04 4 17.479 4.370 18.063 4.516
6.50 7.50 6.88 18 13.987 1.648 14.279 1.683
7.50 8.50 8.10 17 14.608 1.771 11.919 1.445
9.00 11.00 9.75 15 7.561 0.976 7.358 0.950
18.00 20.00 18.61 33 2.558 0.223 2.550 0.222
20.00 25.00 20.58 19 2.442 0.280 2.369 0.272
25.00 30.00 28.28 15 1.781 0.230 1.732 0.224
30.00 35.00 32.75 84 1.623 0.089 1.543 0.084
35.00 40.00 37.38 36 1.368 0.114 1.419 0.118
40.00 50.00 42.35 66 1.003 0.062 1.102 0.068
50.00 60.00 55.94 38 0.627 0.051 0.739 0.060
60.00 70.00 64.94 6 0.289$^b$ 0.059 0.285$^b$ 0.058
70.00 90.00 77.04 54 0.450 0.031 0.551 0.038
90.00 120.00 107.91 14 0.318 0.042 0.375 0.050
120.00 140.00 136.56 83 0.345 0.019 0.347 0.019
140.00 160.00 152.45 49 0.284 0.020 0.271 0.019
160.00 180.00 168.04 4 0.334 0.083 0.269 0.067
---------------- --------------- ---- ----------- ----------- ----------- -----------
: Average Normal Equivalent Width estimates for D68.[]{data-label="dtab3"}
$^a$ Error computed assuming a 50% uncertainty in each measurement.
$^b$ Measurements questionable (see text).
---------------- --------------- ---- ----------- ----------- ------------- -----------------
$\theta$ Range Ave. $\theta$ N Contrast Error$^a$ Scatter$^b$ Norm. Error$^c$
(deg) (deg) $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-6}$
0.60 0.90 0.80 1 2046.485 46.831 — 1023.243
0.90 1.10 1.00 1 1265.638 35.206 — 632.819
1.10 1.30 1.20 3 840.770 9.916 — 242.709
1.30 1.50 1.40 1 614.805 5.447 — 307.403
1.50 1.70 1.60 3 534.560 1.879 — 154.314
1.70 1.90 1.80 3 402.540 1.536 — 116.203
1.90 2.10 2.00 5 341.047 1.380 126.893 76.260
2.10 2.30 2.20 5 247.076 1.558 60.313 55.248
2.30 2.50 2.40 6 190.889 1.133 36.065 38.965
2.50 2.90 2.68 8 137.794 0.717 23.698 24.359
2.90 3.50 3.20 9 73.509 0.342 20.430 12.251
3.50 3.90 3.65 4 41.069 0.473 14.701 10.267
3.90 4.50 4.18 8 26.370 0.332 7.590 4.662
4.50 4.90 4.72 5 19.201 0.270 6.476 4.293
4.90 5.90 5.48 8 14.154 0.143 4.346 2.502
5.90 7.00 6.37 6 11.195 0.112 1.922 2.285
7.00 8.00 7.50 6 9.504 0.111 1.328 1.940
8.00 10.00 8.70 4 7.608 0.154 0.981 1.902
10.00 14.00 12.00 3 4.952 0.072 0.000 1.429
17.50 18.50 18.12 15 2.789 0.017 0.064 0.360
18.50 19.50 18.92 18 2.696 0.014 0.123 0.318
19.50 20.50 19.95 16 2.484 0.015 0.084 0.310
45.00 54.00 50.86 9 0.721 0.008 0.035 0.120
54.00 65.00 56.46 22 0.551 0.002 0.033 0.059
70.00 90.00 80.41 4 0.447 0.032 0.024 0.112
90.00 110.00 99.09 1 0.371 0.017 — 0.185
110.00 130.00 121.59 26 0.371 0.003 0.057 0.036
130.00 150.00 136.47 9 0.271 0.001 0.030 0.045
150.00 165.00 159.02 10 0.286 0.008 0.046 0.045
165.00 180.00 166.09 5 0.311 0.003 0.016 0.070
---------------- --------------- ---- ----------- ----------- ------------- -----------------
: Average Normal $I/F$ contrast across the G-ring’s inner edge.[]{data-label="gtab3"}
$^a$ Error computed by propagating errors on individual measurements.
$^b$ $rms$ scatter of measurements.
$^c$ Error computed assuming a 50% uncertainty in each measurement.
The above procedures yielded 751 estimates of D68’s Normal Equivalent Width and 225 estimates of the brightness contrast across the G-ring’s inner edge. These estimates, along with the relevant geometrical parameters, are provided in Tables \[dtab1\]-\[gtab2\]. These individual estimates of the rings’ brightness are plotted as a function of phase angle as the small data points in Figures \[d68phase\] and \[gphase\]. For both the G ring and the D68 Fit measurements, the scatter of these data points around the mean trend is less than 50% (the D68 Int measurements having somewhat larger scatter). This dispersion, while small enough for the purposes of this analysis, is larger than the expected statistical errors in these parameters based on the relevant fits, indicating that systematic uncertainties dominate the scatter. For D68, the observed scatter is comparable to the known longitudinal asymmetries in this ring feature [@hedman2014], so real brightness variations within the ringlet are probably responsible for most of the observed dispersion in the D68 Fit brightness estimates. The larger scatter in the D68 Int measurements probably arise because this method is more sensitive to structures in the background under the ring. Similarly, the scatter in the G-ring data is probably associated with systematic errors in the brightness contrast estimates due to varying background signals in the images.
To further facilitate the analysis of these brightness data, we define a series of bins in scattering angle and compute the average brightness of the relevant ring feature in each bin, along with the corresponding uncertainty in this average brightness level. This uncertainty is estimated by assuming each data point has an independent systematic error of 50%. This should provide a conservative estimate of the uncertainty in the SPF because the $rms$ dispersion of the measurements within each bin (which is predominantly due to the systematic phenomena discussed above) rarely exceeds 50%. These estimates of the rings’ brightness are provided Tables \[dtab3\] and \[gtab3\] and are shown as points with error bars in Figures \[d68phase\] and \[gphase\]. Both sets of brightness data span a broad range of scattering angles between 0.5$^\circ$ and 170$^\circ$, and the brightness decreases over four orders of magnitude with increasing scattering angle. The one notable exception to this generally decreasing trend are the six estimates of the D68’s brightness between scattering angles of 60$^\circ$ and 70$^\circ$, which fall about a factor of two below the measurements at slightly higher and lower scattering angles. Thus far, we have been unable to identify any instrumental or analytical phenomenon that could explain these discrepant results, but such a narrow dip in the phase function seems unphysical, and so we will treat these measurements with suspicion for the remainder of this investigation.
Our brightness estimates of D68 and the G ring are reasonably consistent with the limited Voyager observations of these features [@showalter1993; @showalter1996; @hedman2007d]. The observed trends in the brightness of these ring features are also similar to the previously published phase curves of Jupiter’s main ring and Saturn’s F ring [@throop2004; @french2012]. All these rings contain a strong forward-scattering peak together with a relatively flat phase function for scattering angles above about 60$^\circ$. However, the extensive data for D68 and the G ring compiled here now enable us to examine these brightness variations in more detail.
The brightness profiles shown in Figures \[d68phase\] and \[gphase\] are proportional to the average scattering phase function of these rings, with the proportionality constant set by the optical depth of the ring and bond albedo of the ring particles. Ideally, the SPF is normalized so that the integral of the function over all solid angles is either unity or $4\pi$. Unfortunately, even though our observations cover almost the entire range of possible scattering angles, we cannot compute the relevant integral reliably because the brightness increases so rapidly in the core of the forward-scattering peak. Even if the rings’ brightness scales like $1/\theta$ between 0$^\circ$ and 0.5$^\circ$, that last half-degree would still contribute an order of magnitude more to the integral than all other solid angles. The derived integral is therefore extremely sensitive to the assumed shape of the last 0.5$^\circ$ of the forward scattering peak.
Since we cannot robustly determine the integral under the phase curve, we will instead normalize the SPFs to be unity where the scattering angle is 90$^\circ$. These normalized SPFs are shown in Figure \[dgphase\], which reveals that the SPFs of the G ring and D68 are remarkably similar to each other. Another way to look at these data is shown in the bottom panel of Figure \[dgphase\], where we plot the local value of the power-law index in the SPF as a function of scattering angle. To generate this plot, we fit the SPF in the vicinity of each data point[^2] to a simple power-law function (i.e. SPF$\propto\theta^{Q}$). The plot shows how the index $Q$ varies with scattering angle and reveals that both SPFs can be divided into roughly four regimes:
- [**Scattering angles above 50$^\circ$**]{}, where the power law index is close to zero, indicating that the brightness of both features does not depend much on scattering angle.
- [**Scattering angles between 10$^\circ$ and 50$^\circ$**]{}, where the ring’s brightness scales inversely proportionally with scattering angle.
- [**Scattering angles between 2$^\circ$ and 10$^\circ$**]{}, where the scattering function becomes dramatically steeper, with a power-law index approaching -3.5.
- [**Scattering angles less than 2$^\circ$**]{}, where the scattering angle becomes shallow again.
Fits to the measured scattering phase functions {#fits}
===============================================
In order to further quantify the trends observed in the above data and clarify their implications, we compare the measurements to various models. For these studies, we use the normalized, binned data displayed in Figure \[dgphase\] and only consider the estimates of D68’s brightness derived using the Fit method. We also exclude the anomalously low D68 measurement at scattering angles around 65$^\circ$.
First, we use Henyey-Greenstein functions to derive simple analytical expressions that reproduce the relevant observations. Next, we compare the ring data to laboratory measurements of real particle populations. These comparisons demonstrate that the basic shapes of our SPFs are reasonable, but their detailed form is different from currently-available laboratory samples. Finally, we consider physically-motivated models based on Fraunhofer and Mie theories in an attempt to gain some insights into the particle size distributions of these rings.
Henyey-Greenstein functions
---------------------------
[lllllllll]{} G & $12.3$ & $<10^{-16}$ & $0.525^{+0.475}_{-0.525}$ & $1.0$\
D68 & $22.8$ & $<10^{-16}$ & $0.393^{+0.430}_{-0.384}$ & $1.0$\
G & $2.5$ & 0.00045 & $0.985^{+0.015}_{-0.045}$ & $0.447^{+0.342}_{-0.221}$ & $0.345^{+0.255}_{-0.305}$ & $0.553^{+0.221}_{-0.342}$\
D68 & $4.6$ & $1.2\times10^{-12}$ & $0.995^{+0.005}_{-0.055}$ & $0.779^{+0.07}_{-0.443}$ & $0.325^{+0.165}_{-0.155}$ & $0.221^{+0.443}_{-0.07}$\
G & $0.71$ & 0.84 & $0.995^{+0.005}_{-0.015}$ & $0.643^{+0.076}_{-0.306}$ & $0.665^{+0.085}_{-0.185}$ & $0.176^{+0.266}_{-0.106}$ & $0.035^{+0.225}_{-0.035}$ & $0.181^{+0.291}_{-0.101}$\
D68 & $1.5$ & 0.15 & $0.995^{+0.005}_{-0.015}$ & $0.754^{+0.055}_{-0.247}$ & $0.585^{+0.165}_{-0.165}$ & $0.151^{+0.166}_{-0.096}$ & $0.005^{+0.255}_{-0.005}$ & $0.095^{+0.146}_{-0.045}$\
$^a$ Probability that the $\chi^2/\nu$ parameter for the best-fit model would be at least as large as its observed value if the selected model were correct.
The Henyey-Greenstein (HG) function is the function most commonly used to describe the SPF of debris disks, and can be expressed as follows: $$p\!\left(g,\theta\right) = \frac{1}{4\pi}\frac{1-g^2}{[\,1+g^2-2g\cos{\theta}\,]^{3/2}},$$ where $\theta$ is the scattering angle and $g$ is the HG asymmetry parameter,which ranges from -1 for perfect backscattering to 1 for perfect forward scattering [@henyey1941]. We fit the normalized brightnesses using a single HG function, examining all possible values of $g$ in steps of $0.001$ and varying the amplitude of the fit to minimize the $\chi^2$ metric. The best fits, shown as blue lines in Figure \[HG\_fits\], correspond to $g=0.525$ for the G ring and $g=0.393$ for D68. However, these fits have $\chi^2/\nu = 12.3$ and $22.8$, respectively ( $\nu=$ degrees of freedom for the fit), and the probability of the best-fit $\chi^2$ values being at least this large just due to random chance is extremely low (i.e. $p_> <10^{-16}$). A single HG function is therefore a very poor fit to the measured SPFs.
In an attempt to achieve an acceptable match to the data (i.e. $\chi^2/\nu \sim 1$ and $p_> \sim 0.5$), we fit the measured SPFs using a 2-component HG function of the form $$p_2\!\left(g_1,w_1,g_2,\theta\right) = w_1\; p\!\left(g_1,\theta\right) + w_2\; p\!\left(g_2,\theta\right),$$ where $w_2=(1-w_1)$ and $w_1$ ranges from 0 to 1. We examined all possible combinations of $g_1$ and $g_2$ using step sizes of $0.01$, and all possible values of $w_1$ in steps of $0.005$. The best fit parameters are listed in Table \[HG\_fits\_table\] and result in $\chi^2/\nu = 2.5$ and $4.6$ for the G ring and D68, respectively. These best fits are shown as red lines in Figure \[HG\_fits\]. While these fits are much better than the single HG solutions, the probabilities that the $\chi^2$ statistics would be this large are still below 0.001, indicating that the data deviate significantly from this model.
Finally, we examined 3-component HG functions of the form $$\label{hg3comp_equation}
p_3\!\left(g_1,w_1,g_2,w_2,g_3\theta\right) = w_1\; p\!\left(g_1,\theta\right) + w_2 \; p\!\left(g_2,\theta\right)+ w_3 \; p\!\left(g_3,\theta\right),$$ with the requirements $w_3 = (1-w_1-w_2)$ and $w_3 \ge 0$. To make a brute force $\chi^2$ calculation more tractable, we restricted $0.5 < g_1 < 1$, $0.25 < g_2 < 0.75$, and $0 < g_3 < 0.5$ using step sizes of $0.01$. Given the results of the 2-component fit, and the lack of a back-scattering feature in the measured SPF, we don’t expect these limitations to impact the best derived fit. We examined all possible, valid combinations of $w_1$ and $w_2$ in steps of $0.005$. The best fit parameters are listed in Table \[HG\_fits\_table\]. The 3-component fits have significantly better $\chi^2/\nu$ statistics, with values of $0.71$ and $1.5$ (and corresponding $p_<$ values of 0.84 and 0.15 ) for the G ring and D68, respectively. These best fit 3-component functions are shown as black lines in Figure \[HG\_fits\] and reproduce the data well. Hence, If one’s goal is to quickly approximate the SPFs of the G ring or D68, we recommend using these three-component fits. However, we do not regard the parameters of these fits as having any specific physical significance because a HG function only provides a rough approximation of any given system’s light-scattering properties.
Empirical comparisons
---------------------
While the HG functions can reproduce the observed brightness data, the HG formalism provides little information about the physical properties of the material in these rings. Insights into the rings’ particle properties can potentially be obtained by comparing our data to the Amsterdam Light Scattering Database [@munoz2012]. This database contains high-precision laboratory measurements of scattering phase functions for particles with a wide variety of compositions, including liquid water droplets, clays, olivine, and fluffy aggregates of synthetic circumstellar dust. Each sample has an estimated size distribution, with effective grain radii typically falling between $0.1$ and $10$ $\mu$m. We included only the 36 non-hydrosol samples that were measured at a wavelength of $0.63$ $\mu$m.
To compare these laboratory measurements with our astronomical SPFs, we interpolated the laboratory SPF measurements and their fractional uncertainties in log-log space. We added the uncertainty of both data sets in quadrature when calculating a best fit. Our comparison of the data sets was limited to $\theta>3^{\circ}$ due to the limitations of the Amsterdam Light Scattering Database experiment [@munoz2012]. Note that we are unable to adjust the grain size distribution of the laboratory measurements, and so better fits may be possible with grain size distributions that are not currently available.
We were unable to achieve $\chi^2/\nu < 3.4$ for any of the compositions in the database. In particular, most of the 7 different samples of fluffy synthetic circumstellar dust aggregates were poor fits to our data. Large grain olivine samples ($s_{\rm eff}>4$ $\mu$m), Sahara sand samples, and volcanic ash samples fit best with $\chi^2/\nu \lesssim 7$ for the G ring and $\chi^2/\nu \lesssim 8$ for D68. Figure \[empirical\_fits\] shows three examples of these SPFs overlaid on our data. While the matches are not particularly good, these laboratory SPFs do at least demonstrate that the overall shape of the rings’ SPFs are not unreasonable for $\theta>3^\circ$. They also suggest that the rings’ SPFs are not particularly sensitive to the particles’ compositions.
Fraunhofer diffraction
----------------------
In lieu of more extensive laboratory measurements, we can fit our data to theoretical models based on Mie theory and Fraunhofer diffraction. Ideally, such theories would allow us to translate features in the observed SPF into information about the particle size distribution. However, in practice the approximations inherent in these theories and the large parameter space of possible particle size distributions complicates efforts to find unique and well-defined solutions. Thus the following analyses should be considered initial explorations that can provide some insights into these rings’ particle size distributions and guide future efforts to model the photometric properties of these systems.
Fraunhofer diffraction theory provides the simplest light-scattering model that can translate a ring’s observed SPF into information about its particle size distribution. This model assumes that the light scattered by each particle in the ring has a forward-scattering peak with a characteristic width set by the ratio of the particle’s radius $s$ to the wavelength of the scattered light $\lambda$. Strictly speaking, classical Fraunhofer diffraction is only valid in the far-field limit of small opaque particles, but at very small scattering angles ($\theta \ll \lambda/s$) Fraunhofer theory can provide a useful approximation of light-scattering properties for even the ice-rich particles found in Saturn’s rings [@hedman2009]. In this limit, the brightness of the ring material should be given by the following expression: $$\mu I/F = \mathcal{C}\int \frac{\pi s^2}{\sin^2\theta}\frac{dN}{ds}J_1^2(k s\sin\theta) ds,$$ where $\mathcal{C}$ is a constant numerical coefficient, $k=2\pi/\lambda$ is the wavenumber of the observed radiation and $dN/ds$ is the number of particles in a small range of particle size $ds$. Note that if the differential size distribution is a pure power law (i.e. $dN/ds \propto s^{q}$), then this expression implies that the combined light from all the particles would be a power-law function of scattering angle (i.e. the observed brightness is $\propto \theta^{-(q+5)}$). Note that a shallower particle size distribution (i.e. less negative $q$) will produce a steeper SPF (i.e. more positive $q+5$). Furthermore, since the square of the Bessel function $J_1^2(x)$ has a peak at $x \simeq 2$, the measured brightness at a given scattering angle $\theta$ will be predominantly due to particles with a size $s \simeq \pi^{-1}\lambda/\theta$ (assuming $q$ is close to $-3$).
The observed SPFs for the G ring and D68 do not follow a simple power law for small scattering angles. Instead, the power-law index of the scattering function becomes dramatically less steep when the scattering angles falls below 2$^\circ$ (see Figure \[dgphase\]). This suggests that the particle size distribution does not follow a perfect power law. More specifically, the transition to a lower phase slope at smaller scattering angles implies that particle size distribution becomes steeper (or is cut off) above some critical size. In this situation, we can define an effective particle size $s_{\rm eff}$, which corresponds to the average effective size of the particles in the ring. We can even estimate this parameter by fitting the observed SPF to a Fraunhofer model of the scattering function for a single particle, because the width of the forward scattering peak of a particle with size $s_{\rm eff}$ will be close to the width of the forward scattering peak of a particle size distribution with that effective average size.
We considered a range of possible effective grain sizes from $0.1$–$100$ $\mu$m. To satisfy the Fraunhofer criterion of small scattering angles, we selected only the data at $\theta < 0.1\lambda/s_{\rm eff}$ for each effective grain size. We fit this portion of the forward scattering peak with an Airy function (appropriate for spherical grains) and considered only those fits that had at least 1 degree of freedom. We found a best fit effective grain size of $s_{\rm eff}=2.2^{+0.3}_{-0.7}$ $\mu$m for the G ring, with $\chi^2/\nu = 1.8$ ($p_>=0.40$), and a best fit effective grain size for D68 of $s_{\rm eff}=2.5^{+0.2}_{-0.3}$ $\mu$m, with $\chi^2/\nu = 0.28$ ($p_>=0.96$). The high probabilities for the $\chi^2$ to exceed the observed values suggests that the uncertainties on the data points in the forward-scattering peak may be overestimated. When relaxing the small angle constraint to $\theta < 0.25 \lambda/s$, best fit grain sizes roughly doubled while $\chi^2/\nu$ remain relatively unchanged. Thus our estimate of $s_{\rm eff}$ is likely less certain than the above error bars suggest, and so we can only conclude that the typical particle size in this ring is probably of order a few microns.
Mie theory
----------
Mie theory provides exact analytical expressions for the amount of light scattered in all directions by perfect dielectric spheres, and it is commonly used to constrain the composition of debris disks by modeling the disk thermal emission in the infrared, along with the optical and near-IR photometry [e.g. @li1998; @lebreton2012; @rodigas2015]. However, Mie theory does not accurately predict the SPF for non-spherical particles [e.g. @pollack1980]. The data shown in Figure \[dgphase\] provide new opportunities to examine the applicability of Mie theory to real systems of fine debris.
@rodigas2015 calculated the scattering phase functions for 8407 unique combinations of materials, stemming from 19 “root" data sets containing the complex indices of refraction as a function of wavelength. Here, we fit the measured SPFs using a subset of these materials, requiring that each material contain some fraction of water ice, which is known to be prevalent in Saturn’s rings [@cuzzi2009 and references therein]. Each “root" water ice data set was mixed with other combinations of materials according to the rules of @rodigas2015, resulting in 19 porous water mixtures, 144 2-component mixtures, and 720 porous 2-component mixtures per water ice data set. We examined three different “root" water ice data sets, for a total of 2652 mixtures.
We calculated the scattering efficiency and SPF of each mixture at $0.63$ $\mu$m using Mie theory and Bruggeman effective medium theory as described in @rodigas2015. We calculated the SPF from 0–$180^{\circ}$ in $0.5^{\circ}$ increments for 50 values of grain size logarithmically spaced from $0.1$–$500$ $\mu$m. We assumed a power law grain size distribution of the form $$\frac{dN}{ds} \propto s^{q},$$ between the sizes $s_{\rm max}$ and $s_{\rm min}$ and calculated the size-integrated SPF by weighting each grain size by the size distribution power law, grain cross section, and scattering efficiency calculated by Mie theory. We investigated 51 values of $q$ ranging from $-7$ to $-2$. We truncated the size distribution at minimum and maximum grain sizes that spanned the full range of investigated sizes. We interpolated each model SPF in log-log space for comparison with the measured values and calculated a reduced $\chi^2$ value for each mixture.
The G ring was best fit by a shallow size distribution ($q=-2.2$) of grains ranging from 4 to 460 $\mu$m, with $\chi^2/\nu=0.7$. The single best fit composition was a mixture of 10% water ice and 90% olivine by volume, with an additional porosity of 10%. However, this best fit is not unique. Figure \[mie\_theory\_fits\_chi2contours\_G\] shows the projected $\Delta\chi^2$ contours as a function of $s_{\rm min}$ and $q$. To create this plot, we projected the minimum $\chi^2$ from all other parameters. The three contours, labeled as 1, 2, and 3$\sigma$, correspond to the appropriate 2 parameter joint confidence levels for $\Delta\chi^2 = 2.3$, $6.17$, and $14.2$, respectively. Two 1$\sigma$ “islands" exist at the top of the plot. The leftmost of these islands contains shallow size distributions with minimum grain sizes $\sim$few $\mu$m, and includes the best fit. A total of 221 unique mixtures with a wide range of compositions and porosities are included in this portion of phase space, with water ice volumetric fractions typically $<30\%$. This suggests that a flat size distribution can reproduce the observations well, regardless of the details of the scattering and optical properties. The rightmost 1$\sigma$ island requires larger minimum grain sizes $\sim40$ $\mu$m, and includes only three compositions, all of which are roughly an 80% mixture of iron-rich olivine or iron-rich orthopyroxene mixed with 20% water ice.
Figure \[mie\_theory\_fits\_G\] shows the SPFs for three sample fits. The red curve shows the model SPF for the best fit. The green curve corresponds to an $s_{\rm min} = 18$ $\mu$m and $q=-2.8$ fit, which falls within the 2$\sigma$ contours shown in Figure \[mie\_theory\_fits\_chi2contours\_G\]. The blue curve corresponds to an $s_{\rm min} = 35$ $\mu$m and $q = -3.2$ fit. All three of these fits predict that the forward-scattering peak continues to rise steeply, such that the SPF is four orders of magnitude brighter at $\theta=0^{\circ}$ than at $\theta \approx 1^{\circ}$.
Similar to the G ring, the D68 ringlet was best fit by a shallow size distribution ($q=-2.5$) of grains ranging from 5 to 390 $\mu$m, with $\chi^2/\nu=0.9$. The single best fit composition was a mixture of 10% water ice and 90% iron-rich orthopyroxene by volume, with an additional porosity of 10%. Figure \[mie\_theory\_fits\_chi2contours\_D\] shows the $\Delta\chi^2$ contours for the D68 ringlet. Here the 1$\sigma$ best fits are well isolated around the overall best fit and 2 distinct regions of parameter space are included within 3$\sigma$. Like the G ring fits, the SPF of the D68 ringlet is best fit by either a shallow size distribution with minimum grain size $\sim$few $\mu$m, or a size distribution with minimum grain size $\sim$few tens of $\mu$m. Within the 1$\sigma$ limit, a wide range of compositions and porosities are included, typically with volumetric ice fractions $<30\%$.
Figure \[mie\_theory\_fits\_D\] shows the SPFs for two sample fits to the D68 SPF. The red curve shows the model SPF for the best fit. The blue curve was drawn from the right 2$\sigma$ island and corresponds to $s_{\rm min} = 35$ and $q = -5.1$. Both of these fits predict that the forward-scattering peak is 2–3 orders of magnitude brighter at $\theta=0^{\circ}$ than at $\theta \approx 1^{\circ}$.
Two aspects of these best-fit solutions are surprising and may reflect limitations of Mie-theory-based models: the relatively low fractions of water ice and the relatively high fraction of particles with radii greater than 10 $\mu$m. Let us consider the compositional issues first. Saturn’s dense main rings are known to be composed of relatively pure water ice based on both strong water-ice absorption bands in the near infrared [@cuzzi2009 and references therein], as well as the ring’s high reflectivity and low emissivity at radio wavelengths [e.g. @Pollack1975]. The situation for the dusty rings is less clear, but the F ring and outer D ring both exhibit water-ice absorption bands in the near-infrared [@hedman2007; @cuzzi2009; @hedman2011; @vahidinia2011], and these bands are also visible in spectra of D68 and the G ring (Hedman et al. in prep). Thus we might have expected that the bet-fit solutions would be ones with a higher fraction of water ice than those found above. There is evidence for at least two non-icy contaminants in Saturn’s rings and moons [@cuzzi1998; @poulet2003]. One is spectrally neutral and probably corresponds to carbon-rich cometary materials. The other absorbs strongly at wavelength shorter than 0.5 microns and could be either organic compounds or nano-phase iron-rich grains [@cuzzi2009; @clark2012]. The concentration of these contaminants in D68 and the G ring are currently unknown, and so the lower water-ice fractions in the best-fit models may be due to these contaminants. However, it is also possible that models with substantial non-icy components are favored because these materials suppress structures in the SPF at high scattering angles that Mie theory predicts for ice-rich spheres (e.g. those responsible for rainbows) that are not seen in the data because the particles have irregular shapes. Studies of the rings’ near-infrared spectra should help clarify these issues. For example, Figure \[ir\_spec\_G\] shows the absorption coefficient $Q_{abs}$ versus wavelength for few of the better-fitting G-ring models. The large variations in the $Q_{abs}$ curves should produce obvious features in the rings’ near-infrared spectra, and so spectral measurements should reveal whether any of these compositional models are sensible. In the meantime, both these Mie-theory based calculations and the previous comparisons with empirical measurements indicate that the SPFs alone cannot provide strong constraints on the particle composition of these dusty systems.
Turning to the particle-size distributions, we may note that none of the best-fit models have an effective average particle size in the range of a few microns, which would be consistent with the Fraunhofer-diffraction-based analysis. instead, all the best-fit models require a substantial population of particles that are several tens of microns across. These large particles have forward-scattering lobes that are less than a degree wide according to standard diffraction theory, which is too narrow to be easily detectable in the available data. Thus a substantial population of such large particles would be essentially invisible to the Fraunhofer-based calculations presented above. While Cassini has detected large particles in the G ring [@hedman2007], there are reasons to be skeptical of the idea that most of the particles in these rings are greater than 50 microns across.
Recall that according to classical Fraunhofer diffraction theory, a particle size distribution with power-law index $q$ should have an SPF that scales like $\theta^{-(q+5)}$. Hence the steepest parts of the forward-scattering lobe, where the brightness goes like $\theta^{-3}$ (see Figure \[dgphase\]), are consistent with the shallow size distribution ($q\simeq-2$) favored by the best-fitting Mie-theory models. However, the observed SPFs also become much less steep for scattering angles less than 1$^\circ$ or 2$^\circ$, which is most naturally explained as a cut-off in the size distribution. The best-fitting Mie-theory-based models do not show a sharp reduction in the slope of the SPF at small scattering angles, so these pure power-law models may not be capturing this aspect of the particle size distribution. Furthermore, this difference between the models and the data indicates that Mie theory calculations are not favoring models with many large particles because this improves the fit near the top of the forward-scattering peak. Instead, these models probably favor a substantial population of large grains because such particles can alter the shape of the SPF at larger scattering angles. In particular, large particles will make significant contributions at large scattering angles while making only a small change to the forward-scattering peak between $\theta=0.5^\circ$ and 10$^\circ$. Thus the large particle population probably improves the fit by increasing the flux at large scattering angles relative to the forward-scattering peak. If this is the case, then the amount of large particles could well be overestimated because Mie theory tends to underestimate the amount of light scattered at moderate to high scattering angles by irregular particles [@pollack1980].
We expect that a more consistent picture of the particle-size distribution will be obtained is we allow for more complex particle size distributions and/or non-spherical particles. Such calculations are beyond the scope of this work, but based on the above considerations, we expect that the final particle size distribution will be rather shallow up to some critical particle size between a few and a few tens of microns.
Implications of the SPFs for exoplanetary debris disks. {#discussion}
=======================================================
Even if we cannot yet derive a fully consistent physical model for the SPFs of the G ring and D68, our 3-component HG fits still provide an accurate description of these rings’ scattering phase functions, and even these simple phenomenological models are useful for evaluating data from exoplanetary disks. For example, HG functions are commonly used to estimate the degree of forward scattering of a debris disk. Reported values of $g$ typically range from $0.0$ to $0.3$ for debris disks [e.g. @kalas2005; @schneider2006; @debes2008; @thalmann2011]. However, these estimates are commonly made by fitting the flux ratio along the projected minor axis of the disk and are limited to a small range of scattering angles. Therefore, they may be a poor representation of the shape of the SPF and true degree of forward scattering. Recent fits to the shape of the derived SPF suggest significantly more forward scattering [@stark2014].
If a debris disk’s true SPF resembled the measured G ring SPF shown in Figure \[dgphase\], what might we expect to observe? A debris disk inclined by 30 degrees from face-on (e.g., HD 181327), would enable observations ranging in scattering angle from 60$^{\circ}$–120$^{\circ}$. Figure \[observable\_SPF\] shows the variation in the best-fit 3-component HG SPF for the G ring over this range of scattering angles, properly normalized such that the integral of the scattering phase function over all angles is unity. We adopt this as our G ring SPF “model."
The left panel of Figure \[observable\_SPF\] shows that when fitting the observable portion of the model SPF with a single HG function, one can roughly reproduce the flux ratio at the smallest and largest scattering angles, but the fit to the shape of the SPF is poor. The best fit $g=0.17$ is similar to $g$ values reported in the literature for a number of debris disks. We conclude that over the range of observable scattering angles, the measured SPFs of Saturn’s G ring and D68 are roughly consistent with typical debris disk observations.
The left panel of Figure \[observable\_SPF\] also shows the best-fit 2-component HG function, which does very well at reproducing the model over the observable scattering angles. However, as shown in the right panel, neither HG function fit accurately predicts the model SPF at $\theta<10^{\circ}$. We conclude that fits to debris disk SPFs, over typical ranges of observable scattering angles, cannot accurately predict the degree of forward scattering. *The reported degrees of forward scattering for debris disks therefore may be greatly underestimated.*
If the model G ring SPF is in fact representative of debris disk SPFs, what would this imply about debris disks? First, we note that the model G ring SPF near $\theta\approx90^{\circ}$ is $\sim0.02$, when properly normalized such that its integral over all solid angles is unity. In comparison, isotropic scatterers have a SPF equal to $1/4\pi \sim 0.08$ at all scattering angles. Thus, debris disks would appear a factor of $\sim4$ dimmer than their isotropically-scattering counterparts, potentially explaining the low apparent albedo of some disks [e.g., @krist2010; @golimowski2011; @lebreton2012].
Second, the very forward-scattering model G ring SPF suggests that edge-on disks may be problematic for future exoEarth imaging missions. @stark2015 showed that cold debris disks may produce a “pseudo-zodiacal" haze of forward-scattered starlight if oriented within a few tens of degrees of edge-on. At a projected separation of 1 AU, this “pseudo-zodi" becomes non-negligible under the assumption of a single HG SPF with $g>0.7$. The G ring model SPF satisfies this forward-scattering criterion, as the dominant component of the model fit has $g=0.995$.
Conclusions
===========
- The Cassini spacecraft provides measurements of the dusty rings’ relative brightness over a large range of scattering angles from 0.5$^\circ$ to 170$^\circ$. These data are consistent with previous studies of dusty rings and debris disks, but also provide new information about the SPF of dusty debris systems at small scattering angles.
- The brightness data for these rings is well described by a combination of 3 Henyey-Greenstein functions. If one’s goal is to quickly approximate the SPFs of the G ring or D68, we recommend using this fit.
- None of the best fitting compositions from the Amsterdam Light Scattering Database resemble the icy material that we’d expect to find in the G ring or D68. Also, Mie-thoery-based calculations with a range of compositions fit the phase curves equally well. This suggests that a wide variety of compositions can produce similar SPFs. Thus, the SPF by itself is probably not a good discriminator of debris disk composition.
- Our Fraunhofer fits suggest effective grain radii on the order of a few microns, while the Mie-theory based calculations assuming a power-law size distribution favor typical particles of a few tens of microns. The applicability of Fraunhofer diffraction theory and Mie Theory to these systems of irregular grains mean that these estimates may be off by factors of a few.
- The strong forward scattering peaks of these dusty systems suggest that the degree of forward scattering in extrasolar debris disks may be greatly underestimated, which could have implications for the albedos of these systems and for future exo-Earth searches.
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[^1]: “Rev" is a designation for one of Cassini’s orbit around Saturn.
[^2]: Each fit includes all data obtained at scattering angles within $\pm50$% of the selected data point, excluding the discrepant point in D68’s SPF around $\theta=65^\circ$.
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